FPTAS for Mixed-Strategy Nash Equilibria in Tree Graphical Games and Their Generalizations

TL;DR

This paper introduces the first fully polynomial time approximation scheme (FPTAS) for computing approximate mixed-strategy Nash equilibria in tree-structured graphical multi-hypermatrix games, generalizing several game types.

Contribution

It presents the first FPTAS for tree polymatrix and graphical games with bounded actions, extending the tractability of approximate equilibria in complex game classes.

Findings

FPTAS for tree-structured GMhGs established

Approximate equilibria computed efficiently in polynomial time

Quasi-PTAS available when actions grow logarithmically with players

Abstract

We provide the first fully polynomial time approximation scheme (FPTAS) for computing an approximate mixed-strategy Nash equilibrium in tree-structured graphical multi-hypermatrix games (GMhGs). GMhGs are generalizations of normal-form games, graphical games, graphical polymatrix games, and hypergraphical games. Computing an exact mixed-strategy Nash equilibria in graphical polymatrix games is PPAD-complete and thus generally believed to be intractable. In contrast, to the best of our knowledge, we are the first to establish an FPTAS for tree polymatrix games as well as tree graphical games when the number of actions is bounded by a constant. As a corollary, we give a quasi-polynomial time approximation scheme (quasi-PTAS) when the number of actions is bounded by the logarithm of the number of players.