1. Games Search Neil Heffernan Some of these slides are screen shots from the the slides my professor at CMU (Andrew Moore) used. (Sorry for the low resolution)

4. Lets Play some Nim-II! Turn to you partner

5. Who Wins? Can you prove it? Can we come up with an algorithm for any came?

6. Draw the complete search space for NIM-II Label the terminal states with a pay off

10. Questions About Mini-max Last week when looking at search algorithms we saw that they al returned a path to the goal. Why doesn’t mini-max? Is minimax recursive? Is it efficient? What if loops in the search space are possible?

14. How can you save?

15. Can you use your cut-off tricks if you don’t know the range of possible values for the payoff function?

16. What leafs would you not have to explore in this example?

18. General Rule: We can be sure a node will not be visited if either player has a better alternative at any ancestor to that node.

19. 2 If min(V2,V4,V6,V7)

20. Alpha-Beta Pruning Effectiveness Does keeping track of alpha and beta cost much? Hard to analyze. Depends on how lucky you are. In practice, Alpha-beta pruning can allow you to search twice as deep as compared to mini-max for the same amount of time.

24. Can we deal with this?

25. State of the Art Chess- Easy Othello- Easy Go- very hard Checkers- world champions Why? What is the average branching factor

26. Backgammon What is the complexity if we add chance nodes? –B- average branching factor –N-number of distinct chance outcomes –M-the average number of moves needed. –Backgammon n=21, m~20 (sometimes 4000) Turns out search is prohibitive and better to get a good evaluation function (using a neural network)

27. General Principle When uncertainty enters the picture, we have many more possibilities. Can you do alpha-beta trick with backgammon?

28. Discussion When playing tournament chess, players get 2 hours for the first ~moves. How does that effect our search?

29. What is wrong with all this search? Is this how humans reason? Its all searching forward.