I was thinking about what properties action pairs in the support of Nash equilibrium strategies have. Specifically, let $(x, y)$ be a mixed-strategy Nash equilibrium of some general-sum game $(A,B)$ with $A, B \in \mathbb{R}^{m\times k}$. What I am wondering about is whether the following is true?
For any $(i,j) \in supp((x,y))$ at least one of the following statements holds:
- $A_{i,j} \geq x^\top A y$
- $B_{i,j} \geq x^\top B y$
On the one hand, it would be weird to me if this is not true, since that would mean that an action pair, which has positive weight, is not beneficial for either player. On the other hand, I do feel like there could be a counterexample similar to the prisoner's dilemma (the prisoner's dilemma is not a counterexample, I think, since the Nash eqquilibrium is deterministic).