A Continuation Method for Nash Equilibria in Structured Games

TL;DR

This paper introduces efficient continuation algorithms for computing exact Nash equilibria in structured multi-agent games, including graphical games and MAIDs, leveraging game structure for improved performance.

Contribution

It presents the first efficient exact equilibrium algorithms for graphical games with arbitrary topology and exploits structural properties of MAIDs, advancing computational methods in multi-agent game theory.

Findings

Algorithms are guaranteed to find at least one equilibrium.

Graphical game algorithm matches or outperforms previous approximate methods.

MAID algorithm can solve larger games than prior approaches.

Abstract

Structured game representations have recently attracted interest as models for multi-agent artificial intelligence scenarios, with rational behavior most commonly characterized by Nash equilibria. This paper presents efficient, exact algorithms for computing Nash equilibria in structured game representations, including both graphical games and multi-agent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normal-form and extensive-form games due to Govindan and Wilson; they follow a trajectory through a space of perturbed games and their equilibria, exploiting game structure through fast computation of the Jacobian of the payoff function. They are theoretically guaranteed to find at least one equilibrium of the game, and may find more. Our approach provides the first efficient algorithm for computing exact equilibria in graphical games with arbitrary topology, and the first algorithm to exploit fine-grained structural properties of MAIDs. Experimental results are presented demonstrating the effectiveness of the algorithms and comparing them to predecessors. The running time of the graphical game algorithm is similar to, and often better than, the running time of previous approximate algorithms. The algorithm for MAIDs can effectively solve games that are much larger than those solvable by previous methods.