The document describes alpha-beta pruning in minimax search. It keeps track of alpha, the highest value seen by maximizing nodes, and beta, the lowest value seen by minimizing nodes. Pruning occurs when a node value is lower than alpha (for maximizing nodes) or higher than beta (for minimizing nodes). The document walks through 14 steps of applying alpha-beta pruning to a sample search tree.

1. An Alpha-Beta Primer
Stephen D. Bay
Minimax search with alpha-beta pruning involves depth rst search of a game tree, keeping
track of
Alpha: highest value seen so far on a maximizing level
Beta: lowest value seen so far on a minimizing level
Note that when we look at alpha and beta values we look only at nodes along the path from
our current node to the root.
Pruning is done according to the following two rules:
When on a maximizing level, do not expand any more sibling nodes once a node has
been seen whose evaluation is lower than or equal to Alpha.
When on a minimizing level, do not expand any more sibling nodes once a node has
been seen whose evaluation is greater than or equal to Beta.
For this primer, we will consider the following search tree:
A MAX
B C MIN
D E F G MAX
H I J K L M N O MIN
2 -3 8 5 -2 1 5 4
This is the full search tree to depth 3. The numbers beneath the leaves are values returned
by the evaluation function.
1

2. Step 1
A MAX
MIN
MAX
MIN
alpha = unde ned, beta = unde ned, Expand A
Step 2
A MAX
B C MIN
MAX
MIN
alpha = unde ned, beta = unde ned, Expand B
2

3. Step 3
A MAX
B C MIN
D E MAX
MIN
alpha = unde ned, beta = unde ned, Expand D
Step 4
A MAX
B C MIN
D E MAX
H I MIN
2
alpha = unde ned, beta = unde ned, Evaluate H and propagate values upwards.
3

4. Step 5
A MAX
B C MIN
>= 2
D E MAX
H I MIN
-3
alpha = 2, beta = unde ned, Evaluate I and propagate values upwards.
Step 6
A MAX
<= 2
B C MIN
2
D E MAX
H I MIN
alpha = unde ned, beta = 2, Expand E. Note that alpha is unde ned since we are at E and
we only consider alpha-beta values from E to the root A i.e. values at B and A.
4

5. Step 7
A MAX
= 2
B C MIN
D E MAX
H I J K MIN
8
alpha = unde ned, beta = 2, Evaluate J and propagate values upwards.
Step 8
A MAX
= 2
B C MIN
= 8
D E MAX
H I J K MIN
alpha = 8, beta = 2, Prune K as evalj 8 2. Once K has been pruned we can
propagate values up to A.
5

6. Step 9
= 2
A MAX
2
B C MIN
D E MAX
H I J K MIN
alpha = 2, beta = unde ned, Expand C
Step 10
= 2
A MAX
B C MIN
D E F G MAX
H I J K MIN
alpha = 2, beta = unde ned, Expand F
6

7. Step 11
= 2
A MAX
B C MIN
D E F G MAX
H I J K L M MIN
-2
alpha = 2, beta = unde ned, Evaluate L and propagate values upward.
Step 12
= 2
A MAX
B C MIN
= -2
D E F G MAX
H I J K L M MIN
1
alpha = 2, beta = unde ned, Evaluate M and propagate values upward.
7

8. Step 13
= 2
A MAX
= 1
B C MIN
1
D E F G MAX
H I J K L M MIN
alpha = 2, beta = 1, Prune G as evalF 1 2. Once G has been pruned we can
propagate values up to A.
Step 14
2
A MAX
1
B C MIN
D E F G MAX
H I J K L M MIN
The computer choses the move that leads to B.
8