Mafia: Strength in Numbers

1024px-Louis_Capone_and_Emanuel_WeissWhen you manage to gather a group of seven or more friends for an afternoon of idle amusement, the first thing that comes to mind, naturally, is to uncover old rivalries, trade loud barbs, and destroy the remaining fragile pretenses of camaraderie. Only that, perhaps, explicates the popularity of the party game Mafia; but knowledge of probability and social psychology might lend us some insight into the game.

For the blissfully uninitiated, here are the rules. Jurisdictions vary, but there are five essential roles. The God moderates the game and secretly assigns the players their roles: a few Mafia, one Sheriff, and the rest as Townspeople. The game begins with the Night phase, when everyone is asked to close their eyes. The Mafia are asked by the God to “wake up” by opening their eyes, silently agree to “murder” a player by pointing, then close their eyes. Then the Sheriff is called upon to silently accuse a player as a Mafioso, and receives a thumbs-up or down as confirmation of the player’s identity.

After this, everyone opens their eyes and the Day phase proceeds. The narrator reveals the Night’s victim, and the group must vote to kill off a player whom they agree is a Mafioso. The “dead” players leave the game and are prevented from further communication. All players close their eyes and the cycle restarts. If the Mafia outnumbers the Townspeople and the Sheriff, the Mafia win; if all the Mafia are dead, they lose.

An interesting question: How many players should be Mafia to provide both groups comparable chances of winning? A joint paper – given that the co-authors “were inspired by gameplay”, quite a feat that they could maintain working relationships – in 2008 established two conjectures [1]. Firstly, that the optimum number of Mafia in a game with N players, without a Sheriff, would be of order N^.5. Secondly, with the addition of a Sheriff, the optimum number of mafia escalates to of order N. A second paper in response refines the proof of the first conjecture, and claims that the second is not necessarily true [2]. A third paper derives the exact formula of optimum number of Mafia, and finds that the Mafia’s probability of winning increases by around (pi)^0.5/2 if an odd player is added to a even-numbered game [3]. Counter-intuitively, one mafia member is posed to win more frequently with a game of 9 total players than with 4. A concise takeaway is displayed in a graph below:

Figure 1. Optimal number of mafia members for a given number of players. Points show numerical results, that is, number of mafia members mopt for which w(n, mopt) is the closest to 1/2.

The shortcomings of a computational analysis lies in its gross assumptions: that the townspeople have no useful sources of information to evaluate deception, that the moderator’s and all players’ choices are completely random, and that the players vote in privacy during the Day phase, to name a few. In addition, there is little work on how game extensions – for example, the addition of special roles and items – affects gameplay. Thus quantitative modeling remains mostly theoretical, only letting us approximate a baseline probability for which to evaluate human gameplay.

Most essentially, humans appear to be better at determining a liar than lying. A study of a Mafia variant, Werewolf, shows that all the players were able to detect deception more successfully than random guessing, with the top classifier exhibiting an 87% improvement over the baseline [4]. Analyzing gameplay videos, they noted that the most major tip-offs were sudden changes of pitch and a sudden decrease of body movements and facial tics. They conceded, however, that the players’ individual personalities and motivations had to be taken into account in the analyses of these clues.

The last factors rang especially true from my personal experience – the game is often more chaotic than its rules already deem. The accusations during the first round usually revolve subtle indices during the assignment of roles: “I heard something stirring at that corner when the Mafia were called,” for example. Many moderators fall into predictable patterns of assignment choices. Players cite their real-life bonds with accusers as proof of innocence: “You know I’m not lying – I’m your roommate!” Some players even use cunning meta-game strategies: in a particularly interesting game where I moderated, a Mafioso managed to avoid detection by silently poring through his problem set throughout the game, leading everyone to assume he was not a player.

Fundamentally, Mafia is a game to be taken half-heartedly. If you are the Mafia, it is more important within a circle of friends, old or new, to be known for honesty than deception; yet were you to play the Mafia completely honestly, you’d be chided for making a lousy game. My personal conclusion, unguided by scientific research: The only way to win is to not play. Or play God, and narrate your so-called-friends’ grisly deaths as you chuckle evilly from your pulpit.

References

1. Mark Braverman, Omid Etesami and Elchanan Mossel, “Mafia: A Theoretical Study of Players and Coalitions in a Partial Information Environment”, The Annals of Applied Probability, Vol. 18, No. 3 (Jun., 2008), pp. 825-846
2. Erlin Yao, “A Theoretical Study of Mafia Games”, (Apr., 2008)
3. Migdał, Piotr, “A mathematical model of the Mafia game” (Jul., 2011)
4. G. Chittaranjan, H. Hung, “Are you a werewolf? Detecting deceptive roles and outcomes in a conversational role-playing game” International Conference on Acoustics, Speech and Signal Processing, 2010

Wing Kui Brian Ng is a second-year student at the University of Chicago majoring in Economics and English. His strategy in playing Mafia is to stay very still and pretend to be inanimate. (Shh, don’t tell!) Follow The Triple Helix Online on Twitter and join us on Facebook.

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