Class #9 Highlights

Whew. At long last we finally get to signaling. Here, we studied an education game with two types where education is a pure costly signal--it has no productivity value. In this class, we looked for a pooling equilibrium, this is an equilibrium in which all types choose the same signal and therefore the posteriors of the receiver are the same as the priors.

To find such an equilibrium, we needed to specify strategies as well as beliefs on the part of the receiver. To determine the full set of such equilibria, we relied on pessimistic beliefs. In our setting, these beliefs are a bit like a grim trigger strategy in repeated games--anything not supportable by these beliefs cannot be supported by any other beliefs.

With this specification, we determined that the upper bound occurred where the low type was indifferent between the equilibrium education level, e*, and choosing e = 0 and being branded as a low type hence making r_L.

These beliefs are a WPBE and, with an appropriate specification of perturbations, a sequential equilibrium as well. To support these as an SE, suppose that L types choose a random e with probability epsilon. H types follow the same strategy with probability epsilon^2. Taking limits as epsilon -> 0, it then follows that the posterior beliefs of the receiver put infinitely more weight on low types for all off equilibrium choices of e. But these are precisely the pessimistic beliefs we specified.

In a sense, these beliefs are goofy. Why would low types be infinitely more likely to choose higher, and hence more costly, education levels than high types?

To deal with this, one refinement that is commonly used is the intuitive criterion. It asks the following question: Suppose a given type chose an education level (i.e. a signal) e and got to pick what it would like the receiver to believe. Given these beliefs, the receiver would then choose an optimal action. If the result is that this type still earns less than her equilibrium payoffs, then the receiver should never believe that such an e came from that type.

In terms of our game, choose an education level e' such that, if the receiver R believed that it came from a high type for sure, an L type would be indifferent between and the equilibrium e*. Formally, e' solves:

r_H - e' theta_L = r_bar - e* theta_L.

Rewriting:

r_H - r_bar = theta_L (e' - e*)

Clearly, all education levels above e' could not have come from low types. But notice that, for a high type,

r_H - r_bar > theta_H (e' - e*)

since theta_H < theta_. But this means H types could profitably deviate from e* if they are recognized as H types.

The intuitive criteria requires that beliefs following choices e > e' must place all weight on H types. As a consequence, none of the pooling equilibria satisfy the intuitive criterion.


Key things should be able to do after this class:

1. Know how to construct a pooling equilibrium.
2. Make an argument using the intuitive criterion to rule out any pooling equilibrium.