Adverse Selection

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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 cars quality than buyers Adverse selection
is often a feature in these settings - it
arises when an informed individuals decisions
depend on his privately held information in a way
that adversely affects uninformed market
participants
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Lemons problem
A classic example Alerlofs lemons used car
market. Nobel 2001 Akerlof, Spence, Stiglitz
- sellers of used cars have private
information on vehicle quality, which buyers
lack - suppose we have two types of cars
high and low quality only sellers observe type
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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
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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
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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
- senders optimal strategy depends on
receivers strategy 3) Receiver observes
signal -chooses action, optimal strategy depends
on a signal Equilibrium definition
WPBE 1) s is sequentially rational, given
beliefs µ 2) system of beliefs µ is
consistent with s (eqm-on-the-path)
determined using Bayes rule whenever possible
PBE adds 3) all players have the same
beliefs off-the-path
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Qual Question Fall 2002 (75)
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Two types of equilibria
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General Procedure
Start analyzing a Signaling game by deciding
whether to look for a Separating or Pooling
Equilibrium. Well start with a Separating
Equilibrium 1) Along the equilibrium path,
type is revealed. This gives us a place to
start. 2) Assess the receivers BR, under the
assumption that the receiver knows the senders
type 3) Back-up the game tree determine the
senders 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 receivers beliefs - to obtain
the widest range of equilibrium paths, make the
receiver pessimistic about the senders 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
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Separating equilibrium (75)
1) Separating equilibrium path, type is
revealed 2) Cs BR (sequentially rational
strategy), given that it knows Ps 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 Hs payoff is Wh (T -
th) therefore in eqm. th 0 Ls payoff is Wl
(T - tl) d Proof of th 0 Suppose
not. Then th 0. Consider th 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 Wh (T th) Wh (T - th)
If C pays, H is better off because Wh (T th)
d Wh (T - th) QED.
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Incentive compatibility
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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 senders 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
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Efficiency
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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
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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) Cs
best response on observing t, Cs payoff from
paying is (1-q) A-d Cs payoff from not paying
is 0 Cs 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
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Further Steps

• Both types must prefer t to any other t
• 1st Case (1-q) A-d
• In this case, the relevant condition is Wi (T -
  t) Wi (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 Wi (T -
  t) d Wi (T- t) for all t ? t,
• which yields d/Wh t
• t cannot be too high or P would prefer
  choosing t 0 and not receiving d

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The WPBE
WPBE th tl t 1st Case (1-q) A-d 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/Wh 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
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