Westpark Gamers
Brettspiele in Mnchen

Bluff Endspiel-Strategie

Eine (fast) optimale Strategie für das Endspiel von Bluff!

Wir haben uns hier schon häufiger mit dem Endspiel von Bluff befasst. Dieses Mal kommen wir der optimalen Strategie (Nash Gleichgewicht) des Spieles tatsächlich schon recht nahe! Ich möchte eine Strategie für Spieler 2 vorstellen, die mit mindestens 50% Wahrscheinlichkeit gewinnt; außerdem noch eine Strategie für Spieler 1 (Startspieler), die mit mindestens 48% Wahrscheinlichkeit gewinnt!

Vermutlich kann man beide Strategien noch etwas verbessern, sodass bei optimaler Strategie der Startspieler mit maximal 2% Wahrscheinlichkeit im Nachteil ist – für ein reales Spiel sind diese kleinen Unterschiede aber kaum bemerkbar.

Hier also die recht anschauliche Strategie für Spieler 2: Spieler 1 macht das erste Gebot!

1.Runde, Spieler 2:

hört Pasch: zw (=Anzweifeln)

hört 1: hat 1,* -> 11 (Spieler 2 hört ein Gebot von 1; er hat entweder eine 1 oder einen * gewürfelt und bietet dann Pasch-1) 2,3 -> zw 4 -> 4 5 -> 5

hört 2: hat 2,* -> 22 1,3 -> zw 4 -> 4 5 -> 5

hört 3: hat 3,* -> 33 1,2 -> zw 4 -> 4 5 -> 5

hört 4: hat 4,* -> 44 1,2,3 -> zw 5 -> 5

hört 5: hat 5,* -> 55 1,2,3,4 -> zw

hört *: hat 1,3,4,5 -> zw 2 -> 22 * -> **

Alle weiteren Runden: Anzweifeln!

Beweis: Als Beweis dafür, dass hier Spieler 2 mit mindestens 50% Wahrscheinlichkeit gewinnt, betrachte man folgende Tabelle, welche die minimalen Gewinnwahrscheinlichkeiten(in Einheiten von 1/36) von Spieler 2 enthält, wenn Spieler 1 X(Zeile) geworfen hat und dann Y(Spalte) als Gebot ansagt! Zu beachten ist, dass Spieler 2 in der zweiten Runde immer anzweifelt!

1 2 3 4 5 * min 1 4 4 4 4 4 4 4 2 4 4 4 4 4 5 4 3 4 4 4 4 4 4 4 4 3 3 3 3 4 4 3 5 2 2 2 3 2 4 2 * 1 1 1 1 1 2 1 (Pasch-Gebote werden hier nicht aufgeführt, da Spieler 2 dabei immer mit mindestens 4/36 gewinnt)

In der rechten Spalte ist das Minimum der Zeile aufgeführt, da Spieler 1 natürlich versucht, die Gewinnwahrscheinlichkeit für Spieler 2 durch sein Gebot zu minimieren. Die Summe der Min-Spalte ergibt jetzt 18/36 – also 50% Gewinnwahrscheinlichkeit für Spieler 2! q.e.d.

Jetzt noch die angekündigt Strategie von Spieler 1 mit 173/360 = ca. 48% Gewinnwahrscheinlichkeit:

1.Runde, Spieler 1:

* -> 4 (3/10) 5 (7/10) (Spieler 1 hat * gewürfelt und bietet eine 4 mit 3/10 und eine 5 mit 7/10 Wahrscheinlichkeit) 1 -> 4 (1/2) 5 (1/2) 2,3->4 (2/5) 5 (3/5) 4 -> 4 5 -> 5

2.Runde, Spieler 1:

Hat 1 und 4 gesagt(in erster Runde):

Hört * : 11 (3/5) zw (2/5) Hört 5 : * (1/5) 11 (3/5) zw (1/5) Hört Pasch -> zw

Hat 1 und 5 gesagt:

Hört * : 11 Hört Pasch -> zw

Hat 2 und 4 gesagt:

Hört * : 22 (7/8) zw (1/8) Hört 5 : * (1/4) 22 (3/4) Hört Pasch -> zw

Hat 2 und 5 gesagt:

Hört * : 22 (5/6) zw (1/6) Hört Pasch -> zw

Hat 3 und 4 gesagt:

Hört * : 33 (7/8) zw (1/8) Hört 5 : * (1/4) 33 (3/4) Hört Pasch -> zw

Hat 3 und 5 gesagt: Immer : zw

Hat 4 und 4 gesagt:

Hört * : 44 (7/20) zw (13/20) Hört 5 : * (1/10) 44 (3/10) zw (6/10) Hört Pasch -> zw

Hat 5 und 5 gesagt:

Immer : zw

Hat * und 4 gesagt:

Hört * : ** (5/6) 11 (1/6) Hört 5 : * (anschließend in 3.Runde immer **) Hört Pasch -> **

Hat * und 5 gesagt:

Immer : **

Alles weitere Anzweifeln !

Beweis: Die Berechnung der Wahrscheinlichkeiten wurde per PC durchgeführt mittels der gtf-Bibliothek; Siehe http://www.daimi.au.dk/~trold/gtf.html

Nachtrag (30.12.08): Eine komplette spieltheoretische Berechnung des Spieles mittels der performanceoptimierten PCx Bibliothek ergibt eine Gewinnwahrscheinlichkeit für den Startspieler von 48,6% (gerundet). Die dabei berechneten Strategien sind aber nicht mehr so anschaulich und können daher hier auch nicht mehr aufgeführt werden. Die obige Strategie für Spieler 1 ist also nur 0,6% vom Optimum entfernt! Ein paar weitere Erkenntnisse: • Setzt man voraus, dass der Startspieler in der ersten Runde nur eine 4 oder 5 bieten soll, so ist die obige Startspieler-Strategie sogar optimal! • Setzt man voraus, dass der Spieler 2 in der zweiten Runde immer anzweifelt, so ist die obige Strategie für den Spieler 2 mit der Gewinnwahrscheinlichkeit ½ sogar optimal ! • „Immer 4“ Strategie: Die im letzten Artikel beschriebene Gegenstrategie des Spielers 2 ist optimal(best response); die „Immer 4“ Strategie hat also eine Gewinnwahrscheinlichkeit von 15/36 (41,7%). • „Immer 5“ Strategie: Diese Strategie hat dieselbe Gewinnwahrscheinlichkeit wie die „Immer 4“ Strategie: 15/36 ! (Unser kleiner running gag bei den WPGlern hat damit eine versöhnliche mathematische Lösung erhalten :) • „Immer X“ Strategie: Tja, wer hätte das gedacht; für X=1,2,3,4,5 ergibt sich hier immer die (nicht optimale) Gewinnwahrscheinlichkeit 15/36! Für X=* erhält man 10/36.

©2008, Günther Rosenbaum

 

Flaschenteufel

 

Flaschenteufel cover
Autor Günter Cornett
Verlag Bambus Spieleverlag
erschienen 2003 (1995)
Spielerzahl 3-4
Spieldauer 45 Minuten
Wertung pic pic pic pic pic pic pic pic pic pic

Flaschenteufel

Spieltipps von Walter Sorger

Vom ersten Tag an in seinem nunmehr 10 jährigen Leben hat dieses kleine Kartenspiel die Kritiker begeistert. „Außergewöhnliches Stichspiel“ – „für anspruchsvolle Spieler“ – „eine der besten und originellsten Kartenspielideen der letzten Jahre“, so tönen die Lobeshymnen.

Jeder Spieler bekommt eine Handvoll Karten in den Farben rot, blau und gelb ausgeteilt, auf denen die Zahlen 1 bis 37 stehen. Wie in Stichspielen üblich spielt jeder reihum eine Karte zu einem Stich aus. Die relativ höchste Zahl gewinnt den Stich und bringt Siegpunkte ein. Die Spiegpunkte mehrerer Spiele werden zusammengezählt; am Ende gewinnt, wer zuerst 300 oder 500 Siegpunkte erzielt hat.

Im Begriff „relativ höchste“ liegt der ganze Witz der Spiels. Auf dem Tisch liegt eine „Teufelskarte“ mit dem Anfangswert 19. Ist keine der ausgespielten Karten kleiner als die Teufelskarte, so gewinnt die absolut höchste Karte den Stich. Ist eine der ausgespielten Karten kleiner, so gewinnt die höchste Karte, die kleiner ist als die Teufelskarte, den Stich. Im zweiten Fall wird nach dem Stich diese Karte zur neuen Teufelskarte. Der Grenzwert, der Hoch von Niedrig unterscheidet, sinkt also ständig nach unten, und es wird immer schwieriger, die Teufelskarte zu unterbieten.

Wer den letzten Stich mit einer Karte kleiner als die Teufelskarte gewonnen hat, hat allerdings Pech gehabt: alle Siegpunkte in seinen Stichen zählen nicht; im Gegenteil er bekommt stattdessen eine Anzahl von Minuspunkten aufgebrummt.

Von den üblichen Stichspielen unterscheidet sich der „Flaschenteufel“ dadurch, dass es zwar verschiedene Kartenfarben gibt, aber keine davon eine Trumpffarbe darstellt; die Kartenfarben dienen lediglich dazu, die Auswahlmöglichkeiten der Spieler beim Bedienen eines Stichs einzuschränken und damit den Ablauf überschaubarer zu machen. Auch an den Mechanismus, dass sich das Stichpotential einer Karte während des Spiels ständig ändert, muss sich ein erfahrener Skatbruder erst einmal gewöhnen. In den ersten Runden kommt einem der Spielablauf zuweilen wie ein unkalkulierbares Chaos vor. Und wer oft genug auf der Teufelskarte sitzen geblieben ist, oder wer mit den ausgeteilten „schlechten“ Karten in seiner Hand keinen einzigen punkte-trächtigen Stich gewonnen hat, der lädt Frust über Frust auf seine Seele und hadert mit der ungerechten Schicksalsgöttin, die die Karten verteilt. Dann äußert sich gleich die abwertende Kritik: „Dieses Spiel ist absolut unbeherrschbar!“

Wie falsch, wie falsch! Natürlich spielt die Kartenverteilung eine große Rolle. Es gibt einfach bessere und schlechtere Kartenblätter. Aber das ist in jedem Stichkartenspiel der Fall; in keinem Kartenspiel der Welt gewinnt der beste Spieler jede Kartenausteilung! Auch nicht beim göttlichen Bridge; doch hier tauchen in den Titellisten der Weltmeisterschaften immer die gleichen Namen auf. In einem beherrschbaren Spiel wird wenigstens auf Dauer immer der bessere Spieler die besten Ergebnisse erzielen.

Welche Qualifikationen gehören nun beim „Flaschenteufel“ zu diesem „besseren Spieler“? Wie in jedem guten Stichkartenspiel: Ein gutes Gedächtnis, eine sicheres Einmaleins, eine scharfe Logik und eine ausgeprägte Phantasie für die Entwicklung des Spielablaufs. Ohne besondere Sortierreihenfolge möchte ich im Folgenden ein paar lose Regeln aufzählen, die im „Flaschenteufel“ für ein gutes Ergebnis beherrscht und beherzigt sein müssen.

    • Karten mit sehr geringem Stichpotential sind diejenigen mit Werten zwischen 20 und 30 Punkten. Wenn du nicht gerade mit kleinen Karten überhäuft bist, solltest du diese Karten weiter schieben oder in den Teufelsstich geben. Mit kleineren und mit größeren Werten kannst du viel stärker in das Geschehen eingreifen, Stiche mit Siegpunkten bekommen und damit auch das Abspiel der Farben beeinflussen.
    • Ausnahmen von dieser Regel sind die Mittelkarten von roter Farbe, diese solltest du behalten, um deine hohen roten Werte vor dem Zugebenmüssen zu schützen.
    • Schiebst du zwei kleine Karten weiter, so gib deinem rechten Nachbarn die niedrigere und dem linken Nachbarn die höhere. Damit sorgst du für den Fall vor, dass dein rechter Nachbar unter Druck die geschobene Karte zu einem Stiches ausspielt oder zugibt, und du nicht weißt, ob du drunterbleiben darfst oder eine höhere Karte abwerfen musst. In jedem Fall kann dein linker Nachbar jetzt nicht risikolos die andere geschobene Karte loswerden, weil er sonst den Stich übernehmen muss. (Schwierig, schwierig, ich kämpfe selber immer noch mit dieser Logik!)
    • Merke dir, welche Karte du an welchen Mitspieler weiter geschoben hast. In der Regel werden es kleine Karten sein. Solange diese Karten noch im Spiel sind, kannst du deine Stich-Taktik darauf aufbauen und du brauchst mit deinen eigenen kleinen Karten den Teufelsstich nicht zu fürchten.
    • Spiele dich in einzelne Farben frei; dann gewinnst du größere Freiheitsgrade beim späteren Zugeben von Karten und kannst öfter frei entscheiden, ob du einen Stich gewinnen oder eine gefährliche Karten loswerden willst.
    • Nutze niedrige rote Karten um alle hohen roten Werte in den gegnerischen Händen heraus zu treiben. Umgekehrt, wirst du deine liebevoll gehüteten roten 36 und 37 blitzschnell los, wenn die Gegner zur rechten Zeit ihre roten 11er, 14er und 16er Karte ausspielen.
    • Merke dir, welche Karten gefallen sind: Das gilt vor allem für hohe Karten mit Stichpotential. Solange die höchsten Karten noch da sind, solltest du keine fast-höchsten Karten ausspielen, um Stiche zu kassieren. Erst musst du mit mittleren Karten die gegnerischen hohen Karten herauslocken.
    • Kassiere rechtzeitig die rote 37, solange noch alle Spieler die rote Farbe bedienen können und sie nicht mit einer niedrigeren nicht-roten Karte überstechen können. Dazu solltest du natürlich wissen, ob es überhaupt noch niedrigere Karten gibt. Hat die Teufelskarte bereits ihren endgültigen niedrigsten Wert, dann solltest du deine hohen Karten mit maximaler Punkteausbeute abspielen. Letzteres ist zwar auch nicht trivial, aber Standard-Weisheit in jedem anständigen Stichkartenspiel.
    • Hast du in der Startausteilung viele hohe Karten bekommen, so muss du versuchen, den Flaschenteufel möglichst schnell auf einen niedrigen Stand bringen. Erst dann machen überwiegend die hohen Karten ihre Stiche und deine guten Werte zahlen sich aus.
    • Wenn du in der Endphase mit zwei benachbarten kleinen Karten in deiner Hand fürchten musst, auf dem Teufelsstich sitzen zu bleiben, musst du zuerst den kleineren Wert spielen. Wenn du damit den Stich bekommst, dann kannst du wenigstens noch hoffen, im letzten Stich den Flaschenteufel wieder loszuwerden. Hast du aber den Stich mit der höheren der beiden Karten gewonnen, so gehört unweigerlich auch der letzte Stich und damit der Flaschenteufel dir.
    • Bist du Startspieler, so kann die gelbe 18 in deiner Hand mit absoluter Sicherheit den ersten Stich machen. Die blaue 17 ist ebenfalls gut, doch kann sie von der gelben 18 überstochen werden, wenn sich ein Spieler beim Schieben von allen seinen blauen Karten befreit hat. Auch die rote 16 ist eine vorzügliche Karte für den Startspieler; sie kann selten mit der 17 oder 18 überstochen werden, weil in der Regel die Mitspieler ihre (hohen) roten Karten behalten haben und die Farbe bedienen müssen. Die rote 16 besitzt dazu noch den Vorteil, dass die Mitspieler beim Farbe-Bedienen kaum niedrige Karten loswerden können und so die Gefahr, mit deinen niedrigen Karten später den Teufelsstich zu bekommen, nicht nennenswert größer wird.
    • Wenn du als Startspieler alle drei Werte von 16 bis 18 bekommen hast, dann spiele sie von oben nach unten! So machst Du damit drei sichere Stiche. Allerdings musst du dabei im Auge behalten, ob dabei die Gegner dabei sehr viele niedrige Kartenwerte abwerfen. Wenn du selbst viele kleine Karten hast, wird das Risiko, dass du darauf sitzen bleibst, immer größer.

Eine kleine theoretische Frage an den Logiker: Mit welcher Kartenhand kannst du alle Stiche machen? Ganz einfach: In einer 4er Runde brauchst du dazu die Karten von 11 bis 18. Du spielst die Karten konsequent von oben nach unten, d.h. 18, 17, 16 usw. und gewinnst alle Stiche, einschließlich Teufelstich. Am Ende hast du zwar keinen einzigen Punkt gewonnen, die Mitspieler aber auch nicht. Ein Nullsummenspiel im wahrsten Sinne des Wortes (von den Minuspunkten für den Teufelsstich einmal abgesehen).

Eine kleine praktische Frage an den Logiker: Wenn du in einer 4er Runde tatsächlich die wunderschöne Kartensequenz von 11 bis 18 bekommen hast, wie spielst du das Blatt? Ganz einfach: du spielst deine Karten von der 17 an von oben nach unten: 17, 16, 15 usw. und bleibst jedes Mal bei Stich. Das machst du so lange, wie – nach dem Stich! – noch mindestens eine niedrigere Karte im Spiel ist. (Achtung: Die Karten im Teufelsstich musst du hierbei natürlich berücksichtigen!) Dann gehst du mit der 18 vom Stich, lehnst dich zurück und lässt deine Mitspieler um Rest-Punkte und Rest-Stiche streiten!

Noch eine kleine theoretische Frage an den Logiker: Bei welcher Kartenverteilung bekommst du garantiert den Teufelsstich? Mit den Karten von 1 bis 8! Egal in welcher Reihenfolge du deine Karten ausspielst, am Ende gehört die Teufelskarte dir. Was ist das beste Ergebnis, das du mit diesem Blatt erzielen kannst? Theoretisch kannst du alle Stiche bekommen, nämlich dann, wenn man dir mit deiner 8 den ersten Stich überlässt und du die dann die restlichen Karten von oben nach unten abziehst. Glück gehabt!

Diese Erkenntnis weist auf ein realistisches (und zugleich defaitistisches) Vorgehen bei einer Überzahl von niedrigen Karten in der Hand hin, mit denen sich der Teufelsstich ziemlich sicher nicht vermeiden lässt: Gewinne mit einer möglichst hohen-niedrigen Karten einen der ersten Stiche und spiele dann die Restkarten von oben nach unten. Damit werden deine Mitspieler relativ wenige Siegpunkte kassieren.

Jetzt eine kleine theoretische Frage an den Mathematiker: Welche ist die kleinste Karte, die du in einer 4er Runde als Startspieler ausspielen darfst, ohne damit gleich beim ersten Ausspiel todsicher für immer und ewig den Teufelsstich zu bekommen? Das ist 7 plus die Anzahl niedrigerer Karten in deiner Hand, ggf. minus 1, falls du eine niedrigere Karte in den Teufelsstich gelegt hast. Bei dieser Lösung gibt es – falls alle drei Mitspieler niedrige Karten zugeben und alle drei unbekannten Karten im Teufelsstich niedrig sind – mindestens noch eine niedrige Karte im Spiel, die die Teufelskarte ablösen muss. (Ein anderes Problem in dieser Situation ist natürlich, wie du nach diesem Manöver deine eigenen niedrigern Karten noch loswirst.)

Aber keine Angst, nur in einer Anfängerrunde läufst du Gefahr, gleich mit dem ersten Stich in den endgültigen Besitz der tödlichen Teufelskarte zu gelangen. Nur Anfänger sitzen gebannt wie das Kaninchen vor der Schlange vor ihrer Kartenhand, wollen unter allen Umständen ihre kleinen Karten loswerden und sind erst beruhigt, wenn endlich der Wert der Teufelskarte niedriger ist als die kleinste Karte in ihrer Hand. Erfahrene Spieler wuchern mit ihren kleinen Karten: Sie nutzen sie dafür, Stiche und Siegpunkte zu machen. Da wird kein Mitspieler mit einer gelben 1 den – ansonsten tödlichen – ersten Stich bekommen. Denn damit wäre die Teufelskarte für das gesamte restliche Spiel festgelegt. Spieler mit Mittelkarten hätten ihre Chance leichtfertig vergeben und bekämen keinen einzigen Stich, während sich der Spieler mit vielen roten 30er Karten schon auf den Punkte-Segen freuen kann, der notwendigerweise über ihn hereinbrechen wird.

Diese Liste von „Tips for Tops“ ist keinesfalls erschöpfend. Ich will hier aber abbrechen. Langer Rede kurzer Sinn: Im „Flaschenteufel“ gibt es eine unüberschaubare Menge von guten Spielzügen, die ein Spitzenspieler alle kennen und anwenden muss, um gute Ergebnisse zu schreiben. Nicht in jedem einzelnen Spiel, aber in der Summe.

Und wer erst einmal am Rechner üben will, bevor er sein Können im Freundeskreis unter Beweis stellen will, dem sei die Internet-Seite des Spiele-Autors Günter Cornett empfohlen. Unter „http://www.flaschenteufel-online.de/“ kann man jederzeit sein Wissen erproben und vertiefen. Das Spiel ist beherrschbar und gewinnt von Mal zu Mal an Tiefe und Raffinesse. Es kann sogar süchtig machen.

Mini-Mäxchen und Mikro-Bluff mit 2 Personen(Teil 2)

Mini-Mäxchen und Mikro-Bluff mit 2 Personen(Teil 2)
Von Günther Rosenbaum , 11.07.2005

Untersuchung einiger Strategien beim Spiel Mikro-Bluff für 2 Spieler

Wie im Teil 1 dieses Artikels beschrieben, ist mit „Mikro-Bluff“ hier das Endspiel von Bluff mit 2 Spielern und je einem Würfel gemeint.

Vor der Diskussion einer speziellen Strategie möchte ich aber noch 2 Zitate aus Teil II von Ferguson, Thomas. S.: Game Theory erläutern, da wir diese Argumente weiter unten nutzen werden.

Zitat 1:

The Minimax Theorem:

Every finite game has a value, and both players have minimax strategies.

We note one remarkable corollary of this theorem. If the rules of the game are changed so that Player II is required to announce her choice of a mixed strategy before Player I makes his choice, then the apparent advantage given to Player I by this is illusory. Player II can simply announce her minimax strategy.

Any p ∈X*  that achieves the maximum of (3) is called a best response or a Bayes strategy against q. In particular, any row i that achieves the maximum of (1) is a (pure) Bayes strategy against q. There always exist pure Bayes strategies against q for every q ∈Y*  in finite games.  (Aus Teil II, Kapitel 4)

Wir sehen:

    • Wenn ein Spieler seine optimale Minimax-Strategie wählt, so kann er diese ruhig veröffentlichen; sein Gegenspieler kann hieraus keinen Vorteil ziehen !
    • Wenn wir zu einer vorgegebenen Strategie eines Spielers eine optimale Gegenstrategie suchen (best response), so findet man hier auch immer eine reine (nicht gemischte) Strategie! Eine solche optimale Gegenstrategie zu einer vorgegebenen Minimax-Strategie realisiert zwar den gleichen Spielwert wie die optimale Minimax-Strategie des Gegenspielers, ist allerdings im allgemeinen keine optimale Minimax-Strategie!

