Prerequisites
Almost essential
Risk
SIGNALLING
MICROECONOMICS
Principles and Analysis
Frank Cowell
July 2015
Frank Cowell: Signalling
1
Introduction
 A key aspect of hidden information
 Information relates to personal characteristics
• hidden information about actions is dealt with under “moral hazard”
 But a fundamental difference from screening
• informed party moves first
• opposite case (where uninformed party moves first) dealt with under
“adverse selection”
 Nature of strategic problem
• uncertainty about characteristics: game of imperfect information
• updating by uninformed party in the light of the signal
• equilibrium concept: perfect Bayesian Equilibrium (PBE)
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Signalling
 Agent with the information makes first move:
• subtly different from other “screening” problems
• move involves making a signal
 Types of signal
• could be a costly action (physical investment, advertising, acquiring an
educational certificate)
• could be a costless message (manufacturers' assurances of quality,
promises by service deliverers)
 Message is about a characteristic
• this characteristic cannot be costlessly observed by others
• let us call it “talent”
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Talent
 Suppose individuals differ in terms of hidden talent τ
 Talent is valuable in the market
• but possessor of τ cannot convince buyers in the market
• without providing a signal that he has it
 If a signal is not possible
• may be no market equilibrium
 If a signal is possible
• will there be equilibrium?
• more than one equilibrium?
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Frank Cowell: Signalling
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Overview
Signalling
Costly signals:
model
An educational analogy
Costly signals:
equilibrium
Costless signals
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Frank Cowell: Signalling
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Costly signals
 Suppose that a “signal” costs something
• physical investment
• forgone income
 Consider a simple model of the labour market
 Suppose productivity depends on ability
• ability is not observable
 Two types of workers:
• the able – ta
• the basic – tb
• ta > tb
 Single type of job
• employers know the true product of a type t-person
• if they can identify which is which
 How can able workers distinguish themselves from others?
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Signals: educational “investment”
 Consider the decision about whether acquire education
 Suppose talent on the job identical to talent at achieving
educational credentials
• assumed to be common knowledge
• may be worth “investing” in the acquisition of credentials
 Education does not enhance productive ability
• simply an informative message or credential
• flags up innate talent
• high ability people acquire education with less effort
 Education is observable
• certificates can be verified costlessly
• firms may use workers'’ education as an informative signal
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Signalling by workers
 “Nature” determines worker’s type
0
p
1-p
[LOW]
[HIGH]
h
[NOT
INVEST]
 Workers decide on education
 Firms make wage offers
 Workers decide whether to accept
h
[INVEST]
[NOT
INVEST]
[INVEST]
f1
[low]
[high]
[low]
[high]
simultaneous offers:
Bertrand competition
f2
[low]
[high]
[low]
investment involves
time and money
[high]
[low]
[high]
[low]
[high]
[accept 1]
h
July 2015
…
…
…
Frank Cowell: Signalling
Examine stages
1-3 more closely
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A model of costly signals
 Previous sketch of problem is simplified
• workers only make binary decisions (whether or not to invest)
• firms only make binary decisions (high or low wage)
 Suppose decision involve choices of z from a continuum
 Ability is indexed by a person’s type t
 Cost of acquiring education level z is C(z, t) ≥ 0
• C(0, t) = 0 Cz(z, t) > 0
• Czz(z, t) > 0 Czt(z, t) < 0
 Able person has lower cost for a given education level
 Able person has lower MC for a given education level
 Illustrate this for the two-type case
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Costly signals
(education, cost)-space
Cost function for an a type
Cost function for a b type
Costs of investment z0
MC of investment z0
C
C(•,tb)
C(z0,ta)
C(•,ta)
C(z0,tb)
0
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z
z0
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Payoffs to individuals
18
y
C(z, t) = (1/t) z2
16
low t
14
12
 Talent does not enter the utility function directly
•
•
•
•
10
8
6
individuals only care about income
4
2
measure utility directly in terms of income:
0
v(y, z; t) := y - C(z, t)
0
0.5
v depends on τ because talent reduces the cost of net income
high t
z
1
1.5
2
2.5
 Shape of C means that ICs in (z, y)-space satisfy single-crossing
condition:
•
•
•
•
•
•
IC for a person with talent t is: y = u + C(z, t)
slope of IC for this type is: dy/dz = Cz(z, t)
for person with higher talent (t'>t) slope of IC is: dy/dz = Cz(z, t')
but Czt(z, t) < 0 so IC(t') is flatter than IC(t) at any value of z
so, if IC(t') and IC(t) intersect at (z0, y0)
IC(t') lies above original IC(t) for z < z0 and below IC(t) for z > z1
 This is important to simplify the structure of the problem
Example
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3
3.5
Rational behaviour
 Workers:
• assume income y is determined by wage
 Wage is conditioned on “signal” that they provide
• through acquisition of educational credentials
 Type-τ worker chooses z to maximise
• w(z) - C(z, t)
• where w(⋅) is the wage schedule that workers anticipate will be offered by
firms
 Firms:
• assume profits determined by workers’ talent
 Need to design w(⋅) to max profits
• depends on beliefs about distribution of talents
• conditional on value of observed signal
 What will equilibrium be?
