Journal of Mathematical Economics 51 (2014) 187–196

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Journal of Mathematical Economics

journal homepage: www.elsevier.com/locate/jmateco

Proportional individual rationality and the provision of a public goodin a large economyKang Rong ∗,1

School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, ChinaKey Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai 200433, China

a r t i c l e i n f o

Article history:Received 1 May 2013Received in revised form18 November 2013Accepted 20 November 2013Available online 4 December 2013

Keywords:Public goodsα proportional individual rationalityAsymptotic efficiencyAsymptotic inefficiencyInfluenceConvergence rates

a b s t r a c t

This paper studies the public good provision problem in which a non-excludable public good can beprovided and payments can be collected from agents only if the proportion of agents who obtainnonnegative interim expected utilities from the public good provision mechanism weakly exceeds aprespecified ratio α. We call this requirement ‘‘α proportional individual rationality’’. We identify a keythreshold such that if α is less than this threshold, then efficiency obtains asymptotically. If α is greaterthan the threshold, then inefficiency obtains asymptotically. In addition, we obtain the convergence rateof the probability of provision to its efficient/inefficient level. Moreover, as a methodological contributionof this paper, we propose the standard deviation of an agent’s interim expected provision as a measure ofthe agent’s influence in a mechanism. We find that as the economy becomes large, an agent’s influencein any sequence of anonymous mechanisms converges to zero, and thus any sequence of anonymousfeasible mechanisms must converge to a constant mechanism. We obtain uniform bounds for those ratesof convergence.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

This paper studies the provision of a (non-excludable) publicgood. It is well-known that the voluntary provision of publicgoods may not attain the first-best efficient level due to the free-rider problem. How to design a budget-balanced mechanism2

that can achieve first-best efficiency is thus a fundamental topicin research in public economics. In the literature, if the interimindividual rationality constraint is not required for any agent, thenfirst-best efficiency can be achieved (d’Aspremont and Gérard-Varet, 1979). If interim individual rationality is required for allagents, then first-best efficiency cannot be achieved (Mailath andPostlewaite, 1990). This paper considers a model that connectsthese two cases. Specifically, we consider a model in whichthe public good is provided and payments are collected fromagents only if the proportion of agents who obtain nonnegativeinterim expected utilities from the mechanism weakly exceedsa prespecified required agreement rate α ∈ [0, 1]. We call thisconstraint α proportional individual rationality.

∗ Tel.: +86 21 65902905.E-mail address: rong.kang@shufe.edu.cn.

1 This paper is based on a chapter of my Ph.D. dissertation at the University ofIllinois at Urbana–Champaign.2 In the case where the budget balance constraint is not required, Groves (1973)

and Clarke (1971) showed that first-best efficiency can be achieved by a dominant-strategy mechanism.

0304-4068/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jmateco.2013.11.004

Theα proportional individual rationality constraint is appropri-ate in a setting where players vote on the adoption of the mech-anism under an α majority rule, and vote sincerely because thechance of being pivotal is negligible. Consequently,we consider thepublic goodprovisionproblem in a large finite economy.Wearenotonly interested in whether first-best efficiency can be achieved ornot in a large economy, but also interested in the speed at whichthe efficiency or inefficiency is reached as the economy becomeslarge. Thus, two basic research questions of this paper are as fol-lows. First, for any given α, can the first-best provision level beachieved in a large economy? Second, how rapidly does the prob-ability of provision approach its efficient or inefficient level as theeconomy becomes large?

We assume agents’ valuations are i.i.d. and the total provisioncost of the public good in the n-agent economy is nc , where c isthe constant per capita cost of provision. We assume that c is lessthan the expected value of an agent’s valuation. This assumptionensures that the public good should be provided with probability 1in a large economy. Our results are:

(i) If α is less than a threshold α∗, then there exists a sequenceof mechanisms that satisfy ex post incentive compatibility, expost budget balance, and α proportional individual rationality.As n goes to infinity, the probabilities of provision in thissequence of mechanisms approach 1 at an exponential rate.

(ii) If α is greater than α∗, then for any sequence of anonymousmechanisms satisfying interim incentive compatibility, ex ante

188 K. Rong / Journal of Mathematical Economics 51 (2014) 187–196

budget balance, and α proportional individual rationality, theprobabilities of provision approach 0 as n goes to infinity. Inaddition, the convergence speed is not slower than 1/n

13 .

The above results are summarized in Fig. 1.The threshold α∗ equals the probability of an agent’s valuation

being higher than the per capita cost of provision, i.e., 1 −

F(c), where F is the distribution function of an agent’s valuation.The intuition is as follows. As the economy becomes large,the probability that an agent is pivotal in any anonymousmechanism becomes small. As a result, the agent’s expectedpayment must be nearly constant, independent of his valuation,if he is to report honestly. Ex ante budget balance then requiresthis constant payment to equal the per capita cost of provision.Thus, in a large economy, the proportion of agents who obtainnonnegative interim expected utilities from the mechanism mustbe approximately 1− F(c). Whether or not the public good can beprovided then depends on whether or not the required agreementrate α is less than 1 − F(c).

The results in this paper can be better understood ifwe comparethem with those of Mailath and Postlewaite (1990). They showedthat if we require unanimity, then the public good will not beprovided in a large economy. This implies that some coercion isnecessary if the principal wants the public good to be providedefficiently in a large economy. However, it is not clear fromthe literature exactly how much coercion is needed. Our resultsprovide an answer to this question. That is, if we want the publicgood to be provided efficiently in a large economy, then theproportion of agents that are allowed to be hurt in the mechanismshould be at least F(c).

Furthermore, if we fix a required agreement rateα, and let v∗ bethe valuation such that 1−F(v∗) = α, then efficiency obtains if andonly if v∗ is greater than the per capita cost c . This result shares thesame spirit as the result in Mailath and Postlewaite (1990), whoshowed that efficiency obtains if and only if the lowest type v isgreater than c . In other words, our results can be understood inthe following manner. For any given α, we can simply chop off thelower end of an agent’s distribution of valuation by a proportionof 1 − α and obtain v∗ as the new lowest type. Whether or notefficiency obtains then depends onwhether or not this new lowesttype v∗ is greater than c.

