William – Critical Play: Social Game

I played a 2 player variant of Mafia called Confrontation Mafia (since we couldn’t find a third person) but it was super interesting!

It seems to be created by some random people on the internet, and can be played without any technology.

The target audience seems to be people who have a mathematical/probabilistic/game theory background, as on the surface, the game can seem quite dry.

The rules and mechanics of the game are very simple: each player is assigned 1 of 3 mafia families at the start of the game. The players try to figure out each other’s family without revealing their own. By the end of the game, each player needs to decide whether to shoot the other player or not. If both players are of the same family, then they want to not shoot each other. If both players are of different families, they want to shoot each other. If one player shoots and the other doesn’t, then it’s really bad for the player who shoots if they’re the same family, and really bad for the player who doesn’t shoot if they’re different families.

Social deduction games are all probabilistic games to some extent, but this version really pushes it to its extreme. The game starts with a seeming deadlock, since neither side wants to be the first to give up any information. This downside can be controlled, however, by giving information probabilistically. For example, if I’m family A, I could choose to give mostly correct information by saying A 80%, B 10%, and C 10%. I could choose to give absolutely no information by saying A 33%, B 33%, and C 33%. I could choose to give incorrect information by saying A 10%, B 45%, C 45%.

My hypothesis is that giving incorrect information is the best / most popular choice, since if your partner believes you, you’re well set-up for a backstab, and if your partner doesn’t believe you, you’re both a bit closer to figuring out whether you are both from the same family. This heavily relies on your belief of your partner’s probability of choosing incorrect information over correct information.

This reveals a meta-game. Let’s say your partner chooses to give non-zero information, and you correct guess which strategy your partner is using (which can probably be discovered and exploited in repeated play, since I hypothesize humans are bad at switching play styles in a game theory-optimal way). Without the loss of generality, let’s say your partner chooses to give incorrect information. Also without the loss of generality, let’s say your partner says they are family B (this is without the loss of generality since your partner is A, B, or C uniformly randomly). Now you know your partner is more likely to be family B, and equally probable to be family A and C. At this point, you still cannot make a +EV decision.

Wait is it even possible to get to a +EV position? Now I’m not so sure. I suppose this might actually be a flawed game, but would require a more detailed analysis.

I would perhaps change the game to experiment with 2 families instead of 3. This would make the default state a 50/50, so it’s more conceivable that information would push it into +EV territory.

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