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Overview
Signalling
Costly signals:
model
Costly signals
discriminate
among agents
Costly signals:
equilibrium
•Separating equilibrium
•Out-of-equilibrium behaviour
•Pooling equilibrium
Costless signals
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Separating equilibrium (1)
 Start with a separating Perfect Bayesian Equilibrium
 Both type-a and type-b agents are maximising
• so neither wants to switch to using the other's signal
 Therefore, for the talented a-types we have
• f(ta) - C(za, ta) ≥ f(tb) - C(zb, ta)
• if correctly identified, no worse than if misidentified as a b-type
 Likewise for the b-types:
• f(ta) - C(za, tb) ≤ f(tb) - C(zb, tb)
 Rearranging this we have
• C(za, tb) - C(zb, tb) ≥ f(ta) - f(tb)
• positive because f(⋅) is strictly increasing and ta > tb
• but since Cz > 0 this is true if and only if za > zb
 So able individuals acquire more education than the others
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Frank Cowell: Signalling
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Separating equilibrium (2)
 If there are just two types, at the optimum zb = 0
• everyone knows there are only two productivity types
• education does not enhance productivity
• so no gain to b-types in buying education
 So, conditions for separating equilibrium become
• C(za, ta) ≤ f(ta) - f(tb)
remember that
• C(za, tb) ≥ f(ta) - f(tb)
C(0, t)=0
 Let z0, z1 be the critical z-values that satisfy these conditions
with equality
• z0 such that f(tb) = f(ta) - C(z0, tb)
• z1 such that f(tb) = f(ta) - C(z1, ta)
 Values z0, z1 set limits to education in equilibrium
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Bounds to education
IC for a b type
IC for an a type
critical value for an a type
critical value for a b type
possible equilibrium z-values
y
v(•,ta)
both curves pass through (0, f(tb))
f(ta) = f (tb) - C(z1, ta)
f(ta) = f (tb) - C(z0, tb)
f(ta)
v(•,tb)
f(tb)
0
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z0
z1
z
Frank Cowell: Signalling
Separating eqm:
Two examples
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Separating equilibrium: example 1
“bounding” ICs for each type
possible equilibrium z-values
wage schedule
y
max type-b’s utility
max type-a’s utility
v(•,ta)
both curves pass through (0, f(tb))
f(ta)
•
w(•)
determines z0, z1 as before
low talent acquires zero education
v(•,tb)
f(tb)
0
July 2015
high talent acquires education
close to z0
•
za
z
Frank Cowell: Signalling
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Separating equilibrium: example 2
possible equilibrium z-values
a different wage schedule
max type-b’s utility
y
max type-a’s utility
v(•,ta)
f(ta)
•
 just as before
low talent acquires zero
education (just as before)
w(•)
high talent acquires
education close to z1
v(•,tb)
f(tb)
•
0
July 2015
za
z
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Overview
Signalling
Costly signals:
model
More on beliefs
Costly signals:
equilibrium
•Separating equilibrium
•Out-of-equilibrium behaviour
•Pooling equilibrium
Costless signals
July 2015
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Out-of-equilibrium-beliefs: problem
 For a given equilibrium can redraw w(⋅)-schedule
• resulting attainable set for the workers must induce them to
choose (za, f(ta)) and (0, f(tb))
 Shape of the w(⋅)-schedule at other values of z?
• captures firms' beliefs about workers’ types in situations that
do not show up in equilibrium
 PBE leaves open what out-of-equilibrium beliefs may
be
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Perfect Bayesian Equilibria
 Requirements for PBE do not help us to select among the
separating equilibria
• try common sense?
 Education level z0 is the minimum-cost signal for a-types
• a-type's payoff is strictly decreasing in za over [z0, z1]
• any equilibrium with za > z0 is dominated by equilibrium at z0
 Are Pareto-dominated equilibria uninteresting?
• important cases of strategic interaction that produce Pareto-dominated
outcomes
• need a proper argument, based on the reasonableness of such an
equilibrium
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Out-of-equilibrium beliefs: a criterion
 Is an equilibrium at za > z0 “reasonable”?
