Minimax Trees: Utility Evaluation, Tree Evaluation, PruningTwo-Person Perfect Information Deterministic GameMinimaxMinimax treeMinimax Tree EvaluationSlide 6Slide 7Slide 8Slide 9Slide 10Minimax EvaluationPruning the Minimax Treea Cutsa Cut exampleSlide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23b cutsb Cut examplea-b PruningSlide 27Pruning at MAX nodePruning at MIN nodeIdea of a-b PruningUtility Evaluation FunctionUtility EvaluationSlide 33Evaluation functionsSlide 35Evaluation Functions for OrderingMinimax Trees:Utility Evaluation, Tree Evaluation, PruningCPSC 315 – Programming StudioSpring 2009Project 2, Lecture 2Adapted from slides of Yoonsuck ChoeTwo-Person Perfect Information Deterministic GameTwo players take turns making movesBoard state fully known, deterministic evaluation of movesOne player wins by defeating the other (or else there is a tie)Want a strategy to win, assuming the other person plays as well as possibleMinimaxCreate a utility functionEvaluation of board/game state to determine how strong the position of player 1 is.Player 1 wants to maximize the utility functionPlayer 2 wants to minimize the utility functionMinimax treeGenerate a new level for each moveLevels alternate between “max” (player 1 moves) and “min” (player 2 moves)Minimax treeMaxMinMaxMinMinimax Tree EvaluationAssign utility values to leavesSometimes called “board evaluation function”If leaf is a “final” state, assign the maximum or minimum possible utility value (depending on who would win)If leaf is not a “final” state, must use some other heuristic, specific to the game, to evaluate how good/bad the state is at that pointMinimax treeMaxMinMaxMin100-24-8-14-73-100-5-70-12-370412-3212823Minimax Tree EvaluationAt each min node, assign the minimum of all utility values at childrenPlayer 2 chooses the best available moveAt each max node, assign the maximum of all utility values at childrenPlayer 1 chooses best available movePush values from leaves to top of treeMinimax treeMaxMinMaxMin100-24-8-14-73-100-5-70-12-370412-321282328 -312 70 -3 -73 -14-8Minimax treeMaxMinMaxMin100-24-8-14-73-100-5-70-12-470412-321282321 -312 70 -4 -73 -14-8-3-4-73Minimax treeMaxMinMaxMin100-24-8-14-73-100-5-70-12-470412-321282321 -312 70 -4 -73 -14-8-3-4-73-3Minimax EvaluationGiven average branching factor b, and depth m:A complete evaluation takes time bmA complete evaluation takes space bmUsually, we cannot evaluate the complete state, since it’s too bigInstead, we limit the depth based on various factors, including time available.Pruning the Minimax TreeSince we have limited time available, we want to avoid unnecessary computation in the minimax tree.Pruning: ways of determining that certain branches will not be usefulGiven the assumption that players will always choose their best option, prune subtrees that cannot be their best option. CutsIf the current max value is greater than the successor’s min value, don’t explore that min subtree any more Cut example10021 -3 12 70 -4 -73 -14-3 -4-3-73MaxMaxMin Cut exampleDepth first search along path 110021 -3 12 -70 -4 -73 -14MaxMaxMin Cut example21 is minimum so far (second level)Can’t evaluate yet at top level10021 -3 12 -70 -4 -73 -14MaxMaxMin21 Cut example-3 is minimum so far (second level)-3 is maximum so far (top level)10021 -3 12 -70 -4 -73 -14MaxMaxMin-3-3 Cut example12 is minimum so far (second level)-3 is still maximum (can’t use second node yet)10021 -3 12 -70 -4 -73 -14MaxMaxMin-3-312 Cut example-70 is now minimum so far (second level)-3 is still maximum (can’t use second node yet)10021 -3 12 -70 -4 -73 -14MaxMaxMin-3-3-70 Cut exampleSince second level node will never be > -70, it will never be chosen by the previous levelWe can stop exploring that node10021 -3 12 -70 -4 -73 -14MaxMaxMin-3-3-70 Cut exampleEvaluation at second level is again -7310021 -3 12 -70 -4 -73 -14MaxMaxMin-3-3-70-73 Cut exampleAgain, can apply  cut since the second level node will never be > -73, and thus will never be chosen by the previous level10021 -3 12 -70 -4 -73 -14MaxMaxMin-3-3-70-73 Cut exampleAs a result, we evaluated the Max node without evaluating several of the possible paths10021 -3 12 -70 -4 -73 -14MaxMaxMin-3-3-70-73 cutsSimilar idea to  cuts, but the other way aroundIf the current minimum is less than the successor’s max value, don’t look down that max tree any more Cut exampleSome subtrees at second level already have values > min from previous, so we can stop evaluating them.1021 -3 12 70 -4 73 -14MinMinMax212170 73- PruningPruning by these cuts does not affect final resultMay allow you to go much deeper in tree“Good” ordering of moves can make this pruning much more efficientEvaluating “best” branch first yields better likelihood of pruning later branchesPerfect ordering reduces time to bm/2i.e. doubles the depth you can search to!- PruningCan store information along an entire path, not just at most recent levels!Keep along the path:: best MAX value found on this path (initialize to most negative utility value): best MIN value found on this path (initialize to most positive utility value)Pruning at MAX node is possibly updated by the MAX of successors evaluated so farIf the value that would be returned is ever > , then stop work on this branchIf all children are evaluated without pruning, return the MAX of their valuesPruning at MIN node is possibly updated by the MIN of successors evaluated so farIf the value that would be returned is ever < , then stop work on this branchIf all children are evaluated without pruning, return the MIN of their valuesIdea of - PruningWe know  on this path is 21So, when we get max=70, we know this will never be used, so we can stop here10021 -312 70 -4212170Utility Evaluation FunctionVery game-specificTake into account knowledge about game“Stupid” utility1 if player 1 wins-1 if player 0 wins0 if tie (or unknown)Only works if we can evaluate complete treeBut, should form a basis for other evaluationsUtility EvaluationNeed to assign a numerical value to the stateCould assign a more complex utility value, but then the min/max determination becomes trickierTypically