A Theoretical Study of Mafia Games

TL;DR

This paper critically examines and corrects a key theorem about the probabilities of winning in mafia games, and confirms a conjecture that extends the understanding of the game's dynamics to broader scenarios.

Contribution

It corrects a previous theorem on mafia game probabilities and proves a conjecture to generalize the phenomenon to more extensive frameworks.

Findings

The main theorem by Mossel et al. does not guarantee the original conclusion.

A new theorem validates the original conclusion about winning probabilities.

The conjecture by Mossel et al. is proven, extending the phenomenon to broader settings.

Abstract

Mafia can be described as an experiment in human psychology and mass hysteria, or as a game between informed minority and uninformed majority. Focus on a very restricted setting, Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2] showed that in the mafia game without detectives, if the civilians and mafias both adopt the optimal randomized strategy, then the two groups have comparable probabilities of winning exactly when the total player size is R and the mafia size is of order Sqrt(R). They also proposed a conjecture which stated that this phenomenon should be valid in a more extensive framework. In this paper, we first indicate that the main theorem given by Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2] can not guarantee their conclusion, i.e., the two groups have comparable winning probabilities when the mafia size is of order Sqrt(R). Then we give a theorem which validates the correctness of their conclusion. In the last, by proving the conjecture proposed by Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2], we generalize the phenomenon to a more extensive framework, of which the mafia game without detectives is only a special case.