  We study new classes of games, called zero-sum equivalent games and zero-sum
equivalent potential games, and prove decomposition theorems involving these
classes of games. We say that two games are "strategically equivalent" if, for
every player, the payoff differences between two strategies (holding other
players' strategies fixed) are identical. A zero-sum equivalent game is a game
that is strategically equivalent to a zero-sum game; a zero-sum equivalent
potential game is a zero-sum equivalent game that is strategically equivalent
to a common interest game. We also call a game "normalized" if the sum of one
player's payoffs, given the other players' strategies, is always zero. We show
that any normal form game can be uniquely decomposed into either (i) a zero-sum
equivalent game and a normalized common interest game, or (ii) a zero-sum
equivalent potential game, a normalized zero-sum game, and a normalized common
interest game, each with distinctive equilibrium properties. For example, we
show that two-player zero-sum equivalent games with finite strategy sets
generically have a unique Nash equilibrium and that two-player zero-sum
equivalent potential games with finite strategy sets generically have a
strictly dominant Nash equilibrium.
