lecture4b - CMSC 498T Game Theory 4b Game-Tree Search Dana...

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Nau: Game Theory 1 Updated 9/27/11 CMSC 498T, Game Theory 4b. Game-Tree Search Dana Nau University of Maryland
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Nau: Game Theory 2 Updated 9/27/11 Finite perfect-information zero-sum games Finite : Ø f nitely many agents, actions, states, histories Perfect information : Ø Every agent knows all of the players’ utility functions all of the players’ actions and what they do the history and current state Ø No simultaneous actions – agents move one-at-a-time Constant sum (or zero-sum ): Ø Constant k such that regardless of how the game ends, Σ i =1,…, n u i = k Ø For every such game, there’s an equivalent game in which k = 0
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Nau: Game Theory 3 Updated 9/27/11 Examples Deterministic : Ø chess, checkers Ø go, gomoku Ø reversi (othello) Ø tic-tac-toe, qubic, connect-four Ø mancala (awari, kalah) Ø 9 men’s morris (merelles, morels, mill) Stochastic : Ø backgammon, monopoly, yahtzee, parcheesi, roulette, craps For now, we’ll consider just the deterministic games
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Nau: Game Theory 4 Updated 9/27/11 Outline A brief history of work on this topic Restatement of the Minimax Theorem Game trees The minimax algorithm α - β pruning Resource limits, approximate evaluation Most of this isn’t in the game-theory book For further information, look at one of the following Ø The “private materials” page Ø Russell & Norvig’s Arti f cial Intelligence: A Modern Approach There are 3 editions of this book In the 2 nd edition, it’s Chapter 6
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Nau: Game Theory 5 Updated 9/27/11 Brief History 1846 (Babbage) designed machine to play tic-tac-toe 1928 (von Neumann) minimax theorem 1944 (von Neumann & Morgenstern) backward induction 1950 (Shannon) minimax algorithm ( f nite-horizon search) 1951 (Turing) program (on paper) for playing chess 1952–7 (Samuel) checkers program capable of beating its creator 1956 (McCarthy) pruning to allow deeper minimax search 1957 (Bernstein) f rst complete chess program, on IBM 704 vacuum-tube computer could examine about 350 positions/minute 1967 (Greenblatt) f rst program to compete in human chess tournaments 3 wins, 3 draws, 12 losses 1992 (Schaeffer) Chinook won the 1992 US Open checkers tournament 1994 (Schaeffer) Chinook became world checkers champion; Tinsley (human champion) withdrew for health reasons 1997 (Hsu et al) Deep Blue won 6-game match vs world chess champion Kasparov 2007 (Schaeffer et al) Checkers solved: with perfect play, it’s a draw 10 14 calculations over 18 years
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Nau: Game Theory 6 Updated 9/27/11 Restatement of the Minimax Theorem Suppose agents 1 and 2 use strategies s and t on a 2-person game G Ø Let u ( s,t ) = u 1 ( s,t ) = – u 2 ( s,t ) Ø Call the agents Max and Min (they want to maximize and minimize u ) Minimax Theorem : If G is a two-person f nite zero-sum game, then there are strategies s * and t * , and a number v called G ’s minimax value , such that Ø If Min uses t * , Max’s expected utility is v , i.e., max s u ( s , t * ) = v Ø If Max uses s * , Max’s expected utility is v , i.e., min t u ( s * , t ) = v Corollary 1 : u ( s * , t * ) = v ( s * , t * ) is a Nash equilibrium s * (or t *
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This note was uploaded on 01/13/2012 for the course CMSC 498T taught by Professor Staff during the Fall '11 term at Maryland.

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lecture4b - CMSC 498T Game Theory 4b Game-Tree Search Dana...

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