For the case where α is less than the threshold α∗, themechanism that we construct is actually quite simple. Each agentis required to report his valuation to the principal simultaneously.The public good is provided if and only if the proportion ofagents whose reported valuations exceed the per capita costis greater than α, and the cost of provision is distributedequally among all agents whenever the public good is provided.Asymptotic efficiency can be attained by this mechanism becauseas the economy becomes large, the proportion of agents whosevaluations exceeds the per capita cost of provision approaches1 − F(c), which is greater than the required agreement rate α byassumption.

For the casewhereα is greater than the thresholdα∗, in order toobtain the convergence rate of the probability of provision towardszero, we need to characterize the limiting behavior of anonymousfeasiblemechanisms,where amechanism is feasible if and only if itsatisfies all the constraints (i.e., interim incentive compatibility, exante budget balance andα proportional individual rationality). Ourapproach used to characterize the limiting behavior of anonymousfeasible mechanisms is very general and may be regarded asa methodological contribution of the paper. In particular, wepropose the standard deviation of an agent’s interim expectedprovision3 as a measure of the agent’s influence in a mechanism.

3 An agent’s interim expected provision is the expected value of the provision ofthe public good conditional on the agent’s own report/valuation. It is a function ofthe agent’s own valuation.

We find that, as the economy becomes large, an agent’s influencein any sequence of anonymous mechanisms decreases to zero ata speed not slower than 1/

√n. Based on this result, we find that

for any sequence of anonymous feasible mechanisms, an agent’sinterim expected provision function must converge to a constantfunction at a speed not slower than 1/

√n, and the agent’s interim

expected payment function must converge to a constant functionat a speed not slower than 1/n

13 . Roughly speaking, we obtain

bounds for the convergence speed of any sequence of anonymousfeasible mechanisms towards a ‘‘constant mechanism’’.4

Our result about the convergence rate of an agent’s influence issimilar to the result obtained byAl-Najjar and Smorodinsky (2000),who showed that in any sequence of mechanisms, the averageinfluence of agents converges to zero at a speed not slower than1/

√n as the economy becomes large. Themain difference between

the two results is that the measures of influence are different.5 Inparticular, Al-Najjar and Smorodinsky (2000) use the differencebetween the maximum and the minimum of an agent’s interimexpected provision to measure the agent’s influence, while we usethe standard deviation.

In the special case where α = 1, Mailath and Postlewaite(1990) showed that the probabilities of provision in any sequenceof mechanisms that satisfy interim incentive compatibility, exante budget balance, and interim individual rationality convergeto zero at a speed not slower than 1/n

14 . Our convergence rate

result improves this bound to 1/n13 . It should be noted that Al-

Najjar and Smorodinsky (2000) also improved the bound obtainedby Mailath and Postlewaite (1990) to 1/n

13 .6 However, our result

is more general in the sense that they rely on the assumptionthat the distribution of an agent’s valuation has an atom at thatagent’s lowest possible value, while our result does not requirethis assumption. In addition, Al-Najjar and Smorodinsky (2000)focus on the case where α = 1, while our result holds for anyα > 1 − F(c).

Our paper also relates to Norman (2004) in the sense thatboth papers have an asymptotic threshold characterization resultfor the public good provision. The difference between the twopapers is that the public good considered in Norman (2004) isexcludable, but it is non-excludable in our paper. In addition, ourpaper allows some agents to be hurt, while in Norman (2004), noagent can be coerced into participation. Norman (2004) also showsthat if agents’ valuations are i.i.d., then a fixed fee mechanism isalmost constrained-optimal in a large economy. This result sharesthe same feature with ours because we show that any sequenceof anonymous feasible mechanisms must converge to a constantmechanism as the economy becomes large, which implies that aconstant mechanismmust be asymptotically constrained-optimal.However, Norman (2004) does not characterize the speed of suchconvergence.

This paper is organized as follows. The next section presentsthe model. Section 3 discusses the asymptotic efficiency and

4 By ‘‘constant mechanism’’, we refer to the mechanism where any agent’sinterim expected provision and payment functions are constant.5 Another difference between the two results is that our result is based on

the assumptions that agents’ valuations are identically distributed and that themechanism is anonymous, while Al-Najjar and Smorodinsky (2000) do not makesuch assumptions. However, this difference is not crucial, because our result (inparticular, Lemma 2) can be easily generalized to the case where agents’ valuationsare not identically distributed and the mechanism is non-anonymous.6 Al-Najjar and Smorodinsky (2000) showed that when the distribution is

continuous on (v, v] and is discrete at v with P(v) ≥ ϵ for some ϵ > 0, theprobability of provision in any mechanism is smaller than C1

1√

η1

√n + C2η for any

0 < η < ϵ for some constants C1 and C2 . It can be verified that this result impliesthat the bound for the probability of provision is O(1/n

13 ), instead of O(1/

√n) as

stated by Al-Najjar and Smorodinsky (2000) (see Appendix B for a detailed proof).

K. Rong / Journal of Mathematical Economics 51 (2014) 187–196 189

Fig. 1. Probability of provision under various α as the economy becomes large.

asymptotic inefficiency results. Section 4 discusses the case wherethe cost per capita converges to zero as the economy becomeslarge. Concluding remarks are offered in Section 5.

