Hidden-Role Games: Equilibrium Concepts and Computation

TL;DR

This paper introduces a formal framework for analyzing hidden-role games, defines a new equilibrium concept suitable for such games, and demonstrates efficient computation in some cases while proving hardness in others.

Contribution

It provides the first rigorous equilibrium definition for hidden-role games and analyzes the computational complexity of finding optimal equilibria.

Findings

Optimal equilibria can be computed efficiently in certain recreational games.

Computing an equilibrium is NP-hard or coNP-hard in most other cases.

Exact equilibria were computed for large Avalon game instances with over 10^56 information sets.

Abstract

In this paper, we study the class of games known as hidden-role games in which players are assigned privately to teams and are faced with the challenge of recognizing and cooperating with teammates. This model includes both popular recreational games such as the Mafia/Werewolf family and The Resistance (Avalon) and many real-world settings, such as distributed systems where nodes need to work together to accomplish a goal in the face of possible corruptions. There has been little to no formal mathematical grounding of such settings in the literature, and it was previously not even clear what the right solution concepts (notions of equilibria) should be. A suitable notion of equilibrium should take into account the communication channels available to the players (e.g., can they communicate? Can they communicate in private?). Defining such suitable notions turns out to be a nontrivial task with several surprising consequences. In this paper, we provide the first rigorous definition of equilibrium for hidden-role games, which overcomes serious limitations of other solution concepts not designed for hidden-role games. We then show that in certain cases, including the above recreational games, optimal equilibria can be computed efficiently. In most other cases, we show that computing an optimal equilibrium is at least NP-hard or coNP-hard. Lastly, we experimentally validate our approach by computing exact equilibria for complete 5- and 6-player Avalon instances whose size in terms of number of information sets is larger than $10^{56}$.