# What is the expected payoff in the following matching pennies game

So we have the following matching penny game

$$\begin{bmatrix} \ & L & R \\ T & (1,-1) & (-1,1)\\ B & (-1,1) & (1,-1) \end{bmatrix}$$

Let $$p_1$$ denote the row player, $$p_2$$ denote the column player.

Now suppose there is an event, $$D$$, which players assess with distinct subjective probabilities: $$p_1(D) = 2/3; p_2(D) = 1/3$$. Suppose players bindingly agree to the following strategies: $$2$$ plays $$L$$ and $$1$$ plays $$T$$ if $$D$$ occurs and they play $$(B,R)$$ otherwise. What is the expected payoff for both players in this case. I am expecting it to be $$1/3$$, but I cant figure it out.

Any helps or insight is appreciated.

• Why is this tagged with nash-equilibrium? The question doesn't seem to have anything to do with that. – joriki Feb 17 at 20:53

Since $$(T,L)$$ and $$(B,R)$$ yield the same payoffs $$(1,-1)$$, the probabilities are irrelevant – a strategy pair with payoffs $$(1,-1)$$ is certain to be played, so the payoffs will be $$(1,-1)$$.