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