<p>I'm trying to figure out why in a signaling game with more than 2 types, multiple equilibria with different equilibrium strategies satisfy the intuitive criterion.</p>
<p>Take the classic Spence job market signaling game as an example. We know when there are 2 types, only one kind of equilibria satisfies the intuitive criterion. However, if why multiple equilibria survive given more than 2 types?</p>
<p>Suppose a worker with productivity <span class="math-container">$\theta\in\{\theta_L,\theta_H\}$</span>, prior the entering the labor market, the worker can obtain education <span class="math-container">$e\geq 0$</span> at cost <span class="math-container">$c(e,\theta)$</span>. Education is less costly for high type workers; that is, <span class="math-container">$c_1(e,\theta_H)&lt;c_1(e,\theta_L)$</span> with <span class="math-container">$c_1(e,\theta)&gt;0$</span>. Two firms observe the education level of the worker but not his productivity. Firms believe that the probability of <span class="math-container">$\theta_H$</span> is <span class="math-container">$\lambda\in (0,1)$</span>. Following any level of education <span class="math-container">$e$</span>, the firms have the same belief <span class="math-container">$\mu(e)$</span> that the worker's type is <span class="math-container">$\theta_H$</span>. Given belief <span class="math-container">$\mu(e)$</span>, the firms must offer the same wage: <span class="math-container">$w(e)=\mu(e)\theta_H+[1-\mu(e)]\theta_L$</span>. We can show that there is a separating equilibrium with <span class="math-container">$e^*(\theta_H)=e$</span> iff and only if <span class="math-container">$e\in [\hat{e},\bar{e}]$</span>, and there is a pooling equilibrium with <span class="math-container">$e^*(\theta_H)=e$</span> iff and only if <span class="math-container">$e\in [0,\underline{e}]$</span>, where</p>
<ul>
<li><span class="math-container">$\hat{e}$</span> be such that the low type is indifferent between <span class="math-container">$(\theta_L,0)$</span> and <span class="math-container">$(\theta_H,\hat{e})$</span>, where <span class="math-container">$(\cdot,\cdot)$</span> is (wage, education)</li>
<li>Let <span class="math-container">$\bar{e}$</span> be such that the high type is indifferent between <span class="math-container">$(\theta_L,0)$</span> and <span class="math-container">$(\theta_H,\bar{e})$</span></li>
<li>Let <span class="math-container">$\underline{e}$</span> be such that the low type is indifferent between <span class="math-container">$(\theta_L,0)$</span> and <span class="math-container">$(E(\theta),\underline{e})$</span></li>
</ul>
<p>The only equilbria that survives the intuitive criterion are the separating equilibria with <span class="math-container">$e^*(\theta_H)=\hat{e}$</span>. How do I show that it is not the case when there are more than 2 types?</p>

CC BY-SA 4.0
2024-04-20T04:38:48.860Z
1. Please provide the source for the claim that you want to prove.
Giskard
CC BY-SA 4.0
2024-04-20T04:40:05.250Z
2. Please provide the **exact** claim that you want to prove. There are many kind of signalling games, and it is easy to produce some (e.g. with finite action spaces) which don't even have multiple equilibria.
Giskard
CC BY-SA 4.0
2024-04-20T15:16:06.137Z
@Giskard I'm very sorry. I've added more context
Ludwig Gershwin
CC BY-SA 4.0
2024-04-20T15:20:03.053Z
Please provide a reference for what you mean by "classic Spence job market signaling game"! Most people encounter this in a lecture or textbook, where the game is usually refined and/or simplified from the "classic" version that appeared in the original paper.
Giskard
CC BY-SA 4.0
2024-04-20T15:35:12.383Z
For both clarity and self-containedness it would be best if you just **defined the exact game** in the body of your question. If you look around the SE, you will see that there are several posters who put a lot of effort into their questions, to make them clear and easily accessible for other people.
Giskard
CC BY-SA 4.0
2024-04-20T18:11:01.247Z
@Giskard I added more context.
Ludwig Gershwin
CC BY-SA 4.0
2024-04-21T05:11:08.870Z
Much better. Still, where do you get the claims? Giving an exact quatition and reference would still significantly improve the question.
Giskard
CC BY-SA 4.0
2024-04-22T23:21:52.060Z
@Giskard Actually, the claim is from my problem set. Please see here https://drive.google.com/file/d/1qaKJJtglg6HWkr_cYCWxV5K-SN6Et-uU/view?usp=drive_link
Ludwig Gershwin
CC BY-SA 4.0
2024-04-23T06:44:11.943Z
Excellent, thank you for the referrence. Next time please start with this.
Giskard
CC BY-SA 4.0
2024-04-23T06:47:14.057Z
"*Make sure to reference all sources (if any) you used in solving the questions*" hey, so, if someone provides an answer to your Q, will you reference it?
Giskard
CC BY-SA 4.0
2024-04-23T12:21:49.310Z
@Giskard Yes I will, and I have already referenced it quite a few times.
Ludwig Gershwin
<game-theory><signaling><equilibrium-selection>
How do I show that if there are more than 2 types in a signaling game, then multiple equilibria satisfy the intuitive criterion?
Ludwig Gershwin
# How do I show that if there are more than 2 types in a signaling game, then multiple equilibria satisfy the intuitive criterion?

