Sequential equilibrium

Infobox equilibrium
name=Sequential Equilibrium
subsetof=Subgame perfect equilibrium, perfect Bayesian equilibrium
supersetof = extensive-form trembling hand perfect equilibrium,Quasi-perfect equilibrium
discoverer=David M. Kreps and Robert Wilson
usedfor=Extensive form games

Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. A sequential equilibrium specifies not only a strategy for eachof 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 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.

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 and even
perfect Bayesian equilibrium. It is itself refined by extensive-form trembling hand perfect equilibrium. Strategies of sequential equilibria (or even extensive-form trembling hand perfect equilibria) are not necessarily
admissible. A refinement of sequential equilibrium thatguarantees admissibility is quasi-perfect equilibrium.

References

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