Zitat 2:

In general, the dimension of the space of behavioral strategies is much smaller than the dimension of the space of mixed strategies. The question arises – Can we do as well with behavioral strategies as we can with mixed strategies? The answer is we can if both players in the game have perfect recall . The basic theorem, due to Kuhn in 1953 says that in finite games with perfect recall, any distribution over the payoffs achievable by mixed strategies is achievable by behavioral strategies as well.  (Aus Teil II, Kapitel 5)

Wir befassen uns hier immer mit der expliziten Darstellung des Spieles als „Kuhn-Baum“, also genau so, wie die Spieler ihre Züge abwechselnd durchführen … und nicht mit der Matrix-Darstellung  des Spieles. Der obige Satz sagt nun, dass wir unsere Strategien als Verhaltensstrategien beschreiben dürfen – also zu jedem Zeitpunkt als Wahrscheinlichkeitsverteilung über die verfügbaren Alternativen! (Denn Bluff hat die Eigenschaft „perfect recall“, d.h. die Spieler vergessen keine Infos über frühere Züge und Erkenntnisse).

Nun aber Schluss mit der Theorie, die hier an dieser Stelle doch nie vollständig sein kann !

Wir wollen uns jetzt mit einer ganz speziellen Strategie bei Mikro-Bluff beschäftigen, die lange Zeit bei einigen Westparkgamern als ziemlich gut – ja sogar optimal galt!

Die folgende Behauptung ist falsch:

Der 1. Zug des Startspielers „Konstant 4 ansagen, unabhängig vom eigenen Würfelzug“ ist optimal beim Bluff-Endspiel mit 2 Personen mit jeweils 1 Würfel.

Leider ist das ja keine komplette Beschreibung einer Strategie, sondern nur der ERSTE Zug von vielen möglichen Strategien. Die Behauptung heißt also eigentlich nur: Es gibt eine optimale Minimax- Strategie (A) mit Startzug „Konstant 4“.

Diese Behauptung hat dann zu einer schnellen (und unbedachten) Wette geführt – weshalb ich diese Behauptung hier also widerlegen werde (und damit die Wette gewonnen habe J).

Der Gegenbeweis muss also zeigen, dass keine Strategie mit „Konstant 4 “ als Startzug (egal welche Folgeansagen gemacht werden), optimal sein kann! (Dies ist leider bedeutend schwieriger zu beweisen, als wenn diese Strategie komplett ausformuliert wäre mit allen Reaktionen in den Folgerunden!)

Wir führen den Beweis in 2 Schritten:

1)     Wir definieren eine Gegenstrategie (B1) für Spieler 2 und zeigen damit, dass die „Konstant 4 Strategie“ mit Wahrscheinlichkeit <= 15/36 gegen (B1) gewinnt.
Es mag noch eine bessere Gegenstrategie des Spielers 2 geben, aber wir wissen zu mindestens, dass die Gewinnwahrscheinlichkeit von Spieler 1 bei optimaler Gegenwehr von Spieler 2 höchstens 15/36 ist !

2)     Wir definieren dann eine „bessere“ Strategie (A2) als „Konstant 4“, von welcher wir zeigen werden, dass sie bei optimaler Gegenwehr von Spieler 2 eine Gewinnwahrscheinlichkeit von mehr als 15/36 garantiert!

Da die optimale Minimax-Strategie von Spieler 1 gerade so definiert ist, dass sie die beste Gewinnwahrscheinlichkeit unter allen möglichen Strategien von Spieler 1 bei optimaler Gegenwehr von Spieler 2 garantiert, kann „Konstant 4“ also nicht optimal sein – denn es gibt ja noch eine bessere Strategie (A2).

Beginnen wir also mit Schritt 1:

Sei die Strategie A1 für den Startspieler definiert durch:

1. Runde:

Ungesehen immer 4 ansagen !

2. Runde:

„optimal“ (mit vollständiger Kenntnis der Strategie B1, was dem Spieler 1 gegebenenfalls noch einen Vorteil verschafft)

Sei die Strategie B1 für den Spieler 2 definiert durch:

1. Runde:

4 gehört: Hat 1,2,3 : anzweifeln

: Hat 4,5 oder * : auf 5 erhöhen

2. Runde:

Hört *  und hat * : Auf Pasch * erhöhen;

sonst immer anzweifeln

Behauptung 1: Gewinnwahrscheinlichkeit der Strategie A1 ist <= 15/36

Beweis:

Wir berechnen die Gewinnwahrscheinlichkeit von A1 bei fest vorgegebenem B1 und erhalten damit eine obere Schranke für die Gewinnwahrscheinlichkeit von A1.

Berechnung:

1. Sp2 hat 1,2,3 und hört 4:

Anzweifeln: 2/6 Gewinnwahrscheinlichkeit für Sp2,  1/6 für Sp1

2. Sp2 hat 4,5,* und hört 4:

Spieler 2 erhöht auf 5;

2.1 Sp1 hat 1..3:

Sp1 weiß: Spieler 2 hat 4,5 oder *;

Alternativen: Anzweifeln oder auf * oder Pasch 1..3 erhöhen ergibt 1/3 Chance;

Es ergeben sich also folgende Gewinnwahrscheinlichkeiten:

Sp2: ½ * ½ * 2/3=  6/36                     Sp1: 3/36

2.2 Sp1 hat 4:

Ansage Pasch 4 und gewinnen mit 2/3 (anzweifeln oder * liefern nur 1/3)

Sp2: ½ * 1/6 * 1/3 =1/36                    Sp1:  1/2*1/6*2/3 = 2/36

2.3 Sp1 hat 5:

Anzweifeln geht nicht; Erhöhen auf Pasch 5 liefert 2/3 ( * liefert nur 1/3)

Sp2: 1/2 * 1/6*1/3 = 1/36                Sp1: ½ * 1/6 * 2/3 =  2/36

2.4 Sp1 hat *:

Ansage *, Pasch 5/4 liefern 2/3; Pasch 1,2,3, * liefert 1/3

Sp2: ½* 1/6 * 1/3 = 1/36                 Sp1: ½ * 1/6 * 2/3 = 2/36

In Summe ergeben sich also folgende Gewinnwahrscheinlichkeiten:

Sp2 = 12/36 +  6/36 + 1/36 +1/36 + 1/36   = 21/36

Sp1 = 6/36 + 3/36 + 2/36 + 2/36 + 2/36 = 15/36

Anbei noch die Matrix der Gewinnwahrscheinlichkeiten für Spieler 2 bei den 36 möglichen Würfelergebnissen (bei der oben fett markierten Strategie von Spieler 1; *1/36)::

Sp1/Sp212345*Summe/6
1112/32/32/315/6
2112/32/32/315/6
30011114/6
40001113/6
50000111/3
*001/31/31/32/35/18
Summe/61/31/34/911/187/917/1862/108

q.e.d.

Nun folgt der Schritt 2:

Sei die Strategie A2 für den Startspieler definiert durch:

1. Runde

1,2,*     -> Zu je 1/3 Wahrscheinlichkeit 3,4,5 sagen

3,4,5     -> die Wahrheit sagen

2. Runde

Hört 4:

–         hat 1,2,3: anzweifeln

–         hat 4,5,*: auf Pasch 4 erhöhen

Hört 5:

–         hat 1,2,3,4: anzweifeln

–         hat 5,*:  auf Pasch 5 erhöhen

Hört *:

–         hat 1,2,3,4,5: anzweifeln

–         hat *: Pasch * ansagen

Hört beliebigen Pasch : Anzweifeln

weitere Runden: Anzweifeln

Behauptung 2: Gewinnwahrscheinlichkeit der Strategie A2 ist >= 46/108>15/36

Beweis:

Spieler 2 kennt die Strategie von Spieler 1 und nutzt die optimale Gegenstrategie!

(Spieler 1 hat nach seinem zweiten Gebot alle Informationen über seinen Wurf preisgegeben!)

Die optimale Gegenstrategie von Spieler 2 ist jeweils fett markiert;

z.B. „hat 4: hört 3 : 4                 : 5/6“ bedeutet:

Spieler 2 hat eine 4 geworfen; er hört das Gebot „3“ von Spieler 1 und antwortet mit einem Gebot „4“ und gewinnt in diesem Fall mit einer bedingten Wahrscheinlichkeit von „5/6“ . Die alternativen (schlechteren) Strategien sind in den Folgezeilen aufgeführt um zu überprüfen, dass keine besseren Alternativen vorhanden sind

Spieler 2
hat *: hört 3 : 5             : 1

Anzweifeln           : 0

4                     : 1

5                     : 1

*                     : 5/6

Pasch 1,2             : 1/3

Pasch 3               : 2/3

Pasch 4,5             : 1/6

Pasch *               : 1/6

 

hört 4 : 5              : 1

(Berechnung dito)

 

hört 5 : *              : 5/6

Anzweifeln           : 0

*                     : 5/6

Pasch 1,2             : 1/3

Pasch 3,4             : 1/6

Pasch 5               : 2/3

Pasch *               : 1/6

 

hat 5: hört 3 : 5             : 5/6

Anzweifeln           : 1/3

4                     : 1/6

5                     : 5/6

*                     : 1/6

Pasch 1,2,3,4         : 0

Pasch 5               : 1/6

Pasch *               : 0

 

hört 4 : 5             :5/6

(Berechnung dito)

 

hört 5 : Pasch 5       : 2/3

Anzweifeln           : 0

*                     : 1/6

Pasch 1,2,3,4         : 0

Pasch 5               : 2/3

Pasch *               : 0

 

hat 4: hört 3 : 4             : 5/6

Anzweifeln           : 1/3

4                     : 5/6

5                     : 1/6

*                     : 1/6

Pasch 1,2,3           : 0

Pasch 4               : 1/6

Pasch 5               : 0

Pasch *               : 0

 

 

hört 4 : Pasch 4       : 2/3

Anzweifeln           : 0

5                     : 1/6

*                     : 1/6

Pasch 1,2,3           : 0

Pasch 4               : 2/3

Pasch 5               : 0

Pasch *               : 0

 

hört 5 : Anzweifeln    : 1/3

Anzweifeln           : 1/3

*                     : 1/6

Pasch 1,2,3           : 0

Pasch 4               : 1/6

Pasch 5               : 0

Pasch *               : 0

 

 

hat 3: hört 3 :Pasch 3       : 2/3

Anzweifeln           : 0

4                     : 1/6

5                     : 1/6

*                     : 1/6

Pasch 1,2             : 0

Pasch 3               : 2/3

Pasch 4,5             : 0

Pasch *               : 0

 

hört 4 :Anzweifeln     : 1/3

Anzweifeln            : 1/3

5                     : 1/6

*                     : 1/6

Pasch 1,2             : 0

Pasch 3               : 1/6

Pasch 4,5             : 0

Pasch *               : 0

 

hört 5 :Anzweifeln     : 1/3

Anzweifeln            : 1/3

*                     : 1/6

Pasch 1,2             : 0

Pasch 3               : 1/6

Pasch 4,5             : 0

Pasch *               : 0

 

hat 2: Anzweifeln             : 1/3

hört 3: anzweifeln            : 1/3

4                     : 1/6

5                     : 1/6

*                     : 1/6

Pasch 1,3,4,5         : 0

Pasch 2               : 1/3

Pasch *               : 0

Hört 4: anzweifeln            : 1/3

5                     : 1/6

*                     : 1/6

Pasch 1,3,4,5         : 0

Pasch 2               : 1/3

Pasch *               : 0

Hört 5: anzweifeln            : 1/3

*                     : 1/6

Pasch 1,3,4,5         : 0

Pasch 2               : 1/3

Pasch *               : 0

 

hat 1: Anzweifeln             : 1/3

(Berechnung wie bei 2)

 

 

In Summe ergeben sich also folgende Gewinnwahrscheinlichkeiten für Spieler 2:

1/6 * ( 2/3 + 1/3*5/6 + 2/3*5/6 + 1/3*2/3 + 1/3*5/6 + 1/3*2/3 + 1/3*1/3 +

+ 1/3*2/3 + 2/3*1/3 + 1/3 + 1/3)

= 1/6 * 1/18 *(12+5+10+4+5+4+2+4+4+6+6)

= 1/6 * 1/18 * 62 = 62/108

 

Sp1 = 46/108 > 45/108 = 15/36.

 

Anbei noch die Matrix der Gewinnwahrscheinlichkeiten für Spieler 2 bei den 36 möglichen Würfelergebnissen (bei der oben fett markierten Strategie von Spieler 1; *1/36)::

Sp1/Sp212345*Summe/6
1112/32/32/315/6
2112/32/32/315/6
30011114/6
40001113/6
50000111/3
*001/31/31/32/35/18
Summe/61/31/34/911/187/917/1862/108

q.e.d.

Die Untersuchung der „Konstant 4 Strategie“ ist damit abgeschlossen –

Die echten, optimalen Minimax-Strategien bei Mikro-Bluff kennen wir aber immer noch nicht ! Die obigen Untersuchungen liefern ein paar Hinweise darauf – vielleicht gibt es die Lösung dann in einem späteren Artikel ?!

Als Abschluss dieses Artikels möchte ich noch die Berechnung der Gewinnwahrscheinlichkeiten von einigen Strategien anfügen, die jeweils in der zweiten Runde generell „Anzweifeln“ !

Strategie A = „Konstant 5“:

Falls 1,2,3,4,5,* gewürfelt , sage 5 an

In Folgerunden immer anzweifeln !

Gewinnwahrscheinlichkeit = 1/3

Berechnung Strategie A:

1) Spieler 2 hört 5:

1a)er hat 1,2,3,4

Zweifeln liefert:

Spieler 2:      2/3 Wahrscheinlichkeit zu gewinnen

Alles andere ist schlechter.

1b) er hat 5

Pasch 5 liefert:

Spieler 2:   1/3

1c) er hat *

Ansage * liefert:     1

In Summe ergibt das:

Gewinnwahrscheinlichkeit für Spieler 2:

2/3 * 2/3  +  1/6 * 1/3  +  1/6 * 1 =

= 8/18 + 1/18 + 3/18

= 24/36

Strategie A: Spieler 1 hat also nur 12/36 = 1/3 Gewinnwahrscheinlichkeit !

Strategie B (Die modifizierte 4er Strategie; 0<=p<=1)

1,2,3,4,*          -> 4

5 -> mit Wkeit p   -> 5

mit Wkeit 1-p -> 4

Folgerunden anzweifeln.

Gewinnwahrscheinlichkeit = 10/36 + 3p/36
Berechnung Strategie B:

1)Spieler 2 hört 4

1a)er hat 1,2,3

Anzweifeln liefert: (4-p)/(6-p)

1b) hat 4

Pasch 4 liefert:       2/(6-p)

1c) hat 5

Ansage 5 liefert:      1

1d) hat *

Ansage * liefert:      1

In Summe ergibt Fall 1) also:

Gewinnwahrscheinlichkeit für Spieler 2:

(6-p)/6 *

( 1/2 *   (4-p)/(6-p) +

1/6 *       2/(6-p)  +

1/3 *   (6-p)/(6-p)           )

= 1/6 ( 2 – p/2 + 1/3 + 2 – p/3 )

= 1/6 (13/3 – p*5/6)

= 26/36  –  p* 5/36

2) Spieler 2 hört 5

2a) er hat 1,2,3,4

Verloren:    0

2b) er hat 5,*

Ansage Pasch 5 liefert:    1

In Summe ergibt Fall 2) also:

Gewinnwahrscheinlichkeit für Spieler 2:

p/6 * ( 2/3 * 0  + 1/3 *1)

= p/18

1)+2) zusammen liefern also folgende Gewinnwahrscheinlichkeit für Spieler 2:

26/36 – P *5/36 + p/18

= 26/36 – p * 3/36

Sp1 minimiert dies mit p=1 zu 23/36 und garantiert sich mindestens 13/36 !!

Die Strategie „Konstant 4“ (entspricht p=0) liefert nur 1 – 26/36 = 10/36 Gewinnwahrscheinlichkeit !

In diesem Sinne ist die Strategie „Konstant 4“ (10/36)  schlechter als „Konstant 5“ (12/36).

Aber noch etwas besser ist mit 13/36 die folgende Strategie (p=1):

1,2,3,4,*  ->    4

5          ->    5

Folgerunden anzweifeln.

All diese Aussagen gelten natürlich nur, falls in Runde 2 immer angezweifelt wird !!

Strategie C

1         -> Pasch 1

2,3,4,5   -> die Wahrheit sagen

*         -> Pasch 5

Folgerunden anzweifeln

Gewinnwahrscheinlichkeit = 13/36

Berechnung Strategie C:

1) Spieler 2 hört 2

hat 1 -> verloren

2 -> Pasch 2, gewinnt

3,4,5,* -> 3,4,5,* gewinnt

Wkeit: 1/6 * 5/6 = 5/36

2) hört 3

Wkeit: 1/6 * 4/6 = 4/36

3) hört 4

Wkeit: 1/6 * 3/6 = 3/36

4) hört 5

Wkeit: 1/6 * 2/6 = 2/36

5) hört Pasch 1

Anzweifeln liefert

Wkeit 1/6 * 2/3 = 1/9

6) hört Pasch 5

hat 1,2,3,4 -> anzweifeln, gewinnt

hat 5  -> verliert

hat *  -> Pasch * , gewinnt

Wkeit 1/6 * 5/6 = 5/36

In Summe ergibt das also:

Gewinnwahrscheinlichkeit für Spieler 2:

1/36 * (5+4+3+2+4+5) = 23/36

Für Spieler 1 bleiben also noch 13/36 !

Dies ist genauso viel, wie die verbesserte „konstant 4“ Strategie !

A Winning Strategy For „Sankt Petersburg“

This is a very successful strategy that I have developed over dozens of test games, and which you might try out. Basically there are two „schools of thought“ in „St. Petersburg“: The players who go for buildings (and early victory points), and the players who go for Nobles (late victory points).

This strategy tries to concentrate mostly on upgrades (which are usually underestimated as something to concentrate on), and tries to improve the chances of getting the most of them.

Generally it is clear that an upgrade is MUCH better than the same cost paid for nobles or buildings. A „17“ building gives you 4 VPs, whereas an upgraded „18“ building can give you 4 VP’s AND 3 GP. The same is to be said about the noble and craftsmen upgrades, although the latter are perhaps the least important. If you get your upgrades early, you will get a very good head start against the other players, a fact that is mostly overlooked. Also upgrades, if played evenly on nobles and buildings, will give you more options in every phase, as you will always have money to spend.

Your first turn should look like this:

  1. Craftsmen phase: Pick the two lowest costing cards, as everybody else.
  2. Buildings: Buy only ONE building, and only the cheapest one possible (preferably the 2/6 Potemkin village or the 5 building). If the only available buildings cost 14 or more, don’t buy one, rather save your money for a noble (this is important for the endgame).
  3. Nobles: Buy only ONE noble, and only the cheapest one (preferably the 4 noble). If you haven’t bought a building the turn before you definitely MUST buy a noble now, even if it is expensive.
  4. Upgrades: You now should have 21 Gold (if you have been very lucky = 25-(3+2+2+4)+7) or less. Choose upgrades in the following order:
    1. Blue upgrades that bring a lot of money
    2. Blue upgrades that bring both money and VP’s
    3. Blue upgrades that bring more VP’s than money
    4. Red upgrades that bring VP’s AND money (Never buy a red upgrade that only brings VP if it costs you more than 6 Gold)
    5. Green upgrades that bring additional money or VP’s (NEVER buy Green upgrades that bring you discounts, they are good, but not for this strategy).

Basically you should hoard upgrades as much as possible, but only bring your hand up to three if you are absolutely positive you can build at least one upgrade in the next Craftsmen or Building phase.

IMPORTANT: Don’t build the upgrade yet, not even if it’s a green one. Never build an upgrade BEFORE the phase you really need it, as having more money always gives you more options. ALWAYS build the upgrade(s) as the last thing you do in a phase WHERE THEY APPLY, there is never any hurry, you can always play as long you want until the game continues.

Your next turns (until the last turn of the game) should look like this:

  1. Craftsmen phase: Buy everything you can, as long as you still make a profit (Usually it is clear if this or next round will be the last: if there is a possibility to „force“ the game end, always assume that somebody will do it).
  2. Buildings: First look if you can buy a cheap (or even several) building/s AND upgrade the building you bought last round. If this is possible, by all means do so (in this order)! If you can only upgrade one (or even more) buildings, do so as well. If you don’t have a building you can upgrade, buy as many of the cheapest buildings as you can. Regardless of what you do take care to have AT LEAST 7 Gold to spend for the Nobles phase (INCLUDING the income you make this round). If you have room in your hand you can stock up on cheap buildings*, but only to the limit of three if you are sure you can play a card out of your hand in the next phase. You should keep fully invested here: don’t spend too little, as you need the VP income, don’t spend to much, you want to be fully active in the Nobles phase. The most common nobles are the 4’s and 7’s, so having 7 gold saved will make it pretty safe for you.
  3. Nobles: First look if you can buy a cheap (or even several) noble/s AND upgrade a Noble. If this is possible, by all means do so! If you can only upgrade one or several nobles, by all means do so. If you don’t have a noble to upgrade, buy the cheapest (or several) cheap nobles. If you have room in your hand, stock up on the cheapest nobles*, but only up to three if you are sure that you can play a card out of your hand in the next round (Upgrades). The rule about keeping fully invested also applies here, but this time you want to keep AT LEAST 5 Gold instead of 7 (with 5 you are usually sure to buy a craftsman in the craftsmen phase, and that is the next phase that you want to spend money in, not in he upgrade phase where you only HOARD the cards).
  4. Upgrades: Stock up on upgrades THAT YOU WILL BE ABLE TO PLAY IN THE NEAR FUTURE (meaning you own cards that can be upgraded). Stock up even up to three, but only if you are sure to be able to play something out of your hand in the next BLUE round.This is the order of acquisition preference UP TO TURN THREE OF THE GAME:
    1. Blue upgrades that bring a lot of money
    2. Blue upgrades that bring both money and VP’s
    3. Blue upgrades that bring more VP’s than money
    4. Red upgrades that bring VP’s AND money, if not available: only VP’s, if not available: only money (never buy a red upgrade that only brings VP, if it’s more expensive than paying 6 Gold)
    5. Green upgrades that bring additional money or VP’s (NEVER buy Green upgrades that bring you discounts, they are good, but not for this strategy).

    AFTER TURN THREE the order changes as follows:

    1. Blue upgrades that bring a lot of VP’s
    2. Blue upgrades that bring VP’s AND money
    3. Red upgrades that bring both VP’s and money, if not available: only VP’s (after turn three, NEVER buy red upgrades that only bring money, only if you’re desperate)
    4. Green upgrades that bring VP’s.

