Public-Good Provision in a
Large Economy
Felix Bierbrauer and Martin F. Hellwig
Max Planck Institute for Research
on Collective Goods
Bonn
October 24, 2007
1
1.1
Motivation
Large Economies versus Small
The standard mechanism design approach to public
good provision (dominant strategies or Bayesian) focuses on how to calibrate people’s payments to their
expressions of preferences so that they have no wish
either to understate their preferences for the public
good (to reduce their payments) or to overstate their
preferences (to get a greater provision level at other
people’s expense).
This analysis considers small economies where each
person’s expression of preferences can have a noticeable e¤ect on the level of public-good provision. The
model is one of people in a condominium deciding
how much to spend on gardening services, rather
than people in the US deciding on how much to pay
for national defense or for the judicial system.
For private goods, the large-economy paradigm where
no one person is able to a¤ect private-goods prices is
deemed to provide the proper framework for studying what happens in a multimillion-people system in
the absence of market power.
For private goods, the large-economy paradigm serves
as conceptual idealization and normative standard.
What about public goods? Should we not think
about models of public-good provision where no one
person is able to a¤ect the level of aggregate percapita spending on the public good, i.e., no one person is ever “pivotal”?
If no one person is ever “pivotal”, the problem of
calibrating payments to expressed preferences has a
trivial solution: make payments independent of expressed preferences.
If payments are independent of expressed preferences,
each person pays the per-capita provision cost if the
public good is provided.
If participation is voluntary, nothing is provided
(Mailath & Postlewaite 1990).
Samuelson’s (1954) anti-contractarian discussion is
fully vindicated.
If coercion is possible, there is no problem about
reaching …rst-best.
1.2
An Example
A public good comes as a single invidivisible unit,
Q = 0 or Q = 1.
Per capita provision costs are equal to 4.
The individual valuations of the public good are
0, 3, or 10.
Population shares are
for people with valuation
0
p
for people with valuation
3
:3
for people with valuation
10:
:7
p
An e¢ cient provision rule requires provision if
3p + 10:3 > 4, i.e.,if p > 1=3;
and non-provision if
3p + 10:3 < 4, i.e.,if p < 1=3:
If the public good is provided, everybody pays 4.
In this case, payo¤s are
4
for people with valuation 0;
1
for people with valuation 3;
+6
for people with valuation 10:
Does this make sense?
Individual incentive compatibility and feasibility hold.
If coercion is feasible, individual rationality does not
matter.
If p is common knowledge, there is no further concern. This is, e.g., the case, if we think of the large
economy as a limit of models with independent private values. In the limit, the information problem
of how to …nd out the e¢ cient level of public-good
provision has disappeared.
If p is not common knowledge, the mechanism relies
on communication from people with valuations 0 and
3. All these people are hurt by public-good provision.
Why should they communicate in such a way that the
mechanism learns the value of p?
Why should they not all communicate valuations of
zero so that the mechanism believes that p = 0?
Answer: Honest reporting is individually incentive
compatible because everybody believes that he or
she has no e¤ect on the aggregate outcome.
Our approach: Introduce a concept of coalition proofness in order to articulate the concern that people
with valuations below 4 collectively have an incentive to distort the mechanism’s understanding of p.
The above mechanism fails coalition proofness.
How general is this?
1.3
Correlations and Robustness
What does it mean to say that p is not common
knowledge?
In a Bayesian framework with a common prior, we
might consider p to be the realization of a random
variable P such that, conditional on the event fP =
pg, population shares are .7 – p, p, and .3.
We might also suppose that, conditional on the event
{P=p}, each person’s individual valuation is a random variable which takes the values 0, 3, and 10
with conditional probabilities .7-p, p, and .3.
This brings us into a setting with correlated private
values.
In small economies, generically, correlations can be
exploited to implement …rst-best with budget balance (Johnson et al. (1990), d’Aspremont et al.
(2004).
Can correlations also be used to deal with coalition
proofness?
1.4
The Example Continued
Suppose that p is the realization of a random variable
with possible values
p = :2 (leading to non-provision) and p = :6 (leading to provision).
Consider the following payment rule:
o People who claim that they value the public good
at 3 pay 0 if the public good is provided, pay 8 if it
is not provided.
o People who claim that they value the public good
at 0 or 10 pay 10 if the public good is provided,
receive 2 if it is not provided.
