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