NBER WORKING PAPER SERIES
OPTIMAL PROVISION OF MULTIPLE EXCLUDABLE PUBLIC GOODS Hanming Fang Peter Norman Working Paper 13797 http://www.nber.org/papers/w13797
NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 February 2008
This paper substantially generalizes and supersedes Fang and Norman (2003), a manuscript that was circulated under the title "An Efficiency Rationale for Bundling of Public Goods." We thank Mark Armstrong, Ted Bergstrom, Martin Hellwig, Larry Samuelson, Stephen Morris and seminar participants at many seminars and conferences for comments and helpful discussions. The usual disclaimer applies. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2008 by Hanming Fang and Peter Norman. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.
Optimal Provision of Multiple Excludable Public Goods Hanming Fang and Peter Norman NBER Working Paper No. 13797 February 2008 JEL No. H41 ABSTRACT This paper studies the optimal provision mechanism for multiple excludable public goods when agents' valuations are private information. For a parametric class of problems with binary valuations, we demonstrate that the optimal mechanism involves bundling if a regularity condition, akin to a hazard rate condition, on the distribution of valuations is satisfied. Bundling alleviates the free riding problem in large economies in two ways: first, it may increase the asymptotic provision probability of socially efficient public goods from zero to one; second, it decreases the extent of use exclusions. If the regularity condition is violated, then the optimal solution replicates the separate provision outcome.
Hanming Fang Department of Economics Duke University 213 Social Sciences Building Box 90097 Durham, NC 27708-0097 and NBER
[email protected] Peter Norman Department of Economics University of North Carolina at Chapel Hill 107 Gardner Hall Chapel Hill, NC 27599-3305
[email protected]
1
Introduction This paper studies the optimal provision mechanism for multiple excludable public goods. We
brie‡y consider a somewhat more general setup where we obtain some characterization results, but most of the paper focuses on a parametric version of the model where valuations are binary. In the binary valuation case, we demonstrate that there is a considerable degree of bundling in the optimal solution if a regularity condition, akin to a hazard rate condition, on the distribution of valuations is satis…ed. If the regularity condition is violated, which happens when valuations are too strongly positively correlated across goods, the optimal solution replicates the separate provision outcome. To motivate the importance of a better understanding of bundling of non-rival goods, we note that many goods that are provided in bundles are close to fully non-rival. The most striking example is the access to electronic libraries, for which the typical contractual arrangement is a site license that allows access to every issue of every journal in the electronic library. Another example is cable TV, where the basic pricing scheme consists of a limited number of available packages. Other examples include computer software and digital music …les. For several of these cases, the pros and cons of bundling for the consumer have been frequently debated by the media, legal scholars, and in the courtroom. Still, there is no normative benchmark that explicitly considers the non-rival nature of these goods in the economics literature. We consider a model with M excludable public goods, meaning that all goods are fully non-rival, but consumers can be excluded from usage. Each consumer is characterized by a valuation for each good, and the willingness to pay for a subset of goods is the sum of the individual good valuations. In addition, the cost of provision for each good is independent of which other goods are provided. Under these separability assumptions, the …rst best benchmark is to provide good j if and only if the sum of valuations for good j over all consumers exceeds its provision cost and to exclude no consumer from usage. Under perfect information, there is thus no role for either bundling or use exclusions. However, when preferences are private information, consumers must be given appropriate incentives to truthfully reveal their willingness to pay. Together with self-…nancing and participation constraints, it is then impossible to implement the (non-bundling) perfect information social optimum. Bundling is then potentially useful because it improves possibilities to extract surplus from the consumers, which will then relax a binding constraint on the problem. The …rst part of the paper considers a relatively general setup, and characterizes the form of optimal provision mechanisms in symmetric environments. We then apply these results in the special case with binary valuations for which we obtain an exact characterization of the constrained e¢ cient mechanism. To make the problem tractable, we impose symmetry conditions on costs and type distributions in addition to the restriction that valuations take on only two values for each of the M excludable public goods. The solution to the problem is rather striking. When the economy is large in the sense that
1
the number of consumers goes to in…nity, the optimal mechanism will either provide all goods with probability close to one, or provide all goods with probability close to zero. Which of the two scenarios applies depends on whether or not a monopolist pro…t maximizer that provides the goods for sure could break even. If a regularity condition, akin to a hazard rate condition, on the distribution of valuations is satis…ed, the optimal mechanism also prescribes a very simple rule for user access to the public goods once they are provided. All consumers will fall into one of three groups: the …rst group, consisting of those whose numbers of goods for which they have high valuations strictly exceed a threshold, will be given access to the grand bundle consisting of all goods; the second group, consisting of those whose number of goods for which they have high valuations are strictly lower than the threshold, will be given access to only those goods for which they have a high valuation; the third group, consisting of those with exactly the threshold-level number of high valuation goods, will always be given access to their high valuation goods, and will also be given access to their low valuation goods with some probability. Note that the third group of consumers for which some randomization is applied will be quite rare when there are many public goods, i.e., as M gets large. For the special case with two goods we solve the problem also for the case where the regularity condition discussed above is violated. With two goods, the regularity condition is violated when the valuations are too positively correlated. This invalidates the standard approach to solve screening problems using only downwards adjacent constraint, as there are simply too few “mixed types” (those with one high and one low valuation) to justify giving them a better treatment than the type with two low valuations. We show that the optimal solution in fact is identical to the solution when both goods must be provided separately. It is interesting to relate this to the seminal contribution by Adams and Yellen [1]. Their explanation of bundling was that the monopolist seller knows more about the willingness to pay for the bundle than for the components provided that there is negative correlation. It has been shown that the relevant comparison is the willingness to pay for the components versus the willingness to pay for the average, implying that essentially the same explanation also works when valuations are independent.1 Our result here shows that this logic falls apart when valuations are too positively correlated. The analytical tractability for the multidimensional mechanism design problem in our paper comes from exploiting some important similarities to unidimensional problems. In particular, in the unidimensional case, it is known that maximizing social surplus subject to budget and participation constraints leads to a Lagrangian characterization which can be interpreted as a compromise between pro…t and welfare maximization (see Hellwig [10] and Norman [17]). In our model we cannot collapse the constraints to a single integral constraint, but we are able to use the optimality conditions to link the values of the multipliers associated with various constraints so that the optimal solution can be understood analogously. 1
See, e.g., McAfee, McMillan and Whinston [15] and Fang and Norman [9].
2
The remainder of the paper is structured as follows. Section 2 presents the model and some characterization results. Section 3 introduces the special case when valuations are binary and derives the optimal mechanism in the regular case. In Section 4 we use some special cases to better interpret the characterization in Section 3, in particular, we characterize the optimal mechanism for the two good case when the regularity condition is violated. Section 5 contains a brief discussion of the relevance of our analysis with respect to concrete anti-trust issues. Appendix A contains all the proofs of results in Section 3.2
2
The Model This section lays out a fairly general model (Section 2.1). The set of randomized direct mech-
anisms is represented in a somewhat nonstandard (but useful) way (Section 2.2), before we set up the mechanism design problem (Section 2.3). We then gradually show, sometimes with additional restrictions on the environment, that it is without loss of generality to consider a smaller and more tractable class of simple, anonymous and symmetric mechanisms (Sections 2.4 and 2.5). The main results of this section are Propositions 1 and 2, which are used in later Sections to reduce the dimensionality of the design problem.
2.1
The Environment
There are M excludable public goods, labeled by j 2 J = f1; :::; M g and n consumers, indexed
by i 2 I = f1; :::; ng. Each public good is indivisible, and the cost of providing good j, denoted
C j (n), is independent of which of the other goods are provided. Since n is the number of consumers in the economy, not the number of user s, all goods are fully non-rival. The rationale for indexing
cost by n is to be able to analyze large economies without making the public goods a “free lunch” in the limit. We therefore allow for the existence of cj > 0 such that limn!1 C j (n) =n = cj > 0. There is no need to give this assumption any economic interpretation. It is best viewed as a way to ensure that the provision problem remains “signi…cant” with a large number of agents. Consumer i is described by a valuation for each good j 2 J ; so that her type is given by a
vector
i
=
1 M i ; :::; i
2
RM : Agent i has preferences represented by the utility function, X j j Ii i t i ; (1) j2J
where Iji is a dummy variable taking value 1 when i consumes good j and 0 otherwise, and ti is the quantity of the numeraire good transferred from i to the mechanism designer. Preferences over lotteries are of expected utility form. The type
i
is private information to the agent. While we allow valuations across goods to be
correlated for the individual, it is essential that we assume independence across agents. We denote 2
An appendix available at the authors’websites contains the proofs for the more technical results in Section 2.
3
by F the joint cumulative distribution over
i.
For brevity of notation, we let
which will be referred to as a type pro…le. In the usual fashion, we let
2.2
i
( 1 ; :::;
= ( 1 ; ::;
n)
2
n;
i 1 ; i+1 ; ::: n ) :
Randomized Direct Mechanisms
An outcome in our environment has three components: (1). Which goods, if any, should be provided; (2). Who are to be given access to the goods that are provided; and (3). How to share the costs. The set of feasible pure outcomes is thus A=
|
f0; 1gM {z }
provision/no provision for each goods j
|
f0; 1gM {z
n
}
inclusion/no inclusion for each agent i and good j
n | R {z } :
(2)
“taxes” for each agent i
By the revelation principle, we restrict attention to direct mechanisms for which truth-telling is a Bayesian Nash equilibrium. A pure direct mechanism is a map from
n
to A: We represent
a randomized mechanism in analogy with the representation of mixed strategies in Aumann [5]. That is, let
[0; 1] ; and think of # 2
as the outcome of a …ctitious lottery, where, without
loss of generality, # is uniformly distributed and independent of : A random direct mechanism is then a measurable mapping G :
n
! A. A conceptual advantage of this representation
is that it allows for a useful decomposition. That is, we may write G as a (2M + 1)-tuple, G =
(
j
j2J
; !j
j2J
; ) where,
Provision Rule:
j
:
n
Inclusion Rule:
!j
:
n
:
n
Cost-sharing Rule: We refer to
j
! f0; 1g
! f0; 1gn
! Rn :
as the provision rule for good j, and interpret E
provision for good j given announcements : The rule
!j
(3)
=
j
( ; #) as the probability of
! j1 ; :::; ! jn
is the inclusion rule for
good j; and E ! ji ( ; #) is interpreted as the probability that agent i gets access to good j when announcements are ; conditional on good j being provided. Finally, sharing rule, where
i(
= ( 1 ; :::;
n)
is the cost-
) is the transfer from agent i to the mechanism designer given announced
valuation pro…le . In principle, transfers could also be random, but the pure cost-sharing rule in (3) is without loss of generality due to risk neutrality.
