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

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

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

Game trees

Classical algorithms MiniMax O(w d ) Alpha-Beta O(w d/2 ) √√√ΧΧ√√√ΧΧ max min max

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

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

Game tree statistical evaluation Minimax value Tree count

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

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

Cumulative probability function by level

Probability density function by level

Relative performance (Leaf nodes visited)

Hey! That's My Fish!

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

Iterative deepening

Number of positions searched

Relative number of positions searched

Relative time elapsed

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

Questions ?