Signaling game and intuitive criterion

Question

I need some help in understanding the intuitive Kreps criterion. Consider the following game, where Nature acts with probability $\frac{1}{2}$

Here's what I found (prove me wrong if I made a mistake)

1. weak challenger prefers U regardless of the incubent action

2. if strong challenger chooses R in equilibrium then the incumbent knows that a challenger taht chooses R is strong and he knows that a challenger that chooses U weak. So the incumbent chooses A after observing R and F after observing U

3. Given that the incumbent will choose A after observing R and F after observing U, a strong challenger will not deviate from R; a weak challenger will not deviate from U

4. Therefore there is a weak sequential equilibrium in which a weak challenger chooses U, a strong challenger chooses R, and the incumbent chooses A after observing R and chooses F after observing U.

5. If a strong challenger chooses U too in equilibrium, then by consistence of the belief, so as nature move with probability of 1/2 A is optimal for the incumbent.

6. Strong challenger gets 5 by sticking to the strategy U; if he deviates to R, he cannot get higher payoff regardless of the incumbent's action. So it's weak sequential equilibrium

So there is a weak sequential equlibrium in which both types of challenger choose U, the incumbent chooses A upon observing U.

Is there another sequential equilibrium and does it satisfies intuitive criterion?

Answer 1

The first equilibrium you identify through steps 1-4 is not merely a weak sequential equilibrium (SE), but SE proper. It also survives the intuitive criterion, since this is a separating equilibrium in a game where the informed player has two types and two actions.

The second "equilibrium" you propose is not fully specified. It can be a (proper) SE if the incumbent has belief $\mu_I(\text{Strong}\mid R)=1$ and chooses $A$ after seeing $R$. Nevertheless, the equilibrium could be a weak PBE (or weak SE, in your language) if the incumbent holds belief $\mu_I(\text{Weak}\mid R)>0$ and best responds accordingly (i.e. $A$ when $\mu_I(\text{Strong}\mid R)\ge \frac14$ or $F$ when $\mu_I(\text{Weak}\mid R)\ge\frac34$). This class of equilibria also survives the intuitive criterion since both types already get the highest possible payoff of $5$.

There is also a SE that is semi-separating:

• Strong-type mixes between $U$ and $R$ with $(\frac13,\frac23)$
• Weak-type plays $U$
• Incumbent holds belief $\mu_I(\text{Strong}\mid U)=\frac14$ and plays $(\frac12,\frac12)$ after seeing $U$
• Incumbent holds belief $\mu_I(\text{Strong}\mid R)=1$ and plays $A$ after seeing $R$

This equilibrium also survives intuitive criterion since the Strong-type's lowest payoff from deviating ($3$) is not higher than their equilibrium payoff ($4$)