374 CHAPTER 11. INFORMATION
always necessary to introduce quite strong assumptions about the structure of
preferences and technology. In virtually every case we have used the “single-
crossing condition” for di¤erent families of indi¤erence curves in order to …nd
a tractable solution and to be able to draw interpretable conclusions from th e
analysis.
Finally, let us remind ourselves of some common curiosities that emerge from
imperfect-information models.
 The possible multiplicity of equilibria –as in the signalling mo d els (section
11.3). It is not c lear that intellectual devices to reduce this plethora are
entirely convincing.
 More disturbing perhaps is the possible lack of equilibrium in some cases:
see the model of the insurance market (section 11.2.6) and some signalling
models (Exercise 11.5).
 The use of rationing and price distortions to f orce a second-best solu-
tion where imperfect information means that “…rst best“ just can not be
implemented.
We will see that some of these features will be particularly relevant for our
discussion of the problem of economic design.
11.6 Reading notes
Good introductions to the economics of information and the theory of contracts
are provided in Macho-Stadler and Pérez-Castrillo (1997) and in Salanié (1997).
An overview of the issues is provided by Arrow (1986). The classic reference on
adverse selection, screening and the economics of insurance markets (on which
subsection 11.2.6 is based) is Rothschild and Stiglitz (1976).
The classic papers on the economics of signalling are Akerlof (1970) and
Spence (1973). The intuitive criterion is attributable to Cho and Kreps (1987).
The case of costless signals –so called “cheap-talk”models –is treated in Craw-
ford and Sobel (1982). A good introduction is in Salanié (1997), pages 95¤ on
which the example in section 11.3.2 is based.
For an introduction to the Principal-and-Agent model see Ross (1973) and

11.2 An employee’s type can take the value 
1
or 
2
, where 
2
> 
1
. The
bene…t of the employee’s services to his employer is proportional to z, the amount
of education that the employee has received. The cost of obtaining z years of
education for an employee of type  is given by
C (z; ) = ze

:
The employee’s utility function is
U(y; z) = e
y
 C (z; )
where y is the payment received from his employer. The risk-neutral employer
designs contracts contingent on the observed gross bene…t, to maximise his ex-
pected pro…ts.
1. If the employer knows the employee’s type, what contracts will be o¤ered?
If he does not know the employee’s type, which type will self-select the
“wrong” contract?
2. Show how to determine the second-best contracts. Which constraints bind?
How will the solution to compare with that in part 1?
11.3 A large risk-neutral …rm employs a number of lawyers. For a lawyer of
type  the required time to produce an amount x of legal services is given by
z =

2. Where individuals’risk types are unknown to the monopolist.
11.5 Good second-hand cars are worth 
a
1
to the buyer and 
a
0
to the seller
where 
a
1
> 
a
0
. Bad cars are worth 
b
1
to the buyer and 
b
0
to the seller where

b
1
> 
b
0
. It is common knowledge that the proportion of bad cars is . There is
a …xed stock of cars and e¤ectively an in…nite number of potential buyers
1. If there were perfect information about quality, why would cars be traded

ductivity 2 and type-b workers have productivity 1. Workers productivities are
unobservable by …rms but workers can spend their own resources to acquire edu-
cational certi…cates in order to signal their productivity. It is common knowledge
that the cost of acquiring an education level z equals z for type-b workers and
1
2
z for type-a workers.
1. Find the least-cost separating equilibrium.
11.7. EXERCISES 377
2. Suppose the proportion of type-b workers is . For what values of  will
the no-signalling outcome dominate any separating equilibrium?
3. Suppose  =
1
4
. What values of z are consistent with a pooling equilibrium?
11.7 A worker’s productivity is given by an ability parameter  > 0. Firms
pay workers on the basis of how much education, z, they have: the wage o¤ered
to a person with education z is w (z) and the cost to the worker of acquiring an
amount of education z is ze

