  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.