    After turn 3 you should never buy any upgrade that only brings money if possible, you have already built them! Hoard the upgrades, as always playing them at the last moment where they apply!

 

The last turn of the game should look like this:

  1. Craftsmen: only buy one if you still make a profit (= pay 2 or less).
  2. Buildings: Now buy the MOST expensive building you see on the board (of course preferring the discounted one, if there is one), IF you can still be sure to be able to play all the cards you have hoarded in this round, of course. Also play all Blue upgrades you have (this should be top priority, of course). Invest all your money; you now should have a good income of the upgraded buildings for the next round.
  3. Nobles: You should have a lot of money to spend now. Buy/play out of your hand AS MANY DIFFERENT NOBLES AS YOU CAN! You want to have the most! This is where all the cheap nobles that you have hoarded might come in handy. What nobles you buy depends on your hand and on the money you are able to spend, but you want to have as many different ones as possible in the end.
  4. UPGRADES: With the money from your nobles, BUY MORE NOBLES! There might be some on the board which you haven’t been able to pay yet, or there might be more in your hand, deviously hidden.

*) About hoarding: if more than one other player plays the „building“ strategy, you should hoard buildings, to foil them. The same is to be said about hoarding nobles if more than one player plays the „nobles“ strategy. In the last turn of the game you might even consider hoarding nobles you can’t actually play anymore, if they take away more than 5 points from a nobles-collector (very tricky, this one).

And this is it: If you follow these guidelines, you will usually win or come in second in a game that has the other players collect buildings or nobles. Try it out – you will be surprised how well it works. Of course there is always the possibility that the „cards are not right“- this is a luck-driven game as well…

Cards you can consider buying:

  • „Sternwarte“ (to be able to select an additional card in the next round)
  • The Schenke card (use it in the last two rounds)
  • Cards you should NEVER buy:
  • Cards giving discounts only, but no VP’s or Gold
  • The expensive noble upgrades that only give VP’s

Cards you should ALWAYS buy:

  • Potemkin village

 

Mini-Mäxchen und Mikro-Bluff mit 2 Personen

Günther Rosenbaum

Mäxchen und Bluff(von Richard Borg, FX Schmid, 1993)/Liars Dice(MB, 1974) sind bekannte Würfelspiele. Mäxchen habe ich schon als Jugendlicher gespielt und Bluff/Liar´s Dice ist auch heute an geselligen Spieleabenden immer wieder auf dem Tisch.

In der amerikanischen mathematischen Literatur findet man auch Analysen des Spieles Liar´s Dice – aber Vorsicht – hier ist das bei uns unter dem Namen Mäxchen (oder Meier) bekannte Spiel gemeint. Literatur über Bluff habe ich nicht gefunden.

In unseren Spielrunden kommt immer wieder mal die Diskussion hoch, welches denn nun die beste Strategie im Endspiel bei Bluff wäre und der eine oder andere behauptet dann auch mal inbrünstig, dieses oder jenes Verhalten wäre das optimale!

Den (mathematischen) Beweis schuldig geblieben sind bisher jedoch alle – und auch eine plausible, einfache Strategie hat bisher noch keiner angegeben!

Gerade wegen dieser angeregten Diskussionen stellt sich dem ingefleischten Spielefreak da natürlich die Frage nach einer optimalen Strategie für diese Spiele. Diese ist leider nicht so einfach zu finden – die Theorie bestätigt uns hier allerdings zumindest die Existenz einer solchen (gemischten) optimalen Strategie für 2 Personen.

Daher betrachten wir hier zuerst mal eine vereinfachte Version des Spieles Mäxchen – Mini-Mäxchen für 2 Personen:

Es wird mit einem Würfel gespielt. Spieler 1 würfelt verdeckt, sagt eine Würfelzahl an und behauptet, dass er mindestens diese Zahl gewürfelt hat.

  1. a) Spieler 2 kann dieses anzweifeln – in diesem Fall wird aufgedeckt und falls die behauptete Zahl größer als die gewürfelte Zahl ist gewinnt Spieler 2 ansonsten Spieler 1.
  2. b) Spieler 2 kann diese Behauptung auch glauben – dann muss er erneut Würfeln und einen höheren Wert behaupten. Spieler 1 kann dann wieder glauben oder anzweifeln, u.s.w.

Dieses Spiel sieht sehr einfach aus – trotzdem ist die optimale Strategie nicht trivial.

Zu beachten ist hierbei, dass das Spiel ja das Zufallselement „Würfel“ beinhaltet – wir können (im Allgemeinen) also keine Strategie angeben mit der wir immer gewinnen. Stattdessen suchen wir eine Strategie, bei welcher wir bei häufiger Wiederholung des Spieles möglichst oft gewinnen!

Zuerst betrachten wir folgende (nicht optimale, aber einfache) Strategie:

Spieler 1: Sagt in der ersten Runde immer die Wahrheit und zweifelt bei einer Folgerunde immer an.

Spieler 2: Glaubt die Behauptung von Spieler 1 immer (außer die 6) und versucht besser zu würfeln.

Analyse:

Was ist die optimale Gegenstrategie für Spieler 2?

Da Spieler 1 immer die Wahrheit sagt, braucht Spieler2&xnbsp; also nie anzuzweifeln; da Spieler 1 in der zweiten Runde immer anzweifelt/aufdeckt, ist die Gewinnwahrscheinlichkeit für Spieler 2: {Spieler1 sagt 1 und Spieler 2 würfelt 2..6} + {Spieler 1 sagt 2 und Spieler 2 würfelt 3..6} + … = 1/6 * {5/6 + 4/6 + 3/6 + 2/6 + 1/6 + 0} = 15/36 .

Spieler 1 hat also eine Gewinnwahrscheinlichkeit von mindestens 21/36!

Was ist nun die optimale Gegenstrategie für Spieler 1?

Da Spieler 2 ja immer alles glaubt (außer die 6), ist es für Spieler 1 natürlich das Beste, die Chancen für das Nachwürfeln des Spielers 2 zu minimieren – er behauptet also bei 1..5 immer, mindestens eine 5 gewürfelt zu haben. Spieler 2 kann nun mit 1/6 Wahrscheinlichkeit noch eine 6 würfeln und gewinnen. Spieler 1 gewinnt also mit Wahrscheinlichkeit von höchstens 1/6 * (5 * 5/6 +1) = 31/36!

Ergebnis:

Wir sehen, die Strategie von Spieler 2 ist die optimale Gegenstrategie zu Spieler 1 aber nicht umgekehrt! Die beiden Strategien stehen also nicht im gegenseitigen Gleichgewicht zueinander – sie sind also nicht optimal!

Nun zur optimalen Strategie:

Spieler 1 gewinnt mit Wahrscheinlichkeit 41/60 (=Wert des Spieles).

Strategie Spieler 1:

Wurf 1 -> er behauptet 3 bzw. 4 bzw. 5 mit den Wahrscheinlichkeiten 3/10, 5/10, 2/10.

(d.h. bei 1 lügt er immer!)

Wurf 2 -> er behauptet 2 bzw. 3 mit den Wahrscheinlichkeiten 3/10, 7/10

Wurf 3, 4, 5, 6 -> er sagt die Wahrheit

In der zweiten Runde zweifelt Spieler 1 alles an!

Strategie Spieler 2:

Vorgabe 1 oder 2 ->  immer glauben und nachwürfeln

Vorgabe 3  -> mit 1/3 Wahrscheinlichkeit anzweifeln, sonst glauben

Vorgabe 4-> mit 1/2 Wahrscheinlichkeit anzweifeln, sonst glauben

Vorgabe 5 -> mit 3/5 Wahrscheinlichkeit anzweifeln, sonst glauben

Vorgabe 6 -> immer anzweifeln

Falls Spieler 2 glaubt und nachwürfelt, so sagt er dann die nachgewürfelte Zahl an, sofern diese genügend hoch ist. Andernfalls erhöht er das Gebot des Vorgängers um 1.

In einer zweiten Runde zweifelt Spieler 2 alles an!

Beweis:

1) Wir nehmen jetzt an, Spieler 1 nutzt die angegebene Strategie und wir suchen jetzt die optimale Gegenstrategie.

  1. a)Wenn Spieler 2 die 6 hört, dann hat Spieler1 auch die 6 und Spieler 2 verliert beim zweifeln.
  2. b)Spieler 2 hört als Ansage die 5. Die Wahrscheinlichkeit, dass Spieler 1 wirklich die 5 hat ist P(Spieler1 hat 5 | Spieler 1 sagt 5) = P(Spieler 1 hat 5 und sagt 5)/P(Spieler 1 sagt 5) = (1/6 ) / (1/6 + 1/6 * 2/10) = (1/6) / (1/5) = 5/6 Wenn Spieler 2 anzweifelt, so gewinnt er also mit Wahrscheinlichkeit 1/6; glaubt Spieler 2 das Gebot von 5, so kann er noch mit Wahrscheinlichkeit 1/6 eine 6 nachwürfeln und gewinnen. Beide Wahlmöglichkeiten sind für Spieler 2 gleich schlecht – dieses Prinzip der „Indifferenz“ zwischen den sinnvollen Wahlmöglichkeiten ist charakteristisch für optimale Strategien!
  3. c)Spieler 2 hört als Ansage die 4. Die Wahrscheinlichkeit, dass Spieler 1 wirklich die 4 hat ist P(Spieler1 hat 4 | Spieler 1 sagt 4) = P(Spieler 1 hat 4 und sagt 4)/P(Spieler 1 sagt 4) = (1/6) / ( 1/6 + 1/6 * ½) = (1/6) / (1/4) = 2/3 Wenn Spieler 2 anzweifelt, so gewinnt er also mit Wahrscheinlichkeit 1/3; glaubt Spieler 2 das Gebot von 4, so kann er noch mit Wahrscheinlichkeit 1/3 eine 5 oder 6 nachwürfeln und gewinnen – denn Spieler 1 wird auf jeden Fall anzweifeln und ein Bluffen wird also nicht wirken.
  4. d)Spieler 2 hört als Ansage die 3. Die Wahrscheinlichkeit, dass Spieler 1 wirklich die 3 hat ist P(Spieler1 hat 3 | Spieler 1 sagt 3) = P(Spieler 1 hat 3 und sagt 3)/P(Spieler 1 sagt 3) = (1/6) / ( 1/6 + 1/6 * 3/10 + 1/6 * 7/10) = (1/6) / (1/3) = 1/2 Wenn Spieler 2 anzweifelt, so gewinnt er also mit Wahrscheinlichkeit 1/2; glaubt Spieler 2 das Gebot von 3, so kann er noch mit Wahrscheinlichkeit 1/2 eine 4, 5 oder 6 nachwürfeln und gewinnen.
  5. e)Spieler 2 hört als Ansage die 2. Spieler 1 sagt also die Wahrheit, Spieler 2 glaubt und würfelt mit Wahrscheinlichkeit 2/3 etwas Besseres.
  6. f)Die Ansage 1 wird von Spieler 1 nie angesagt.
  7. g)Was ist nun die Gewinnwahrscheinlichkeit von Spieler 1 bei optimaler Gegenstrategie von Spieler 2? Hört Spieler 2 eine Ansage von 3 bis 6 von Spieler 1, so können wir annehmen, dass Spieler 2 anzweifelt, da er ja gleich hohe Gewinnwahrscheinlichkeiten beim Zweifeln und Glauben hat. Spieler 1 gewinnt also, wenn er 3 .. 6 sagt und nicht gelogen hat. Dies ist genau dann der Fall, wenn er 3, 4, 5 oder 6 gewürfelt hat (Wahrscheinlichkeit 2/3). Zusätzlich ist noch der Fall zu betrachten, dass Spieler 1 eine 2 würfelt (1/6) und auch eine 2 ansagt (3/10) und dann noch Spieler 2 nichts besseres würfelt (1/3). Also ist die Gewinnwahrscheinlichkeit für Spieler 1 mindestens = 2/3 + 1/6 * 3/10 * 1/3 = 2/3 + 1/60 = 41/60 !!

2) Als zweites nehmen wir jetzt an, Spieler 2 nutzt die angegebene Strategie und wir suchen jetzt die optimale Gegenstrategie von Spieler 1. (D.h. wir suchen die maximale Gewinnwahrscheinlichkeit von Spieler 1.)

  1. a)Nehmen wir an, Spieler 1 wirft eine 1 oder 2. Blufft er und sagt eine 3, 4 oder 5 an, so gewinnt er, wenn Spieler 2 dies glaubt und nichts besseres würfelt, also mit Wahrscheinlichkeiten (bei 3): 2/3 * ½ = 1/3 (bei 4): ½ * 2/3 = 1/3 (bei 5): 2/5 * 5/6 = 1/3 Bei einer Ansage von 6 würde er sofort verlieren und auch eine Ansage von 1 ist mit einer Gewinnwahrscheinlichkeit von 1/6 für Spieler 1 nicht sinnvoll. Was geschieht jetzt noch bei einer Ansage von 2? Spieler 2 glaubt diese und würfelt nach – womit Spieler 1 wiederum mit Wahrscheinlichkeit 1/3 gewinnt. Auch hier ist also wieder das Prinzip der Indifferenz zu beobachten! Spieler 1 gewinnt hierbei also bestenfalls mit Wahrscheinlichkeit 1/3.
  2. b)Nehmen wir an, Spieler 1 wirft eine 3. – Würde er mit einer 1 oder 2 bluffen, so gewinnt er nur mit 1/6 oder 1/3 Wahrscheinlichkeit. – Sagt er die Wahrheit – also die 3 – an, so gewinnt er, falls Spieler 2 anzweifelt oder wenn Spieler 2 glaubt und nichts Besseres würfelt. Er gewinnt also mit Wahrscheinlichkeit 1/3 + 2/3 * ½ = 2/3. – Blufft er mit 4, so gewinnt er nur, wenn Spieler 2 dieses glaubt und nichts besseres würfelt; also mit Wahrscheinlichkeit ½ * 2/3 = 1/3 – Blufft er mit 5 so gewinnt er mit Wahrscheinlichkeit 2/5 * 5/6 = 1/3 – Blufft er mit 6, so verliert er immer. Also: Die beste Strategie ist hier, die Wahrheit zu sagen mit einer Gewinnwahrscheinlichkeit 2/3!!
  3. c)Nehmen wir an, Spieler 1 wirft eine 4. Wie unter b) sieht man wieder, dass man auch jetzt am besten die Wahrheit sagt mit einer Gewinnwahrscheinlichkeit von ½ + ½ * 2/3 = 5/6
  4. d)Nehmen wir an, Spieler 1 wirft eine 5. Wie unter b) sieht man wieder, dass man auch jetzt am besten die Wahrheit sagt mit einer Gewinnwahrscheinlichkeit von 3/5 + 2/5 * 5/6 = 14/15
  5. e)Wirft Spieler1 eine 6, so gewinnt er sicher wenn er diese auch ansagt.
  6. f)Mit welcher Wahrscheinlichkeit kann Spieler 1 also höchstens gewinnen bei der vorgegebenen Strategie von Spieler 2? Wie oben berechnet erhalten wir 1/6 * ( 1/3 + 1/3 + 2/3 + 5/6 + 14/15 + 1) = 1/6 * (10/30 + 10/30 + 20/30 + 25/30 + 28/30 + 30/30) = 1/6 * (123/30) = 123/180 = 41/60

3) Die beiden beschriebenen Strategien für Spieler 1 und 2 sind also optimal im Sinne des Minimax Theorems der Spieltheorie und der Spielwert des Spieles beträgt 41/60. Spieler 1 gewinnt also bei Anwendung der vorgegebenen Strategie mit einer Wahrscheinlichkeit von mindestens 41/60 und Spieler 2 verliert höchstens mit einer Wahrscheinlichkeit von 41/60 bei Anwendung seiner optimalen Strategie!

Nachdem wir oben gesehen haben, dass selbst das sehr einfache Minimäxchen schon eine komplizierte optimale Strategie besitzt, betrachten wir nun das Spiel Bluff/Liar´s Dice und versuchen uns der optimalen Strategie des Endspieles bei Bluff (2 Personen mit je einem Würfel) langsam zu nähern…

Daher betrachten wir hier zuerst mal eine vereinfachte Version des Spieles Bluff 

Micro-Bluff/Micro Liars Dice für 2 Personen:

Es wird mit je&xnbsp; einem Würfel pro Spieler gespielt. Beide Spieler würfeln verdeckt, Spieler 1&xnbsp;&xnbsp; sagt eine Würfelzahl und eine Häufigkeit an&xnbsp; und behauptet, dass beide Spieler gemeinsam diese Zahl in mindestens dieser Häufigkeit gewürfelt haben.

  1. a) Spieler 2 kann dieses anzweifeln – in diesem Fall wird aufgedeckt und falls die Behauptung falsch war gewinnt Spieler 2 ansonsten Spieler 1.
  2. b) Spieler 2 kann diese Behauptung auch glauben und das Gebot erhöhen – dann muss er(ohne erneutes Würfeln) eine höhere Würfelzahl bei gleicher Häufigkeit oder eine beliebige Würfelzahl mit höherer Häufigkeit behaupten. Spieler 1 kann dann wieder glauben oder anzweifeln, u.s.w.
  3. c) Falls auf dem Würfel auch ein Stern vorhanden ist, gilt dieser als höchste Zahl und als Joker für jede andere Zahl. Die Ordnung der Gebote ist dann so, dass direkt vor 2k Einsen das Gebot von k Sternen einzuordnen ist.

Nano Liars Dice (I):

Mit Würfel W6 und den Geboten 1<2<3<4<5<6 , also ohne Pasch und ohne Stern!

Optimale Strategie(für Nano Liars Dice (I)):

Spieler 1 gewinnt mit Wahrscheinlichkeit 7/12&xnbsp; (=Wert des Spieles).

Strategie Spieler 1:

Spieler 1 sagt immer die gewürfelte Zahl an, also die Wahrheit.

In 2-ter Runde immer anzweifeln!

Strategie Spieler 2:

Ist der eigene Wurf von Spieler 2 höher als die Ansage von Spieler 1,&xnbsp; dann erhöht er auf die gewürfelte Zahl;&xnbsp; ansonsten zweifelt er an.

In 2-ter Runde immer anzweifeln!

Analyse/Beweis:

  1. a)Wir nehmen jetzt an, Spieler 1 nutzt die angegebene Strategie und wir suchen jetzt die optimale Gegenstrategie. (zu beachten ist hier, dass Spieler 1 ja in der zweiten Runde immer anzweifelt).
  2. b)Wenn Spieler 2 die 6 hört, dann hat Spieler1 auch die 6 und Spieler 2 verliert beim Zweifeln.
  3. c)Spieler 2 hört als Ansage ein x mit x=1,2,3,4 oder 5. Spieler 1 sagt immer die Wahrheit und zweifelt in der 2-ten Runde. Anzweifeln macht also keinen Sinn und Bluffen auch nicht. Also gewinnt Spieler 2 genau dann, wenn er besser als die Ansage von Spieler 1 gewürfelt hat und dieses dann ansagt! Bei einer Ansage von x gewinnt Spieler 2 also mit Wahrscheinlichkeit (6-x)/6, wohingegen Spieler 1 mit Wahrscheinlichkeit x/6 gewinnt.
  4. d)Was ist nun die Gewinnwahrscheinlichkeit von Spieler 1 bei optimaler Gegenstrategie von Spieler 2? (Spieler 2 hört 6&xnbsp; + Spieler 2 hört 5 und verliert + Spieler 2 hört 4 und verliert + ….) P(Spieler 1 gewinnt) = 1/6 * (1 + 5/6 + 4/6 + 3/6 + 2/6 + 1/6) = 21/36 = 7/12 .

2) Als zweites nehmen wir jetzt an, Spieler 2 nutzt die angegebene Strategie und wir suchen jetzt die optimale Gegenstrategie von Spieler 1. (D.h. wir suchen die maximale Gewinnwahrscheinlichkeit von Spieler 1.) Beachte: Spieler 2 zweifelt in der 2-ten Runde immer an!

  1. a)Spieler 1 kann hier prinzipiell auch bluffen. Nehmen wir an, Spieler 1 blufft mit einer Ansage y < x , wenn er x gewürfelt hat. => Spieler 2 habe a gewürfelt; nach seiner Strategie gilt: (I)&xnbsp;&xnbsp;&xnbsp;&xnbsp; a < y&xnbsp; , Spieler 2 zweifelt . Spieler 1 verliert (außer, wenn a = y) (II)&xnbsp;&xnbsp; y < a < x , Spieler 2 glaubt und behauptet a. Spieler 1 erhöht auf x und gewinnt. (III) a > x , Spieler 2 glaubt und behauptet a. Spieler 1 verliert. Hätte Spieler 1 nicht geblufft und gleich x behauptet, so hätte er im Fall (I) immer gewonnen, im Fall (II) genauso gewonnen und im Fall (III) noch bei a = x gewonnen. In allen Fällen wäre er also ohne Bluff („tiefstapeln“) besser gewesen ! Da wir die optimale Gegenstrategie für Spieler 1 suchen, brauchen wir also den Fall, dass wir eine kleinere Zahl ansagen als wir gewürfelt haben, nicht mehr zu betrachten!
  2. b)Wir nehmen an, Spieler 1 hat x gewürfelt und y > x angesagt. Spieler 2 hat a gewürfelt. Ist a = y (Wahrscheinlichkeit 1/6), so gewinnt Spieler 1. Ist a < y (Wahrscheinlichkeit (y-1)/6 ), so gewinnt Spieler 1, falls er nicht gelogen hat.(Falls er gelogen hat, also y > x, so würde ein Bluff mit y+1 sofort erkannt werden, da Spieler 2 in der zweiten Runde ja immer anzweifelt. Hier benötigen wir Aussage a), denn bei y < x wäre eine Erhöhung auf x möglich). Ist a > y, so gewinnt Spieler 2, da ja Spieler 1 wegen y > x nicht mehr ohne zu bluffen erhöhen kann! Mit P(a | b) = P(a und b) / P(b) erhalten wir also: P(Spieler 1 gewinnt) = = P(Spieler 1 sagt 1) * ( 1/6 ) + + P(Spieler 1 sagt 2) * ( 1/6 + (2-1)/6 *P(Spieler 1 hat 2 | Spieler 1 sagt 2) ) +…+ P(Spieler 1 sagt 6) * (1/6 + (6-1)/6 * P(Spieler 1 hat 6 | Spieler 1 sagt 6) ) = 1/6 * ( P(Spieler 1 sagt 1) + …+ P(Spieler 1 sagt 6) ) + + 1/6 * P(Spieler 1 hat 2 und sagt 2) + 2/6 * P(Spieler 1 hat 3 und sagt 3)+ + … + 5/6 * P(Spieler 1 hat 6 und sagt 6) = 1/6 * ( 1 ) + 1/6 * P(Spieler 1 hat 2 und sagt 2) + + 2/6 * P(Spieler 1 hat 3 und sagt 3) +…+ 5/6 * P(Spieler 1 hat 6 und sagt 6)

Nun ist&xnbsp; P(Spieler 1 hat y und sagt y) < P(Spieler 1 hat y) = 1/6. Damit erhalten wir P(Spieler 1 gewinnt) < < 1/6 + 1/6 * 1/6 + 2/6 * 1/6 + … + 5/6 * 1/6 = 1/36 * (1 +2 +3 + 4 +5 + 6) = 7/12. Die optimale Gegenstrategie von Spieler 1 gewinnt also mit einer Wahrscheinlichkeit von höchstens 7/12; wenn Spieler 1 bei einem Wurf von 2 bis 6 immer die Wahrheit sagt, so erreicht er diese 7/12 auch!

