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Matching pennies is a non-cooperative game studied in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even wins and keeps both pennies. If the pennies do not match, then Odd wins and keeps both pennies.
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, a Perfect Bayesian Equilibrium (PBE) is a solution with Bayesian probability to a turn-based game with incomplete information. More specifically, it is an equilibrium concept that uses Bayesian updating to describe player behavior in dynamic games with incomplete information. Perfect Bayesian equilibria are used to solve the outcome of games where players take turns but are unsure of the "type" of their opponent, which occurs when players don't know their opponent's preference between individual moves. A classic example of a dynamic game with types is a war game where the player is unsure whether their opponent is a risk-taking "hawk" type or a pacifistic "dove" type. Perfect Bayesian Equilibria are a refinement of Bayesian Nash equilibrium (BNE), which is a solution concept with Bayesian probability for non-turn-based games.
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In game theory, trembling hand perfect equilibrium is a type of refinement of a Nash equilibrium that was first proposed by Reinhard Selten. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability.
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.
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In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem justifies a puzzling aspect of mixed strategy Nash equilibria: each player is wholly indifferent between each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.
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Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff dominant if it is Pareto superior to all other Nash equilibria in the game.1 When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction. This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.
In game theory, an epsilon-equilibrium, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.
A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe or checkers (draughts). In other words, it extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite.
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The concept of coalition-proof Nash equilibrium applies to certain "noncooperative" environments in which players can freely discuss their strategies but cannot make binding commitments. It emphasizes the immunization to deviations that are self-enforcing. While the best-response property in Nash equilibrium is necessary for self-enforceability, it is not generally sufficient when players can jointly deviate in a way that is mutually beneficial.
In game theory, Mertens stability is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens. This solution concept is now called Mertens stability, or just stability.
M equilibrium is a set valued solution concept in game theory that relaxes the rational choice assumptions of perfect maximization and perfect beliefs. The concept can be applied to any normal-form game with finite and discrete strategies. M equilibrium was first introduced by Jacob K. Goeree and Philippos Louis.
The Berge equilibrium is a game theory solution concept named after the mathematician Claude Berge. It is similar to the standard Nash equilibrium, except that it aims to capture a type of altruism rather than purely non-cooperative play. Whereas a Nash equilibrium is a situation in which each player of a strategic game ensures that they personally will receive the highest payoff given other players' strategies, in a Berge equilibrium every player ensures that all other players will receive the highest payoff possible. Although Berge introduced the intuition for this equilibrium notion in 1957, it was only formally defined by Vladislav Iosifovich Zhukovskii in 1985, and it was not in widespread use until half a century after Berge originally developed it.
References
- ↑ R. Aumann (1959), Acceptable points in general cooperative n-person games in "Contributions to the Theory of Games IV", Princeton Univ. Press, Princeton, N.J..
- 1 2 B. D. Bernheim; B. Peleg; M. D. Whinston (1987), "Coalition-Proof Equilibria I. Concepts", Journal of Economic Theory, 42: 1–12, doi:10.1016/0022-0531(87)90099-8.
- 1 2 3 4 Nessah, Rabia; Tian, Guoqiang (2014-06-15). "On the existence of strong Nash equilibria". Journal of Mathematical Analysis and Applications. 414 (2): 871–885. doi: 10.1016/j.jmaa.2014.01.030 . ISSN 0022-247X.
- ↑ D. Moreno; J. Wooders (1996), "Coalition-Proof Equilibrium", Games and Economic Behavior, 17: 80–112, doi:10.1006/game.1996.0095, hdl: 10016/4408 .