4
2.3
The Design Problem
Let E
i
denote the expectation operator with respect to (
i ; #) :
A mechanism is incentive
compatible if truth-telling is a Bayesian Nash equilibrium in the revelation game induced by G; 2 3 2 3 X X j j b b E i4 ( ; #)! ji ( ; #) ji E i4 ( i ; i ; #)! ji (bi ; i ; #) ji i ( )5 i( i; i )5 ; j2J
j2J
8i 2 I; 2
n
; bi 2
: (IC)
We also require that the project be self-…nancing. For simplicity, this is imposed as an ex ante balanced-budget constraint: 3 0 X E@
i(
X
)
i2I
j
j2J
1
( ; #) C j (n)A
0:
(BB)
Finally, we require that a voluntary participation, or individual rationality, condition is satis…ed. Agents are assumed to know their own type, but not the realized types of the other agents, when deciding on whether to participate. Individual rationality is thus imposed at the interim stage as, 2 3 X j E i4 ( ; #)! ji ( ; #) ji 0; 8i 2 I; i 2 : (IR) i ( )5 j2J
A mechanism is incentive feasible if it satis…es (IC), (BB) and (IR). Utility is transferable, implying that constrained ex ante Pareto e¢ cient allocations may be characterized by solving a utilitarian
planning problem, where a …ctitious social planner seeks to maximize total surplus in the economy, subject to the constraints (IC), (BB) and (IR).4 Thus a mechanism is thus constrained e¢ cient if it maximizes
X
E j ( ; #)
j2J
over all incentive feasible mechanisms.5
"
X
! ji ( ; #)
j i
#
C j (n) ;
i2I
It is ex post e¢ cient to provide good j if and only if
P
i2I
j i
(4)
C j (n), and to never exclude
any agent from usage, which is the same rule as the …rst best rule for a single public good. This is implementable if and only if a non-excludable public good can be e¢ ciently provided under (IC), (BB) and (IR). (See Mailath and Postlewaite [17].) Our setup is thus a second best problem. 3
As shown in Borgers and Norman [9] it is without loss of generality to consider a resource constraint in ex ante
form. Given independence and two or more agents, transfers can be adjusted so as to satisfy ex post budget balance without changing the interim expected payo¤ for any individual. 4 The quali…er ex ante is crucial for the equivalence. See Ledyard and Palfrey [18] for a characterization of interim e¢ ciency. 5 All these constraints are noncontroversial if the design problem is interpreted as a private bargaining agreement. If the goods are government provided, the participation constraints (IR) may seem questionable. One defense in this context is that the participation constraint is a reduced form of an environment where agents may vote with their feet. Another defense is to view this as a reduced form for inequality aversion of the planner. See Hellwig [15].
5
2.4
Simple Anonymous Mechanisms
To simplify the analysis, we …rst exploit the symmetry, as well as the linearity, of the constraints and the objective function. This allows us to reduce the dimensionality of the problem: De…nition 1 A mechanism is called a simple mechanism if it can be expressed as (2M + 1)-tuple g=(
where
j j2J
j
;
j j2J
; t) such that for each good j 2 J ; Provision Rule:
j
:
Inclusion Rule:
j
:
Cost-sharing Rule:
t:
is the provision rule for good j,
j
n
! [0; 1]
! [0; 1]
(5)
! R;
is the inclusion rule for good j (same for all agents),
and t is the transfer rule (also same for all agents). There are a number of simpli…cations in (5) relative to (3). First, inclusion and transfer rules j
are the same for all agents; second, conditional on ; the provision probability
( ) is stochasti-
cally independent from all other provision probabilities, and all inclusion probabilities; third, the inclusion and transfer rules for any agent i are independent of the realization of
i;
and fourth,
all agents are treated symmetrically in terms of the transfer and inclusion rules. But (5) allows provision rules to treat agents asymmetrically. We therefore need a de…nition to express what it means for the name of an agent to be irrelevant: De…nition 2 A simple mechanism is called anonymous if for every j 2 J ; every
;
0
2
n
n
such that
0
can be obtained from
j
( ) =
j
0
for
by permuting the indices of the agents.
We now show that focusing on simple anonymous mechanisms is without loss of generality: Proposition 1 For any incentive feasible mechanism G of the form (3), there exists a simple
anonymous incentive feasible mechanism g of the form (5) that generates the same social surplus.
The idea is roughly that risk neutral agents care only about the perceived probability of consuming each good and the expected transfer. Therefore, there is nothing to gain from conditioning transfers and inclusion probabilities on
i,
or by making inclusion and provision rules condition-
ally dependent. Mechanisms of the form (5) are therefore su¢ cient. Moreover, given an incentive feasible mechanism, permuting the roles of the agents leaves the surplus unchanged and all constraints satis…ed. An anonymous incentive feasible mechanism that generates the same surplus as the initial mechanism can therefore be obtained by averaging over the n! permuted mechanisms.6 6
The actual proof is a bit more complex than simply randomizing with equal probabilities over the n! permutations.
The reason is that inclusion and provision probabilities are potentially correlated.
6
2.5
Symmetric Treatment of the Goods
Our next result, on which we rely heavily in Sections 3 and 4, identi…es conditions under which it is without loss of generality to treat goods symmetrically. Obviously, the underlying environment must be symmetric, and we formalize this by assuming that random variable, that is F ( i ) = F C (n) such that
Cj
let
0 i
is a permutation of
;
M i i;
is an exchangeable
and that there exists
and a one-to-one permutation mapping P : J ! J of the set of goods,
denote the permutation of agent i0 s type by changing the role of the goods in accordance to P i
P : that is, write
whenever
1 i;
=
(n) = C (n) for all j:
Given valuation pro…le P i
0 i
i
P
= P 1 ; :::;
P 1 (1) P 1 (2) P 1 (M ) ; i :::; i ; i P n as the valuation pro…le
where P
1
denote the inverse of P . For simplicity,
obtained when the role of the goods is changed in
accordance to P for every i 2 I. De…nition 3 Mechanism g is symmetric if for every 1.
P
1 (j)
P
=
j
( ) for every j 2 J ;
2.
P
1 (j)
P i
=
j
( i ) for every j 2 J ;
P i
3. t
and every permutation P : J ! J :
= t ( i) :
In de…ning a symmetric mechanism, the same permutation of goods must be applied for all agents. As an example, suppose that there are two agents and two goods, and that the valuation for each good is either h or l: In this case = ( 1;
2)
2:
= ((h; l) ; (l; h)) 2
= f(h; h) ; (h; l) ; (l; h) ; (l; l)g. Consider the type pro…le
Applying the only non-identity permutation of the goods, i.e.,
P (1) = 2 and P (2) = 1; to all agents generates a type pro…le
P
=
P P 1; 2
= ((l; h) ; (h; l)) :
De…nition 3 requires that the allocations for type pro…le ((l; h) ; (h; l)) is the same as the allocation for ((h; l) ; (l; h)) with goods relabeled, and that transfers are unchanged.7 The result is: Proposition 2 Suppose that that
Cj
i
is an exchangeable random variable and that there exists C (n) such
(n) = C (n) for all j 2 J : Then, for any simple anonymous incentive feasible mechanism
g; there exists a simple anonymous and symmetric incentive feasible mechanism that generates the same surplus as g: The idea is similar to that of Proposition 1, except that it is the identities of the goods that are permuted. Consider the case with two goods, and suppose that the two goods are treated 7
If we were to apply di¤erent permutations for the two agents, e.g., applying the identity permutation for agent
1 and the non-identiy permutation for agent 2, then we would obtain a pro…le ((h; l) ; (h; l)), which is a qualitatively di¤erent from either ((h; l) ; (l; h)) or ((l; h) ; (h; l)) : In the pro…le ((h; l) ; (h; l)) ; both agents have low valuations for good 2 and high valuations for good 1, whereas, in the pro…les ((h; l) ; (l; h)) or ((l; h) ; (h; l)) ; one and only one agent has high valuation for both goods.
7
asymmetrically. Reversing the role of the goods, an alternative mechanism that generates the same surplus is obtained. Averaging over the original and the reversed mechanism creates a symmetric mechanism where surplus is unchanged.8 Incentive feasibility of the new mechanism follows from incentive feasibility of the original mechanism. Proposition 2 generalizes this procedure by permuting the goods (M ! possibilities) and creating a symmetric mechanism by averaging over these permuted mechanisms.
3
The Case with Binary Valuations In the remainder of the paper we assume that the valuation for each good j is either high or
low, so that denoted
j i
2 fl; hg for each i 2 I and j 2 J : The probability of any type
i ( i) ;
and using independence
the probabilities of type pro…le j2
J :10
For any
i.e.,
i
2
and
i
n i=1 i ( i ) and respectively:9 Moreover,
( )=
i(
i)
=
suppose that
n m ( i) = # j 2 J :
Maintaining the assumption that
i
M! m!(M m)!
types of
j i i
o =h :
2
such that
j i
i0 6=i i0 ( i0 ) denote C j (n) = cn for all
j i
= h;
= h for exactly m goods.
is an exchangeable random variable, we therefore have that the
probability that an agent has a high valuation for exactly m goods, denoted by m
i
fl; hgM is
2
= fl; hgM ; write m ( i ) 2 f0; :::; M g as the number of goods for which
Given any m 2 f0; :::; M g there are
where
i
=
M! m! (M m)!
m;
is given by
i ( i) ;
(6)
is any type with m high valuations. This formulation allows the willingness to pay between
goods to be correlated. In the simplest case where valuations for di¤erent goods are i.i.d., for any i
such that m ( i ) = m,
i ( i)
=
m (1
)M
m
and
m
=
M! m!(M m)!
m (1
)M
m
; where
is
j i
the probability that = h: Appealing to Proposition 1, we only consider mechanisms in the form of (5), implying that it is without loss of generality to consider:11 8
Provision probabilities and taxes are given by straightforward averaging, but since inclusion and provision prob-
abilities may be correlated the procedure is somewhat more involved for the inclusion rules. 9 Note that the probability of type i ; i ( i ) does not depend on i: The subscript i is used in
i
so that we can
use
to denote the probability of the valuation pro…le = ( 1 ; :::; n ) : 10 Keeping the per capita costs constant simpli…es notation, but is not necessary. 11 Note that the dimensionality of the problem could be reduced further by using Propositions 1 and 2, but it is
notationally convenient to impose these additional symmetry restrictions at a later stage. As such, we are indexing j i
and ti by i; even though we know from Propositions 1 and 2 that they do not need to depend on i in the optimal
mechanism.
8
max
f ; ;tg
X
0
i2
X
0
i2
i
( )
(
i)
M X
( )
where
2
[0; 1] ; i
j i
j i
( i)
#
j
j i
( )
( i)
j i
0 i
i;
2
(
i)
M X
j
(
i ; l)
j i
2
X
4
ti ( i )
(l) l
(7)
cn
i=1
i2
i
(
i)
M X
j
i;
0 i
0 i
j i
j i
ti
j=1
i
0 i
3 5
(8)
ti (l)
(9)
( ) cn
(10)
j=1
i
i ( i ) ti ( i )
i=1 i 2
j
" n X
j=1
i
n X X
0
j
j=1
i
for every X
( )
2 n
M X
X 2
j i
( )
n
M X
j
j=1
( i ) 2 [0; 1] ;
denotes the set of possible pro…les for all agents except i: In words, the planner maximizes
the social surplus subject to the incentive compatibility constraints (8), the participation constraint for type l = (l; ::::; l) (9), and the resource constraint (10). To simplify our discussions in the analysis below, we use the following terminology: De…nition 4 An incentive constraint (8) is referred to as: j i
1. a downwards incentive constraint if 2. an upwards incentive constraint if
j i
0j i 0j i
for all j 2 J ;
for all j 2 J ;
3. a diagonal incentive constraint if there exists j; k 2 J so that
j i
>
0j i
and
k i
0 2
1
0
is used, the ex ante
probability of getting access to low valuation good is 0
( (
+ 1+
1
2 ) (h
+ l) 2 ) (h + l)
2c : 2l
Some algebra along the lines discussed in connection with (26) shows that the ex ante probability of getting access and therefore also the social surplus is actually smaller using the bundling mechanism in this case.