3) Die beiden beschriebenen Strategien für Spieler 1 und 2 sind also wieder optimal im Sinne des Minimax Theorems der Spieltheorie und der Spielwert des Spieles beträgt 7/12!

Bemerkung: Der Spielwert des Spieles ist 7/12 und eindeutig; die optimalen Strategien sind im Allgemeinen nicht eindeutig! Z.B. kann Spieler 1 im obigen Beispiel beim Wurf einer 1 auch jeden anderen Wert ansagen; oder auch mit Wahrscheinlichkeiten 0,7 und 0,3 die Werte 3 und 4 ansagen . Seine Gewinnwahrscheinlichkeit bleibt weiterhin 7/12 bei optimalem Gegenspiel von Spieler 2! Spieler 2 kann seine Strategie noch dahingehend verbessern, dass er auf nicht optimales Spiel seines Gegners reagiert – z.B. im Fall, dass er genau die Ansage seines Gegners auch gewürfelt hat (und damit beim Zweifeln automatisch verliert), kann er bluffen, die Ansage erhöhen und hoffen, dass sein Gegner nicht anzweifelt !

Nano Liars Dice (II):

Mit Würfel Wn und den Geboten 1<2< … < n-1 < n , also mit einem n-seitigen Würfel und ohne Pasch und ohne Stern!

Optimale Strategie(für Nano Liars Dice (II)):

Spieler 1 gewinnt mit Wahrscheinlichkeit ½ + 1/(2n) (=Wert des Spieles).

Strategie Spieler 1:

Spieler 1 sagt immer die gewürfelte Zahl an, also die Wahrheit.

In 2-ter Runde immer anzweifeln!

Strategie Spieler 2:

Ist der eigene Wurf von Spieler 2 höher als die Ansage von Spieler 1,&xnbsp; dann erhöht er auf die gewürfelte Zahl; ansonsten zweifelt er an.

In 2-ter Runde immer anzweifeln! 

Beweis:

Der Beweis ist identisch zu Nano Liars Dice (I).

Der Spielwert berechnet sich aus

(1/n2) * (n + (n-1) + … + 1) =

= (1/n2) * (n + 1) * (n/2)

= (1/n2) * (n2/2&xnbsp; + n/2)

= ½ + 1/(2n)

Bemerkung:

Würde man noch den „Stern“ zulassen, als höchste Zahl – aber auch als Joker, so ändert sich die Strategie nicht. Würfelt jemand den Stern, so würde man diesen sofort ansagen und gewinnen.

Nano Liars Dice (III):

Mit Würfel W6 und den Geboten 1<2<3<4<5<6<11<22<33<44<55<66, also mit Pasch aber ohne Stern!

Wie ändern sich nun die Strategien der Spieler?

Hier ein paar Vorüberlegungen:

Kann Spieler 1 schon einen Pasch vorhersagen in seiner ersten Runde?

Wenn Spieler 1 eine 1 gewürfelt hat, dann könnte er 11 (Pasch 1) ansagen (und sich damit seine Gewinnwahrscheinlichkeit von 1/6 nach einer geworfenen 1 bewahren) – aber in allen anderen Fällen würde er sich seine Gewinnwahrscheinlichkeit ja auf 1/6 reduzieren!

Das Einführen des Pasches ist also wohl ein Vorteil für Spieler 2.

Allenfalls, wenn Spieler 1 in der ersten Runde „tiefstapelt“, könnte er in der zweiten Runde sinnvoller weise auf einen Pasch erhöhen.

Wie reagiert man, wenn der Gegner einen Pasch angesagt hat?

Wenn die Ansage falsch ist, so kann man ruhig anzweifeln – wenn die Ansage richtig ist, so kann man sowieso nicht mehr erhöhen und damit auch gleich anzweifeln.

Also: Ein vom Gegner angesagter Pasch sollte sofort angezweifelt werden.

Beispiel:

Spieler 1 sagt 6 an.

  1. a)Spieler 2 würfelt a zwischen 1 und 5. Er kann Anzweifeln oder Pasch a ansagen. Will er aber bei Pasch a gewinnen, so müsste Spieler 1 auch ein a<6 gewürfelt haben und hätte somit bei der Ansage von 6 auch gelogen. Damit sollte Spieler 2 also direkt anzweifeln
  2. b)Spieler 2 würfelt auch eine 6. Wenn er Anzweifelt, so hätte er verloren. Also kann er genauso gut gleich Pasch 6 ansagen und gewinnt, falls Spieler 1 nicht gelogen hat.

Korrekturen, Anregungen und insbesondere weiterführende Vermutungen und Lösungen werden gerne entgegengenommen!

Literatur:

 Ferguson, Thomas. S.: Game Theory. Class notes for Math 167, Fall 2000

Models for the game of Liar´s dice. Stochastic games and related topics. In honor of Prof. L. S. Shapley, Proc. Workshop, Chicago/IL (USA) 1987, Theory Decis. Libr., Ser. C 7, 15-28 (1991).

The tactics of liar dice. R. Stat. Soc.,Ser.C 38, No.3, 507-516 (1989). [ISSN 0035-9254]

Ponssard, Jean-Pierre; Sorin, Sylvain: Optimal behavior strategies in 0-sum games with almost perfect information. Math. Oper. Res. 7, 14-31 (1982). [ISSN 0364-765X]

Kings & Castles Strategietipps

Kings & Castles1. Der erste Spieler

Wie von uns bei den Spielen bisher bemerkt wurde, ist der Anfangsvorteil immens: Man kann nicht nur bei großzügigem Einsatz seiner Truppen viele Punkte machen (eine Wertung ist fast Pflicht als Startspieler), nein, man ist auch DER EINZIGE der punktet, da die anderen Spieler noch nicht auf dem Feld sind. Daher sollte man ruhig versuchen, hierauf zu bieten. Viele Spieler sind bei ihrem ersten Spiel dabei sehr konservativ und zaudernd, ich persönlich würde für den Startspielervorteil bis zu 6 Armeepunkte bieten, vielleicht sogar mehr.

2. Die kleinen Länder

Die meiste „Action“ des Spiels wird in England und Frankreich geschehen, wobei Frankreich vielleicht sogar die meisten Kämpfe sieht, da hier die neutrale Farbe Schwarz nie ausgerottet werden kann. Danach kommen, in dieser Reihenfolge, Schottland, Irland und schliesslich Wales. Man sollte versuchen in mindestens einem dieser Länder einmal König zu werden (wobei Wales und Irland besonders interessant sind), denn dort platzierte Armeen leben länger. Gerade Wales ist sehr klein und nur zweimal Königsland, und daher besonders geeignet. Selten wird Wales auch als zweites Aktionsland genutzt, da es zu wenige Möglichkeiten bietet, unerwünschte andersfarbige Armeen fies einzusetzen.

3. Die schwarzen Armeen

Grundsätzlich gilt, daß man möglichst immer schwarze Armeen in einem Regierungsland leben lassen sollte. Ideal sollte man dann die Felder, in die die Schwarzen angreifen können, gut verteidigen. Die anderen Spieler sind dann entweder gezwungen, die Schwarzen erst auszurotten, bevor sie gegen einen selber aktiv werden können, oder sie müssen sehr viele schwarze Armeen haben, was nicht unbedingt häufig ist.

4. Die „uninteressanten“ Länder

Kings and Castles Das Geheimnis des Sieges bei „Kings & Castles“ sind nicht die Punkte, die man bei seinen eigenen Taxierungsrunden macht, sondern die, DIE MAN BEI DENEN DER ANDEREN SPIELER macht. Daher sind viele der Strategien, die auch bei dem anderen Ragnar-Brothers-Spiel „History of the World“ (deutsch: „Empire“) funktionieren, auch hier gültig.

Die wertvollen Felder wie London sind meistens heiss umkämpft, selten macht es Sinnn hier viel zu investieren. Interessant sind vor allem die Felder, die nicht den Weg zu einem wertvollen Feld versperren, und die nur einen Punkt wert sind, sogenannte „uninteressante“ Felder.

England sieht gerade im Norden wenig Kämpfe (da es bald keine schwarzen Armeen mehr gibt, wird meistens direkt im Süden um London oder Bristol gekämpft) – ein Feld wie Carlisle wird einem eventuell das ganze Spiel erhalten bleiben, dasselbe gilt für Toulouse oder Rennes in Frankreich, Perth oder Stirling (nicht beide besetzen!) in Schottland, Cardigan oder Denbigh (nicht beide besetzen) in Wales, Limerick in Irland.

Eine Garantie des Überlebens gibt es nie, aber man sollte diese Felder auf jeden Fall zu erobern versuchen, wenn man es kann.

Das beste 2er-Feld im Spiel ist übrigens York (im Norden Englands). Dieses Feld lohnt es sich eventuell stark zu verteidigen!

5. Festungen

Auch bei „History of the World“ wird oft der Fehler begangen, wertvolle Felder mit Festungen zu verteidigen. Dies ist nur in den seltensten Fällen richtig, man sollte dagegen gerade die uninteressanten Felder so stark wie möglich verbarrikadieren. Eine 3er Armee und eine 3er Festung werden in den meisten Fällen gegnerische Spieler von der Eroberung eines Feldes abhalten, das nur einen Punkt bringt, und nicht durchquert werden muß, um wertvollere Felder zu erreichen. In einem Feld wie Limerick kann eine solche Festung das ganze Spiel überleben, und bei 4 Spielern bis zu 17 Punkten machen (3×2 eigene Wertung, 1×2 Schlusswertung, 3×3 Wertungen anderer Spieler)!!!

6. Jammern

Ein bißchen Jammern und gelegentlich auch geschicktes Verhandeln schadet hier, wie bei vielen anderen Spielen, nicht. Aber nicht übertreiben, manchmal kann diese Taktik auch extreme Agressionen bei den anderen Spielern hervorrufen!

Moritz Eggert, 19.4.2003

 

5 Tips for winning 6 nimmt

6 nimmt

This supposedly light game has more strategy than most people think. Here are some tested tips to lower your average score (scoring is negative for the player in “6 nimmt” – just for those of you who don’t know)

1. Play the card with the least risk of taking a row

This is so obvious that I feel embarassed pointing it out, but some players actually DO forget this and rather play “from the gut”. Look at your cards, and if there is a card that is the most safe to play, choose it, as easy as it is. Of course this is not always obvious, but sometimes it is very obvious that playing a certain card will most definitely have you take a row. Avoid this card. Understood? Very good, than we can move on to more subtle strategy…

2. Get rid of extreme cards as quickly as possible

It is clear to the experienced player that the very high and very low cards are your greatest danger. Usually players are more scared of the low cards, as they are more difficult to play without taking a row, but that can be deceiving. In fact a high card can become much more dangerous – a “100” is a bigger risk keeping than a “1”, for example.

The low cards are also not attractive, BUT… even a “1”card can come handy late in the game, when you rather take a small row with one or 2 cards than risking more with a medium or high card. You might even have your closest competitor take a high-value row because you mess up the cards with taking an unexpected row – one of the few events in the game when you can actively play against someone! Some players insist that it is good to keep at least one pretty low card until the last 3 rounds or so.

Most of the time players will play their low cards in the first round – do the opposite and play your highest cards, when it’s still safe (at least with 4 players), and THEN play your low cards, when the other players have established nice and safe low rows for your perusal…

3. Don’t always try to “set up a series”, especially when playing high cards

Very often it is tempting to play the first card of a series of cards in your hand (for example playing the 95 when you have 95, 96 and 98), hoping that you can then deliver the other cards pretty safely.

But most of the time what will happen is that a 5-card row gets blocked, with your card being the last card of 5. Now you sit on your 96 or 98, and at some point you will be forced to take the row, if you want it or not.

If you have high cards you want to get rid of try to do this in the first rounds, when it’s still safe. Begin with your highest cards. Immediately play a “104” for example, if you have one, even if you have the “103” as well. If all your other cards are lower, you’ll never have to take this card, and blocking one row will raise the chances of other people taking rows. If there are more than 2 players chances are better that it is NOT you who will suffer from this, although you might, of course. But “no risk no fun”….

4. The art of playing the “lowest highest card”

In every game there will come a situation when no card you have is safe, and all rows are pumped up with 4 or 5 cards. Somebody WILL have to take a row. What to do now?

If there is a low-scoring row you might consider taking it deliberately with a low card, better than ending up with lots of “oxens” otherwise. But if there is no such row, play the “highest lowest” card, as we call it at the “Westpark Gamers”. You don’t want to play too high, as you might be too close for comfort to one of the “sitting duck” rows. You don’t want to play too low, as you might end up with a forced-on-you row as well.

What should you do? Check the “margins”: 1) what is the lowest card that will definitely take a row? 2) What is the highest card in this low row? The best card for you would then be the card which is as close as possible to the HIGHEST card of the LOWEST row. If that card is not available for you, the truth lies in the exact middle – it might look as if this card is a sure “taker” as well, but chances are that another player is playing a lower card, starting a new row where you suddenly can play the card safely.

This is why keeping your “medium value” cards is so important, as very often the ideal card in this situation will be something around 40-60 or so. There is no sure-fire tactic here, you might STILL end up with a row, but playing the “lowest highest” card is the best – and only – hope to avoid this.

5. Wait a little before you play your card and watch the other players

You might learn some interesting things. Let’s say you have 56, and there is a row with only a 5th possible card left that ends with 54. Everybody knows that there is a certain risk involved in playing a 56 – if somebody else plays the 55, you’re busted! But of course the 55 is not necessarily in the game (or has already been played, but in “6 nimmt” it is more difficult than usual to remember played cards, because everything happens so quickly).

If one of the other players VERY QUICKLY plays a card, s/he most probably will have played a “no-brainer” card, meaning that it is a “series” card which is absolutely safe to play. If you’re in the above situation and nobody is playing quickly you might be safe with the 56, who knows. Of course the other players might have read these strategy tips :-)

But most players I know, including myself, will play “6 nimmt” very quick, so sometimes you’ll forget to “hold back”. If you remember watching the others sometimes it will increase your chances, believe me!

Moritz Eggert, 19.4.2003

A Bed of Steel

Strategy in 1830

By Alan Applebaum

The 1830 game system is one of the finest multiplayer systems around. As Edward Fahrmeier noted in his article „All Aboard“ (Vol. 23, No. 6 of The GENERAL), it is possible for players to recover from a bad start; yet it is difficult to „gang-up“ on the leader and reduce the game to a wild scramble with an arbitrary winner. Moreover, it is as close to a game of pure skill as a multi-player game can get, especially if all the players are competent so that their actions are reasonably predictable. Careful long-term planning is usually rewarded, while errors are mercilessly punished. Best of all, unlike CIVILIZATION or DIPLOMACY, the game is self-balancing because the cleverly-designed private company auction forces players to pay more for the most desirable private companies. Avalon Hill is to be congratulated for improving the already excellent 1829 (1830’s forerunner) system by increasing player interaction while simultaneously reducing playing time to a comfortable evening.

„Alliances“ can be useful in 1830, but diplomacy should not play a major role. Because play is sequential rather than simultaneous, one member of an alliance will usually have an opportunity to betray another without fear of retaliation. Furthermore, most alliances typically benefit one member much more than the other, and that inequality will be fairly obvious. For these reasons, alliances are generally not a large factor in the strategy, beyond the point of two players tacitly or explicitly cooperating to develop a particular section of the mapboard.

Like Mr. Fahrmeier, I too think the game divided into three phases; but I define each of them slightly differently. The „opening“ is the time before the sale of the first „4“ train; the primary features of this phase are the development of the players‘ stock portfolios and the struggle for control of the most desirable railroad corporations. In the second phase (or „middle“ game), the primary focus is on the second stage of corporation start-ups and the acquisition of „permanent“ trains. (I include the „4“ train purchases here since the decision whether to purchase a „4“ train usually determines when a railroad will acquire a permanent train and what kind it will get.) This second is the phase in which several railroads will be running for capital rather than for dividends, and one or more players may face bankruptcy. In the end game, which begins when all railroads have a permanent train, players turn their attention to maximizing their railroads‘ runs while impeding opponents by selling stock in their railroads, placing tokens to block routes, or looting and dumping jointly owned corporations. Many games are so close that one corporation’s gain or loss of one row or column on the stock chart can determine the winner.

In this article, I will discuss each phase of the game separately. In the course of the discussion I will try to review and comment on the ideas expressed by Messers. Shelley and Fahrmeier in their articles in the 1830 feature issue of The GENERAL.

The Opening

The first task facing the players is to bid for the private companies. In general, I agree with the comments of both Fahrmeier and Shelley. The Baltimore & Ohio (B&O) provides the presidency of a good railroad corporation, but because it does not generate recurring revenue and may not be sold to a railroad corporation, it is not worth much more than its face value. The Camden & Amboy (C&A) is unquestionably the most valued private company in relation to its base price, often selling for more than $200. The fewer the players in the game, the more the C&A is worth, for in a three- or four-player game purchase of the C&A will not prevent its owner from becoming president of a corporation in the first stock round. I generally don’t show much interest in the other private companies, but will buy any of them at face value. The Delaware & Hudson (D&H) is probably worth a bit more than its face value of $70, but our experience has been that most attempts to use its special power of „teleportation“ to hex F16 end in failure (more on this later). I believe Mr. Fahrmeier is incorrect in stating that the Mohawk & Hudson (M&H) cannot be sold to a corporation; the option to exchange it for a share of New York Central (NYC) or sell it to a corporation makes it worth more than $110, but it is still a poor cousin of the C&A. While I am no fan of the Canadian Pacific (CP) in general, it is poor strategy to run this railroad early in the game without possession of the Champlain & St. Lawrence (C&StL).

In a game of fewer than six players, therefore, if not buying the C&A, a player should not spend so much on the private companies that he cannot start a corporation on the first round. This means you need $402 , or $335 if you can start the Pennsylvania Railroad (PRR) because of the free share which accompanies the C&A. In a five-player game, you may buy a private company if you are sure that you will have enough cash by the third stock round to start a company.

These calculations may seem cumbersome at first, but are necessary for good play of 1830; if at any point in the game you find yourself or your railroad just a few dollars short of taking an important action, you are in trouble and it’s probably your fault. The only time this type of analysis really fails is if someone unexpectedly sells shares ahead of you that you were planning to dump, thus lowering the price. (This brings me to rules beef #1: I think all player and railroad holdings should be public. Since all transactions are public, keeping cash holdings secret simply rewards the players with better memories and slows down the game by requiring players to recap long sequences to figure out how much money a player or a corporation may have. Public disclosure also allows the players to catch each other if they – accidentally, of course – make an error in a transaction with the bank. With full disclosure, analysis and psychology, rather than memory, become the paramount virtues – a benefit, in my view.)

In conclusion, the only private company worth getting really excited about owning is the C&A; the others should not fetch more than $10 above face value. Buying a private company also affects the turn order, of course, and can cost you the chance to start the corporation of your choice.

After all the private companies have been sold, the next decision facing a player is whether or not to attempt to become president of a railroad. When we first began playing 1830, only two or three corporations would „float“ (begin operating) on the first stock round. When the presidents of those first two or three consistently won most of the games, we realized that it is usually better to start a poorly capitalized railroad than none at all. There are several reasons for this.

Becoming a minority stockholder in someone else’s corporation benefits the president a lot more than it does you. You help drive the stock market token upward on the chart (if all shares are in player’s hands); you reduce the number of shares the president must buy to float the railroad; and you give the president the option of dumping the railroad whenever the priority card is favorably placed for him. Even if you sell your minority petition after three or four operating rounds, the president doesn’t greatly mind. You have temporarily driven the stock price down, but this is offset by the creation of bank pool shares which can be used to increase the railroad’s treasury without running for capital. Alternatively, the president (or other players) may purchase the shares from the bank pool at a discount which is unavailable to you (since you cannot sell and buy the same type of shares in the same stock round).

Another major reason to start your own company rather than help someone else do so is that it gives you an opportunity to sell your private companies. Typically, the players who do well in a game have sold their private companies into their own corporations at a handsome profit and then managed to rescue the corporation they „looted“ by various stratagems we will discuss later. Those who play an „honest game“, operating their railroads to maximize the railroad’s success rather than their own, will find themselves falling behind, especially in games with fewer than five players.

Third, starting a railroad in the first stock round reduces the number of non-productive operating rounds for you. If you invest in a minority position in a corporation at the beginning of the game, the shares will not produce revenue on the first operating round simply because the railroad does not have a train. If, a few stock rounds later, you sell your minority holding and start your own company, you will not get revenue from your six new shares on the next OR for the same reason. You would have been better off starting a railroad of your own at the beginning, even at a lower price. Even if you must capitalize an extra time in the middle game because of the low railroad treasury, you are still better off than if you had started with minority positions planning to begin your own railroad later.

Finally, starting a railroad early gives you a head start on track development, as well as rightward movement of your corporation’s token on the stuck chart. Early in the game, the revenue generated by a share of stock is more important than its share price, so a $76 or $71 railroad is as good as a $100 one. In some ways, it is better, because a lower starting price enables you to invest in more shares of stock as well as allowing you to profit from upward and rightward movement on the stock chart. It is hardly ever worth waiting an extra stock round simply to start your first railroad at a higher price.

The only time it is really attractive to be a minority shareholder early in the game is if all the outstanding shares will be bought so that the token will moved up a row on the stock chart each round. If you sell at the right moment, you will have a handsome profit since the shares will have appreciated significantly in price in addition to generating revenue each operating round. If you miss-guess the president’s intentions, however, you may be trapped in a no-win situation. He will either begin running for capital, shutting off your income and driving the stock price lower, or will loot the railroad and dump it onto you.

Buying the presidency without enough money to float the railroad is even worse than taking a minority position. It is true that some other players may not be able to start a railroad of their own and will have to buy a minority position in some railroad. If you can’t buy enough shares to float alone, you are at their mercy. If they invest in a different railroad, you are stuck with dead shares. Probably one president will be bailed out in this situation – usually the B&O because of its high potential early revenue – but a second hardly ever will be (except in a six-player game perhaps). Once one non-president buys a minority position, later-playing non-presidents will tend to buy the same stock to benefit from the row rise on the stock chart and to avoid helping any other corporations float. If you buy only the president’s certificate and then wait to see what the other players will do before committing yourself further, you won’t be as badly burned if no one helps you out. But you may lose the chance to buy all the shares you want. Moreover, you will end up buying stock after everyone else is done and thus place the priority deal card to your immediate left, ensuring that you will buy last in the next stock round. In summary, unless starting the B&O (or there are no „live“ shares to buy), don’t even consider starting a railroad with insufficient funds to float it unless the other players have previously shown a tendency toward generosity. The only exception, of course, is in a six-player game, where no railroad other than the PRR can be started by a single player in any case.

