Adverse Selection
Asymmetric information is feature of many markets
- some market participants have information that the others do not have
1) The hiring process – a worker might know more about his ability than the firm does
- the idea is that there are several types of workers
- some are more productive than others are
2) Insurance – insurance companies do not observe individual characteristics such as
driving skills
3) Project financing – entrepreneurs might have more information about projects than
potential lenders
4) Used cars – sellers know more about the car’s quality than buyers
Adverse selection is often a feature in these settings
- it arises when an informed individual’s decisions depend on his privately held information
in a way that adversely affects uninformed market participants
Lemon’s problem
A classic example: Alerlof ’s “lemons” used car market. Nobel 2001: Akerlof, Spence, Stiglitz
- sellers of used cars have private information on vehicle quality, which buyer’s lack
- suppose we have two types of cars high and low quality; only sellers observe type
Two types: high and low quality: denote buyer's value Vi , and seller's value U i
Assume that VH > VL , and that VH  U H , VL  U L
Assume the high type occurs with probablity p , low occurs with prob. (1- p )
Willingness to pay for a car, given that type is unknown is
 p VH  (1- p ) VL 
1) There is a market failure if U H   p VH  (1- p ) VL  since trade will not take place.
Sellers of high quality cars will take them off the market, so only lemons are left
2) There is an incentive for sellers of U H cars to signal their quality to buyers
Signaling
In the example above,
-sellers with high quality cars would want to convince buyers that the car is high quality.
-the signaling device has to be one that sellers with bad cars cannot use effectively.
A “This is a Good Car” sign is ineffective: every type of seller will use it, and it will provide
no new info
Rule 1: in order to be effective, signaling must be costly
Rule 2: costs must differ for different “types,” with the cost structure favoring high types
Signaling devices can generate efficient outcomes, but need not always
- Since buyers have positive surplus for both types of cars and sellers prefer to sell
at any price above their reservation price, efficiency requires that both good and
bad cars be offered for sale
- An effective signal might be a guarantee, since people with bad cars would be unwilling
to guarantee their cars, or let a mechanic inspect the car
- Since employing signaling devices requires incurring costs (otherwise the devices are
ineffective) they can actually make agents worse off
Perfect Bayesian Equilibrium
In order to analyze signaling games, we sometimes use a refinement of WPBE
Recall that given the equilibrium an information set is on-the-path if it will be reached with
positive probability if the game is played according to the equilibrium strategies, and
it is off-the-path if the information set is certain not to be reached with positive
probability with play of the equilibrium strategies
Perfect Bayesian Equilibrium (PBE)
-Requires that at info sets off-the-path, all players agree on what their beliefs are
Signaling Game
Dynamic game of incomplete information
Two players:
- one with private info : Sender
- another without private info : Receiver
Structure of signaling game:
1) Nature moves first - determines type of sender
2) Sender moves – sender observes his own type and sends a signal
- sender’s optimal strategy depends on receiver’s strategy
3) Receiver observes signal -chooses action, optimal strategy depends on a signal
Equilibrium definition:
WPBE
1) σ is sequentially rational, given beliefs μ
2) system of beliefs μ is consistent with σ (eqm-on-the-path)
determined using Bayes’ rule whenever possible
PBE adds
3) all players have the same beliefs off-the-path
Qual Question Fall 2002 (#75)
C: charity, P: person seeking funds. The charity prefers to fund low productivity types
P is either a high productivity type Wh , or low productivity type Wl , where Wh > Wl
The ex ante probability that P is the high type is q , low type is 1 -q 
.
After observing his type, P chooses an amount of time, t , to devote to lobbying C for funds
C observes t and updates his beliefs about P's type and then either pays P payment d or not
Payoffs: If C does not pay
does pay
If C does not pay
does pay
P gets (determined by productivity) Wi  T - t 
P gets
Wi  T - t  +d
C gets 0
 if P is high type C gets  -d 