.
1. Find the …rst-order condition for a type  person and show that it must
satisfy
 = log

dw (z

)
dz


1=2
z, and his reservation utility  = 0.
1. Solve for the full-information contract.
2. Con…rm that the owner would like to induce the manager to take action
z.
3. Solve for the second-best contracts in the event that the owner cannot
observe the manager’s action.
4. Comment on the implications for risk sharing.
11.9 The manager of a …rm can exert an e¤ort level z =
4
3
or z = 1 and gross
pro…ts are either 
1
= 3z
2
or 
2
= 3z. The outcome 
1
occurs with probability
 =
2
3
if action z is taken, and with probability  =
1
3
otherwise. The manager’s
utility function is u(y; z) = log y z, and his reservation utility is  = 0. The
risk neutral owner designs contracts which specify a payment y

> x
1
> 0

1
> 
2
> 0:
The project requires credit from a monopolistic, risk-neutral bank. There is
limited liability, so that the bank gets nothing if the project fails.
1. If the bank stipulates repayment y from any successful project what is the
expected payo¤ to the …rm and to the bank if the …rm selects project i?
2. What would be the outcome if there were perfect information?
3. Now assume that the bank cannot monitor which project the …rm chooses.
Show that the …rm will choose project 1 if y  y where
y :=

1
x
1
 
2
x
2

1
 
2
4. Plot the graph of the bank’s expected pro…ts against y. Show that the bank
will set y = y if 

(z) > 0 i = 1; 2. The tax
11.7. EXERCISES 379
authority o¤ers the inspector a wage rate w
i
= w(x), contingent on the result
achieved and obtains the bene…t B(x w). The inspector’s utility function is
U(w; z) = u(w)  v(z)
and his reservation level of utility is . Assume
B
0
() > 0; B
00
()  0; u
0
() > 0; u
00
()  0; v
0
() > 0; v
00
()  0:
Information is symmetric unless otherwise speci…ed.
1. For each possible e¤ort level …nd the …rst-order conditions characterising
the optimal contract w
i
i = 1; :::; n.
2. What is the form of the optimal contract when the tax-authority is risk-
neutral and the inspector is risk-averse? Comment on your solution and
illustrate it in a box diagram.
3. How does this optimal contract change if the inspector is risk-neutral and

introduced in the discussion of Principal and Agent.
The key problem can be summarised thus. In most of our previous work we
have assumed the existence of an economic institution that sets and administers
the rules of economic transactions: usually this was the market in some form.
Occasionally we have noted cases where the shortcomings of the institution are
evident –for example in the allocation of goods characterised by “nonrivalness”
or in the presence of externalities (see pages 245¤). Now we want to turn
this mental experiment around. Can we establish the principles which would
381
382 CHAPTER 12. DESIGN
underpin a well-functioning economic system and thereby provide guidelines for
designing such a system?
12.2 Social choice
If we are to consider the problem of economic design from scratch then we had
better be clear about the objectives of the exercise. What is it that the economic
system is supposed to achieve? We need a representation of the workings of the
economy that it is su¢ ciently ‡exible to permit general modelling of a variety
of individual and social objectives.
We can do this simply and powerfully by revisiting the ideas that underlay
the concepts of social welfare discussed in chapter 9. First we will reuse the
very general description of a social state  and the concept of a “pro…le” of
preferences de…ned over , the set of all possible social states: remember that
a pro…le is just an ordered list of preference relations, one for each household
in the economy under consideration (see page 228). However, we will …nd it
more convenient to work with the notation of utility functions rather than with
the weak preference symbol 
h
as in chapter 9, although this tweak is little
more than cosmetic. In particular let us use the “reduced-form” representation
of the utility function that expresses utility of household (agent) h as a direct

v
1
; v
2
; :::

(12.2)
A few points to note about the social-choice function :
 As a true function (rather than a correspond enc e) it selects a single mem-
ber of  once a given pro…le of preferences is plugged in.
 The arguments of  are utility functions, not utility levels: this is like the
constitution  that we de…ned in chapter 9 (page 228).
12.2. SOCIAL CHOICE 383
 social state
 set of all so cial states
v
h
() “reduced-form”utility function for agent h
[v] =

v
1
; v
2
; v
3
; :::

pro…le of utility functions
V set of all p ossib le pro…les

h
(). Then the social-choice function  is Paretian if


= 

v
1
; v
2
; :::

