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 These measures involve understanding game positions, possible outcomes, and computation required for various game scenarios.
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