Chapter 6Chapter 6

Adversarial SearchAdversarial Search

Adversarial Search ProblemAdversarial Search Problem

Initial StateInitial State Successor FunctionSuccessor Function Terminal TestTerminal Test Payoff FunctionPayoff Function

Numeric value for terminal statesNumeric value for terminal states Usually Usually + for one player+ for one player

-- for the other player-- for the other player

Mini-Max AlgorithmMini-Max Algorithm

function maxValue(state)function maxValue(state)if (terminal) return value;if (terminal) return value;for s in succ(state)for s in succ(state)

maxValue = max(minValue(s), maxValue)maxValue = max(minValue(s), maxValue)

return maxValuereturn maxValue

function minValue(state)function minValue(state)if (terminal) return value;if (terminal) return value;for s in succ(state)for s in succ(state)

minValue = min(maxValue(s), minValue)minValue = min(maxValue(s), minValue)

return minValuereturn minValue

Mini-Max ExampleMini-Max Example

Mini-max AlgorithmMini-max Algorithm

Complete Depth-First SearchComplete Depth-First Search Space ComplexitySpace Complexity

O(bd)O(bd) Time ComplexityTime Complexity

O(bO(bdd))

Need Some method of pruning Need Some method of pruning the search treethe search tree

Alpha-Beta PruningAlpha-Beta Pruning

Some parts of the tree clearly do not Some parts of the tree clearly do not need to be searchedneed to be searched

Alpha-Beta PruningAlpha-Beta Pruning

alpha (alpha (αα) : ) : best choice so far for Maxbest choice so far for Max beta (beta (ββ) :) : best choice so far for Minbest choice so far for Min

alpha cut-off ifalpha cut-off if min on a branch is < alphamin on a branch is < alpha

beta cut-off ifbeta cut-off if Max on a branch is > betaMax on a branch is > beta

Alpha-Beta AlgorithmAlpha-Beta Algorithm

function maxValue(state,A,B)function maxValue(state,A,B)if (terminal) return value;if (terminal) return value;for s in succ(state)for s in succ(state)

v = max(minValue(s,A,B), v)v = max(minValue(s,A,B), v)if (v >= B) then return vif (v >= B) then return vA = max(A,v)A = max(A,v)

return vreturn v

function minValue(state,A,B)function minValue(state,A,B)if (terminal) return value;if (terminal) return value;for s in succ(state)for s in succ(state)

v = min(maxValue(s,A,B), v)v = min(maxValue(s,A,B), v)if (v <= A) then return vif (v <= A) then return vB = min(B,v)B = min(B,v)

return vreturn v

Alpha-Beta PruningAlpha-Beta Pruning

EffectivenessEffectiveness Highly dependent on which node is Highly dependent on which node is

expanded firstexpanded first However, if a good ordering can be However, if a good ordering can be

foundfound O(bO(bd/2d/2) instead of O(b) instead of O(bdd)) Effective branching factor is sqrt(b)Effective branching factor is sqrt(b)

Chess 6 vs 35Chess 6 vs 35

Imperfect Real-time decisionsImperfect Real-time decisions

Initial StateInitial State Successor FunctionSuccessor Function Terminal TestTerminal Test Payoff FunctionPayoff Function

Numeric value for terminal statesNumeric value for terminal states Usually Usually + for one player+ for one player

-- for the other player-- for the other player

Imperfect Real-time decisionsImperfect Real-time decisions

Initial StateInitial State Successor FunctionSuccessor Function Terminal TestTerminal Test Heuristic Evaluation FunctionHeuristic Evaluation Function

Numeric value for terminal statesNumeric value for terminal states Usually Usually + for one player+ for one player

-- for the other player-- for the other player

Imperfect Real-time decisionsImperfect Real-time decisions

Initial StateInitial State Successor Function Successor Function Terminal TestTerminal Test Heuristic Evaluation FunctionHeuristic Evaluation Function

Numeric value for terminal statesNumeric value for terminal states Usually Usually + for one player+ for one player

-- for the other player-- for the other player

Evaluation FunctionsEvaluation Functions Properties of a good evaluation functionProperties of a good evaluation function

AccurateAccurate FastFast

FeaturesFeatures Important aspects of the GameImportant aspects of the Game ChessChess

Material AdvantageMaterial Advantage Positional AdvantagePositional Advantage

Pawn StructurePawn Structure King SafetyKing Safety

Checkers?Checkers?

Evaluation FunctionEvaluation Function

Weighted Linear FunctionWeighted Linear Function Assumes Feature IndependenceAssumes Feature Independence