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

2. 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.

3. 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)

4. MinMax first Example

5. 1 1 -2 -1

6. 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)

7. 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.

8. Alpha Beta First Example

9. 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

10. 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

11. 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)

12. 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

13. 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

14. 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

15. #2b Left->Right
= -, = +

16. #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

17. 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

18. 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

19. #2b Right->Left
= -, = +

20. #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, =+