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Adversarial Search Chapter 6 Section 1 – 4

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Types of Games

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Deterministic games in practice Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply. Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions. Othello: human champions refuse to compete against computers, who are too good. Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.

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Outline Optimal decisions α-β pruning Imperfect, real-time decisions

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Games vs. search problems "Unpredictable" opponent specifying a move for every possible opponent reply Time limits unlikely to find goal, must approximate

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Game tree (2-player, deterministic, turns)

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Game setup Two players: MAX and MIN MAX moves first and they take turns until the game is over. Winner gets award, looser gets penalty. Games as search: –Initial state: e.g. board configuration of chess –Successor function: list of (move,state) pairs specifying legal moves. –Terminal test: Is the game finished? –Utility function: Gives numerical value of terminal states. E.g. win (+1), loose (-1) and draw (0) in tic-tac- toe (next) MAX uses search tree to determine next move.

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Optimal strategies Find the contingent strategy for MAX assuming an infallible MIN opponent. Assumption: Both players play optimally !! Given a game tree, the optimal strategy can be determined by using the minimax value of each node: MINIMAX-VALUE(n)= UTILITY(n)If n is a terminal max s successors(n) MINIMAX-VALUE(s) If n is a max node min s successors(n) MINIMAX-VALUE(s) If n is a max node

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Minimax Perfect play for deterministic games Idea: choose move to position with highest minimax value = best achievable payoff against best play E.g., 2-ply game:

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Minimax algorithm

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Properties of minimax Complete? Yes (if tree is finite) Optimal? Yes (against an optimal opponent) Time complexity? O(b m ) Space complexity? O(bm) (depth-first exploration) For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible

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α-β pruning example

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Properties of α-β Pruning does not affect final result Good move ordering improves effectiveness of pruning With "perfect ordering," look ahead twice as fast as minimax in the same amount of time.

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Why is it called α-β? α is the value of the best (i.e., highest- value) choice found so far at any choice point along the path for max If v is worse than α, max will avoid it prune that branch Define β similarly for min

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The α-β algorithm

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Resource Limits Suppose we have 100 secs, explore 10 4 nodes/sec 10 6 nodes per move Standard approach: cutoff test: e.g., depth limit evaluation function = estimated desirability of position

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Evaluation functions For chess, typically linear weighted sum of features Eval(s) = w 1 f 1 (s) + w 2 f 2 (s) + … + w n f n (s) e.g., w 1 = 9 with f 1 (s) = (number of white queens) – (number of black queens), etc.

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Cutting off search MinimaxCutoff is identical to MinimaxValue except 1.Terminal? is replaced by Cutoff? 2.Utility is replaced by Eval Does it work in practice? b m = 10 6, b=35 m=4 4-ply lookahead is a hopeless chess player! –4-ply ≈ human novice –8-ply ≈ typical PC, human master –12-ply ≈ Deep Blue, Kasparov

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Summary Games are fun to work on! They illustrate several important points about AI perfection is unattainable must approximate good idea to think about what to think about

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