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Adversarial Search and Game Playing
Examples
Game Tree
MAX’s play
MIN’s play
Terminal state(win for MAX)
Here, symmetries have been used to reduce the branching factor
MIN nodes
MAX nodes
Game Tree
MAX’s play
MIN’s play
Terminal state(win for MAX)
In general, the branching factor and the depth of terminal states are largeChess:• Number of states: ~1040
• Branching factor: ~35• Number of total moves in a game: ~100
Example: Tic-tac-Toe
e(s) = number of rows, columns, and diagonals open for
MAX number of rows, columns, and diagonals open for MIN
88 = 0 64 = 2 33 = 0
Backing up Values
6-5=1
5-6=-15-5=0
5-5=0 6-5=1 5-5=0 4-5=-1
5-6=-1
6-4=25-4=1
6-6=0 4-6=-2
-1
-2
1
1Tic-Tac-Toe treeat horizon = 2 Best move
Continuation
0
1
1
1 32 11 2
1
0
1 1 0
0 2 01 1 1
2 22 3 1 2
Example
Example
= 2
2
The beta value of a MINnode is an upper bound onthe final backed-up value.It can never increase
Example
The beta value of a MINnode is an upper bound onthe final backed-up value.It can never increase
1
= 1
2
Example
= 1
The alpha value of a MAXnode is a lower bound onthe final backed-up value.It can never decrease
1
= 1
2
Example
= 1
1
= 1
2 -1
= -1
Example
= 1
1
= 1
2 -1
= -1
Search can be discontinued belowany MIN node whose beta value is less than or equal to the alpha valueof one of its MAX ancestors
Search can be discontinued belowany MIN node whose beta value is less than or equal to the alpha valueof one of its MAX ancestors
An example of Alpha-beta pruning
0
5 -3 3 3 -3 0 2 -2 3
max
max
max
min
min
00
0
0
0
-3-3
0
00
0
0
3
0
00
0
0
Final tree
0
5 -3 3 3 -3 0 2 -2 3
max
max
max
min
min
0
0
0
0
3
0
00
0
Example of Alpha-beta pruning
An example of Alpha-beta pruning
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0 -3
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0 -3
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0 -3
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0 -3 3
3
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0 -3 3
3
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
5
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
5
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
0
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
0
1
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
2
2
2
2
1
1
Example
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
1
2
2
2
2
1
Nondeterminstic games are the games with both an element of chance and Add chance nodes to tree
2 4 7 4 6 0 5 -2
0.5 0.5 0.5 0.5children
Expected value for chance node
P(child)utility(child)
Example with coin flip instead of dice
Example with coin flip instead of dice (cont.)
3
2
2 4
3
4
7 4
0
6 0
-2
5 -2
-1
0.5 0.5 0.5 0.5
44
Alpha-Beta prunning in Tic-Tac-Toe
