5. Two-Stage Games with Incomplete Information
Georg Nöldeke
Wirtschaftswissenschaftliche Fakultät, Universität Basel
Advanced Microeconomics (HS 10)
Two-Stage Games with Incomplete Information
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1. Signaling Games
1.1 Definition
A signaling game is a two-player two-stage game with observable
actions and incomplete information with the following structure
Players i = 1, 2. Player 1 is called the sender, player 2 the receiver
At stage 0 nature draws a type t for the sender from a finite set T
according to a probability distribution p on T satisfying p(t) > 0 for
all t ∈ T . The sender observes t, the receiver does not.
At stage 1 the sender chooses a message m ∈ M. The game
continues to stage 2.
At stage 2 the receiver, having observed m but not t chooses a
response r ∈ R. Thereafter the game stops.
Payoffs are given by u1 (t, m, r) for the sender and u2 (t, m, r) for the
receiver.
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1. Signaling Games
1.2 Strategies, Strategic Form, and Bayesian Nash Equilibrium
A strategy for the sender is a function m̂ : T → M specifying a
message m̂(t) for every type of the sender. Let M̂ denote the set of
all strategies for the sender.
A strategy for the receiver is a function r̂ : M → M specifying a
response r̂(m) for every message he might receive. Let R̂ denote
the set of all strategies for the receiver.
Given a strategy profile (m̂, r̂) the expected payoff of the sender is
U1 (m̂, r̂) =
∑ p(t)u1 (t, m(t), r(m(t)))
t∈T
whereas the expected payoff of the receiver is
U2 (m̂, r̂) =
∑ p(t)u2 (t, m(t), r(m(t))).
t∈T
A strategy profile (m∗ , r∗ ) is a Bayesian Nash equilibrium of the
signaling game if for all (m̂, r̂) ∈ M̂ × R̂ the inequalities
U1 (m∗ , r∗ ) ≥ U1 (m̂, r∗ ) and U2 (m∗ , r∗ ) ≥ U2 (m∗ , r̂) hold.
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1. Signaling Games
1.2 Strategies, Strategic Form, and Bayesian Nash Equilibrium
The equilibrium condition for the sender is equivalent to requiring
that m∗ (t) solves
max u1 (t, m, r∗ (m))
m∈M
for all t, that is each type of the sender chooses a message which
is optimal given the receiver’s strategy.
Rewriting the equilibrium condition for the seller as requiring
optimality conditional on the message he has received is more
cumbersome (and reveals a problem):
Let T ∗ (m) = {t ∈ T | m∗ (t) = m}. T ∗ (m) 6= 0/ means that there is some
type of the sender who uses signal m.
The equilibrium condition for the seller is equivalent to requiring that
r∗ (m) solves
max ∑ p(t)u2 (t, m, r)
r∈R
t∈T ∗ (m)
for all m such that T ∗ (m) 6= 0.
/
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1. Signaling Games
1.2 Strategies, Strategic Form, and Bayesian Nash Equilibrium
The problem is that for those messages with T ∗ (m) = 0/ Bayesian
Nash equilibrium imposes no restrictions on r∗ (m).
For instance, even if there exists r0 ∈ R such that
u2 (t, m, r0 ) > u2 (t, m, r) holds for all r 6= r0 , Bayesian Nash equilibrium
does not require the receiver to choose the response r0 after having
observed m.
The concept of a perfect Bayesian (Nash) equilibrium addresses
this problem by adding an optimality condition restricting the
receiver’s response against the unused messages.
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1. Signaling Games
1.3 Perfect Bayesian Equilibrium
Definition (Belief)
A belief for the receiver in a signaling game is given by a function
µ : T × M → [0, 1] satisfying
∑ µ(t | m) = 1
t∈T
for all m ∈ M.
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1. Signaling Games
1.3 Perfect Bayesian Equilibrium
Definition (Perfect Bayesian Equilibrium)
A perfect Bayesian equlibrium for a signaling game is given by a
strategy profile (m∗ , r∗ ) and a belief µ ∗ such that
1
for all t ∈ T the message m∗ (t) solves
max u1 (t, m, r∗ (m)).
m∈M
2
For all m ∈ M the response r∗ (m) solves
max ∑ µ(t | m)u2 (t, m, r).
r∈R t∈T
3
For all m such that T ∗ (m) 6= 0/ the belief satisfies
∗
µ (t | m) =
p(t)
∑t 0 ∈T ∗ (m)
p(t 0 )
.
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1. Signaling Games
1.4 Remarks
Perfect Bayesian equilibrium is a refinement of Bayesian Nash
equilibrium: If (m∗ , r∗ , µ ∗ ) is a perfect Bayesian equilibrium of a
signaling game then (m∗ , r∗ ) is a Bayesian Nash equilibrium of the
signaling game.
The concept of perfect Bayesian equilibrium can be defined for
arbitrary games in extensive form and then refines both Bayesian
Nash equilibria and subgame perfect equilibria.
The definition of a perfect Bayesian equilibrium for a signaling
game extends to models with a continuum of types in the same
way that the definition of a Bayesian Nash equilibrium does.
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1. Signaling Games
1.5 Separating and Pooling Equilibria
Definition (Separating Equilibrium)
A perfect Bayesian equilibrium (m∗ , r∗ , µ ∗ ) of a signaling game is
separating if different types of the sender choose different messages:
t 6= t 0 ⇒ m∗ (t) 6= m∗ (t 0 ).
Definition (Pooling Equilibrium)
A perfect Bayesian equilibrium (m∗ , r∗ , µ ∗ ) of a signaling game is
pooling if all types of the sender choose the same message: there
exists mP ∈ M such that m∗ (t) = mP holds for all t ∈ T .
If there are only two possible types of the sender every equilibrium
is either separating or pooling.
If there are more than two types of the sender, there may also be
semi-separating or partially pooling equilibria, i.e., perfect
Bayesian equilibria that are neither separating or pooling.
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2. Application: Job Market Signaling
Michael Spence
Winner of the Nobel Prize in
Economics 2001
Photo: Robert Scoble,
http://upload.wikimedia.
org/wikipedia/commons/a/
a6/A_Michael_Spence.jpg
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2. Application: Job Market Signaling
2.1 The Spence Model
A variant of a signaling game in which there are two receivers:
The role of the sender is taken by a worker who has either high or
low ability. The corresponding types are denoted by H and L. To
simplify notation let p(H) = q > 0 and p(L) = 1 − q > 0.
The signal chosen at stage 1 is an education level e ∈ R+ .
In stage 2 two firms j = 2, 3 observe that worker’s eduction level
and then simultaneously make wage offers w j ∈ R+ . The worker is
assigned to whichever firm offers the higher wage. (If both firms
offer the same wage, the worker is assigned with probability 1/2 to
each firm.
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2. Application: Job Market Signaling
2.1 The Spence Model
The payoff of the worker is
u1 (t, e, w) = w − c(t, e),
where w is the higher of the wages offered by the two firms and
c(t, e) is the cost of the worker of obtaining education e if his type
is t ∈ {L, H}.
The payoff of a firm is 0 if it does not obtain the worker and
u j (t, e, w) = y(t, e) − w
otherwise, where y(t, e) is the value of the output of a worker with
ability t who has obtained eduction e.
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2. Application: Job Market Signaling
2.1 The Spence Model
Assumptions:
High ability workers produce more output than low ability workers
with the same education level: y(H, e) > y(L, e) for all e.
Education is not detrimental to productivity: y(t, e) is
non-decreasing in e for t ∈ {L, H}.
An interesting special case is the one of unproductive education in
which y(t) does not depend on e.
The cost of education is increasing in the education level: c(t, e) is
increasing in e for t ∈ {L, H}.
The cost of education has increasing differences, also known as
the single crossing property or sorting condition:
e0 > e ⇒ c(L, e0 ) − c(L, e) > c(H, e0 ) − c(H, e).
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2. Application: Job Market Signaling
2.2 Perfect Bayesian Equilibrium
Assuming that both firms hold identical beliefs µ ∗ they will make
identical wage offers w∗ (e) in a perfect Bayesian equilibrium
satisfying
w∗ (e) = µ ∗ (H | e)y(H, e) + (1 − µ ∗ (H | e))y(L, e) for all e.
(1)
Optimality of the worker’s strategy e∗ requires that for t ∈ {L, H}
the education level e∗ (t) solves the problem
max w∗ (e) − c(t, e).
e
(2)
If e∗ (L) 6= e∗ (H) the equilibrium is separating and the remaining
equilibrium condition is
µ ∗ (H | e∗ (H)) = 1 and µ ∗ (H | e∗ (L)) = 0.
(3)
If e∗ (L) = e∗ (H) = eP the equilibrium is pooling and the remaining
equilibrium condition is
µ ∗ (H | eP ) = q.
(4)
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2. Application: Job Market Signaling
2.2 Perfect Bayesian Equilibrium
To describe the outcome of a perfect Bayesian equilibrium is suffices
to specify (e∗ (L), e∗ (H)):
If the equilibrium is separating, the corresponding wages are
given by y(L, e∗ (L)) for the worker of type L and y(H, e∗ (H)) for the
worker of type H.
If the equilibrium is pooling, both types of workers receive the
wage q · y(H, eP ) + (1 − q) · y(L, eP ).
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2. Application: Job Market Signaling
2.3 Pooling Equilibria Outcomes
Proposition
There is a pooling equilibrium with e∗ (L) = e∗ (H) = eP if and only if
q · y(H, eP ) + (1 − q) · y(L, eP ) − c(t, eP ) ≥ max [y(L, e) − c(t, e)]
e
holds for t ∈ {L, H}.
When education is unproductive, the condition
q · [y(H) − y(L)] ≥ c(L, eP ) − c(L, 0)
is necessary and sufficient for the existence of a pooling
equilibrium in which both types choose eP .
Note that this condition is satisfied for eP = 0, so the existence of a
pooling equilibrium is assured.
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2. Application: Job Market Signaling
2.4 Separating Equilibria Outcomes
Proposition
There is a separating equilibrium in which the sender’s strategy is
given by e∗ (L) and e∗ (H) if and only if e∗ (L) solves
max [y(L, e) − c(L, e)]
e
and e∗ (H) satisfies
y(H, e∗ (H)) − c(H, e∗ (H)) ≥ max [y(L, e) − c(H, e)] ,
e
as well as
y(L, e∗ (L)) − c(L, e∗ (L)) ≥ y(H, e∗ (H)) − c(L, e∗ (H)).
When education is unproductive, e∗ (L) = 0 holds in every
separating equilibrium, whereas there usually is a range of values
e∗ (H) that may occur in a separating equilibrium.
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