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Game Trees

Game Trees. Ryan Wilson. Chapter 6. Trees – A Review. Trees are a collection of nodes A Tree has a root node and zero or more sub-trees. Root Node. A. B. E. G. C. D. F. H. I. An Example Tree. Common Traversal Algorithm. Function TraverseTree( node ) visit( node )

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Game Trees

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  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 B E 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 O X X X O O X X O X O X

  8. Game Trees • 3 Components to Implementing a Game Tree: • Tree Generator • Position Evaluator • Minimax Method

  9. X Position Evaluator O O X X X … X O O O X X 0 X O X X X … O X O X X 1 = Computer Wins 0 = Draw -1 = Human Wins O O X O O O -1 1

  10. REMEMBER! Higher Score is better for the computer! 40 MAX Computer’s Turn 40 20 MIN Human’s 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 > 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 < bestValue ) bestValue = value end for End function

  13. Demonstration • Tic-Tac-Toe!

  14. Conclusions • In Tic-Tac-Toe, there’s only about 27 legal moves for the computer to consider • That’s about 30,000 moves • But it’s still kind of slow… • Almost 45 seconds to make the first move! • Experts estimate 10100 legal moves in Chess! • That’s 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 • Don’t 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! >40 MAX Computer’s Turn 40 <20 MIN Human’s Turn 40 30 60 20

  18. Beta-Pruning REMEMBER! Higher Score is better for the computer! <100 MIN Human’s Turn 100 >120 MAX Computer’s 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. O O X X X X O O X X X X

  23. Transposition Tables • Once we’ve calculated the value of a given position, we save it in a table so it’s 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++

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