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Games can be a single round or repetitive. The approach a player takes in making their moves constitutes their strategy. Rules govern the outcome for the moves taken by the players, and outcomes produce payoffs for the players; rules and resulting payoffs can be expressed as decision trees or in a payoff matrix. Classical theory requires the ...
Normal form or payoff matrix of a 2-player, 2-strategy game The normal (or ... game theory is used to model the decision-making process of players, such as investors ...
An example prisoner's dilemma payoff matrix. William Poundstone described this "typical contemporary version" of the game in his 1993 book Prisoner's Dilemma: Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other.
"A best response to a coplayer’s strategy is a strategy that yields the highest payoff against that particular strategy". [9] 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.
Suppose a zero-sum game has a payoff matrix M where element M i,j is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column).
The game of Wald's maximin model is also a 2-person zero-sum game, but the players choose sequentially. With the establishment of modern decision theory in the 1950s, the model became a key ingredient in the formulation of non-probabilistic decision-making models in the face of severe uncertainty.
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 ...
The two pure strategy Nash equilibria are unfair; one player consistently does better than the other. The mixed strategy Nash equilibrium is inefficient: the players will miscoordinate with probability 13/25, leaving each player with an expected return of 6/5 (less than the payoff of 2 from each's less favored pure strategy equilibrium).