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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern