Sequential equilibrium is a refinement of

Nash Equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, 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 equ ...

for extensive form games due to David M. Kreps 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 conta ...

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 In game theory, a subgame perfect equilibrium (or subgame perfect Nash 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 ...

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

References

David M. Kreps and Robert Wilson. "Sequential Equilibria", ''Econometrica'' 50:863-894, 1982.

Roger B. Myerson Roger Bruce Myerson (born March 29, 1951) is an American economist and professor at the University of Chicago. He holds the title of the David L. Pearson Distinguished Service Professor of Global Conflict Studies at The Pearson Institute for the ...

. ''Game Theory: Analysis of Conflict'', 1991. {{game theory Game theory equilibrium concepts