How to understand the intuitive criterion

Question

I am studying for my exam in MicroEconomics 2 which involves game theory and I have trouble with understanding the intuitive criterion and how to use it. Consider the down below signalling game.

First, I am asked to specify whether or not a separating PBE exists in which the smart applicant chooses MSc while the dump chooses BSc. Here I have said that this does not exist because there is an incentive for the dumb applicant to deviate because, given the beliefs of player 2, player 2 will think that he is a smart type thus choosing hire. This gives the dumb applicant a higher payoff.

Next I have to specify all pooling PBE on BSc degree. Here I the following

$$\sigma^* = ((smart \mapsto BSc, dumb \mapsto Bsc),(BSc \mapsto refuse, MSc \mapsto refuse))$$

with the believes that $\mu^*=(\mu_B^*,\mu_M^*=(1/3,x)$ where $x \in [0,2/3]$.

Then I have to say which PBE satisfies the intuitive criterion. I have written that as there is an incentive for player 1, given he is a dumb applicant, to MSc, this PBE does not satisfy the intuitive criterion. However, I have said that the pooling PBE satisfies the intuitive criterion because there is no incentive for the applicant, either smart or dumb, to deviate because he will either get -1 instead of 0 or -2 instead of 0. Is this correct? And if so is this how you in general use the intuitive criterion?

Thanks.

Does this answer your question? How to intuitively understand the 'Intuitive criterion'? — Bayesian, Jun 18 '21 at 09:01
Hi. Thanks for the comment. I would probably get a better understanding if someone would use my example and guide me through how to solve it (how to use the intuitive criterion, i.e. what payoffs to compare and why). — Mathias, Jun 18 '21 at 09:14

score 2 · Answer 1 · answered Jun 18 '21 at 09:41

First, you also have to check all possible separating equilibria. You only checked (MSc,BSc), not (BSc,MSc) with all possible reactions of the receiver. No separating equilibrium exists here.

The pooling eq you find is correct. There is no pooling eq in which both types play MSc.

The intuitive criterion is a refinement to rule out "unreasonable" off-path beliefs. The belief $x$ that supports your equilibrium says that the receiver thinks it is not too likely that an MSc is the smart type. Is that reasonable? Yes, it is. According to the intuitive criterion, it would be an unreasonable belief if the dumb type was worse off by deviating to MSc for any belief and the smart type was better off for some belief. This is not the case here, because for $x>2/3$ the dumb type would be better off from being hired: MSc-> hired leads to payoff 1 whereas the equilibrium path BSc->refused leads to payoff 0<1. If the opposite was true, it would be unreasonable to attach positive probability to "MSc is a dumb type" because no dumb type would ever get an MSc in such a world. As this is not the case here, the off-path belief is reasonable.

How to understand the intuitive criterion

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