5
Conclusion and Discussion This paper studies the optimal provision mechanism for multiple excludable public goods when
agents’valuations are private information. For a parametric class of problems with M goods whose valuations take binary values, we fully characterize the optimal mechanism and demonstrate that it involves bundling if a regularity condition, akin to a hazard rate condition, on the distribution of valuations is satis…ed. Bundling alleviates the free riding problem in large economies in two ways: 20
…rst, it may increase the asymptotic provision probability of socially e¢ cient public goods from zero to one; second, it decreases the extent of use exclusions. For the case of two goods, we also show that if the regularity condition is violated, then the optimal solution replicates the separate provision outcome. Our Model as a Positive Theory of Bundling.
Bundled discounts are common in many
markets. In fact, some have argued that almost any commodity is best viewed as a bundle of characteristics.13 However, once we take a narrower perspective of only considering bundles of commodities that are viable “stand-alone goods” in the sense that a consumer has a substantial willingness to pay for each component even if no other goods in the bundle are consumed, it is arguable that commonly bundled commodities mostly have non-rival properties, or, more generally, goods with a substantial …xed costs. Software, music, electronic libraries, and TV programming are obvious examples, and these kinds of goods are growing more important. As such, our paper may be interpreted as a positive theory of bundling of goods with no-rival properties. It is worth noting that in this paper we focused on the harder problem of maximizing social welfare instead of maximizing pro…ts; but the pro…t-maximizing selling strategy is a by-product of our analysis. It is easy to see that the only change in the asymptotic characterization of the pro…tmaximizing selling mechanism is that the threshold number of high valuation goods a consumer needs in order to get access to her low valuation goods will increase. Or, put di¤erently, the price of the grand bundle will be higher than in Proposition 3 because a pro…t-maximizing monopolist will seek as high a pro…t as possible rather than to only break even. The rest of Proposition 3 is unchanged for the pro…t-maximization problem. The key di¤erence between our model of bundling for non-rival goods and the standard bundling model for private goods with constant unit costs is that the decision for whether to provide a good becomes non-trivial. This in turn makes it necessary to go beyond the standard setup with a single consumer. As a result the characterization of the optimal selling mechanism in our setup is more complicated when there is a …nite number of consumers. A Regulatory Benchmark for Bundling of Goods with Large Fixed Costs.
Bundling
may under some circumstances violate current U.S. anti-trust legislation; indeed the legality and desirability of bundling of software (a non-rival good) have been a critical point of debate in several recent cases. While economists have been involved in this discussion, to the best of our knowledge there is not yet a clear-cut normative benchmark in the economics literature. The analysis in this paper is a small step towards the development of a useful normative benchmark for bundling in natural monopoly situations. 13
For example, a car can be considered as a “bundle” consisting of a chassis, an engine, cup holders, a stereo
system, air conditioning etc.
21
Though our model is highly stylized, it has enough ‡exibility to generate a non-trivial trade-o¤. On the one side, it is a possible that bundling is required for the monopolist to break even. In this case the pro…t-maximizing outcome with bundling is better for the consumer than the welfare maximizing outcome without bundling; thus a requirement to “unbundle” is strictly worse for the consumers, regardless of other budget-balancing remedies that may be combined with the decision to unbundle.14 On the other side, the fact that bundling makes it easier for the monopolist to extract consumer surplus can make consumers worse o¤ for the obvious reasons. Our model still makes many restrictive assumptions on the cost and preference side. In particular, while the assumption that preferences exhibit no complementarities highlights the role of bundling as a screening instrument, it is counter-factual in many cases. Moreover, arguments based on alleged complementarities are often used to justify bundling. However, it should be rather clear that in itself, complementarities are not su¢ cient to provide a rationale for bundling: if many consumers like to consume a particular computer program jointly with their computer, they may choose to do so regardless of whether the computer program is bundled with the computer.15 Even though the argument that “bundling is good for the customer because the customers like the goods together”is suspect, complementarities in preferences may matter if combined with some other potential justi…cation for bundling, such as the considerations about …xed costs of the current paper. We intend to study this question in future research.
References [1] Adams, W.J. and Janet L. Yellen (1976). “Commodity Bundling and the Burden of Monopoly.” Quarterly Journal of Economics, Vol. 90, 475 –98. [2] Armstrong, Mark (1996). “Multiproduct Non-Linear Pricing.” Econometrica, Vol. 64, 51-76. [3] Armstrong, Mark (1999). “Price Discrimination by a Many-Product Firm.” Review of Economic Studies, Vol. 66,151-168. [4] Armstrong, Mark and Jean-Charles Rochet (1999). “Multi-dimensional screening: A User’s Guide.” European Economic Review, Vol. 43, No. 4, 959-979. [5] Aumann, Robert J. (1964). “Mixed and Behavioral Strategies in In…nite Extensive Games.” Annals of Mathematical Studies, Vol. 52, 627-650. 14
This line of reasoning was an important part of the motivation in the decision by the O¢ ce of Fair Trading [24]
in the U.K. on alleged anti-competitive mixed bundling by the British Sky Broadcasting Limited. 15 Indeed, for computers bought on the internet, the buyer already has a long array of options on how to customize the computer, and it would be easy to allow the consumer to opt out of the standard software bundle (a choice which is usually not available).
22
[6] Bakos, Yannis and Erik Brynjolfsson (1999). “Bundling Information Goods: Pricing, Pro…ts and E¢ ciency.” Management Science, Vol. 45, 1613-1630. [7] Bergstrom, Carl T and Theodore C. Bergstrom (2004). “The Costs and Bene…ts of Library Site Licenses to Academic Journals.” Proceedings of the National Academy of Sciences in the United States of America, Vol. 101, 897-902. [8] Bolton, Patrick and Mathias Dewatripont (2005). Contract Theory, MIT Press: Cambridge, MA. [9] Borgers, Tilman and Peter Norman (2007). “A Note on Budget Balance under Interim Participation Constraints: The Case of Independent Types.” manuscript, University of Michigan and University of North Carolina. [10] Cremer, Jacques and Richard P. McLean (1988). “Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions.” Econometrica, Vol. 56, 1247-1257. [11] Dana, James D. (1993). “The Organization and Scope of Agents: Regulating Multiproduct Industries.” Journal of Economic Theory, Vol. 59, 288-309. [12] Fang, Hanming and Peter Norman (2003). “An E¢ ciency Rationale for Bundling of Public Goods.” Cowles Foundation Discussion Paper No. 1441. [13] Fang, Hanming and Peter Norman (2006). “To Bundle or Not to Bundle.” RAND Journal of Economics, Vol. 37, No. 4, 946-963. [14] Hellwig, Martin F. (2003). “Public Good Provision with Many Participants.” Review of Economic Studies, Vol. 70, 589-614. [15] Hellwig, Martin F. (2005). “A Utilitarian Approach to the Provision and Pricing of Excludable Public Goods.” Journal of Public Economics, Vol. 89, 1981-2003. [16] Jackson, Matthew O. and Hugo F. Sonnenschein (2007). “Overcoming Incentive Constraints by Linking Decisions.” Econometrica, Vol. 75, No. 1, 241-258. [17] Mailath, George and Andrew Postlewaite (1990). “Asymmetric Information Bargaining Problems with Many Agents.” Review of Economic Studies, Vol. 57, 351-368. [18] John O. Ledyard and Thomas R. Palfrey (2007). “A General Characterization of Interim E¢ cient Mechanisms for Independent Linear Environments.” Journal of Economic Theory, Vol. 133, No. 1, 441-466. [19] Matthews, Steven, and John Moore (1987). “Monopoly Provision of Quality and Warranties: An Exploration in the Theory of Multidimensional Screening.” Econometrica, Vol. 55, No. 2, 441-467. 23
[20] McAfee, R. Preston, John McMillan and Michael D. Whinston (1989). “Multiproduct Monopoly, Commodity Bundling, and Correlation of Values.”Quarterly Journal of Economics, Vol. 104, 371–84. [21] Myerson, Roger (1981). “Optimal Auction Design.”Mathematics of Operations Research, Vol. 6, 58-73. [22] Myerson, Roger and Mark A. Satterthwaite (1983). “E¢ cient Mechanisms for Bilateral Trading.” Journal of Economic Theory, Vol. 29, 265-281. [23] Norman, Peter (2004). “E¢ cient Mechanisms for Public Goods with Use Exclusion.” Review of Economic Studies, Vol. 71, 1163-1188. [24] O¢ ce of Fair Trading (2003). BSkyB decision dated 17 December 2002: rejection of applications under section 47 of the Competition Act 1998 by ITV Digital in Liquidation and NTL, London, UK. Available at: http://www.oft.gov.uk/business/competition+act/decisions/bskyb.htm
A
Appendix: Proofs of Results in Section 3
Proof of Lemma 1. Proof. The only variables that are not automatically in a compact set are the transfers. However, ti (l)
M l < M h and ti ( i )
ti ( i jlk )
m ( i ) h + [M
m ( i )] l < M h. Recursive application
of (12) therefore implies that we may bound ti ( i ) from above by M 2 h: Since this is also an upper bound for the di¤erence between ti ( i ) and ti ( i jlk ) ; it follows from (14) that we may bound ti ( i ) M 2 h: Hence, existence of a solution to (11) follows from Weierstrass maximum
from below by theorem.
Proof of Lemma 2. Proof. Let m ( i ) = m bi = m, let
i
0 i; i
bi ; b0 denote the multiplier associated with i b 1 downwards adjacent constraints for type i . Proposition 2 ensures that provision
downwards adjacent constraints for type
one of the m
denote the multiplier associated with one of the m 1
i
and let
i
and inclusion rules are symmetric and by use of strong duality in linear programming we …nd that 0 it is without loss to assume that i i ; 0 = i bi ; b . i
Proof of Lemma 3. Proof. Fix m and let involving i : 0
X |
i
i
(
i)
M X j=1
j
i
2
( ) {z
Term A
j i
i
with m ( i ) = m: Consider the incentive compatibility conditions
( i)
j i
ti ( i ) }
X
|
i
(
i)
M X j=1
i
j
(
j i ; i jlk ) i
{z
Term B
24
( i jlk )
j i
ti ( i jlk ) (A1) }
We note that ti ( i ) enters in Term A for m conditions: there are m ways to manipulate a single coordinate so as to announce a pro…le with one h replaced with a l: From Lemma 2, the multiplier associated with each of these conditions is (m) : In addition, ti ( i ) enters in Term B for M m downwards incentive constraints for types with m + 1 high valuations. Finally, it also enters the resource constraint (14). The …rst order condition with respect to ti ( i ) can thus be written as m (m) + (M
m) (m + 1) +
i
( i) =
m (m) + (M
m! (M m)! M!
m) (m + 1) +
m
= 0; (A2)
where we used (6) for the …rst equality. For m = M; condition (A2) reads M (M ) = M , implying that Lemma 3 is also true for m = M: Now suppose that (m) is given by the expression in (17) for some m M . The 0 0 optimality condition (A2) with respect to ti i where m i = m 1 then reads: 0
=
(m
1) (m
1) + (M
=
(m
1) (m
1) +
=
(m
1) (m
1) +
=
(m
1) (m
1) +
1)! (M m + 1)! m 1 M! (m 1)! (M m + 1)! m + 1) m (m) + m 1 m M! M m + 1) m! (M m)! X (m 1)! (M m + 1)! j + m M! M! j=m m + 1) (m) +
(M (M
(m
(m
1)! (M m + 1)! M!