Once you decide to start a railroad, you must choose which one to start. The predominant view among my circle of players is that the PRR is far and away the best choice for an early start. The PRR/C&A combination is extremely powerful; players who hold these seem to win more than their share of our games (unless the C&A auction soared far over the $200 level). Even if the PRR and C&A are in different hands, we find that the C&A is quickly sold by its owner into another corporation so he can start a second new corporation early. If this does happen, hex H18 is freed quickly and the PRR runs for excellent revenue. Even if H18 is tied up, the PRR and B&O presidents can cooperate to generate fairly good runs between H16 and Baltimore. Only the very first PRR dividend (usually $3 per share) will be substandard.

Not only does the PRR have excellent access to the lucrative Baltimore-New York corridor, but it is unique among the companies in that: 1) it only requires the President to purchase five shares to begin operation; and 2) it does not need to build any bridges or tunnels to achieve an optimal run with a „5“ or „6“ train. This means that it needs to run for capital fewer times than other railroads, and can so be started at a lower price, enabling its owner to invest in more shares of stock. In fact, it is not unrealistic to construct a budget for the PRR that requires no capitalization at all:

„2“ train – $ 80
„3“ train – $180
„5“ train – $450
H16 token – $ 40
total – $750

Thus the PRR can be started as low as $76 per share (an initial investment of only $380 for the president) and still have an excellent chance to become the highest-valued stock by the end of the game. If the B&O is started early and its president cooperates (as is in his interest to do), the PRR can expect to run for up to $200 with a „2“ and „3“ train in the early game (I15-H14-H16 and I15-I17-H16-H18) and eventually have a permanent run for a „5“ train of $270 (New York-H18-H16-H10-Chicago).

Of course, this is the ideal scenario. It requires that the C&A owner and the B&O president do not go out of their way to impede the PRR’s development, and that the PRR gets an opportunity to buy a „5“ train. Nonetheless, the prospect of ending the game with six shares worth $350 each and generating $27 each per operating round usually makes the PRR the first choice for being floated. An added incentive, of course, is that the PRR has four station tokens.

The best other early-start railroad is the B&O. As owner of the B&O private company, I always start with the B&O if possible, even at a low price. While the B&O is generally required to construct bridges (J14 and I17), the huge revenue it can generate early in the game gives its president a big boost. This high revenue is necessary for the B&O president to keep pace with other players who will sell their private companies to their own railroads. Fortunately, the B&O has by far the best potential runs of any corporation during the early operating rounds. A sample development will show this: on the first operating round, the B&O plays J14 and buys three „2“ trains. On the second round, if green tiles are available, it upgrades J14, builds a station there and runs for $170; if green tiles are not yet available, it plays a straight track in I17, builds a station in J14 and runs for $140. This latter course burns an alarming $440 from the treasury, but one or two runs for capital at $250 or so (after a „3“ train has been bought at the earliest opportunity) will restore the treasury to within purchase price of a permanent train. The B&O tends to dominate the game if other players buy into it during the opening stage; if they don’t, and bring out the „4“ trains early starting their own railroads – thus killing the „2“ trains – the B&O is still an above-average investment.

Some players prefer to keep the B&O railroad dormant and collect their $30 private company revenue each round, but that appears to be a losing decision. The following chart shows an ordinary sequence, with the consequences for Player A who starts the B&O railroad at first opportunity and for Player B who doesn’t. The figures refer to the accumulated cash generated by the private company or the two shares of B&O stock that it brings.

 

 Operating Round  Player A  Player B
 1  $30 $30
 2(run for $140)  $ 58  $ 60
 3 (run for $210)  $100  $ 90
 4 (run for $250)  $150  $120

Even if the „3“ trains are delayed or if the „2“ trains disappear after the fourth operating round (and the B&O’s run reduced to $110 or $120), the amount of cash generated by the two active shares in Player A’s hand will not fall far below that generated by an active B&O private company. Add in Player A’s increasing stock value, the benefits of corporation ownership, and the head start the B&O will get in track development, and the clear advantage lies with Player A’s strategy. The player starting the B&O late will likely find himself shut out of H16 (and probably New York as well); the B&O will have lost its geographical advantage. On the other hand, allowing another player to take control of the B&O sacrifices the value of the presidency for which you have paid a premium. The fact that two expensive shares of stock are tied up far outweighs the short-term revenue gain (if any) provided by the private company.

The only variation of the „B&O Delay“ which appears viable is to pass during the first stock round, until either all other players have spent all their money or someone has bought B&O stock or started the Chesapeake & Ohio (C&O). If neither of the latter two events occur, it is reasonably safe to delay floating the B&O because your control of the B&O corporation and hex H16 are not immediately threatened. Another factor to consider is whether you expect the PRR president to work with or against the B&O. If the PRR president wants to be nasty and the B&O should be delayed three or four operating rounds before starting, the B&O may be permanently prevented from getting a decent run. On the other hand, a cooperative PRR president can do much of the B&O’s early track development for you.

After the PRR and the B&O, the next „tier“ of starting railroads consists of the abovementioned C&O, and the New York, New Haven and Hartford (NYNH). The Erie is out of the question, of course.

The New York Central (NYC) runs too poorly with „2“ trains and yellow tiles only; the Boston & Maine (B&M) is just a poor cousin of the NYNH; and the Canadian Pacific (CPR) has too little potential.

The C&O is a railroad full of promise. Unfortunately, it rarely delivers on that promise. Like the PRR, it has no immediate need to waste money on bridges or tunnels; but unlike the PRR, it will always be shut out of the Baltimore-New York corridor unless the B&O is not started early. I have seen very few games in which H16 was not occupied by a PRR and B&O token at the earliest possible opportunity. Moreover, the C&O has one less token than the PRR.

There are at least three basic strategies which can be employed by the C&O president. If the B&O is not started early, the obvious choice is to sprint for hex H16. This option includes buying only one „2“ train and playing tiles on G7, H8 and H10. Unfortunately, even if the B&O is started a round or two late, you can still lose this race. The decision to follow this path depends upon how committed you think the B&O president is to keeping the corporation dormant. Even if the race to H16 is lost, playing tile #2 on G7 prevents its use on G17 (where it has unique value in developing highly profitable runs for the B&O, PRR, NYNH and/or NYC).

The C&O’s second strategy is the high-spending, high early-revenue course similar to that suggested for the B&O above. Build a sharp curve in G5 toward F4 during the first operating round. Then build a bridge in F4 and buy a second train during the next operating round. On the third and fourth, build two F4-Chicago runs by upgrading F4 with tile #15 (when available), placing a token in F4 and building a sharp curve in E3. On the fourth operating round, if the green tiles are available, you will be running for at least $170 with a „2“ and a „3“ train. On the next round, you can play another sharp curve in G3 and boost the run to $240 if you have two „2s“ and a „3“. Unfortunately, this is pretty slow development in comparison with others, and costs $420 from the treasury. It is unlikely that the other players will allow you to run for $200+ more than once before buying a „4“ train and spoiling your fun. Your long-term development prospects are now fairly poor as you will never reach the East Coast; expect to survive with a permanent run of $230 or less. Your permanent run can be slightly improved by developing E5, but this costs an additional $80 and damages your short-term prospects as you will probably wish to play tile #14 rather than #15 on hex F4 to reach E5 (reducing your access to Chicago).

The third feasible strategy for C&O is that of the „D&H teleportation“. On operating round #1, play a tile in G7 so that, should green tiles be available on the next round, you can play at F16 and still have a run. On the first operating round possible, buy the D&H with the C&O and spend $120 to place a tile at F16. The tile can be pointed either EW or NW-SE. Eventually you hope to establish a „4“ train run of New York-F18-F16-G17-New York (worth $210) or a „5“ train run extending to H18 (worth $260). This is a very nice set up, but takes a long time to establish; other nearby railroads may interfere with your plans, and your runs will be poor until New York is upgraded. If you are forced to do this yourself, you will have spent $495 (at least – $80 + $180 + $35 minimum + $80 + $120). If you also decide to buy a „4“ train, you will get some great runs but will likely use them for capital since, even if you started the C&O at $100 per share, you will have but $215 left in its treasury. If you opt not to buy the „4“, then you will have gained little from the D&H play. Your „3“ train could run for $130 from Cleveland to Chicago, the same as F16-G17-New York when fully developed. But you can manage a „5“ train run of $250 out west at less expense (Chicago-E5-F4-H4-Gulf). The play in hex F16 does help the NYNH and NYC however, so if you own one of those corporations or expect to, you may have more to gain than this.

The NYNH president also has several viable options. The short-term strategy is to build toward Boston and purchase two „2“ trains and a „3“ train, placing a token in F22 and quickly running New York-F20, F21-F22 and F22-F24-Boston (for $140 with yellow tiles or $200 for green). This will involve a bridge on F22 ($80), bridge in New York ($80), token in F22 ($40), two „2“ trains ($160) and a „3“ train ($180) – or about $540. Your best „3“ run will probably be New York-F20-F22 for $100, so to stay within one round of a „5“ train you must start the railroad at $90 per share. The permanent NYNH run will likely be confined to New York – Boston ($190 when fully developed), so a „5“ train is all you can use. One advantage of this is that your play in hex F20 seriously impairs the development of the NYC, especially if all the sharp curves are in play so that it cannot establish a run with only two tile lays.

A longer-term strategy for the NYNH is to play tile #69 toward Albany, eventually hoping to cooperate with the NYC and establish a coast-to-coast route. Your hope is that the NYC president will pick up the tab for upgrading New York, while the NYNH plays its station in E19 or perhaps F16. The NYNH thus lays out only $300 for a „2“ train, a „3“ train and a token in Albany in the early going, and can afford a „4“ later with a view towards acquiring a diesel. The best version of this strategy is for the NYNH president to acquire the NYC also, of course, but this will usually demand bankrupting the NYNH by selling it private companies. Otherwise, another player will surely grab the NYC first as it will have a head start on track development.

The NYC itself is an endless source of fascination in our gaming group. Its four tokens and proximity to New York make it extremely attractive as a long-term investment, but it tends to run very poorly early in the game. Its initial run is only $30, and it cannot conveniently run a second train until several green tiles have been played. When the „3“ trains are available, it is the overwhelming favorite among players starting a railroad at that time, and is devastating to control in combination with one of its neighbors.

Obviously, the Erie cannot be started at the beginning of the game because it cannot even build track in its home hex until the green tiles are in play. When the green tiles do appear, prospects will be limited because there are only two „OO“ tiles in the game (one of which invariably appears on H18 at the first opportunity). Thus, early development for the Erie consists only of a gentle curve into Rochester and a connection to Albany, for a lackluster run of $90. This railroad is therefore viable only as a second corporation, probably for the owner of a nearby line such as the NYC which can aid its progress. The Erie’s sole advantage is its ability to monopolize the high-scoring cities in the Buffalo-Toronto area (because the OO upgrade tiles allow a single token to block passage through the hex). Thus, once the brown tiles are available, the Erie becomes pretty reasonable – especially if nearby track has been also developed.

The B&M has all the disadvantages of the NYNH, with none of the advantages. It is not guaranteed passage through New York, so it will often have to expend its only token there. Unless a tunnel is cut in D22, the B&M cannot reach Albany early enough to avoid being blocked there. If tile #56 is played on F20 and/or the NYNH plays its token in E19, the B&M’s dim prospects become dismal.

Many players feel the CP an attractive railroad to start at the beginning of the game. As a potential D&H „teleporter“, it is superior to the C&O simply because it has more tokens and its initial run between B16 and A19 is worth $60. If a player can contrive to hold both the C&StL and the D&H, the CP becomes very attractive; but this is extremely difficult to do because they are sold consecutively on the first stock round.

With the C&StL alone, the CP can run well in the early going: on operating round #1, place a sharp curve in B18 connecting A17 and buy two „2“ trains; on #2, play B16 and put a token there, running the trains for $90, and buy a „3“ train if available; on #3 (assuming the green tiles are available), buy the C&StL, play a sharp curve in B20 or upgrade B16 and run for $160; in the fourth round, upgrade B18 with tile #18 and run for $190. Unfortunately, this is probably your last run with the „2“ trains, and now you will run for only $80 with a „3“. You have spent some $480 to reach this point. Thus, unless the CP started at $90 or $100 a share, you are more than one turn of capitalization away from a „5“ train and will run many rounds with your „3“ train. A „4“, even if you could afford it, is wasted because you don’t have the rail net to take full advantage of two trains. The only decent prospects for track development lie towards Toronto; but building there costs money and the Erie president will certainly thank you just before buying a token to freeze out the CP. Other than the Erie, no other railroad is likely to help the CP’s track development much once Albany is occupied.

Building the CP towards Albany, however, is even more futile. It requires the laying of at least two bridges (because the C&StL tile cannot be played right away), so you need to play B18 toward A16 to guarantee a run on the CP’s second operating round. The chance of reaching Albany before two of the three other nearby railroads occupy it with tokens is slight. In my view, then, the best early alternative for the CP (and the only one with a chance for profitable success) is the F16 play with the D&H. It is far better not to start the CP until the green tiles are sure to be available, however. Otherwise, you will be forced to spend $80 extra to establish a run through B18 in addition to your „2“ train costs and the $120 for F16.

In summary, the CP is only the sixth-best railroad to start early in the game (bettering only the isolated B&M and the Erie). Late in the game, it proves even worse, since it is the only railroad on the map likely to gain no benefit whatsoever from track construction by the others. In our games, the CP is often either the last railroad started-or it is never begun at all.

Now that you have some idea of which railroad to start, what do you do with the rest of your money? If you have played the first stock round correctly, you shouldn’t have much left. It almost always pays to start your railroad at the highest price you can afford, planning to extract any extra cash it has by selling your private companies to it. By the third or fourth operating rounds, however, you should be ready to decide whether to start a second corporation or invest in an already existing line. Starting a second railroad is usually preferable, unless all the remaining corporations have very poor prospects because of unfavorable track construction by early existing lines. The second railroad not only allows you to hold an extra share within your limit of certificates, but also to cooperate with your primary railroad in track construction and station building. And, most importantly, a second one allows you to control the timing of your train purchases and to distribute cash efficiently between the two. It also avoids the necessity to invest in another player’s railroad (which may be looted and dumped upon you) in order to reach your certificate limit.

Ideally your second railroad should be started at $100 per share. This is not so much because it needs more cash to operate (it actually needs less, especially if the „4“ trains are already available); rather the extra cash is needed to transfer to your older corporation so it doesn’t have to run for capital as much. Too, a high-priced second railroad has a fighting chance to reach the „high-rent“ section of the stock chart, while a low-priced one doesn’t.

In many cases, unfortunately, the player seeking to start a second railroad cannot conveniently raise $600 alone, and must give up on it or start it at a lower share-price. If the older railroad has enough money to get a permanent train with but one capitalization round, the second can be started at a low price because no cash needs to be shifted. In most cases, however, the second round of railroad starts will involve those players who have sold private companies to their first for large amounts; such first companies usually need some help at this stage. Paradoxically, if your first railroad is in really terrible shape, you can raise the cash to start your second by selling all but the president’s certificate. No one else will want to take on the railway from you if it is broke. On the next turn of operating rounds, the railroad may get cash from the shares in the bank pool without being forced to run for capital. If, by selling your shares, you now have enough money to start a second company at $100 per share, you should be able to bail out the first if necessary.

If a really attractive railroad company is available to be started, you don’t have enough money to „float“ it until the next stock round, and you think another player yet to purchase during this stock round will grab it, you could buy the president’s share only (or even one additional share) and set the price at $67. Other players will realize that if they take control from you, you are quite liable to sell your two or three shares and drive the market token down two or three rows on the chart making it a rather unattractive investment. The two or three shares are deadwood in your hand, of course. You have to weigh carefully whether you have more profitable short-term purchases and whether you are willing to run the railroad at $67 per share later. This is one of the few situations where it is quite proper to buy shares that are not certain to be „live“ during the next operating round. Unless it’s a six-player game, I would assume that no one will throw me a lifeline if I should purchase only four or five shares of a new company.

In the event that you are unable or unwilling to become president of a second railroad, you will usually have a choice of a number in which to become the second-leading shareholder. The obvious considerations are the track development potential of the railroad, the state of its treasury, the skill of its president, and the state of other railroads owned by that president. The most attractive minority position is one in a railroad with good potential, lots of money, run by a capable player who has no other railroads and no unsold private companies. Other things being equal, I tend to choose one owned by the player nearest my immediate left. This has two advantages. First, the president cannot loot the railroad and stick you with the presidency unless he has the priority card himself. Second, if he has fewer than five shares, you may get a chance to take the railroad away from him if the other outstanding shares are sold piecemeal by the other players, allowing you to pick them up before the president can.

Keep in mind that owning more than one share of any railroad that you are not president of is always a risky proposition if the president has a way of looting it. If the person does loot it and makes you the president, he might not win – but you definitely won’t! Controlling your share or more than your share of corporations removes this risk for the entire game. In a six-player game, you can survive nicely with one railroad as you can own six shares of it and one share of six others, for a total of 11 certificates (your limit). In a five-player game, the winner will almost always come from among the two or three who have two railroads. In a four-player game, you can do quite well as president of only two corporations if you hold six shares of each and can obtain one share of each of the six others (for a total of l6 certificates). With three players, the player with only two railroads is usually the loser (unless both are very good).

One factor which will color your strategy is the tendency of the other players towards what I call a „fast“ or a „slow“ game. A „fast“ game is one in which four or five railroads are started early and the train inventory is run through quickly; in a „slow“ game, the „3“ trains may not emerge for five or six rounds. In a fast game, the „2“ trains will not last long, so it is unwise to invest in too many of them. Except for the B&O (and possibly the C&O), no railroad can profitably use three „2“ trains. Several railroads in a fast game will require their presidents to bail them out from cash in hand, so it doesn’t pay to use a long-term strategy. A bankruptcy is fairly likely, especially in a five- or six-player game.

[Rules beef #2 (my only serious one): the rules covering bankruptcy are terrible! If a bankruptcy occurs, the winner is the richest player at that moment. All long-term planning is useless; the result of the hours spent playing is completely arbitrary. Why not just put all the bankrupt player’s stock (including his presidencies) into the bank pool and send him home early? I have seen games in which one player intentionally saves another from bankruptcy simply because the first was not winning at the time, even though the aid given would not otherwise have been in his best interest. I have even played in games in which players doing poorly were unsporting enough to intentionally bankrupt themselves, so spoiling the game for everyone. This rule requires some revision in future editions.]

In looking at a „fast“ game, the basic principle is to go against the grain – do the opposite of what most of the other players are doing. If you expect a fast game, you should play conservatively to protect your railroads from having their trains stripped away before the treasury is rebuilt. You aim to force others to deplete their capital by buying short-lived, temporary trains.

On the other hand, in a „slow“ game, in which players are reluctant to buy multiple trains and/or start new railroads, you should play aggressively, buying multiple „2“ and „3“ trains, running for dividends every chance and building an early cash lead. If your railroads reach a precarious strait, you should have ample warning if the others decide to bring about your downfall by suddenly speeding through the train deck and killing your „3“ trains since not many competing railroads will be in play. If the game develops slowly enough, you may not even mind paying for a permanent train out of pocket; you will be more than compensated by your early cash and high stock prices.

Always buy that „3“ train at the first opportunity. A railroad without a „3“ train will lose a run when the „4“ trains appear (unless you were lucky enough to buy the first „4“). As the „3“ trains last through the next seven to 11 bought, they get a lot of use. Don’t buy two „3“ trains, however, unless you can definitely sell one to a second railroad, or you expect a very slow game. Once a „5“ is bought, the limit on trains is two and you cannot buy a „5“ or „6“ yourself until the first „6“ train is bought by some other player (thus killing your „3“ trains and forcing you to lose a run).

Although it seems counter to one’s instincts, it is better to capitalize on your large early runs (the operating round or two immediately preceding the disappearance of the „2“ trains) rather than your small ones later. It is important to run for capital as infrequently as possible to avoid damaging the stock price, since each capitalization costs you two columns on the stock chart. Do capitalize for $200 or more, unless the extra dividends in your hand would enable you to start an additional railroad on the next round which would otherwise be started by a competitor. If, on the other hand, you are but one “3″ run away from having $450 in the company treasury, you can afford to wait and capitalize on the smaller „3“ later when the „5“ trains are available. And be careful not to capitalize if this will shift the operating order so that you lose any opportunity to run with your „2“ trains! If you have a choice, try to capitalize at the last possible moment. This not only enables you to invest your dividends earlier, but also maintains your position in the operating order, possibly gaining you that extra run with your „2“ or „3“ trains before they disappear.

As in real life, it is better in 1830 to have money sooner than later, because you can immediately reinvest early cash and make some additional profits. Therefore, it is almost always good policy to spend company treasury money early rather than later for bridges, tunnels, trains and tokens to improve your run – even if this seriously depletes the reserves. Later you can start a second railroad at a high price and bail out the first if you have accumulated enough cash in your hand to do so. There are a limited number of good railroads to own, and a one-round delay in accumulating the required cash to start one will often be fatal – so get there as quickly as you can. Once you have captured control of the railroads you want, you can ease up on the accelerator and save for some permanent trains.

Looting and then dumping a railroad early in the game is not as clever an idea as it may seem, unless most of the railroad’s treasury ends up in the looter’s hands. This is usually done by selling private companies to the railroad in question for the maximum amount possible and buying trains, bridges, and such to completely empty its treasury, then taking advantage of the favorable position of the priority card to sell enough shares into the bank pool to make another player the president. However, not only does this fail to benefit the looter substantially, but it really doesn’t injure the new president particularly either.

The looter will generate enough cash to start one or even two new railroads at a high price. Unfortunately for him, however, at least one of those will be very undesirable (possibly both). The railroads with the best potential (including, perhaps, the one just looted) will already be owned by others. Secondly, the new railroad(s) will start back in the red rectangle of the stock chart, making them unlikely to reach a high stock price by the end of the game. Third, the looter’s new railroad(s) will not run on the next operating round (because they have no train), costing the looter a turn of revenue. Finally, single shares of the looted railroad may be picked up at a bargain price by other players.

