The game known as the battle of the Bismarck Sea (named for that part of the southwestern Pacific Ocean separating the Bisma rck Arch ipelago from Papua-New Guinea) summarizes a well-known game actually played in a naval engagement between the United States and Japan in World War II. In 1943, a Japanese admiral was ordered to move a convoy of ships to New Guinea; he had to choose between a rainy northern route and a sunnier southern rou te, both of which required 3 days sailing time. The Americans knew that the convoy would sail and wanted to send bombers after it, but they did not know which route it would take. The Americans had to send re­ connaissance planes to scout for the convoy, but they had only enough re­ connaissance planes to explore one route at a time. Both the Japanese and the Americans had to make their decisions with no knowledge of the plans being made by the other side.

If the convoy was on the route explored by the Americans first, they could send bombers right away; if not, they lost a day of bombing. Poor weather on the northern route would also hamper bombing. If the Americans explored the northern route and found the Japanese right away, they could expect only 2 (of 3) good bombing days; if they explored the northern route and found that the Japanese had gone south, they could also expect 2 days of bombing. If the Americans chose to explore the southern route first, they could expect 3 full days of bombing if they found the Japanese right away but only 1 day of bom bing if they found that the Japanese had gone north.

(a) Illustrate this game in a game ta ble.

(b) Identify any dominant strategies in the game and solve for the Nash equilibrium.