M X
m 1
j;
j=m 1
where the third equality follows from induction hypothesis. Thus, (17) holds also for m
1: The
result follows from induction. Proof of Lemma 4. Proof. Let j be some good for which respect to ji ( i ) may be written as X i2
j
( ) i
(m + 1) (M
i2
where j i
( i)
j i
( i)
j i
= h: The optimality conditions for problem (11) with X
( ) h + (m) m X
m) j i
j i
i2
i
(
i)
i)
j
i
(
( )h
j i
( i)
j i
j i
= 0;
(A3)
i
j
( )h +
j i
( i) 1
( i)
=
0
i
( i ) = 0 and
( i)
( i ) are respectively the multipliers for the constraints ji ( i ) P j( ; 0. Since we are assuming that i) i i ) > 0 and since i( i2 i ( i ) and
j i
X i2
( )
j
( )h =
i ( i)
X i2
i
i(
i)
j i
( i)
j
( ) h;
0 and 1
(A4)
i
we may simplify the …rst equality in (A3) and write i ( i) h
+ (m) mh
(m + 1) (M
25
m) h +
j i
( i ) = 0;
(A5)
which, by the use of the complementary slackness conditions in (A3), implies that 8 > 1 if i ( i ) + (m) m (m + 1) (M m) > 0 > < j i
( i) =
> > :
x 2 [0; 1] 0
But, from (15) we have that i ( i)
if
i ( i)
+ (m) m
(m + 1) (M
m) = 0
if
i ( i)
+ (m) m
(m + 1) (M
m) < 0:
(m) m
(m + 1) (M
+ (m) m
(m + 1) (M
Proof of Lemma 5. Proof. Let j be some good for which respect to ji ( i ) are: X i2
(m + 1)
X i2
j i
j i
where
( i ) and
j i
( )
j
j i
m) =
m) = (1 + )
X
i
i2
i
i)
implying that
i ( i)
> 0:
= l: The optimality conditions for problem (11) with
( ) l + (m) m j
i ( i) ;
i
(
( ) [(M
j i
( i ) = 0 and
m
(
j
i)
( )l
(A6)
i
j i
1) l + h] +
j i
( i)
( i) = 0
i
( i)
j i
j i
( i) 1
( i ) = 0;
( i ) are the multipliers for the constraints
j i
( i)
0 and 1
j i
( i)
0
respectively. Using (A4) as in the proof of Lemma 4, we may rearrange the …rst inequality in (A6) as: i ( i) l
+ (m) ml
(m + 1) [(M
1) l + h] + P
m
j i
j i
( i)
i2
i(
i
( i) = 0: j( ) i)
(A7)
Using the relations between the multipliers we …nd that
i
/Lemma 3/
=
i ( i ) l (1 + )
=
= 1+
( i ) l (1 + )
=
=
=
( i ) l + (m) ml
/(15)/
/(6)/
for
i
(m + 1) [(M (m + 1) (h
(h
m! (M
l)
m
1) l + h]
l) (m + 1))! M!
M X
j
j=m+1
M X m! (M m)! m! (M (m + 1))! l (1 + ) (h l) m j; M! M! j=m+1 2 3 M X m! (M (m + 1))! 4 5 m)l (1 + ) (h l) m (M j M! j=m+1
m! (M
(m + 1))! (1 + ) Gm ( ) M!
: Substituting into (A7) and using the complementary slackness conditions in (A6), we
have j i
( i ) = (m) =
8 > > < > > :
0
if Gm ( ) < 0
z 2 [0; 1]
if Gm ( ) = 0
1
26
if Gm ( ) > 0;
as asserted. Proof of Lemma 6. Proof. The optimality conditions for ambiguous notation) be written as
( )
"
M X1
n X
j i
#
j i
( i)
M X
cn +
i=1
j
( ) associated with the problem (11) may (with somewhat
(m) m H j ( ; m)
i
(
i) h
+ Lj ( ; m)
(m + 1) H j ( ; m)
i
(
m) h + Lj ( ; m)
i ) (M
i
(
i)
m=0
j
( ) cn +
j
( )
i
(
i)
(m) l
m=0
(m) f(M
m) l + (h
l)g
( ) = 0;
together with the complementary slackness conditions.16 Recall that H j ( ; m) (respectively, Lj ( ; m)) is the number of agents that has a high (respectively, low) valuation for good j and m high valuations in total, whereas
(m) denotes the probability that an agent gets access to his low valuation
goods when having m high valuations (as characterized in Lemma 5), and
j
( ) and
j
( ) are the
multipliers for the boundary constraints. After using (A4) and the fact that n X
j i
j i
( i)
=
M X
H j ( ; m) h +
m=0
i=1
M X
Lj ( ; m) (m) l;
m=0
we can write the condition as M X
H j ( ; m) h +
m=0 M X1
M X
Lj ( ; m) (m) l +
m=0
(M
m) h + Lj ( ; m) i ( i)
m=0
j
j
( )
m h + Lj ( ; m) ( ) i i
(m) H j ( ; m)
m=0
(m + 1) H j ( ; m)
(1 + ) cn +
M X
( )
( )
1 (m) ((M i ( i)
m (m) l i ( i)
m) l + (h
l))
m) l + (h
l)g]
= 0:
Collecting terms, we get: M X
H j ( ; m) h +
m=0 M X
Lj ( ; m)
h [ (m) m i ( i)
(m) [ (m) ml i ( i)
(m) l +
m=0 j
(1 + ) cn + 16
j
( ) ( )
( )
i
m)]
(m + 1) f(M
= 0:
The notation is somewhat unsatisfactory in that
where
(m + 1) (M
i
(
i)
would be more appropriately denoted by
i
( j i)
would describe a type that would enter in the particular term (and therefore change with each term).
Concretely, H j ( ; m)
i
(
i)
would correspond to
i
( j i ) for
27
i
with
j i
= h and a total of m high valuations.
Using the di¤erence equation for the multipliers in (15), which implies that (m) m i ( i ) ; we can simplify this further [after using (15)] to: M X
m=0
M X
H j ( ; m) fh + hg + j
(1 + ) cn +
(m) l + (m)
(m + 1) (h i ( i)
l
m=0
j
( )
Lj ( ; m)
( )
( )
But using (17) and (6), we have that
(m + 1) (M
l)
m) =
(A8)
= 0: m!(M m 1)! M!
(m + 1) =
1 m! (M m M! i ( i)
(m + 1) /(17)/ = i ( i) =
/(6)/ =
1 m! (M i ( i ) M ! (M
m)! m)
1 (M m
M X
m)
PM
j;
j=m+1
1)!
M X
thus (A9)
j
j=m+1 M X
j
j=m+1 j:
j=m+1
Hence, (m) l + (m)
/(18)/
=
(1 + )
=
(1 + )
=
(1 + ) m (M
=
(m + 1) (h i ( i)
l
l)
2
1 (M m
(m) l + (m) 4 l
=
8 9 2 3> > > > M < = X 1 4 5 (m) l (h l) > > 1+ m) j=m+1 j m (M > > | {z } : ; = 8 2 39 M < = X 1 (m) (1 ) l + 4l l)5 j (h : ; m) j=m+1 m (M 8 2 M X (m) < (1 ) m (M m) l + 4 m (M m) l m) :
j
M X
m)
j
(h
j=m+1
39 = l)5 ;
(h
j=m+1
(1 + ) max f0; Gm ( )g : m) m (M
Substituting this back into (A8) gives us (1 + )
"
M X
j
H ( ; m) h +
m=0
M X
j
L ( ; m)
m=0
1 (M m
m)
max f0; Gm ( )g
#
j
cn +
j
( )
( )
( )
= 0;
which, together with the complementary slackness conditions, gives the result. Proof of Lemma 7. Proof. The solution to (11) violates an incentive constraint in (7) if there exists X i2
i( i
i)
M X j=1
j
( )
j i
( i)
j i
ti ( i )
0 only if
m)
max f0; Gm ( )g
cn
0:
The lemma follows from the fact that n X
j i
( i)
j i
cn =
M X
H ( ; m) h +
m=1
i=1
and 1 m (M
j
m)
Gm ( ) = l
M X
Lj ( ; m) l
cn;
m=0
2
4(h
l)
M X
j=m+1
j
3
5 < l:
(Proof of Lemma 11, continued:) Since we assume that ( ; ; t) is not ex post e¢ cient, it must be the case that (n 1)h > nc. Otherwise, the ex post optimal mechanism is either to implement if and only if ji = h for all i (when (n 1) h < nc < nh), or to never provide any good (when 32
nh nc). In each of these two cases, the ex post optimal mechanism is trivially implementable under the constraints in (11). We therefore assume that (n 1)h > nc in the remainder of the proof. If a downwards adjacent incentive constraint does not bind, there exists some i such that X
i
i2
(
i)
M X
j
( )
j i
j i
( i)
j=1
i
i2
j i
If
X
ti ( i ) >
( i jlk ) < 1; we can increase P 1)h > nc; it follows that i2
j i
i
(
i)
M X
j
j i ; i jlk ) i
(
j=1
i
( i jlk )
j i
ti ( i jlk )
( i jlk ) slightly without violating the constraint. Since (n
i
i(
i)
j
(
i ; i jlk )
> 0 for every j: Hence, the value of
the objective function increases, which contradicts the assumption that ( ; ; t) is an optimum in the …rst place. Suppose instead that ji ( i jlk ) = 1: Then, we can …nd some " > 0 and raise the tax to e ti ( i ) = ti ( i ) + " for i and every 0i with m 0i = m ( i ) : This raises some extra
revenue, implying that the taxes can be lowered slightly for some other types without upsetting the resource constraint. We can therefore proceed inductively as follows. First consider types 0 i
with m 0 i
m
= m ( i)
= m ( i)
2: If
j i
0 i
< 1 we can increase the surplus by increasing
2 and simultaneously decrease ti
0 i
keep the downwards incentive constraint for types with in resource constraint this can be done). If types
0 i
with m
0 i
= m ( i)
j i
0 i
for types with 0 i
= m ( i)
= 1 and m ( i )
0 i
j i
= m ( i)
0 i
for
0 i
0 i
with
1 in order to
1 satis…ed (as we have slack 2
1 repeat the argument for
3: By Claim A1, it follows that the associated increased access is
always desirable from the social welfare viewpoint, so by induction, we conclude that either there exists a feasible allocation with a higher surplus or that
j i
( i ) = 1 for all
i
and j; meaning that
there are no active use of exclusions in the solution. However, as we have slack in the resource constraint we can also gain surplus by increasing
j
( ) for some
unless the solution to (11) is the
ex post optimal rule. Proof of Lemma 12. Proof. By Lemma 7, all “diagonal” incentive constraints hold, and by Lemma 8 all downwards constraints hold. By Lemma 11 we have that all downwards adjacent incentive constraints bind, which, by use of Lemma 10 assures that the upwards adjacent incentive constraints hold. Finally, Lemma 9 guarantees that all upwards incentive constraints hold. Taken together, we have then veri…ed that all incentive constraints in (7) hold if the solution to the relaxed problem is monotonic and di¤erent from the …rst best. Proof of Lemma 13. Proof. (1). First note that (19) is irrelevant when m = M; and it is immediate from (21) that the probability of provision given (h; i
2
1 m (M m)
i:
i)
is always weakly higher than under ( i ;
i)
for any
Next, from (19) and (21), we know that a su¢ cient condition for monotonicity is that
max f0; Gm ( )g is increasing in m on f0; ::::M 1g : But, 2 M X Gm ( ) (h l) 4 =l m) m) m (M m (M
j=m+1
33
j
3 5
(A15)
which proves the …rst part of the Lemma. (2). If valuations are independent across goods, say the probability that an agent has a high valuation for any good j is ; then the probability that an agent has m high valuations is given by m
=
M! m!(M m)!
m (1
m+1
=
)M
m
: This implies that
M! (m + 1)! (M m
m+1
1)!
