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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.
This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the payoff pair of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell – then the cell represents a Nash equilibrium.
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 ...
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).
In game theory, a Bayesian game is a strategic decision-making model which assumes players have incomplete information. Players may hold private information relevant to the game, meaning that the payoffs are not common knowledge. [1] Bayesian games model the outcome of player interactions using aspects of Bayesian probability.
The best-known example of a 2-player anti-coordination game is the game of Chicken (also known as Hawk-Dove game). Using the payoff matrix in Figure 1, a game is an anti-coordination game if B > A and C > D for row-player 1 (with lowercase analogues b > d and c > a for column-player 2). {Down, Left} and {Up, Right} are the two pure Nash equilibria.
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 ...