Transcript Extensive Form - London School of Economics

Prerequisites
Almost essential
Risk
Frank Cowell: Microeconomics
February 2007
Signalling
MICROECONOMICS
Principles and Analysis
Frank Cowell
Introduction
Frank Cowell: Microeconomics
Jump to
“Moral
Hazard”
Jump to
“Adverse
selection”

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A key aspect of hidden information
Information relates to personal characteristics

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But a fundamental difference from screening

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hidden information about actions is dealt with under “moral
hazard”
informed party moves first
opposite case (where uninformed party moves first) dealt with
under “adverse selection”
Nature of strategic problem


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uncertainty about characteristics: game of imperfect information
updating by uninformed party in the light of the signal
equilibrium concept: perfect Bayesian Equilibrium (PBE)
Signalling
Frank Cowell: Microeconomics
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Agent with the information makes first move:
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Types of signal

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subtly different from other “screening” problems
move involves making a 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

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this characteristic cannot be costlessly observed by others
let us call it “talent”…
Talent
Frank Cowell: Microeconomics
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Suppose individuals differ in terms of hidden
talent τ
Talent is valuable in the market
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If a signal is not possible

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but possessor of τ cannot convince buyers in the market
without providing a signal that he has it
may be no market equilibrium
If a signal is possible

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will there be equilibrium?
…more than one equilibrium?
Overview...
Signalling
Frank Cowell: Microeconomics
Costly signals:
model
An educational
analogy
Costly signals:
equilibrium
Costless signals
Costly signals
Frank Cowell: Microeconomics

Suppose that a “signal” costs something
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Consider a simple model of the labour market
Suppose productivity depends on ability
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the able – ta
the bog-standard – tb
ta > tb
Single type of job

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Ability is not observable
Two types of workers:
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physical investment…
forgone income…
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?
Signals: educational “investment”
Frank Cowell: Microeconomics
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Consider the decision about whether acquire education
Suppose talent on the job identical to talent at achieving
educational credentials
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Education does not enhance productive ability

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assumed to be common knowledge
may be worth “investing” in the acquisition of credentials.
simply an informative message or credential
flags up innate talent
high ability people acquire education with less effort
Education is observable

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certificates can be verified costlessly
firms may use workers'’ education as an informative signal
Signalling by workers
“Nature” determines worker’s type
Workers decide on education
0
1-p
[LOW]
[HIGH]
h
[NOT
INVEST]
h
Firms make wage offers
Workers decide whether to accept
[NOT
[INVEST]
INVEST]
[INVEST]
f1
[low]
[high]
[low]
[high]
f2
[low]
[high]
[low]
[high]
[low]
[high]
[low]
[high]
investment involves
time and money
simultaneous
offers: Bertrand
competition
h
[accept 1]
Frank Cowell: Microeconomics
p
…
…
…
Examine stages
1-3 more closely
A model of costly signals
Frank Cowell: Microeconomics
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Previous sketch of problem is simplified
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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
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workers only make binary decisions (whether or not to invest)
firms only make binary decisions (high or low wage)
C(0, t) = 0
Czz(z, t) > 0
Cz(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
Costly signals
Frank Cowell: Microeconomics
(education, cost)-space
Cost function for an a type
C
Cost function for a b type
Costs of investment z0
MC of investment z0
C(•,tb)
C(z0,ta)
C(•,ta)
C(z0,tb)
0
z
z0
C(z, t) = (1/t) z2
y
18
16
low t
14
12
Payoffs to individuals
10
8
6
Frank Cowell: Microeconomics
high t
4
2
z
0
0
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1.5
2
2.5
individuals only care about income
measure utility directly in terms of income:
v(y, z; t) := y - C(z, t)
v depends on τ because talent reduces the cost of net income
Shape of C means that ICs in (z, y)-space satisfy single-crossing
condition:

Example
1
Talent does not enter the worker's utility function directly


0.5
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
3
3.5
Rational behaviour
Frank Cowell: Microeconomics
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Workers:
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Wage is conditioned on “signal” that they provide
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through acquisition of educational credentials
Type-τ worker chooses z to maximise
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assume income y is determined by wage
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

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What will equilibrium be?
Overview...
Signalling
Frank Cowell: Microeconomics
Costly signals:
model
Costly signals
discriminate
among agents
Costly signals:
equilibrium
Costless signals
•Separating equilibrium
•Out-of-equilibrium behaviour
•Pooling equilibrium
Separating equilibrium (1)
Frank Cowell: Microeconomics
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Start with a separating Perfect Bayesian Equilibrium
Both type-a and type-b agents are maximising
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Therefore, for the talented a-types we have
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f(ta) - C(za, tb) ≤ f(tb) - C(zb, tb)
Rearranging this we have
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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:
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so neither wants to switch to using the other's signal
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
Separating equilibrium (2)
Frank Cowell: Microeconomics
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If there are just two types, at the optimum zb = 0
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So, conditions for separating
equilibrium
become
remember
that
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C(za, ta) ≤ f(ta) - f(tb)
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
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everyone knows there are only two productivity types
education does not enhance productivity
so no gain to b-types in buying education
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…
Bounds to education
Frank Cowell: Microeconomics
IC for a b type
IC for an a type
y
critical value for an a type
critical value for a b type
possible equilibrium z-values
v(•,ta)
both curves pass through
(0, f(tb))
f(ta) = f(tb) - C(z1, ta)
f(ta)
f(ta) = f(tb) - C(z0, tb)
v(•,tb)
f(tb)
0
Separating eqm:
Two examples
z0
z1
z
Separating equilibrium: example 1
Frank Cowell: Microeconomics
“bounding” ICs for each type
possible equilibrium z-values
wage schedule
y
max type-b’s utility
v(•,ta) max type-a’s utility
both curves pass through
(0, f(tb))
•
f(ta)
determines z0, z1 as before
w(•)
low talent acquires zero
education
v(•,tb)
f(tb)
high talent acquires
education close to z0
•
0
za
z
Separating equilibrium: example 2
Frank Cowell: Microeconomics
possible equilibrium z-values
a different wage schedule
y
max type-b’s utility
max type-a’s utility
v(•,ta)
•
f(ta)
 just as before
low talent acquires zero
education (just as before)
high talent acquires
education close to z1
w(•)
v(•,tb)
f(tb)
•
0
za
z
Overview...
Signalling
Frank Cowell: Microeconomics
Costly signals:
model
More on
beliefs…
Costly signals:
equilibrium
Costless signals
•Separating equilibrium
•Out-of-equilibrium behaviour
•Pooling equilibrium
Out-of-equilibrium-beliefs: problem
Frank Cowell: Microeconomics
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For a given equilibrium can redraw w(⋅)-schedule
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Shape of the w(⋅)-schedule at other values of z?
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resulting attainable set for the workers must induce
them to choose (za, f(ta)) and (0, f(tb))
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
Perfect Bayesian Equilibria
Frank Cowell: Microeconomics