People who value the public good at 3 now want it
to be provided.
With population shares
:5; :2; :3 when p = :2
and
:1; :6; :3 when p = :6;
the mechanism is feasible.
For instance, when p = :6; a population share :4 has
valuations 0 or 10 and pay 10 each, for a total intake
of 4, which is just the provision cost.
If type-dependent beliefs are derived from a common
prior that has
1
p = :2 with probability
2
and
p = :6 with probability
1
;
2
and that has conditional probabilities given p equal to
:7
p for any person’s valuation being equal to 0;
p for the valuation being equal to 3;
:3
for the valuation being equal to 10;
then the mechanism is also incentive-compatible and even
individually rational.
Under the given common prior, the probability of
public-good provision, i.e., the event p = .6, is assessed at 1/6 by a person with valuation 0, at 34 by
a person with valuation 3, and at 12 by a person with
valuation 10.
These di¤erences in beliefs allow for an incentivecompatible dependence of payments on types. The
resulting payments scheme can be interpreted as a
combination of cost sharing with a system of bets
on the state of the economy. The bets make people
with valuation 10 transfer some of their bene…ts to
the other participants.
However, if the common prior had
p = :2 with probability 1=3
and
p = :6 with probability 2=3;
the given scheme would no longer be incentive compatible. People with valuation 10 would wish to
claim that they have valuation 3.
The incentive compatibility of the given type-dependent
payment scheme is thus not robust to changes in the
speci…cation of beliefs.
We impose a requirement of robust implementation
in the sense of Bergemann and Morris (2005): The
outcome function should be (interim) implementable
in all common-prior type spaces, i.e. for all speci…cations of interim beliefs that can be derived from a
common prior.
1.5
Preview of Results
In a large economy, robust implementation requires
a type-independent payment rule.
In a large economy, robust implementation of …rstbest involves an e¢ cient provision rule in combination with equal cost sharing.
In a large economy, coalition-proof robust implementation of …rst-best is usually not possible.
In a large economy, coalition-proof robust implementation can condition on the population shares of people who “vote”(!) for or against the provision of the
public good, i.e. on the fractions of people with valuations below the per capita provision cost and with
valuations above the per capita provision cost.
In a large economy, coalition-proof robust implementation cannot condition on the distribution of types
within the set of people who are for provision or
within the set of people who are against provision.
Intensities of preferences cannot be made to play a
role.
In the example, this means that one cannot condition on
p.
If p = .2 with 12 and p = .6 with 12 , one may decide
that, with an expected value of p equal to .4, i.e. greater
than 1/3, one may want to provide the public good in all
states.
However, this line of argument makes the provision rule
depend on the common prior. If p = :2 with 43 and p = :6
with 14 , the expected value of p would be .3, i.e., less than
one third, and one may want to provide the public good
in no state.
Is there a way to introduce robustness into the method
by which one conditions on the numbers of people who
“vote” for or against the provision of the public good?
2
Analysis
Description of the Economy
Agents: An atomless measure space (A; A; ) with (A) =
1:
Goods: One public, one private. The public good comes
as a single indivisible unit.
Technology: Installation of the public good requires k
units of the private good.
Payo¤s: Given a public-good provision level Q 2 f0; 1g
and a required contribution of p units of the private good,
agent a obtains the payo¤
u(Q; ; a) = (a)Q
p;
where (a) is a preference parameter.
The map a ! (a) is measurable, with values in a …nite
set = f 1; :::; ng; which does not contain k:
Allocations and Social Choice Functions
An allocation is given a public-good provision level Q 2
f0; 1g and a measurable function p : A ! R
such that, for each a 2 A; p(a) is the number of units
of the private good that agent a must contribute.
An allocation is feasible if
Z
p(a)d (a)
kQ:
A social choice function is a map F = (QF ; pF ) from
preference parameter functions ( ) to feasible allocations.
Restrict attention to social choice functions taking the
form:
QF ( ( ))
pF (aj ( ))
^F (
Q
'
^ F ( (a )j
1)
1 );
so that Q depends only the cross-section distribution
1 of ( ) and p(a) depends only on (a) and the
cross-section distribution of ( ).
This restriction re‡ects anonymity and insensitivity to
null sets.
