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

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


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 $i,j \in \mathbb{N}$.

Where was this from? I might be missing something but I don't see a reference. Thank you in advance for your help.

  • $\begingroup$ Are you asking for a proof or for the name of the theorem? "Well-known" is usually code for unclear origin. $\endgroup$ – Giskard Sep 18 '19 at 22:13
  • $\begingroup$ I've seen more recent papers using this result by referring to this paper, but I can't find a proof or trace its origin. $\endgroup$ – alpeter Sep 19 '19 at 19:06

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