Adversarial Search

Game playing

Multi-agent competitive environment The most common games are deterministic, turn-

taking, two-player, zero-sum game of perfect information

In AI terminology: deterministic, fully observable, two agents act alternately, the utility values at then end are always equal and opposite

2

Tic-tac-toe

X O XX

X O O

3

Two players, max and min Loop:

Max put X Min put O Until one play takes 3 contiguous space or all spaces are filled

+1

X O XO XO-1

X O XO O XX X O

0

4

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

7

8

9

10

11

12

Another example

13

max

min

max

4

[4, +∞]

[-∞, 3]

(3)

[-∞, 3]

4

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

Move ordering

Best ordering leads to O(bm/2) time complexity

16

17

The perfect order is not known until all child nodes are exploredRandomized order works reasonably well

Summary Deterministic, turn-taking, two-player, zero-sum

game of perfect information Minimax algorithm, optimal for optimal player Not all nodes need to be explored, alpha-beta

pruning

18

HW1 due next Monday,Exam 1 on next Monday