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Self-confirming equilibrium | |
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A solution concept in game theory | |

Relationship | |

Subset of | Rationalizability |

Superset of | Nash equilibrium |

Significance | |

Proposed by | Drew Fudenberg and David K. Levine |

Used for | Extensive-form games |

In game theory, **self-confirming equilibrium** is a generalization of Nash equilibrium for extensive form games, in which players correctly predict the moves their opponents make, but may have misconceptions about what their opponents *would* do at information sets that are never reached when the equilibrium is played. Informally, self-confirming equilibrium is motivated by the idea that if a game is played repeatedly, the players will revise their beliefs about their opponents' play if and only if they *observe* these beliefs to be wrong.

**Game theory** is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, 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, the **Nash equilibrium**, named after the late mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

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. If the game has perfect information, every information set contains only one member, namely the point actually reached at that stage of the game. Otherwise, it is the case that some players cannot be sure exactly what has taken place so far in the game and what their position is.

**Consistent self-confirming equilibrium** is a refinement of self-confirming equilibrium that further requires that each player correctly predicts play at all information sets that can be reached when the player's opponents, but not the player herself, deviate from their equilibrium strategies. Consistent self-confirming equilibrium is motivated by learning models in which players are occasionally matched with "crazy" opponents, so that even if they stick to their equilibrium strategy themselves, they eventually learn the distribution of play at all information sets that can be reached if their opponents deviate.

In game theory, a **solution concept** is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

In game theory, the **centipede game**, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.

In game theory, a player's **strategy** is any of the options which he or she chooses in a setting where the outcome depends *not only* on their own actions *but* on the actions of others. A player's strategy will determine the action which the player will take at any stage of the game.

An **extensive-form game** is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature".

In game theory, a **Perfect Bayesian Equilibrium** (PBE) is an equilibrium concept relevant for dynamic games with incomplete information. A PBE is a refinement of both Bayesian Nash equilibrium (BNE) and subgame perfect equilibrium (SPE). A PBE has two components - *strategies* and *beliefs*:

In game theory, a **Bayesian game** is a game in which the players have incomplete information on the other players, but, they have beliefs with known probability distribution.

In game theory, **strategic dominance** occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.

In game theory, **rationalizability** is a solution concept. The general idea is to provide the weakest constraints on players while still requiring that players are rational and this rationality is common knowledge among the players. It is more permissive than Nash equilibrium. Both require that players respond optimally to some belief about their opponents' actions, but Nash equilibrium requires that these beliefs be correct while rationalizability does not. Rationalizability was first defined, independently, by Bernheim (1984) and Pearce (1984).

In game theory, **folk theorems** are a class of theorems about possible 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 equilibrium.

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 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 the recommended strategy, the distribution 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 if the players played any smaller game that consisted of only one part of the larger game, their behavior would represent a Nash equilibrium of that smaller game. Every finite extensive game has a subgame perfect equilibrium.

**Quantal response equilibrium** (**QRE**) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

**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. 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, the **traveler's dilemma** is a type of non-zero-sum game in which two players attempt to maximize their own payoff, without any concern for the other player's payoff.

**The intuitive criterion** is a technique for equilibrium refinement in signaling games. It aims to reduce possible outcome scenarios by first restricting the type group to types of agents who could obtain higher utility levels by deviating to off-the-equilibrium messages and second by considering in this sub-set of types the types for which the off-the-equilibrium message is not equilibrium dominated.

**Cognitive hierarchy theory** (CHT) is a behavioral model originating in behavioral economics and game theory that attempts to describe human thought processes in strategic games. CHT aims to improve upon the accuracy of predictions made by standard analytic methods, which can deviate considerably from actual experimental outcomes.

- Fudenberg, Drew; Levine, David K. (1993). "Self-Confirming Equilibrium".
*Econometrica*.**61**(3): 523–545. JSTOR 2951716.

* Econometrica* is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief is Joel Sobel.

**JSTOR** is a digital library founded in 1995. Originally containing digitized back issues of academic journals, it now also includes books and other primary sources, and current issues of journals. It provides full-text searches of almost 2,000 journals. As of 2013, more than 8,000 institutions in more than 160 countries had access to JSTOR; most access is by subscription, but some of the site's public domain and open access content is available at no cost to anyone. JSTOR's revenue was $86 million in 2015.

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