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A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly.This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory or computer assistance.
The strategy-stealing argument shows that the second player cannot win, by means of deriving a contradiction from any hypothetical winning strategy for the second player. The argument is commonly employed in games where there can be no draw, by means of the law of the excluded middle .
A player's strategy set is the set of pure strategies available to that player. A mixed strategy is an assignment of a probability to each pure strategy. When enlisting mixed strategy, it is often because the game does not allow for a rational description in specifying a pure strategy for the game.
A straightforward example of maximizing payoff is that of monetary gain, but for the purpose of a game theory analysis, this payoff can take any desired outcome—cash reward, minimization of exertion or discomfort, or promoting justice can all be modeled as amassing an overall “utility” for the player.
In theory, they examine all positions / nodes, where each move by one player is called a "ply". This search continues until a certain maximum search depth or the program determines that a final "leaf" position has been reached.
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists.
In 1994, L. Victor Allis raised the algorithm of proof-number search (pn-search) and dependency-based search (db-search), and proved that when starting from an empty 15×15 board, the first player has a winning strategy using these searching algorithms. [28] This applies to both free-style gomoku and standard gomoku without any opening rules.
There exist pairs of games, each with a higher probability of losing than winning, for which it is possible to construct a winning strategy by playing the games alternately. Parrondo devised the paradox in connection with his analysis of the Brownian ratchet , a thought experiment about a machine that can purportedly extract energy from random ...