If the looted railroad is in really terrible shape, the new president will have some anxious moments, but will usually be able to recover before having to pay substantial amounts out of his own pocket. The game mechanisms provide for this in several respects. The looted railroad can collect revenue, not only from the private companies now owned, but also from any shares of its stock which remain in the bank pool during the operating rounds. Thus it can rebuild its treasury without running for capital, and its stock price will recover quickly. Other players besides the new president may be reluctant to buy the shares out of the bank, fearing that this would force the new president to capitalize. And the looted railroad may be able to sell its surplus trains to another railroad owned by the new president, thus benefiting both. And, if it retains those „extra“ trains it bought, it will run for high revenue (at least temporarily). The new president will eventually be grateful to you for acquisition of a railroad with a good stock price and well-developed track. Alternatively, he may turn the railroad into a „feeder line“, driving its stock price into the colored zone of the chart.

Note that in criticizing this „loot and dump“ tactic, I refer to the plunder of a player’s only railroad early in the game. Looting a second railroad corporation in the middle or late stage is much more profitable for the perpetrator and more damaging to the victim.

Another destructive tactic is „churning“ – the buying and selling of large numbers of shares in railroads owned by others in order to lower the stock price. If two or three players cooperate, they can lower the stock market token to the very bottom of the chart. It is even more effective if one additional player remains uninvolved to buy up the now-cheap shares so that the railroad will not get any benefit from having shares in the bank pool. If done fairly (that is, equally to everyone), this tactic will not have much effect on the game except to reduce the importance of the early starting railroads. Our local group tends to avoid this tactic as it is time-consuming, and either unimportant or unfair in the long run. In a cut-throat game, however, it can be very effective when the victim cannot retaliate.

The Middle Game

After six or seven railroad corporations have been „floated“, the most critical phase of the game begins. Players strive to reach their share limit at a time when many of them will be short of cash because they hold many shares in railroads running for capital. Railroad presidents jockey for position to insure that their railroads can smoothly acquire a permanent train. Crucial tiles and tokens will be placed, determining which railroads will have profitable runs and which will not. In short, the game is usually won or lost in the middle game.

Near the start of this stage, the question of railroad ownership will be pretty much settled (unless a „loot and dump“ entrepreneur arises). The cash-rich players will try to acquire at least two railroads, and possibly even three. Those with more railroads will, however, find that their shares have a lower average price than the holdings of other players, and they may be in danger of paying out of their pocket for a permanent train. If one railroad is deemed so inferior that no one will start it, only 63 certificates will be available; so some players may not be able to reach their certificate limit in a four-, five- or six-player game.

Four strategic questions face players in the middle game: 1) start a second railroad or invest in established ones; 2) run for capital or dividends; 3) buy a „4“ train; 4) aim for a „5“ or „6“ train, or a diesel? We’ve touched upon the first question in the discussion above. The last three are intimately connected, as the following discussion will show.

In general, one should not run for capital for any of the following purposes: to get a better permanent train, to build a bridge or tunnel, to place a token, to buy a second permanent train (unless it’s for a second railroad you own), to move the stock token leftward into the yellow zone. Capitalizing costs you a lot of money. If your stock is one of the higher-priced ones, the two columns you lose may cost you $5O per share at the end of the game; added to the lost dividends, that’s $400 to $500 lost per operating round if you hold six shares. Even worse, the immediate dividends you forego may prevent you from buying stock on the next round, adding to that loss. On the other hand, do capitalize when your run is at least $200 and there is no other way to obtain a permanent train without spending at least $300 from your own cash reserve.

Experience shows that you should almost always buy the first permanent train available; saving for a diesel is rarely a good investment. As Mr. Fahrmeier indicated in his article, the game will usually last only six – nine operating rounds after the first diesel purchase. For example, if the diesel allows your railroad to run for $400 while a „5“ train would run for $230, you gain $17 per share per operating round if you own a diesel. If you have six shares, your gain is therefore $102 per round. After seven rounds, you have gained $714. Therefore, if you capitalized more than once to get the diesel rather than the „5“ train, you were a loser – because the two extra capitalizations have cost you $800-$1000. Moreover, unless you have the opportunity to buy the first diesel, you will have no trains after another railroad buys one and so will lose an additional run. Note also that other players may cut down your eventual big diesel run by placing tokens, reducing your ability to recover your investment.

In deciding whether or not to go for a diesel train, you should consider the fact that in order to get a diesel, you need to buy a „4“ train. Without a „4“, you will have no trains after the first „6“ is bought-so you will have to buy the remaining „6“ train. If the second „6“ is gone, you will be forced to buy a diesel (with a lot of your own money). „4“ trains cost $300 and have a notoriously short life span. Many a „4“ train has not provided its railroad with a single dollar of revenue, as a diesel is bought before the railroad runs again! Moreover, many railroads do not get much benefit out of the „4“ train: with only green tiles in play, the rail net is too limited to allow both their „3“ and „4“ trains to run fully. Worst of all, owning a „3“ and a „4“ will put your corporation at its limit for trains as soon as a „5“ is bought. Therefore, unless you happen to have $450 in your treasury and have a chance to run at the exact moment the first „5“ train is available, your railroad cannot get a „5“ without help from a second railroad. In fact, it will probably not get a „6“ either, since only one „6“ (at most) will be left in the bank when your „3“ disappears, leaving room for the second train. With both a „3“ and a „4“, your timing must be perfect to get a „5“ or a „6“ train. Thus, if you buy a „4“ train, you are committed to buying a diesel unless a second railroad can help by buying the permanent train and selling it to the first railroad. Unless you are the first corporation to build your treasury to $800 with a „4“ train, you must either run for capital an unacceptable number of times or pay a substantial amount out of pocket (after losing a run to boot).

As a consequence of these grim facts, you should buy a „4“ train only in certain specific situations. If other owners are profiting more than you from the operation of „2“ trains, buying a „4“ will put them out of business. If it is the first train a railroad is buying, then, assuming you have any choice at all, you can afford a „4“ because the railroad will still have enough money for at least a „5“ (unless it started at less than $76 a share). If it is a „slow“ game and you envision that there will be a long delay before a „5“ train is bought, buy it quickly, enabling you to get four or five runs with your „4“. If you have enough cash in the railroad fund and good enough track to ensure you reach $800 first, invest in a „4“. And if you have two railroads, one of them will need a „4“ since it is unlikely that you will be able to get a „5“ or „6“ with each of your railroads. Remember that your railroad can be the first to accumulate $800 in its treasury and still not be able to trade the „4“ for a diesel if no „6“ has yet been bought before it is your turn, leaving you with no train. If the second „6“ is gone, you will not only lose a run (getting no revenue and losing another two columns on the stock chart), but will also have to pony up $300 in spite of your „conservative“ play. Never buy the last remaining „4“ train (we dub it the „Poisoned Four“) unless the railroad can also buy a „5“ or your second railroad runs soon after and will be able to buy one or two „5“ trains. If you do decide to buy a „4“, at least buy it at first opportunity; nothing is worse than dithering and finally buying one when it will only run once before vanishing.

Of course, even if you don’t buy a „4“, you can still come to grief if all the „5“ and „6“ trains disappear before you have a chance to purchase any. The $450 (or $630) you saved in the corporation coffers so carefully will leave you to dish out $470 to $650 from your pocket for a diesel. It would seem, then, that whichever route you take is risky, and that you have little control over what kind of train you get and how much you pay for it out of personal funds. This is quite true … if you own only one railroad.

With two railroads under control, the situation is a lot more stable and predictable. Naturally, each railroad has one „3“ train by this time (not two – you don’t want to be prevented from buying a „5“ by the train limit and then lose a run when the first „6“ is bought). One of your railroads (not both) buys a „4“ train at first opportunity. Now you have the desired flexibility. You can shift the money and the open slot for a train around to your best advantage, positioning yourself to get a „5“ or „6“ with one railroad and a „6“ or „D“ with the other. If, on top of this, you can get the diesel without losing a run and get it to the railroad with the better route, you are probably a game winner. Don’t try to switch permanent trains after both are bought, however; this is very expensive.

This phase of the game can involve some tricky maneuvering. Players will often spend the stock round preceding the critical set of operating rounds attempting to manipulate the stock prices of the various railroads so their own will be able to buy a „5“ and avoid the „Poisoned Four“. One of the best situations is to start a railroad for $90 or $100 per share at a time when it can purchase two „5“ trains on its first operating round. This solves the problem of a permanent train for both its owner’s corporations (since one of the „5“ trains can be sold to the older for a nominal sum). Starting the new railroad just to buy the „Poisoned Four“ can also be a good move, so long as one or both your other railroads will be positioned to buy a „5“. Losing revenue by capitalizing several times is not good, but losing a run entirely because you have no train is even worse. If you get through this phase of the game without losing a run or paying out of your pocket, you will almost certainly be a contender (even if your railroads have poor runs).

Occasionally, however, it can be a good tactic to purposely deprive one of your railroads of all its trains. Suppose the situation is this: you own two railroads, one of which has a „4“ train or better (the „good“ railroad) and one which has only a „3“. The railroad with the „3“ cannot accumulate enough money in its treasury to buy a „5“ even if it runs for capital once, but you know that it will run first when a „5“ or „6“ is available. You can help this railroad by having the good one buy the „3“ from it on its turn. Then, when the bad railroad runs, it can buy the permanent train with the help of your personal funds. This avoids having the poor railroad capitalize, and then either capitalize again or be trainless during the following operating round. You gain two columns on the stock chart at the cost of the $50 or $100 you would have raised for the railroad by capitalizing with the „3“ train – usually a good bargain.

My local group has discovered an extremely cruel but effective tactic to punish poorly-run railroads (and, by the way, prevent the tactic just discussed). The perpetrator owns a railroad with a „3“ and a „4“, and this railroad now has the opportunity to buy the first „5“ train. It does so, and now must turn a train in to the bank to meet the new train limit. Instead of turning in the „3“ (as one would ordinarily expect), it turns in the „4“! The victim’s railroad has only a „3“ and not enough in the treasury to buy a „4“, „5“ or „6“. After the first „6“ is bought, the victim’s railroad has no train. Even if the second „6“ is available, the victim’s railroad may not buy it; instead it must buy the least expensive train available-namely, that „4“ the perpetrator turned in to the bank. The next time around, the first diesel has almost certainly been bought, so the victim is trainless again.

In conclusion, your goal in the middle game is to acquire permanent trains for all your corporations without losing runs for lack of trains, or paying heavily out of pocket, or running for capital multiple times. Achieving these goals will usually involve owning at least two railroads (unless you are willing to trust to fate). You must always plan to acquire a „5“ or „6“ train with at least one of your railroads. Do not feel unhappy if you wind up with no diesels; they fail to pay off as often as not. In but one specific situation is it wise to capitalize an extra round to grab a diesel: your railroad has only a „4“ train, one „6“ remains in the bank, the railroad has between $630 and $799 in its treasury and would have over $800 if you run for capital on the current operating round. Now you face a simple decision between running for dividends and buying a „6“, or capitalizing and buying a diesel. Normally it would be worth the effort to capitalize and buy a diesel, especially if it slows other players who need their „4“ trains. This situation is an exception, however. In the majority of cases, one should accept the cheapest permanent train available.

The End Game

The last stage begins when all active railroad corporations have a permanent train. In the end game, play focuses on tile and token play on the board rather than on stock transactions. Typically, all players will have reached their stock certificate limit around the beginning of this phase. Stock transactions are now focused upon damaging the value of competitors‘ portfolios and acquiring shares of such companies as may be in the colored zones on the chart. If a railroad remains inactive, at least one player will be faced with a decision as to whether to float it.

Several decisions arise in the end game: Should a railroad with a permanent train but empty treasury run for capital in order to obtain funds for bridges, tunnels or tokens? Should a player sell shares of a competitor’s railroad to lower its value? Should a railroad token be placed so as to damage another’s run? Should a railroad be looted and dumped? At this stage in the game, each player’s reactions can be predicted with such a fair degree of certainty that one can calculate whether any of the above actions will gain or lose ground. Unfortunately, it will often take more time to perform these calculations than the other players will permit you, so some general guidelines are in order.

It is rarely profitable to run for capital in the end game. If one is certain that by building a particular tunnel or bridge the railroad can improve its run substantially (at least by $15 per share), it may be worth considering. The lower the railroad’s stock price, the less costly it is to capitalize (because the two lost columns make less of a difference to the price when the token is near the left or bottom of the chart). Remember also that if a really big profit is in the offing, other players may conspire to deprive you of the tiles you need to complete your projected run.

The question of whether to attempt to reduce a competitor’s stock price by selling your shares usually arises on the very last stock round of the game. While it is extremely time-consuming to attempt to calculate precisely whether the game will end after the ensuing three operating rounds, a rule of thumb is that in the late game $30 per operating round will be removed from the bank for each share of stock in play. If all 80 shares are in play, about $7200 will be removed from the bank on three operating rounds. In rare circumstances, someone will be able to delay the end of the game by building unnecessary bridges and tunnels (or even by going to the extreme of purchasing an extra diesel). But selling shares in other companies should be directed at the players you believe are doing the best, since selling any high-priced stock will cost you money. You must be sure it will hurt your competitor more than it hurts you. If you can sell only one share, and the stock is already at the top of the chart, your move is futile since someone else will simply buy the share causing the token to rise back to its original row.

The question of token placement is more one of psychology than tactics. It is pretty easy to determine whether a token play hurts or helps you. But if you plan on playing with the same group again in the future, you might consider that a surprisingly large number of gamers view aggressive token play as bordering on the unethical. I personally prefer a „cut-throat“ type of game in which each player is committed to furthering his own self interest by harming the positions of others, but some disagree with me. If you think that the player you damage by your placement of a token will take seven-fold revenge in future games, that is a factor to consider.

The end game presents the classic loot-and-dump opportunity described by Mr. Fahrmeier. The situation is such that during an operating round, you own two railroads, each with a permanent train. The railroad with the lower stock price has at least $1 remaining in its treasury. The priority card is located so that on the next stock round, you will play before a player to whom you can pass control of your more expensive railroad. On the third operating round of the current set, your cheap railroad buys the permanent train from the other, leaving that railroad with a depleted treasury and no train. On the subsequent stock round, you sell enough shares of the expensive railroad to transfer the presidency to your victim. He will have to come up with almost the full $1100 for a new diesel on the following operating round. This play costs you some profits, but you can make some or all of it back in dividends because your remaining railroad now has two permanent trains. If there is other stock left to buy, your total income per round may actually increase. And you are now protected from the same ploy by someone else because you have an „extra“ permanent train should a railroad be dumped on you.

This tactic is so effective, in fact, that the threat of it tends to dominate play in the later stages of our games. Players will maneuver to avoid owning two or more shares of another railroad unless that president is to their immediate left (so the minority holder cannot be victimized). The best antidote, of course, is to be president of enough railroads that you can complete your portfolio without owning more than one share of any other.

Players occasionally find it profitable to operate what has been called a „feeder line“ – a railroad with very low stock price which capitalizes often so that its market value remains in the colored zone of the stock chart. The advantages of this are that the shares do not count against their owner’s certificate limit and that the extra treasury money accumulated can be used to assist the track development of his other nearby railroads or place tokens so as to injure close competitors. In order for this situation to be profitable, however, other players must have contributed to driving down the stock price. It is a losing strategy to capitalize a railroad five or six times simply to drive down the stock. What you gain by creating larger runs, you lose in early revenue and in stock appreciation of the feeder line. Too, the ability to exceed your share limit only benefits you for the one or two operating rounds out of three that you can run for dividends. And, by the time you get the feeder line’s price into the yellow zone, there may well be other worthwhile shares to buy. Thus, I believe one can be pushed into a successful feeder-line strategy by the actions of others, but cannot set out to create such an opportunity; it’s simply too costly.

Conclusion

1830 is one of a very small number of multi-player games which are almost pure tests of skill. Unlike DIPLOMACY and its brethren, however, 1830 is unique in that one’s fate is somewhat less dependent upon the actions of other players if one so chooses. The game lends itself so well to analysis, in fact, that it suffers from the same problem as chess unless a time limit is set, the game can drag on far too long to be enjoyable. I think the game is best enjoyed if each player is practiced enough so that only three or four important decisions will require extensive thought and calculation. Our local gaming group plays a very sophisticated game, but usually completes a match (no bankruptcies) in three to four hours. This is a comfortable evening’s entertainment.

This article is intended to provide the beginner with some examples of the issues he must think about when making these crucial decisions. Profound hex-by-hex, dollar-by-dollar analysis of a large number of specific situations is beyond my scope, but I will be glad to respond to future discussion or criticism of the points raised here in future issues of The GENERAL.

(this article first appeared in issue #6 volume #26 of Avalon Hill’s „GENERAL“. It is reproduced here with the permission of the author)

Civilization Strategy Seminar

Steve Jones dons his toga and provides a comprehensive guide to the great game of aspiring cultures.

CIVILIZING INFLUENCES

In purely abstract game terms, Civilization is a steeplechase race game; I shall elaborate upon this theme as the article develops. Looked at in this light, the game may seem quite ordinary. What elevates it to a classic is a combination of two things: the subject matter (or the ‚chrome‘ aspect of the game), and the wonderfully balanced and rich structure of the rules. The latter moulds the game into one with elevated player interaction in ways which are complex, subtle and shifting, and yet are always under the players‘ collective control. In my opinion, Civilization is quite simply the best multi-player board game ever invented.

The subject matter of the game is an attractive and evocative one. It concerns the growth and development of the ancient civilisations around the shores of the Mediterranean Sea, and in the Middle East.

Each player starts off in control of a nomadic Stone Age tribe which then proceeds to develop a highly advanced classical civilisation over the course of millennia. The object of the game could not be simpler: it is to gain a state of overall advancement, involving cultural, economic and political factors, faster than anyone else. Although this makes the game into a race, it is vitally important to realise that it is one conducted on several different levels.

It is also important to realise that it is a game which megalomaniacs always lose. War is available as an instrument of policy, but the game is so structured as to make it into a blunt and relatively inefficient one. The more efficient means available to combat your opponents are, to a greater or lesser extent, subtle, hidden, indirect, sophisticated, and ‚civilised‘.

THE KNOWN WORLD

The game consists of a map board, an Archaeological Succession Table (AST), eleven sets of trade cards, sixteen sets of civilization cards and seven sets (nine in the Avalon Hill version) of faction counters. The map board consists of three leaves (West, Central and East) depicting a region stretching from Sardinia in the west to the head of the Persian Gulf in the east, and from the upper Nile Valley in the south to the southern Ukraine in the north. Hartland have recently released an expansion kit containing a fourth leaf (WXB) which extends the map westwards to take in Iberia.

The map is divided into land and sea zones in much the same way as Diplomacy and Risk. The sea zones are subdivided into two types: coastal and open. The land zones each have a population carrying capacity indicated by a number. Some of the land zones also contain natural city sites. The map features, in addition, four flood plains and three volcanoes; there is a fifth flood plain on the WXB extension leaf.

The faction counters each consist of fifty six tokens, nine cities and four ships. One token is used as a marker on the AST, and the remainder may be used to represent either population on the board or money in the player’s treasury. Depending upon the number of players in the game, the players are limited to a specific number of tokens each. This last is a deliberate design feature in the game. The main problem that the players continually face is one of limited resources. The subdivision of the each player’s finite number of tokens among population on the board, money in his/her treasury, and his/her stock (tokens currently not in play) is a brilliant design mechanism for creating this shortage.

During the course of the game, there is a constant flow of tokens between a player’s stock on one hand, and the board and his/her treasury on the other. Furthermore, this flow is more or less under the player’s control. Nevertheless, players will find themselves facing problems, and occasionally windfalls, depending upon where their tokens are currently located. There are many subtle aspects to the management of the token flow, and players should acquaint themselves with these.

COMMODITIES

There are seventy four trade cards in the game. Of these, sixty six represent eleven commodities which have values from one to nine. The remaining eight cards represent calamities (more commonly known as disasters), and these all have value zero. The commodity cards are initially stacked, face downwards, in nine decks, each deck corresponding to the value of the commodities in it, and the calamity card of the appropriate type is placed at the bottom of each deck except the first.

The eleven commodities are Ochre, Hides, Papyrus, Iron, Salt, Grain, Cloth, Bronze, Spice, Gems and Gold; the first two have value one, the second two have value two, and the remaining seven have the values ranging from three to nine in order. When more than one card of a given commodity is held the net value of the collection is greatly increased. The formula for finding the total value of several cards of the same commodity is to multiply the square of the number of cards held by the commodity value; for example, five grain are worth (5×5) x 4 = 100.

During the course of the game, each player takes the top card from each trade deck for each city he/she has, and uses these to construct packages for trade with the other players. Through trade, the aim is to increase the net value of the commodities held in hand, and to use these to purchase civilization cards. The trade system is one of barter involving trade cards alone. Any number of trade deals may be made between the players, two at a time. In each trade deal, each player must offer at least three trade cards, correctly quote the total value of the package and correctly give the commodity of one of them; outside of these restrictions, anything can be said about the package. This last is important because trading does carry risks: some of the disaster cards can be passed on in a trade deal.

DISASTERS

The eight disasters are Volcano/Earthquake, Famine, Civil War, Flood, Epidemic, Civil Disorder, Iconoclasm & Heresy, and Piracy. The first four are so-called red-backed disasters and affect the players who pick them up. The last four, on the other hand, are so-called black-backed disasters and can be passed on to another player through a trade deal; the disaster then affects that player!

As the names suggest, it does not help your aspiring civilisation to be affected by one of these disasters. Players will find it essential to acquaint themselves with the effects of the disasters, how to lessen their effect, how to direct them at their opponents, and how to reduce the chances of receiving one. In practice, it is impossible to avoid disasters completely. This is because an individual disaster does not affect only the player who received the card; that player may also, depending upon the exact nature of the disaster, ‚distribute‘ its effects among the other players! This is just another one of the wonderful balancing mechanisms in the game.

If trade cards constitute the major currency in Civilization, civilization cards must be considered to be the raison d’etre of the game, and the focus of all actions. There are seventy two civilization cards representing sixteen different aspects of civilisation. They are divided into four colour groups representing broad areas of human knowledge/endeavour. Most cards are members of only one group, but some are members of two. The groups are Arts (blue), Crafts (orange), Sciences (green) and Civics (red). The sixteen card types are Mysticism, Cloth Making, Pottery, Drama & Poetry, Music, Architecture, Astronomy, Metalworking, Agriculture, Coinage, Literacy, Engineering, Medicine, Law, Democracy, and Philosophy. They each have a value of between 30 and 240; the purchase price is equal to the card value but, in practice, it may be decreased by cards already held.

Each type also has an effect on play, usually by modifying some aspect of the normal rules. They are obtainable by exchanging trade cards and treasury of at least the same value. They can be relatively costly, and there may be a shortage of the various types depending upon the number of players in the game. If the latter is the case, you will find yourself competing with your opponents for a very limited resource, particularly so in the case of the more expensive cards. This has a considerable effect upon long term strategy, of which more anon.

