Linear Game Theory : Reduction of complexity by decomposing large games into partial games

TL;DR

This paper introduces a method to reduce the complexity of large games by decomposing them into smaller partial games using an interaction graph identified through eigenvalue analysis, enabling more efficient computation of Nash equilibria.

Contribution

The study proposes a novel approach to decompose large games into partial games by identifying interaction graphs via eigenvalue problems, improving computational efficiency.

Findings

Successfully identified interaction graphs from utility dependencies.

Decomposed games into smaller components using linear combinations.

Verified the method with experiments on example games.

Abstract

With increasing game size, a problem of computational complexity arises. This is especially true in real world problems such as in social systems, where there is a significant population of players involved in the game, and the complexity problem is critical. Previous studies in algorithmic game theory propose succinct games that enable small descriptions of payoff matrices and reduction of complexities. However, some of the suggested compromises lose generality with strict assumptions such as symmetries in utility functions and cannot be applied to the full range of real world problems that may be presented. Graphical games are relatively promising, with a good balance between complexity and generality. However, they assume a given graph structure of players' interactions and cannot be applied to games without such known graphs. This study proposes a method to identify an interaction graph between players and subsequently decompose games into smaller components by cutting out weak interactions for the purpose of reducing complexity. At the beginning, players' mutual dependencies on their utilities are quantified as variance-covariance matrices among players. Then, the interaction graphs among players are identified by solving eigenvalue problems. Players' interactions are further decomposed into linear combinations of games. This helps to find a consistent equilibrium, which is a Nash equilibrium specified by the decomposition, with reduced computational complexity. Finally, experiments on simple example games are shown to verify the proposed method.