Games CSE 473 – Autumn 2003

Alpha-Beta Pruning

Alpha-Beta MiniMax MinVal(state, alpha, beta){ if (terminal(state)) return utility(state); for (s in children(state)){ child = MaxVal(s,alpha,beta); beta = min(beta,child); if (alpha>=beta) return child; } return beta; alpha = highest value choice along any path for MAX beta = lowest value choice along any choice for min

Alpha-Beta MiniMax MaxVal(state, alpha, beta){ if (terminal(state)) return utility(state); for (s in children(state)){ child = MinVal(s,alpha,beta); alpha = max(alpha,child); if (alpha>=beta) return child; } return alpha; alpha = highest value choice along any path for MAX beta = lowest value choice along any choice for min

Alpha-Beta Pruning pruned (1) call MaxVal(B,10,…) (3) (2) MinVal(C1…) returns 20 (6) beta=15 alpha=20 alpha>=beta true! immediately return to B B (4) call MinVal(C2,20,…) pruned (5) call MaxVal(D,20,…) returns 15

Effect of Alpha-Beta Pruning Best case: reduces number of nodes searched from O(bd) to O(bd/2) Can double depth of search Using good static evaluation function to order children gets close to best case in practice

Evaluation Functions eval(s) = w1 * material(s) + w2 * mobility(s) + w3 * king safety(s) + w4 * center control(s) + ... In practice MiniMax improves accuracy of heuristic eval function But one can construct pathological games where more search hurts performance! (Nau 1981)

Learning Weights

Learning New Features Search for common patterns Logistello (Buro 1997)

End-Game Databases Ken Thompson – all 5 piece end-games Lewis Stiller – all 6 piece end-games Refuted common chess wisdom: many positions thought to be ties were forced wins – 90% for white! Is perfect chess a win for white?

The MONSTER White wins in 255 moves (Stiller, 1991)

Deep Blue wins - 1997 (3 wins, 1 loss, 2 draws) Kasparov vs. Deep Blue IBM team led by JC Tan 32 RISC processors + 256 VLSI chess engines 200 million positions per second 16 plies Deep Blue wins - 1997 (3 wins, 1 loss, 2 draws) Kasparov Speaks!

Kasparov vs. Deep Junior Israeli programmers Amir Ban & Shay Bushinsky 2M positions/sec 8 CPU, 8 GB RAM Win2000 15 plies Strongest program ever! Buy it for $100! Deep Junior – August 2, 2003 Match ends in a 3 / 3 tie!