AbOrigineMundi

When starting a game of werewolf, it is useful to know the probability that wolves win, given the number of werewolves and villagers. The closer chances are to 50-50, the more fun the game.

The actual probabilities that werewolves win are difficult if not impossible to calculate since this depends on several factors. The most important ones are:

• the existence of special roles, such as the seeer, the witch, the possessed,…
• how well the players (especially the werewolves and special characters like the seeer) can play their role, i.e. how well they can prevent the other players from having information on who they are.
• the interaction between the above factors.

If we make some assumptions, calculating the probabilities is possible (see figure):

• no special roles: only werewolves and villagers are allowed. This boils down to using the vanilla rules.
• no information: villagers do not have any information on who the wolves are, so they vote randomly.
• optimal play for werewolves (i.e. werewolves don’t vote for each other)

How to choose the number of werewolves?

If both parties, werewolves and villagers, have an equal chance to win, the game is most fun. Up to 17 players, a lone wolf is the best choice. From 18 players on, two wolves are needed. From 44 players, 3 werewolves have about an equal chance to win. Four wolves are needed to play in a game with 86 players, five in a game with 140 players and six in a rather massive game with 210 players.

Download the free wwprob calculator here. File is zipped (800kb). Unzip and run wwprob.exe. No installation required.

Probability of a win for the werewolves, given their number and the number of villagers, under the vanilla rules and assuming no information and perfect play.

The sawtooth-effect

An astonishing result is what I call the sawtooth-effect. Increasing the number of villagers by one can actually decrease the probability that they win the game. How is that possible?

Suppose we have one werewolf and three villagers. The probability that the lone wolf wins the game is 2/3. Why?

During the first night, the werewolf will eat one victim. In the morning, the number of villagers is 2. They will hang one person. Chances are 1 in 3 that this is a werewolf. If the two villagers hang the werewolf, they win the game. If not, a daytime massacre occurs and the werewolf wins. So the probability of win for a lone wolf against three villagers is 1-(1/3)=2/3.

Now we increase the number of villagers to 4. The probability of win for the wolves is then 3/4, which is considerably higher than the chance to win with one villager less. Why?

During the first night, the wolf will eat a victim. In the morning, only three villagers and one wolf are left. The villagers hang one player randomly, so they have 1 chance out of 4 to hang the werewolf.

If they do, they win (probability = 1/4). If they don’t, they hang a fellow citizen and the wolf will eat one victim during the night. Only one villager and one werewolf survive. The sun rises, and the wolf will organize a daytime massacre…

So the source of the sawtooth-effect is this: the probability of hanging the wrong guy is higher when there are more innocent persons around.

Ten quasi-perfect games of werewolf

With 100 players or less, the perfect game (probabilities of winning for both parties exactly 50-50) does not exist.

These ten games come closest (indicated before the colon is the probability of werewolves winning):

• 0,5016: 24 players of which 2 werewolves
• 0,5040: 35 players of which 2 werewolves
• 0,5059: 52 players of which 3 werewolves
• 0,4952: 54 players of which 3 werewolves
• 0,5060: 73 players of which 3 werewolves
• 0,5040: 75 players of which 3 werewolves
• 0,4941: 77 players of which 3 werewolves
• 0,5045: 92 players of which 4 werewolves
• 0,4996: 94 players of which 4 werewolves
• 0,4947: 96 players of which 4 werewolves