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Alpha-Beta Example

Understand Alpha-Beta pruning, Heuristic Evaluation Functions, and applied examples in Chess, Tic-Tac-Toe, and Checkers using the Alpha-Beta algorithm.

danielkelley
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Alpha-Beta Example

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  1. Alpha-Beta Example Do DF-search until first leaf Range of possible values [-∞,+∞] [-∞, +∞]

  2. Alpha-Beta Example (continued) [-∞,+∞] [-∞,3]

  3. Alpha-Beta Example (continued) [-∞,+∞] [-∞,3]

  4. Alpha-Beta Example (continued) [3,+∞] [3,3]

  5. Alpha-Beta Example (continued) [3,+∞] This node is worse for MAX [3,3] [-∞,2]

  6. Alpha-Beta Example (continued) , [3,14] [3,3] [-∞,2] [-∞,14]

  7. Alpha-Beta Example (continued) , [3,5] [3,3] [−∞,2] [-∞,5]

  8. Alpha-Beta Example (continued) [3,3] [2,2] [3,3] [−∞,2]

  9. Alpha-Beta Example (continued) [3,3] [2,2] [3,3] [-∞,2]

  10. Comments about Alpha-Beta Pruning • Pruning does not affect final results • Entire subtrees can be pruned • Alpha-beta pruning can look twice as far as minimax in the same amount of time

  11. Heuristic Evaluation Function (EVAL) • Idea: produce an estimate of the expected utility of the game from a given position. • Performance depends on quality of EVAL. • Must be able to differentiate between good and bad board states • Exact values not important

  12. Heuristic Evaluation Function (EVAL) • Must be consistent with the utility function • values for terminal nodes (or at least their order) must be the same • should reflect the actual chances of winning • Frequently weighted linear functions are used • E = w1 f1 + w2 f2 + … +wn fn • combination of features, weighted by their relevance • Example in chess • Weights: Pawn=1, knight=bishop=3, rook=5, queen=9

  13. Example Chess Score • Black has: • 5 pawns, 1 bishop, 2 rooks • Score = 1*(5)+3*(1)+5*(2) = 5+3+10 = 18 White has: • 5 pawns, 1 rook • Score = 1*(5)+5*(1) = 5 + 5 = 10 Overall scores for this board state: black = 18-10 = 8 white = 10-18 = -8

  14. Example: Tic-Tac-Toe • simple evaluation function E(s) = (rx + cx + dx) - (ro + co + do) where r,c,d are the numbers of row, column and diagonal lines still available; x and o are the pieces of the two players • 1-ply lookahead • start at the top of the tree • evaluate all 9 choices for player 1 • pick the maximum E-value • 2-ply lookahead • also looks at the opponents possible move • assuming that the opponents picks the minimum E-value

  15. Tic-Tac-Toe 1-Ply E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4 E(s11) 8 - 5 = 3 E(s12) 8 - 6 = 2 E(s13) 8 - 5 = 3 E(s14) 8 - 6 = 2 E(s15) 8 - 4 = 4 E(s16) 8 - 6 = 2 E(s17) 8 - 5 = 3 E(s18) 8 - 6 = 2 E(s19) 8 - 5 = 3 X X X X X X X X X

  16. Tic-Tac-Toe 2-Ply E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4 E(s1:1) 8 - 5 = 3 E(s1:2) 8 - 6 = 2 E(s1:3) 8 - 5 = 3 E(s1:4) 8 - 6 = 2 E(s1:5) 8 - 4 = 4 E(s1:6) 8 - 6 = 2 E(s1:7) 8 - 5 = 3 E(s1:8) 8 - 6 = 2 E(s1:9) 8 - 5 = 3 X X X X X X X X X E(s2:41) 5 - 4 = 1 E(s2:42) 6 - 4 = 2 E(s2:43) 5 - 4 = 1 E(s2:44) 6 - 4 = 2 E(s2:45) 6 - 4 = 2 E(s2:46) 5 - 4 = 1 E(s2:47) 6 - 4 = 2 E(s2:48) 5 - 4 = 1 O O O X X X O X X O X X X O O O E(s2:9) 5 - 6 = -1 E(s2:10) 5 -6 = -1 E(s2:11) 5 - 6 = -1 E(s2:12) 4 - 6 = -2 E(s2:13) 6 - 6 = 0 E(s2:14) 5 - 6 = -1 E(s2:15) 6 -6 = 0 E(s2:16) 5 - 6 = -1 O X X O X X X X X X O O O O O O E(s21) 6 - 5 = 1 E(s22) 5 - 5 = 0 E(s23) 6 - 5 = 1 E(s24) 4 - 5 = -1 E(s25) 6 - 5 = 1 E(s26) 5 - 5 = 0 E(s27) 6 - 5 = 1 E(s28) 5 - 5 = 0 X O X O X X X X X X O O O O O O

  17. Checkers Case Study • initial board configuration • Black single on 20 single on 21 king on 31 • Redsingle on 23 king on 22 • evaluation functionE(s) = (5 x1 + x2) - (5r1 + r2) where x1 = black king advantage, x2 = black single advantage, r1 = red king advantage, r2 = red single advantage 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

