1. Mini-Max search Alpha-Beta pruning General concerns on games
Game Playing
Mini-Max search
Alpha-Beta pruning
General concerns on games

2. Why study board games ?
One of the oldest subfields of AI (Shannon and Turing, 1950)
Abstract and pure form of competition that seems to require intelligence
Easy to represent the states and actions
Very little world knowledge required !
Game playing is a special case of a search problem, with some new requirements.

3. Types of games Chance Deterministic Imperfect information
Chess, checkers, go, othello
Backgammon, monopoly
Sea battle
Bridge, poker, scrabble, nuclear war

4. Why new techniques for games?
“Contingency” problem:
We don’t know the opponents move !
The size of the search space:
Chess : ~15 moves possible per state, 80 ply
1580 nodes in tree
Go : ~200 moves per state, 300 ply
nodes in tree
Game playing algorithms:
Search tree only up to some depth bound
Use an evaluation function at the depth bound
Propagate the evaluation upwards in the tree

5. MINI MAX Restrictions: 2 players: MAX (computer) and MIN (opponent)
deterministic, perfect information
Select a depth-bound (say: 2) and evaluation function
- Construct the tree up till
the depth-bound
MAX
MIN
Select
this move 3
- Compute the evaluation
function for the leaves 2 1 3
- Propagate the evaluation
function upwards:
- taking minima in MIN
- taking maxima in MAX 2 5 3 1 4

6. The MINI-MAX algorithm:
Initialise depthbound;
Minimax (board, depth) =
IF depth = depthbound
THEN return static_evaluation(board);
ELSE IF maximizing_level(depth)
THEN FOR EACH child child of board
compute Minimax(child, depth+1);
return maximum over all children;
ELSE IF minimizing_level(depth)
return minimum over all children;
Call: Minimax(current_board, 0)

7. Alpha-Beta Cut-off Generally applied optimization on Mini-max.
Instead of:
first creating the entire tree (up to depth-level)
then doing all propagation
Interleave the generation of the tree and the propagation of values.
Point:
some of the obtained values in the tree will provide information that other (non-generated) parts are redundant and do not need to be generated.

8. Alpha-Beta idea: Principles:
generate the tree depth-first, left-to-right
propagate final values of nodes as initial estimates for their parent node.
MIN
MAX 2 2
- The MIN-value (1) is already
smaller than the MAX-value of
the parent (2) 1
- The MIN-value can only
decrease further, 2 =2 1
- The MAX-value is only allowed
to increase, 5
- No point in computing further
below this node

9. Terminology: - The (temporary) values at MAX-nodes are ALPHA-values
- The (temporary) values at MIN-nodes are BETA-values
MIN
MAX 2 2 5 =2 2 1 1
Alpha-value
Beta-value

10. The Alpha-Beta principles (1):
- If an ALPHA-value is larger or equal than the Beta-value of a descendant node:
stop generation of the children of the descendant
MIN
MAX 2 2 5 =2 2 1 1
Alpha-value
Beta-value 

11. The Alpha-Beta principles (2):
- If an Beta-value is smaller or equal than the Alpha-value of a descendant node:
stop generation of the children of the descendant
MIN
MAX 2 2 5 =2 2 3 1
Alpha-value
Beta-value 

12. Mini-Max with  at work:8 7 3 9 1 6 2 4 5
 4 16
 5 31 39
= 5
MAX 6
 8
 5 23 15
= 4 30
= 5
 3 38
MIN 33
 1 2
 8 10
 2 18
 1 25
 3 35
 2 12
 4 20
 3 5
= 8 8
 9 27
 9 29
 6 37
= 3 14
= 4 22
= 5
MAX 1 3 4 7 9 11 13 17 19 21 24 26 28 32 34 36
11 static evaluations saved !!

13. “DEEP” cut-offs - For game trees with at least 4 Min/Max layers:
the Alpha - Beta rules apply also to deeper levels. 4
 4
 4 2
 2

14. The Gain: Best case:
- If at every layer: the best node is the left-most one
MAX
MIN
Only THICK is explored

15. Example of a perfectly ordered tree
MAX
MIN 21

16. How much gain ? # (static evaluations) = - Alpha / Beta : best case :
2 bd/ (if d is even)
b(d+1)/2 + b(d-1)/ (if d is odd)
- The proof is by induction.
- In the running example: d=3, b=3 : !

17. Best case gain pictured:10
100
1000
10000
100000 1 2 3 4 5 6 7
# Static evaluations
Depth
No pruning
b = 10
Alpha-Beta
Best case
- Note: a logarithmic scale.
- Conclusion: still exponential growth !!
- Worst case??
For some trees alpha-beta does nothing,
For some trees: impossible to reorder to avoid cut-offs

18. The horizon effect. Because of the depth-bound
Queen lost
Pawn lost
horizon = depth bound
of mini-max
Because of the depth-bound
we prefer to delay disasters, although we don’t prevent them !!
solution: heuristic continuations

19. Time bounds: How to play within reasonable time bounds?
Even with fixed depth-bound, times can vary strongly!
Solution:
Iterative Deepening !!!
Remember: overhead of previous searches = 1/b
Good investment to be sure to have a move ready.

20. Games of chance
Ex.: Backgammon:
Form of the game tree:

21. State of the art Drawn from an article by Mathew Ginsberg,
Scientific American, Winter 1998, Special Issue on Exploring Intelligence

22. State of the art (2)

23. State of the art (3)

24. Win of deep blue predicted:
Computer chess ratings studied around 90ies:
1500
2000
2500
3000
3500 2 4 6 8 10 12 14
Chess Rating
Depth in ply
Kasparov ?
Further increase of depth was likely to win !