Rational Generating Functions and Integer Programming 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.

Symmetric games with piecewise linear utilities

“…The simplicity by which they were used to compute important information on the structure of these games, again reinforces the power of generating functions as an analytical and computational tool for game theory. This confirms the hope of the previous study which used generating functions in games [26] had that these methods can be used to enlighten further investigations in algorithmic game theory, and only adds weight to the case for future use of the method.…”

“…al. [26] on using rational generating functions to find equilibria in integer programming games, in particular both build on the theory of rational generating functions, largely from the…”

“…The current paper relates to some recent work by one of the authors [26]. This work introduced rational generating function methods to the algorithmic study of games, showing that they can be used to compute pure-strategy Nash equilibria of games in which the action sets are represented by fixed-dimensional polyhedra and the utilities are given by piecewise linear functions.…”

Nash equilibria in the two-player kidney exchange game

“…Note also that in our formulation of N -KEG, players' strategies are lattice points inside polytopes described by systems of linear inequalities. Thus, according to [15], N -KEG and, in particular, 2-KEG belongs to the class of integer programming games.…”

Computing pure strategy nash equilibria in compact symmetric games

“…The current paper relates to some recent work by one of the authors [24]. This work introduced rational generating function methods to the algorithmic study of games, showing that they can be used to compute pure-strategy Nash equilibria of games in which the action sets are represented by fixed-dimensional polyhedra and the utilities are given by piecewise linear functions.…”