(12.3)
De…nition 12.3 Suppose there are two pro…les [v] and [~v] such that


= 

v
1
; v
2
; :::

and, for all h :
v
h
(

)  v

states . The plain-language interpretation of monotonicity (De…nition 12.3) is
that the chosen social state is never dropped unless it becomes less attractive
for some individual agent h. De…nition 12.4 is intuitive: for example, if person
1 is a dictator then, when we replace the functions v
2
; v
3
; :::; v
h
; :::in (12.2) by
any other utility functions and leave the function v
1
unchanged we will …nd that
 remains u nchanged. The dictatorship property seems as unappealing in the
context of a social-choice function as it did in the context of a constitution.
A comparison of de…nitions 12.1–12.4 and the discussion of the constitution
(page 229) suggests that there may be a counterpart to the Arrow Impossibility
Theorem (Theorem 9.1) that applies to social choice functions. This is indeed
the case:
Theorem 12.1 (Dictatorial social choice functions) Suppose the number
of social states is more than two and the social-choice function  is de…ned for
all logically possible utility functions. Then, if  is Paretian and monotonic, it
must be dictatorial.
The ‡avour of Theorem 12.1 is similar to Theorem 9.1 and, indeed, the proof
is similar (check the reading notes to this chapter and Appendix C). But its
implication may not be immediately striking. To appreciate this more f ully let
us introduce a crucial property that will enable us to build a bridge between the
welfare-economic discussion of the constitution and the behavioural analysis of
our discussion of the economics of information:
De…nition 12.5 A social choice function  is manipulable if there is a pro…le

; x
h
2

-space draw th e “better-than”(actually, “no-worse-than”) set B (

; v) when
ind ividual preferences are given by the pro…le

v
1
() ; v
2
() ; v
3
() ; :::

:
(b) Suppose agent h’s preferences change from v
h
() to ~v
h
(): interpret condition (12.4)
using B (

; v) and B (

; ~v)
(c) State the monotonicity condition using this diagram.
12.3. MARKETS AND MANIPULATION 385

For a manipulable social-choice function there may be a premium on false in -
formation for some agents in the economy: the form of the utility function is
of course the quintessentially private information. If there were a way for h to
reveal the false utility function ^v
h
then the economic system would resp ond in
such a way that h would b e genuinely better o¤ –notice that the inequality in
expression (12.5) uses the genuine utility function v
h
.
However, monotonicity implies that the social-choice function cannot be ma-
nipulable.
2
This leads us on to a key result that is really no more than just a
corollary of Theorem 12.1:
Theorem 12.2 If there are at least three social states and, for each house-
hold, any strict ranking of these alternative states is permissible then the only
Paretian, non-manipulable social choice function is dictatorial.
Theorem 12.2 is a …rst attempt at capturing an essential concept that car-
ries over from our consideration of information in chapter 11. It has profound
consequences for the way in which economic systems can be designed if there is
less than full information.
12.3 Markets and manipulation
To illustrate the power of misrepresentation and manipulation in a familiar
setting let us rework the standard mod el of an exchange economy.
12.3.1 Markets: another look
Take the particularly interesting example of a social-choice function from chap-
ter 7. Specify the details of the following:
 The technology of the …rms;
 The resource endowments;

may be unwarranted: that each individual agent is e¤ectively too small to mat-
ter. Let us look more closely at the market system in the context of the el-
ementary model of a two-commodity exchange economy: this is illustrated in
the Figure 12.1 which represents a standard Edgeworth diagram box f or the
two-person case.
The initial property distribution is R
a
= (0; R
2
), R
b
= (R
1
; 0): Alf has all
the commodity 2 and Bill all the commodity 1. Each person could survive on
his endowment, but would bene…t from trade with the other. Alf’s indi¤erence
curves are represented by the contour map with broken lines with origin at O
a
;
Bill’s indi¤erence curves are those with origin O
b
. The set of all Pareto-e¢ cient
allocations – the locus of is drawn in as the irregularly-shaped line joining O
a
and O
b
. The core of the two-person game is represented by the subs et of this that
is bounded by points [x
a
] and