2. Model

Assume that a non-excludable public good can be provided inthe quantity of either 0 unit or 1 unit. The cost of providing 1unit of the public good is nc , where n is the number of agents inthe economy, or the size of the economy, and c is a constant.7We denote agent i’s valuation of the public goods by vi. Agent i’svaluation vi is knownonly to agent i.We assume that v1, . . . , vn areindependent and identically distributed.8 The distribution functionof vi is denoted by F , which is commonknowledge among all agentsand the principal. The support of F is denoted by [v, v] ⊂ R+.The density function of F is denoted by f . We assume that f iscontinuous and strictly positive on [v, v]. Finally, we assume v <c < ve < v where ve

= Evi. The assumption that an agent’sexpected value of valuation ve exceeds the per capita cost c ismade in order to ensure that in a large economy, the average socialbenefit of the public good is greater than the average social cost ofthe public good. As a result, in a large economy, first-best efficiencyrequires that the public good be provided with probability 1.9

We consider direct anonymous mechanisms in this paper. Adirect mechanism is a function pair

qn, tni

ni=1

where qn :

[v, v]n

→ 0, 1 indicates whether or not the public good isprovided and tni : [v, v]

n→ R is the payment collected from

agent i. Note that qn and tni are functions of reported valuations(v1, . . . , vn) of all agents. The anonymity of the mechanism re-quires qn and tni to be functions that depend exclusively on thereported valuations and not on the identities of agents.10 Givenagents’ reported valuations (v1, . . . , vn), agent i’s ex post utilityunder the mechanism

qn, tni

ni=1

is given by viqn(v1, . . . , vn) −

tni (v1, . . . , vn). Define Uni (vi, vi) = Ev−i [viqn(vi, v−i) − tni (vi, v−i)]

and Uni (vi) = Un

i (vi, vi). Then Uni (vi, vi) represents agent i’s

interim expected utility when he reports vi and Uni (vi) repre-

sents agent i’s interim expected utility when he reports truthfully,both conditional on all other agents reporting truthfully. Sincethe prior distribution of vi is the same for all i and the mecha-nism qn, tni

ni=1 is anonymous by assumption, the functionUn

i (vi)

7 Here, the cost function is such that the cost of provision increases in proportionto the number of agents. This assumption is in the spirit of Mailath and Postlewaite(1990), in which the per capita cost of provision is bounded away from zero.8 The i.i.d. setting in the public good provision problem also appeared in Rob

(1989) and in Ledyard and Palfrey (2002).9 If instead, ve is less than c , then first-best efficiency requires that the public good

be provided with probability 0 in a large economy. As a result, for any given α, first-best efficiency can always be achieved in a large economy by the trivial mechanismwhere the public good is not provided and payments are not collected from agentsfor any reported valuations.10 Formally, the anonymity of the mechanism requires (i) q(v1, . . . , vn) =

q(σ (v1, . . . , vn)) for any permutation σ(v1, . . . , vn) of (v1, . . . , vn), (ii)tni (vi, v−i) = tni (vi, σ (v−i)) for any permutation σ(v−i) of v−i , and (iii)tni (vi, v−i) = tnj (vj, v−j) where vi = vj and v−j is a permutation of v−i .

(similarly, Uni (vi, vi)) must be the same for all i andwill be denoted

by Un(vi) (Un(vi, vi), respectively) whenever there is no confusion.We impose three constraints on the mechanism. The first

constraint is the interim incentive compatibility constraint INTIC.With INTIC, the truthful strategy is a Bayesian Nash equilibriumof the mechanism

(INTIC) : Un(vi) ≥ Un(vi, vi) for all vi, vi ∈ [v, v]. (1)

A stronger version of the interim incentive compatibilityconstraint is the ex post incentive compatibility constraint EXPIC

(EXPIC) : viqn(v) − tni (v) ≥ viqn(vi, v−i) − tni (vi, v−i)

for any vi, vi ∈ [v, v] and v−i ∈ [v, v]n−1. (2)

For a direct mechanism, the truthful strategy is a weaklydominant strategy of the mechanism if and only if the mechanismsatisfies EXPIC. Obviously, if a mechanism satisfies EXPIC, then itmust satisfy INTIC.

The second constraint is the ex ante budget balance constraintEXABB

(EXABB) : E

tni (v)

n− cqn(v)

≥ 0. (3)

A stronger version of the ex ante budget balance constraintis the ex post budget balance constraint EXPBB, which requiresthat the mechanism be budget-balanced for any realization ofvaluations11

(EXPBB) :

tni (v)

n− cqn(v) ≥ 0 for all v ∈ [v, v]

n. (4)

The last constraint is the α proportional individual rationalityconstraint. This constraint reflects the requirement that the publicgood can be provided and payments can be collected from agentsonly if the proportion of agents who obtain nonnegative interimexpected utilities from themechanism is at least α. In other words,the α-PIR constraint reflects the requirement that a proportionα of the agents need to approve the mechanism in order for themechanism to be enforced.12

With the α-PIR constraint, as long as the mechanism isapproved, all agents are forced to participate regardless of theirIR constraints. However, it should be noted that even if themechanism is approved, it does not mean the public good will be

11 Although EXPBB seems to be a stronger requirement than EXABB, it is wellknown in the literature that EXABB and EXPBB are actually equivalent in the sensethat for any mechanism satisfying EXABB, there exists a mechanism that satisfiesEXPBB and yields the same interim expected utility for any agent (see, e.g., Börgersand Norman, 2009).12 We can imagine that there exists a voting procedure before the implementationof the public goodprovisionmechanism. That is, a public goodprovisionmechanismis proposed by a principal, and a group of agents then vote on the mechanism,using the α majority rule. We assume that agents vote sincerely so that an agentvotes for the mechanism if and only if the agent obtains a nonnegative expectedutility from the mechanism. The sincere-voting assumption is appropriate for ourmodel because we consider a large economy, where the probability that any agentis pivotal is small.

190 K. Rong / Journal of Mathematical Economics 51 (2014) 187–196

provided. That is, agents are voting on the public good provisionmechanism, instead on whether to provide the public good.

We will introduce some notation before we formally define theα proportional individual rationality constraint. Define the agree-ment set V n as the set of valuations for which an agent’s interimexpected utility is nonnegative, i.e., V n

= vi ∈ [v, v]|Un(vi) ≥ 0.

Define rn(v) =

ni=1 1

vi∈V n

n where 1• represents the indicatorfunction. Then, rn(v) is the proportion of agents who obtain non-negative interim expected utilities from the mechanism. Now, theα proportional individual rationality constraint is:

(α-PIR) : qn(v) = 0 and tni (v) = 0 (for all i) if rn(v) < α.

