Game Trees

1
Game Trees

• Ryan Wilson

Chapter 6
2
Trees A Review

• Trees are a collection of nodes
• A Tree has a root node and zero or more sub-trees

3
Root Node
A
E
B
G
C
D
F
H
I
An Example Tree
4
CommonTraversalAlgorithm

• Function TraverseTree( node )
• visit( node )
• for each child of node
• TraverseTree( child )
• end for
• End Function
• TraverseTree( root )

5
Computer Games

• Problem How do we make an intelligent opponent
  for a human player to compete against?
• One Possibility
• Game Trees

6
Game Trees

• Treat each state of the game as a node in a tree
• Search the tree to find the best move

7
X
X
X
X
O
O
O
X
X
O
O
X
X
8
Game Trees

• 3 Components to Implementing a Game Tree
• Tree Generator
• Position Evaluator
• Minimax Method

9
X
Position Evaluator
X
X
O
O
X

O
X
O
O
X
X
0
X
X
X
O
X

O
O
X
X
X
1 Computer Wins 0 Draw -1 Human Wins
X
O
O
O
O
O
1
-1
10
REMEMBER! Higher Score is better for the computer!
40
MAX Computers Turn
40
20
MIN Humans Turn
40
30
60
20
11
Minimax - Computer

• Function FindComputerMove(GameStateNode)
• var bestValue
• if( terminal )
• return evaluate(GameStateNode)
• for each child of GameStateNode
• value FindHumanMove(child)
• if( value gt bestValue )
• bestValue value
• end for
• End function

12
Minimax - Human

• Function FindHumanMove(GameStateNode)
• var bestValue
• if( terminal )
• return evaluate(GameStateNode)
• for each child of GameStateNode
• value FindHumanMove(child)
• if( value lt bestValue )
• bestValue value
• end for
• End function

13
Demonstration

• Tic-Tac-Toe!

14
Conclusions

• In Tic-Tac-Toe, theres only about 27 legal moves
  for the computer to consider
• Thats about 30,000 moves
• But its still kind of slow
• Almost 45 seconds to make the first move!
• Experts estimate 10100 legal moves in Chess!
• Thats approx. 2333 legal positions!

15
What can we do?

• Apply the position evaluator to non-terminal
  nodes
• Limit the depth of your search through the tree
• Have to estimate the value of a position
• Alpha-Beta Pruning
• Trim un-needed portions of the tree
• Transposition Tables

16
Alpha-Beta Pruning

• Dont need to look at all subtrees of a given
  node if you can already know which one is best

17
Alpha-Pruning
REMEMBER! Higher Score is better for the computer!
gt40
MAX Computers Turn
40
lt20
MIN Humans Turn
40
30
60
20
18
Beta-Pruning
REMEMBER! Higher Score is better for the computer!
lt100
MIN Humans Turn
100
gt120
MAX Computers Turn
100
40
120
30
19
Demonstration

• Tic-Tac-Toe!
• With Alpha-Beta Pruning

20
Conclusions

• A definite improvement!
• First move only takes about 4 seconds (about 2000
  moves)

21
Transposition Tables

• In many games, there are multiple ways to arrive
  at the same board position

22
X
X
O
X
O
X
X
X
O
X
O
X
23
Transposition Tables

• Once weve calculated the value of a given
  position, we save it in a table so its easy to
  look up.
• For each position in the tree, look that position
  up in the table, if it exists, return the value
  stored in the table.

24
Demonstration

• Tic-Tac-Toe!
• With Alpha-Beta Pruning
• With Transposition Tables

25
Conclusions

• Another improvement!
• First move only takes about 3 seconds (918 nodes)
• Almost ½ as many nodes!

26
Game Trees

• How-To of Game Trees
• Alpha-Beta Pruning
• Transposition Tables

27
References

• DewdneyThe New Turing Omnibus
• WeissAlgorithms, Data Structures, and Problem
  Solving with C
• WeissData Structures Algorithm Analysis in C