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Science , — CrossRef Google Scholar. Hare, F. Harrison, N. Hendrick, D. 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. 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.
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. 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.
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.
And correspondingly for positions with B to move. Decision complexity of a game is the number of leaf nodes in the smallest decision tree that establishes the value of the initial position. 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. 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.
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:. 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.
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. 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 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. 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 of space or computer 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 be PSPACE-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.
There are three states for each cell and 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, To bound the game tree, there are 9 possible initial moves, 8 possible responses, and so on, so that there are at most 9! However, games may take less than 9 moves to resolve, and an exact enumeration gives , possible games. When rotations and reflections of positions are considered the same, there are only 26, possible games. The computational complexity of tic-tac-toe depends on how it is generalized. A natural generalization is to m , n , k -games: played on an m by n board with winner being the first player to get k in a row.
Due to the large size of game complexities, this table gives the ceiling of their logarithm to base 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.
Anonymous Not logged in Create account Log in. Hand W iki. From HandWiki. Namespaces Page Discussion. More More Languages. Short description : Notion in combinatorial game theory. The bridge table can be regarded as having one slot for each player and trick to play a card in, which corresponds to board size Game-tree complexity is a very weak upper bound: 13! State-space complexity is for one given deal; likewise regardless of legality but with many transpositions eliminated.
Note that the last 4 plies are always forced moves with branching factor 1. Acta Informatica 13 1 : 59— While the solar lectures build on each other to some extent, the lunar lectures are episodic and can be read independently of each other. Written in a relaxed style, the author uses his didactic expertise to guide the reader through the theory in an insightful and enjoyable manner. No background in game theory is assumed, making the whole text informative and accessible to a wide audience.
This monograph gives the reader an excellent introduction to the basics of the subject and highlights some of the most recent breakthroughs in research. It provides the reader with a launch pad for further research. Export citation Select the format to use for exporting the citation. Computational complexity , Algorithmic game theory.
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