Introduction
Asymmetric information
„ example: seller and buyer of used car
„ rational responses
Manipulation of information
„ example: persuasive advertising
„ signalling
Representation
„ extensive form
„ beliefs
Solutions: Refinement of Nash Equilibrium
„ Perfect Bayesian Equilibrium
„ updating of beliefs
Perfect Bayesian Equilibrium
Figure: Extensive-form game in figure 4.1.1.
Note: (L,L’) and (R,R’) are both Nash and Subgame Perfect Equilibria.
However, (R,R’) depends on a non-credible threat.
Intuitively, the Perfect Bayesian Equilibrium merges the ideas of Baysian Nash
Equilibrium and Subgame Perfection; in particular, the equilibrium concept puts restrictions on players’ beliefs.

Four requirements (cf. Gibbons):
1) At each information set, the player with the move must have a belief about which node has be reached, i.e. a probability distribution over nodes in the information set.
2) Given beliefs, strategies must be sequentially rational, i.e. optimal given beliefs and subsequent play.
3) At information sets on the equilibrium path – i.e. which are reached by positive probability given equilibrium play – beliefs are determined by
Bayes’ rule and players’ equilibrium strategies.
4) At information sets off the equilibrium path – i.e. which are not reached given equilibrium play – beliefs are updated by Bayes’ rule and players’ equilibrium strategies where possible.
A Perfect Bayesian Equilibrium consists of strategies and beliefs that satisfy the above four requirements.
Adverse Selection and Moral Hazard
Adverse selection refers to situations where one party cannot observe the
“type” of the other(s)
„ games of hidden information
„ examples: quality (experience or credence goods), hiring of workers
Moral hazard refers to situations where one party cannot observe the actions of the other(s)
„ games of hidden action
„ example: insurance and incentives
Adverse Selection: The Market for Lemons
Two types of cars: “lemons” and “plums”.
Sellers know their type of cars, but buyers do not.
Reservation prices
„ sellers: 1000 for lemons, 2000 for plums
„ buyers: 1200 for lemons, 2400 for plums
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Note: if quality of cars was verifiable – to buyers as well as sellers – the market would be efficient; lemons would trade for prices between 1000 and
1200 and plums for prices between 2000 and 2400.
Extensive-form representation:
1) Nature determines, with a 50-50 probability, whether a given car is a lemon or a plum.
2) Seller sets offer price.
3) Buyer decides whether to accept the offer.
Suppose there exists an equilibrium in which half of the cars are plums and the other half lemons. Buyers would then expect that the chance that any given car is a plum is 50%, and so (risk-neutral or risk-averse) buyers would be willing to pay at most 0.5·1200 + 0.5·2400 = 1800 for a car. Given that cars sell for at most 1800, sellers of plums would not offer them for sale; consequently, only lemons would be offered. In other words; there does not exist an equilibrium in which half of the cars offered for sale are plums.
Note that how the rationality of beliefs (i.e. updating using Bayes’ rule) determine strategies and hence equilibrium.
One can show that the only equilibrium is one in which only lemons are offered and they trade for a price between 1000 and 1200.
The following set of strategies form an equilibrium:
„ seller: for a lemon, ask 1100; for a plum ask 2100
„ buyer: irrespective of the offer price, believe the car is a lemon and only accept an offer no higher than 1100.
Note that the buyer’s belief is rational given the strategy of the seller, and that the buyer’s strategy is rational given his belief. The seller’s strategy is similarly rational given the buyer’s belief and strategy.
Market failure: good used cars are not traded. The reason is an informational externality between sellers of bad and good cars. When a seller offers a car on the market he affects buyers’ perceptions of the quality of the average car on the market – the very act of offering to sell something sends a signal to buyers. This lowers the price they are willing to pay, which hurts sellers of good cars.
This is an example of Adverse Selection: bad items crowd out good items.
Other examples include insurance (high-risk individuals are more likely to want insurance).
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Moral Hazard: Insurance
Consider the case of bicycle insurance.
The likelihood of a bicycle being stolen may depend on actions taken by the owner, such as where the bicycle is placed when not in use and whether it is locked.
Whether the owner takes precautionary actions – which may be costly – may depend on whether or not the bicycle is insured:
„ no insurance: the owner bears the full cost of theft
„ complete reimbursement: the owner bears no cost
Consequently, an insurance company who offers high reimbursement must expect having to pay out more often; there is a trade off between reimbursement rates and costs of offering insurance because reimbursement rates affect owners’ incentives.
Therefore, insurance companies will generally not offer complete insurance; in particular, insurance contracts generally include a “deductible”, an amount that the insured party has to pay in any claim.
This is an example of Moral Hazard: contractual relations affect incentives to undertake costly actions. Other examples include employment contracts
(fixed-wage contracts lead to little effort).
Extensive-form representation
1) Insurance company offer bicycle-insurance policy, where K is the amount paid out in case the bicycle is lost and
γ
K is the price of the insurance policy.
