A soldier can hide in one of five foxholes (1, 2, 3, 4, or 5) (see Figure 5).
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A gunner has a single shot and may fire at any of the four spots A, B, C, or D. A shot will kill a soldier if the soldier is in a foxhole adjacent to the spot where the shot was fired. For example, a shot fired at spot B will kill the soldier if he is in foxhole 2 or 3, while a shot fired at spot D will kill the soldier if he is in foxhole 4 or 5. Suppose the gunner receives a reward of 1 if the soldier is killed and a reward of 0 if the soldier survives the shot.

a Assuming this to be a zero-sum game, construct the reward matrix.

b Find and eliminate all dominated strategies.

c We are given that an optimal strategy for the soldier is to hide 1/3 of the time in foxholes 1, 3, and 5. We are also told that 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. Determine the value of the game to the gunner.

d Suppose the soldier chooses the following nonoptimal strategy: ½ of the time, hide in foxhole 1; 1/4 of the time, hide in foxhole 3; and ¼ of the time, hide in foxhole foxhole 5. Find a strategy for the gunner that ensures that his expected reward will exceed the value of the game.

e Write down each player's LP and verify that the strategies given in part (c) are optimal strategies.