511 Week 3
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This document discusses different aspects of game complexity, including state-space complexity (the number of possible positions in a game), game tree size (the number of possible end-states reached through different paths), and decision complexity (how difficult it is to determine the best move). It provides examples of these measures for simple games like tic-tac-toe (765 possible positions) and complex games like chess (around 10^43 possible positions). The document also discusses game tree complexity and computational complexity in more depth.
511 Week 3
- 1. Games and Game Complexity Alan Nochenson IST 511 Sept 8, 2012
- 2. Combinatorial game theory Sequential games Perfect information Win, lose, or draw (ordinal utility) Examples include chess, checkers, tic tac toe, nim
- 3. State-space complexity The number of legal positions possible in a game For tic-tac-toe, it is 765 (removing rotations) For chess, it is ~10^43
- 4. Game tree size The number of leaf nodes in the tree representation (end-states reached in different ways are counted twice) 26, 830 in tic-tac-toe (removing rotations)
- 5. Decision complexity How “hard” it is to decide on a given move, or how many moves deep in the tree you need to look to be sure of the outcome
- 6. Game tree and Computational complexity Game tree complexity The number of leaf nodes of a full-width game (equivalently how many leaf nodes are in the level you need to reach to get a minimax score) GTC = b^d where b is the average branching factor, and d is the game tree depth (in plys) Computational complexity Asymptotic difficulty of a game (big-O notation) Play a generalized version of the game (n x n)
- 7. Works Cited General information from http://en.wikipedia.org/wiki/Game_complexity State-space complexity: Facts from http://www.methodshop.com/games/play/tictactoe/index.shtml Picture from http://www.gameideasforkids.com/images/tictactoe.JPG Game tree size picture from http://scienceblogs.com/goodmath/wp- content/blogs.dir/476/files/2012/04/i- 30601eb10fe21a4ce5a4f2f92e80eb10-tic-tac-toe.png Decision complexity picture from http://www.cs.berkeley.edu/~ddgarcia/teaching/CS3Gamesman/assignment/ttt branch.gif Fact from http://www.nature.com/news/2007/070716/full/news070716-13.html