ITCS 3153 Artificial IntelligenceGamesMinimaxWhat data do we need to play?Minimax StrategySlide 6Slide 7Minimax AlgorithmFeasibility of minimax?Pruning minimax treeAlpha-beta pruningSlide 12Realtime decisionsEvaluation FunctionTruncating minimax searchBenefits of truncationGames with chanceExpectiminimaxPruningHistory of GamesDeep BlueCheckersChinook, Jonathan Schaeffer, 1990OthelloBackgammonDiscussionBuilding alpha-beta treeMetareasoningGoal-directed reasoning / planningSlide 30ITCS 3153Artificial IntelligenceLecture 9Lecture 9Adversarial SearchAdversarial SearchChapter 6Chapter 6Lecture 9Lecture 9Adversarial SearchAdversarial SearchChapter 6Chapter 6Games““Shall we play a game?”Shall we play a game?”Let’s play tic-tac-toeLet’s play tic-tac-toe““Shall we play a game?”Shall we play a game?”Let’s play tic-tac-toeLet’s play tic-tac-toeMinimaxWhat data do we need to play?Initial StateInitial State•How does the game start?How does the game start?Successor FunctionSuccessor Function•A list of legal (move, state) pairs for each stateA list of legal (move, state) pairs for each stateTerminal TestTerminal Test•Determines when game is overDetermines when game is overUtility FunctionUtility Function•Provides numeric value for all terminal statesProvides numeric value for all terminal statesInitial StateInitial State•How does the game start?How does the game start?Successor FunctionSuccessor Function•A list of legal (move, state) pairs for each stateA list of legal (move, state) pairs for each stateTerminal TestTerminal Test•Determines when game is overDetermines when game is overUtility FunctionUtility Function•Provides numeric value for all terminal statesProvides numeric value for all terminal statesMinimax StrategyOptimal StragtegyOptimal Stragtegy•Leads to outcomes at least as good as any other strategy Leads to outcomes at least as good as any other strategy when playing an infallible opponentwhen playing an infallible opponent•Pick the option that most (max) minimizes the damage your Pick the option that most (max) minimizes the damage your opponent can do opponent can do –maximize the worst-case outcomemaximize the worst-case outcome–because your skillful opponent will certainly find the most because your skillful opponent will certainly find the most damaging movedamaging moveOptimal StragtegyOptimal Stragtegy•Leads to outcomes at least as good as any other strategy Leads to outcomes at least as good as any other strategy when playing an infallible opponentwhen playing an infallible opponent•Pick the option that most (max) minimizes the damage your Pick the option that most (max) minimizes the damage your opponent can do opponent can do –maximize the worst-case outcomemaximize the worst-case outcome–because your skillful opponent will certainly find the most because your skillful opponent will certainly find the most damaging movedamaging moveMinimaxAlgorithmAlgorithm•MinimaxValue(n) =MinimaxValue(n) =Utility (n)Utility (n)if n is a terminal stateif n is a terminal statemax MinimaxValue(s) max MinimaxValue(s) of all successors, sof all successors, s if n is a MAX nodeif n is a MAX nodemin MinimaxValue(s) min MinimaxValue(s) of all successors, sof all successors, s if n is a MIN nodeif n is a MIN nodeAlgorithmAlgorithm•MinimaxValue(n) =MinimaxValue(n) =Utility (n)Utility (n)if n is a terminal stateif n is a terminal statemax MinimaxValue(s) max MinimaxValue(s) of all successors, sof all successors, s if n is a MAX nodeif n is a MAX nodemin MinimaxValue(s) min MinimaxValue(s) of all successors, sof all successors, s if n is a MIN nodeif n is a MIN nodeMinimaxMinimax AlgorithmWe wish to identify minimax decision at the rootWe wish to identify minimax decision at the root•Recursive evaluation of all nodes in game treeRecursive evaluation of all nodes in game tree•Time complexity = O (bTime complexity = O (bmm))We wish to identify minimax decision at the rootWe wish to identify minimax decision at the root•Recursive evaluation of all nodes in game treeRecursive evaluation of all nodes in game tree•Time complexity = O (bTime complexity = O (bmm))Feasibility of minimax?How about a nice game of chess?How about a nice game of chess?•Avg branching = 35 and avg # moves = 50 for each playerAvg branching = 35 and avg # moves = 50 for each player–O(35O(35100100) time complexity = 10) time complexity = 10154154 nodes nodes10104040 distinct nodes distinct nodesMinimax is impractical if directly applied to chessMinimax is impractical if directly applied to chessHow about a nice game of chess?How about a nice game of chess?•Avg branching = 35 and avg # moves = 50 for each playerAvg branching = 35 and avg # moves = 50 for each player–O(35O(35100100) time complexity = 10) time complexity = 10154154 nodes nodes10104040 distinct nodes distinct nodesMinimax is impractical if directly applied to chessMinimax is impractical if directly applied to chessPruning minimax treeAre there times when you know you need not Are there times when you know you need not explore a particular move?explore a particular move?•When the move is poor?When the move is poor?•Poor compared to what?Poor compared to what?•Poor compared to what you have explored so farPoor compared to what you have explored so farAre there times when you know you need not Are there times when you know you need not explore a particular move?explore a particular move?•When the move is poor?When the move is poor?•Poor compared to what?Poor compared to what?•Poor compared to what you have explored so farPoor compared to what you have explored so farAlpha-beta pruningAlpha-beta pruning–the value of the best (highest) choice so far in search of the value of the best (highest) choice so far in search of MAXMAX–the value of the best (lowest) choice so far in search of the value of the best (lowest) choice so far in search of MINMIN•Order of considering successors matters (look at step f in Order of considering successors matters (look at step f in previous slide)previous slide)–If possible, consider best successors firstIf possible, consider best successors first–the value of the best (highest) choice so far in search of the value of the best (highest) choice so far in search of MAXMAX–the value of the best (lowest) choice so far in search of the value of the best (lowest) choice so far in search of