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Game Playing

Game Playing. Pruning. Pruning is used to narrow the search down e.g. if we find a branch where we will win no matter what, we don’t need to search that part of the tree anymore And if we find a guaranteed losing line we don’t need to search that part of tree any more. MiniMax.

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Game Playing

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  1. Game Playing

  2. Pruning • Pruning is used to narrow the search down • e.g. if we find a branch where we will win no matter what, we don’t need to search that part of the tree anymore • And if we find a guaranteed losing line we don’t need to search that part of tree any more

  3. MiniMax • NIM Search tree 6 Max moves 1-5 2-4 3-3 Min moves 1-1-4 1-2-3 2-2-2 Max moves 1-1-1-3 1-1-2-2 Min moves 1-1-1-1-2

  4. MiniMax • NIM Search tree 6(?) Max moves 1-5 2-4 3-3 Min moves 1-1-4 1-2-3 2-2-2 Max moves 1-1-1-3 1-1-2-2 Min moves 1-1-1-1-2

  5. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4 1-2-3 2-2-2 Max moves 1-1-1-3 1-1-2-2 Min moves 1-1-1-1-2

  6. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(?) 1-2-3(?) 2-2-2 Max moves 1-1-1-3 1-1-2-2 Min moves 1-1-1-1-2

  7. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(?) 1-2-3(?) 2-2-2 Max moves 1-1-1-3(?) 1-1-2-2(?) Min moves 1-1-1-1-2

  8. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(?) 1-2-3(?) 2-2-2 Max moves 1-1-1-3(?) 1-1-2-2(?) Min moves 1-1-1-1-2(?)

  9. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(?) 1-2-3(?) 2-2-2 Max moves 1-1-1-3(?) 1-1-2-2(?) Min moves 1-1-1-1-2(-1)

  10. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(?) 1-2-3(?) 2-2-2 Max moves 1-1-1-3(-1) 1-1-2-2(?) Min moves 1-1-1-1-2(-1)

  11. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(?) 1-2-3(?) 2-2-2 Max moves 1-1-1-3(-1) 1-1-2-2(1) Min moves 1-1-1-1-2(-1)

  12. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(1) 1-2-3(?) 2-2-2 Max moves 1-1-1-3(-1) 1-1-2-2(1) Min moves 1-1-1-1-2(-1)

  13. MiniMax • NIM Search tree 6(?) Max moves 1-5(?) 2-4(?) 3-3(?) Min moves 1-1-4(1) 1-2-3(1) 2-2-2 Max moves 1-1-1-3(-1) 1-1-2-2(1) Min moves 1-1-1-1-2(-1)

  14. MiniMax • NIM Search tree 6(?) Max moves 1-5(1) 2-4(?) 3-3(?) Min moves 1-1-4(1) 1-2-3(1) 2-2-2 Max moves 1-1-1-3(-1) 1-1-2-2(1) Min moves 1-1-1-1-2(-1)

  15. MiniMax • NIM Search tree 6(1) Max moves X X 1-5(1) 2-4(?) 3-3(?) Min moves X 1-1-4(1) 1-2-3(1) 2-2-2 Max moves 1-1-1-3(-1) 1-1-2-2(1) Min moves 1-1-1-1-2(-1)

  16. Alpha Beta Pruning • A generalized pruning procedure • During our depth-first search we remember • The score of the best choice so far for Max, alpha • And the best score so far for Min, beta • If we find a point where MIN can choose a move with the utility ≤ alpha. Then we can stop search there since • MAX will not choose a move since there are a alternative route that is better. (Where alpha was found)

  17. MAX MIN ≤5 2 MAX 2 3 12 8 14 5 2 Alpha-Beta (-) Pruning Example ≥ 3 3 ≤ 2 ≤ 14

  18. Alpha Beta Pruning • What can it do for us? • Allows deeper searches without penalty • Allows deeper searches at same time cost • The result is always the same as without pruning, but a lot faster

  19. Do Exact Values Matter?

  20. MiniMax and Heuristics • What about more complex games? • If the search space is too large • We could use a heuristic function to determine which nodes to expand first. (Not very useful though) • Limit our depth and when we reach a certain depth we evaluate the current standing there • After evaluating we use MiniMax as usual

  21. MiniMax and heuristics • Practically, rather than implement MiniMax, we implement MiniCutoff • Don’t build the whole tree. Takes lots of memory. • Limit the depth of “look ahead” • Use a utility function rather than an evaluation function.

  22. Cutting Off Search

  23. Cutting Off Search • Issues • Quiescence • Play has “settled down” • Evaluation function unlikely to exhibit wild swings in value in near future • Horizon effect • “Stalling for time” • Postpones inevitable win or damaging move by opponent • See: Figure 6.9 R&N • Solutions? • Quiescence search: expand non-quiescent positions further • No general solution to horizon problem at present

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