2) Bicycle owner – who owns a bicycle of value L – decides whether or not to take out insurance.
3) Bicycle owner decides whether or not to buy a lock, at cost c .
4) Nature determines whether bicycle is lost. With no lock, the probability of loss is p ; with lock the probability of loss is
β p , 0
β
1 .
Assuming risk-neutrality, payoffs are (first number refers to bicycle owner, second to insurance company):
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State
No insurance, no lock
Insurance, no lock
Insurance, lock
0,0
− γ
K ,
γ
K
− γ
K
− c ,
γ
K
−
L ,0
K
− − γ
K ,
γ
K
−
K
K
− − γ
K
− c ,
γ −
K
If the bicycle owner takes out insurance, he will be willing to buy a lock if c
< p
[
1
− β ][
−
]
.
It follows that the bicycle owner does not buy a lock if he is fully insured, i.e.
K
=
L .
The gain to the insurance company from the bicycle owner buying a lock is p
[
1
− β ]
K .
The following set of strategies constitute an equilibrium:
„ insurance company: offer a policy with K
= − p c
[
1
− β ] and
γ β p
„ bicycle owner: accept insurance offer and buy a lock
Signalling
Sometimes the market offers solutions to overcome market imperfections caused by asymmetric information.
For example: in the used-car example, sellers may offer warranties, i.e. a payback if the car turns out to be bad. The incentive to offer warranties is greater for sellers of good cars than for sellers of bad cars. Thus, a warranty may provide a “signal” by which sellers of good cars may distinguish themselves from sellers of bad cars.
This is an example of Signalling: “good” players distinguish themselves by undertaking costly actions, but less costly to them than to “bad” players. Other examples include education as a signal in labour markets (less difficult, or costly, for able persons to get an education).
Job-Market Signalling, Michael Spence (1973)
Extensive-form representation:
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1) Nature determines a worker’s productive ability, which is a
L
(or low) with probability q and a
H
(or high) with probability 1q , where a
L
< a
H
.
2) The worker learns his or her ability and then chooses a level of education e
≥
0 .
3) Two firms observe the worker’s education (but not ability) and then simultaneously make wage offers to the worker.
4) The worker decides which offer to accept.
Note: the worker’s strategy at Stage 4 is trivial, i.e. to accept the highest offer
(or pick a firm randomly if the offers are the same).
It is assumed that the cost to the worker of education is c
H
< c
L
. Note that education does not affect ability. i
,
=
, , where
The worker’s payoff is w
− c e , where w is the wage accepted wage. The payoff of the firm that hires the worker is a i
− w , while the other firm’s payoff is
0.
Consider the following set of beliefs and strategies:
„ worker: if ability is a
L
, choose no education, i.e. e
=
0 ; if ability is choose education e
= e * . a
H
,
„ firms: if education is e
≥ e * , believe the worker has high ability and offer wage w
= a
H
; if e
< e * , believe that the worker has low ability and offer wage w
= a
L
;
This set constitutes a Perfect Bayesian Equilibrium if
(*) a
H
− a
L c
L
< e *
< a
H
− a
L c
H
To see this, note first that given firms’ beliefs and strategies, the worker does not benefit from deviating from his strategy:
„ if of high ability, equilibrium payoff is a
H
− c e * , whereas the payoff from choosing e
> e * is certainly lower and the payoff from choosing e
< e * is at most a
L
, which is less than the equilibrium payoff by (*);
„ if of low ability, equilibrium payoff is a
L
, whereas the payoff from choosing
0
< < most a
H e
−
* is certainly lower and the payoff from choosing c e * e
, which is less than the equilibrium payoff by (*).
≥ e * is at
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Furthermore, the firms’ beliefs are clearly rational given the worker’s strategy
(in particular, Requirement 3 above is satisfied). At equilibrium, both firms receive a payoff of zero. Offering a higher wage would lead to a loss, while offering a lower wage would result in a payoff of zero also.
Note that in this (separating) equilibrium, signalling is a pure waste, since it is costly but does not raise output. Nevertheless, signalling is privately profitable since it allows high-ability workers to distinguish themselves from low-ability workers. If, instead, education was assumed to increase ability, the result would be less straightforward.
There may also be pooling equilibria, in which the worker chooses the same level of education independently of ability. Consider for example the following set of strategies and beliefs:
„ worker: choose no education, i.e. e
=
0 .
„ firms: if education is e
=
0 , believe the worker has low ability with probability q and high ability with probability 1
− q and offer wage w
= qa
L
[ ]
H
; if e
>
0 , believe that the worker has low ability and offer wage w
= a
L
.
Given strategies and beliefs of firms, the worker can do no better than choose no education; a higher education would be costly and would decrease wage.
Given the worker’s strategy, firms’ beliefs are rational (although Bayes’ rule has no bite) and payoff cannot be increased by a different strategy; offering a higher wage would lead to an expected loss, while offering a lower wage would result in a payoff of zero.
Other examples:
„ durable goods and warranties
„ corporate investment and capital structure
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