Similar remarks apply to the second-most commonly used complexity measure, the amount of space or computer memory used by the computation. It will be upper-bounded by the complexity of any particular algorithm that works for the family of games. The asymptotic complexity is defined by the most efficient (in terms of whatever computational resource one is considering) algorithm for solving the game 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. (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.) This concept doesn't apply to particular games, but rather to games that have been generalized so they can be made arbitrarily large, typically by playing them on an n-by- n board. The computational complexity of a game describes the asymptotic difficulty of a game as it grows arbitrarily large, expressed in big O notation or as membership in a complexity class. It is hard even to estimate the game-tree complexity, but for some games an approximation can be given by raising the game's average branching factor b to the power of the number of plies d in an average game, or: This is an estimate of the number of positions one would have to evaluate in a minimax search to determine the value of the initial position. A full-width tree includes all nodes at each depth. The game-tree complexity of a game is the number of leaf nodes in the smallest full-width decision tree that establishes the value of the initial position. And correspondingly for positions with B to move.)ĭecision complexity of a game is the number of leaf nodes in the smallest decision tree that establishes the value of the initial position. (Terminal positions can be labelled directly a position with player A to move can be labelled 'player A wins' if any successor position is a win for A, or labelled 'player B wins' if all successor positions are wins for B, or labelled 'draw' if all successor positions are either drawn or wins for B. The next two measures use the idea of a decision tree, which is a subtree of the game tree, with each position labelled with 'player A wins', 'player B wins' or 'drawn', if that position can be proved to have that value (assuming best play by both sides) by examining only other positions in the graph.

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.įor 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. 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 a tic-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). The game tree size is the total number of possible games that can be played: the number of leaf nodes in the game tree rooted at the game's initial position. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game. The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. Measures of game complexity State-space complexity 3 Complexities of some well-known games.2 Example: tic-tac-toe (noughts and crosses).

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