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# What is meant by (a) Zero-sum game? (b) Payoff matrix?

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## Reward matrix, Dominated strategies, Optimal strategies

Please state all your assumptions and show all your work. Define your decision variables clearly. Briefly explain your constraints and objective functions. Define all units of measure (e.g., hours, \$, \$/hour, etc.). Explain what software package you used (e.g., LINDO, LINGO, EXCEL solver, etc.). For EXCEL solver, be sure to give a separate statement of the formulation /input. Use an equation editor for the equations. If you can not get one, then use subscripts to indicate indexing. Graphs should be detailed and easy to read.

A soldier can hide in one of five foxholes (1, 2, 3, 4, or 5) as shown below. 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.

[1] A [2] B [3] C [4] D [5]

a) Assuming this to be a zero-sum game, construct the reward matrix (payoff table).

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 non-optimal strategy: 1/2 of the time hide in foxhole 1; 1/4 of the time hide in foxhole 3, and 1/4 of the time, hide in 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.

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