Requirements for PBE do not help us to select among the
separating equilibria
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Education level z0 is the minimum-cost signal for a-types
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try common sense?
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?
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important cases of strategic interaction that produce Paretodominated outcomes
Need a proper argument, based on the reasonableness of such an
equilibrium
Out-of-equilibrium beliefs: a criterion
Frank Cowell: Microeconomics
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Is an equilibrium at za > z0 “reasonable”?
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Imagine someone observed choosing z′
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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:
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requires w(•) that sets w(z′) < f(ta) for z0 < z′ < za
so firms must be assigning the belief π(z′)>0
π(z′) = 0 for any z′  (z0, za)
So only separating equilibrium worth considering is where

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a-types are at (z0, f(ta))
b-types are at (0, f(tb)).
Overview...
Signalling
Frank Cowell: Microeconomics
Costly signals:
model
Agents appear to
be al the same
Costly signals:
equilibrium
Costless signals
•Separating equilibrium
•Out-of-equilibrium behaviour
•Pooling equilibrium
Pooling
Frank Cowell: Microeconomics
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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
Example
 Ef(t):=[1 - p]f(ta) + pf(tb)
Being paid this wage might be in interests of all workers…
No signals: an example
Frank Cowell: Microeconomics
possible z-values with signalling
outcome under signalling
y
outcome without signalling
v(•,tb)
highest a-type IC under
signalling
v(•,ta)
f(ta)
Ef(t)
both pass through (0, Ef(t))
the type-b IC must be
higher than with signalling
but, in this case, so is the
type-a IC
•
f(tb)
0
zz00
z
z1
should school be
banned?
Pooling: limits on z?
Frank Cowell: Microeconomics
b-type payoff with 0 education
critical IC for a b-type
y
expected marginal productivity
critical z-value for b-type to accept
pooling payoff
v(•,tb)
viable z-values in pooling eqm
Ef(t) = [1-p]f(ta) + pf(tb)
f(ta)
[1-p]f(ta) + pf(tb) - C(z2, tb)
= f(tb)
Ef(t)
f(tb)
z
0
z2
Pooling equilibrium: example 1
Frank Cowell: Microeconomics
expected marginal productivity
viable z-values in pooling eqm
v(•,tb)
y
v(•,ta)
wage schedule
utility maximisation
equilibrium education
f(ta)
w(•)
Ef(t)
f(tb)
0
z*
z
Pooling equilibrium: example 2
Frank Cowell: Microeconomics
expected marginal productivity
viable z-values in pooling eqm
v(•,tb)
y
v(•,ta)
wage schedule
utility maximisation
equilibrium education
f(ta)
but is pooling consistent with
out-of-equilibrium behaviour?
w(•)
Ef(t)
f(tb)
0
z*
z
Intuitive criterion again
Frank Cowell: Microeconomics
a pooling equilibrium
a critical z-value z'
y
wage offer for an a-type at z0 > z'
max b-type utility at z0
v(•,ta)
v(•,tb)
f(ta)
Ef(t)
Ef(t) - C(z*, tb) = f(ta) - C(z′,tb)
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
z with the intuitive criterion
f(tb)
0
max a-type utility at z0
z* z' z0
Overview...
Signalling
Frank Cowell: Microeconomics
Costly signals:
model
An argument by
example
Costly signals:
equilibrium
Costless signals
Costless signals: an example
Frank Cowell: Microeconomics

Present the issue with a simplified example
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N risk-neutral agents share in a project with output
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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
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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)
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general treatments can be difficult
q - ch
Consider this as a simultaneous-move game
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what is the NE?
improve on NE by making announcements before the game starts?
Example: NE without signals
Frank Cowell: Microeconomics
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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
h
 −c
otherwise
Expected payoff to h is a[p]N−1 − ch
Probability that expected payoff is positive is a[p]N−1

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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
Example: introduce signals
Frank Cowell: Microeconomics
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Introduce a preliminary stage to the game
Each agent has the opportunity to signal his intention:
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Then there is an equilibrium in which the following occurs
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each agent announces [YES] or [NO] to the others
each agent then decides whether or not to participate
each h announces [YES] if and only if ch < α
h selects zh = 1 iff all agents have announced [YES]
In this equilibrium:
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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
Signalling: summary
Frank Cowell: Microeconomics
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Both costly and costless signals are important
Costly signals:
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separating PBE not unique?
intuitive criterion suggests out-of-equilibrium beliefs
pooling equilibrium may not be unique
inconsistent with intuitive criterion?
Costless signals:
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a role to play in before the game starts