Signaling
Economics 302 - Microeconomic Theory II: Strategic Behavior
Shih En Lu
Simon Fraser University
(with thanks to Anke Kessler)
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Most Important Things to Learn Today
1
Understand the economic intuition of signaling.
2
Know how to …nd signaling equilibria.
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Quick Review of Probabilities
Let P (X ) be the probability of an event X occurring.
Let P (XY ) be the probability both events X and Y occurring.
Let P (X jY ) be the probability of event X occurring given that event
Y has occurred.
Example: X = Canucks win; Y = Your friend is happy
What is P (X jY ) in terms of P (X ), P (Y ) and P (XY )?
What is P (X ) in terms of P (Y ), P (X jY ) and P (X jnot Y )?
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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.
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Education as a Signal
This model is due to Spence (1973).
Two types of workers: skilled (s = 1) and unskilled (s = 0).
Fraction p of the worker pool is skilled.
A …rm …nds it optimal to pay workers their expected skill level:
w = E [s ].
Education (e = 1 if educated, e = 0 if not) does NOT enhance skills.
Skilled workers have utility w ce and unskilled ones have utility
w ke: education imposes cost c on skilled workers and k > c on
unskilled ones.
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Equilibrium in a Signaling Game
Each type of the informed side (here skilled and unskilled workers)
must act optimally.
The uninformed side has beliefs µ: probabilities on the types of the
informed side, conditional on the observables. It acts optimally based
on those beliefs.
Example: "Educated workers are skilled with probability 0.7, and
non-educated workers are skilled with probability 0.2." This can be
denoted as µ(s = 1je = 1) = 0.7, µ(s = 1je = 0) = 0.2.
Beliefs must be consistent with what the informed side is doing.
Example: If fraction q of the skilled and fraction q 0 of the unskilled
get educated, what must the …rms’beliefs be?
Note: beliefs after unexpected behaviour can be anything. So if
nobody gets educated, then the …rm is free to have any belief if,
hypothetically, it sees an educated person.
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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.
This is only sustainable if skilled workers don’t …nd it worthwhile to
be educated. So we must have w1 c p.
So any pro…le of the following form is a pooling equilibrium: "Nobody
gets educated. w0 = µ(s = 1je = 0) = p,
w1 = µ(s = 1je = 1) 2 [0, c + p ]."
Exercise (to do at home): …nd all equilibria where everyone gets
educated. When do such equilibria exist?
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Separating Equilibrium
In a separating equilibrium, each type does something di¤erent.
Here, it must be that the skilled get educated, and the unskilled
don’t. (Why can’t it be the opposite?)
As a result, w0 = µ(s = 1je = 0) = 0 and w1 = µ(s = 1je = 1) = 1.
For this equilibrium to exist, it must be that:
1
2
the skilled …nd an education worthwhile: c 1; and
the unskilled don’t …nd an education worthwhile: k 1.
Thus, the signaling device must be not too costly for some and costly
enough for others.
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Semi-Separating (or Hybrid) Equilibrium
In a semi-separating equilibrium, one type plays a mixed strategy.
For example, fraction q of skilled workers get an education.
Thus, skilled workers must be indi¤erent between e = 1 and e = 0,
so c = w1 w0 .
Hence, unskilled workers will strictly prefer being uneducated, so
w1 = µ(s = 1je = 1) = 1.
Thus, w0 = 1 c. Also, we must have
p (1 q )
w0 = µ(s = 1je = 0) = 1 pq .
Thus, 1
c=
p (1 q )
1 pq ,
so q =
p +c 1
cp .
Since q 2 (0, 1), this equilibrium exists if and only if c 2 (1
p, 1).
Exercise (to do at home): …nd all equilibria where the unskilled type
plays a mixed strategy.
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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.
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