Types and Beliefs
Information is modelled by an abstract type space T =
[(T; T ); t; ; ]:
(T; T ) is a measurable space, t is a measurable map from
(A; A) into (T; T );
is a measurable map from T into
;
is a measurable map from T into the space M(M(T ))
of probability distributions over measures on T:
We refer to t(a) as the "abstract type" of agent a; and
to
(a) = (t(a))
as his payo¤ type and
b(a) = (t(a))
as his belief type: The belief type b(a) is a probability
measure on the space M(T ) of cross-section distributions of types in the economy.
Incentive Compatibility and Implementation
For a given type space, a direct, anonymous, i.n.s. mechanism, is a mapping
^ f (d); p^f ( jd))
f : d ! (Q
from cross-section distributions of abstract types into feasible allocations, with payments taking the form
p^f ( jd) = '
^ f ( ( )jd):
^f ; '
The mechanism f = (Q
^ f ) is incentive-compatible
on the type space T if it satis…es
Z
Z
^ f (d)
[ (t)Q
'
^ f ( (t)jd)]d (djt)
^ f (d)
[ (t)Q
'
^ f ( (t0)jd)]d (djt)
for all t and all t0 in T:
The mechanism achieves the social choice function F
on the type space T if
f (d) = F (d
1)
for all d 2 M(T ): The mechanism f implements the
social choice function F on the type space T if it is
incentive-compatible on T and, moreover, it achieves F
on T.
Robust Implementation
A social choice function F is said to be robustly implementable if, for every common-prior type space T, there
exists a direct, i.n.s., anonymous mechanism f that implements F on T:
The social choice function F is said to be ex post implementable if
^ F (s )
Q
for all
and 0 in
'
^ F ( js)
^ F (s )
Q
and all s 2 M( ):
'
^ F ( 0js)
^F ; '
Proposition 1 A social choice function F = (Q
^F )
is robustly implementable if and only if it is ex post implementable.
^F ; '
Corollary 2 A social choice function F = (Q
^ F ) is
robustly implementable if and only if payments are independent of individual payo¤ types, i.e., '
^ F takes the
form
'
^ F ( js) = '
^ F (s )
for all
2
and all s 2 M( ):
Robust Implementation of First-Best Allocations
An allocation is said to be …rst-best for the preference
pro…le ( ) if it maximizes the aggregate surplus
Z
A
[ (a )Q
p(a)]d (a)
over the set of feasible allocations.
A social choice function yield …rst-best outcomes if, for
every preference pro…le ( );
the allocation (QF ( ( )); pF ( j ( ))) is …rst-best.
^F ; '
Lemma 3 A social choice function F = (Q
^ F ) yields
…rst-best outcomes if and only if, for any s 2 M( );
where
Z
^ F (s );
'
^ F ( js)ds = kQ
^ F (s) = 0 if
Q
(s ) < k
^ F (s) = 1 if
Q
(s) > k;
and
where
(s) :=
Z
ds( ):
Proposition 4 A robustly implementable social choice
^F ; '
function F = (Q
^ F ) yields …rst-best outcomes if and
only if, for all s 2 M( ) and all 2 ;
^ F (s )
'
^ F ( js) = kQ
^ F (s) is zero or one depending on whether
where Q
is less than k or greater than k:
(s )
Collective Manipulation Mechanisms
Given: a robustly implementable social choice function
^F ; '
^F :
F = (Q
^ F ) with '
^ F = kQ
Also given: a common-prior type space T = [(T; T ); t; ; ]
^f ; '
and a direct, i.n.s., anonymous mechanism f = (Q
^f )
that implements F on T:
A collective manipulation mechanism asks people for
messages m 2 T [ f;g.
m(a) = ; means that agent a does not join the coalition
of deviators, m(a) 2 T means that he does.
