I have a little doubt in the Job Market Signalling Game. I am referring to A Primer in Game Theory: Gibbons, Chapter 4, Signaling Games The attached paragraph refers to a Separating Equilibrium case wherein, the High Ability type player chooses to signal $e_s>e^*(H)$, so that the Low Ability type has no incentive to mimic High Ability type, since he/she will mimic only when $e$ is between $e^*(H)$ and $e_s$. I understand the reasoning behind this. However, I cannot seem to understand why a High Type player would choose to signal thus is he/she gets a lower payoff by doing so (since it is not optimal). Is it the case that the 'loss' due to Low Ability Type mimicing him/her is greater than the 'loss' ny not playing optimal action? Does this have anything to do with sequential rationality?
I don't have a copy of Gibbons handy, so I cannot speak to the specific model presented there, but only generally. The intuition of the conclusion is based on the combination of the following factors:
- Whenever the firm can tell the high and low types apart, it's willing to pay a high wage to high type and a low wage to low type
- If the firm cannot tell the two types apart, it will pay both types the same wage equal to their average productivity
- Low type will try to pretend to be high type whenever possible to get a higher wage (note that even the average wage is higher than the low wage)
- High type, in order to get the high wage as opposed to the average wage, will try to distinguish themselves from the low type whenever possible
Hence, if $e^*(H)$ is a sufficiently small cost for the low type to pay to pretend to be high type, then the high type choosing $e^*(H)$ will only result in them getting the average wage instead of the high wage.
As a consequence, high type would be willing to pay a bigger cost $e_s>e^*(H)$ --- a cost so big for the low type that they won't find it profitable to pay to get the average wage, but at the same time small enough for the high type so that their gains from getting the high wage instead of the average wage outweigh such a cost --- to distinguish themselves from the low type.
The key here is to take into account the presence of the low type (and asymmetric info). They are the ones who cause the reduction of the high type's wage to the average level at cost $e^*(H)$.