In this paper, α is exogenously fixed and can be any numberbetween 0 and 1. If α = 0, then the α proportional individualrationality constraint is automatically satisfied by anymechanism.If α = 1, then the α proportional individual rationality constraintis satisfied if and only if the interim expected utility is nonnegativefor all agents, i.e., if and only if the interim individual rationalityconstraint is satisfied for all agents.13

The α-PIR constraint actually describes what is happening ifthe proportion of the agents who approve the mechanism is lessthan the required agreement rate α. One may think that the α-PIRconstraint should be defined in a manner such that a mechanismsatisfies the constraint if at least a proportion α of the agents arebetter off than the outside option (i.e., a proportion α of the agentsobtain nonnegative interim expected utilities). This definition isproblematic for the following reason. A mechanism is designedby the principal at the ex ante stage. For a given mechanism,whether there exists a proportionα of the agentswho are better offthan the outside option depends on the distribution of the realizedvaluations of all agents. In other words, at the ex ante stage, whenthe principal designs the mechanism, the principal cannot requirethe mechanism to be such that at least α proportion of the agentsare better off than the outside option because the principal doesnot know the realized valuations of the agents.

The following lemma will be used to simplify the α-PIRconstraint. Define qni (vi) = Ev−iq

n(v) as agent i’s interim expectedprovision and tni (vi) = Ev−i t

ni (v) as agent i’s interim expected

payment. We have:

Lemma 1. If the mechanism qn, tni ni=1 satisfies INTIC, then:

(i) qni (vi), tni (vi) and Un(vi) are nondecreasing on [v, v];(ii) Un(vi) is continuous on [v, v].

Lemma 1 is well-known in the mechanism design literature(see, e.g., Myerson, 1981). Lemma 1(i) essentially says that inorder to make an agent truthfully reveal his information, theagent’s interim expected provision function, interim expectedpayment function and interim expected utility function must benondecreasing in the agent’s valuation. Lemma 1(ii) implies thatUn(vi) is either (i) equal to zero at some point in [v, v], (ii) greaterthan zero for all vi ∈ [v, v], or (iii) less than zero for all vi ∈ [v, v].We define vn as follows:

vn=

minvi|vi ∈ [v, v]

and Un(vi) = 0 if Un(vi) = 0 for some vi ∈ [v, v];

v if Un(vi) > 0 for all vi ∈ [v, v];

∞ if Un(vi) < 0 for all vi ∈ [v, v].

13 The ‘‘if’’ part is obvious. To see the ‘‘only if’’ part, if there is a v∗

i ∈ [v, v] suchthat Ui(v

∗

i ) < 0, then we must have rn(v∗

i , v−i) < 1 for all v−i , which impliesqn(v∗

i , v−i) = 0 and tni (v∗

i , v−i) = 0 for all v−i by the α proportional individualrationality constraint. Thus, Ui(v

∗

i ) must equal zero, which is a contradiction withUi(v

∗

i ) < 0.

Using the definition of vn and the fact thatUn(vi) is nondecreas-ing in vi, we have

V n= vi ∈ [v, v]|Un(vi) ≥ 0 = vi ∈ [v, v]|vi ≥ vn

.

Thus,

rn(v) =

ni=1

1vi∈Vn

n=

ni=1

1vi≥vn

n.

The α-PIR constraint can be rewritten as:

(α-PIR) : qn(v) = 0 and tni (v) = 0

(for all i) if

ni=1

1vi≥vn

n< α. (5)

We now define the first-best mechanism. The first-bestmechanism qFB(n), tFB(n)i

ni=1 is any ex ante budget-balanced

mechanism that satisfies the Lindahl–Samuelson provision rule:

qFB(n) =

1 if

vi

n≥ c;

0 otherwise.

By the weak law of large numbers, as n → ∞, the per capitabenefit

vi

n of providing the public good in the first-best mech-anism approaches ve, which is greater than the per capita cost cby assumption. Thus, the probability that the public good will beprovided in the first-best mechanismmust approach 1 as n goes toinfinity, i.e., P(qFB(n)(v) = 1) → 1 as n → ∞.

Finally, we define the per capita welfare of the mechanismqn, tni

ni=1

as the per capita expected value of the net benefit

of the mechanism, i.e., W (qn) =1nE

(

vi − nc)qn(v)

=

E(

vi

n − c)qn(v). Obviously, the per capita welfare of the first-

best mechanism,W (qFB(n)), approaches ve− c as n → ∞.

3. Analysis

3.1. Asymptotic efficiency result

In this section, wewill explore whether first-best efficiency canbe achieved by mechanisms that are constrained by INTIC, EXABBand α-PIR.

In this subsection, wewill show that, if α < 1−F(c), then first-best efficiency can be achieved asymptotically by the sequence ofmechanisms Mn

α∞

n=1, where Mnα = qn,α, tn,αi

ni=1 is constructed

so that qn,α(v) = 1 and tn,αi (v) = c for any i ifn

i=1 1vi≥cn ≥ α, and

qn,α(v) = 0 and tn,αi (v) = 0 for any i ifn

i=1 1vi≥cn < α. Simply

speaking, Mnα requires that, if there exists at least α proportion of

agentswhose reported valuations of the public good exceed c , thenthe public good will be provided and the cost will be distributedequally among all agents. Otherwise, the public good will not beprovided and no payment is collected from any agent. This mech-anism is equivalent to the voting mechanism in which agents canvote on the provision of the public good and the public goodwill beprovided (with taxes equally distributed among agents) if and onlyif the proportion of agents who vote for the provision of the pub-lic good is greater than α. We thus callMn

α the α-referendum of the

K. Rong / Journal of Mathematical Economics 51 (2014) 187–196 191

n-agent economy.14 We have the following asymptotic efficiencyresult for the α-referendum.

Theorem 1. (i) Mnα satisfies EXPIC, EXPBB and α-PIR;

(ii) If α < 1−F(c), then the probability of provision inMnα converges

to 1 at a speed not slower than 2 exp−2n(β − α)2, whereβ = 1 − F(c) > α.

Proof. See Appendix A.

