There exists a two$-$player game with the property that $“$the payoff for Alice if Alice plays a

strategy S$1$ and Bob plays a strategy S$2”$ is the same as $“$the payoff for Bob if Bob plays S$1$

and Alice plays S$2”.$

$1.$ Show the given property holds in the above two$-$player game.

$2.$ Give a case to show that there is one Pareto efficient and such a Pareto efficient is also

a Nash equilibrium $($justify your answers$)$ in the above two$-$player game.

$3.$ If you think the above case is unique, prove it$.$ Otherwise, give a counter$-$example.

$4.$ Prove or disprove that there does not exist two Nash equilibria in the above two$-$player

game.