How to intuitively understand the 'Intuitive criterion'?

The intuitive criterion by Cho and Kreps is a refinement to minimise the set of perfect Bayesian equilibria in signalling games. What would a simple and intuitive example to explain this criterion be? Assume any undergrad student should be easily able to appreciate the refinement through the example.

A concise, completely informal way of putting it is this: The intuitive criterion rules-out any out-of-equilibrium beliefs that can only be correct if some player did something stupid.

Below is a slightly more long-winded explanation with an informal example.

In many signalling games (that is, games in which one player—the sender—can communicate information to another—the receiver), there are often a lot of implausible equilibria. This happens because the Perfect Bayesian solution concept does not specify what the receiver's beliefs must be when the sender deviates; we can therefore support a lot of equilibria simply by saying that if the sender deviates from those equilibria then he will be "punished" with very bad beliefs. Such punishment will usually be enough to make the sender play a strategy that would otherwise not be a best response.

For example, in Spence's classic job market signalling paper there is an equilibrium in which high-ability individuals invest in education (learning is easy for them) whilst low-ability individuals do not (because they find it too costly to do so). Education is then a signal of ability. We might ask: is there also an equilibrium of this game in which nobody chooses to get an education and no information is transmitted to the receiver? The answer is 'yes'. We can support such an equilibrium by saying that a deviation in which a sender is educated causes the receiver to adopt the belief that the sender is certainly low-ability. If education has the effect of signalling low-ability then, of course, everyone is happy to play along with the putative equilibrium and not get educated.

It is also clear that this equilibrium is not very plausible: the receiver knows that it is less costly for a high-ability agent to get an education than a low-ability one, so it doesn't make much sense for him to think of an education as signalling low-ability. The intuitive criterion rules out this kind of equilibrium by requiring beliefs to be "reasonable" in the following sense:

Suppose the receiver observes a deviation from the equilibrium. The receiver should not believe that the sender is of type $$t_{\text{bad}}$$ if both of the following are true:

1. the deviation would result in type $$t_{\text{bad}}$$ being worse off then if he has stuck to the equilibrium for any beliefs.
2. there is some type $$t_{\text{good}}$$ who is better off by playing the deviation than by sticking to the equilibrium for some belief other than $$t_{\text{bad}}$$.

Returning to the education signalling model: Suppose that the equilibrium is that nobody gets an education and that the receiver believes that a deviation to getting education signals low ability. Anticipating these beliefs, a low ability worker is made worse-off by deviating because he not only incurs the cost of the education but is then thought of as a bad type as a result. Thus, condition 1. is satisfied.

Can we find some alternative belief such that the high-ability worker would like to deviate to getting education? The answer is yes: if the receiver believes that education signals high ability then this deviation is indeed profitable for the high-type. Thus, condition 2 is also satisfied.

Since both conditions are satisfied, the intuitive criterion rules-out the implausible pooling equilibrium.

• I'm sorry this is so wordy. Let me know if it is unclear or if you want something more formal and I will edit accordingly. Nov 22 '14 at 17:57
• Thanks for the detailed answer. I will be happy if you could add a simple model as well (as another answer, if you wish), where the 'off-equilibrium' removal is clear mathematically. Nov 23 '14 at 18:53
• I added a (very) stylised model in a separate answer. I also edited point 1 in the quote box for this answer to fix an error I made in the definition of the Intuitive Criterion. Nov 23 '14 at 21:45

Here's a simple model to complement my less formal answer:

A worker is of (privately known) type $H$ or $L$, each with probability $1/2$. The marginal product of the two types is $\pi_H>\pi_L$. The labour market is competitive so that workers are paid their (expected) marginal product. The worker can invest in education; doing so costs type $i$ $c_i$, with $\pi_H-c_L<\pi_L$ and $\pi_H-c_H>(\pi_H/2)+(\pi_L/2)$.

The game is as follows: the worker observes his type and decides whether to invest in education. Employers then observe wether the worker invested or not and make competitive wage offers based on their beliefs about his productivity.

Consider the following two perfect Bayesian equilibria (PBE) of the game.

1. (Separating equilibrium) Type $H$ invests; type $L$ does not invest. If employers observe investment then they update their beliefs to $\Pr(H)=1$ and offer a wage of $\pi_H$. If they observe no investment then they update to $\Pr(H)=0$ and offer wage $\pi_L$.

