Game Trees The Minimax Method

1
Game Trees The Minimax Method

• Anthony Brown
• COT 4810
• January 25, 2008

2
Minimax

• Minimize your maximum losses
• Works best on
• Complete information games
• Two Person games
• Zero-Sum games
• Opponent is logical

3
Minimax History

• Money Money Money
• John Von Neumann
• Chess Chess Chess
• Claude Shannon

4
Minimax

• Generate all nodes in a game tree
• Score each leaf node
• Score each MIN node with min(children)
• Score each MAX node with max(children)

MAX (Player)
MIN
MAX
5
Minimax Leaves evaluated
MAX
MIN

MAX
3
0
9
5
0
2
3
6
4
0
1
6
Minimax Max(children)
MAX
MIN
3

9
5
2
4
6
1
MAX
3
0
9
5
0
2
3
6
4
0
1
7
Minimax Min(children)
MAX
3
5
2
1
MIN
3

9
5
2
4
6
1
MAX
3
0
9
5
0
2
3
6
4
0
1
8
Minimax final Max(Children)
5
MAX
3
5
2
1
MIN
3

9
5
2
4
6
1
MAX
3
0
9
5
0
2
3
6
4
0
1
9
Prisoners Dilemma non-zero sum
MAX
Rat out Prisoner B
Keep Quiet
MIN
Rat out Prisoner A
Keep Quiet
Rat out Prisoner A
Keep Quiet
6 months/ 6 months
10 years/ Go free
Go free/ 10 years
5 years/ 5 years
A B
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Minimax pseudo code

• If N is a leaf
• return eval(N)
• If this level is a MAX
• return max(children)
• If this level is a MIN
• return min(children)

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OPTIMIZATIONS

• Variable cutoff levels
• Alpha Beta Pruning
• Heuristics Check power moves first
• Transposition Tables

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Alpha beta pruning

• Post-order traversal of the tree
• Alpha recorded as the largest value scanned by
  a MAX node
• Beta recorded as the smallest value scanned by
  a MIN node
• Prune when Alpha rises above Beta
• Or
• Prune when Beta falls below Alpha

Max
?
Min
A
D
Max
7
B
4
E
C
9
13
Large Alpha Beta Tree
Max
5
Min
2
5
4
Max
7
2
5
8
4
9
Min
2
7
3
2
9
1
8
2
5
4
8
4
3
9
7
2
8
2
6
5
9
1
6
2
4
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Alpha Beta Efficiency

• Pruning can reduce the number of leaf nodes
  scanned by the square root of non-pruned Minimax
  trees.
• Best Case
• O(bd/2)
• b of children
• d of recursions (turns)

15
Transposition Tables

• Used when many moves can reach the same position
• Positions are stored in a hash table for speed
• Takes up much more memory

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Homework Questions

• Find the path to the best position using the
  Minimax algorithm on this tree.
• How could Minimax fail in the real world?
• Why is tic-tac-toe solvable and chess unsolvable
  with Minimax?

1
2
3
4
0
3
1
0
1
7
9
7
5
8
3
1
17
Bibliography

• Dewdney, A. K. The New Turing Omnibus. New York
  Henry Holt, 1989. 36-41.
• Shah, Rajiv Bakulesh "Minimax", in Dictionary of
  Algorithms and Data Structures online, Paul E.
  Black, ed., U.S. National Institute of Standards
  and Technology. 10 January 2007. (accessed TODAY)
  Available from http//www.nist.gov/dads/HTML/mini
  max.html