may no longer hold.
Figure 12.1: Manipulated trading
12.3.3 Manipulation: power and misrepresentation
Consider now a story about market power. Suppose Alf knows the trades that
Bill is to make at each price and has the power to dictate the price. We can
imagine an exercise in which various prices are tried out on Bill, and Bill’s
desired consumptions. Using this information Alf can exploit his position as
monopolist of commodity 2 to force up the price. The outcome would be at a
point such as [^x] with prices ^p where the terms of trade have been moved in
388 CHAPTER 12. DESIGN
favour of Alf.
5
Alternatively we can see this as a story of misrepresentation in which Alf lies
and reveals a false indi¤erence to his trading partner. The story runs as follows.
Each day of the week each trader comes to the market with the endowments
represented by point [R]. But there is an apparent change of tastes during the
week:
 On Monday preferences are publicly declared to be as description of the
indi¤erence curves above. Haggling takes place between the two traders,
with each telling the truth, and revealing to the other his demand func-
tions. A competitive equilibrium is agreed upon, possibly by each sid e
agreeing to abide by the rulings of an impartial arbitrating auctioneer. So
each trader is acting as though he were a price-taker at prices p

– the
equilibrium is at point [x

] in the accompanying …gure .
 On Tuesday each trader arrives again with stocks [R], but Alf has now
decided to lie –purely for material advantage of course. He realises that


v
1
; v
2
; v
3
; :::

pro…le of utility functions
V set of all p ossib le pro…les
s
h
strategy of agent h

s
1
; s
2
; s
3
; :::

pro…le of strategies
S set of all strategy pro…les
 outcome function
 social-choice function
Table 12.3: Mechanism: Notation
So our next step to examine the engine that drives this general class of
economic problem. To do this it is useful to pick up on the essentials of a game,

2
; s
3
; :::

.
 The speci…cation of the players’objectives. This cons ists of a pro…le of
preferences

v
1
; v
2
; v
3
; :::

. So, once the outcome (social state)  has been
determined, this leads to utility payo¤s v
1
() ; v
2
() ; v
3
() ; :::.
If all three items in the above list are speci…ed in detail then the game is
fully described. Now the …rst two of these components give us exactly what
is needed for a general description of the “engine” that is at the core of this
chapter:
De…nition 12.6 A mechanism consists of the strategy sets S and an outcome


s
1
; s
2
; s
3
; :::

:
 The outcome function determines the social state in the light of the pro…le
of strategies


= 

s
1
; s
2
; s
3
; :::

: (12.8)
 Is this 

the one that the designer would have wished from the social-
choice function  in (12.2)?
But this begs a number of important questions about the way in which

There is a mechanism
1. for whi ch a ll the Nash equilibria yield 

.
12.4. MECHANISMS 391
Drawing together this discussion for an important, but special interpretation
of the concept, we may summarise thus:
De…nition 12.7 The mechanism (S;  ()) weakly implements the social-choice
function  in dominant strategies if there is a dominant-strategy equilibrium of
the mechanism,

s
1
() ; s
2
() ; s
3
() ; :::

such that


s
1

v
1

; s
2

everything that there is to be known about motivation in playing the game. It is
a game of messages akin to those discussed in section 11.3 of chapter 11. In this
game the strategy space –th e message space –S is exactly the space of all the
possible utility pro…les V;
7
the outcome function maps announced announced
preferences directly into social states such that, for all pro…les in V,


v
1
; v
2
; v
3
; :::

= 

v
1
; v
2
; v
3
; :::

:
In other words, the mechanism is so simple that the outcome function is the
social-choice function itself; unsurprisingly this device is conventionally known

