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1 Game Playing

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2 Outline Perfect Play Resource Limits Alpha-Beta pruning Games of Chance

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3 Games vs Search problems Unpredictable opponent –use contingency plan Time limits –only approximate solution Approaches –algorithm for perfect play –finite horizon, approximation –pruning to reduce costs

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4 Types of Games

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5 Tic-Tac-Toe Example

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6 Structure of a Game Initial State –starting configuration and whose move Operators –legal moves Terminal Test –Checks if game is over Utility Function –Evaluates who has won and by how much

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7 Minimax

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8 Minimax Algorithm function MINIMAX-DECISION(game) returns an operator for each op in OPERATORS[game] do VALUE[op] <- MINIMAX-VALUE(APPLY(op, game), game) end return the op with the highest VALUE[op] function MINIMAX-VALUE(state,game) returns a utility value if op TERMINAL-TEST[game](state) then return UTILITY[game](state) else if MAX is to move in state then return the highest MINIMAX-VALUE of SUCCESSORS(state) else return the lowest MINIMAX-VALUE of SUCCESSORS(state)

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9 Properties of MiniMax Complete ? –Yes (if tree finite) Optimal ? –Yes, against optimal opponent Time Complexity ? –O(b m ) Space Complexity ? –O(bm) (depth first exploration)

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10 Resource Limits Time complexity means given time limits –limited choice of solution Approach –cutoff test (depth limit) –evaluation function (estimate of desirability of position)

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11 Evaluation Functions Normally weighted linear sum of features –Eval(s) = w 1 f 1 (s) + w 2 f 2 (s) +.. + w n f n (s)

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12 Cutting off search MinimaxCutoff identical to MinimaxValue except –TERMINAL? Replaced by CUTOFF? –UTILITY replaced by EVAL In practice if b m = 10 6, b = 35 => m = 4 4-ply - human novice 8-ply - typical PC or human master 12-ply - Deep Blue, grand master

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13 Alpha-Beta pruning Example MAX MIN A1A1 A2A2 A3A3 3 3 A 11 A 13 A 12 3128 X <=2 A 21 A 23 A 22 2XX A 31 A 33 A 32 14 <=14 XX 5X <=5 2 X<=2 5

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14 Properties of Alpha-Beta pruning doesn’t effect final result Ordering improves efficiency of pruning “perfect ordering”, time complexity O(b m/2 ) –doubles depth of search In practice, time complexity O(b 3m/4 )

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15 Alpha-Beta Algorithm function MAX-VALUE(state, game, alpha, beta) returns the minimax value of a state inputs: state, current state in game game, game description alpha, the best score for MAX along the path to state beta, the best score for MIN along the path to state if CUT-OFF(state) then return EVAL(state) for each s in SUCCESSORS(state) do alpha <- MAX(alpha, MIN-VALUE(s, game, alpha, beta)) if alpha >= beta then return beta end return alpha

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16 Alpha-Beta cont. function MIN-VALUE(state, game, alpha, beta) returns the minimax value of a state if CUT-OFF(state) then return EVAL(state) for each s in SUCCESSORS(state) do beta <- MIN(beta, MAX-VALUE(s, game, alpha, beta)) if alpha >= beta then return alpha end return beta

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17 Deterministic Games Checkers:- –Chinook, 1994 Chess –Deep Blue, 1997 Othello –computers too good Go –computers too bad

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18 Non-deterministic Games Chance adds difficulty –dice roll, deal of cards, flip of coin ExpectiMax, like MiniMax –with additional chance nodes

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19 Backgammon

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20 ExpectiMiniMax expectimax(C) = sum i (p(d i ) max s (utility(s))) –C: chance node, d i : dice roll –p(d i ): probability of roll occurring –max s (utility(s)): max utility possible after dice roll Time Complexity –O(b m n m ) makes problems even harder to solve

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