1. Adversarial Search

2. Game playing u Multi-agent competitive environment u The most common games are deterministic, turn- taking, two-player, zero-sum game of perfect information u In AI terminology: deterministic, fully observable, two agents act alternately, the utility values at then end are always equal and opposite 2

3. Tic-tac-toe XOX X XOO 3 u Two players, max and min u Loop: u Max put X u Min put O u Until one play takes 3 contiguous space or all spaces are filled +1 XOX OX O XOX OOX XXO 0

4. 4

5. 5 Higher values are good for max, but bad for min MAX assumes that min plays optimally What if MIN does not play optimally? Is MAX still optimal?

6. 6

7. 7

8. 8

9. 9

10. 10

11. 11

12. 12

13. Another example 13 max min max 4 [4, +∞] [-∞, 3] (3) [-∞, 3] 4

14. 14 Assuming a left-to-right evaluation of the tree, give the most general relation between x, y, z, w, k, l and m that must hold such that the tree in the dotted part would be pruned by alpha-beta pruning

15. 15

16. Move ordering u Best ordering leads to O(b m/2 ) time complexity 16

17. 17 The perfect order is not known until all child nodes are explored Randomized order works reasonably well

18. Summary u Deterministic, turn-taking, two-player, zero-sum game of perfect information u Minimax algorithm, optimal for optimal player u Not all nodes need to be explored, alpha-beta pruning 18 HW1 due next Monday, Exam 1 on next Monday