Secret Hitler Game Theory: Testing the Chancellor
https://secrethitler.tartanllama.xyz/
Preface: Secret Hitler is a secret identity and social deduction board game. Played by 5 to 10 players, Secret Hitler assigns each player to either the fascist (bad) or liberal (good) team. The fascists win by playing 6 fascist policies and the liberals win by playing 5 liberal policies. During each round of play, two players (one president and one chancellor) get voted onto an executive board. The president gets to select a chancellor. If the election doesn’t get a majority vote or when the next round comes, the person to the left of the last president or presidential candidate becomes the next presidential candidate. If elected, the president gets to secretly view three cards, each of which is either a fascist or a liberal policy. The president secretly discards one card and then hands the remaining two to the chancellor, who secretly discards one card and places (plays) the remaining card on the board for all the players to see. The biggest challenge of the game is that the fascists get to find out who their teammates are before the game begins, but the liberals do not get that luxury. Yet, to be successful, fascists must not let their evil nature be too obvious, for liberal players always outnumber fascist players and can thus vote down any executive board with a known fascist on it. Thus the game Secret Hitler often involvers every player acting liberal with fascist players discarding liberal policies when they think they can do it without being caught (ex. When a fascist is president, only he/she gets to see the three policies, so the president could discard a liberal in a 1L2F draw and claim to have gotten 3F).
The website above is a collection of unofficial Secret Hitler strategy. One of the strategies included is “testing.” When elected as president, a liberal player can test his/her chancellor’s party allegiance by giving them one liberal (1L) and one fascist (1F) policy. The idea is that a liberal chancellor will play the liberal policy while a fascist player may every so often play a fascist policy and blow his/her cover. This, of course, can only happen when the president receives 2F1L or 2L1F, but these combinations happen quite often because there are 7 liberal policies and 11 fascist policies in the deck.
Is testing a good strategy? Since liberal players always want to pass liberal policies, we can disregard the 1L2F draw—no liberal would discard a liberal in that situation. But with the 2L1F draw, game theory is an important consideration. Game theory suggests that liberal presidents should never test their chancellors. When the chancellor receives a split and plays the liberal, the president knows nothing more about them. The chancellor might be a real liberal or an undercover fascist. If the chancellor plays the fascist, the president can be sure that their chancellor is fascist, but they have no means of truly proving it because the two discarded cards cannot be viewed and the chancellor can just claim that they got two fascist policies and thus had to choice but to play one of them. At this point, the liberal president has concrete information about one fascist player, but every other liberal player has no way of knowing which of the two players in the argument is liberal and which one is fascist. Usually these two players will be voted down in all elections because the liberal don’t know who is a liberal. Assuming that the fascist player is playing optimally, we can conclude that this fascist player would only choose to the play the fascist policy card when given a split if that player determines that it is optimal. Thus we can conclude that a liberal president should never test his/her chancellor and discard a liberal in a 2L1F draw. If the president discards the fascist card, a liberal card is guaranteed to be played. If the president discards a liberal to test the chancellor, the president is only risking that a fascist chancellor would find it optimal to play the fascist.