low type C gets  A - d  0 

Two types of equilibria
μ
(1-μ)
1) Separating Eqm: Different types choose different strategies. t h  t l
type is revealed along the equilibrium path ; after seeing signal t h    1; t l    0
this makes Bayesian updating simple since Pr t h P is L  Pr  P is L   0
=
Pr t h P is H  Pr  P is H 
Pr t h P is H  Pr  P is H   Pr t h P is L  Pr  P is L 
2) Pooling Eqm: Different types choose the same strategy: t h = t l . The signal gives no new info
after seeing signal t h = t l    q ; Posterior beliefs are identical to the prior belief
General Procedure
Start analyzing a Signaling game by deciding whether to look for a Separating or Pooling
Equilibrium. We’ll start with a Separating Equilibrium:
1) Along the equilibrium path, type is revealed. This gives us a place to start.
2) Assess the receiver’s BR, under the assumption that the receiver knows the sender’s type
3) Back-up the game tree; determine the sender’s optimal choice given how the receiver
plays in response to the signal
4) Then apply the incentive compatibility condition:
high type must prefer sending high signal; low type must prefer sending low signal
5) Finally, check off-the-path beliefs
- here the equilibrium concept does not specify the receiver’s beliefs
- to obtain the widest range of equilibrium paths, make the receiver pessimistic about
the sender’s type off-the-path
- then check that the high type prefers the high signal to any off-the-path signal, and
that the low type prefers the low signal to any off-the-path signal
Separating equilibrium (#75)
1) Separating equilibrium path, type is revealed
2) C’s BR (sequentially rational strategy), given that it knows P’s type
on seeing th , C believes that P is H with prob. 1; C does not pay
on seeing tl , C believes that P is L with prob. 1; C pays
3) Back-up the game tree
H’s payoff is W h (T - th) therefore in eqm. th = 0
L’s payoff is Wl (T - tl) + d
Proof of th = 0: Suppose not. Then th > 0. Consider th’ < th. . C cannot have more
pessimistic
beliefs, so the worse that can happen to P is that C does not pay. Even in this case,
P is better off because he wastes less time lobbying. Thus, th cannot be optimal.
If C does not pay given th’ , H is better off because W h (T – th’) > W h (T - th)
If C pays, H is better off because W h (T – th’) + d > W h (T - th)
QED.
Incentive compatibility
4)
The high type must prefer sending the high signal: compare t h = 0 to t l
Wh  T   Wh  T  t l   d
tl  d W
h
The t l signal has to be costly enough to deter H from pretending to be L
The low type must prefer sending the low signal: compare t l to t h = 0
Wl  T  t l   d  Wl  T 
tl  d W
l
In equilibrium,
d
d
Wh  t l  Wl
Check Off-the-path beliefs
5) Here the equilibrium concept does not specify off-the-path beliefs, so we choose them
- usually, the easiest way to specify these beliefs is to assume that the receiver is highly
pessimistic about sender’s type off-the-path.
- if any t other than th or tl is received, C believes with prob. 1 that the sender is H
- then C does not pay
Separating Perfect Bayesian Equilibrium (SPBE)
H chooses t h = 0 ; L chooses t l where d W  t l  d W
h
l
On observing t l , C updates to believe P is L with prob. 1   =0
On observing t h , C updates to believe P is H with prob. 1   =1
On observing t  t l , t h , C updates to believe P is H with prob. 1   =1
If  =0, C pays ; If  =1, C does not pay
Efficiency
The separating equilibria can be Pareto-ranked
In all of them if P is L,  =0, C pays ; if P is H,  =1, C does not pay
H gets Wh  T  ; C gets 0 if H and  A  d  if L
but L's payoff Wl
T  t l  
d  is decreasing in t l *

The efficient separating equilibrium is H chooses t h = 0; L chooses t l *  d W
h

Typically in a separating equilibrium the worst type does not incur costs to signal that it is the
worst type
With n types we can order the cost of signaling by type and again the worst type does not
incur costs to signal that it is the worst type
Pooling equilibrium (#75)
Pooling equilibria exist under general conditions and are often not very interesting
1) Along the equilibrium path, type is not revealed: th = tl = t*. Posteriors equal priors: μ = q
2) C’s best response on observing t*,
C’s payoff from paying is (1-q) A-d; C’s payoff from not paying is 0
C’s optimal choice:
if (1-q) A-d < 0 then C does not pay
if (1-q) A-d > 0 then C pays
There are no incentive compatibility conditions because both types choose the same t
3) Check off-the-path beliefs: assume pessimisms for t ≠ t*,
On observing any t ≠ t*, C updates beliefs that P is H w/ prob. 1
Given this, μ =1 and C does not pay
Further Steps
4) Both types must prefer t* to any other t
1st Case: (1-q) A-d < 0 and C does not pay
In this case, the relevant condition is W i (T - t*) ≥ W i (T- t) for all t ≠ t*,
which yields t ≥ t*
Given this, t* = 0 This is intuitive: why spend anything to get nothing?
2nd Case: (1-q) A-d > 0 and C pays
In this case, the relevant condition is W i (T - t*) + d ≥ W i (T- t) for all t ≠ t*,
which yields d/W h ≥ t*
t* cannot be too high or P would prefer choosing t= 0 and not receiving d
The WPBE
WPBE: th = tl = t*
1st Case: (1-q) A-d < 0; th = tl = t* = 0
On observing 0, C updates to μ = q and does not pay
On observing t ≠ 0 , C updates to μ = 1 and does not pay
2nd Case: (1-q) A-d ≥ 0 and d/W h ≥ t*
On observing t*, C updates to μ= q and pays
On observing t ≠ t*, C updates to μ = 1 and does not pay
Note that in the second case, the pooling equilibrium makes the senders worse off than in
the case where no signal is possible. This is an example of a general result. Pooling
equilibria generate no new information for the receiver, so if signaling is costly then the
senders would be better off if no signaling is possible