Theorem 1(i) is obvious. Mnα is ex post budget balanced by

construction. It satisfies the EXPIC constraint because any agentwith valuation above c has the incentive to report truthfullyto increase the probability of provision and any agent withvaluation below c has the incentive to report truthfully to decreasethe probability of provision; the truthful strategy is actually adominant strategy for each agent. Finally, noting that the interimexpected utility under Mn

α equals zero at vi = c , it can be verifiedthatMn

α satisfies the α-PIR constraint.The intuition of the proof for Theorem 1(ii) is as follows. In the

α-referendum Mnα , the public good will be provided if and only

if the proportion of agents whose valuations exceed c is greaterthan α. By the weak law of large numbers, as n → ∞, the pro-portion of agents whose valuations exceed c approaches P(vi ≥

c) = 1 − F(c), which is greater than α by assumption. Hence,the probability of the public good being provided under Mn

α ap-proaches 1 as n goes to infinity. Using Hoeffding’s inequality forsums of bounded random variables, we can show that the conver-gence of P(qn,α(v) = 1) towards 1 is at the exponential rate.

We have the following result regarding the per capita welfareof the α-referendum.

Corollary 1. Assume that α < 1 − F(c), then the difference be-tween the per capita welfare of the first-best mechanism and the percapita welfare of the α-referendum (i.e., W (qFB(n)) − W (qn,α)) ap-proaches 0 as n → ∞.

Remark. Assume that α < 1 − F(c) and fix any α′∈ [α, 1 −

F(c)), then it can be verified that the per capita welfare of the α′-referendum also approaches the per capita welfare of the first-bestmechanism (in addition, the α′-referendum satisfies EXPIC, EXPBBand α-PIR). However, the gap between the per capita welfare ofthe α′-referendum and the per capita welfare of the first-bestmechanism is usually the smallest when α′

= α.

Corollary 1 implies that the per capita welfare losses from usingα-referenda instead of first-bestmechanisms vanish as n increases.The performance of α-referenda are illustrated in the followingexample.

Example 1. Assume that vi is uniformly distributed on [0, 1], andthat c = 0.4 and α = 0.5.

Table 1 lists the probabilities that the public good will beprovided in 0.5-referenda as n increases from 5 to 320. It alsolists the per capita welfare of 0.5-referenda, the per capitawelfare of first-best mechanisms and the per capita welfare losses(i.e., 1 −

W (qn,0.5)W (qFB(n))

) from using 0.5-referenda instead of first-bestmechanisms.

14 The α-referendum defined in this paper is equivalent with the j∗-referendumdefined in Ledyard and Palfrey (2002) with α =

j∗

n . While Ledyard and Palfrey(2002) focused on whether or not interim efficiency can be achieved by referendaasymptotically, we focus on whether or not ex post efficiency can be achieved byreferenda asymptotically. In addition, there is no individual rationality constraint inthemodel of Ledyard and Palfrey (2002),whilewe requireα proportional individualrationality.

Table 1Probability of provision and relative efficiency of 0.5-referenda.

n 1 − P(qn,0.5 = 1) W (qn,0.5) W (qFB(n)) 1−W (qn,0.5)

W (qFB(n))

5 0.3169 0.1097 0.1165 0.058110 0.1662 0.1034 0.1062 0.026620 0.1273 0.0990 0.1016 0.026440 0.0747 0.0981 0.1002 0.020980 0.0267 0.0990 0.1001 0.0095

160 0.0041 0.0997 0.1000 0.0021320 0.0001 0.0999 0.0999 0.0001

The threshold in our example is 1 − F(c) = 0.6. Since α = 0.5is less than the threshold, first-best efficiency can be achievedasymptotically. In particular, Table 1 shows that as n increases,1 − P(qn,0.5) approaches 0 rapidly. It also shows that the percapita welfare loss from the 0.5-referendum vanishes rapidly as nincreases. Actually, when the size of the economy is greater than80, the per capita welfare loss is less than 1%.

3.2. Asymptotic inefficiency result

This subsection considers the case where the required agree-ment rate is large. We will show that, if α > 1 − F(c), then theprobabilities that the public good is provided in any sequence ofanonymous mechanisms satisfying INTIC, EXABB and α-PIR ap-proach 0 as n → ∞ and the convergence speed is not slower than1/n

13 .The next lemma states that the variance of any agent’s interim

expected provision in an n-agent anonymous economy mustconverge to 0 at a speed not slower than 1/n as n → ∞. LetVar(qni (vi)) =

[v,v]

[qni (vi) − Eqni (vi)]2dF(vi). We have:

Lemma 2. For any sequence of anonymous mechanisms qn,tni

ni=1

∞

n=1, we have Var(qni (vi)) = O(1/n).

Proof. See Appendix A.

Although Lemma 2 is not necessary if our purpose is just toobtain the asymptotic inefficiency result, it is essential for usto establish the convergence rate of the probability of provisiontowards zero.

The intuition of Lemma 2 is as follows. Using the assumptionthat v1, . . . , vn are independent, one can show that

i Var(q

ni (vi))

≤ Var(qn(v)). Since Var(qn(v)) ≤ 1 and v1, . . . , vn are identicallydistributed, we thus have Var(qni (vi)) = O(1/n).

We define agent i’s influence relative to the mechanismqn, tni

ni=1

by

Var(qni (vi)).

Var(qni (vi)) measures the impact

on the interim expected provision as agent i’s reported valua-tion changes. A small

Var(qni (vi)) indicates a small impact. If

Var(qni (vi)) is zero, then qni (vi) equals a constant function almostsurely,which implies that agent i essentially has no influence in themechanism

qn, tni

ni=1

. Lemma 2 says that as n goes to infinity,

an agent’s influence in any sequence of anonymous mechanismsdecreases to zero at a speed not slower than 1/

√n.

The next lemma is a direct result of Lemma 2. It says thatfor any v ∈ (v, v), the differences between qni (v) and Eqni (vi)under any sequence of interim incentive compatible anonymousmechanisms vanish as n → ∞ and are O(1/

√n). This implies

that for any sequence of interim incentive compatible anonymousmechanisms, as the economy becomes large, an agent’s interimexpected provision function must converge to a constant function,and the convergence speed is not slower than 1/

√n.

For any given v ∈ (v, v), definem(v) = min(P(vi ≥ v), P(vi ≤

v)) (notice thatm(v) > 0 because the probability density functionof vi is positive on [v, v]). We have:

192 K. Rong / Journal of Mathematical Economics 51 (2014) 187–196

Lemma 3. Let qn, tni ni=1

∞

n=1 be any sequence of anonymousmechanisms where, for each n, qn, tni

ni=1 satisfies INTIC. Then, for

any given v ∈ (v, v), we have:qni (v) − Eqni (vi) ≤

1

2m(v)n

.

