| Combinatorial Game Theory |
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Applying CGT to a ''position'' attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is tortuously difficult unless the game is very simple. CGT should not be confused with another mathematical theory, traditionally called Game Theory , used in the theory of economic competition and cooperation. Game theory includes games of chance, games of imperfect knowledge and games in which players move simultaneously. HISTORY CGT arose in relation to Nim , which can be solved completely. Nim is an Impartial Game for two players, and subject to the ''normal play condition'', which means that a player who cannot move loses. In the 1930s , the Sprague–Grundy Theorem showed that all impartial games are equivalent to heaps in nim, thus showing that major unifications are possible in games considered at a Combinatorial level (in which detailed strategies matter, not just pay-offs). The theory introduced in the 1960s of Partizan Game s extended the impartial theory, by relaxing the condition that a play available to one player be available to both. It was pioneered by Elwyn R. Berlekamp , John H. Conway and Richard K. Guy in their book '' Winning Ways For Your Mathematical Plays ''. Some of the inspiration (for the use in particular of disjunctive sums of games) was based on Conway's observation of the play in Go endgames. His book '' On Numbers And Games '', which introduces the concept of Surreal Number and its generalization to games, was published ahead of ''Winning Ways'', though based in part on the same collaboration. EXAMPLES The introductory text ''Winning Ways'' introduced a large number of games, but the following were used as motivating examples for the introductory theory:
The classic game Go was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and ''temperature'' theory for it (see references). Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way. OVERVIEW A game, in its simplest terms, is a list of possible "moves" that two players, called ''left'' and ''right'', can make. Every move is in fact, another game, such that each game can be considered a single state that the game can exist in. | ||
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