As a function of the message pro…le m( ) and as a function of a randomization indicator ( a(! ); (! )), the collective manipulation mechanism determines:
a report
`~(aj!; m( )) = `^(m(a)j a(! ); (! );
m 1)
that agent a with m(a) 2 T is going to make to the
mechanism f ; for agent
a side payment
z~(aj!; m( )) = z^(m(a)j a(! ); (! );
to agent a with m(a) 2 T:
m 1)
Implications for the overall mechanism
Given manipulation mechanism with message pro…le m
and truthtelling by the other agents, the overall mechanism receives reports
m 1)
t~`(aj!; m( )) = `^(m(a)j a(! ); (! );
if m(a) 6= ;
and
t~`(aj!; m( )) = t(a) if m(a) = ;;
with a cross-section distribution
d^`(!;
m 1) =
m 1
m 1 ):
t^`( j!;
^ f (d^`(!;
Public-good provision level: Q
m 1))
Payment requirement for agent a: '
^ f (d^`(!;
m 1))
Payo¤s:
Payo¤ realization:
VM(m(a); (a); !;
m 1)
^ f (d^`(!;
= (a )Q
m 1))
'
^ f (d^`(!;
m 1))
+^
z (m(a)j a(! ); (! );
m 1 ):
Interim expected payo¤ of agent a:
WM(mj (a); b(a);
=
Z Z
T
VM(m; (a); !;
m 1)
m 1)d (! )db(a):
Interim expected payo¤ of an agent of type t :
WM(mjt; ) = WM(m(a)j (t(a)); (t(a));
where
assigns messages to types.
1 );
Conditions on Manipulation Mechanisms and Message Pro…les
The approach taken is axiomatic. A collective manipulation mechanism is admissible if it satis…es:
Undetectability: The pro…le of reports that results from
coalition members doing what the mechanism tells them
is a possible pro…le of reports when there is no manipulation mechanism.
Feasibility:
Z
A
z^(m(a)j a(! ); (! );
m 1 )d (a )
for all ! and all messages pro…les m( ):
0
Incentive Compatibility: The type-dependent message
pro…le is an interim Nash equilibrium.
A version of the revelation principle implies that w.l.o.g.,
one may assume that
(t) 2 ft; ;g for all t:
Incentive compatiblity then involves:
Internal incentive compatibility: For coalition members, truthtelling is a best response.
External incentive compatibility: For coalition members, getting in is a best response, for non-members, staying out is a best response.
Attractiveness to Type Set C: A feasible manipulation
mechanism is attractive to type set C if a strategy pro…le
involving
(t) = t for t 2 C
and
(t) = ; for t 2
=C
is incentive compatible and induces the types in C to be
strictly better o¤ than they would be if the manipulation
mechanism did not exist.
Remark: We avoid the problems of strategic interdependences of decisions to join the collective manipulation
mechanism, with th epossibility that nobody is joining
because nobody else is joining.
The overall mechanism f is said to be coalition proof
if, given this mechanism, there is no feasible collective
manipulation mechanism and no type set C that …nds
the collective manipulation mechanism attractive.
Lemma 5 If the overall mechanism involves equal cost
sharing, there is no loss of generality in assuming that the
manipulation mechanism involves zero side payments.
In the absence of side payments of the manipulation mechanism, individual payo¤ assessments are
^ f (d^`(!;
(a )Q
m 1)) ;
or
[ (t)
^ f (d^`(!;
k ]Q
1 )):
People with (a) < k …nd a manipulation mechanism
attractive if it reduces the probability of public-good provision.
People with (a) > k …nd a manipulation mechanism
attractive if it increases the probability of public-good
provision.
Coalition proofness of the overall mechanism implies:
There is no manipulation mechanism that reduces
the probability of public-good provision from the perspective of people with (t) < k:
There is no manipulation mechanism that increases
the probability of public-good provision from the perspective of people with (t) > k:
In particular, regardless of what the preference pro…le of
people with (t) < k may be, truthtelling by the manipulation mechanism organizer minimizes the conditional
probability of public-good provision over the set of possible report pro…les.
Also, regardless of what the preference pro…le of people
with (t) > k may be, truthtelling by the manipulation
mechanism organizer maximizes the conditional probability of public-good provision over the set of possible report
pro…les.
If misreporting of the preference pro…le of one of the two
indicated coalitions is undetecatble, then, in a commonprior type space, these conditions are the equilibrium
conditions of a strictly competitive game of incomplete information, which may be thought of as being
played between the managers of the two coalitions of
people with (t) < k and of people with (t) > k: The
game has the following structure:
Nature chooses a type distribution.
Each coalition manager observes the size of his own
clientele and infers the size of his opponents clientele.
Each coalition manager also observes the type distribution of his own clientele.
