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Games: Efficiency

Games: Efficiency. Andrea Danyluk September 23, 2013. Announcements. Programming Assignment 1: Search Autograding done Please sign up for a code review Programming Assignment 2: Games Remove your print statements before turning it in

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Games: Efficiency

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  1. Games:Efficiency Andrea Danyluk September 23, 2013

  2. Announcements • Programming Assignment 1: Search • Autograding done • Please sign up for a code review • Programming Assignment 2: Games • Remove your print statements before turning it in • Need help with buggy code? If I’m not in the office, email me your code and a description of the problem.

  3. Today’s Lecture • Heuristic evaluation functions for minimax • Making the search more efficient: • α-β pruning

  4. Minimax Reality • Can rarely explore entire search space to terminal nodes. • Choose a depth cutoff – i.e., a maximum ply • Need an evaluation function • Returns an estimate of the expected utility of the game from a given position • Must order the terminal states in the same way as the true utility function • Must be efficient to compute • Trading off plies for heuristic computation • More plies makes a difference • Consider iterative deepening

  5. Evaluation Functions • Ideal: returns the utility of the position • In practice: typically weighted linear sum of features: • Eval(s) = w1f1(s) + w2f2(s) + … + wnfn(s)

  6. Exercise • Evaluation function for Connect Four?

  7. An earlier example revisited 7 6 2 3 0 -2 6 2 5 6 9 2

  8. An earlier example revisited 7 7 6 2 3 0 -2 6 2 5 6 9 2

  9. An earlier example revisited 7 3 7 6 2 3 0 -2 6 2 5 6 9 2

  10. An earlier example revisited 3 7 3 7 6 2 3 0 -2 6 2 5 6 9 2

  11. An earlier example revisited Wow! 0 is a great score! 3 7 3 0 7 6 2 3 0 -2 6 2 5 6 9 2

  12. An earlier example revisited I wonder if I can do even better! 3 7 3 0 7 6 2 3 0 -2 6 2 5 6 9 2

  13. An earlier example revisited Too bad the maximizer won’t ever let me get here! 3 7 3 0 7 6 2 3 0 -2 6 2 5 6 9 2

  14. An earlier example revisited Better not to waste my time! 3 7 3 0 7 6 2 3 0 -2 6 2 5 6 9 2

  15. An earlier example revisited 3 7 3 0 7 6 2 3 0 -2 6 2 5 6 9 2

  16. An earlier example revisited 3 0 7 3 0 7 6 2 3 0 -2 6 2 5 6 9 2

  17. An earlier example revisited 3 0 7 3 0 6 7 6 2 3 0 -2 6 2 5 6 9 2

  18. An earlier example revisited 3 0 Wow! 9! 7 3 0 6 9 7 6 2 3 0 -2 6 2 5 6 9 2

  19. An earlier example revisited 3 0 Oh, wait! 7 3 0 6 9 7 6 2 3 0 -2 6 2 5 6 9 2

  20. An earlier example revisited 3 0 Min won’t let me go here! 7 3 0 6 9 7 6 2 3 0 -2 6 2 5 6 9 2

  21. An earlier example revisited 3 0 6 Min won’t let me go here! 7 3 0 6 9 7 6 2 3 0 -2 6 2 5 6 9 2

  22. An earlier example revisited 3 0 6 Min won’t let me go here! 7 3 0 6 9 7 6 2 3 0 -2 6 2 5 6 9 2

  23. α-β Pruning • If something looks too good to be true, it probably is. • One example of the class of branch and bound algorithms with two bounds • α : the value of the best choice for Max • β : the value of the best choice for min

  24. α-β Pruning • Given these two bounds • α : the value of the best choice for Max • β : the value of the best choice for min • Basic idea of the algorithm • On a minimizing level, if you find a value < α, cut the search • On a maximizing level, if you find a value > β, cut the search

  25. functionαβSEARCH(state) returns an action a v = MAX-VALUE(state, -infinity, +infinity) return action a in ACTIONS(state) with value v function MAX-VALUE(state, α, β) returns a utility value v if TERMINAL-TEST(state) then return UTILITY(state) v = -infinity for each ainACTIONS(state) do v = MAX(v, MIN-VALUE(RESULT(state, a), α, β)) ifv ≥ βthen returnv α = MAX(α, v) returnv function MIN-VALUE(state, α, β) returns a utility value v if TERMINAL-TEST(state) then return UTILITY(state) v = infinity for each ainACTIONS(state) do v = MIN(v, MAX-VALUE(RESULT(state, a), α, β)) ifv ≤ αthen returnv β = MIN(β, v) returnv

  26. α = -inf Β = +inf 2 8 7 9 1 6 11 19 8 3 4 5

  27. α = -inf Β = +inf α = -inf Β = +inf 2 8 7 9 1 6 11 19 8 3 4 5

  28. α = -inf Β = +inf α = -inf Β = +inf α = -inf Β = +inf 2 8 7 9 1 6 11 19 8 3 4 5

  29. α = -inf Β = +inf α = -inf Β = +inf α = 8 Β = +inf 2 8 7 9 1 6 11 19 8 3 4 5

  30. α = -inf Β = +inf α = -inf Β = +inf 2 8 8 7 9 1 6 11 19 8 3 4 5

  31. α = -inf Β = +inf α = -inf Β = 8 2 8 8 7 9 1 6 11 19 8 3 4 5

  32. α = -inf Β = +inf α = -inf Β = 8 α = -inf Β = 8 2 8 8 7 9 1 6 11 19 8 3 4 5

  33. α = -inf Β = +inf α = -inf Β = 8 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  34. α = -inf Β = +inf 8 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  35. α = 8 Β = +inf 8 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  36. α = 8 Β = +inf α = 8 Β = +inf 8 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  37. α = 8 Β = +inf 2 8 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  38. α = 8 Β = +inf α = 8 Β = +inf 2 8 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  39. α = 8 Β = +inf α = 8 Β = +inf 2 8 α = 8 Β = +inf 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  40. α = 8 Β = +inf α = 8 Β = +inf 2 8 α = 8 Β = +inf 2 8 9 8 7 9 1 6 11 19 8 3 4 5

  41. α = 8 Β = +inf α = 8 Β = +inf 2 8 2 8 9 8 8 7 9 1 6 11 19 8 3 4 5

  42. Is there a problem here??? α = 8 Β = +inf α = 8 Β = +inf 2 8 2 8 9 8 8 7 9 1 6 11 19 8 3 4 5

  43. functionαβSEARCH(state) returns an action a v = MAX-VALUE(state, -infinity, +infinity) return action a in ACTIONS(state) with value v function MAX-VALUE(state, α, β) returns a utility value v if TERMINAL-TEST(state) then return UTILITY(state) v = -infinity for each ainACTIONS(state) do v = MAX(v, MIN-VALUE(RESULT(state, a), α, β)) ifv ≥ βthen returnv α = MAX(α, v) returnv function MIN-VALUE(state, α, β) returns a utility value v if TERMINAL-TEST(state) then return UTILITY(state) v = infinity for each ainACTIONS(state) do v = MIN(v, MAX-VALUE(RESULT(state, a), α, β)) ifv ≤ αthen returnv β = MIN(β, v) returnv

  44. Properties of α-β • Pruning does not affect the final result (but be careful) • Good move ordering improves effectiveness of pruning • If search depth is d, what is the time complexity of minimax? • O(bd) • With perfect pruning, get down to O(bd/2) • Doubles solvable depth [Adapted from Russell]

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