Proof. See Appendix A.

The intuition for the proof of Lemma 3 can be explained asfollows. Lemma 2 insures that the variance of qni (vi) approacheszero as n goes to infinity. Hence, as n becomes large, qni (vi) becomes‘‘flat’’. This implies that for almost every point vi ∈ (v, v), qni (vi)converges to E(qni (vi)). That is, the set of points in (v, v) for whichthe convergence does not hold has measure zero. Since qni (vi) isnondecreasing in (v, v) (which is true for any mechanism thatsatisfies INTIC), we can show that actually for all of the points in(v, v), we have qni (vi) converging to E(qni (vi)).15 Using the resultin Lemma 2 and Chebyshev’s inequality, we can show that theconvergence speed of qni (vi) to E(qni (vi)) is not slower than 1/

√n.

There is a natural connection between Lemma 3 and the resultabout influence in Al-Najjar and Smorodinsky (2000). To see this,assume that vi follows a discrete distribution on V , where V isa finite set and each element in V has a probability of at leastϵ. Lemma 3 then implies that

qni (vi) − qni (v′

i) ≤

1√

ϵn for any

vi, v′

i ∈ V .16 This implies that maxvi,v′i∈V (qni (vi) − qni (v

′

i)) ≤1

√ϵn .

On the other hand, Al-Najjar and Smorodinsky (2000) proposemaxvi,v

′i∈V (qni (vi) − qni (v

′

i)) as the measure of the influence of anagent, and show that an agent’s influence is roughly boundedby 1

√π

√ϵn when n is large. So, roughly speaking, for the case

where the agents’ valuations are discrete, ourmeasure of influencecan be transformed to the measure of influence in Al-Najjar andSmorodinsky (2000), and the bound obtained by our paper andthat obtained by Al-Najjar and Smorodinsky (2000) differ only bya constant.

In the casewhere the distributions of agent’s valuations are con-tinuous, the comparison between our measure of influence andthat of Al-Najjar and Smorodinsky (2000) is less straightforward.The reason is that in Al-Najjar and Smorodinsky (2000), it is prob-lematic to directly use maxvi,v

′i∈V (qni (vi) − qni (v

′

i)) as a measureof influence in the continuous case, because in doing so, the in-fluence of an agent may then be very large even if qni (vi) is aconstant on a set with a measure of one. Al-Najjar and Smorodin-sky (2000) thus propose to use infA⊂[v,v]:P(A)<ϵ supvi∈A qni (vi) −

supA⊂[v,v]:P(A)<ϵ infvi∈A qni (vi) as a measure of influence.However, this measure of influence depends on an exogenous pa-rameter ϵ, while our measure of influence (

Var(qni (vi))) does

not depend on any exogenous parameter. We thus cannot directlycompare those two measures in the continuous case.

A good measure of an agent’s influence in a mechanism shouldrely only on themechanism, and not on any exogenous parameter.It is in this sense that our measure of influence seems to be moresuitable than that of Al-Najjar and Smorodinsky (2000) for the casewhere agents’ valuations are continuous.

Now, we define

Mn= qn, tni

ni=1|q

n, tni ni=1 satisfies INTIC, EXABB,

and α-PIR in the n-agent economy.

15 The reason here is that if there is an ‘‘outlier’’ point at which the convergencedoes not hold, then given that qni (vi) is nondecreasing, there must exist a set ofpoints with positive measure for which the convergence does not hold.16 Although we assume that vi follows a continuous distribution, the results inLemmas 2 and 3 also hold for the case where vi is discrete.

We call any mechanism in Mn a feasible mechanism of the n-agent economy. The next lemma characterizes the convergencespeed at which the interim expected payment function inany sequence of anonymous feasible mechanisms approaches aconstant function. Recall that m(v) = min(P(vi ≥ v), P(vi ≤

v)) > 0 for any v ∈ (v, v). We have:

Lemma 4. Let qn, tni ni=1

∞

n=1 be any sequence of anonymousmechanisms where qn, tni

ni=1 ∈ Mn for each n. Then, for any

v ∈ (v, v) and any 0 < η < m(v), we havetni (v) − Etni (vi)

≤

v 1√

ηn + 2vη.

Proof. See Appendix A.

Notice that m(v) does not depend on n. The bound obtainedin Lemma 4 is the tightest when η = 4−

23 n−

13 . This implies that

the bound obtained in Lemma 4 is on the order of 1/n13 (instead

of 1/√n). Thus, an agent’s interim expected payment function

converges to a constant function at a speed not slower than 1/n13

as n → ∞.The intuition of the proof of Lemma 4 is as follows. The

constraint INTIC on the mechanism qn, tni ni=1 implies that

v′

i(qni (vi) − qni (v

′

i)) ≤ tni (vi) − tni (v′

i) ≤ vi(qni (vi) − qni (v′

i)) forany vi, v

′

i ∈ [v, v]. That is, the variation of tni (vi) is boundedby the variation of qni (vi). By Lemma 3, the interim expectedprovision function qni approaches a constant function as n goes toinfinity. Thus, the interim expected payment function tni must alsoconverge to a constant function as the economybecomes large. Thebound v 1

√ηn + 2vη in Lemma 4 can be explained as follows. Fix

v ∈ (v, v), and consider the difference between tni (v) and tni (vi) forany given vi ∈ [v, v]. By Lemma 3,

qni (v) − qni (vi) is bounded by

1√

ηn , whereη = minm(v),m(vi). So, using INTIC,tni (v) − tni (vi)

is bounded by v 1

√ηn . Notice that the threshold minm(v),m(vi)

goes to zero if vi is close to v or v. However, in the case where vi isclose to v or v, we can use the fact that

tni (vi) is bounded by v to

get a bound ontni (vi) − tni (vi)

. Combining the above two boundsand taking the expectation of tni (v) − tni (vi) over vi, we then havethat

tni (v) − Etni (vi) is bounded by v 1

√ηn + 2vη.

