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Minimax Trees: Utility Evaluation, Tree Evaluation, Pruning

Minimax Trees: Utility Evaluation, Tree Evaluation, Pruning. CSCE 315 – Programming Studio Spring 2019 Project 2, Lecture 2 Robert Lightfoot. Adapted from slides of Yoonsuck Choe , John Keyser. Two-Person Perfect Information Deterministic Game. Two players take turns making moves

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Minimax Trees: Utility Evaluation, Tree Evaluation, Pruning

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  1. Minimax Trees:Utility Evaluation, Tree Evaluation, Pruning CSCE 315 – Programming Studio Spring 2019 Project 2, Lecture 2 Robert Lightfoot Adapted from slides of YoonsuckChoe, John Keyser

  2. Two-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 possible

  3. General Strategy • Player 1 wants to maximize his chance of win • Player 2 wants to maximize his (minimize Player 1’s) … (9 possibilities) … … (8 possibilities) …

  4. Minimax Tree • Use Minimax tree to emulate human’s forward thinking process • In Chess, players think a couple of steps (e.g., 3-4 steps) ahead in contingency of opponent moves • Similar idea applies to other such games

  5. Minimax tree Max • Minimax tree • Generate a new level for each move • Levels alternate between “max” (player 1 moves) and “min” (player 2 moves) Min Max Min

  6. Minimax Tree Evaluation • Assign utility values to leaves • Sometimes called “board evaluation function” • If a state is a leaf state where Player 1 wins (looses), assign the maximum(minimum) possible utility value • Otherwise, keep expanding (make the next move) Board State 1 Board State 2 Board State 3 Utility: -100 Utility: 0 Utility: +100

  7. Minimax tree Max Min Max 100 Min 100 -100 -100

  8. Minimax Tree Algorithm functionMAX-VALUE(state) ifTERMINAL-TEST (state) returnUTILITY (state)foreachsinSUCCESSORS (state)v = MAX (v, MIN-VALUE(s))returnv functionMIN-VALUE(state) ifTERMINAL-TEST (state) returnUTILITY (state)foreachsinSUCCESSORS (state)v = MIN (v, MAX-VALUE(s))returnv

  9. Minimax 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 tree

  10. Problem?? • What if the node we are trying to judge is NOT a LEAF node • For Chess, a LEAF node could be many levels down • Could take years(!!!) to compute • It would be best if there was a Oracle saying: “this is promising state, go for this move” • The answer to this problem: Utility Function

  11. Utility Function • Create a utility function • Evaluation of board/game state to determine how strong the position of player 1 is. • Assume that “x” is Player 1 in the following Tic-Tac-Toe • Example: • You have all of Queen, Rooks, Knights, Bishops and some Pawns, while your opponent has only 1 Knight, you have a high chance of winning • Utility function will assign a numeric value to this board state (say 90/100)

  12. Minimax tree Max Min Max 28 -3 -8 12 70 -4 100 -73 -14 Min 23 28 21 -3 12 4 70 -4 -12 -70 -5 -100 -73 -14 -8 -24

  13. Minimax tree Max Min -4 -3 -73 Max 28 -3 -8 12 70 -4 100 -73 -14 Min 23 28 21 -3 12 4 70 -4 -12 -70 -5 -100 -73 -14 -8 -24

  14. Minimax tree Max -3 Min -4 -3 -73 Max 28 -3 -8 12 70 -4 100 -73 -14 Min 23 28 21 -3 12 4 70 -4 -12 -70 -5 -100 -73 -14 -8 -24

  15. Minimax 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.

  16. Utility 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 evaluations

  17. Utility 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 assign numerical values to lots of individual factors • a = # player 1’s pieces - # player 2’s pieces • b = 1 if player 1 has queen and player 2 does not, -1 if the opposite, or 0 if the same • c = 2 if player 1 has 2-rook advantage, 1 if a 1-rook advantage, etc.