)M
(1
m 1
(M m) (m + 1) (1
=
)
m:
(A16)
Using (A16), we have that 1 (M m m+1
M X
1) j=m+2
j
=
=
=
/j
m + 1 and
M
M X1
1 (M m m+1
> 0/
0 is implied by Gm ( ) is implied by Gm ( )
(A17)
j+1
0; and that Gm
1(
)
1. conditional on provision, all consumers get access to all their high valuation goods and the inclusion rule for low valuation goods is given by: 8 > 0 if Gm (1) < 0 > < j z 2 [0; 1] if Gm (1) = 0 i ( i ) = (m) > > : 1 if Gm (1) > 0;
(A19)
2. there exists some 0 such that the provision rule for good j satis…es 8 PM h j maxf0;Gm (1)g j > 0 if 1 + > m=0 H ( ; m) h + L ( ; m) > m (M m) < h PM maxf0;Gm (1)g j j j z 2 [0; 1] if 1 + ( )= m=0 H ( ; m) h + L ( ; m) m (M m) > h > P > maxf0;Gm (1)g M j j : 1 if 1 + m=0 H ( ; m) h + L ( ; m) (M m) m
i cn < 0 i cn = 0 i cn > 0 (A20)
Proof of Claim A2: The derivation of the inclusion rules follow the analysis of the constrained welfare problem step by step and is omitted. To derive the provision rule, note that the optimality conditions for problem (A18) may be written as ( )+
M X
(m) m H j ( ; m)
i
(
i) h
j
( ) associated with the
+ Lj ( ; m) (m) l
m=0 M X1
(m + 1) H j ( ; m)
i
(
i ) (M
m) h + Lj ( ; m)
i
(
i)
(m) [(M
m) l + (h
l)]
m=0
j
( ) cn +
j
( )
( ) = 0;
together with the complementary slackness conditions. By noting that write the condition as 1+
M X
m h + Lj ( ; m) ( ) i i
(m) H j ( ; m)
m=0 M X1
(m + 1) H j ( ; m)
(M
cn +
j
j
( )
( )
( )
( )
i)
=
1 (m) [(M i ( i)
m) l + (h
= 0:
Collecting terms, we get 1+
M X
h [ (m) m i ( i)
H j ( ; m)
m=0
+
M X
Lj ( ; m)
m=0
cn +
j
(m) f (m) ml i ( i) j
( ) ( )
( )
= 0:
35
1 i( i)
m (m) l i ( i)
m) h + Lj ( ; m) i ( i)
m=0
i(
(m + 1) (M
(m + 1) [(M
m)]
m) l + (h
l)]g
l)]
; we can
Using the di¤erence equation for the multipliers in (15), we can simplify this further to: M X
1+
M X
H j ( ; m) h +
m=0
Lj ( ; m)
m=0
Using (A9), we can eliminate 1+
M X
H j ( ; m) h +
m=0
M X
Lj ( ; m)
Furthermore, (m) l
l)
M X
j
8
>
> :
0
if 1 +
z 2 [0; 1]
if 1 +
1
Observe that
i
j i=1 Z ( i ) Pn j i=1 Z ( i ) Pn j i=1 Z ( i )
if 1 +
h
if
maxf0;Gm (1)g m (M m)
j i
if
The point with this formulation is that Zj
Pn
k i
= l and Zj
cn = 0 cn > 0;
=h
= h for exactly m goods k 2 J :
n i=1
i
j i
cn < 0
is a sequence of i.i.d. random variables.
is bounded below by 0 and above by h, so the variance is bounded. This
allows us to use the central limit theorem to establish a “generalized paradox of voting”, which simply states the intuitively obvious fact that the in‡uence of a single agent approaches zero as the number of agents goes out of bounds. To state this, we will now need to be careful about the fact that the solution depends on the number of agents in the economy, and a mechanism with n agents will therefore now be denoted by ( denoted by
n ; n ; tn )
and the multiplier on the resource constraint will be
n:
(Proof of Lemma 15, continued): Consider a mechanism that solves (A18) …rst. As Z j for every i 2 ; it follows that for any pair 0i ; 00i we have E
j n
( )j
0 i
E
j n
( )j
2
Pr 41 +
00 i
=
Pr
2
41
n
cn n
0
@h + h
X k6=i
Z j ( k )A
X k6=i
36
1
Zj ( k)
3
05
cn 1
3 cn 5
n
:
2
Pr 41 +
n
0
@0 +
i
X k6=i
2 [0; h] 1
Z j ( k )A
3
cn > 05
= VAR Z j
Let
where kn
i
: We can then rewrite the probability statement as " # P j EZ j ( i ) h k6=i Z ( k ) p Pr kn ; kn + p n 1 n 1
1 pcn n 1
ph n 1
n
EZ j ( i ) p : n 1
is …nite, E Z j ( k )
As
n i=1 P j j k6=i Z ( k ) EZ ( i ) p n 1 h p ! 0; which n 1
EZ j ( i ) = 0; and Z j
is an i.i.d. sequence, we know that the central limit theorem is applicable, thus is asymptotically distributed as a standard normal distribution. Moreover,
i
together with the convergence in distribution implies that for every real number k and any " > 0; there exists N < 1 such that " P j EZ j ( i ) k6=i Z ( k ) p Pr k n 1
h k+ p n
1
#
1 p 2
Z
k+"
exp
k
y2 2
dy + "
for n
N: But, the standard normal is symmetric and single-peaked, so for n N; " # P Z kn +" j( ) j( ) Z EZ i h 1 y2 k k6=i p p kn + p dy + " Pr kn exp 2 n 1 n 1 2 kn Z " 2 1 y2 p exp dy + " ! 0 " 2 2 2
as " ! 0: The result follows. For the case in which f
above with
Znj ( i )
=
(
1 n ; n ; tn gn=1
is a sequence from the solution to (11), we replace Z j ( i )
h
if
maxf0;Gm ( n )g m (M m)
if
By choice of subsequences such that Gm ( maxf0;Gm g : m (M m)
j i n)
= l and if
k i
j i
=h
= h and m ( i ) = m:
! Gm ; we can approximate
maxf0;Gm ( n )g m (M m)
with
The rest of the argument follows the one above step by step.
Proof of Proposition 3. Proof. [PART 1] For contradiction, let there be a subsequence of feasible mechanisms with h i limn!1 E jn ( ) = and limn!1 n (m) e = ; where m e is the threshold number of high goods
in the sense of Lemma 14, which without loss is taken to be constant along the sequence by choice
of converging subsequence. We use the following notations below: h i E jn ( ) jm; ji = h denotes the conditional provision probability for good j perceived by agent i given that
j i
= h and
k i
= h for exactly m goods k 2 J (including j). Notice that
this probability is the same for all goods by Proposition 2. h i E jn ( ) j m; ji = l denotes the conditional provision probability for good j perceived by agent i given that
j i
= l and
k i
= h for exactly m goods k 2 J. Again, this probability is
the same for all goods by Proposition 2.
37
Note that Propositions 1 and 2 jointly imply that for every m 2 f0; ::; M g ; there exists some
tn (m) such that the transfer tn (m) is paid by every agent i with m high valuations. The utility (from truth-telling) is,
Hence,
h i mhE jn ( ) jm; ji = h tn (m) if m < m e h i h i j j j j mhE e e i = h + (M m) e n (m) e E n ( ) jm; e i = l l tn (m) e if m = m e n ( ) jm; h i h i mhE jn ( ) jm; ji = h + (M m)E jn ( ) jm; ji = l l tn (m) if m > m: e tn (m)
mhE
h
j n(
i = h for
j i
) jm;
with m < m; e
i
(A21)
since otherwise these agents would be better o¤ not to participate.17 . Similarly, tn (m) e
mhE e
h
j n
( ) jm; e
j i
i = h + (M
m) e
n
(m) e E
h
j n
j i
( ) jm; e
i = l l for
i
with m = m; e
(A22)
again immediately from the participation constraint (or the downwards adjacent incentive constraint combined with (A21)). Using (A22), we …nd h i h i (m e + 1)hE jn ( ) jm e + 1; ji = h + (M (m e + 1))E jn ( ) jm e + 1; ji = l l tn (m e + 1) h i h i j mhE e e 0j e E jn ( ) jm; e 0j (m e + 1)] l + h) tn (m) e n ( ) jm; i = h + n (m) i = l ([M h i h i 0j j j j /(A22)/ ( m) e E ( ) j m; e = l ([M ( m e + 1)] l + h) (M m) e ( m) e E ( ) j m; e = l l n n n n i i h i = e E jn ( ) jm; e 0j l) : n (m) i = l (h
Hence,
tn (m e + 1)
(m e + 1)hE n
(m) e E
h
h
j n j n
i = h + (M i = l (h l) :
( ) jm e + 1;
( ) jm; e
0j i
j i
m e
1)E
h
j n
( ) jm e + 1;
j i
i =l l
(A23)
Finally, for every for every m > m e + 1; (A23) implies that h
i h i ( ) jm; ji = h + (M m)E jn ( ) jm; ji = l l tn (m) h i h i e + 1; ji = h + E jn ( ) jm e 1) h] (m e + 1)hE jn ( ) jm e + 1; ji = l [(M m) l + (m m h i h i E jn ( ) jm e + 1; ji = l [(m m e 1) (h l)] + n (m) e E jn ( ) jm; e 0j l) ; i = l (h
mhE
j n
tn (m e + 1)
hence,
tn (m)
mhE h E
h
i = h + (M i j e + 1; ji = l [(m n ( ) jm j n
( ) jm;
j i
m)E m e
From from Lemma 15, we have lim E
n!1
17
h
j n
( ) jm;
j i
i h = h = lim E n!1
j n
( ) jm;
j i
h
j n
( ) jm;
1) (h
j i
l)]
i h = l = lim E n!1
i =l l
n
j n
(m) e E
( ) jm; e
(A24) h 0j i
j n
( ) jm; e
0j i
i
= l (h
i = l = lim E n!1
j n
l) :
( ) =
:
(A25)
Strictly speaking, we only impose the participation constraint on type l: However, the downwards adjacent
incentive constraint together with the participation constraint for type l imply that all participation constraints hold.
38
Combining (A25) with (A21), (A22), (A23) and (A24), it follows that, for every " > 0 there is some N such that, whenever n
N;
mh + " if m < m e
1. tn (m) 2. tn (m) e
3. tn (m e + 1) 4. tn (m)
[mh e + (M
m) e l] + "
[(m e + 1)h + (M
[(m e + 1)h + (M
m e
m e
1)l
(h
1)l
l)] + "
l)] + " if m > m e + 1:
(h
Summing over m, we …nd that the expected per capita transfer revenue satis…es: M X
m tn
"
(m)
m=0
+
=
M X
"
(1 " (1 "
"
f(m e + 1)h + (M
m)l e
M X
"
m
M X
m)l e +
M X
m
m=m+1 e
m=m e
m
!
!
M X
i=1
i2
i
( i ) tin ( i )
X 2
n
( )
M X
j n
m
m=m+1 e
!
(h
!
P
m)l) e +h
2
n
X m
j=1
f(1 |
mm
#
(A26)
l) + "
m e
m e
m e X1
j n(
m tn
(m)
mm
m=0
m)l) e +h
1)lg + h
mm
m=0
( )
m e X
1)lg + h
(mh e + (M
R (m)g e + ":
m e X
m=0
#
f(m e + 1)h + (M
( ) cn = n
We conclude that corresponding to any
!
1)lg + h
f(m e + 1)h + (M
Moreover, for any "; we can …nd N such that M junction with (A26) means that n X X
m
(mh e + (M
) R (m e + 1) +
f(1
M X
m e
m=m+1 e
m=m+1 e
m e (M
)
+ =
m e (M
)
+
=
m
m=m+1 e
!
