Game playing

1
Game playing
2
Type of games
Here we consider Games with Perfect Information
3
Example - Tic-Tac-Toe
0
-10
10
0
-10
10
10
-1
10
0
0
0
4
Minimax search
MAX node
MIN node
value computed by minimax
f value
5
Minimax search
6
Minimax algorithm
7
Properties of minimax

• Complete? Yes (if tree is finite)
• Optimal? Yes (against an optimal opponent)
• Time complexity? O(bm)
• Space complexity? O(bm) (depth-first exploration)
  
• For chess, b 35, m 100 for "reasonable"
  games? exact solution completely infeasible

8
Alpha-beta pruning

• We can improve on the performance of the minimax
  algorithm through alpha-beta pruning.

• We dont need to compute the value at this node.
• No matter what it is it cant effect the value of
  the root node.

2
1
2
7
1
?
9
Alpha-beta pruning
10
Alpha-beta pruning
11
Alpha-beta pruning
12
Why is it called a-ß?

• a is the value of the best (i.e., highest-value)
  choice found so far at any choice point along the
  path for max If v is worse than a, max will avoid
  it ? prune that branch Define ß similarly for min

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The Alpha-Beta Procedure

• There are two rules for terminating search
• Search can be stopped below any MIN node having a
  beta value less than or equal to the alpha value
  of any of its MAX ancestors.
• Search can be stopped below any MAX node having
  an alpha value greater than or equal to the beta
  value of any of its MIN ancestors.

14
The a-ß algorithm
15
The a-ß algorithm
16
Properties of a-ß

• Pruning does not affect final result
• Good move ordering improves effectiveness of
  pruning
• With "perfect ordering," time complexity
  O(bm/2)
• ? doubles depth of search

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The Alpha-Beta Procedure
Example
max
min
max
min

























18
The Alpha-Beta Procedure
Example
max
min
max
min
? 4







4

















19
The Alpha-Beta Procedure
Example
max
min
max
min
? 4

5





4

















20
The Alpha-Beta Procedure
Example
max
min
max
? 3
min
? 3

5





4
3
















21
The Alpha-Beta Procedure
Example
max
min
max
? 3
min
? 3
? 1

5





4
3
1















22
The Alpha-Beta Procedure
Example
max
min
? 3
max
? 3
min
? 3
? 1
? 8

5





4
3
1

8













23
The Alpha-Beta Procedure
Example
max
min
? 3
max
? 3
min
? 3
? 1
? 6

5

6



4
3
1

8













24
The Alpha-Beta Procedure
Example
max
min
? 3
max
? 3
? 6
min
? 3
? 1
? 6

5

6



4
3
1

8
7












25
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
max
? 3
? 6
min
? 3
? 1
? 6

5

6


4
3
1

8
7










26
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
max
? 3
? 6
min
? 3
? 1
? 6
? 2

5

6


4
3
1

8
7
2









27
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
max
? 3
? 6
? 3
min
? 3
? 1
? 6
? 2

5

6


4
3
1

8
7
2









28
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
max
? 3
? 6
? 3
min
? 3
? 1
? 6
? 2
? 5

5

6


4
3
1

8
7
2

5







29
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
max
? 3
? 6
? 3
min
? 3
? 1
? 6
? 2
? 4

5

6

4
4
3
1

8
7
2

5







30
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
? 4
max
? 3
? 6
? 4
min
? 3
? 1
? 6
? 2
? 4

5

6

4
4
3
1

8
7
2

5
4






31
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
? 4
max
? 3
? 6
? 4
min
? 3
? 1
? 6
? 2
? 4
? 6

5

6

4
4
3
1

8
7
2

5
4


6



32
The Alpha-Beta Procedure
Example
? 3
max
min
? 3
? 4
max
? 3
? 6
? 4
min
? 3
? 1
? 6
? 2
? 4
? 6

5

6

4
4
3
1

8
7
2

5
4
7

6



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The Alpha-Beta Procedure
Example
? 4
max
min
? 3
? 4
max
? 3
? 6
? 4
? 6
min
? 3
? 1
? 6
? 2
? 4
? 6

5

6

4
4
3
1

8
7
2

5
4
7

6
7


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The Alpha-Beta Procedure
Example
? 4
max
Done!
min
? 3
? 4
max
? 3
? 6
? 4
? 6
min
? 3
? 1
? 6
? 2
? 4
? 6

5

6

4
4
3
1

8
7
2

5
4
7

6
7


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Games that include chance

• Possible moves (5-10,5-11), (5-11,19-24),(5-10,10-
  16) and (5-11,11-16)

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Games that include chance
chance nodes

• Possible moves (5-10,5-11), (5-11,19-24),(5-10,10-
  16) and (5-11,11-16)
• 1,1, 6,6 chance 1/36, all other chance 1/18

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Games that include chance

• 1,1, 6,6 chance 1/36, all other chance 1/18
• Cannot calculate definite minimax value, only
  expected value

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Expected minimax value

• EXPECTED-MINIMAX-VALUE(n)
• UTILITY(n) If n is a terminal
• maxs ? successors(n) MINIMAX-VALUE(s) If n
  is a max node
• mins ? successors(n) MINIMAX-VALUE(s) If n
  is a max node
• ?s ? successors(n) P(s) . EXPECTEDMINIMAX(s)
  If n is a chance node
• These equations can be backed-up recursively all
  the way to the root of the game tree.

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Position evaluation with chance nodes

• Left, A1 wins
• Right A2 wins
• Outcome of evaluation function may change when
  values are scaled differently.
• Behavior is preserved only by a positive linear
  transformation of EVAL.

40
Alpha-beta pruning in games with chance

• We can improve on the performance of the minimax
  algorithm for games with chance through
  alpha-beta pruning.

• We dont need to compute the value at these nodes.

? 1
? 2.5
? 0.5
Suppose -5 ? e(n) ? 5

• 3 0.5 (- 1) 0.5 1.0