  18. Utility Evaluation • The individual factors are combined by some function • Usually a linear weighted combination is used • u = aa + bb + cc • Different ways to combine are also possible • Notice: quality of utility function is based on: • What features are evaluated • How those features are scored • How the scores are weighted/combined • Absolute utility value doesn’t matter – relative value does.

  19. Evaluation functions • If you had a perfect utility evaluation function, what would it mean about the minimax tree?

  20. Evaluation functions • If you had a perfect utility evaluation function, what would it mean about the minimax tree? You would never have to evaluate more than one level deep! • Typically, you can’t create such perfect utility evaluations, though.

  21. 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

  22. a Cuts • If the current max value is greater than the successor’s min value, don’t explore that min subtree any more

  23. a Cut example Max -3 Min -3 -4 -73 Max 21 -3 12 70 -4 100 -73 -14

  24. a Cut example Max • Depth first search along path 1 Min Max 21 -3 12 -70 -4 100 -73 -14

  25. a Cut example Max • 21 is minimum so far (second level) • Can’t evaluate yet at top level Min 21 Max 21 -3 12 -70 -4 100 -73 -14

  26. a Cut example Max -3 • -3 is minimum so far (second level) • -3 is maximum so far (top level) Min -3 Max 21 -3 12 -70 -4 100 -73 -14

  27. a Cut example Max -3 • 12 is minimum so far (second level) • -3 is still maximum (can’t use second node yet) Min 12 -3 Max 21 -3 12 -70 -4 100 -73 -14

  28. a Cut example Max -3 • -70 is now minimum so far (second level) • -3 is still maximum (can’t use second node yet) Min -70 -3 Max 21 -3 12 -70 -4 100 -73 -14

  29. a Cut example Max -3 • Since second level node will never be > -70, it will never be chosen by the previous level • We can stop exploring that node Min -70 -3 Max 21 -3 12 -70 -4 100 -73 -14

  30. a Cut example Max -3 • Evaluation at second level is -73 Min -70 -3 -73 Max 21 -3 12 -70 -4 100 -73 -14

  31. a Cut example Max -3 • Again, can apply a cut since the second level node will never be > -73, and thus will never be chosen by the previous level Min -70 -3 -73 Max 21 -3 12 -70 -4 100 -73 -14

  32. a Cut example Max -3 • As a result, we evaluated the Max node without evaluating several of the possible paths Min -70 -3 -73 Max 21 -3 12 -70 -4 100 -73 -14

  33. b cuts • Similar idea to a 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

  34. b Cut example Min 21 • Some subtrees at second level already have values > min from previous, so we can stop evaluating them. Max 21 70 73 Min 21 -3 12 70 -4 100 73 -14

  35. a-b 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!

  36. a-b Pruning • Can store information along an entire path, not just at most recent levels! • Keep along the path: • a: best MAX value found on this path (initialize to most negative utility value) • b: best MIN value found on this path (initialize to most positive utility value)

  37. Pruning at MAX node • a is possibly updated by the MAX of successors evaluated so far • If the value that would be returned is ever > b, then stop work on this branch • If all children are evaluated without pruning, return the MAX of their values

  38. Pruning at MIN node • b is possibly updated by the MIN of successors evaluated so far • If the value that would be returned is ever < a, then stop work on this branch • If all children are evaluated without pruning, return the MIN of their values

  39. Idea of a-b Pruning 21 • We know b on this path is 21 • So, when we get max=70, we know this will never be used, so we can stop here 21 21 -3 70 12 70 -4 100

  40. Idea of a-b Pruning • Pruning gives the exact same result that you would have gotten without pruning • It just allows you to go deeper • Pruning and searching the minimax tree are independent of the particular game being studied • The game influences: • The utility function • The branching ratio/options at any one level

  41. Evaluation Functions for Ordering • As mentioned earlier, order of branch evaluation can make a big difference in how well you can prune • A good evaluation function might help you order your available moves • Perform one move only • Evaluate board at that level • Recursively evaluate branches in order from best first move to worst first move (or vice-versa if at a MIN node)

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