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Game Trees – The Minimax Method. Anthony Brown COT 4810 January 25, 2008. Minimax. Mini mize your max imum losses Works best on: Complete information games Two Person games Zero-Sum games Opponent is logical. Minimax History. Money Money Money John Von Neumann Chess Chess Chess

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game trees the minimax method

Game Trees – The Minimax Method

Anthony Brown

COT 4810

January 25, 2008

minimax

Minimax

Minimize your maximum losses

Works best on:

Complete information games

Two Person games

Zero-Sum games

Opponent is logical

minimax history

Minimax History

Money Money Money

John Von Neumann

Chess Chess Chess

Claude Shannon

minimax4

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

minimax leaves evaluated

Minimax – Leaves evaluated

MAX

MIN

MAX

$

3

0

9

5

0

2

4

0

3

6

1

minimax max children

Minimax – Max(children)

MAX

MIN

MAX

3

$

9

5

2

4

6

1

3

0

9

5

0

2

4

0

3

6

1

minimax min children

Minimax – Min(children)

MAX

MIN

3

5

2

1

MAX

3

$

9

5

2

4

6

1

3

0

9

5

0

2

4

0

3

6

1

minimax final max children

Minimax – final Max(Children)

MAX

5

MIN

3

5

2

1

MAX

3

$

9

5

2

4

6

1

3

0

9

5

0

2

4

0

3

6

1

prisoner s dilemma non zero sum

Prisoner’s Dilemma – non-zero sum

MAX

Keep Quiet

Rat out Prisoner B

MIN

Keep Quiet

Rat out Prisoner A

Keep Quiet

Rat out Prisoner A

A

B

6 months/

6 months

10 years/

Go free

Go free/

10 years

5 years/

5 years

minimax pseudo code

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)

optimizations

OPTIMIZATIONS

Variable cutoff levels

Alpha Beta Pruning

Heuristics – Check power moves first

Transposition Tables

alpha beta pruning

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

9

C

large alpha beta tree

Large Alpha Beta Tree

Max

5

Min

2

5

4

Max

7

2

5

8

4

9

Min

3

7

2

5

2

8

4

9

1

2

4

3

8

7

9

2

5

6

2

8

4

9

1

2

6

alpha beta efficiency

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)

transposition tables

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

homework questions

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

bibliography

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/minimax.html

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