NationMaster - Encyclopedia: Correlated equilibrium

In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by the Robert Aumann (1974). The ideas is that a strategy set is chosen at random by some outside force. If neither player would want to deviate from the declared strategy, that strategy is correlated equilibrium. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, the Nash equilibrium (named after John Nash who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. ... Israel Robert John Aumann (×™×©×¨××œ ××•×ž×Ÿ) (born June 8, 1930) is an Israeli mathematician and a member of the United States National Academy of Sciences. ... [[|Diversity]] Binomial name Gallus gallus Trinomial name Type Species [[Image: ]] Synonyms A chicken (Gallus gallus) is a type of domesticated bird which is often raised as a type of poultry. ... Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, the Nash equilibrium (named after John Nash who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. ... Israel Robert John Aumann (×™×©×¨××œ ××•×ž×Ÿ) (born June 8, 1930) is an Israeli mathematician and a member of the United States National Academy of Sciences. ...

An example

Consider the game of chicken (pictured to the right). In this game two individuals are challenging each other to a contest where each can either dare or chicken out. If one is going to Dare, it is better for the other to chicken out. But if one is going to chicken out it is better for the other to Dare. This leads to an interesting situation where each wants to dare, but only if the other might chicken out. The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious harm unless one of them backs down. ...

In this game, there are three Nash equilibria. The two pure strategy Nash equilibria are (D, C) and (C, D). There is also a mixed strategy equilibrium where each player Dares with probability 1/3. In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... A pure strategy is a term used to refer to strategies in Game theory. ... A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ...

Now consider a third party (or some natural event) that draws one of three cards labeled: (C, C), (D, C), and (C, D). After drawing the card the third party informs the players of the strategy assigned to them on the card (but not the strategy assigned to their opponent). Suppose a player is assigned D, he would not want to deviate supposing the other player played their assigned strategy since he will get 7 (the highest payoff possible). Suppose a player is asigned C. Then the other player will play C with probability 1/2 and D with probability 1/2. The expected utility of Daring is 0(1/2) + 7(1/2) = 3.5 and the expected utility of chickening out is 2(1/2) + 6(1/2) = 4. So, the player would prefer to Chicken out. The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. ...

Since neither player has an incentive to deviate, this is a correlated equilibrium. Interestingly, 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.

Formal defintion

A probability distribution, $p(cdot)$ is a correlated equilibrium if for all strategies

s i

such that

p (s i) > 0

and for every alternative strategy

s i'

where $u_i(cdot)$ is i's utility function and

S - i

is the set of all possible strategies that i's opponents might take.

References

Category: Game theory Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ... In game theory, normal form is a way of describing a game. ... An extensive form game is a specification of a game in game theory. ... A cooperative game is a game wherein two or more players do not compete, but rather strive toward a unique objective and therefore win or lose as a group. ... In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed so far. ... A strategy in game theory is a sequence of activities and reactions, that fully determine an agents bahaviour in a game or a business situation. ... A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ... In game theory, dominance (also called strategic dominance) occurs when one strategy is better than another strategy for one player, no matter how that players opponents may play. ... In game theory, the Nash equilibrium (named after John Nash who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ... In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ... In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. ... Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ... In game theory, a sequential game is a game where one player chooses his action before the others chooses theirs. ... In game theory, a repeated game (or iterated game) is an extensive form game which consists in some number of repetitions of some base game (called a stage game). ... Signaling games are dynamic games with two players, the sender (S) and the receiver (R). ... Cheap Talk is a term used in Game Theory for pre-play communication which carries no cost. ... Zero-sum describes a situation in which a participants gain (or loss) is exactly balanced by the losses (or gains) of the other participant(s). ... Mechanism design is a sub-field of game theory. ... Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? Many points in this article may be difficult to understand without a background in the elementary concepts of game theory. ... The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious harm unless one of them backs down. ... In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ... The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. ... Matching Pennies is the name for a simple example game used in game theory. ... Minority Game is game proposed by Yi-Cheng Zhang and Damien Challet from the University of Fribourg and is inspired from the El Farol bar problem which is a simple mode that shows how selfish players cooperate with each other in the absence of communication. ... Rock, Paper, Scissors chart Rock, Paper, Scissors (also known by several other names) is a popular hand game most often played by children. ... Game theory studies strategic interaction between individuals in situations called games. ... Minimax is a method in decision theory for minimizing the expected maximum loss. ... In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is feasible if under that outcome the players minimax conditions are satisfied. ... In game theory, the Bishop-Cannings theorem proves that all members of a mixed evolutionarily stable strategy have the same payoff, and that none of these can also be a pure evolutionarily stable strategy. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Economics (from the Greek Î¿Î¯ÎºÎ¿Ï‚ [oikos], house, and Î½Î¿Î¼Î¿Ï‚ [nomos], rule, hence household management) is a social science that studies the production, distribution, trade and consumption of goods and services. ... Nobel Prize in Economics winner Daniel Kahneman, was an important figure in the development of behavioral finance and economics and continues to write extensively in the field. ... Evolutionary biology is a subfield of biology concerned with the origin and descent of species, as well as their change over time, i. ... Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ... Population genetics is the study of the distribution of and change in allele frequencies under the influence of the five evolutionary forces: natural selection, genetic drift, mutation, migration and nonrandom mating. ... Behavioral ecology is the study of the ecological and evolutionary basis for animal behavior, and the roles of behavior in enabling an animal to adapt to its environment (both intrinsic and extrinsic). ... This is a list of notable economists, mathematicians, political scientists, and computer scientists whose work has added substantially to the field of game theory. ...

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Formal defintion

References

An example