Prove two person gerneral sum game , the expected payoff cannot be lower than safety level.

Question

Prove that in a two-person general sum game, the expected payoff of any player at any Strategic Equilibrium (mixed or pure) can not be smaller than the safety level of this player.

How do I prove this?

Answer 1

If player I's safety level is $s$, then by definition there exists a (possibly mixed) strategy $x$ such that whatever player II does, player I gets at least $s$ by playing $x$.

Now suppose $(x',y)$ is a Nash equilibrium in which player I's payoff is $p$ with $p < s$. By definition, since $(x',y)$ is a Nash equilibrium $x'$ must be a best response against $y$. Suppose that $x$ achieves payoff $q$ against $y$. Then we have $q \ge s > p$, which contradicts the fact that $x'$ is a best response. Thus $(x',y)$ is not a Nash equilibrium. This proves the claim.