1. Game Playing

2. Introduction One of the earliest areas in artificial intelligence is game playing. Two-person zero-sum game. Games for which the state space is small enough – generate the entire space. Games for which the entire space cannot be generated.

3. The Game NIM 7 6-15-24-3 5-1-14-2-13-2-23-3-1 4-1-1-13-2-1-12-2-2-1 3-1-1-12-2-1-1 2-1-1-1-1-1

4. NIM- MAX Plays First. 7 6-15-24-3 5-1-14-2-13-2-23-3-1 4-1-1-13-2-1-12-2-2-1 3-1-1-12-2-1-1 2-1-1-1-1-1 MAX MIN MAX MIN 1 0 1

5. NIM- MIN Plays First. 7 6-15-24-3 5-1-14-2-13-2-23-3-1 4-1-1-13-2-1-12-2-2-1 3-1-1-12-2-1-1 2-1-1-1-1-1 MIN MAX MIN MAX 0 1 0

6. Minimax Algorithm Repeat If the limit of search has been reached, compute the static value of the current position relative to the appropriate player. Report the result. Otherwise, if the level is a minimizing level, use the minimax on the children of the current position. Report the minimum value of the results. Otherwise, if the level is a maximizing level, use the minimax on the children of the current position. Report the maximum of the results. Until the entire tree is traversed.

7. Minimax Applied to NIM. 7 6-15-24-3 5-1-14-2-13-2-23-3-1 4-1-1-13-2-1-12-2-2-1 3-1-1-12-2-1-1 2-1-1-1-1-1 MIN MAX MIN MAX0 1 0 0 1 0 10 00 0 0 00

8. Generating the Game Tree to a Depth In some cases the game tree will be too large to generate. In this case the tree is generated to a certain depth or ply. Heuristic values are used to estimate how promising a node is. Horizon effect.

9. Example

10. Heuristic for Tic-Tac-Toe h(n) = x(n) - o(n) where – x(n) is the total of MAX’s possible winning (we assume MAX is playing x) – o(n) is the total of the opponent’s, i.e. MIN’s winning lines h(n) is the total evaluation for a state n.