#
#
mm
m=0
m e X
mm
m=0
+"
#
+"
)c
M
m e X
#
Mc
X 2
) R (m e + 1) + {z
( )
"; which in con-
j n
( )c
(A27)
n
R (m) e
0 such that the right hand side of
(A27) is negative. Hence, the budget constraint (14) is violated for large n along any sequence with h i non-vanishing provision probability. It thus follows that limn!1 E jn ( ) = 0 for any convergent subsequence.
[PARTS 2 and 3] Consider the sequence of mechanisms bn ; bn ; b tn where bjn ( ) = 1 for every
n and ; where all agents get to consume all high valuation goods and the inclusion rule for low
39
valuation goods is
8 > >
> :
0
if m < m
R(m +1) cM R(m +1) R(m )
if m = m
1
if m > m ;
and where the transfer tule is 8 > mh > < b tn (m) = m h + (M m ) l > > : (m + 1) h + (M m 1)l
if m < m if m = m (h
l) if m > m ;
for
R (m + 1) cM : (A28) R (m + 1) R (m ) It is easy to check that truth-telling is incentive compatible and individually rational. The per capita expected transfer under the above mechanism is =
M X
m=0
b m tn (m)
= h
m X1
mm
+
M X
+
m
m=m +1
=
=
(
(1
) h "
+
/(A28)/
m
m=0
h
(1
!
m X
[m h + (M
[(m + 1) h + (M
mm
mm
M X
+
m=0
m X1
m )
M X
+
m
m=m
m=0
m
m
m=m +1
!
l]
!
(A29)
1)l
(h
l)]
[(m + 1) h + (M
[m h + (M
)
m
1)l]
#
m )l]
) R (m + 1) + R (m ) cM R (m ) R (m + 1) cM R (m + 1) + R (m ) = cM: R (m + 1) R (m ) R (m + 1) R (m )
=
The feasibility constraint thus holds exactly. Thus we conclude that bn ; bn ; b tn is incentive feasible
for every n: The associated social surplus with the above mechanism is ( M " #) M M X X X Sn ( ) = h H j ( ; m) + l Lj ( ; m ) + Lj ( ; m) m=1
j=1
cM;
(A30)
m=m +1
where the notations H j ( ; m) and Lj ( ; m) were explained earlier. By Lemma 14, we know that the surplus maximizing mechanism is also characterized by some threshold m and some inclusion probability given m = m denoted by ; so that the associated social surplus is given by Sn ( ) =
M X
j
j=1
"
( ) h
M X
j
H ( ; m) + l
m=1
(
j
L ( ;m ) +
m=m
j
Notice that H (n ;m) converges in probability to M m m : These immediately imply that M lim Pr
n!1
"
Sn ( ) n
h
M X
m=0
m
m
l
"
(M
m )
M X
m
m M
+
m
and M X
m=m +1
40
L ( ; m) +1
Lj ( ;m) n
(M
)
j
#
c :
(A31)
converges in probability to
m)
m
#
#
+ cM > " = 0
(A32)
for any " > 0: Next, note that M X
j
H ( ; m) h +
m=1
mh +
M X M
m=0
M n X m M m=0
=
M X m = M m=1
1 (M m
Lj ( ; m)
m=0
converges in probability to M X m M m=1
M X
mh
mh
m m
M
1 m (M
o + max f0; Gm ( )g M X
+
m=m
(1
) (M
m)
m)
max f0; Gm ( )g
max f0; Gm ( )g
cn
(A33)
cn
(A34)
cn
m) l
m
2 4
+
+1
m
(M
m) l
(h
l)
M X
j
j=m+1
3 5
cn:
Recalling the form of the surplus maximizing provision rule in (21), we see that good j is provided is and only if the expression in (A33) is positive. Now: 1. Suppose that m
> m : Then, R (m ) i h establishes that limn!1 E jn ( ) ! 0.
cM < 0 and an argument identical with part 1
2. Suppose that m
< m and R (m ) cM < 0: Then an argument identical with part 1 i h establishes that limn!1 E jn ( ) ! 0.
3. Suppose that m
< m and R (m ) cM 0: Then, the expression in (A34) is strictly i h j positive and limn!1 E n ( ) ! 1: The social surplus must therefore converge towards the same expression as in (A32), but with m replaced by m : But, if m
< m surplus is
smaller because there are more exclusions. 4. Suppose that m = m : If bn (m ) ! < an argument identical with part 1 establishes h i h i that limn!1 E jn ( ) ! 0. If bn (m ) ! > we have that limn!1 E jn ( ) ! 1 and, again, social surplus converges towards the same expression as in (A32), but with by
: Again, surplus is smaller if
>
replaced
since exclusions are higher.
Proof of Proposition 4. Proof. To prove Proposition 4, we add the (non-adjacent) constraint that the highest type should not have an incentive to pretend to be the lowest type: 0
X i2
i
(
i)
i
to program (11). Let
M X
j
(
i ; h) h
2 4
ti (h)
j=1
X i2
i
(
i)
M X j=1
i
j
(
i ; l)
j i
(l) h
3
ti (l)5
(A35)
denote the multiplier on constraint (A35). The optimality conditions with
respect to t are then 2 (2) (2) 2 (1) +
+
(1) + (0) + 41
1 2
2
= 0
1
= 0
0
= 0;
which are obtained just like (A2) by considering types with 2, 1 and 0 high types respectively in the special case with two high valuations and the additional constraint (A35). It follows immediately that: Lemma A1 Consider m = 2: Then, there are three possibilities in the solution to the program where (A35) has been added to (11): 1. The only binding constraints are the adjacent constraints in (11). In this case the solution is unchanged relative to program (11) and
1 2
(2) =
2;
1 2
(1) =
(
1
+
2)
and
(0) =
2. The only binding constraints are (A35) and the downwards adjacent constraints for types with m = 1: In this case
=
2;
(1) =
1 2 1
and
(0) =
3. All constraints bind, in which case 2 (2)
=
2;
1 2 1;
(1) 2
(0) = :
1 2
(
1
+
2)
and
We now consider these possibilities in turn: Claim A3 Suppose that
1
2
0 if G1 ( ) < 0 > < (1) z 2 [0; 1] if G1 ( ) = 0 > > : 1 if G1 ( ) > 0; 8 > 0 if G0 ( ) < 0 > < (0)
> > :
where
z 2 [0; 1]
if G0 ( ) = 0
1
G1 ( ) =
1l
G0 ( ) =
0 2l
if G0 ( ) > 0;
(h
l)
(h
2
l) (
1
+
2)
and the provision rule (21) simpli…es to j
( )=
8 > > < > > :
0 z 2 [0; 1] 1
Lj ( ;0) 2 0 max f0; G0 ( Lj ( ;0) h+ 2 max f0; G0 ( 0 Lj ( ;0) h+ 2 max f0; G0 ( 0
if H j ( ; 1) + H j ( ; 2) h + j
j
j
j
if H ( ; 1) + H ( ; 2) if H ( ; 1) + H ( ; 2)
)g + )g + )g +
Lj ( ;1) 1
Lj ( ;1) 1
Lj ( ;1) 1
Notice that G1 ( )
= l
(h
1
l)
2
(h
0:
max f0; G1 ( )g
cn = 0
j k j( implying that (1) (0) and j ( i ; i ) i ; ll) if i = l and i = h for k 6= j: Let t (0) ; t (1) and t (2) denote transfers from types with zero, one and two high valuations respectively. Because the downwards adjacent incentive constraint for type hh and the types with one high valuation are binding, it follows that
/binding DAIC for hh/
X i2
|
=
X i2
X
=
i2
+
i
=
i2
+
|
i2
+ |
{z
1
2
i2
i
i
(
i)
1
(
i ; hl) h
2
+
(
1 ; hl)
(1) h
i
(A36)
With slack in the downwards adjacent constraints for hh and a binding (A35) the optimality conditions with respect to 2i (hl) my be written as X i2
( i
i ; hl)
2
(
i ; hl) l
+ (1)
X i2
i
(
i)
2
(
i ; hl) l
+
2 i
(hl)
2 i
(hl) = 0:
i
This implies that 2i (hl) = 1i (lh) = (1) = 1 because a small increase in the low valuation usage probability for the types with one high valuation has no e¤ect on incentives for hh if the downwards adjacent constraints for type hh are nonbinding. After a simpli…cation (that removes the awkwardness of the notation mentioned in footnote 16), the optimality conditions with respect
43
t (1) :
to
1(
) are given by the complementary slackness conditions together with h H 1 ( ; 1) + H 1 ( ; 2) + l L1 ( ; 1) + (0) L1 ( ; 0) cn 2 1 + (1) H 1 ( ; 1) h + L1 ( ; 1) (0) l + H 1 ( ; 2) h 1
2
1
(
i
(0) H 1 ( ; 2)
i)
(0)
(1)
0
1
1
L1 ( ; 0) l
cn +
1
( )
constraint 1 as increasing (1)
1(
) is for the constraint the multiplier 1(
L1 (
)
= 0;
)
0 and
1
1(
i ; lh)
0: It is easy to see from (A37) that
; 1) by a unit and decreasing
(0) ; it follows that X ( i2
( )
( )
0
1(
H 1 ( ; 1) h + L1 ( ; 1) l
0
+ (0)
where
(A37)
1
i)
(
i ; lh)
L1 (
(
i2
i
1(
i ; ll)
for every
i
; 1) by a unit: Together with the fact that
X
(1)
( ) is the multiplier for the
1
i)
(
i ; ll)
(0) ;
i
which means that the perceived probability to consume a low valuation good is weakly higher for a consumer with a high valuation for the other good. Hence,
/(A35)binds/
|
X i2
=
=
| |
+ (h /binding DAIC for lh/
= |
+ (h =
+(h
which contradicts (A36).
|
|
i
(
i)
1
(
i ; hh)
i
X i2
X i2
l) X i2
l) X i2
l)
{z
2
+
(
1 ; hh)
h
t (2)
utility from truth-telling for hh i
(
i)
1
(
i ; ll)
2
(0) h +
i
{z
(
1 ; ll)
}
(0) h
t (0)
(0) h
}
utility if hh announces ll i
(
i)
1
(
i ; ll)
i
i
i2
1 ; ll)
t (0) }
(
1
i)
(
i ; ll)
(0)
i
i
(
i)
1
(
i ; lh)
i
2
(1) l +
{z
(
1 ; lh) h
t (1) }
utility from truth-telling for lh
X
i
i2
(
1
i)
(
i ; ll)
(0)
i
i
(
i)
1
(
i ; lh)
i
i2
{z
(
utility if lh announces ll
X
X
2
(0) l +
{z
(1) h +
2
(
1 ; lh) h
utility if hh announces lh i
(
i)
1
i
(
i ; ll)
(0)
(h {z
l)
X i2
t (1) }
i
(
i)
1
(
i ; lh)
i
0
(Proof of Proposition 4, continued): We conclude that all constraints must bind. This immediately implies that (1) = (0) : Assuming that l < c < h it is easy to check that 0 < (1) = (0) < 1 44
(1); }
for su¢ ciently large n; which can only hold if G1 ( ) = G0 ( ) = 0: That means that the provision rule for good j becomes
j
( )=
8 > > < > > :
0 z 2 [0; 1] 1
if H j ( ; 1) + H j ( ; 2) h
cn < 0
j
j
cn = 0
j
j
cn > 0;
if H ( ; 1) + H ( ; 2) h if H ( ; 1) + H ( ; 2) h
which coincides with the provision rule if good j is provided as a stand alone good. Since the inclusion rule is also the same as that when only good is provided, the result follows.
