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In game theory, a bimatrix game is a simultaneous game for two players in which each player has a finite number of possible actions. The name comes from the fact that the normal form of such a game can be described by two matrices - matrix describing the payoffs of player 1 and matrix describing the payoffs of player 2.
The expected payoff for playing strategy 1 / 2 Y + 1 / 2 Z must be greater than the expected payoff for playing pure strategy X, assigning 1 / 2 and 1 / 2 as tester values. The argument for mixed strategy dominance can be made if there is at least one mixed strategy that allows for dominance.
In game theory, Kuhn's theorem relates perfect recall, mixed and unmixed strategies and their expected payoffs. It is named after Harold W. Kuhn.. The theorem states that in a game where players may remember all of their previous moves/states of the game available to them, for every mixed strategy there is a behavioral strategy that has an equivalent payoff (i.e. the strategies are equivalent).
A payoff function for a player is a mapping from the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility—often cardinal in the normal-form representation) of a player, i.e. the payoff function of a player takes as its input a ...
Coordination games also have mixed strategy Nash equilibria. In the generic coordination game above, a mixed Nash equilibrium is given by probabilities p = (d-b)/(a+d-b-c) to play Up and 1-p to play Down for player 1, and q = (D-C)/(A+D-B-C) to play Left and 1-q to play Right for player 2.
Given a normal form game and a parameter >, a totally mixed strategy profile is defined to be -proper if, whenever a player has two pure strategies s and s' such that the expected payoff of playing s is smaller than the expected payoff of playing s' (that is (,) < (′,)), then the probability assigned to s is at most times the probability assigned to s'.
In this situation, no player can unilaterally change their strategy to achieve a higher payoff, given the strategies chosen by the other players. For a Bayesian game, the concept of Nash equilibrium extends to include the uncertainty about the state of nature: Each player maximizes their expected payoff based on their beliefs about the state of ...
A matrix is used to present the payoff of both players in the game. For example, the best response of player one is the highest payoff for player one’s move, and vice versa. For player one, they will pick the payoffs from the column strategies. For player two, they will choose their moves based on the two row strategies.