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?