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Artificial Intelligence for Games Game playing Patrick Olivier

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Games & search “unpredictable" opponent –specifying a move for every possible reply time limits: unlikely to find goal, approximate

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Brief history of game playing… Computer considers possible lines of play (Babbage, 1846) Algorithm for perfect play (Zermelo, 1912; Von Neumann, 1944) Finite horizon, approximate evaluation (Zuse, 1945; Wiener, 1948; Shannon, 1950) First chess program (Turing, 1951) Machine learning to improve evaluation accuracy (Samuel, ) Pruning to allow deeper search (McCarthy, 1956)

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Game tree: 2-player/deterministic/turns

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Minimax Perfect play for deterministic, perfect-information games Choose position with highest minimax value –best achievable payoff against best play

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Minimax algorithm

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Example

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Solution

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Properties of minimax Complete –Yes, if tree is finite (chess has specific rules for this) 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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3 2 2 Optimisation using α - β pruning 3 3 2 5 2

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α - β pruning algorithm

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Example

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Solution <=3 8 0 <= >=

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Properties of α-β pruning pruning does not affect final result good move ordering improves effectiveness of pruning "perfect ordering," time complexity O(b m/2 ) example of reasoning about which computations are relevant: metareasoning

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Evaluation functions Heuristic for game state: –order the terminal states correctly –fast to compute –good estimate of “chance of winning” Examples: –chess: weighted linear combination of features –e.g. pawn=1, bishop=3, etc. –based on judgements of chess experts

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Applying evaluation functions perform minimax and use the evaluation function at the maximum depth –e.g. fixed depth limit, iterative deepening secondary search: –address nonquiescent states –address the horizon effect –select clearly superior moves singular extensions (b=1)

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Deterministic games in practice Checkers: –Chinook ended 40-year-reign champion (1994) –Perfect play for games of 8 pieces or less (database) Othello: –champions won’t play computers (which are too good) Go: –champions won’t play computers (as not good enough) –branching factor can be as high as > 300 Chess: –Deep Blue defeats Kasparov in 6-game match (1997)

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Deep blue 30 node IBM supercomputer (& 480 single-chip chess search engines). 3 level architecture mixing software and hardware searching million evolutions of board configurations per sec (reached 330 at one point). Searching with alpha-beta search with complex evaluation function In 3 minutes for each move can search full width 12 deep and some paths up to 40 deep. Optimised with knowledge of opening and end games.

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Games with an element of chance expectiminimax algorithm chance nodes: use weighted sum of probabilities

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Samuel’s checker-playing program Arthur Samuel (IBM) Learnt own evaluation function –tuned the weights of a weighted linear function (up to 16 terms) –used comparison with full search Remembered evaluation function values –extends the effective depth of the search

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