hw8 - Math 167 Homework 8 December 9 2008 Game Theory...

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Math 167 Homework 8 December 9, 2008 Game Theory Thomas Ferguson Section II.3.7 Problem 15. Battleship. The game of Battleship, sometimes called Salvo, is played on two square boards, usually 10 by 10 . Each player hides a fleet of ships on his own board and tries to sink the opponent’s ships before the opponent sinks his. (For one set of rules, see http://www.kielack.de/games/destroya.htm, and while you are there, have a game.) For simplicity, consider a 3 by 3 board and suppose that Player I hides a destroyer (length 2 squares) horizontally or vertically on this board. Then Player II shoots by calling out squares on the board, one at a time. After each shot, Player I says whether the shot was a hit or a miss. Player II continues until both squares of the destroyer have been hit. The payoff to Player I is the number of shots that Player II has made. Let us label the squares from 1 to 9 as follows 1 2 3 4 5 6 7 8 9 The problem is invariant under rotations and reflections of the board. In fact, of the 12 possible positions for the destroyer, there are only two distinct invariant choices available to Player I: the strategy, [1 , 2] * , that chooses one of [1 , 2] , [2 , 3] , [3 , 6] , [6 , 9] , [8 , 9] , [7 , 8] , [4 , 7] and [1 , 4] , at random with probability 1 / 8 each, and the strategy, [2 , 5] * , that chooses one of [2 , 5] , [5 , 6] , [5 , 8] and [4 , 5] , at random with probability 1 / 4 each. This means that invariance reduces the game to a 2 by n game where n is the number of invariant strategies of Player II. Domination may reduce it somewhat further. Solve the game. Consider the mis` ere version of the take-away game of Section 1.1, where the last player to move loses. The object is to force your opponent to take the last chip. Analyze this game. What are the target positions (P-positions)? Solution Ferguson II.5.9 Problem 1. The Silver Dollar. Player II chooses one of two rooms in which to hide a silver dollar. Then, Player I, not knowing which room contains the dollar, selects one of the rooms to search. However, the search is not always successful. In fact, if the dollar is in room #1 and I searches there, then (by a chance move) he has only probability 1 / 2 of finding it, and if the dollar is in room #2 and I 1
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searches there, then he has only probability 1 / 3 of finding it. Of course, if he searches the wrong room, he certainly won’t find it. If he does find the coin, he keeps it; otherwise the dollar is returned to Player II. Draw the game tree. Solution 1 0 1 0 2 1 0 1 1 0 2 2 Problem 2. Two Guesses for the Silver Dollar. Draw the game tree for problem 1, if when I is unsuccessful in his first attempt to find the dollar, he is given a second chance to choose a room and search for it with the same probabilities of success, independent of his previous search. (Player II does not get to hide the dollar again.) Solution 1 1 0 1 0 2 1 1 0 1 0 2 2 1 0 1 1 0 2 1 1 0 1 1 0 2 2 2 Problem 10 Find the equivalent strategic form and solve the game of (a) Exercise 1.
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