Based on slides by: Rob Powers Ian Gent Min-Max Trees Yishay Mansour Based on slides by: Rob Powers Ian Gent

Two Players Games One Search Tree for both Players Even layers – Max Player move Odd Layers – Min Player move The state evaluated according to heuristic function.

MinMax search strategy Generate the whole game tree. (Or up to a constant depth) Evaluate Terminal states (Leafs) propagate Min-Max values up from leafs Search for MAX best next move, so that no matter what MIN does MAX will be better off For branching factor b and depth search d the complexity is O(bd)

MinMax first Example

1 1 -2 -1

MinMax evaluation function function MinMax-Decision(game) returns an operator for each op in Operator[game] do Value[op] := MinMax-Value(Apply(op,game),game) end return the op with the highest Value[op] function MinMax-Value(state,game) returns an utility value if Terminal-Test[game](state) then return Utility[game](state) else if MAX’s turn return the highest MinMax-Value of Successors(state) else (MIN’s turn) return the lowest MinMax-Value of Successors(state)

Cuting Off Search We want to prune the tree: stop exploring subtrees with values that will not influence the final MinMax root decision In the worst case, no pruning. The complexity is O(bd). In practice, O(bd/2), with branching factor of b1/2 instead of b.

Alpha Beta First Example

Alpha-Beta search cutoff rules Keep track and update two values so far: alpha is the value of best choice in the MAX path beta is the value of best choice in the MIN path Rule: do not expand node n when beta <= alpha for MAX node return beta for MIN node return alpha

Alpha and Beta values At a Max node we will store an alpha value the alpha value is lower bound on the exact minimax score the true value might be   if we know Min can choose moves with score <  then Min will never choose to let Max go to a node where the score will be  or more At a Min node,  value is similar but opposite Alpha-Beta search uses these values to cut search

Alpha Beta in Action Why can we cut off search? Beta = 1 < alpha = 2 where the alpha value is at an ancestor node At the ancestor node, Max had a choice to get a score of at least 2 (maybe more) Max is not going to move right to let Min guarantee a score of 1 (maybe less)

Alpha and Beta values Max node has  value Min node has  value the alpha value is lower bound on the exact minimax score with best play M x can guarantee scoring at least  Min node has  value the beta value is upper bound on the exact minimax score with best play Min can guarantee scoring no more than  At Max node, if an ancestor Min node has  <  Min’s best play must never let Max move to this node therefore this node is irrelevant if  = , Min can do as well without letting Max get here so again we need not continue

Alpha-Beta Pruning Rule Two key points: alpha values can never decrease beta values can never increase Search can be discontinued at a node if: It is a Max node and the alpha value is  the beta of any Min ancestor this is beta cutoff Or it is a Min node and the beta value is  the alpha of any Max ancestor this is alpha cutoff

Alpha-Beta prunning function Max-Value(state,game,alpha,beta) returns minmax value of state if Cutoff-Test(state) then return Eval(state) for each s in Successor(state) do alpha := Max(alpha,Min-Value(s, game,alpha,beta)) if beta <= alpha then return beta end; return alpha function Min-Value(state,game,alpha,beta) returns beta := Min(beta,Max-Value(s, game,alpha,beta)) if beta <= alpha then return alpha end; return beta

#2b Left->Right = -, = +

#2b Left->Right = 7, = + = -, = 4 = 6, = + = 4, = 6 = -, = + = -, = 4 = 8, = + = 6, = 8 (Alpha pruning) = 4, = + = 8, = 6 = 4, = 3 = 4, = 8 = 5, = 6 (Alpha pruning) = 8, = + = 4, = 8 =5, =8 =5, =6

Beta Pruning 9 = 4, = + = -, = 4 = 4, = 8 = -, = + = 8, = + = 4, = 8 (Alpha pruning) = 4, = + = 8, = 6 = 4, = 3 = 4, = 8 (Alpha pruning) 9 = 8, = + = 4, = 8

Beta Pruning 9 = 4, = + = -, = 4 = 4, = 8 = -, = + = 8, = + = 9, = 8 (Beta pruning) (Alpha pruning) = 4, = + = 8, = 6 = 4, = 3 = 4, = 8 (Alpha pruning) 9 = 8, = + = 4, = 8

#2b Right->Left = -, = +

#2b Right->Left = 7, = + = 7, = 7 = 7, = 6 = -, = + (Alpha pruning) (Alpha pruning) = 7, = + = 7, = + (Alpha pruning) = 7, = -1 = 7, = + = 7, = 6 (Alpha pruning) =7, =+ =7, =+ =7, =+