Lecture 11 - Signalling

SignalingEconomics 302 - Microeconomic Theory II: Strategic Behavior

Instructor: Songzi Du

compiled by Shih En LuSimon Fraser University

March 23, 2015

ECON 302 (SFU) Lecture 11 March 23, 2015 1 / 19

Most Important Things to Learn

1 Conditional probability (Bayes rule).

2 Understand the economic intuition of signaling.

3 Know how to find signaling equilibria.


Quick Review of Probabilities

Let P(X ) be the probability of an event X occurring.

Let P(X ,Y ) be the probability both events X and Y occurring.

Let P(X | Y ) be the probability of event X occurring given that eventY has occurred.

Bayes rule:

P(X | Y ) = P(X ,Y )P(Y )

.

Law of total probability:

P(X ) = P(Y )P(X | Y ) + P(not Y )P(X | not Y )


Example 1

Your neighbor has 2 children. You learn that he has a son, Joe. Whatis the probability that Joes sibling is a brother?


Example 2

40% of population is rich, the other 60% poor. Among the rich, 40%uses iPhone, and 60% uses Samsung Galaxy. Among the poor, 50%uses iPhone, and 50% uses Samsung Galaxy. Suppose you seesomeone uses iPhone, how likely is he/she rich?


Example 3

Suppose a drug test is 99% sensitive and 99% specific. That is, thetest will produce 99% true positive results for drug users and 99%true negative results for non-drug users. Suppose that 0.5% of peopleare users of the drug. If a randomly selected individual tests positive,what is the probability he or she is a user?


Signaling

Sometimes, part of the informed side of a market wants to revealinformation.

Example: smart job applicants.

How to do so credibly?

Idea: certain tasks are easier for some people than others.

Example: going to school is easier for smarter (in an academic sense)people.

If such tasks are worthwhile for some people, but not for others, theycan provide information about peoples type.


Education as a Signal (Spence, 1973)

Two types of workers: skilled (s = 1) and unskilled (s = 0). Both arerisk-neutral.

Fraction p of the worker pool is skilled.

Education (e = 1 if educated, e = 0 if not) does NOT enhance skills.It imposes cost c on skilled workers and k > c on unskilled ones.

So skilled workers have utility we ce and unskilled ones have utilitywe ke, where w is the wage.

The firm observes education (e), but not type (s), pays workers theirexpected skill level conditional on education:

w1 = E[s | e = 1] = (s = 1 | e = 1),w0 = E[s | e = 0] = (s = 1 | e = 0).


Equilibrium in a Signaling Game

Each type of the informed side (here skilled and unskilled workers)must act optimally.The uninformed side (the firm) has beliefs : probabilities on thetypes of the informed side, conditional on the observables(education) . It acts optimally based on those beliefs (i.e., pay wagesbased on the conditional beliefs).

Example: Educated workers are skilled with probability 0.7, andnon-educated workers are skilled with probability 0.2

(s = 1 | e = 1) = 0.7, (s = 1 | e = 0) = 0.2.In equilibrium, beliefs must be consistent with what the informedside is doing (derived from Bayes rule).Example: If fraction q of the skilled and fraction q of the unskilledget educated, what must the firms beliefs be?Note: beliefs after unexpected behaviour can be anything. So ifnobody gets educated, then the firm is free to have any belief if,hypothetically, it sees an educated person.



Each type of the informed side (here skilled and unskilled workers)must act optimally.The uninformed side (the firm) has beliefs : probabilities on thetypes of the informed side, conditional on the observables(education) . It acts optimally based on those beliefs (i.e., pay wagesbased on the conditional beliefs).Example: Educated workers are skilled with probability 0.7, andnon-educated workers are skilled with probability 0.2

(s = 1 | e = 1) = 0.7, (s = 1 | e = 0) = 0.2.

In equilibrium, beliefs must be consistent with what the informedside is doing (derived from Bayes rule).Example: If fraction q of the skilled and fraction q of the unskilledget educated, what must the firms beliefs be?Note: beliefs after unexpected behaviour can be anything. So ifnobody gets educated, then the firm is free to have any belief if,hypothetically, it sees an educated person.