RACE TRACK

The AST is the race track of the steeplechase alluded to earlier on. It is essentially a non-linear timeline consisting of fifteen time steps, plus start and finish positions, and covering the period from 8000 BC to 250 BC. The non-linear nature of the timeline is evident from the fact that the first time step covers a period of three thousand years while the last step covers less than one hundred years. The timeline is further subdivided into nine independent tracks, each associated with a starting area on the map board and named after the various ancient civilisations and/or regions. These are, in the order ‚down‘ the table: Africa, Italy, Illyria, Thrace, Crete, Asia, Assyria, Babylon and Egypt.

The WXB map leaf adds an extra track for Iberia and a modified African track which is only used if the WXB leaf is in play; in addition, Italy is not used when the WXB leaf is in play. Each track is divided into five numbered epochs identified as New Stone Age, Early Bronze Age, Late Bronze Age, Early Iron Age, and Late Iron Age. Furthermore, the epochs are not of the same length on the different tracks; this can have an enormous effect on the choice of strategy of the different countries.

At the end of every turn, the AST tokens are usually moved forward one step; therefore the game is at least 16 turns long. However, entry to a new epoch requires the fulfilment of a minimum condition which depends upon the new epoch. The second epoch requires two cities, the third requires civilization cards representing three groups, the fourth requires seven civilization cards, and the fifth requires civilization cards totalling at least 1000 points. Once in the fifth epoch, the player needs a stated value on the AST to move forward; this time the value of treasury and trade cards may be included in the total. In addition, in any epoch except the first, the token is moved backwards if a player has no cities! It should now be clear why the game can be likened to a steeplechase. Nearly every decision which a player makes during the course of a game should be geared towards keeping that token moving on the AST. One of the things which make the game into a classic is the fact that some of your actions can also be directed to affect the movement of your opponents‘ tokens along the AST.

ORDER OF PLAY

Each game turn is subdivided into thirteen separate phases. These are Taxation and Revolts, Population Expansion, Census, Ship Construction, Movement, Conflict, City Building, Trade Card Acquisition, Trade, Civilization Card Acquisition, Calamity Resolution, Excess Trade Card Surrender, and AST Movement. In many of these phases, the order of play will often change from turn to turn, and it is vital to become thoroughly familiar with all the ramifications of this. This variable order of play is a self-balancing mechanism that has been deliberately built into the game’s structure. It is well worth briefly discussing the eight cases in each game turn where an order of play is important.

Taxation

In the Taxation phase, any players who have insufficient tokens in their stocks to pay the taxation on their cities will lose to a revolt those cities which cannot pay. The cities in revolt go over to the player with the most tokens in his/her stock after taxes have been paid. The order in which revolts occur is in AST order, and if two players both have the same maximum number of tokens in their stocks, the recipient is decided in AST order.

Thus revolts generally occur to empires which have grown too big in some uncontrollable manner, and the recipient of cities in revolt is invariably the smallest empire on the board. However, it should be noted that revolts are, by and large, fairly rare, and usually result from player incompetence. The one exception is that a revolt can be engineered if the player concerned was nominated in a civil war in the previous rum, and this resulted in him/her having too many cities and too few tokens in stock!

Ship construction

In the Ship Construction phase, players should build and/or maintain ships in AST order. In practice, players tend to do this simultaneously as the rule is not explicit, but is implied by Rule [62] on page 16 of the rule book. However, players further down the AST are perfectly within their rights to insist upon strict AST order being observed. There are occasions when a player’s decision of whether or not to build or maintain ships will critically depend upon what other players have decided. If those players are higher up the AST, just watch what they do and make your decision accordingly.

Movement

In the Movement phase, movement is sequential in the order of the largest to the smallest populations, with ties being resolved in AST order. Since combat occurs more or less simultaneously, it is a disadvantage (except in the early turns) to move first, and the player moving last can possess an enormous strategic advantage. This is probably the single most important balancing mechanism in the game. In practice, it makes it virtually impossible to eliminate players, and gives small empires a defensive edge over their larger neighbours.

Conflict

In the Conflict phase, the order of play is only important if a token shortage is possible. Then, a player under attack may insist upon resolving first all conflicts involving only tokens, before dealing with those involving tokens attacking cities.

This is particularly important because conflicts between tokens and cities require the defending cities to convert into at least six tokens; if there are insufficient tokens in the player’s treasury, the city will surrender. The rule governing the surrender of cities implies that the best time to attack another player’s cities is when he/she has few or no tokens in his/her stock, and then, the attack should be a limited one directed only at cities.

City building

In the City Building phase, play is in AST order. As with ship construction, city building is not simultaneous, and a player is perfectly entitled to insist upon players higher up the AST making their decisions first, as to whether to build any more cities. The reasons for insisting on this are subtle.

First, it should be pointed out that the rules do not compel a player to build a city if the conditions for city construction are satisfied; the players have a choice as to whether or not to add to their city numbers. The reason for the players having this choice is to enable them to manipulate the order of play in the Trade Card Acquisition phase so as, on the one hand, to maximise the number of trade cards they acquire, and on the other, to avoid disasters. Only the very best players will be constantly aware of this stratagem during the hurly-burly of play.

Trade card acquisition

In the Trade Card Acquisition phase, the order of play is from the smallest to the largest (in terms of cities), with ties being resolved (yes, you guessed it) in AST order. This is yet another of the wonderful subtleties of the game. It is deliberately designed to ensure that the largest empires pick up their trade cards last and, in so doing, increases the chances that they will end up getting fewer cards than their smaller neighbours.

There is no point in having more cities than anyone else if this results in you gaining fewer trade cards than anyone else simply because the trade decks were depleted prior to your turn to pick up! The good players will be aware of this, and will carefully adjust the size of their empires according to what the other players are doing.

When it comes time to exchange trade cards for civilization cards, the order of play is in reverse AST order. This is the only instance in the game where this occurs. All things being equal, the players at the bottom of the AST normally have a higher chance of finding some of the trade card decks depleted when they pick up their cards. To offset this, they are given the first opportunity to purchase the relatively rare civilization cards.

Calamity resolution

In the Calamity Resolution phase, the order of play goes in ascending order of the disaster number. This can be important for players affected by more than one disaster, since it can mitigate the effect of the later disasters. For instance, if a player has picked up both the Famine and Flood cards, he could lose tokens on his/her flood plain to the famine and end up avoiding the flood completely.

STRATEGY

Now that you have some idea of how the game is played, and how its many components interact with each other, how do you develop a winning strategy? One aspect of this game is that there is no one, single winning strategy. There is, however, a winning approach to how the game should be played, which consists of six maxims.

The first maxim is to achieve the maximum population growth in the first five or six turns without affecting your progress on the AST. This involves delaying building your first cities as long as possible, and spreading out to occupy as much territory as is possible without becoming involved in fruitless border disputes. If you delay building your first city until the fifth turn, you can have six cities by the sixth turn. In contrast, building two cities in the fourth turn leaves you so short of population that you will have only three cities in the sixth turn; this will leave you short of trade cards in the early trading sessions.

The second maxim is to aim to have a medium number of cities throughout most of the second to the fifth epochs; the ideal number is six or seven, depending upon circumstances. Sticking to this should, on average, maximise the number of trade cards picked up every turn. It should also make it easier to manipulate your place in the order of play so as to avoid the worse disasters.

The third maxim is to maintain a healthy distribution of your tokens between the stock, treasury and map, and to avoid where possible having too many tokens in one of these three locations. This should ensure that you are in the best position to avoid the problems associated with over-population and inflation. There are four ways of controlling your token distribution: deliberately killing off population on the board by over-stacking, maintaining four ships on the board, partly paying for civilization cards with treasury tokens, and purchasing the Coinage card as soon as practically possible.

The fourth maxim is to offer mutually profitable trade deals wherever possible. It is not a good idea to squeeze the last drop out of a trade deal. It is also good policy to be truthful about your trade packages (except, of course, when you are attempting to pass on a disaster card). If you follow this maxim consistently, players will tend to want to deal with you in the future because they will know they are going to get a square deal.

The fifth maxim concerns the ideal policy on how to distribute disasters should you be unfortunate enough to get landed with one. In the early stages up until the end of the Bronze Age, the best policy is to distribute disasters evenly amongst the other players. Even so, there are occasions when somebody will be particularly deserving of the maximum permissible dose, and that is perfectly acceptable should it occur. However, later on, from about the start of the Iron Age onwards, disasters should be directed exclusively at the leaders wherever possible in an attempt to slow down, or even stop, their progress on the AST.

The sixth, and final, maxim is to maintain a low profile on the map, and a high one in the trading sessions. In particular, don’t make unnecessary waves such as launching unprovoked attacks on your neighbours. If you have to attack somebody, make sure it is for a good reason, and immediately apologise, explaining your reasons for the attack. And, above all, maintain a flexible approach in your actions at all times.

CALAMITIES

Following on from the general considerations discussed above, I shall continue the discussion of game strategy by considering the eight disasters and the ten starting positions, and finish up with some observations about trading and the best mixes of civilization cards to aim for.

As observed earlier in this article, the disasters are subdivided into two blocks, those which affect the player who picked the card up, and those which affect the player to whom the card was passed in a trade deal. The former consists of Volcano/Earthquake, Famine, Civil War and Flood, while the latter consists of Epidemic, Civil Disorder, Iconoclasm & Heresy and Piracy. Before we discuss each of them in turn, an important point should be made concerning the black-backed disasters in general. If you pick one up, and decide to pass it on to another player in a trade deal, the victim should be chosen carefully. By the time these cards turn up, it should be fairly obvious who is doing well and who is doing badly. Under no circumstances should one be passed on to the players who are doing relatively badly. These disaster cards are potent weapons to be aimed only at the leaders. They are the most active traders in the game, and they always want valuable cards. As soon as it becomes obvious during a trade session what commodity one of the leaders is after, make up a trade package consisting of that commodity and the disaster card and offer it him. And you should make sure the disaster is well disguised, throwing in another relatively high value commodity if necessary.

Volcano/Earthquake is the least damaging of all the disasters. There are only three volcano sites on the map: Vesuvius and Etna in Italy, and Santorini in the Aegean Sea. If any of your tokens or cities are on one of these volcano sites, the volcano erupts and all units are lost. If you do not occupy a volcano site, one of your cities is struck by an earthquake and is reduced instead; this involves replacing the city with a number of tokens equal to the carrying capacity of the area.

The best means of mitigating the effect of this disaster depends upon whether you have a volcano or an earthquake. If it is a volcano, place population tokens only in the volcano sites. Although all the volcano sites are natural city locations, it is not a good idea to set up a city in one of them because it relatively difficult to rebuild it. If, on the other hand, it is an earthquake, it is best to choose a city located in a high population carrying capacity area since this makes it easier to rebuild. The only exception to this is if you have a city on a 3-site, and reduction will lead to a token shortage in your stock and taxation problems in the next turn. If you have a city located adjacent to another player’s city, you have the option of reducing that player’s city as well; whether you do so will depend upon the game position at that time, and it is best to follow a flexible policy on this. You might even be able to force some favour out of that player in exchange for not doing it!

Famine is a mixed disaster. Although it forces you to lose nine units (cities counting as five units equivalent), you can force other players to lose up to twenty units, with a maximum of eleven from any one player. For this reason, it is one of the ‚best‘ of the game-balancing disasters. The effect of the disaster can be mitigated if you possess Pottery and at least one grain; in this case, for every grain card held, you reduce the number of units lost by four. This is obviously the perfect defence against famine, but it does have the double drawback that Pottery is one of the cheapest civilization cards available and it ties up grain cards which might be better used either in obtaining better trade deals or in purchasing another civilization card.

Civil War is one of the most potentially crippling of the disasters. If you have picked it up, you choose a nominee from among the other players. You and your nominee then divide your civilisation into two parts; you choose one, and your nominee gets the other! Usually, one of the parts must contain thirty five units (with cities again counting as five units equivalent), provided you do not hold Philosophy and/or Democracy. If you hold Philosophy, your nominee must be the player with the least number of tokens in his/her stock, (this might be you!) and fifteen units secede to the nominee. If you hold Democracy (and not Philosophy), one of the two parts must contain forty five units.

The one certain way to avoid the effects of Civil War is to keep your civilisation to thirty five units or less (or forty five if you hold Democracy). Another method is to maintain a mental count of the number of grain cards in the grain deck, and thereby know when the Civil War card will turn up. Then, by manipulating the number of cities you own, it is possible to avoid picking the card up.

If, despite all your efforts, you do pick up Civil War, there are two ways to mitigate its effect. If you have a very big civilisation, choose as your nominee a player with an equally big civilisation. Careful calculation may ensure that the nominee will lose most of the seceding units by revolt because of token and city shortage, and you might get them back provided you have the most tokens in your stock.

The second way is to choose as your nominee the player with the weakest position and make him an offer. This will usually involve the nominee choosing those units whose loss is least damaging to your overall position. Whatever way, Civil War is always vicious; do everything you can to avoid it.

Flood is the last of the red-backed disasters, and the easiest to absorb. If you have no flood plain in your civilisation, the only damage is missing out on picking up a valuable cloth card. If you do have a flood plain, the best policy is to keep it lightly populated, avoid building cities in it unless the site is elevated above the plain, and get Engineering as quickly as possible.

If you have been counting the cloth cards, and suspect that you will pick up flood, it might be worthwhile moving some tokens into someone else’s flood plain; that player’s units will be affected by the disaster as well!

Epidemic is the first of the black-backed disasters. The recipient loses sixteen units, and these losses must be taken with no area being completely depopulated; thus, a city is replaced by one token and counts as a loss of four. The recipient may then spread the epidemic among the other players, forcing them to lose up to twenty five units, with no more than ten from any one player. However, the player who passed on the Epidemic card is immune to this spread of the epidemic; the lack of side-effects makes this disaster a good one to pass on. The effect of an epidemic can be mitigated to some effect if you hold Medicine. Should the card be traded to you, you can at least affect your rivals for the lead. Consequently, the Epidemic is probably the least terrifying of the tradable disasters.

The effect of the Civil Disorder disaster increases with the civilisation size. Provided the recipient has more than four cities, and does not hold Law or Democracy, the excess are reduced. If the recipient holds Law, only cities in excess of five are reduced, and if he/she holds Democracy, cities in excess of six are reduced. Civil Disorder is a good card to trade to one of the leaders for two reasons. The first is that the leaders usually have a large number of cities, and this disaster can chop them down in size considerably. The second is that the recipient is the only player to suffer, and the player who traded it suffers no side-effects apart from the victim’s wrath.

Iconoclasm & Heresy is a terrible disaster, particularly for players who do not hold Law or Philosophy. In this case, the recipient loses four cities by reduction. Unlike Civil Disorder, he/she may also order the reduction of two cities owned by the other players, including the player who traded the card! If the recipient holds Law, only three cities are reduced, and if he/she holds Philosophy, only two cities are reduced. Similar decreases apply to the recipient’s counterattack. However, the one to watch out for is the combination of Civil Disorder with Iconoclasm & Heresy. This can be a killer for anyone without Law: the former reduces the recipient to four cities, and the latter knocks these out! This, in turn, forces the recipient backwards on the AST; this combination should be exclusively aimed at the leader if at all possible.

Piracy is potentially the worse disaster of all. The recipient loses one coastal city for every ship owned by the player who traded the card. Consequently, the recipient could lose up to four cities. If this should happen, the effect is worse than Iconoclasm & Heresy since, in that case, the cities are only reduced, whereas piracy removes the city completely, and leaves nothing behind. An obvious motto comes immediately to mind: be wary of ‚Greeks bearing gifts‘ – especially in ships!

STARTING POSITIONS

The ten starting positions are not all equal; the ones with better potential usually have a combination of excellent terrain to expand into and a relatively easy track on the AST. I shall now consider them in AST order.

Africa

Africa has one of the easiest tracks on the AST, as well as a low end-game target total of 1200 points. Its position at the head of the AST also makes it relatively easy to avoid disasters, but is a disadvantage when purchasing civilization cards. This implies that you must look ahead when planning what to purchase.

The AST advantages are offset by Africa’s poor lebensraum, and this is made more critical by the location of the natural city sites in the best population areas. Africa should therefore build at least one of the famous ‚desert‘ cities (twelve tokens converted into a city in one of its 1-areas), claim as much of north Africa as possible, and should be prepared to go to war over Sicily. Agriculture is an essential acquisition given the lack of breathing room. These problems are reduced by the WXB leaf, and this has been offset by the larger end-game target of 1300 points.

Iberia

Iberia has a relatively easy track on the AST, and a low end- game target of 1200 points. This is offset by a lack of natural city sites, one of which is in a flood plain. Although Italy is not in play when the WXB leaf is used, most of the Italian peninsula will have been claimed by the time you get there. Build a couple of ‚desert‘ cities in Iberia, and try to get Agriculture.

Italy

Italy is one of the blue chip starting positions, possessing an easy track on the AST, plenty of natural city sites, and a medium end-game target of 1300 points. Aim to occupy all of the Italian peninsula, and the Balkan coast as far south as Corfu. It really is too easy playing Italy and the other players should be aware of it.

Illyria

Illyria is a very difficult starting position. (‚What should I do in Illyria?‘) This is because its two neighbours Italy and Thrace have easier AST tracks and consequently tend to be chosen first. If this happens there is nowhere to expand into, and the position should not be chosen. However, if the WXB leaf is in use, Illyria effectively takes up Italy’s position in the standard game. Its slightly more difficult AST track and maximum end-game target of 1400 points balances this part of the map out nicely. You can now have a viable game with it.

Thrace

Thrace has an AST track of about the same degree of difficulty as Italy’s. However, it suffers from a lack of natural city sites, and the Danube Delta flood plain. Consequently, it should insist upon a frontier with Italy located as far west as possible, and a frontier with Asia/Assyria as far east in southern Ukraine as possible. It would also be an advantage to acquire Agriculture.

Crete

Crete’s central starting location is a relatively difficult one, offset partly by its low end-game target of 1200. Although there is no shortage of city sites in Greece and Asia Minor, there is a lack of breathing space. This is accentuated by the necessity to build ships to get off the island of Crete in the early stages. Once these ships have been built, Crete always moves last in the race to occupy as much land as possible before the first bout of city building. Consequently, you should concentrate upon creating a constantly mobile population, and the acquisition of both Agriculture and Astronomy are essential.

Asia

Asia has a relatively difficult AST track, which is not helped by the high end-game target of 1400. It also tends to be crowded out by Assyria. Consequently, it is advisable to choose Asia only if Babylon has not been chosen; in this case, Asia and Assyria have plenty of room to expand into. If Assyria and Babylon have already been chosen, avoid Asia like the plague.

Assyria

Assyria is another blue chip starting position to rank along side Italy. However, its slightly harder AST track and higher end-game target of 1400 make it more of a challenge to play. You should aim to expand into Asia Minor and to the Med coast at Antioch, working out sensible frontiers with Crete and Babylon. If Asia is in play in place of Babylon, it is probably best to let Asia have Asia Minor, and take over the normal Baby Ionian position yourself. However, allowing Asia through into the Babylonian position is equally viable.

Babylon

Babylon has an excellent area to expand into on the map, despite the huge Euphrates flood plain. However, its AST track is extremely difficult because of the early first barrier. Against good players, the best policy is probably to go for maximum city expansion in the early stage and take a deliberate stop for one turn at the first barrier. You will then have to attempt to ensure that all the other players make at least one stop later in the game. If Egypt is also in the game, and follows the same policy, it is advisable to cooperate closely with him/her on this. Apart from this, it is advisable to avoid building more than one low lying city on the flood plain unless it is absolutely necessary.

Egypt

Egypt has a similar problem to Babylon, and should follow a similar policy. It is essential to expand into Palestine and claim as many of the city sites there as possible. It is also advisable to agree to a sensible frontier with Africa and thereby avoid trouble on that side. And whatever you do, avoid building cities in the Nile Delta if at all possible.

TRADING

In the trading sessions, your primary aim should be to build up a set of trade cards with a value as large as possible. This generally means concentrating on one commodity at a time, and trying to corner the market in that commodity. The best commodities to concentrate upon are Salt, Grain, Cloth, Bronze and Spice. The first four commodities tend to be used as short change to fill up a trade package, but if you manage to make up a full set they complement your hand very nicely. Although Gems and Gold can make up substantial sets, there are not many of them, and the number of players with eight or nine cities is usually low. Even if you do not have nine cities, it is useful to convert eighteen tokens in your treasury whenever possible. It gains you a useful card to trade with players who do have nine cities, and helps relieve your inflation problems. It is also a good idea to offer trade packages of four or more cards every now and then; this may make it easier to conceal a disaster card in the deal if that opportunity arises.

CARDS

There is no ideal set of civilization cards to collect. If somebody maintained that such a set existed, the maximum limit of eleven cards per player would make things very tight in the five, six and seven player games. However, some general guidelines can be given. Any winning set must contain at least two of the three civic cards Law, Philosophy and Democracy, and preferably all three. The easiest way to acquire Law is to pick up Literacy and Architecture first; together, they knock forty points off the price of Law. Philosophy can purchase very cheaply if you also acquire Music and the five science cards: Mysticism, Astronomy, Coinage, Medicine, and Engineering; this reduces the price of Philosophy to seventy. The purchase price of Democracy can be reduced to one hundred and ten with the additional purchase of the four craft cards Pottery, Cloth Making, Metal Working and Agriculture.

The order in which you purchase your cards should be partly decided by two criteria. The first is to buy them in the sequence which makes the entire set you aim to collect as cheap as possible. The second criterion should be based on whether the purchase advances you past all barriers on the AST in the shortest number of turns. The problem here is that getting past one barrier ‚easily‘ may create problems at the next barrier. The most critical example of this dilemma is evidenced by a decision to purchase lots of cheap cards in order to obtain the seven cards required to pass the third barrier; this may make it extremely difficult to acquire an additional four cards with the values required to push your net total to the one thousand points required to pass the fourth barrier. However, it is sometimes not possible to stick to these criteria. This is usually caused by buying pressure on specific cards.

The trickiest decision to be made is usually connected with the order in which the cards belonging to a particular colour group are bought. The natural order is to buy from cheapest to dearest, largely because this makes it easier to pass the early barriers. However, it can be advantageous to buy some of the expensive ones first in order to create a shortage in them. This is a particularly important consideration in the five, six and seven player games.

The mix of civilization cards to aim for is also critically determined by the number of players in the game. If the number is four or less, there is no card shortage, and it is generally best to aim for the high value cards unless this causes problems during the approaches to the barriers on the AST. With larger numbers of players, it is worthwhile going for some of the cheaper cards if only to ease your passage along the AST. However, you must make sure, firstly, that you pick up a substantial proportion of the highest value cards, and secondly, that your main rivals miss out on more of them.

In conclusion, Civilization is an excellent multi-player game for those who like long games which require considerable thought, concentration and decision making. As a final point, I’ll leave you with a recommendation not to use the Avalon Hill expansion set. The additional trade cards mean that the disasters are less frequent, and this upsets the entire play balance of the game.

Civilization is a game for two to seven players; designed by Francis Tresham and originally published by Hartland Trefoil; it is now also marketed under licence by Avalon Hill and Gibsons Games. Players should be aware of which set they are playing with: there are slight differences, principally in the sequence of play, between the Hartland rules and the Avalon Hill rules.