We can check that this is an equilibrium: type H's payoff is $\pi_H-c_H$. If he deviates to no education then his payoff is $\pi_L$, which is lower. Type $L$'s payoff is $\pi_L$. If he deviates to getting education then his payoff is $\pi_H-c_L<\pi_L$, which is lower. Thus neither type wants to deviate. The wage offers are (trivially) best responses given the beliefs because the labour market is competitive. Lastly, note that the beliefs are consistent with Bayes' rule and the equilibrium play of the game.

2. (Pooling equilibrium) Neither type invests. The employer updates beliefs to $\Pr(H)=0$ if education is observed and offers wage $\pi_L$. The employer sticks with the prior belief of $\Pr(H)=1/2$ and offers wage $(\pi_H/2)+(\pi_L)/2$ if education is not observed.

Let's check that this is also an equilibrium. Since education is costly but adversely affects the employer's beliefs in equilibrium, it is optimal for neither type to get education. Given beleifs and the competitiveness of the labour market, the putative wage offers are optimal. The belief $\Pr(H)=1/2$ is consistent with Bayes' rule if no education is observed (because this observation contains no new information about the worker's type). Lastly, Bayes' rule does not pin down beliefs in the event of (out of equilibrium) investment in education so, according to the definition of a PBE, we are free to specify whatever beliefs we like.

The intuitive criterion rules out equilibrium number 2. Firstly, if type $L$ deviates to getting education then the best payoff he can get is $\pi_H-c_L<\pi_L$ so such a deviation is dominated. Secondly, suppose type $H$ deviates to getting education and the employers adopt some posterior belief $\Pr(H)=1$. The payoff of the deviating $H$-type is then $\pi_H-C_L>\pi_L$. So that the deviation would be profitable. The intuitive criterion therefore rules that the beliefs $\Pr(H)=0$ are not reasonable for a deviation to investing in education and we cannot have any equilibrium that depends of such beliefs.

In fact, this game has other pooling equilibria. For example there is a pooling equilibrium in which the employer sticks with his prior belief irrespective of whether he observes education or not. This (and all other pooling equilibria) is also ruled-out by the intuitive criterion. The reason is that any deviation from an equilibrium in which nobody is educated is dominated for the $L$-type so the intuitive criterion is going to require that the employer never associates education with $L$-types. Given that education will therefore be associated with $H$-types, it is profitable for $H$-types to deviate from the no education equilibrium.

I once wrote an example of Kreps criterion using the canonical signaling model and The Simpsons. I think it goes along the same lines as @Ubiquitous' answer while being much less precise and general. But I thought the Simpsons' context might help in a pedagogical setting.

Suppose that Hank Scorpio must decide of a wage schedule for employees at Globex Corporation depending on observed education. There are two candidates : Martin Prince, a $H$ type (for "high") with an elementary school degree $e_1$, and Homer, a $L$ type (for "low") with a degree from Springfield University $e_2 > e_1$ (cf. Season 5, episode 3}).

A third possible signal would consist in getting a PhD in nuclear physics from MIT, which we denote $e_3 > e_2$.

Suppose Scorpio's believes that the productivity associated with the twho lower education levels are $\rho(e_2) > 0$, and $\rho(e_1) = 0$. Assume that this forms a sequential equilibrium, that is, at this equilibrium, neither Martin nor Homer find it worth it to get a PhD from MIT (I assume that if you are at the point of explaining Kreps criterion, you already covered sequential equilibira).

Martin would not need to exert much effort to get $e_3$ (see children's power plant competition, season 8, episode 23), and he would not mind doing so if it was the case that $\rho(e_3) = 1$. On the other hand, Homer is far better with his $e_2$ than he would be with $e_3$ even if $\rho(e_3)$ was $1$ because getting a PhD from MIT would be a huge pain for him (cf. aforementioned episode).

Because $(e_1,e_2,\rho)$ is an equilibrium, $\rho(e_3)$ must be sufficiently small to deter Martin from obtaining his PhD. This means Scorpio attaches a high probability to the fact that agents choosing $e_3$ are $L$ type. Is this equilibrium supported by reasonable beliefs? Not according to Kreps criterion : under the assumption that Scorpio knows that Homer would never try to get $e_3$ while Martin would not mind getting $e_3$, if Scorpio observes someone getting $e_3$, he could logically infer that this person is Martin, a $H$ type.