Person A and Person B are playing a two-finger morra game, where they each show either one finger or two fingers, at the same time.
If the sum of the fingers is even, person A gets the product of the two fingers. If the sum of the fingers is odd, person B gets the product of the two fingers.
What's the expected payoff for Person A?
Here are the possibilities:
Person A plays 1, person B plays 1 ==> Person A wins 1 dollar
Person A plays 2, person B plays 1 ==> Person B wins 2 dollars
Person A plays 1, person B plays 2 ==> Person B wins 2 dollars
Person A plays 2, person B plays 2 ==> Person A wins 4 dollars
Lets assume that the probability that Person A chooses 1 is $p$, and lets assume that Person B knows $p$ and can choose his strategy based on this information.
How would I find the expected payoff for A?
I want to say that the expected payoff for A is: $p - 2(1-p) - 2p + 4(1-p)$
But I don't know if this is correct
You can model this as the zero-sum game with payoff matrix $$G = \left[ \begin{array}{cc} 1 & -2 \\ - 2 & 4 \end{array} \right]$$ The unique maximin (or equilibrium) strategies are $p=P(A \mbox{ plays } 1) = 2/3$ and $q=P(B \mbox{ plays } 1) = 2/3$. The value of the game is $0$.