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Unformatted text preview: CHAPTER 15: GAME THEORY Example 1: In early 1943, the Japanese controlled the northern half of the island of New Guinea, while the allies controlled the southern half. Intelligence reports indicated that the Japanese were assembling a convoy to reinforce their troops on the island. The convoy would take one of two routes: (1) north of New Britain, where rain and bad visibility were predicted, or (2) south, where the weather was expected to be fair. It was estimated that the trip would take 3 days on either route. Upon receiving these intelligence estimates, the Supreme Allied Commander, General Douglas McArthur, ordered General George C. Kenney, commander of the Allied Air Forces in the Southwest Pacific Area, to inflict maximum possible damage on the Japanese convoy. Kenney had the choice of sending the bulk of his reconnaissance aircraft on either the southern or northern route. His objective was to maximize the expected number of days the convoy could be bombed, so he wanted to have his aircraft find the convoy as quickly as possible. Consequently, Kenney had two choices: (1) use most of his aircraft to search the northern route, or (2) focus his search in the south. The payoff would then be measured by the expected number of days Kenney would have at his disposal to bomb the convoy. The overall situation facing the two commanders in what came to be known as the Battle of Bismarck Sea can be represented as follows, where the entries indicate the expected number of days of bombing by Kenney following detection of the Japanese convoy. Japanese Sail north Sail South Allies Search North 2 days 2 days Search South 1 day 3 days What is the best strategy for each commander? Solution: The rectangular array of numbers 3 1 2 2 is called a payoff matrix . The rows are labeled with the choices available to General Kenney, while the columns represent the alternatives at the disposal of the Japanese commander. By convention, the entries in the matrix are taken to be the payoff to the row player , in this case the Allies. Since the interests of the Allies and the Japanese are directly opposed, the payoffs to the column player , the Japanese, are simply the negatives of these numbers. To analyze the game, let’s imagine General Kenney’s thought process: “If I search south, I may only get to bomb one day whereas if I search north I am guaranteed 2 days of bombing no matter what my opponent does.” The rational thing to do is to look at the worst-case scenario for each option and then pick the does....
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