1. Uncertain Reasoning in Games Dmitrijs Rutko Faculty of Computing University of Latvia LU and LMT Computer Science Days at Ratnieki, 2011

2. Game Tree Search Deterministic / stochastic games Perfect / imperfect information games

3. Finite zero-sum games deterministicchance perfect informationchess, checkers, go, othello backgammon, monopoly, roulette imperfect information battleship, kriegspiel, rock- paper-scissors bridge, poker, scrabble

4. Game trees

5. Classical algorithms MiniMax O(w d ) Alpha-Beta O(w d/2 ) 1274368954 2 789 28 8 √√√ΧΧ√√√ΧΧ max min max

6. Advanced search techniques Transposition tables Time efficiency / high cost of space PVS Negascout NegaC* SSS* / DUAL* MTD(f)

7. Uncertain Reasoning O(w d/2 ) More cut-offs 1274368954 <5<5 ?≥5 <5<5 √√ΧΧΧ√Χ√ΧΧ max min max

8. Game tree statistical evaluation Minimax value Tree count 251 265 2711 2838 29124 30206 31252 32189 33111 3442 3514 367 1000

9. Game tree analytical evaluation FXFX FXFX FXFX FXFX F min F max Probability density Cumulative distribution

10. Game tree analytical evaluation FXFX FXFX FXFX FXFX F min F max

11. Cumulative probability function by level

12. Probability density function by level

13. Relative performance (Leaf nodes visited)

14. Hey! That's My Fish!

15. Evaluation function Fish Amount (player) – Fish Amount (opponent)

16. Iterative deepening

17. Number of positions searched

18. Relative number of positions searched

19. Relative time elapsed

20. Conclusions and Future Work BNS gives a 10 percent performance improvement Transposition tables Different evaluation functions Multi-player game Approximation search

21. Questions ? dim_rut@inbox.lv