  18. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 -8 -8 0 1 29 30 31 32 -8 -8 -4 6 2 6 1 1 1 0 1 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers MiniMax Example 31 -> 27 20 -> 16 MAX 21 -> 17 31 -> 26 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 13 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 27 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  19. -4 -8 0 1 -8 -8 -4 1 6 1 0 2 1 1 6 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 • a 1 • b 6 MAX 31 -> 27 20 -> 16 17 18 19 20 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 27 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  20. -4 -8 0 1 -8 -8 -4 1 6 1 0 2 1 1 6 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 • a 1 • b 1 MAX 31 -> 27 20 -> 16 17 18 19 20 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 27 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  21. -4 -8 0 1 -8 -8 -4 1 0 1 6 2 1 1 6 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b 1 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 b- cutoff: no need to examine further branches 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  22. -8 -8 -4 1 1 0 2 6 6 1 1 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b 1 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 1 0 -4 -8 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  23. -8 -8 -4 1 1 0 2 6 6 1 1 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b 1 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 b- cutoff: no need to examine further branches 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 1 0 -4 -8 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  24. -8 -8 -4 1 1 0 2 6 6 1 1 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b 1 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 1 0 -4 -8 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  25. -8 -8 -4 1 1 0 2 6 6 1 1 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b 0 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 1 0 -4 -8 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 13 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  26. -8 -8 -4 1 1 0 2 6 6 1 1 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b -4 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 1 0 -4 -8 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  27. -8 -8 -4 1 1 0 2 6 6 1 1 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b -4 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 a- cutoff: no need to examine further branches 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 1 0 -4 -8 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  28. -8 -8 -4 1 1 0 6 1 1 2 6 1 1 1 1 2 6 1 1 1 1 1 1 1 1 1 1 6 0 0 0 -4 -4 -8 -8 -8 -8 Checkers Alpha-Beta Example 1 2 3 4 a 1 b -8 5 6 7 8 9 10 11 12 13 14 15 16 MAX 31 -> 27 20 -> 16 17 18 19 20 21 -> 17 31 -> 26 21 22 23 24 25 26 27 28 29 30 31 32 1 0 -4 -8 MIN 23 -> 32 23 -> 30 22 -> 31 22 -> 18 22 -> 26 22 -> 17 22 -> 18 22 -> 25 23 -> 26 23 -> 27 MAX 16 -> 11 16 -> 11 31 -> 22 31 -> 27 31 -> 24 20 -> 16 21 -> 17 31 -> 27 21 -> 14 31 -> 27 20 -> 16 31 -> 27 21 -> 17 16 -> 11 20 -> 16 21 -> 17 20 -> 16 31 -> 26

  29. Horizon Problem • Moves may have disastrous consequences in the future, but the consequences are not visible • Agent cannot see far enough into search space

  30. Games with Chance • In many games, there is a degree of unpredictability through random elements • throwing dice, card distribution, roulette wheel, … • This requires chance nodes in addition to the Max and Min nodes • branches indicate possible variations • each branch indicates the outcome and its likelihood (probability)

  31. Games with Chance chance nodes

  32. Decisions with Chance • The utility value of a position depends on the random element • the definite minimax value must be replaced by an expected value • Calculation of expected values • utility function for terminal nodes • for all other nodes • calculate the utility for each chance event • weigh by the chance that the event occurs • add up the individual utilities

  33. More interesting (but still trivial) game • Deal four cards face up • Player 1 chooses a card • Player 2 throws a die • If it’s a six, player 2 chooses a card, swaps it with player 1’s and keeps player 1’s card • If it’s not a six, player 2 just chooses a card • Player 1 chooses next card • Player 2 takes the last card

  34. Expectiminimax Diagram

  35. Expectiminimax Calculations

  36. Games and Computers • State of the art for some game programs • Chess • Checkers • Othello • Backgammon • Go

  37. Chess • Deep Blue, a special-purpose parallel computer, defeated the world champion Gary Kasparov in 1997 • the human player didn’t show his best game • some claims that the circumstances were questionable • Deep Blue used a massive data base with games from the literature • Fritz, a program running on an ordinary PC, challenged the world champion Vladimir Kramnik to an eight-game draw in 2002 • top programs and top human players are roughly equal

  38. Checkers • Arthur Samuel develops a checkers program in the 1950s that learns its own evaluation function • reaches an expert level stage in the 1960s • Chinook becomes world champion in 1994 • human opponent, Dr. Marion Tinsley, withdraws for health reasons • Tinsley had been the world champion for 40 years • Chinook uses off-the-shelf hardware, alpha-beta search, end-games data base for six-piece positions

  39. Othello • Logistello defeated the human world champion in 1997 • Many programs play far better than humans • smaller search space than chess • little evaluation expertise available

  40. Backgammon • TD-Gammon, neural-network based program, ranks among the best players in the world • improves its own evaluation function through learning techniques • search-based methods are practically hopeless • chance elements, branching factor

  41. Go • Humans play far better • large branching factor (around 360) • search-based methods are hopeless • Rule-based systems play at amateur level • The use of pattern-matching techniques can improve the capabilities of programs • difficult to integrate • $2,000,000 prize for the first program to defeat a top-level player

  42. Chapter Summary • Many game techniques are derived from search methods • The minimax algorithm determines the best move for a player by calculating the complete game tree • Alpha-beta pruning dismisses parts of the search tree that are provably irrelevant • An evaluation function gives an estimate of the utility of a state when a complete search is impractical • Chance events can be incorporated into the minimax algorithm by considering the weighted probabilities of chance events

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