# A mathematical model of the Mafia game

Source: [A mathematical model of the Mafia game](<https://www.sota2.com/research/paper/68fa2c0e1e209131ce84c97c>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T15:06:28.363Z

Authors: Piotr Migda{\\l}

Published: 2010-09-06T11:30:24.000Z

## Abstract

Mafia (also called Werewolf) is a party game. The participants are divided into two competing groups: citizens and a mafia. The objective is to eliminate the opponent group. The game consists of two consecutive phases (day and night) and a certain set of actions (e.g. lynching during day). The mafia members have additional powers (knowing each other, killing during night) whereas the citizens are more numerous.   We propose a simple mathematical model of the game, which is essentially a pure death process with discrete time. We find the closed-form solutions for the mafia winning-chance, w(n,m), as well as for the evolution of the game. Moreover, we investigate the discrete properties of results, as well as their continuous-time approximations.   It turns out that a relatively small number of the mafia members, i.e. proportional to the square root of the total number of players, gives equal winning-chance for both groups. Furthermore, the game strongly depends on the parity of the total number of players.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Traitor win probability estimation on The Traitors Game Televised range configurations 15 \<= n \<= 25 Migdał value based](<https://www.sota2.com/research/sota/traitor-win-probability-estimation-on-the-traitors-game-televised-range-configurations-15-n-25-migda-value-based.md>): Traitor win probability estimation; The Traitors Game Televised range configurations 15 \<= n \<= 25 Migdał value based; Win Probability: 86.4.

- [Traitor win probability estimation on The Traitors Game Small configurations Migdał value based (n \<= 11)](<https://www.sota2.com/research/sota/traitor-win-probability-estimation-on-the-traitors-game-small-configurations-migda-value-based-n-11.md>): Traitor win probability estimation; The Traitors Game Small configurations Migdał value based (n \<= 11); Traitor Win Probability: 98.8.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1009.1031>)

[PDF](<https://arxiv.org/pdf/1009.1031.pdf>)