I am trying to construct a normal-form game satisfying all of the following conditions:
- there are two players;
- each player has a finite number of strategies;
- there is a unique Nash equilibrium, in which
- one of the players uses a pure strategy; and
- the other player uses a non-degenerate mixed strategy.
I have put considerable effort into trying to construct an example, but I keep finding that if a pure–mixed Nash equilibrium exists, then there exist other Nash equilibria as well. I am actually wondering if it is possible to construct an example with a unique such Nash equilibrium.
If this is not possible, I am wondering how to prove the following implication of this:
Let $\Gamma$ be a normal-form game with two players. The strategy sets are $S_1$ and $S_2$, each of which is a non-empty finite set. If $(\sigma_1^*,\sigma_2^*)\in\Delta(S_1)\times\Delta(S_2)$ is a Nash equilibrium of this game such that
$\sigma_1^*(s_1)=1$ for some $s_1\in S_1$; and
$\sigma_2^*(s_2)<1$ for all $s_2\in S_2$,
then there exist other Nash equilibria.
Any feedback is much appreciated.