Use game theory to decide your next action in Werewolf
The game of Werewolf usually consists of twelve players who will belong to two opposing camps in the game: an informed minority (the Mafiosi or the werewolves), and an uninformed majority (the villagers). The villagers, which are larger in number but do not know each other, use exile voting and character skills to eliminate all the werewolves hidden in the crowd to get the final victory. While the werewolves, who are smaller in number and know each other, hide among them and rely on hunting the villagers at night and inducing them to vote incorrectly during the day to win. There are various victory conditions: all werewolves out, villagers win; all villagers out, werewolves win; whether all villagers without skills out or all villagers with skills out, werewolves win.
According to the rules of the game, we assume that each player will take the best option he can to achieve victory for his/her side. Most of the situations are complex and not well-informed, but there are situations where we can use game theory to analyze the possible action patterns of the opponent.
Suppose the situation now is: Sorcerer, guard, hunter, a werewolf that the Sorcerer knows in previous rounds, unknown player ABC.
In the next round, the Sorcerer claims to go to test A. Then two cases: A is a werewolf (probability 1/3), or A is a villager (probability 2/3.) The probability of a villager winning necessarily increases when A is a werewolf, so we only need to discuss the case where A is a werewolf.
Case 1, (probability 1/2, the guard has not guarded A) Since the Sorcerer claims to check A, the guard, therefore, guards A.
1.1 The werewolf kills A and A does not die due to being guarded, the werewolf must be in B and C. B will vote for C and vice versa. The probability of the villagers winning increases by ½*⅔*½=⅙.
1.2 The werewolf kills the Sorcerer, who dies while telling the others that A is a villager by handing over the badge to A. The villager’s probability of winning still increases by ⅙.
Case 2, (1/2 probability that the guard has not guarded the Sorcerer) the guard still guards the Sorcerer even though the Sorcerer claims to check A.
2.1 If the werewolf kills A, the werewolf must be in B and C. B will vote for C, and vice versa. The probability of the villagers winning increases by 1/6 and the probability of the werewolf winning increases by ½*⅔*½=⅙.
2.2 If the werewolf kills the Sorcerer, the Sorcerer will not die because of being guarded. Since the guard can guard known villager A in the next round, and B and C will die during the vote, the werewolf must lose.
Therefore, in the case that the Sorcerer claims to check A, the best choice for the guard is to guard A and the best choice for the werewolf is to kill A. At this point, the villagers have a 2/3 chance of winning and the werewolf has a 1/3 chance of winning.
Reference:
https://en.wikipedia.org/wiki/Mafia_(party_game)