"A soldier can hide in one of five foxholes, and a gunner can hide in four spots: A, B C, and D. The configuration looks like this: 1 (A) 2 (B) 3 (C) 4 (D) 5. If a shot is fired at a location and the soldier is in an adjacent foxhole (ex: shot is fired at B and soldier is in hole #2 or #3), the gunner received a reward of 1. Otherwise, the gunner receives a reward of 0. Assume that this is a zero-sum game
We are given that the optimal strategy for the soldier is to hide 1/3 of the time in foxholes 1, 3 and 5. For the gunner, an optimal strategy is to shoot 1.3 of the time at A, 1/3 of the time at D, and 1/3 of the time at B or C.
I have to determine the value of the game for the gunner. Honestly, I have no idea where to start. I know why the player should hide at holes one and five, but not three. How would I go about solving this>.
HINT: Look at the payoff matrix below. The top line shows the soldier’s options, and the lefthand column shows the gunner’s options. The numbers in the body of the table show the gunner’s payoff for each combination of options chosen by him and the soldier. You’ve been told that the soldier chooses each of his options with probability $1/3$, and that the gunner does the same. Thus, each of the nine outcomes is equally likely. The value of the game to the gunner is his expected payoff, which is simply the weighted average of the $9$ possible payoffs, each weighted according to its probability of occurrence.
$$\begin{array}{r|ccc} &1&3&5\\ \hline A&1&0&0\\ BC&0&1&0\\ D&0&0&1 \end{array}$$