1 / 51

How computers play games with you

How computers play games with you CS161, Spring ‘03 Nathan Sturtevant Outline Historic Examples Classes of Games Algorithms Minimax - pruning Other techniques Multi-Player Games Successful Game Programs Checkers Chinook 1992 Tinsley won 40-game match, 4-2-34

JasminFlorian
Download Presentation

How computers play games with you

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. How computers play games with you CS161, Spring ‘03 Nathan Sturtevant

  2. Outline • Historic Examples • Classes of Games • Algorithms • Minimax • - pruning • Other techniques • Multi-Player Games

  3. Successful Game Programs • Checkers • Chinook • 1992 Tinsley won 40-game match, 4-2-34 • 1994 Tinsley withdrew due to health reasons • 444 billion move end-game database • Chess • Kasparov is currently the best human • 1997 Deep Blue won exhibition match 2-1-3 • 2003 Deep Junior played to a draw

  4. Game Programs (continued) • Othello (Reversi) • 1997, Logistello beat Murakami 6-0 (264/120) • Scrabble • Maven • 1998 played Adam Logan, won 9-5 • Came back from down 98 to win with MOUTHPART • Awari (Mancala) • Solved in 2002 - draw • http://awari.cs.vu.nl/

  5. Overview - Types of Games • Single-Agent Search • 1 player v. a difficult problem • Defined by: • Start state • Successor function • Heuristic function • Goal test

  6. Overview - Types of Games • Game Search (Adversary Search) • Defined by: • Initial State • Successor function • Terminal Test • Utility / payoff function • Similar to heuristics in single agent problems

  7. Chinese Checkers • Based on European game Halma • Americans called it Chinese Checkers • 1 player game? • 2 player game? • Multi-player game?

  8. Classes of Games • Deterministic v. Non-deterministic • Chess v. Backgammon • Perfect Information v. Imperfect information • Checkers v. Bridge • Zero-sum (strictly competitive) • Prisoners dilemna • Non-zero sum

  9. Classes of Games

  10. Classes of Games

  11. Classes of Games

  12. Classes of Games

  13. Classes of Games

  14. How do we simulate games? • Build a game tree • Start state at the root • All possible moves as children

  15. Me Opponent Tic-Tac-Toe

  16. How do we choose our move? • Apply utility function at the leaves of the tree • In tic-tac-toe, count how many rows and columns are occupied by each player and subtract

  17. Me Opponent Tic-Tac-Toe Utility = 3 x: 2r 2c 2d o: 2r 2c 0d x: 2r 2c 2d o: 2r 1c 1d x: 2r 3c 2d o: 2r 1c 1d x: 3r 3c 2d o: 2r 2c 0d Utility = 3 Utility = ∞ Utility = 2 Utility = 3 Utility = ∞ Utility = 2

  18. What is our algorithm? • Apply utility function at the leaves of the tree • In tic-tac-toe, count how many rows and columns are occupied by each player and subtract • Back-up values in the tree • This calculates the “minimax” value of a tree

  19. Maximizer Minimizer 2 3 ∞ 2 Minimizers strategy Minimax 3 1 - ply 3 ∞ 1 - ply

  20. Minimax - Properties • Complete? • Yes - if tree is finite • Optimal? • Yes - against an optimal opponent • Time Complexity? • O(bd) • Space Complexity? • O(bd)

  21. Minimax • Assume our computer can expand 105 nodes/sec • Assume we have 100 seconds to move • 107 nodes/move • Tic-tac-toe • 9! = 362880 (naïve) ways to play a game (b=4) • 39 = 19683 possible states (upper bound) on a board • Chess • b = 35, d = 100, must search 2154 nodes

  22. Minimax - issues • Evaluation function • Where does it come from? • Expert knowledge • Chess: material value • Othello (reversi): positional strength • Learned information • Pre-computed tables • Quiescence

  23. quiescence

  24. Minimax - issues • Quiescence • We don’t see the consequences of our bad choices • quiescence search • Horizon problem • We avoid dealing with a bad situation

  25. Minimax • In Chess • b = 35 • 107 nodes/move • Can search 4-ply into tree (human novice) • Good humans can search 8-ply • Kasparov searches about 12-ply • What to do? • - pruning

  26. Maximizer Minimizer 2 3 ∞ 2 Minimizers strategy Minimax 3 1 - ply 3 ∞ 1 - ply

  27. - pruning •  = lower bound on Maximizer’s score • Start at -∞ •  = upper bound on Minimizer’s score • Start at ∞

  28. Maximizer Minimizer 1  = -∞  = ∞  = -∞  = ∞  = -∞  = ∞ ≥1   -∞ ∞

  29. Maximizer Minimizer 1 2  = -∞  = ∞  = -∞  = ∞  = 1  = ∞ ≥1 2 -∞ ∞

  30. Maximizer Minimizer 1 2  = -∞  = ∞  = -∞  = ∞  = 2  = ∞ 2   -∞ ∞

  31. Maximizer Minimizer 1 3  = -∞  = ∞  = -∞  = 2 ≤ 2  = -∞  = 2 ≥ 3 2 2   -∞ ∞

  32. Maximizer Minimizer 1 3  = -∞  = ∞ ≥ 2  = -∞  = 2 2  = 3  = 2 ≥ 3 2 2   -∞ ∞

  33. Maximizer Minimizer  = 2  = ∞  = 2  = ∞ 5  = 2  = ∞ ≥ 2 2 ≥ 3 ≥ 5 2 1 2 3   -∞ ∞

  34. Maximizer Minimizer  = 2  = ∞  = 5  = ∞ 5 6  = 2  = ∞ ≥ 2 2 ≥ 3 ≥ 5 2 6 1 2 3   -∞ ∞

  35. Maximizer Minimizer  = 2  = ∞ ≥ 2  = 2  = 6 ≤ 6 2  = 2  = 6 ≥ 3 ≥7 2 6 1 2 3 5 6 7   -∞ ∞

  36. Maximizer Minimizer  = 2  = ∞ ≥ 2 6  = 2  = 6 ≤ 6 2 6  = 7  = 6 ≥ 3 ≥7 2 6 1 2 3 5 6 7   -∞ ∞

  37. - pruning • Complete? • Yes - if tree is finite • Optimal? • Computes same value as minimax • Time Complexity? • Best case O(bd/2) • Average case O(b3d/4)

  38. - pruning • Effectiveness depends on order of moves in tree • In practice, we can usually get best-case performance • Chess • Before we could search 4-ply into tree • Now we can search 8-ply into tree

  39. Other Techniques • Transposition tables • Opening / Closing book

  40. Transposition Tables • Only using linear about of memory • Search only takes a few kb of memory • Most games aren’t trees but graphs

  41. Transposition Tables

  42. Transposition Tables • A lot of duplicated effort • Transposition tables hash game states into table • Store saved minimax value in table • Pre-compute & store values • Opening book • Closing book

  43. Multi-Player Games • 2-Player game trees have a single minimax value • Games with ≥ 2 players use a n-tuple of scores • ie (3, 2, 5) • The sum of values in every tuple should be constant

  44. 1 2 2 2 3 3 3 3 3 3 (7, 3, 0) (3, 2, 5) (0, 10, 0) (4, 2, 4) (1, 4, 5) (4, 3, 3) Maxn (3 Players) (7, 3, 0) (7, 3, 0) (0, 10, 0) (1, 4, 5)

  45. Can we prune maxn trees • In minimax we bound the game tree value • In maxn we bound based on sum of values • All scores sum to 10 • If Player 1 gets 7 points… • Player 2-3 will get ≤ 3 points

  46. 1 2 2 2 3 3 3 3 (7, 3, 0) (3, 2, 5) (0, 10, 0) (1, 4, 5) Shallow Maxn Pruning (3 Players) (7, 3, 0) (≥7, ≤3, ≤3) (7, 3, 0) (0, 10, 0) (≤6, ≥4, ≤6)

  47. Shallow Maxn Pruning • Complete? • Yes • Optimal? • Yes* • Time Complexity? • Best-case**: bd/2 • Average-case: bd • Space Complexity? • b•d

  48. Maxn Pruning • Why is maxn weak in practice? • Only compares 2 scores out of n players • Relies on game evaluation properties, not ordering • Last-Branch Pruning • Speculative Pruning

  49. (3, 3, 4) 2 2 3 (1, 4, 5) 3 (2, 4, 4) 1 Last-Branch/Speculative Pruning (3 Players) (3, 3, 4) 1 2

  50. Last Branch/Spec. Pruning • Best case: O(bd·(n-1)/n) • As b gets large • Dependent only on node ordering in tree • http://www.cs.ucla.edu/~nathanst/ for more info

More Related