45
B
Appendix: Proofs for Results in Section 2
Proof of Proposition 1. Claim B1 For any incentive feasible mechanism G of the form (3), there exist an incentive feasible mechanism
j
G=
j j 1 ; :::; n
;
j
that generates the same social surplus, where j i
:
j2J n
:
; (ti )i2I ;
(B1)
! [0; 1] is the provision rule for good j;
! [0; 1] is the inclusion rule for agent i and good j;and ti :
! R is the transfer rule for
agent i:
Proof. Consider an incentive feasible mechanism G. Pick k 2 [0; 1] arbitrarily and de…ne, Z 1 j j ( ) = E j ( ; #) = ( ; #) d# (B2) 0 R1 j R 8 ( ;#)! ji ( ;#)d#dF( i ) j n j R R1 j 0 E ( ;#)! ( ;#) > i i i < R R ( ; #) d#dF ( i ) > 0 if = n 1 j j j E i ( ;#) ( ;#)d#dF( i ) i 0 n 0 ( ) = i i i R R1 j > : ( ; #) d#dF ( i ) = 0 k if n i 0 Z ti ( i ) = E i ( ) = ( ) dF ( i ) ; n
for each
n; j
2
i
2 J and i 2 I: This is a mechanism of the form in (B1), and we will call it
G. Use of the law of iterated expectations on
j
( ) and ti ( i ) shows that (BB) is una¤ected when
switching from G to G: It remains to show that the surplus is unchanged, and that (IC) and (IR) continue to hold under G: The utility of agent i of type i 2 who announces ^i 2 is hP i j ^ j ^ ^i ; i E i ; ; # ! ; ; # in mechanism G (B3) i i i i i j2J i hP i j ^ j ^ ; E i ti ^i in mechanism G: (B4) i i i i j2J i If
R
n
i
R1
j
0
^i ;
i; #
identical, whereas if
E
i
j
^i ;
R
n
i
d#dF ( R1 j 0
i
j i
^i
i)
^i ;
i
= 0; we trivially have that the payo¤s in (B3) and (B4) are i; #
= E
d#dF (
i
j
^i ;
> 0; we have that
i; #
E i ! ji ^i ; E
= E i ! ji ^i ; Trivially, E i ti ( i ) = ti ( i ) = E
i)
i
i; #
j
^i ;
i
i; # j
i; #
^i ;
j
^i ;
i; #
(B5) i; #
i:
( ) ; which combined with (B5) implies that the payo¤s in (B3)
and (B4) are identical. Since the equality between (B3) and (B4) were established for any i; i and ^i ; it follows that all incentive and participation constraints (IC) and (IR) hold for mechanism G given that they are satis…ed in mechanism G. Moreover, [again by (B5)] hP i hP i j j j( ) j( ) E i = E ! ( ; #) ( ; #) i i i i ; j2J j2J i i 1
(B6)
so it follows by integration over E
hP
i2I
P
j
j2J
and summation over i that j i
( )
( i) j
By construction, we also have that j
E
( ; #) C j (n) ; implying that " X X j E j( ) i ( i) i j2J
i
i
hP
=E
i2I
j
( ) = E #
X
C j (n) =
i2I
P
j
j2J
( ; #) ! ji ( ; #)
( ; #) for every
j
E
( ; #)
j2J
"
X
i
i
: Thus E
! ji ( ; #)
i
; j
(B7)
( ) C j (n) = #
C j (n) :
i2I
(B8)
Hence, G and G generate the same social surplus. Claim B2 For every incentive feasible mechanism of the form (B1), there exists an anonymous simple incentive feasible mechanism g of the form (5) that generates the same surplus. Proof. Consider an incentive feasible simple mechanism G on form (B1). For k 2 f1; ::::; n!g ; let
Pk : I ! I denote the k-th permutation of the set of agents I. Note that Pk 1 (i) gives the index
of the agent who takes agent i0 s position in permutation Pk : Moreover, for any given PK
=
Pk 1 (1) ; :::; Pk 1 (n)
n
2
denote the corresponding k-th permutation of
j j j k ; k1 ; :::; kn
k 2 f1; :::; n!g ; let Gk = j k
( ) =
j ki ( i )
=
j
j ( ) Pk 1 (i) i 1
k
ej ; ej1 ; :::; ejn
j=1;2
e ti (
8
(i) ( i )
;e t1 ; :::; e tn
n
2
n;
let
For each
be given by
; j 2 J;
(B9)
8
i
2
; j 2 J ; i 2 I;
8
i
2
; i 2 I;
be given by
j 1 Pn! 8 2 n; j 2 J n!P k=1 k ( ) j j n! k=1 E i [ k ( )] ki ( i ) Pn! 8 i 2 ;i 2 i) = j k=1 E i [ k ( )] P n! 1 8 i 2 ; i 2 I: i ) = n! k=1 tki ( i )
ej ( ) = eji (
; tk1 ; :::; tkn
Pk
tki ( i ) = tP and let g =
j=1;2
.18
2
We now note that: (1) for each j 2 J ; ej ( ) = ej 0 if n n on! j j j (P ( )) n! since the sets ( ) = and k k k k=1 k=1
0
0
I; j 2 J
is a permutation of : This is immediate on! n! = j Pk 0 are the same; k=1 k=1
(2) for j 2 J and each pair i; i0 2 I; eji ( ) = eji0 ( ) : That is, the inclusion rules are the same
for all agents. To see this, consider agent i and i0 , and suppose that i = i0 : We then have n h i on! n h i on! that E i jk ( ) jki ( i ) and E i0 jk ( ) jki0 ( i0 ) are identical and that E i ej ( ) = k=1
18
k=1
To illustrate, suppose n = 3; M = 2;
= ( 1;
2;
3)
= ((1; 2) ; (3; 2) ; (2; 1)) : Consider, for example, pur-
mutation k given by Pk (1) = 2; Pk (2) = 1; Pk (3) = 3: Then Pk 1 (1) = 2; Pk 1 (2) = 1; Pk 1 (3) = 3 and Pk
=
; Pk 1 (1)
; Pk 1 (2)
Pk 1 (3)
= ( 2;
1;
3)
= ((3; 2) ; (1; 2) ; (2; 1)) :
2
E
ej ( ) ; and (3) for each pair i; i0 2 I; e ti ( ) = e ti0 ( ) ; which is obvious since the sets ftki ( i )gn! k=1
i0
n! k=1
0 i
and tki
are identical. Together, (1), (2) and (3) establishes that g is anonymous and
simple.
Now we show that g is incentive feasible and generates the same expected surplus as G. First, since G and Gk are identical except for the permutation of the agents, we have, for k = 1; :::; n!; ( " #) ( " #) X X j X X j j j j j j j E k( ) C (n) = E ( ) C (n) : (B10) i ( i) i ki ( i ) i j2J
i2I
j2J
i2I
Hence, X
j2J
(
j
E e ( ) (
" X i2I
j
e ( i)
#)
j
j i
C (n)
=
X
(
E
j2J
n! 1 X n!
j k
" P X n! E k=1 ( ) Pn!
) " # n! n! 1 X 1 X j j j j j E i E i k ( ) ki ( i ) i E k ( ) C (n) n! n! j2J i2I k=1 k=1 #) ( " " ( n! X X X X X 1 j j j j j C (n) = E ( ) E ki ( i ) i k( ) n! i2I j2J i2I j2J XX
=
=
j i
( )
j ki
E
j i k
( )
k=1
i2I
k=1
j i k
j i
( i)
#)
j
C (n)
k=1
( i)
j i
j
#)
C (n)
;
where the last equality follows from (B10). Hence the surplus generated by g is identical to that by original mechanism G: To show that g is incentive feasible we …rst note that E jk ( ) = E j ( ) P P and E i2I tki ( i ) = E i2I ti ( i ) for all k, since the agents’valuations are drawn from identical
distributions and Gk and G only di¤er in the index of the agents. Thus E
P
P
e i2I ti ( i )
j2J
Eej ( ) C j (n)
=
E
=
E
P
1 i2I n!
P
Pn!
k=1 tki ( i )
i2I ti ( i )
P
j2J
P
j2J
E
j
1 E n!
Pn!
j k=1 k
( ) C j (n)
( ) C j (n) ;
so g satis…es (BB) if G does. Second, (IC) holds for any permuted mechanism, that is, E
i
P
j k(
j2J
for all i 2 I; and E
i
X
j2J
=
=
1 n!
n! X
k=1
)
b 2
i; i j
"
j j ki ( i ) i
j
i
X
j k
j i
( )
e t( i ) = E
j ki
( i)
j i
j2J
n! X 1 X E i n! j2J k=1
E
i
. Hence,
e ( )e ( i )
E
tki ( i )
P
j2J
j b k ( i;
j b i ) ki ( i ;
"
# Pn! n! 1 X j k=1 E i Pn! i k( ) n! k=1 E j2J k=1 # " n! X j 1 X b E i tki ( i ) k( i; n! j2J X
k=1
j b k( i;
j b i ) ki ( i ;
j i) i
n! 1 X tki (bi ; n! k=1
i)
=
X
j2J
j k i
tki (bi ;
j i) i
j ki
( ) j k
( i)
( )
j b i ) ki ( i ;
E i ej (bi ;
j i )ei
j i
j i) i
bi
i)
(B11)
n! 1 X tki ( i ) n! k=1 # b tki ( i ; i ) j i
e t(bi ); (B12)
where the inequality follows from (B11). Hence g is satis…es (IC). Finally, g also satis…es the (IR) because (see the second line in (B12)) all the permuted mechanisms satisfy participation constraints. Proposition 1 follows by combining Claims B1 and B2. Proof of Proposition 2. 3
Notation:
This proof requires us to be explicit about the coordinates of the vector
when per-
muting J . We therefore need some extra notation for this proof (only). We will, with some abuse of notation, write F ( )
i) i2I F ( i ) and F ( k2Ini F j j 1 j+1 1 = ; i ; :::; M i ; :::; i i i
respectively. We write
i
j
been removed. Analogously, j
removed for all agents and j i
all agents. Furthermore,
j
=
1
; :::;
j j 1 ; :::; n
=
j
=
1
n
j
for a type vector where good j has
is the vector collecting the valuations for good j for
used for the vectors obtained respectively from
and
are used also on the distributions, so, for example, F j i:
j i
and
j i
=
j j j j 1 ; :::; i 1 ; i+1 :::; n
j i
conditional on
are
by removing agent i: These conventions denotes the cumulative distribution of
Conditional distributions are denoted in the natural way: for example F
joint distribution of
and
stands for the type pro…le with good j coordinate
j j j i 1 ; i+1 ; :::; n j j
; :::;
( k ) as the joint distribution of
j i
j i:
j
j i
denotes the
Since no integrals are taken over subsets of the range R R of integration, we also conserve space and write h ( ) dF ( ) rather than 2 n h ( ) dF ( ) when
integrating a function h over
and similarly for integrals over various components of :
Proof. Consider a simple anonymous incentive feasible mechanism g. For k 2 f1; :::; M !g let Pk : J ! J
be the k-th permutation of J and
mutation of
i
Pk i
=
Pk 1 (1) P 1 (M ) ; :::; i k i
when the goods are permuted according to Pk : Let
Pk
=
2
be the per-
Pk Pk 1 ; :::; n
2
n
denote the corresponding permutation of :19 For each k 2 f1; :::; M !g de…ne mechanism gk = n o n o ; tk ), where for every 2 n ; ; jk ( jk j2J
j2J
1.
j k
( )=
Pk 1 (j)
2.
j k
( i) =
Pk
Pk 1 (j) Pk i
3. tk ( i ) = t
for every j 2 J ;20
Pk i
for every j 2 J ;21
:
By construction, each gk is simple. Each gk is also anonymous by the anonymity of g: Using the de…nition of gk and manipulating the result by observing that the labeling of the variables is 19
To illustrate, suppose n = 2; M = 3; and
= ( 1;
2)
= ((1; 2; 0) ; (3; 2; 1)) : Consider, for example, pur-
mutation k given by Pk (1) = 2; Pk (2) = 1; Pk (3) = 3: Then Pk 1 (1) = 2; Pk 1 (2) = 1; Pk 1 (3) = 3 and Pk 1
=
Pk 1 (1) ; 1
Pk 1 (2) ; 1
Pk 1 (3) 1
= (2; 1; 0) ;
Pk 2
=
Pk 1 (1) ; 2
((2; 1; 0) ; (2; 3; 1)) : 20 21
This implies that This implies that
Pk 1 (j) k Pk 1 (j) k
Pk Pk i
= =
j
( ) for every j 2 J :
j
( i ) for every j 2 J .
