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What are the pros and cons of the Intuitive Criterion and the Criterion of Divinity in signaling games with three types? I find that separating equilibria often survive the intuitive criterion, whereas only the "Riley"/Pareto-efficient outcome survives divinity. But what about potential pooling equilibria?

For example, I have difficulties investigating whether the asset-pooling equilibria (for positive correlation and no synergies, i.e., section 2.1) in Edmans and Mann's "Financing Through Asset Sales" satisfies the criteria, if I solve for three types.

I hope someone can help me.

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Both of these criteria refine equilibria where there is an unused message or action. Notice that if there is one action that is not used by any of the types, it can be either because they are all pooling on some other action, or because they are all separating, but there are more actions than types, or anything in between (for example some types are pooling and some other are separating, etc.).

Now, unless the unused action is dominated by all types, for most games you will see that no type is using that action because if they do, they anticipate that the other player(s) will react and choose an outcome that no type would rather have.

The idea behind these two criteria is to realize that perhaps the only reason why the other player(s) react in such a bad way is that they have an (off-equilibrium path) belief about the type of the first player that is un-reasonable.

Because no type is actually choosing that action, the second player's beliefs are not restricted in a perfect Bayesian equilibrium. The intuitive criterion, however, imposes the restriction that the second player should not base her strategy after that unused action based on a belief that she is facing a type that would not possibly benefit from deviating to said action.

In the typical signaling game, this means that an employer should correctly believe that if a worker has sufficiently more years of education than what expected in a pooling equilibrium, it is not the low type worker since he has no incentives to signal their type by acquiring more education.

Similarly, the criterion of Divinity imposes stricter restrictions on off-path beliefs. Not only the second player should not believe she is playing against a type that cannot benefit from deviating to this action, but also it should assign a higher probability in her beliefs to be playing against a type who has a higher incentive to deviate.

Due to these extra restrictions, it should not be surprising that while many separating equilibria survive the first criterion (included the pareto-efficient outcome), with stricter restrictions, only the Riley outcome survives. Of course, the result that the efficient outcome is the one that survives is not obvious at all.

I hope this helps you form an intuition about whether the pooling equilibria you are considering survives these criteria or not. I am not familiar with the model you reference, so I can't help much with that. But I'm happy to give further thoughts if you introduce the model here, or expand on what you have been able to figure out.

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