Sequential equilibrium is a refinement of Nash Equilibrium for

extensive form game An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, th ...

s due to

David M. Kreps David Marc "Dave" Kreps (born 1950 in New York City) is a game theorist and economist and professor at the Graduate School of Business at Stanford University (since 1980). The Stanford University Department of Economics appointed Kreps the A ...

and Robert Wilson. 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 of the game belonging to the player, a probability distribution on the nodes in the information set. A profile of strategies and beliefs is called an assessment for the game. Informally speaking, an assessment is a perfect Bayesian equilibrium if its strategies are sensible given its beliefs and its beliefs are confirmed on the outcome path given by its strategies. The definition of sequential equilibrium further requires that there be arbitrarily small perturbations of beliefs and associated strategies with the same property.

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 In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...

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 and even perfect Bayesian equilibrium. It is itself refined by extensive-form

trembling hand perfect equilibrium A tremor is an involuntary, somewhat rhythmic, muscle contraction and relaxation involving oscillations or twitching movements of one or more body parts. It is the most common of all involuntary movements and can affect the hands, arms, eyes, f ...

and

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

. Strategies of sequential equilibria (or even extensive-form trembling hand perfect equilibria) are not necessarily admissible. A refinement of sequential equilibrium that guarantees admissibility is

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 hi ...

References

and Robert Wilson. "Sequential Equilibria", ''Econometrica'' 50:863-894, 1982. Roger B. Myerson. ''Game Theory: Analysis of Conflict'', 1991. {{game theory Game theory equilibrium concepts