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In game theory, fictitious play is a learning rule first introduced by George W. Brown. In it, each player presumes that the opponents are playing stationary (possibly mixed) strategies. In it, each player presumes that the opponents are playing stationary (possibly mixed) strategies.
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.
Matching pennies is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium. [1] This game has no pure strategy Nash equilibrium since there is no pure strategy (heads or tails) that is a best response to a best response. In other words, there is no pair of pure strategies such that neither player ...
The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior, but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of ...
The main problem with these games falls into one of two categories: (1) various mixed strategies of the game are purified by different sequences of perturbed games and (2) some mixed strategies of the game involve weakly dominated strategies. No mixed strategy involving a weakly dominated strategy can be purified using this method because if ...
Set up the inequality to determine whether the mixed strategy will dominate the pure strategy based on expected payoffs. u 1 / 2 Y + u 1 / 2 Z ⩼ u X. 4 + 5 > 5 Mixed strategy 1 / 2 Y and 1 / 2 Z will dominate pure strategy X for Player 2, and thus X can be eliminated from the rationalizable strategies for P2.
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. This allows for a player to randomly select a pure strategy. (See the following section for an illustration.)
The expected payoff for this equilibrium is 7(1/3) + 2(1/3) + 6(1/3) = 5 which is higher than the expected payoff of the mixed strategy Nash equilibrium. The following correlated equilibrium has an even higher payoff to both players: Recommend ( C , C ) with probability 1/2, and ( D , C ) and ( C , D ) with probability 1/4 each.
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