Game Playing Part 2 Alpha-Beta Pruning Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [based on slides from A. Moore http://www.cs.cmu.edu/~awm/tutorials , C. Dyer, J. Skrentny, Jerry Zhu] slide 1
alpha-beta pruning Gives the same game theoretic values as minimax, but prunes part of the game tree. "If you have an idea that is surely bad, don't take the time to see how truly awful it is." -- Pat Winston slide 2
Alpha-Beta Motivation max S A min B 100 C D E F G 200 100 120 20 • Depth-first order • After returning from A, Max can get at least 100 at S • After returning from F, Max can get at most 20 at B • At this point, Max losts interest in B • There is no need to explore G. The subtree at G is pruned. Saves time. slide 3
Alpha-beta pruning function Max-Value (s,α,β) Starting from the root: inputs: Max-Value(root, - , + ) s: current state in game, Max about to play α: best score (highest) for Max along path to s β: best score (lowest) for Min along path to s output: min (β , best-score (for Max) available from s ) if ( s is a terminal state ) then return ( terminal value of s ) else for each s ’ in Succ(s) α := max( α , Min-value(s ’,α,β)) if ( α ≥ β ) then return β /* alpha pruning */ return α slide 4
Alpha-beta pruning function Max-Value (s,α,β) Starting from the root: inputs: Max-Value(root, - , + ) s: current state in game, Max about to play α: best score (highest) for Max along path to s β: best score (lowest) for Min along path to s output: min (β , best-score (for Max) available from s ) if ( s is a terminal state ) then return ( terminal value of s ) else for each s ’ in Succ(s) α := max( α , Min-value(s ’,α,β)) if ( α ≥ β ) then return β /* alpha pruning */ return α function Min-Value (s,α,β) output: max (α , best-score (for Min) available from s ) if ( s is a terminal state ) then return ( terminal value of s) else for each s ’ in Succs(s) β := min( β , Max-value(s ’,α,β)) if (α ≥ β ) then return α /* beta pruning */ return β slide 5
Alpha-beta pruning example 1 α= - max β=+ S A min B 100 C D E F G 200 100 120 20 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 6
Alpha-beta pruning example 1 α= - max β=+ S α= - A min B β=+ 100 C D E F G 200 100 120 20 slide 7
Alpha-beta pruning example 1 α= - max β=+ S α= - A min B β= 200 100 C D E F G 200 100 120 20 slide 8
Alpha-beta pruning example 1 α= - max β=+ S α= - A min B β= 100 100 C D E F G 200 100 120 20 slide 9
Alpha-beta pruning example 1 α= 100 max β=+ S α= - A min B β= 100 100 C D E F G 200 100 120 20 slide 10
Alpha-beta pruning example 1 α= 100 max β=+ S α= 100 α= - A min B β=+ β=100 100 C D E F G 200 100 120 20 slide 11
Alpha-beta pruning example 1 α=100 max β=+ S α= 100 α= - A min B β= 120 β= 100 100 C D E F G 200 100 120 20 slide 12
Alpha-beta pruning example 1 α= 100 max β=+ S α=100 α= - A min B β=20 β=100 100 X C D E F G 200 100 120 20 slide 13
Alpha-beta pruning function Max-Value (s,α,β) Starting from the root: inputs: Max-Value(root, - , + ) s: current state in game, Max about to play α: best score (highest) for Max along path to s β: best score (lowest) for Min along path to s output: min (β , best-score (for Max) available from s ) if ( s is a terminal state ) then return ( terminal value of s ) else for each s ’ in Succ(s) α := max( α , Min-value(s ’,α,β)) if ( α ≥ β ) then return β /* alpha pruning */ return α function Min-Value (s,α,β) output: max (α , best-score (for Min) available from s ) if ( s is a terminal state ) then return ( terminal value of s) else for each s ’ in Succs(s) β := min( β , Max-value(s ’,α,β)) if (α ≥ β ) then return α /* beta pruning */ return β slide 14
Alpha-beta pruning example 1 max S What are the alpha and beta values on S? min A max C B 25 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 15
Alpha-beta pruning example 2 max α= - S β=+ min A max C B 25 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 16
Alpha-beta pruning example 2 max S α= - min A β=+ max C B 25 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 17
Alpha-beta pruning example 2 max S min A max α= - C B 25 β=+ 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 18
Alpha-beta pruning example 2 max S min A max C α=20 B 25 β=+ 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 19
Alpha-beta pruning example 2 max S min A max C α= 20 B 25 β=+ 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 20
Alpha-beta pruning example 2 max S min A max C α= 20 B 25 β=+ 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 21
Alpha-beta pruning example 2 max S min What are the alpha and beta A values on A? max C α= 20 B 25 β=+ 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 22
Alpha-beta pruning example 2 max S min α= - A β= 20 max C B 25 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 23
Alpha-beta pruning example 2 max S min A α= - max C B β= 20 25 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 24
Alpha-beta pruning example 2 max S min A α= -20 max C B β=20 25 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 25
Alpha-beta pruning example 2 max S min A α= 25 max C B β= 20 25 20 X D E F G H 20 -10 -20 25 • Keep two bounds along the path ▪ : the best Max can do ▪ : the best (smallest) Min can do • If at anytime exceeds , the remaining children are pruned. slide 26
Yet another alpha-beta pruning example • Keep two bounds along the path ▪ : the best Max can do on the path ▪ : the best (smallest) Min can do on the path • If at anytime exceeds , the remaining children are pruned. – = – A + = + A max α =- D B C E 0 G H I L F J K M -5 3 8 2 N P Q R S T U V O 4 9 -6 0 3 5 -7 -9 W X -3 -5 [Example from James Skrentny] slide 27
Alpha-beta pruning example • Keep two bounds along the path ▪ : the best Max can do on the path ▪ : the best (smallest) Min can do on the path • If a max node exceeds , it is pruned. • If a min node goes below , it is pruned. A max α =- B D β B C E 0 = min + G H I L F J K M -5 3 8 2 N P Q R S T U V O 4 9 -6 0 3 5 -7 -9 W X -3 -5 [Example from James Skrentny] slide 28
Alpha-beta pruning example • Keep two bounds along the path ▪ : the best Max can do on the path ▪ : the best (smallest) Min can do on the path • If a max node exceeds , it is pruned. • If a min node goes below , it is pruned. A max α =- B D β C E = 0 + min F G H I L F J K M α -5 3 8 2 =- max N P Q R S T U V O 4 9 -6 0 3 5 -7 -9 W X -3 -5 [Example from James Skrentny] slide 29
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