(s = 1 | e = 1) = 0.7, (s = 1 | e = 0) = 0.2.In equilibrium, beliefs must be consistent with what the informedside is doing (derived from Bayes rule).Example: If fraction q of the skilled and fraction q of the unskilledget educated, what must the firms beliefs be?

Note: beliefs after unexpected behaviour can be anything. So ifnobody gets educated, then the firm is free to have any belief if,hypothetically, it sees an educated person.




(s = 1 | e = 1) = 0.7, (s = 1 | e = 0) = 0.2.In equilibrium, beliefs must be consistent with what the informedside is doing (derived from Bayes rule).Example: If fraction q of the skilled and fraction q of the unskilledget educated, what must the firms beliefs be?Note: beliefs after unexpected behaviour can be anything. So ifnobody gets educated, then the firm is free to have any belief if,hypothetically, it sees an educated person.ECON 302 (SFU) Lecture 11 March 23, 2015 9 / 19

Pooling Equilibrium

Let w0 be the wage of uneducated workers, w1 be the wage ofeducated workers.

In a pooling equilibrium, all types do the same thing.

For example, nobody gets educated, and everyone is paid w0 = p.

If everyone gets educated, and everyone is paid w1 = p.


Pooling Equilibrium I

Suppose no-one gets educated.

This is only sustainable if skilled workers dont find it worthwhile tobe educated. So we must have w1 c w0 = p.

So any profile of the following form is a pooling equilibrium:

Nobody gets educated.

w0 = (s = 1 | e = 0) = p,

w1 = (s = 1 | e = 1) c + p


Pooling Equilibrium II

Suppose everyone gets educated.

This is only sustainable if unskilled workers find it worthwhile to beeducated. So we must have w0 w1 k = p k .


Everyone gets educated.

w1 = (s = 1 | e = 1) = p,

w0 = (s = 1 | e = 0) p k

This equilibrium exists only if k p.


Pooling Equilibrium II

Suppose everyone gets educated.

This is only sustainable if unskilled workers find it worthwhile to beeducated. So we must have w0 w1 k = p k .


Everyone gets educated.

w1 = (s = 1 | e = 1) = p,

w0 = (s = 1 | e = 0) p k

This equilibrium exists only if k p.ECON 302 (SFU) Lecture 11 March 23, 2015 12 / 19

Separating Equilibrium

In a separating equilibrium, each type does something different.

Here, it must be that the skilled get educated, and the unskilled dont.

w0 = (s = 1 | e = 0) = 0,

w1 = (s = 1 | e = 1) = 1.


Separating Equilibrium

For the separating equilibrium to exist, it must be that:

1 the skilled find an education worthwhile: c 1; and2 the unskilled dont find an education worthwhile: k 1.

Thus, the signaling device must be not too costly for some and costlyenough for others.


Semi-Separating (or Partially Pooling) Equilibrium

In a semi-separating (also known as partially pooling) equilibrium, onetype plays a mixed strategy.

For example, fraction q of skilled workers get an education.

Thus, skilled workers must be indifferent between e = 1 and e = 0, sow1 c = w0.Hence, unskilled workers will strictly prefer being uneducated, sow1 = (s = 1 | e = 1) = 1.Thus, w0 = 1 c . Also, we must have

w0 = (s = 1 | e = 0) = p(1 q)1 pq .

Thus, 1 c = p(1q)1pq , so q = p+c1cp .Since 0 < q < 1, this equilibrium exists if and only if 1 p < c < 1.


Summary of Signaling Equilibrium

We always assume c < k and 0 p 1.

Equilibrium Parameter

Pooling (no education) no extra condition

Semi-Separating (skilled mixes) 1 p < c < 1Pooling (education) k pSemi-Separating (unskilled mixes) p < k < 1

Separating c 1, k 1


A Separating Equilibrium

Springbok stotting or pronking, signaling to predators that this is a fitand fast animal, not worth chasing.


A Pooling Equilibrium

Mexical Milk Snake (left, non-venomous) and Texas Coral Snake (right,highly venomous)


Signaling Summary

Credible mechanism for (potentially) revealing information.

Three categories of equilibria: pooling (no information is transmitted),separating (type is always revealed) and semi-separating.

Whether (and how many) equilibria exist within each categorydepends on model parameters.

Everywhere in life: gifts, how you dress, idiosyncratic applicationrequirements, etc.


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Lecture 11 - Signalling