4
Pk 1 (2) ; 2
Pk 1 (3) 2
= (2; 3; 1) ;
Pk
=
Pk 1 ;
Pk 2
=
irrelevant, we get:22 E
j k
j j k( i) i
( )
Z
=
Z
=
/relabel/
Z
=
j k
Z
j
P
1 k
j j k ( i ) i dF (
( )
Pk 1 (j)
Pk
(j)
2 4
Z
Pk 1 (j)
(
j Pk
Pk 1 (1)
By exchangeability, we have 0 @
= dF = dF
j
= dF
Pk 1 (j)
; :::;
Pk 1 (j)
j Pk
j
( )
| {z }
j-th (vector) argument
Pk 1 (1) j
; :::;
Pk 1 (j 1)
Pk i
Pk 1 (j)
Pk 1 (j)
;
j j i dF
( i)
;
j
j
Pk 1 (j) dF j i
Pk 1 (j+1)
dFj 0 @
Pk i
j i dF (
)
j
(B13)
Pk
Pk 1 (j)
| {z }
j-th a rg u m e nt
Pk 1 (n)
13
A
Pk 1 (j)
A5 dFj
:
(B14)
1
Pk 1 (j+1)
(B15) Pk 1 (n)
; :::;
jj-th (vector) argument =
Pk 1 (j)
Pk 1 (j)
and Pk 1 (j)
j
; :::;
jPk 1 (j) -th (vector) argument = dFj
Pk 1 (j)
Pk
2 n
Pk 1 (j 1)
jj-th (vector) argument = Pk 1 (j)
Z
)
j Pk
j
Pk 1 (j)
j
where we recall,
dF
) /def of gk / =
= dF
P 1 (j) k
Pk 1 (j)
Pk 1 (j)
;
:
(B16)
Using (B13), (B15) and (B16), we have that
=
=
=
j j k( i) i 2 0 13 Z 1 Pk 1 1 1 1 P (j) Pk (j) A5 Pk (j) j 4 ( ) Pk (j) ( i ) i k dF j @ dFj Pk (j) 1 | {z } P P (j) k k ( j) j-th a rg u m e nt 13 2 0 Z Z 1 1 1 1 1 1 1 P (j) C7 Pk (j) 6 B Pk (j) Pk (j) ( ) Pk (j) ( i ) i k dF Pk (j) @ Pk (j) j dF A 4 5 1 | {z } P (j) k ( j )Pk Pk 1 (j)-th a rg u m e nt Z 1 1 1 P 1 (j) P 1 (j) Pk 1 (j) : ( ) Pk (j) ( i ) i k dF ( ) = E Pk (j) ( ) Pk (j) ( i ) i k
E Z
j k
Moreover, exchangeability implies that Etk ( i ) = Et E
22
(B17)
( )
h P M
i PM t ( ) = E i k j=1 j=1 , , h P Same elements in J and M = E j=1 1 1 Pk (1) ; :::; Pk (M ) j k
( )
j j k ( i) i
Pk i
Pk 1 (j)
= Et ( i ) : The ex ante utility,
Pk 1 (j)
( )
j
j
( )
( i)
Pk 1 (j)
j i
i
( i)
Pk 1 (j) i
Et ( i ) ;
Et ( i ) (B18)
It is important to point out that, in reaching the fourth equality in (B13), we can relabel the integrating varibles
(since they are dummies) but not the integrating functions.
5
is thus unchanged when changing from g to gk . The same steps as in (B13) through (B17) (only j k
somewhat simpler) establishes that E h P j M E ( ) C j (n) j=1 k h PM j ( ) = C (n) E j=1
Pk 1 (j)
( )=E
for every j, implying that
i h PM j t ( ) = C (n) E ( ) i k i j=1 k i h P PM j ( ) C (n) Et ( i ) = E j=1 i
i Et ( ) (B19) i k i i P t ( i) ; i
P
P
so the feasibility constraint is una¤ected when changing from g to gk : Next, write Write U ( i ; 0i ; g) and U ( i ; 0i ; gk ) for the expected utility from announcing
0 i
when the true type is
in mechanisms
i
g and gk respectively. Next, by a calculation in the same spirit as (B13) through (B17): E
j i k
i;
0 i
= =
/relabel/
/exchangeability/
=
= =
Z Z
Z Z
Z
j k i
j i
P
P
"Z 1
k i
1 k i
i;
0 i
dF
i
(
Pk 1 (j)
i;
j i
"Z
(j)
P k i
"Z
(j)
P k i
1
1
Pk 1 (j)
0 Pk i
Pk 1 (j) (j)
Pk 1 (j) (j)
0Pk i
i;
of gk / =
i ) /def
dF
i
dF
j i
i;
0Pk i
i;
0Pk i
(
i)
Z
Pk 1 (j) i
j i
j i
dF
j i
dF
Pk 1 (j) i
=E
i
#
dFj
i;
0 i
the perceived probability of getting good Pk (j) when announcing U ( i ; 0i ; gk ) = E = whereas U ( i;
=
i
PM
j k
j=1
PM
j=1
1
Pk (j) ( k
j 0 j j k ( i) i E i k
j 0 j k ( i) i
0 i; i
U ( i ; 0i ; g)
who announces
0 Pk i
Pk i 0Pk i
0 i= Pk 1 (j)
PM
k
j=1
which establishes that type
i=
0 i; i
i
(
=
tk
(B20)
Pk 1 (j) i
#
dFj
Pk 1 (j) i
Pk 1 (j) i
i
#
dFj
Pk 1 (j) i
i
0Pk i
0 i
t
PM
j=1
j k(
0 i
; so that (B21)
0Pk i; i
t
0 Pk i
;
)
(B22)
1 0 Pk Pk (j) ) i E i jk i
P 1 (j) 0 Pk j ) i E i kk i
who announces
0 Pk i
0 i
1 0 Pk j ) i E i Pk (j) i
j=1
=
i)
in mechanism gk is the same as
1
M X
(
i
That is, the perceived probability of getting j when announcing
0 i ; g)
i
i
Pk 1 (j) i
Pk 1 (j)
dF
j i
Pk
j i
0 Pk i
i;
0Pk i; i
0Pk i; i
t
0 Pk i
t
0 Pk i
= U ( i ; 0i ; gk );
in mechanism gk gets the same utility as type
Pk i
in mechanism g: Hence incentive compatibility and individual rationality
of gk follows from incentive compatibility and individual rationality of g: Now, construct a new
6
mechanism ge = ( ej
; ej j2J ; e t) by letting j Pk 1 PM ! 1 PM ! Pk 1 (j) k=1 k ( ) = M ! k=1 M! PM ! P 1 (j) PM ! j j k k=1 ( ) E ( ) i i k=1 k k = PM ! P j M! k=1 E i k ( ) k=1 E
j2J
ej ( ) = j
e ( i) =
e t ( i) =
1 M ! tk
Pk i
E
i
Pk 1 (j)
i
let P : J ! J be an arbitrary perturbation of the set of goods. Then, P ! P 1 (P 1 (j)) P ! P 1 (j) 1 P Pk k k eP (j) P = M1 ! M = M1 ! M k=1 k=1 since the sets
n
1
Pk
(P
1 (j)
oM !
P Pk
)
e
1 (j)
P i
=
P 1 P k
PM !
k=1
=
PM !
1 (j)
(
Pk 1 (j)
PM !
k
1 (j)
Pk i
PM !
k=1 E
i
)
Pk
( Pi )
E
Pk
Pk
(
E
(
Pk
1 (j)
(
P 1 (P 1 (j)) k i k P 1 (j) P k k i
)
)
= ej ( ) ;
Pk
(B24)
are identical. Furthermore
P 1 P k i k
E
P 1 (j) k
for the same reason. It is obvious that e t
oM !
Pk
k=1
k k=1
P
k=1
n
and
k=1
P
Pk 1 (j)
Pk i
1 M! t
( i) =
(B23)
)
Pk
( P)
Pk
( P)
(B25)
= ej ( i )
= e t ( i ) ; which together with (B24) and (B25)
P i
establishes that ge is symmetric. To complete the proof we need to show that ge is incentive feasible and generates the same surplus as g: We note that PM ! j j P ! j k=1 k ( i )E i k ( ) j PM ! Eej ( ) ej ( i ) ji = M1 ! M j k=1 E k ( ) i k=1 E i k ( ) PM ! j h j P ! PM ! j j k=1 k ( i )E i k ( ) j PM ! = M1 ! E i M = M1 ! E j k=1 E i k ( ) k=1 k ( i ) i E ( ) k=1 i i k hP PM h j P j M ! ) E j=1 e ( ) ej ( i ) ji e ( i ) jk ( ) t ( i ) = M1 ! M k=1 E hP i j=1 k M j( ) j( ) j /(B17) & (B18)/ = E t ( i) ; i j=1 i
j k
( )
j i
j i
i
i tk ( i )
which establishes that the ex ante utility from ge and g are the same for all agents. Moreover, hP i h i Pn PM 1 PM ! j Pn PM ! 1 M j j (n) e E e ( ) C t ( ) = E C (n) ( ) t ( ) i i j=1 i=1 j=1 M ! k=1 k i=1 k=1 M ! k i Pn PM ! h PM j = k=1 E C (n) j=1 k ( ) i=1 tk ( i ) / (B19)/ hP i hP i P Pn Pn ! M M j ( ) C (n) j ( ) C j (n) = M1 ! M E t ( ) = E t ( ) ; i i k=1 j=1 i=1 j=1 i=1
so the budget balance constraint is una¤ected. All incentive compatibility constraints hold since, U ( i ; 0i ; ge) =
= =
/ (B21)/ =
M X
ej ( 0i ) ji E iej
0 i; i
j=1 PM !
j j 0 0 i; i) k=1 k ( i )E i k ( PM ! j 0 E ( ; ) i k i i k=1
1 M! 1 M!
PM ! h k=1
PM !
j k
0 i
E
E
j i k
0 k=1 U ( i ; i ; gk )
By the same calculation, U ( ; ge) =
1 M!
for each k: This completes the proof.
PM !
i
h
e t
1 M! i;
0 i
PM !
j k=1 k
0 i
tk
0 i; i 0 i
i
/ IC for each k/
k=1 U (
; gk )
7
1 M!
i
1 M!
PM !
PM !
k=1 U (
k=1 tk
0 i
; gk ) = U ( ; ge):
0; since all participation constraints hold
