Sequential equilibrium is a refinement of Nash Equilibrium

Nash equilibrium

In game theory, Nash equilibrium is a solution concept of a 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 his own strategy unilaterally...

 for extensive form game

Extensive form game

An extensive-form game is a specification of a game in game theory, allowing explicit representation of a number of important 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 he...

s due to David M. Kreps

David M. Kreps

David Marc "Dave" Kreps is a game theorist and economist and professor at the Graduate School of Business at Stanford University. He is known for his analysis of dynamic choice models and non-cooperative game theory, particularly the idea of sequential equilibrium, which he developed with Stanford...

 and Robert Wilson

Robert B. Wilson

Robert Butler "Bob" Wilson, Jr. is an American economist and the Adams Distinguished Professor of Management, Emeritus at Stanford University. He is known for his contributions to management science and business economics. His doctoral thesis introduced sequential quadratic programming, which...

. A sequential equilibrium specifies not only a strategy for each
of the players but also a belief for each of the players. A belief gives, for each information set

Information set

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...

 of the game belonging to the player, a probability distribution on the nodes in the information set

Information set

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...

. A profile of strategies and beliefs is called an assessment for the game.
Informally speaking, an assessment is a sequential equilibrium if its strategies are sensible given its beliefs and its beliefs are sensible given its strategies.

Consistent assessments

The formal definition of a strategy being sensible given a belief is straigh­tforward; the strategy should simply maximize expected payoff in every information set. It is also straigh­tforward to define what a sensible belief should be for those information sets that are reached with positive probability given the strategies; the beliefs should be the conditional probability distribution on the nodes of the information set, given that it is reached. This entails the application of Bayes' rule.

It is far from straigh­tforward to define what a sensible belief should be for those information sets that are reached with probability zero, given the strategies. Indeed, this is the main conceptual contribution of Kreps and Wilson. Their consistency requirement is the following: The assessment should be a limit point of a sequence of totally mixed strategy profiles and associated sensible beliefs, in the above straigh­tforward sense.

Relationship to other equilibrium refinements

Sequential equilibrium is a further refinement of subgame perfect equilibrium

Subgame perfect equilibrium

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...

 and even
perfect Bayesian equilibrium

Bayesian game

In game theory, a Bayesian game is one in which information about characteristics of the other players is incomplete. Following John C. Harsanyi's framework, a Bayesian game can be modelled by introducing Nature as a player in a game...

. It is itself refined by extensive-form trembling hand perfect equilibrium

Trembling hand perfect equilibrium

There are two possible ways of extending the definition of trembling hand perfection to extensive form games.* One may interpret the extensive form as being merely a concise description of a normal form game and apply the concepts described above to this normal form game. In the resulting perturbed...

 and proper equilibrium

Proper equilibrium

Proper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson.Proper equilibrium further refines Reinhard Selten's notion of atrembling hand perfect equilibrium by assuming that more costly trembles are made with...

. Strategies of sequential equilibria (or even extensive-form trembling hand perfect equilibria

Trembling hand perfect equilibrium

There are two possible ways of extending the definition of trembling hand perfection to extensive form games.* One may interpret the extensive form as being merely a concise description of a normal form game and apply the concepts described above to this normal form game. In the resulting perturbed...

) are not necessarily
admissible

Admissible decision rule

In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....

. A refinement of sequential equilibrium that
guarantees admissibility is quasi-perfect equilibrium

Quasi-perfect equilibrium

Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme. Informally, a player playing by a strategy from a quasi-perfect equilibrium takes observed as well as potential future mistakes of his opponents into account but assumes that he himself...

.