• requires w(•) that sets w(z′) < f(ta) for z0 < z′ < za
• so firms must be assigning the belief π(z′)>0
 Imagine someone observed choosing z′
• b-type IC through (z′, f(ta)) lies below the IC through (0, f(tb))
• a b-type knows he’s worse off than in the separating equilibrium
• a b-type would never go to (z′, f(ta))
• so anyone at z′ out of equilibrium must be an a-type
 An intuitive criterion:
• π(z′) = 0 for any z′  (z0, za)
 So only separating equilibrium worth considering is where
• a-types are at (z0, f(ta))
• b-types are at (0, f(tb))
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Overview
Signalling
Costly signals:
model
Agents appear to
be al the same
Costly signals:
equilibrium
•Separating equilibrium
•Out-of-equilibrium behaviour
•Pooling equilibrium
Costless signals
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Pooling
 There may be equilibria where the educational signal does not work
• no-one finds it profitable to "invest" in education?
• or all types purchase the same z?
• depends on distribution of t
• and relationship between marginal productivity and t
 All workers present themselves with the same credentials
• so they are indistinguishable
• firms have no information to update their beliefs
 Firms’ beliefs are derived from the distribution of t in the population
• this distribution is common knowledge
 So wage offered is expected marginal productivity
• E f(t):=[1 - p]f(ta) + pf(tb)
 Being paid this wage might be in interests of all workers
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Frank Cowell: Signalling
Example
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No signals: an example
possible z-values with signalling
outcome under signalling
y
outcome without signalling
v(•,tb)
v(•,ta)
highest a-type IC under signalling
f(ta)
E f(t)
both pass through (0, E f(t))
the type-b IC must be higher than
with signalling
but, in this case, so is the type-a IC
•
f(tb)
0
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should school be banned?
zz00
z
z1
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Pooling: limits on z?
b-type payoff with 0 education
critical IC for a b-type
expected marginal productivity
y
critical z for b-type to accept pooling payoff
viable z -values in pooling eqm
v(•,tb)
E f(t) = [1-p]f(ta) + pf(tb)
f(ta)
[1-p] f(ta) + pf(tb) - C(z2, tb) = f(tb)
E f(t)
f(tb)
z
0
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z2
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Pooling equilibrium: example 1
expected marginal productivity
viable z-values in pooling eqm
v(•,tb)
y
f(ta)
wage schedule
utility maximisation
equilibrium education
v(•,ta)
w(•)
E f(t)
f(tb)
0
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z*
z
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Pooling equilibrium: example 2
expected marginal productivity
viable z-values in pooling eqm
wage schedule
utility maximisation
v(•,tb)
y
f(ta)
v(•,ta)
equilibrium education
but is pooling consistent with
out-of-equilibrium behaviour?
w(•)
E f(t)
f(tb)
0
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z*
z
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Intuitive criterion again
a pooling equilibrium
a critical z-value z'
wage offer for an a-type at z0 > z'
max b-type utility at z0
y
max a-type utility at z0
v(•,ta)
E f(t) - C(z*, tb) = f(ta) - C(z′,tb)
v(•,tb)
f(ta)
E f(t)
f(tb)
z
0
July 2015
b-type would not choose z0
under intuitive criterion p(z0) = 0
a-type gets higher utility at z0
would move from z* to z0
so pooling eqm inconsistent
with intuitive criterion
z* z' z0
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Overview
Signalling
Costly signals:
model
An argument by
example
Costly signals:
equilibrium
Costless signals
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Costless signals: an example
 Present the issue with a simplified example
• general treatments can be difficult
 N risk-neutral agents share in a project with output
• q = a[z1×z2×z3×...] where 0 < α < 1
• zh = 0 or 1 is participation indicator of agent h
 Agent h has cost of participation ch (unknown to others)
• ch  [0,1]
• it is common knowledge that prob(ch ≤ c) = c
 Output is a public good, so net payoff to each agent h is
• q - ch
 Consider this as a simultaneous-move game
• what is the NE?
• improve on NE by making announcements before the game starts?
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Example: NE without signals
 Central problem: each h risks incurring cost ch while getting
consumption 0
 If π is the probability that any other agent participates, payoff to
h is
• a −ch with probability [p]N−1
• −ch
otherwise
 Expected payoff to h is a[p]N−1 − ch
 Probability that expected payoff is positive is a[p]N−1
• but this is the probability that agent h actually participates
• therefore p = a[p]N−1
• this can only be satisfied if p = 0
 So the NE is zh = 0 for all h, as long as α < 1
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Example: introduce signals
 Introduce a preliminary stage to the game
 Each agent has the opportunity to signal his intention:
• each agent announces [YES] or [NO] to the others
• each agent then decides whether or not to participate
 Then there is an equilibrium in which the following occurs
• each h announces [YES] if and only if ch < α
• h selects zh = 1 iff all agents have announced [YES]
 In this equilibrium:
• agents don’t risk wasted effort
• if there are genuine high-cost ch agents present that inhibit the project
• this will be announced at the signalling stage
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Signalling: summary
 Both costly and costless signals are important
 Costly signals:
• separating PBE not unique?
• intuitive criterion suggests out-of-equilibrium beliefs
• pooling equilibrium may not be unique
• inconsistent with intuitive criterion?
 Costless signals:
• a role to play in before the game starts
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