The following theorem argues that the probabilities that thepublic good is provided in any sequence of anonymous feasiblemechanisms approach 0 at a speed not slower than 1/n

13 . Let vα

be such that P(vi ≥ vα) = α. Note that α > 1 − F(c) impliesvα < c. We have:

Theorem 2. Assume that α > 1 − F(c). Let qn, tni ni=1

∞

n=1 beany sequence of anonymous mechanisms where qn, tni

ni=1 ∈ Mn

for each n, then we have P(qn(v) = 1) = O(1/n13 ).

Proof. Let qn, tni ni=1

∞

n=1 be a given sequence of anonymousmechanisms where qn, tni

ni=1 ∈ Mn for each n. Fix an ϵ ∈

(0, c − vα).17 For any given n, we have the following two cases.(i) vn < vα + ϵ.In this case, since vn < vα + ϵ < ∞, we must have Un(vn) ≥

0 by the definition of vn. The incentive compatibility constraintimplies that Un(vi) is nondecreasing in vi, so Un(vα + ϵ) ≥

Un(vn) ≥ 0. That is:

(vα + ϵ)qni (vα + ϵ) ≥ tni (vα + ϵ). (6)

17 The precise choice of ϵ will not affect the convergence rate we obtain below.

K. Rong / Journal of Mathematical Economics 51 (2014) 187–196 193

Since qni (vi), tni (vi) satisfies INTIC and v < vα + ϵ < c < v,then by Lemma 3, we have:

qni (vα + ϵ) ≤ Eqni (vi) +1

2√m(vα + ϵ)n

. (7)

By Lemma4, for any 0 < η < m(vα+ϵ) (recall thatm(vα+ϵ) =

minP(vi ≤ vα + ϵ), P(vi ≥ vα + ϵ)), we have:

tni (vα + ϵ) ≥ Etni (vi) − v1

√ηn

− 2vη. (8)

(6)–(8) then imply an upper bound for Etni (vi), that is:

Etni (vi) ≤ (vα + ϵ)Eqni (vi) +vα + ϵ

2√m(vα + ϵ)n

+ v1

√ηn

+ 2vη. (9)

Now, by the budget balance constraint, we can get a lowerbound for Etni (vi), that is:

Etni (vi) ≥ cEqni (vi). (10)

Inequality (9) and inequality (10) then imply:

Eqni (vi) ≤1

c − (vα + ϵ)

×

vα + ϵ

2√m(vα + ϵ)

1√n

+ v1

√η

1√n

+ 2vη

. (11)

Since Eqni (vi) = Eqn(v) = P(qn(v) = 1), then we have thefollowing inequality for any 0 < η < m(vα + ϵ):

P(qn(v) = 1) ≤1

c − (vα + ϵ)

vα + ϵ

2√m(vα + ϵ)

1√n

+ v1

√η

1√n

+ 2vη

. (12)

Notice that inequality (12) holds for any sufficiently small η. Itcan be verified that the bound obtained in (12) is the tightest whenη = 4−

23 n−

13 . This implies that the bound obtained in (12) is on the

order of 1/n13 .

(ii) vn≥ vα + ϵ.

Define γ n(v) =

ni=1 1vi≥vα+ϵ

n . Note rn(v) =

ni=1 1vi≥vn

n , thenwe must have γ n

≥ rn. By α-PIR, if rn < α then qn(v) = 0. Thus,we have P(qn(v) = 1) ≤ P(rn ≥ α) ≤ P(γ n

≥ α). Let P(vi ≥

vα + ϵ) := β0. Since 1vi≥vα+ϵni=1 are independent variables and

E(γ n) = E(

ni=1 1vi≥vα+ϵ

n ) = β0, we have P(γ n≥ α) = P(γ n

−

β0 ≥ α − β0) ≤ P(|γ n− β0| ≥ α − β0) ≤ 2 exp−2n(α − β0)

2,

where the last inequality follows from Hoeffding’s inequality forsums of bounded random variables. Hence,

P(qn(v) = 1) ≤ 2 exp−2n(α − β0)2. (13)

For any given n, the probability of provision P(qn(v) = 1) iseither bounded by inequality (12), or bounded by inequality (13).This implies that P(qn(v) = 1) = O(1/n

13 ) for any sequence of

mechanisms qn, tni ni=1

∞

n=1 where qn, tni ni=1 ∈ Mn

α for eachn and α > 1 − F(c).

We now explain the intuition of the proof of Theorem 2. Fixn and fix an ϵ ∈ (0, c − vα), we have the following two cases:(i) vn > vα + ϵ, and (ii) vn

≤ vα + ϵ.In the former case, the break-even valuation vn is relatively

high. This implies that the proportion of agents who obtainnonnegative interim expected utilities is relatively low. Under

the assumption that vn > vα + ϵ, it can be verified that theabove proportion is less than the required agreement rate α withprobability one when n is large. Thus, by α-PR, the public goodwillnot be provided in a large economy.

In the latter case, using Lemmas 3, 4 and the fact that Un(vα +

ϵ) ≥ Un(vn) ≥ 0, we can obtain the following upper bound for theexpected payment per capita (see also inequality (9) in the proofof Theorem 2):

Etni (vi) ≤ (vα + ϵ)P(qn(v) = 1) + C(n) (14)

where C(n) approaches zero as n goes to infinity. The aboveinequality reflects the fact that the expected payment that theprincipal could collect from each agent is not greater than the basepayment of the agent (which is bounded by (vα + ϵ)P(qn(v) = 1))plus the expected virtual benefit of the agent (which is bounded byC(n)). The base payment is the tax that the mechanism designercan impose on all agents, and the expected virtual benefit reflectshowmuch additional tax themechanismdesigner can extract froma high valuation agent under the information constraint.18 SinceC(n) goes to zero as n goes to infinity, the expected paymentthat the principal could collect from each agent is at most (vα +

ϵ)P(qn(v) = 1) in a large economy. However, the assumption thatthe required agreement rate α is greater than 1− F(c) implies thatvα +ϵ < c. Thus, the budget balance constraint cannot be satisfiedunless P(qn(v) = 1) = 0.

