Equilibria in Multiplayer Graph Games: An Algorithmic Study

TL;DR

This paper investigates the computational complexity of finding various types of equilibria in multiplayer graph games, extending game-theoretic verification methods to more complex multi-agent scenarios.

Contribution

It provides complexity results for the constrained existence problem of five equilibrium notions in multiplayer graph games.

Findings

Complexity results for five equilibrium concepts.

Decidability of equilibrium existence with payoff constraints.

Extension of game-theoretic verification to multiplayer settings.

Abstract

To verify the robustness of a program or protocol, it is common in the computer science community to rely on the theoretical framework of game theory. In particular, if one seeks to enforce a desired property, or specification, despite an unpredictable environment, a useful abstraction is to model the situation as a two-player zero-sum game. The goal is then to find a strategy for the system that guarantees the specification against any strategy of the environment. However, to model more complex situations, such as multiple systems with different objectives or an environment composed of various agents, the richer framework of multiplayer games must be considered. In this setting, a natural question is to identify equilibria, i.e., strategy profiles that are robust in the sense that no player has an incentive to deviate. The most well-known equilibrium concept is the Nash equilibrium, but several alternatives exist. We study five such notions and, for each of them, we provide complexity results for the constrained existence problem, which consists of deciding whether a given game contains an equilibrium that ensures each player a payoff within a specified interval.