Rational Generating Functions and Integer Programming Games

TL;DR

This paper introduces efficient algorithms for computing pure Nash equilibria in a new class of integer programming games with piecewise linear convex payoffs, leveraging rational generating functions.

Contribution

It develops algorithms based on rational generating functions for enumerating equilibria in integer programming games with fixed dimensions and payoff complexity.

Findings

Efficient enumeration of pure Nash equilibria using generating functions.

Algorithms for computing pure price of anarchy and threat points.

Extension to Stackelberg--Nash sequential games.

Abstract

We explore the computational complexity of computing pure Nash equilibria for a new class of strategic games called integer programming games with difference of piecewise linear convex payoffs. Integer programming games are games where players' action sets are integer points inside of polytopes. Using recent results from the study of short rational generating functions for encoding sets of integer points pioneered by Alexander Barvinok, we present efficient algorithms for enumerating all pure Nash equilibria, and other computations of interest, such as the pure price of anarchy, and pure threat point, when the dimension and number of "convex" linear pieces in the payoff functions are fixed. Sequential games where a leader is followed by competing followers (a Stackelberg--Nash setting) are also considered.