Equilibria in Multiplayer Graph Games: An Algorithmic Study

2026-05-19

AI Summary

This study addresses a fundamental challenge in the formal verification of multi-agent systems: determining whether equilibrium strategies exist in multi-player graph games that satisfy given payoff constraints. The work provides a systematic investigation of the constrained existence problem under five distinct equilibrium concepts, integrating computational complexity theory, formal methods, and game theory to deliver a complete characterization of the associated complexity classes. In contrast to classical two-player zero-sum games, this research substantially extends the analytical framework by precisely delineating the computational boundaries of constrained equilibrium existence across different solution concepts, thereby establishing a rigorous theoretical foundation for verifying robustness in multi-agent systems.

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.

Problem

• multiplayer games
• equilibria
• Nash equilibrium
• constrained existence
• graph games

Innovation

• multiplayer graph games
• Nash equilibrium
• constrained existence problem
• computational complexity
• equilibrium concepts