Hidden-Role Games: Equilibrium Concepts and Computation
Luca Carminati, Brian Hu Zhang, Gabriele Farina, Nicola Gatti, Tuomas Sandholm
TL;DR
This work formalizes hidden-role games where privately assigned roles create deception and coordination challenges, and communication channels shape strategic possibilities. It introduces hidden-role equilibria (HRE) via split-personality representations (USplit and CSplit) and analyzes their computation through mediated games and secure multi-party computation, yielding a polynomial-time algorithm under a constant-number-of-players, private-communication, minority-but-coordinated adversary setting. The study characterizes computational boundaries with strong lower bounds and defines the price of hidden roles, showing how hiding roles can drastically affect value relative to public-role variants. The authors validate their framework experimentally on Avalon, solving exact equilibria for up to six players and highlighting nontrivial equilibria values and the impact of correlated randomness. The work thus bridges game theory and cryptographic techniques to provide a foundational theory and practical tools for strategic interactions with hidden roles in both games and real-world multi-agent systems.
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 $10^{56}$.