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In game theory, Zermelo's theorem is a theorem about finite two-person games of perfect information in which the players move alternately and in which chance does not affect the decision making process. It says that if the game cannot end in a draw, then one of the two players must have a winning strategy (i.e. can force a win).
The most commonly quoted number for the number of possible games, 10 700 [14] is derived from a simple permutation of 361 moves or 361! = 10 768. Another common derivation is to assume N intersections and L longest game for N L total games. For example, 400 moves, as seen in some professional games, would be one out of 361 400 or 1 × 10 1023 ...
In economics and game theory, the decisions of two or more players are called strategic complements if they mutually reinforce one another, and they are called strategic substitutes if they mutually offset one another. These terms were originally coined by Bulow, Geanakoplos, and Klemperer (1985).
For the purposes of the Sprague–Grundy theorem, a game is a two-player sequential game of perfect information satisfying the ending condition (all games come to an end: there are no infinite lines of play) and the normal play condition (a player who cannot move loses). At any given point in the game, a player's position is the set of moves ...
The ingredients of a stochastic game are: a finite set of players ; a state space (either a finite set or a measurable space (,)); for each player , an action set (either a finite set or a measurable space (,)); a transition probability from , where = is the action profiles, to , where (,) is the probability that the next state is in given the current state and the current action profile ; and ...
In game theory a move by nature is a decision or move in an extensive form game made by a player who has no strategic interests in the outcome. The effect is to add a player, "Nature", whose practical role is to act as a random number generator. For instance, if a game of Poker requires a dealer to choose which cards a player is dealt, the ...
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This game is finite, and the total number of moves (and thus the game's winner) is predetermined by the initial number of crosses: the players cannot affect the result by their play. Thus, this variant may be termed, after Conway's categorisation of mathematics itself, a "one player game". Each move removes two free ends and introduces two more.