511 week 3
TRANSCRIPT
Games and Game
ComplexityAlan Nochenson
IST 511
Sept 8, 2012
Combinatorial game theory Sequential games
Perfect information
Win, lose, or draw (ordinal utility)
Examples include chess, checkers, tic tac toe, nim
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
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)
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
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)
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/tttbranch.gif
Fact from http://www.nature.com/news/2007/070716/full/news070716-13.html