Mafia: A theoretical study of players and coalitions in a partial information environment

Abstract

In this paper, we study a game called ``Mafia,'' in which different players have different types of information, communication and functionality. The players communicate and function in a way that resembles some real-life situations. We consider two types of operations. First, there are operations that follow an open democratic discussion. Second, some subgroups of players who may have different interests make decisions based on their own group interest. A key ingredient here is that the identity of each subgroup is known only to the members of that group. In this paper, we are interested in the best strategies for the different groups in such scenarios and in evaluating their relative power. The main focus of the paper is the question: How large and strong should a subgroup be in order to dominate the game? The concrete model studied here is based on the popular game ``Mafia.'' In this game, there are three groups of players: Mafia, detectives and ordinary citizens. Initially, each player is given only his/her own identity, except the mafia, who are given the identities of all mafia members. At each ``open'' round, a vote is made to determine which player to eliminate. Additionally, there are collective decisions made by the mafia where they decide to eliminate a citizen. Finally, each detective accumulates data on the mafia/citizen status of players. The citizens win if they eliminate all mafia members. Otherwise, the mafia wins. We first find a randomized strategy that is optimal in the absence of detectives. This leads to a stochastic asymptotic analysis where it is shown that the two groups have comparable probabilities of winning exactly when the total population size is $R$ and the mafia size is of order $\sqrt{R}$. We then show that even a single detective changes the qualitative behavior of the game dramatically. Here, the mafia and citizens have comparable winning probabilities only for a mafia size linear in $R$. Finally, we provide a summary of simulations complementing the theoretical results obtained in the paper.Comment: Published in at http://dx.doi.org/10.1214/07-AAP456 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org