NationMaster - Encyclopedia: Rationalizability

In game theory, rationalizability or rationalizable equilibria is a solution concept which generalizes Nash equilibrium. The general idea is to provide the weakest constraints on players while still requiring rational players. It was first discovered independently by Bernheim (1984) and Pearce (1984). 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. ... Matching Pennies is the name for a simple example game used in 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 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. ... In economics and game theory, the participants are sometimes considered to have perfect rationality: that is, they always act in a rational way, and are capable of arbitrarily complex deductions towards that end. ...

Constraints on beliefs

Consider a simple coordination game (the payoff matrix is to the right). The row player can play a if she can reasonably believe that the column player could play A, since a is a best response to A. She can reasonably believe that the column player can play A if it is reasonable for column to believe that the row player could play a. He can believe that she will play a if it is reasonable for him to believe that she could play a, etc. 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. ... 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. ... It has been suggested that this article or section be merged with normal form game. ... In game theory, the best response is the strategy in a single period that creates the most favorable immediate outcome for the current player, taking other players strategies as given. ...

This provides a infinite chain of consistent beliefs that result in the players playing (a, A). This makes (a, A) a rationalizable equilibrium. A similar process can be repeated for (b, B). Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In Game Theory, the prisoners dilemma is a type of non-zero-sum game. ...

Not every strategy in every game is rationalizable. Consider a prisoner's dilemma pictured to the left. Row player would never play c, since c is not a best response to any strategy by the column player. This is an example of a more general fact, that a strategy which is strictly dominated cannot be part of a rationalizable equilibrium. Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In Game Theory, the prisoners dilemma is a type of non-zero-sum game. ... In game theory, dominance occurs when one strategy is better or worse than another regardless of the strategies of a players opponents. ...

Conversely, for two-player games, the set of all rationalizable strategies can be found by iterated elimination of strictly dominated strategies. In games with more than two players, however, there may be strategies that are not strictly dominated, but which can never be the best response. By the iterated elimination of all such strategies one can find the rationalizable strategies for a multiplayer game.

Rationalizability and Nash equilibria

It can be easily proved that every Nash equilibria is a rationalizable equilibria, however the converse is not true. Some rationalizable equilibria are not Nash equilibria. This makes the rationalizability concept a generalization of Nash equilibrium concept.

As an example, consider the game matching pennies pictured to the right. In this game the only Nash equilibrium is row playing h and t with equal probability and column playing H and T with equal probability. However, all the pure strategies in this game are rationalizable. Matching Pennies is the name for a simple example game used in game theory. ... A pure strategy is a term used to refer to strategies in Game theory. ...

Consider the following reasoning: row can play h if it is reasonable for her to believe that column will play H. Column can play H if its reasonable for him to believe that row will play t. Row can play t if its reasonable for her to believe that column will play T. Column can play T if it reasonable for him to believe that row will play h (beginning the cycle again). This provides a infinite set of consistent beliefs that results in row playing h. A similar argument can be given for row playing t, and for column playing either H or T.

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. ... The Winners curse is a phenomenon akin to a Pyrrhic victory that occurs in common value auctions with incomplete information. ... Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In Game Theory, the prisoners dilemma is a type of non-zero-sum game. ... 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 damage 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 (called Roshambo later in this article and also known by 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. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... U.S. Economic Calendar Economics at the Open Directory Project Economics textbooks on Wikibooks The Economists Economics A-Z Daily analysis of economics in the news (UK focus) Institutions and organizations Bureau of Labor Statistics - from the American Labor Department Center for Economic and Policy Research (USA) National Bureau... 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. ...