Rational Generating Functions and Integer Programming Games

Authors:
Matthias Köppe;Christopher Thomas Ryan;Maurice Queyranne
Affiliations:
Department of Mathematics, University of California, Davis, Davis, California 95616;Booth School of Business, University of Chicago, Chicago, Illinois 60637;Sauder School of Business, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Venue:
Operations Research
Year:
2011

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Abstract

We explore the computational complexity of computing pure Nash equilibria for a new class of strategic games called integer programming games, with differences of piecewise-linear convex functions as 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.