# Cross derivatives of payoff functions in potential games

In Monderer and Shapley (1996) (pdf) there is the following theorem.

The next theorem is well-known (and very useful).

THEOREM 4.5

Let $$\Gamma$$ be a game in which the strategy sets are intervals of real numbers. Suppose the payoff functions are twice continuously differentiable. Then $$\Gamma$$ is a potential game iff $$\frac{\partial^2{u^i}}{\partial{y^i}{y^j}}=\frac{\partial^2{u^j}}{\partial{y^i}{y^j}}$$ for every $$i,j \in \mathbb{N}$$.