Suppose each the t as te parameter 
h
for agent h is a numbe r in [0; 1]. Write down the
exact ex pression for the combined strategy space [Hint: check the de…nition on page 486].
392 CHAPTER 12. DESIGN
12.4.3 The revelation principle
The direct mechanism –or direct-revelation mechanism –is of mild interest in
its own right: it is at least intriguing to think up tricks that will cause rational
agents to reveal all the personal information that would otherwise be hidden
from a designer. However direct mechanisms are of fundamental importance
in terms of the general problem of implementation. In the following, note that
the pair (S; ) represents any mechanism that you might think up, while (V; )
represents the direct mechanism just discussed in section 12.4.2:
Theorem 12.3 (Revelation principle) If the social-choice function  is weakly
implementable in dominant strategies by the mechanism (S; ) then  is truth-
fully implementable in dominant strategies using the direct mechanism (V; ).
Figure 12.2: The revelation principle
The idea of this is illustrated in Figure 12.2. Th e implementation story can
be told in one of two ways:
1. The mechanism (S; ) works this way. Given a particular choice of pref-
erence pro…le [v] from V the agents select strategies

s
1

v
1

; s
2

However, the direct-revelation mechanism is not necessarily the one that
would be used in practice to resolve a design problem and the above result does
nothing to clear up whe ther there are multiple equ ilibria in a mechanism that
is used to implement , or, indeed whether there are any equilibria at all.
12.5 The design problem
Equipped with the concept of the mechanism as a basic tool we can now continue
the discussion we left in section 12.2: the issue of designing an economic system
in order ful…l a speci…c set of social objectives. We can build upon the results
about social-choice functions by applying the concept of truthful implementation
in section 12.4.
In particular, by combining the result on dictatorial social-choice func tions
and the revelation principle (Theorems 12.1 and 12.3) we have the following:
Theorem 12.4 (Gibbard-Satterthwaite) If (i) the set of social states 
contains at least three elements; (ii) the social choice function  is de…ned for the
set V of all logically possible pro…les of utility functions and (iii)  is truthfully
implementable in dominant strategies, then  must be dictatorial.
This is a key result. We can better understand the strength of it if we use
the concept of m anipulability of a social-choice function. By extension we can
consider a mechanism to be manipulable if it is not one that ensures truthful
revelation in dominant strategies. Having a mechanism that is non-manipulable
or strategy-proof seems like a particularly attractive property when we try to
design a method of implementing the social objectives. But Theorem 12.4 makes
clear that if all types of tastes are admissible and if the set of social choices is
large enough to be interesting then the only way to achieve this is to allow one
of the agents to act as dictator.
Another plain language interpretation of the result can be seen in terms of
cheating. We have already encountered particular situations in chapter 11 where
individuals have an incentive to misrepresent information about themselves:
high valuation customers might want to pass themselves o¤ as low-valuation in
order to take advantage of a more favourable fee schedule; an Agent would try

but where the outcome is very unattractive.
9
Accordingly we should consider
the possibility of complete implementation using Nash equilibrium. Here each
person knows his own preferen ces and the preferences of all the other players;
8
We characterised the do minant -strategy version of truth-telling (page 262) as “honesty
is always the best-policy.” What is the plain-language expression of the Nash-equilibrium
version of truthtelling?
9
(a) Take the game represented in strategic form by Table 10.2 where th ere are two players
Alf and Bill and exactly two strategies for each player. Suppose the payo¤ (3; 3) is the social
state that is the outcome of the social-choice function that we want t o implement. Let s
h
1
and
s
h
2
represent the strategy of truth-telling and of lying for h = a; b. Explain why

s
a
2
; s
b
2

is an
equilibrium, but is unsatisfactory.

