Lectue_Week1

# Lectue_Week1 - Introduction Game Theory can be called...

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Introduction Game Theory can be called “Interactive Decision Theory”. It studies the competition or cooperation between rational and intelligent decision makers. It has its origin in the entertaining games that people play, such as chess, tic-tac-toe, bridge etc. Zermelo wrote a paper on chess in the beginning of the 20 th century. John von Neumann proved the Minimax Theorem for 2- person 0-sum games in the 1920’s.

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John von Neumann and Oskar Morgenstern published the book “Theory of Games Theory and Economic Behavior” in the 1940’s setting the foundation of Game Theory. The Princeton School made many important contributions to Game Theory in the 1950’s. Game Theory is an essential tool in economics but it also finds many important applications in a wide range of subjects such as political science, philosophy, law, biology, engineering. These interactions further enrich Game Theory.
There are basically three mathematical models to describe a game: Extensive Form, Strategic Form, Coalition Form. They differ in the amount of detail on the play of the game built into the model. We will study various mathematical models of games and create a theory or structure of the phenomena that arise. In some cases, we will be able to suggest what courses of actions should be taken by the players.

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Combinatorial Games Combinatorial Games References: www.math.ucla.edu/~tom/math167.html http://sps.nus.edu.sg/~limchuwe/
1. A Simple Take-Away Game. (1) There are two players. We label them I and II. (2) There is a pile of 21 chips in the center of a table. (3) A move consists of removing one, two, or three chips from the pile. At least one chip must be removed, but no more than three may be removed. (4) Players alternate moves with Player I starting. (5) The player that removes the last chip wins. (The last player to move wins. If you can’t move, you lose. )

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How can we analyze this game? Can one of the players force a win in this game? Which player would you rather be, the player who starts or the player who goes second? What is a good strategy?
We analyze this game from the end back to the beginning. This method is sometimes called backward induction . If there are just one, two, or three chips left, the player who moves n ext wins simply by taking all the chips. Suppose there are four chips left. Then the player who moves next must leave either one, two or three chips in the pile and his opponent will be able to win. So four chips left is a loss for the next player to move and a win for the p revious player, i.e. the one who just moved.

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With 5, 6, or 7 chips left, the player who moves n ext can win by moving to the position with four chips left. With 8 chips left, the next player to move must leave 5, 6, or 7 chips, and so the p revious player can win. We see that positions with 0
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## This note was uploaded on 03/13/2012 for the course MATH 4321 taught by Professor Cheng during the Spring '12 term at HKUST.

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Lectue_Week1 - Introduction Game Theory can be called...

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