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Mafia party game probabilities for optimal play

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I know this is directly anathema to what you're asking, but I actually would encourage you not to get too bogged down in the metagaming of Mafia. My friends and I, largely engineer/high level gaming type people did, and while mafia was a blast to play at first, it becomes almost fantastically easy for the villagers to win in most variants if everyone knows what they're doing. Took a lot of the magic out of the game for us until people started deliberately throwing to keep the games interesting
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Not really. You'd need to know the probabilities of civilians voting for civilians/mafia, this is not random but dependent on player behavior and judgment. Same with Mafia voting Mafia

(Btw, Reddit seems to no longer be broken now, you can delete your double posts)
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You can do Bayesian updating. You start with a set of hypotheses, and the probabilities you think you should attach to them at the start (aka the prior), then there's a procedure that tells you how to update those based on new evidence, depending on the (known) relative chance of said evidence occuring under each hypothesis. That gives you the new (posterior) probabilities.

Odds notation is very useful for such calculations. 1 : 1 : 3 means 20%, 20%, 60%. What are the new probabilities if the second hypothesis becomes 4 times more likely? Simply 1 : 4 : 3. Getting 12.5%, 50%, 37.5% with the usual approach of using Bayes' law is a chore, compared to a single multiplication.

In this case, it could get very complicated - a reasonably simple treatment would be to have one hypothesis for each 2 possible mafia members of the 7 - a total of 21 - plus some assumed values like "chance of mafia killing each other" to account for their strategies.

Let's say, as an example, that you know mafia are 4 times more likely to kill a civilian than another mafia, independent of anything else like number of living mafia or players or current round. Let's also say you have zero clues about anything, so all starting possibilities are equiprobable. Then you start with odds 1 : 1 : 1 : 1 : 1 ... for each possible assignment of {M,M,C,C,C,C,C}, {M,C,M,C,C,C,C}, ...

Then if person 3 dies by mafia, the odds of all cases of the 3rd position being civilian get multiplied by 4, others stay constant. (Or, equivalently, divide the others by 4.) So 4 : 1 : ...

To calculate the chance of each person being mafia, sum up the odds of the cases where he's mafia and divide by the sum of all the odds. It is possible to get conditional probabilities, too - if you want to know the probability of X being true given Y is true, sum up the "X and Y" cases and divide by the sum of the Y cases.

The power of this approach is that you can do all sorts of updates, independent from each other: e.g. someone accusing someone else could raise the odds that they are of opposite alignment a bit (so multiply all those assignments by some number >1), even if you don't directly change the accuser's or accusee's odds towards being mafia or innocent at all. You can treat someone being voted out as weak evidence they're mafia, if you assume the players have slightly more skill than pure luck (and no anti-skill). If someone is verified 100% to be mafia/civ, you can zero out all contradictory possibilities and make further calculations easier. "Someone seems sus so they're 20% more likely to be mafia" is also a valid update.

The weaknesses are that you have to just guess or make up some of these numbers with your intuition if you don't have loads of data already, and that it takes a lot of calculation to get anywhere. Plus the combinatorial explosion of hypotheses - 11 players 4 mafia gives 330 cases to keep track of.

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