(This article was originally published in „Games International“, issue #12 (January 1990) and is reproduced here with the permission of Brian Walker, the former editor of GI)

1830 – Sharp Practices

by Steve Jones

1830 is a game for two to six players, designed by Francis Tresham and published by Avalon Hill. Like its classic predecessor 1829 (Hartland Trefoil), 1830 is a railroad business game.

For those familiar with 1829, there are two major differences in 1830. The first is the locale: whereas 1829 covers the Railroad Era in 19th century Britain, 1830 covers the growth of the railroads in the north-eastern United States and south-eastern Canada from the early part of the 19th century to the present day. The second difference is one of emphasis. 1829 concentrates more on the mechanics of running the railroad companies for high profits and less on manipulating the stock market. In contrast, 1830 concentrates more on manipulating share prices and less on running the railroad companies; indeed, the opportunities for swindle and sharp practices are many and various for the unscrupulous investor. It is not an easy game to play. While its mechanics are fairly straightforward, there are many elements involved during play, and the interplay between them is both complex and subtle. If one were searching for an appropriate single sentence to describe 1830, it would be „This is a sharp game for sharp players“.

PORTFOLIOS

The game consists of a sturdy fold-up board of which about one-half depicts the north-eastern United States and part of south-eastern Canada. This map has superimposed upon it a grid of 4cm hexagonal spaces; hexagonal railroad track tiles are placed in these spaces during the course of play, and a railroad network is gradually built up. Around the edge of the map are marked-out spaces for the Stock Market (where the current corporation stock prices are indicated), the

Bank Pool (for sold stock and trains), the initial stock offerings, the Par Stock Price list, the trains on offer, and various explanatory tables.

The object of the game is to make money; when the game ends, the player with the most money wins. The players achieve this by buying and selling stock in an attempt to build up a stock portfolio with a high revenue and profit potential. The value of the stock certificates in an individual portfolio is determined by two things: the revenue per certificate, and the current Stock Market price per certificate. The six private railroads pay a fixed revenue, and are not quoted on the Stock Market; they have poor long term prospects, but have considerable short term value, of which more anon. The eight railroad corporations, on the other hand, have both variable revenue and Stock Market prices. There are many subtle undercurrents associated with the valuation of the available stock certificates, which become more apparent with experience.

OPERATING ROUNDS

There are no player turns as such in the game. Instead, play proceeds in a sequence of stock rounds and operating rounds until either the Bank runs out of money or a bankruptcy occurs.

In the stock rounds, the players buy and sell stock certificates; there is one certificate per private railroad, and nine certificates per corporation (one president’s certificate worth two shares, and eight single share certificates).

In the operating rounds, the private railroads earn revenue for their owners, and the president (or the largest stock holder) of each corporation performs its operations. These consist of laying a track tile, buying corporation tokens (or garrisons, as they are popularly called), conning train services (and thereby earning revenue), buying (and sometimes selling) trains, and choosing whether or not to pay a dividend to the stockholders. This last decision affects the stock price of the corporation’s stock: the price rises upon payment of dividends, and falls otherwise.

A corporation earns revenue if it has at least one train and one route between two or more cities (the revenue earned is equal to the sum of the city values of the city stops on the route); this route must include at least one city occupied by the corporation’s token.

The number of operating rounds between stock rounds depends upon whether the game is in Stage One, Two or Three, and is equal to the current stage number at the end of each stock round. The game starts in Stage One, and proceeds to Stages Two and Three respectively when the first type 3 and type 5 trains are purchased by any corporation.

CORPORATIONS

To those readers who have played 1829, this is all familiar stuff. However, five major rules changes were introduced into 1830 which have touches of genius and fumed what would have been a good game into a great game. The first rule change is that the eight corporations can be floated at any time, and in any order. Furthermore, the player who buys the presidency certificate gets to set the par value of the shares; the allowable par values lie between $67 and $100.

The second rule change concerns the corporation garrisons. In both 1829 and 1830, garrisons enable corporations to diversify their routes. A garrison in 1829 prevents rival corporations from using the station. However, a garrison in 1830 only prevents rival corporations from passing through the station; they can still run services to the station. This difference introduces a whole new dimension to the planning of routes in 1830. Basically, short routes are relatively secure, and it is the longer routes which are in danger of being chopped.

The third rule change is connected to the dividend payments. Dividends are also paid to shares in the Bank Pool, with the payments being made into the corporation’s treasury. This implies that there are times when the corporation president will feel less pressure to plough back in order to increase its treasury. This rule can be particularly helpful to the weaker companies. There are subtle implications to this rule which will become more obvious after several playings.

The fourth rule change concerns the purchasing of trains by corporations: if a corporation has a route and no trains, it must buy one. If it has not got sufficient funds in its treasury, the president must make up the shortfall out of his own resources. This requirement can easily break a player’s financial resources, and will give any aspiring corporation president nightmares in the later stages of the game. These nightmares are partly due to the technological obsolescence built into the trains, and partly due to the fact that there is no official receiver, as in 1829.

TRAINS

There are six classes of trains available in the game: types 2, 3, 4, 5 and 6, and the diesels. The type N trains run between N consecutive city stops along a specified route; the diesel is unique in that there is no upper limit to the number of stops along its specified route. Initially, there are only type 2 trains available. Each successive class becomes available when the previous class runs out. The catch is that when the first type 4 is sold, the type 2 Trains are removed from the game. The type 3 trains are removed when the first type 6 is sold, and the type 4 trains are removed when the first diesel is sold.

Consequently, a corporation might look healthy, with several trains and good revenue earnings, only to find itself foundering with no trains when another corporation buys a new type of train that has just become available. And if you should be so unfortunate as to find yourself running a corporation possessing no trains, you cannot just sell out. To sell out of a corporation, there must be another player with at least two shares to whom the presidency can be passed; unfortunately, this ‚easy‘ route out is not always available. Clearly, these corporations are not limited liability companies! The good player will be aware of this, and should plan his long term strategy accordingly.

This awareness should include becoming thoroughly familiar with what happens when each new type of train comes into play, and the implications of this on long term strategy. The solution of the problems associated with this unlimited liability rule tends to dominate play in the middle and end game. Particularly critical periods requiring great care and attention occur when there is one type 3 left, and when only one type 4 remains; the former event signals the imminent demise of the type 2 trains, and the latter event signals the imminence of Stage 3 and the consequent maneuvering which occurs when the players attempt to secure the limited number of permanent trains in the end game.

STOCK MARKET

The fifth rule change affects the Stock Market and its operation, which is where the game is won and lost, so it is well to discuss the Stock Market in some detail. The Stock Market, unlike the linear track found in 1829, is a two dimensional matrix of stock price slots printed on the game board. Low prices are found in the bottom left hand comer, and high prices are found in the top right hand comer. Within it both vertical and horizontal movements representing stock price changes are possible; the price increases if the movement is vertically upwards or a horizontal shift to the right, and decreases if the movement is vertically downwards or a horizontal shift to the left. The change in price associated with each movement is variable, with relatively high changes occurring at the extreme ends, and relatively small ones in the middle.

Vertical share price movement takes place during the stock rounds. Every time a player sells a share, the price marker of that corporation moves down one slot. At the end of each stock round, the share price of every fully subscribed corporation moves up one slot. Horizontal movement takes place during each operating round. When the president of a corporation declares a dividend, the share price marker of that corporation shifts one slot to the right; if he/she ploughs the money back into the company, the share price marker shifts one slot to the left.

FINANCIAL KILLING

The positions of the corporations on the Stock Market also serve one very important function: they determine the order of play in each operating round, from highest to lowest. This use of the Stock Market to determine the order of play in the operating rounds is of particularly vital importance in the middle game just before the first type 5 train comes out. The good player will deliberately buy and sell specific stock in an attempt to manipulate the order of play to his advantage; if he gets it right, he can ensure that corporations in which he has a good holding will be in the best positions to acquire the limited number of types 5 and 6 trains available.

Another important feature associated with the Stock Market is the priority deal: the possessor of this gets the first option to buy and sell in the next stock round. This can be very important, particularly when a player president has asset stripped his corporation and has somebody to dump it on. It is also important early on when the first corporations to be floated have increased their market values by a healthy amount. The player who can ditch his shares first has a double advantage: he will make a financial killing at the expense of his fellow shareholders by selling out just before the share price crashes, and he can choose which corporation to float next.

There are many other subtle features associated with the Stock Market, but there is no room to include them all in this article; players new to the game should look upon it as a challenge to pick them up in as few playings of the game as possible.

STRATEGY

Now that you have an idea of how the game is played, how do you develop a winning strategy? One of the beauties of 1830 is that there is no single winning strategy as such. The closest thing to one is to gain control of two corporations, but the hitch is that they both need to be good, strong corporations; if you achieve that, you will be well on the way to winning. However, there are plenty of things that your opponents can do to upset your plans. Another useful strategy is gaining control of a corporation with a potentially good diesel route, and get a diesel train into it; if you can achieve this without excessive expenditure, you could be on to a winner.

One of the tricks to playing 1830 is anticipating problems before they arise and taking the appropriate evasive actions; equally useful is knowing what to do to limit the damage if you find yourself in trouble. The most useful overall advice is to be flexible, and ruthless. Don’t form any fond attachment to your corporations. Many players invite disaster because they let themselves become emotionally attached to the corporations) they founded. They should be viewed as a means to an end, and nothing more. If you have a corporation that is headed for the rocks and you have a chance to get out, do so. Don’t stay around because you think you can rescue it; leave that dubious distinction to someone else.

There are a number of useful rules of thumb to be aware of in general play. Always be aware of who is sitting where, and take the appropriate action. For instance, don’t buy more than one share of any corporation whose president is sitting to your immediate right unless it has a ’safe‘ train, and even then, think hard about it. The reason for this is that, in a later stock round, your chances of getting rid of the excess ‚dangerous‘ shares before he dumps the corporation on you are virtually non-existent. Contrariwise, it is much ’safer‘ to purchase more than one share in a corporation if its president is sitting to your immediate left. In this case, he can only dump the corporation on you if you let him have the priority deal.

Another useful dictum is to avoid being excessively ‚greedy‘. The number of times are legion when players have bought that extra second share on the expectation of a quick short-term profit, only to find the corporation dumped on them later, the risk is usually not worth it.

DIVIDENDS

On the question of declaring dividends, the best policy is almost invariably to declare. This applies even in the case where your corporation has a vulnerable train and a low treasury. The reason is that declarations will generally increase the values of your portfolio, and this will improve your chances of weathering a bankruptcy crisis. In addition, if your corporation is that weak, it will probably have shares in the Bank Pool, in which case the treasury will increase in value even when you declare; this reduces the pressing need to plough back. The only time in which it is cost effective to plough back is when this will gain your corporation an additional train in the end game, or will ensure that it obtains a ’safe‘ train. For example, you may have a type 5 train, and a plough-back will give you enough money in the treasury to purchase a type 6 train; if your corporation has the routes on which to run both trains, it can be a winning move. Equally, a plough-back which gives you enough money to trade in a type 4 train for a diesel is always worth doing.

One other general consideration worth mentioning is to try to build up as large a portfolio as possible; if you have more shares than anyone else, your chances of winning are that much better. One way of achieving this is to get one of your corporations into the yellow, orange or brown sections of the Stock Market where its shares do not count towards the share limit for each player; if you do manage to achieve this, you will have to keep the extra Shares in the yellow at least, and that is as good an excuse to go for a diesel train as any.

The good player will also become thoroughly familiar with the different types of track tiles available, and the attacking and defensive options associated with upgrades. In particular, get to know the different types of station tiles and how they can be placed to build up highly profitable routes.

One tactic with which you should become familiar is the laying of certain strategically important tiles in order to deny another player their use. Included in this category are the yellow type #57 and the green type #59 station tiles. There are only four of the former and two of the latter in the game. They are particularly important when corporations are attempting to build their first routes; removing these tiles from play can delay that first route and cripple the affected corporation.

THE PRIVATES

Following on from the general considerations discussed above, I shall continue the discussion of game strategy by considering the six private railroads and the eight corporations, and finish up with some observations about the beginning, middle and end games.

Although the private railroads have a limited lifetime, they are vital because of their potential to provide a large influx of cash to the players later on in the game. This potential exists because of the rule which allows the players to sell the private railroads to the corporations for between half and double their face value; the only exception to this is the B&O private. This rule effectively allows the president of a corporation to embezzle funds from the corporation’s treasury.

If he times his move correctly, a player can strip the corporation of its cash and dump it on another player possessing at least two shares in it; often, when this ploy is pulled off, the corporation possesses plenty of trains which are in danger of being made obsolete in the near future. In addition to throwing a potentially disastrous problem into someone else’s lap, the large infusion into the player’s cash reserves puts him in a good position to float another corporation which will be in a better position to obtain the new trains.

Let us now consider the six private companies in ascending order of price.

Schuylkill Valley Railroad & Navigation Co.

The Schuylkill Valley Railroad & Navigation Co. is a bit of a joke. Its low face value and revenue make it of little value to anyone. Even its location in a mountain hex (G15) to the west of New York possesses little strategic value apart from being an obstacle. Its one asset is a relatively high revenue for little outlet in the early stages. Don’t break your neck in the lush to pick it up.

Champlain & St. Lawrence Railway

The Champlain & St Lawrence Railway is a slightly better prospect. Its main asset arises after it has been sold to a corporation: the owning corporation can then lay two tiles in one turn, its normal lay and one in the C&StL hex located at B20. This can be of particular value to the CanPac in its efforts to break through to the lucrative routes near New York; but it should be pointed out that this is a long shot, and if it is to be played at all, it should be early.

Delaware & Hudson Railroad

The Delaware & Hudson Railroad is starting to hit the big money. In addition to its trade in value being as high as $140, it has a potentially high strategic value in its location near to New York in hex F16. The owning corporation can lay a station tile and garrison in F16 even if this does not connect into its network. This gives corporations like the C & O, the Erie, and the CanPac the potential to set up lucrative routes to New York. This one should be watched.

Mohawk & Hudson Railroad

The Mohawk and Hudson has a potential double value. In addition to its relatively high revenue of $20 and trade in value of up to $220, it can be traded at any time for a NYC certificate. This last capability means that even if you never get the chance to use it to asset-strip a corporation, you will never lose its capital value since it can be traded in for an NYC share at any time before the first type 5 train is brought in. But don’t hold on to it too long, particularly if the NYC comes out fairly early on.

Camden & Amboy Railroad

The Camden & Amboy Railroad is a potential gold mine. Worth up to $320 cash in an asset-strip, it also has attached to it a free Penn certificate. This latter feature makes it easier to float the Penn early on, and makes an asset-strip even more likely. Even if you don’t get to use it to asset-strip a corporation, its location in H18 is potentially valuable in the struggle to control the approaches to south New York. Its high potential value will mean that there will be a lot of competition in the rush to acquire it; whoever gets it should be made to pay well above the asking price for it.

Baltimore & Ohio Railroad

The Baltimore & Ohio Railroad is a bit of a mixture. On the plus side is its high revenue of $30 and the automatic acquisition of the B&O corporation presidency. On the negative side is the fact that it automatically closes down when the B&O corporation buys its first train. Therefore, it is not as valuable as the Camden & Amboy, and as such, it is rarely worth paying more than $S over the odds to purchase it. The most critical decision which the purchaser has to make is the par value of the B&O corporation. In most cases, it is best to set it to the maximum possible value of $100; this value ensures that the capital loss from the relatively fast disappearance of the B&O private is a minimum. The par value should only be set at a lower value if you have a partner who is willing to help buy out all the shares in the first stock round, thereby ensuring that the price rises steadily in the early stock rounds; even then the decision is marginal, and better terms might need to be set.

The eight railroad corporations are not all equal; some have better potential than others.

Pennsylvania Railroad

The Pennsylvania Railroad (PRR) is potentially one of the best long term corporations, partly because of its relatively close starting position near New York, and partly because of its five tokens. The latter also implies that the PRR is a good corporation to put a diesel train into, if only because all those tokens give it the capability to defend the long routes which a diesel requires to be cost effective.

The PRR is usually one of the first corporations to be floated, because only five shares need to be bought instead of the usual six; the sixth share is already owned by the owner of the C&A. The one drawback of the PRR is its low revenue growth rate; even with two trains in it, it rarely gets above $6 a share early on.

New York Central Railroad

The New York Central (NYC) is potentially the best long term corporation, better even than the PRR. This is because its home base has a better revenue and track switching potential than the PRR’s. However, it also suffers from poor initial revenue growth rates, and because of this it rarely comes out early on. It usually comes out in the middle period of the game. Like the PRR, it has excellent potential as a diesel corporation, particularly if it can get through to south New York via G 17, and into the PRR network. However, its one weakness is that it needs a yellow type #57 tile to build its home base; therefore, any prospective president should make sure that there are plenty of these about when he floats the corporation.

Canadian Pacific Railroad

The Canadian Pacific (CPR) has a poor long-term potential despite its five tokens. Its starting position in Canada is isolated by river and mountain barriers from the lucrative routes in the centre of the board If we add to this the relatively poor revenue growth rate, it is not difficult to conclude that it is probably a lame duck corporation. The only chance it can have to become a good long term prospect is either through early access to New York via a garrison in the D&H hex in F16, or through outside help in route building.

Baltimore & Ohio Railroad

The Baltimore & Ohio Corporation (B&O) is another potential diesel corporation. This is particularly so if it gets

staved early, and gets to H16 before the PRR; in this case, the world is its oyster. Even if it cannot break out of its comer to become a diesel corporation, it still has a good potential to be one of the strongest of the medium corporations. But its president must be prepared to make things happen. And there is no harm done if it is not floated early in the game; just sit back and rake in the revenue from the B&O private. Another advantage associated with a late flotation is that the par value of $100 ensures that there is plenty of cash in its treasury with which to buy a good Train.

Chesapeake & Ohio Railroad

The Chesapeake & Ohio (C&O) is another good medium corporation with an outside chance of gaining good diesel routes. It has excellent revenue growth potential because of its close proximity to Chicago. Consequently, it is a popular choice for flotation in the first stock round Its one drawback is its relative isolation at the western edge of the board This can be partly overcome by its early flotation, which can give it the time to build up its routes.

Erie Railroad

The Erie (‚Erie‘) is one of the poorer corporations. It should never be floated early on because its home base can never be built until Stage 2; the only legal tile that can be placed in E11 is a green type #59 tile. Great care should also be exercised in laying its home base tile and token; it is dangerous to locate its home base on the north-west edge of E11 facing D10 because there are only two green type #59 tiles in the game, and if the second one goes into H18, the Erie will be crippled until Stage 3. These disadvantages usually lead to the Erie being one of the last corporations to be floated. It is strongest as the second corporation in a player’s portfolio.

New York, New Haven & Hartford Railroad

The New York, New Haven & Hartford (NYNH) is an excellent New York based corporation which possesses good revenue growth potential. While it is not common to see it floated early on, it can do well when it is. The one drawback to the corporation is its three tokens; this low number of tokens means that it is a poor choice to put a diesel train into. As one player observed, ‚If the Hartford had an extra token it would be a cracking company!‘

Boston & Maine Railroad

The Boston & Maine (B&M) is a corporation with mixed potential. If it is floated early on, it can do very well. It suffers from two drawbacks. The first is its geographical isolation from the bulk of the map by the Appalachian mountains and the river in F22. The second is its three tokens, which give it very little route flexibility. It is therefore hardly surprising to see it usually doing well only when it is floated early on. If it is one of the last to be floated, it will need friends if it is to do better than moderately well.

There is no ideal strategy to be pursued at the beginning of the game apart from being flexible. The one thing to keep in mind is that it is not vital to buy a presidency and float a corporation in the first stock round. However, if you are going to do so, the vital figure to keep in mind is $402; this is the amount of cash you need to float a corporation on your own. The only exception is the PRR for which you need only $335 in order to float it by yourself, because of the free share that goes with the C&A private. If you do have this amount available after all the private railroads have been sold, well and good; pick the corporation you want and get going. If you don’t have the cash, you will need to strike a deal with another player in order to get his help to float a corporation The one thing to keep in mind when striking such a deal is not to be dogmatic. Certainly, try to strike the best deal that you can, but remember that the other guy should be offered something to entice him into the deal in the first place. And when you make such a deal, stick to it; as a general policy, it is counter-productive to welsh on agreed deals.

The only other thing to keep in mind in the first stock round is to make sure that the other players don’t get their private railroads too cheaply. This applies particularly in the case of the C&A; it is criminal to let a player get this for $165. A useful rule of thumb to keep in mind is whether the price reduces the player’s cash reserves to less than $402, or in the case of the bidder for the C&A, to less than $335.

It is difficult to pinpoint exactly when the beginning game ends, and the middle game starts. Probably the best definition is when the second set of corporations is floated. Generally speaking, the players who are floating these corporations are doing so out of their own resources, and so it is probably a good bet that they are doing well. If a player is floating his second corporation, it is invariably a sign that he is doing very well indeed, and the other players should be thinking of ways of tripping him up. There are many ways of doing this, but the most common means involves hitting his share prices, and attempting to remove from circulation specific tiles which he may require.

Once into the middle game, you should be devoting all your efforts to ensuring that your corporation gets at least one of those valuable type 5 trains. This will usually involve keeping a close watch on the share price trends on the Stock Market, and attempting to predict which corporation is likely to be able to buy the first type 5 train. This includes taking whatever steps are required to manipulate the share prices so that the trends go the way you want them to go. The trick then is to ensure that your corporation buys the first type 5 train, or failing that, has its operating round immediately afterwards. In the former instance, you should be trying to ensure that your closest rivals do not get the good trains.

The middle game usually ends and the end game begins when the first type 5 train is brought. This usually triggers a mad rush for the type 5 and type 6 trains, and more often than not the game settles down to an uneasy period during which someone is usually saving up for a diesel train. If your corporations are ’safe‘ with their type 5 or type 6 train, and your prospects for a diesel are poor, you should be thinking about closing up the game, and making sure that any diesel routes are not very lucrative. On the other hand, if you are saving up for a diesel, you should be building, or have already built, your diesel route, and be taking whatever steps are required to protect it. The question of going for a diesel can be critical, especially since it will usually cost you something to get it. If you think you are already in front, it is probably not worth going for it. On the other hand, if you are coming second it is certainly worthwhile going for it; the only consideration that you might need to take into account is whether there will be enough time left once you have the diesel for its extra revenue to wear down the lead. The only other reason to go for a diesel is to force a bankruptcy, and in the ensuing share price chaos, to come out in front.

In conclusion, 1830 is an excellent business game which requires considerable thought and quickness of mind, and provides endless hours of enjoyment. I’ll leave you with a final recommendation: do not play with the optional extra type 6 train; that version is for wimps!

(This article was originally published in „Games International“, issue #9 (September 1989) and is reproduced here with the permission of the author)