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arXiv:1009.1031math.PR

A mathematical model of the Mafia game

Piotr Migdał

Early discovery. Linked by one identifiable commenter. This paper is outside the main ranking; a mention is not an endorsement.

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.

arXiv comments
12 pages, 4 figures; after corrections
Journal reference
DOI
Subjects
math.PRProbability cs.GTComputer Science and Game Theory physics.soc-phPhysics and Society

On Hacker News

Linked by 1 identifiable commenter in 5 comments across 5 threads, from Nov 2015 to Jan 2026. Found in 5 HN submissions from Mar 2015 to Aug 2022. Open submission discussions.

Comments linking this paper, per year.

Discussed in comments5 threads

  1. Are two heads better than one? 1 comment

    • In voting parity matters. In some cases, effects are interesting - like in some cases if the mafia/werewolf game, when adding a citizen/villager decreses their winning chance by sqrt(pi/2), vide https://arxiv.org/abs/1009.1031 When it comes to the wisdom of crowds, see…

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  2. Convolutions, Polynomials and Flipped Kernels 1 comment

    • Saw that! Sometimes it makes things simpler (quite a a lot of things in combinatorics), other times it is a tools for nice tricks (I have no idea how I would solved these equations if it were not for generating functions, see the appendix from a Mafia game paper…

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  3. Is Werewolf Killing the Conference Hackfest? (2007) 1 comment

    • I miss Mafia. I used to play it in high school on some math camps, then during my first year of undergraduate. Later - not so much. I would love to participate in a conference and "being killed by Mafia". I loved the game so much that my Bachelor's thesis was "A mathematical…

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  4. The CIA uses board games to train officers 1 comment

    • Out of geeky (but impractical) things, here is my "A mathematical model of the Mafia game": https://arxiv.org/abs/1009.1031 :)

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  5. New derivation of pi links quantum physics and pure math 1 comment

    • The game of mafia (aka Werewolf) is linked to PI (my paper: http://arxiv.org/abs/1009.1031, for a summary: http://boardgamegeek.com/thread/1157558/polish-paper-werewol...), also via the Wallis formula. But so what? Pi appears in physics roughly as often as 2 in finance. Even to…

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