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I have asked a similiar question before, but I would very much appreciate if someone would say if my reasoning in this particular case is correct.

Consider the down below:

enter image description here

For part a) I have found only one pooling PBE, namely

$$s^{PBE} = ((bike,bike),(no,no))$$

with beliefs that $\mu_B=1/4$ and $\mu_P \leq 1/2$. Then for part b) my answer is:

The intuitive criterion is a refinement to rule out "unreasonable" off-path beliefs. The belief $\mu_p$ that supports the PBE says that the bank thinks it is not too likely that a Porsche driver is a rich customer. According to the intuitive criterion, it would be an unreasonable belief if the poor customer was worse off by deviating to Porsche for any belief and that rich customer was better of for some belief. This is actually the case. For $1/2 > \mu_P$, the poor customer would be worse off by deviating to Porsche since the bank would choose yes after observing this signal, i.e. yielding him a payoff off $-1$ compared to 0.

Therefore, we can rule out this equlibrium as a possible pooling PBE by the intuitive criterion.

Is this correct?

TIA for any help.

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We cannot judge if your answer is correct because we don't see the game tree.

First, I would not say that "we can rule out this equlibrium as a possible pooling PBE by the intuitive criterion" because the intuitive criterion is simply a refinement. The PBE is still an equilibrium - it's just that we can say that it appears "unreasonable" according to a formal definition.

For some it is helpful to think about the intuitive criterion with the following speech by the rich type to the bank: "I will come by Porsche. I know that you will believe that I am likely poor. However, this is unreasonable. From the payoffs you can see that, in this pooling equilibrium, the poor type is better off by staying on path and coming by bike. That is, the poor type cannot improve their payoff by this deviation no matter what your belief is. Hence, you should not believe that the poor type comes by Porsche. Given this belief and your best response to it, I, as the rich type that I truly am, would be better off by deviating. Hence, I will send the off-path message and come by Porsche, and you should reasonably believe that I am rich."

Whenever one type can honestly hold such a speech, the beliefs are unreasonable according to the intuitive criterion. See the beer-and-quiche game by Cho and Kreps who introduce the criterion in this paper.

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