Game Trees

1
Game Trees

• William Dotson

2
Overview

• Definitions
• Minimax
• Alpha-Beta Pruning
• More Efficient Pruning
• Deep Blue Deep Fritz
• Conclusion

3
Definitions

• Directed Graph
• Nodes
• Paths
• Position Evaluator
• Easy vs Hard

4
Game Tree Complexity

• Tic-Tac-Toe 9! 362,280 states
• Connect Four 1013 states
• Checkers 1018 states
• Chess 1050 states
• Go 10170 states

5
Minimax

• 2 Players Max and Min
• Generate Graph to depth D
• Assign Values to Final Nodes
• Build Up Tree Alternating Max/Min

6
Static Position Evaluator
3
1
2
2
1
1
2
1
2
7
Static Position Evaluator
X
0
0
X
Evaluate from Xs Point of View State Value of 4
8
Minmax Example

• 3 Looks Ahead D 3

9
Minmax Example

• 3 Looks Ahead D 3

3
10
5
4
7
Max
10
Minmax Example

• 3 Looks Ahead D 3

4
5
Min
3
3
10
5
4
7
Max
11
Minmax Example

• 3 Looks Ahead D 3

Max
5
4
5
Min
3
3
10
5
4
7
Max
12
Minimax Pseudocode

• MinMax (GamePosition game)
  MinMove (GamePosition game)
• return MaxMove (game)
  best_move lt-
• 
  moves lt-
  GenerateMoves(game)
• 
  ForEach moves
• MaxMove (GamePosition game)
  move lt- MaxMove(ApplyMove(game))
• if (GameEnded(game))
  if(Value(move) lt
  Value(best_move))
• return EvalGameState(game)
  best_move lt- move
• 
  
• else
  
• best_move lt-
  return best_move
• moves lt- GenerateMoves(game)
  
• ForEach moves
• move lt- MinMove(ApplyMove(game))
• if (Value(move) gt Value(best_move))
• best_move lt- move
• return best_move

13
Alpha-Beta Pruning

• Modification of Minimax
• Next move needs consideration
• If worse then best, first move which
• opposition could take will be last move
• we have to look at.

14
Alpha-Beta Diagram
Player Opponent . .
. Player Opponent
M
n
General Case If M is is better than N for
Player, we will never get to N.
15
Minimax w/ Alpha Beta

• MinMax (GamePosition game)
  MinMove (GamePosition game)
• return MaxMove (game)
  best_move lt-
• 
  moves lt-
  GenerateMoves(game)
• 
  ForEach moves
• MaxMove (GamePosition game)
  move lt- MaxMove(ApplyMove(game))
• if (GameEnded(game))
  if(Value(move) lt
  Value(best_move))
• return EvalGameState(game)
  best_move lt- move
• 
  beta lt-
  Value(move)
• else
  
• best_move lt-
  
• moves lt- GenerateMoves(game)
  if(alpha gt beta)
• ForEach moves
  return best_move
• move lt- MinMove(ApplyMove(game))
  
• if (Value(move) gt Value(best_move))
  return best_move
• best_move lt- move
  
• alpha lt- Value(move)
  
• if(beta gt alpha)
• return best_move

16
Other Algorithms

• Try to avoid horizon affect
• Ignore paths that can be known to be wrong
• CCNS Controlled Conspiracy Node Search
• Target driven.
• Used in Ulysses a 1988 Chess Program.

17
Deep Blue/Deep Fritz

• Prunes Minimax Tree more intelligently
• Both use Targets like Ulysses
• Deep-Fritz uses Pattern Recognition
• Deep-Fritz has 1.3 brute power of Deep Blue, but
  plays at roughly the same level.

18
Conclusion

• Game Trees
• Minimax
• Alpha-Beta
• Moving Ahead

19
Resources

• Alpha Beta Pruning Nodes. http//sern.ucalgary.ca/
  courses/CPSC/533/W99/presentations/L2_5B_Lima_Neit
  z/search.html
• Stuart Russel, Peter Norvig. 1995.
• Minimax Trees. http//www.generation5.org/content/
  2001/minimax.asp James Matthews. 2001.
• Minimax Explained. http//ai-depot.com/LogicGames/
  MiniMax.html Paulo Pinto.
• Deep Fritz Draws Are Humans Getting Smarter, or
  Are Computers Getting Stupider?
  http//www.kurzweilai.net/articles/art0527.html?pr
  intable1 Ray Kurzweil 2002.