(Asked by: username_0 on 2024-04-20)

I'm trying to figure out why in a signaling game with more than 2 types, multiple equilibria with different equilibrium strategies satisfy the intuitive criterion.

Take the classic Spence job market signaling game as an example. We know when there are 2 types, only one kind of equilibria satisfies the intuitive criterion. However, if why multiple equilibria survive given more than 2 types?

Suppose a worker with productivity $\theta\in\{\theta\_L,\theta\_H\}$, prior the entering the labor market, the worker can obtain education $e\geq 0$ at cost $c(e,\theta)$. Education is less costly for high type workers; that is, $c\_1(e,\theta\_H) with $c\_1(e,\theta)>0$. Two firms observe the education level of the worker but not his productivity. Firms believe that the probability of $\theta\_H$ is $\lambda\in (0,1)$. Following any level of education $e$, the firms have the same belief $\mu(e)$ that the worker's type is $\theta\_H$. Given belief $\mu(e)$, the firms must offer the same wage: $w(e)=\mu(e)\theta\_H+[1-\mu(e)]\theta\_L$. We can show that there is a separating equilibrium with $e^\*(\theta\_H)=e$ iff and only if $e\in [\hat{e},\bar{e}]$, and there is a pooling equilibrium with $e^\*(\theta\_H)=e$ iff and only if $e\in [0,\underline{e}]$, where

* $\hat{e}$ be such that the low type is indifferent between $(\theta\_L,0)$ and $(\theta\_H,\hat{e})$, where $(\cdot,\cdot)$ is (wage, education)
* Let $\bar{e}$ be such that the high type is indifferent between $(\theta\_L,0)$ and $(\theta\_H,\bar{e})$
* Let $\underline{e}$ be such that the low type is indifferent between $(\theta\_L,0)$ and $(E(\theta),\underline{e})$

The only equilbria that survives the intuitive criterion are the separating equilibria with $e^\*(\theta\_H)=\hat{e}$. How do I show that it is not the case when there are more than 2 types?

**Tags:** game-theory, signaling, equilibrium-selection

> **Comments:**
> **username_1**: 1. Please provide the source for the claim that you want to prove.  
> **username_1**: 2. Please provide the **exact** claim that you want to prove. There are many kind of signalling games, and it is easy to produce some (e.g. with finite action spaces) which don't even have multiple equilibria.  
> **username_0**: @username_1 I'm very sorry. I've added more context  
> **username_1**: Please provide a reference for what you mean by "classic Spence job market signaling game"! Most people encounter this in a lecture or textbook, where the game is usually refined and/or simplified from the "classic" version that appeared in the original paper.  
> **username_1**: For both clarity and self-containedness it would be best if you just **defined the exact game** in the body of your question. If you look around the SE, you will see that there are several posters who put a lot of effort into their questions, to make them clear and easily accessible for other people.  
> **username_0**: @username_1 I added more context.  
> **username_1**: Much better. Still, where do you get the claims? Giving an exact quatition and reference would still significantly improve the question.  
> **username_0**: @username_1 Actually, the claim is from my problem set. Please see here https://drive.google.com/file/d/1qaKJJtglg6HWkr_cYCWxV5K-SN6Et-uU/view?usp=drive_link  
> **username_1**: Excellent, thank you for the referrence. Next time please start with this.  
> **username_1**: "*Make sure to reference all sources (if any) you used in solving the questions*" hey, so, if someone provides an answer to your Q, will you reference it?  
> **username_0**: @username_1 Yes I will, and I have already referenced it quite a few times.  

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