Signaling
Economics 302 - Microeconomic Theory II: Strategic Behavior
Instructor: Songzi Du
compiled by Shih En Lu
Simon Fraser University
March 15, 2016
ECON 302 (SFU)
Lecture 10
March 15, 2016
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.
ECON 302 (SFU)
Lecture 10
March 15, 2016
2 / 19
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 event
Y 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 )
ECON 302 (SFU)
Lecture 10
March 15, 2016
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Example 1
Your neighbor has 2 children. You learn that he has a son, Joe. What
is the probability that Joe’s sibling is a brother?
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Lecture 10
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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 see
someone uses iPhone, how likely is he/she rich?
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Lecture 10
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Example 3
Suppose a drug test is 99% sensitive and 99% specific. That is, the
test will produce 99% true positive results for drug users and 99%
true negative results for non-drug users. Suppose that 0.5% of people
are users of the drug. If a randomly selected individual tests positive,
what is the probability he or she is a user?
ECON 302 (SFU)
Lecture 10
March 15, 2016
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Signaling
Sometimes, part of the informed side of a market wants to reveal
information.
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, they
can provide information about people’s type.
ECON 302 (SFU)
Lecture 10
March 15, 2016
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Education as a Signal (Spence, 1973)
Two types of workers: skilled (s = 1) and unskilled (s = 0). Both are
risk-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 utility
we − ke, where w is the wage.
ECON 302 (SFU)
Lecture 10
March 15, 2016
8 / 19
Education as a Signal (Spence, 1973)
Two types of workers: skilled (s = 1) and unskilled (s = 0). Both are
risk-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 utility
we − ke, where w is the wage.
The firm observes education (e), but not type (s), pays workers their
expected skill level conditional on education:
w1 = E[s | e = 1] = µ(s = 1 | e = 1),
w0 = E[s | e = 0] = µ(s = 1 | e = 0).
ECON 302 (SFU)
Lecture 10
March 15, 2016
8 / 19
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 µ: probabilities on the
types of the informed side, conditional on the observables
(education) . It acts optimally based on those beliefs (i.e., pay wages
based on the conditional beliefs).
ECON 302 (SFU)
Lecture 10
March 15, 2016
9 / 19
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 µ: probabilities on the
types of the informed side, conditional on the observables
(education) . It acts optimally based on those beliefs (i.e., pay wages
based on the conditional beliefs).
Example: “Educated workers are skilled with probability 0.7, and
non-educated workers are skilled with probability 0.2”
µ(s = 1 | e = 1) = 0.7, µ(s = 1 | e = 0) = 0.2.
ECON 302 (SFU)
Lecture 10
March 15, 2016
9 / 19
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 µ: probabilities on the
types of the informed side, conditional on the observables
(education) . It acts optimally based on those beliefs (i.e., pay wages
based on the conditional beliefs).
Example: “Educated workers are skilled with probability 0.7, and
non-educated workers are skilled with probability 0.2”
µ(s = 1 | e = 1) = 0.7, µ(s = 1 | e = 0) = 0.2.
In equilibrium, beliefs must be consistent with what the informed
side is doing (derived from Bayes’ rule).
Example: If fraction q of the skilled and fraction q 0 of the unskilled
get educated, what must the firms’ beliefs be?
ECON 302 (SFU)
Lecture 10
March 15, 2016
9 / 19
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 µ: probabilities on the
types of the informed side, conditional on the observables
(education) . It acts optimally based on those beliefs (i.e., pay wages
based on the conditional beliefs).
Example: “Educated workers are skilled with probability 0.7, and
non-educated workers are skilled with probability 0.2”
µ(s = 1 | e = 1) = 0.7, µ(s = 1 | e = 0) = 0.2.
In equilibrium, beliefs must be consistent with what the informed
side is doing (derived from Bayes’ rule).
Example: If fraction q of the skilled and fraction q 0 of the unskilled
get educated, what must the firms’ beliefs be?
Note: beliefs after unexpected behaviour can be anything. So if
nobody gets educated, then the firm is free to have any belief if,
hypothetically, it sees an educated person.
ECON 302 (SFU)
Lecture 10
March 15, 2016
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Pooling Equilibrium
Let w0 be the wage of uneducated workers, w1 be the wage of
educated workers.
In a pooling equilibrium, all types do the same thing.
ECON 302 (SFU)
Lecture 10
March 15, 2016
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Pooling Equilibrium
Let w0 be the wage of uneducated workers, w1 be the wage of
educated workers.