Each coalition manager reports a type distribution
for his own clientele so as to maximize or minimize
^f .
his expectation of Q
Example: As in the introduction, except that we now
have four possible valuations,
0 with population share :7
p;
3 with population share p;
7 with population share q;
10 with population share :3
q:
The manager of the coalition with (t) < 4 observes p;
but does not know q:
The manager of the coalition with (t) > 4 observes q;
but does not know p:
Undetectability:
Conditional on coalition size, the misreporting of preferences by a coalition manager cannot be detected by
looking at the pro…le of messages from people outside
the coalition.
In the example, if we think of p and q as the realizations of
random variables p~ and q~; then, undetectability holds, if
the support of the distribution of p~, conditional on q~ = q;
is independent of q and similarly for q~:
Undetectability fails, e.g., if p~ and q~ are perfectly correlated.
The Saddlepoint Theorem for Strictly Competitive
Games can be extended to this game.
The requirement that truthtelling be an equilibrium therefore implies that truthtelling be a saddlepoint.
For "generic" common priors, it follows that, conditional
^ f must be independent of prefon coalition size, Q
erence pro…les.
Theorem 6 If T = [(T; T ); t; ; ] is a common-prior
type space such that, conditional on coalition size, the
misreporting of preference pro…les is undetectable, then,
"generically", coalition proofness implies that public-good
provision can be conditioned on the size of the coalition
of people with (t) < k and, by implication, the size
of the coalition with (t) > k; but not on preference
pro…les within these coalitions.
Coalition proofness also implies that the incidence of public good-provision must be a non-increasing function of
the share of the population with (t) < k:
"Generic":
In the example, suppose that p~ and q~ are independent,
with p~ taking the values :2 and :5 with probability 21 each
and q~ taking the values 0 and :2, also with probability 12
^f ; '
each. A mechanism (Q
^ f ) that involves
^ f = 0 if (~
Q
p; q~) 2 f(:2; :2); (:5; 0)g
and
^ f = 1 if (~
Q
p; q~) 2 f(:2; 0); (:5; :2)g;
^ f , is coalition proof. Coalition proofas well as '
^ f = kQ
ness breaks down if the prior is slightly changed.
Robust Coalition-Proof Implementation
A social choice function F is said to admit Robust CoalitionProof Implementation if, for every common-prior type
space T, there exists a direct, i.n.s., anonymous, and
coalition-proof mechanism f that implements F on T:
^F ; '
Theorem 7 A social choice function F = (Q
^ F ) admits robust coalition-proof implementation if and only if
^ F and '
Q
^ F condition only on the size of the coalition
of people with < k and, by implication, the size of the
coalition with > k; but not on preference pro…les within
these coalitions, i.e., if there exist functions QF ; 'F such
that, for any s 2 M( ) and any 2 , one has
^ F (s) = QF (s(f < kg)
Q
and
'
^ F ( js) = '
^ F (s(f < kg):
Coalition proofness also implies that the incidence of public good-provision must be a non-increasing function of
the share of the population with (a) < k:
Optimal Robust Coalition-Proof Implementation
If the mechanism designer has a prior distribution B
^F ; '
for the state s, choose F = (Q
^ F ) to maximize
expected welfare
Z Z
[ (s )
^ F (d(s))dB (s)
k ]Q
subject to the constraint that, for any two states
s1 ; s 2 ;
s1(f < kg) s2(f < kg)
implies
^ F (d(s1)) Q
^ F (d(s2)):
Q
This yields …rst-best if the possible preference pro…les
are ordered by …rst-order stochastic dominance, but
not in general.
Essentially, the problem is to choose a threshold level
^ F (d(s)) =
S such that, for s(f < kg) > S; one sets Q
^ F (d(s)) =
0; and, for s(f < kg) < S; one sets Q
1:
Majority voting is optimal if the mechanism designer’s prior is uninformed in the sense that
(i) ex ante, he indi¤erent whether to provide the
public good or not, i.e.,
and
Z Z
^ F (d(s))dB (s) = 0;
k ]Q
[ (s )
(ii) the conditional expectations of the per capita
payo¤ parameters of the proponents and the opponents of the public good, given the observation of
the voting outcome s(f < kg) = S;
Z Z Rk
and
1[
Rk
k]ds( )
1 ds(
)
^ F (d(s))dB (sjs(f < kg) = S );
Q
Z Z R1
k]ds( ) ^
k R[
QF (d(s))dB (sjs(f < kg) = S );
1 ds( )
k
are independent of S: