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.