Decompositions and potentials for normal form games

We introduce a method of decomposing a p-player normal form game into 2p simultaneously-played component games, each distinguished by the set of “active” players whose choices influence payoffs. We then prove that a normal form game is a potential game if and only if in each of the component games, all active players have identical payoff functions, and that in this case, the sum of these shared payoff functions is the original game’s potential function. We conclude by discussing algorithms for deciding whether a given normal form game is a potential game.