In a pooling equilibrium, all types do the same thing.
For example, nobody gets educated, and everyone is paid w0 = p.
ECON 302 (SFU)
Lecture 10
March 15, 2016
10 / 19
Pooling Equilibrium
Let w0 be the wage of uneducated workers, w1 be the wage of
educated 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.
ECON 302 (SFU)
Lecture 10
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Pooling Equilibrium I
Suppose no-one gets educated.
This is only sustainable if skilled workers don’t find it worthwhile to
be educated. So we must have w1 − c ≤ w0 = p.
ECON 302 (SFU)
Lecture 10
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Pooling Equilibrium I
Suppose no-one gets educated.
This is only sustainable if skilled workers don’t find it worthwhile to
be educated. So we must have w1 − c ≤ w0 = p.
So any profile of the following form is a pooling equilibrium:
Nobody gets educated.
w0 = µ(s = 1 | e = 0) = p,
w1 = µ(s = 1 | e = 1) ≤ c + p
ECON 302 (SFU)
Lecture 10
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Pooling Equilibrium II
Suppose everyone gets educated.
This is only sustainable if unskilled workers find it worthwhile to be
educated. So we must have w0 ≤ w1 − k = p − k.
ECON 302 (SFU)
Lecture 10
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Pooling Equilibrium II
Suppose everyone gets educated.
This is only sustainable if unskilled workers find it worthwhile to be
educated. So we must have w0 ≤ w1 − k = p − k.
So any profile of the following form is a pooling equilibrium:
Everyone gets educated.
w1 = µ(s = 1 | e = 1) = p,
w0 = µ(s = 1 | e = 0) ≤ p − k
This equilibrium exists only if k ≤ p.
ECON 302 (SFU)
Lecture 10
March 15, 2016
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 don’t.
ECON 302 (SFU)
Lecture 10
March 15, 2016
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Separating Equilibrium
In a separating equilibrium, each type does something different.
Here, it must be that the skilled get educated, and the unskilled don’t.
w0 = µ(s = 1 | e = 0) = 0,
w1 = µ(s = 1 | e = 1) = 1.
ECON 302 (SFU)
Lecture 10
March 15, 2016
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Separating Equilibrium
For the separating equilibrium to exist, it must be that:
1
the skilled find an education worthwhile: c ≤ 1; and
2
the unskilled don’t find an education worthwhile: k ≥ 1.
Thus, the signaling device must be not too costly for some and costly
enough for others.
ECON 302 (SFU)
Lecture 10
March 15, 2016
14 / 19
Semi-Separating (or Partially Pooling) Equilibrium
In a semi-separating (also known as partially pooling) equilibrium, one
type plays a mixed strategy.
For example, fraction q of skilled workers get an education.
ECON 302 (SFU)
Lecture 10
March 15, 2016
15 / 19
Semi-Separating (or Partially Pooling) Equilibrium
In a semi-separating (also known as partially pooling) equilibrium, one
type 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, so
w1 − c = w0 .
Hence, unskilled workers will strictly prefer being uneducated, so
w1 = µ(s = 1 | e = 1) = 1.
Thus, w0 = 1 − c. Also, we must have
w0 = µ(s = 1 | e = 0) =
Thus, 1 − c =
p(1−q)
1−pq ,
so q =
p(1 − q)
.
1 − pq
p+c−1
cp .
Since 0 < q < 1, this equilibrium exists if and only if 1 − p < c < 1.
ECON 302 (SFU)
Lecture 10
March 15, 2016
15 / 19
Summary of Signaling Equilibrium
We always assume c < k and 0 ≤ p ≤ 1.
Equilibrium
Pooling (no education)
Semi-Separating (skilled mixes)
Pooling (education)
Semi-Separating (unskilled mixes)
Separating
ECON 302 (SFU)
Lecture 10
Parameter
no extra condition
1−p <c <1
k ≤p
p<k <1
c ≤ 1, k ≥ 1
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A Separating Equilibrium
Springbok “stotting” or “pronking,” signaling to predators that this is a fit
and fast animal, not worth chasing.
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Lecture 10
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A Pooling Equilibrium
Mexical Milk Snake (left, non-venomous) and Texas Coral Snake (right,
highly venomous)
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Lecture 10
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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 category
depends on model parameters.
Everywhere in life: gifts, how you dress, idiosyncratic application
requirements, etc.
ECON 302 (SFU)
Lecture 10
March 15, 2016
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