The Vote-Left Equilibrium: A Deterministic Coordination Strategy for the Faithful in The Traitors
2026-05-11 • Computer Science and Game Theory
Computer Science and Game Theory
AI summaryⓘ
The authors study The Traitors, a game where a small informed group tries to trick a larger uninformed group. They show that random voting helps the uninformed group when all votes are cast simultaneously, but not when votes happen one by one. They introduce a new voting system called Vote-Left where each player votes for the next alive player in a fixed cycle, making any cheating easy to spot and punish. This method helps the uninformed group win more often when certain conditions are met, especially during televised matches where the informed group can't safely collude.
social deduction gameTraitorsFaithfulrandom votingVote-Left protocolPerfect Bayesian Equilibriumcollusionbanishment distributionsimultaneous-signal protocolsgame theory
Authors
Vince Knight
Abstract
The Traitors is a social deduction game in which an informed minority of Traitors face an uninformed majority of Faithful, and the recurring question facing the Faithful is how to vote. Random voting is known to be optimal for the uninformed majority under simultaneous-signal protocols [Braverman, Etesami and Mossel, 2008], but when votes are cast individually, random votes are indistinguishable from strategic ones and the Faithful remain exposed to coordinated Traitor collusion. We introduce the Vote-Left protocol, a deterministic rule under which every player votes for the next surviving player in a fixed cyclic ordering. Under full compliance every surviving player receives exactly one vote, so the banishment distribution coincides with random voting; since prescribed votes are deterministic functions of public information, any deviation is immediately identifiable. Combined with a simple punishment rule, Vote-Left constitutes a Perfect Bayesian Equilibrium for every state with $n_t > 2m_t + 2$, a region that contains every televised configuration. We characterise the Traitors' best response in the late-game phase ($n_t \leq 2m_t + 2$): deviate via collusion once the Faithful no longer have enough votes to guarantee punishment. Across the configurations played on television, Vote-Left raises the Faithful's winning probability by a factor of approximately three over random voting under collusion.