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Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. In the 21st century, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.
In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a larger piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally.
The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it "prisoner's dilemma", presenting it as follows:
Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:
In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players and no player has anything to gain by changing only his own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who applied it to competing firms choosing outputs.
In economics and game theory, a participant is considered to have superrationality if they have perfect rationality but assume that all other players are superrational too and that a superrational individual will always come up with the same strategy as any other superrational thinker when facing the same problem. Applying this definition, a superrational player playing against a superrational opponent in a prisoner's dilemma will cooperate while a rationally self-interested player would defect.
In game theory, a cooperative game is a game with competition between groups of players ("coalitions") due to the possibility of external enforcement of cooperative behavior. Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing.
In game theory, a non-cooperative game is a game with competition between individual players, as opposed to cooperative games, and in which alliances can only operate if self-enforcing.
Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.
In game theory, battle of the sexes (BoS) is a two-player coordination game. Some authors refer to the game as Bach or Stravinsky and designate the players simply as Player 1 and Player 2, rather than assigning sex.
In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions, payoffs, strategies and "types" of players are thus common knowledge. Complete information is the concept that each player in the game is aware of the sequence, strategies, and payoffs throughout gameplay. Given this information, the players have the ability to plan accordingly based on the information to maximize their own strategies and utility at the end of the game.
In cooperative game theory, the core is the set of feasible allocations that cannot be improved upon by a subset of the economy's agents. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle that is part of an aggregate consumption bundle that can be constructed from publicly available technology and the initial endowments of each consumer in the coalition.
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. If one can change the identities of the players without changing the payoff to the strategies, then a game is symmetric. Symmetry can come in different varieties. Ordinally symmetric games are games that are symmetric with respect to the ordinal structure of the payoffs. A game is quantitatively symmetric if and only if it is symmetric with respect to the exact payoffs. A partnership game is a symmetric game where both players receive identical payoffs for any strategy set. That is, the payoff for playing strategy a against strategy b receives the same payoff as playing strategy b against strategy a.
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.
In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. Single stage game or single shot game are names for non-repeated games.
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 mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their private observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from their strategy, the distribution from which the signals are drawn is called a correlated equilibrium.
In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.
In game theory, a subgame perfect equilibrium is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game, no matter what happened before. Every finite extensive game with perfect recall has a subgame perfect equilibrium.
Rabin fairness is a fairness model invented by Matthew Rabin. It goes beyond the standard assumptions in modeling behavior, rationality and self-interest, to incorporate fairness. Rabin's fairness model incorporates findings from the economics and psychology fields to provide an alternative utility model. Fairness is one type of social preference.
Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division of payoffs to choose. Such surplus-sharing problems are faced by management and labor in the division of a firm's profit, by trade partners in the specification of the terms of trade, and more.
Jean-François Mertens was a Belgian game theorist and mathematical economist.
References
Myerson, Roger B. (1991). Game Theory: Analysis of Conflict. Cambridge, Massachusetts: Harvard University Press. p. 568. ISBN 0-674-34116-3.