Question

Blue holds two military positions which are subject to attack. He values installation A at one unit and installation B at three units. He is capable of successfully defending either installation but not both. His opponent, Red, is capable of attacking either unit but not both. Blue desires to analyze the situation from the viewpoint of game theory (let Blue be the Row player and let Red be the column player).

a. Set up the game matrix representing the situation.
b. State the measure of effectiveness you used.
c. What strategy guarantees Blue at least the maximum payoff?

Answer 1

The game matrix for Blue and Red's situation includes two strategies for Blue (Defend A, Defend B) and two strategies for Red (Attack A, Attack B). The measure of effectiveness is based on the utility of the installations. Blue's dominant strategy that guarantees the maximum payoff is to 'Defend B', ensuring at least a 0 payoff.

Step-by-step explanation:

To construct the game matrix for Blue and Red given the situation, let's denote Blue's strategy of defending installation A as 'Defend A' and defending installation B as 'Defend B'. Similarly, Red's strategies of attacking installation A or B will be denoted as 'Attack A' and 'Attack B', respectively.

The game matrix will have two rows representing Blue's strategies and two columns representing Red's strategies. The entries in the matrix will represent the outcomes for Blue in terms of the value of installations. If Red successfully attacks an installation, Blue loses its value.

Game Matrix

• If Blue defends A and Red attacks A, the outcome is 0 for Blue since A is defended successfully.
• If Blue defends A and Red attacks B, the outcome is -3 for Blue since B, valued at three units, is lost.
• If Blue defends B and Red attacks A, the outcome is -1 for Blue since A, valued at one unit, is lost.
• If Blue defends B and Red attacks B, the outcome is 0 for Blue since B is defended successfully.

The game matrix would look like this:

Attack AAttack BDefend A0-3Defend B-10

The measure of effectiveness in this game is the utility, or the net value of installations after any potential attacks. The strategy that guarantees Blue the maximum payoff, in this case, is to defend the installation of the highest value. Therefore, 'Defend B' guarantees Blue a payoff of at least 0 (since -1 is worse than 0), which is the best Blue can assure itself regardless of Red's actions.