Class #8 Highlights

In this class, we studied equilibrium concepts in sequential games with private information. The logical extension of Nash equilibrium to private information settings is Bayes-Nash equilibrium. This is simply a best response to the expectation of a rival's strategy, where the expectation is formed using Bayes' Rule where possible. The two key difficulties with this solution concept are:

1. It places no restriction on out of equilibrium actions.
2. It places no restriction on out of equilibrium beliefs.

Problem 1 is a familiar one from sequential games of complete information. The usual issue is that non-credible out of equilibrium actions can be used to affect on equilibrium actions. We saw a private information version of this idea in the entry game used to motivate the study of solution concepts. The point here was that the incumbent could threaten to fight out of equilibrium even though this was a dominated strategy.

Weak Perfect Bayesian Equilibrium takes care of the first objection. It requires that all strategies be sequentially rational; that is, on and off equilibrium actions must be best responses to beliefs, which are again formed using Bayes' rule where possible.

But this does not deal with the second objection. One could still use "goofy" beliefs to support sequentially rational out of equilibrium actions.

Sequential Equilibrium attempts to address this concern. This is a bit like the private information analog of trembling hand equilibrium. The idea here is to consider perturbations of the equilibrium strategies that are completely mixed. This forces the use of Bayes' rule on and off the equilibrium path. If there exists a sequence of such perturbations that converges to a WPBE, then we have found a sequential equilibrium.

In terms of our entry game, if the entrant deviated and entered the game, we saw that it would rationally choose a fight strategy and hence one could not justify the beliefs supporting the fight strategy by the incumbent following a sequence of such perturbed strategies.

As we will see in class #9, even sequential equilibrium does not rule out all goofy beliefs. We might still be clever in constructing sequences of deviations satisfying this equilibrium concept yet not satisfying common sense (or empirical testing in the lab or the field).

Technical footnote: Sequential equilibrium is only formally defined for games with finite actions, types, etc. For games with continuous types, the right extension s a matter of some debate. The notion of perfect Bayesian equilibrium, which is a strengthening of WPBE is one way to deal with this.


Key things should be able to do after this class:


1. Know what a BNE, WPBE, and SE are and use these concepts to analyze sequential games of private information.
2. Recognize that the "correct" solution concept depends on the game being studied. For many games, WPBE works just fine. Use your judgment rather than being dogmatic.