Game Trees – The Minimax Method

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# Game Trees - PowerPoint PPT Presentation

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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Presentation Transcript

Anthony Brown

COT 4810

January 25, 2008

### Minimax

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

Claude Shannon

### 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

MAX

MIN

MAX

\$

3

0

9

5

0

2

4

0

3

6

1

MAX

MIN

MAX

3

\$

9

5

2

4

6

1

3

0

9

5

0

2

4

0

3

6

1

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

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

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

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

Variable cutoff levels

Alpha Beta Pruning

Heuristics – Check power moves first

Transposition Tables

### 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

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

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

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

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

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