12.6 Design: applications
The other approaches to dealing with the challenge of Theorem 12.4 can be
usefully illustrated with a number of key economic applications. These are all
of the type of Bayesian games of incomplete information that were modelled
in chapter 11: in particular all of the applications can be seen as versions of
the “adverse selection” class of problems involving hidden characteristics –see
pages 333 ¤.
Remember that the second of the list of three mentioned on page 394 involved
restricting the class of admissible utility functions. Accordingly we will simplify
the representation of individuals’preferences by using the same general form of
utility function as was used in the adverse-selection models. We assume that all
the economic agents in the game have the same general shape of utility function,
but that they di¤er in some “type”or “taste”parameter , a real number. The
various values of parameter  that may be imputed to an individual completely
characterise the di¤erent objectives that the agent may have.
12.6.1 Auctions
An auction can be regarded as an exercise in posing the question “tell me what
your valuation is.”Someone sets up an event or an institution to extract payment
from one or more potential buyers of an object, a collection of goods, ownership
rights, How do the mechanics work? How can the principles of design help
us to understand the rules and likely outcomes?
Of course the problem that makes the analysis of auctions economically
interesting is the nature of the concealed information: the seller usually does
not know the characteristics of individual potential buyers, in particular their
willingness to pay. In view of this it is appropriate to formulate the problem
in terms of a Bayesian game and to use the revelation principle to simplify the
analysis. There is a great variety of types of auction that di¤er in terms of the
information available to participants, the timing, and the rules of conduct of
the auction. We will …rst discuss the informational issues and then the rules.
396 CHAPTER 12. DESIGN

round the table starting from the bottom left-hand corner:
Open bid Sealed bid
Dutch –descending price …rst price
English –ascending price second price
Table 12.4: Types of auction
 The English auction involves public announcements of bids that are grad-
ually increased until only one bidder is left in the auction who wins the
auction and pays the last price bid.
10
Provide a brief argument that this is the case in the auction of a painting.
12.6. DESIGN: APPLICATIONS 397
 The Dutch auction goes in the other direction. Starting from a high value,
the announced price is gradually adjusted downwards until someone is
ready to claim the object at that price.
 In the sealed-bid …rst-price auction all agents submit their bids in a way
that is hidden from the others: the object goes to the agent who submitted
the highest bid; the winner pays exactly the price that he or she bid.
 In th e sealed-bid second-price counterpart the object again go es to the
highest bidder; but the winner is required to pay the price that the “runner
up”had bid –the next highest price.
Fortunately we can simplify matters further by noting that in some cases
these four possibilities e¤ectively reduce to j ust two, corresponding to the two
rows of the table. The Dutch open auction and the …rst-price sealed-bid auction
are essentially equivalent mechanisms; for our information mo del the English
open auction and the second price sealed-bid produce the same results. We will
establish these assertions in each of the next two subsections before moving on
to a more general approach to the auction mechanism.
First price
In s trategic terms Dutch auction is equivalent to the …rst-price auction with
sealed bids: each bidder chooses a critical value at which to claim the object as

a
) := Pr


b
 
1
(p
a
)

= F


1
(p
a
)

(12.9)
where 
1
denotes the inverse function. Because it is a …rst-price auction, the
price you bid is the price you pay, if you win. Therefore, if Alf’s bid succeeds
and he gets the good, his bene…t is 
a
 p
a
; otherwise he gets no net bene…t.
398 CHAPTER 12. DESIGN

a
) (12.11)
where p
a
is the optimal value of p
a
. Because the problem is symmetric, in the
Nash equilibrium each person has the same function ; so
p
a
=  (
a
) (12.12)
and, from (12.9)–(12.11) Alf’s expected net bene…t is:
12
 (
a
) =
Z

a
0
F () d (12.13)
Alf’s expected net bene…t at the optimum can also be written as
 (
a
) =  (p
a
) [
a

individual values are as depicted in Figure 12.4.
Second price auction: a truth-telling mechanism?
Now take the English open-bid auction. In the case of the private-values infor-
mation model, the dominant strategy in such an auction is to carry on bidding
until the bid has reached one’s true value of the object and then, if the price
11
Why is this true?
12
Fill in the missing two lines to establish this point.
13
Explain why, using (12.10).
14
Take a popul at ion of size N > 2. How does the above reasoning change for this case?