Game-Playing Read Chapter 6 Adversarial Search

State-Space Model Modified States the same Operators depend on whose turn Goal: As before: win or win amount Search: somewhat different

Game Types Two-person games vs multi-person –chess vs monopoly vs poker Perfect Information vs Imperfect –checkers vs card games Deterministic vs Non-deterministic –go vs backgammon

Simplest First: Two-Person Games: Perfect Information BF = branching factor (average) Chess: BF ~36 –expert level Checkers: BF ~ 8, world champion Othello: BF ~10, better world champion Go: BF ~200 –$2 million prize

MiniMax Algorithm (perfect information, 2 person game) Assume: evaluation of terminal position Win = +1, Loss = -1, Draw = 0. Descendants of max node is min node, etc. Algorithm: recursive –Value Max Node = max(descendants of node) –Value Min Node = min(descendants of node) –Value of terminal node: by evaluation function Applies to any tree with values assigned to leaves. NOTE: If tree to end of game, guaranteed best move, else no one knows.

MiniMax Example

Optimal Play Make move that yields highest minimax score. Computation: search: depth-first Time = b^d Memory= b*d

Applied to Chess Average game is 40+ moves Tree to large to reach terminal positions Static board evaluation of worthiness Uses Partial Tree MiniMax yields optimal value for restricted tree, with values assigned by evaluation. No theorems connecting valuation on partial tree to estimates for complete tree.

Alpha-Beta Algorithm Yields exactly same value as minimax Knuth analyzed: time or nodes = O(b^d/2) Doubles depth of search with same time. Constant depends on ordering of nodes Iterative deepening alpha/beta achieves better ordering. (reorder after depth)

Alpha-beta Algorithm Each node is assigned a range of values: [alpha,beta]. The real value will lie between. The root is assigned [-inf,+inf]. For any max node N with values [A,B] –if a son has value >=C, then N has new range [C,B]. –If interval is empty, all nodes below cut. For any min node N with values [A,B] –if son has value <=D, then N updated to [A,D]. Formal code in text. eta/alphabeta.html Applet illustrationhttp:// eta/alphabeta.html

Alpha-Beta Example

(1,2,2) Nim

Multi-player Games Extension of minimax –assign a vector of values to each position –vector has value relative to each player –Each player maximizes choice –Equals minimax for 2 person game No variations like alpha-beta

Games with Uncertainty Card games like hearts or bridge Backgammon (roles of dice) Expectimax –Does it work? –Theoretically nice, but where’s the meat – for what games was it successful?

Certainty from Uncertainty Simulation –Replace unknown world by specific world –simulate (or use alpha-beta) –Each simulation yields a play –Vote Works for hearts and bridge play –bridge high level card play can’t make information gathering plans

What about War Games are games – restricted uncertainty What are the operators in war? –unknown effects –unknown number What is the state? –unknown