Combinatorial game theory measuresgame complexity in several ways:

1. State-space complexity (the number of legal game positions from the initial position)
2. Game tree size (total number of possible games)
3. Decision complexity (number of leaf nodes in the smallest decision tree for initial position)
4. Game-tree complexity (number of leaf nodes in the smallest full-width decision tree for initial position)
5. Computational complexity (asymptotic difficulty of a game as it grows arbitrarily large)

These measures involve understanding the game positions, possible outcomes, andcomputational complexity of various game scenarios.

Thestate-space complexity of a game is the number of legal game positions reachable from the initial position of the game.^[1]

When this is too hard to calculate, anupper bound can often be computed by also counting (some) illegal positions (positions that can never arise in the course of a game).

Thegame tree size is the total number of possible games that can be played. This is the number ofleaf nodes in thegame tree rooted at the game's initial position.

The game tree is typically vastly larger than the state-space because the same positions can occur in many games by making moves in a different order (for example, in atic-tac-toe game with two X and one O on the board, this position could have been reached in two different ways depending on where the first X was placed). An upper bound for the size of the game tree can sometimes be computed by simplifying the game in a way that only increases the size of the game tree (for example, by allowing illegal moves) until it becomes tractable.

For games where the number of moves is not limited (for example by the size of the board, or by a rule about repetition of position) the game tree is generally infinite.

Adecision tree is a subtree of the game tree, with each position labelled "player A wins", "player B wins", or "draw" if that position can be proved to have that value (assuming best play by both sides) by examining only other positions in the graph. Terminal positions can be labelled directly—with player A to move, a position can be labelled "player A wins" if any successor position is a win for A; "player B wins" if all successor positions are wins for B; or "draw" if all successor positions are either drawn or wins for B. (With player B to move, corresponding positions are marked similarly.)

The following two methods of measuring game complexity use decision trees:

Decision complexity of a game is the number of leaf nodes in the smallest decision tree that establishes the value of the initial position.

Game-tree complexity of a game is the number of leaf nodes in the smallestfull-width decision tree that establishes the value of the initial position.^[1] A full-width tree includes all nodes at each depth. This is an estimate of the number of positions one would have to evaluate in aminimax search to determine the value of the initial position.

It is hard even to estimate the game-tree complexity, but for some games an approximation can be given by${\displaystyle GTC\ge b^d}$, whereb is the game's averagebranching factor andd is the number ofplies in an average game.

Thecomputational complexity of a game describes theasymptotic difficulty of a game as it grows arbitrarily large, expressed inbig O notation or as membership in acomplexity class. This concept doesn't apply to particular games, but rather to games that have beengeneralized so they can be made arbitrarily large, typically by playing them on ann-by-n board. (From the point of view of computational complexity, a game on a fixed size of board is a finite problem that can be solved in O(1), for example by a look-up table from positions to the best move in each position.)

The asymptotic complexity is defined by the most efficient algorithm for solving the game (in terms of whatevercomputational resource one is considering). The most common complexity measure,computation time, is always lower-bounded by the logarithm of the asymptotic state-space complexity, since a solution algorithm must work for every possible state of the game. It will be upper-bounded by the complexity of any particular algorithm that works for the family of games. Similar remarks apply to the second-most commonly used complexity measure, the amount ofspace orcomputer memory used by the computation. It is not obvious that there is any lower bound on the space complexity for a typical game, because the algorithm need not store game states; however many games of interest are known to bePSPACE-hard, and it follows that their space complexity will be lower-bounded by the logarithm of the asymptotic state-space complexity as well (technically the bound is only a polynomial in this quantity; but it is usually known to be linear).

• Thedepth-firstminimax strategy will use computation time proportional to the game's tree-complexity (since it must explore the whole tree), and an amount of memory polynomial in the logarithm of the tree-complexity (since the algorithm must always store one node of the tree at each possible move-depth, and the number of nodes at the highest move-depth is precisely the tree-complexity).
• Backward induction will use both memory and time proportional to the state-space complexity, as it must compute and record the correct move for each possible position.

Fortic-tac-toe, a simple upper bound for the size of the state space is 3⁹ = 19,683. (There are three states for each of the nine cells.) This count includes many illegal positions, such as a position with five crosses and no noughts, or a position in which both players have a row of three. A more careful count, removing these illegal positions, gives 5,478.^[2][3] And when rotations and reflections of positions are considered identical, there are only 765 essentially different positions.

To bound the game tree, there are 9 possible initial moves, 8 possible responses, and so on, so that there are at most 9! or 362,880 total games. However, games may take less than 9 moves to resolve, and an exact enumeration gives 255,168 possible games. When rotations and reflections of positions are considered the same, there are only 26,830 possible games.

The computational complexity of tic-tac-toe depends on how it isgeneralized. A natural generalization is tom,n,k-games: played on anm byn board with winner being the first player to getk in a row. This game can be solved inDSPACE(mn) by searching the entire game tree. This places it in the important complexity classPSPACE; with more work, it can be shown to bePSPACE-complete.^[4]

Due to the large size of game complexities, this table gives the ceiling of theirlogarithm to base 10. (In other words, the number of digits). All of the following numbers should be considered with caution: seemingly minor changes to the rules of a game can change the numbers (which are often rough estimates anyway) by tremendous factors, which might easily be much greater than the numbers shown.

• Go and mathematics
• Solved game
• Solving chess
• Shannon number
• list of NP-complete games and puzzles
• list of PSPACE-complete games and puzzles

• David Eppstein'sComputational Complexity of Games and Puzzles

• Combinatorial game theory
• Game theory

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