Stochastic game matrix

Question

I am trying to solve this exercise and I can't find all Nash Equilibrium.

I posted as picture because I couldn't make this matrix!.

Answer 1

$x$ can just be treated like $E(x)=\frac{2}{3} (12)+\frac{1}{3} (0)$. Since neither player knows $x$, they can't condition their strategies on it, and player 1 evaluates the action T based on the expected payoff he'd receive from playing T, given what he expects player 2 to do.

The two pure strategy equilibria are BR and TL.

BR is a nash equilibrium because if player 1 deviates to T he gets 3 instead of 8, and if player 2 deviates to L instead of R he gets 0 instead of 9.

Similarly, TL is a nash equilibrium. Player 1 is receives a payoff of $E(x)=8$ from playing T and only receives 6 from playing B. Similarly, player 2 receives 6 from deviating and currently receives 9.

Finally, there is a mixed strategy equilibrium. Let $p$ be the probability player 1 plays T, and $q$ be the probability that player 2 plays L. $q$ must make player 1 indifferent between T and B, i.e. $$qE(x)+3(1-q)=6.$$ Solving for $q$, we get $q=3/5$. Similarly, $p$ must solve $$9p+0(1-p)=6p+9(1-p),$$ so $p=3/4$.