The convergence speed of P(qn(v) = 1) towards zero dependson two factors. One is the speed at which the proportion of agentswho obtain nonnegative interim expected utilities goes to a valuethat is smaller than α in the first case mentioned above. Theother is the speed at which C(n) goes to zero in the second casementioned above. It turns out that the first speed is on the order ofan exponential rate byHoeffding’s inequality and the second speedis on the order of 1/n

13 as a result of Lemma 4. Thus, P(qn(v) = 1)

must converge to zero at a speed not slower than 1/n13 .

4. The role of production technologies

In the previous sections, we assumed that the cost of provisionis proportional to the number of agents in the economy. In thissection, we are going to consider a situation where the per capitacost of provision is decreasing in the number of agents.19 Inparticular, we assume that the cost of provision in the n-agenteconomy is C(n) = nγ c , where γ ∈ [0, 1) and c > 0 are constants.The per capita cost of provision is thus c/n1−γ . In addition, weassume that v = 0.20

The purpose of this section is to explore the impact ofproduction technologies on the provision of the public good. Asthe economy becomes large, there are two opposite effects on theprovision of the public good. One is the decreasing per capita costof provision due to the scale efficiency (measured by 1 − γ ) inproduction. The other is the increasing severity of the free-riderproblem due to the increase of the number of agents.Wewill showthat, for the case where α is less than 1, no matter how small

18 The fact that C(n) goes to zero as n goes to infinity implies that as the economybecomes large, an agent’s expected virtual benefit goes to zero (see also Hellwig,2003, Norman, 2004, etc., for this property).19 We will not consider the case in which the per capita cost of provision isincreasing in the number of agents, because in that case, as the number of agentsincreases, the total cost will eventually exceed themaximumpossible total benefitsof the public good and thus the probability of provision in the first-best mechanismapproaches zero. As a result, first-best efficiency can always be achieved.20 If v > 0, then since the per capita cost approaches 0 < v, first-best efficiencycan always be achieved by a mechanism in which the public good is provided withprobability 1 and each agent is charged v.

194 K. Rong / Journal of Mathematical Economics 51 (2014) 187–196

1 − γ might be, the positive effect of decreasing cost per capitawill eventually outweigh the negative effect of increasing severityof the free-rider problem and first-best efficiency will be achieved.For the case where α equals 1, assuming that vi is discrete at 0, wefind that inefficiency obtains if 1 − γ is less than 1/3.

The analysis for the case where α is less than 1 is straightfor-ward. Observing that C(n)

n approaches zero as n goes to infinity,the threshold 1 − F(C(n)/n) approaches 1. This implies that therequired agreement rate α must be less than the threshold for suf-ficiently large n. As a result, for any γ ∈ [0, 1), first-best effi-ciency can be achieved in a large economy by mechanisms such asα-referendum.

We now analyze the case where α = 1. For simplicity, weassume that the distribution of vi is continuous on (0, v] but isdiscrete at 0 with P(vi = 0) > 0.21 For any n, since the requiredagreement rate equals 1, vn must equal zero. In addition, vα = 0.Thus, we have vn < vα + ϵ = ϵ for any ϵ ∈ (0, c − vα).Letting ϵ → 0, inequality (9) then implies that for any 0 < η <m(vα + ϵ) = m(ϵ) = minP(vi ≤ ϵ), P(vi ≥ ϵ), we have:

Etni (vi) ≤ v1

√ηn

+ 2vη. (15)

Notice that η is independent of n. Inequality (15) implies thatthe expected payment per capita decreases to zero at a speednot slower than 1/n

13 (this is because the right hand side of

inequality (15) is the smallest when η = 4−23 n−

13 ). Thus, if the

cost per capita decreases to zero at a speed slower than 1/n13 (i.e.,

γ ∈ (2/3, 1)), then the cost per capita will eventually exceedthe expected payment per capita and the probability of provisionmust converge to zero. In particular, for γ ∈ (2/3, 1), using theex ante budget balance constraint C(n)Eqni (vi) ≤ nEtni (vi) andinequality (15), we have: Eqni ≤ O(1/nγ−

23 ), i.e., P(qn(v) = 1) ≤

O(1/nγ−23 ). Since 1/nγ−

23 =

1/n13

1/n1−γ , the convergence rate of the

probability of provision (i.e., 1/nγ−23 ) is thus determined by the

convergence rate of the expected payment per capita (i.e., 1/n13 )

and the convergence rate of the cost per capita (i.e., 1/n1−γ ).Finally, if α = 1 and γ ∈ [0, 1/2), then first-best efficiency can

always be achieved in a large economy (see Hellwig, 2003).22In summary, we have:

Theorem 3. Assume the cost of provision in the n-agent economy isnγ c, then,(i) if α ∈ [0, 1) and γ ∈ [0, 1) or α = 1 and γ ∈ [0, 1/2), the first-

best provision level can be achieved asymptotically by a sequenceof mechanisms that satisfy INTIC, EXABB and α-PIR;

(ii) if α = 1 and γ ∈ (2/3, 1), assuming that vi is continuouson (0, v] and is discrete at 0 (i.e., vi has an atom at 0),the probabilities of provision in any sequence of anonymousmechanisms that satisfy INTIC, EXABB and α-PIR approach zeroas n → ∞ at a speed not slower than 1/nγ−

23 .

21 The assumption that the distribution of vi has an atomat 0 helps the inefficiencyresult below (Theorem3(ii)). In particular, this assumption ensures that vα+ϵ

√m(vα+ϵ)

=

ϵ√m(ϵ)

goes to zero as ϵ goes to zero, and thus the upper bound of Etni (vi)weobtain inthe proof of Theorem 2 (in particular, inequality (9)) can be simplified to inequality(15) below.22 Actually, Hellwig (2003) assumed that the total cost of provision is independentof the number of agents (for the case where the first-best provision level isbounded). He constructed a sequence of mechanisms and showed that first-bestefficiency can be achieved by this sequence of mechanisms (Proposition 3, Hellwig,2003). It can be easily verified that the sequence of mechanisms constructed byHellwig (2003) also attains first-best efficiency as soon as the cost per capitaconverges to zero at a speed that is faster than 1/

√n.

The case where α = 1 and γ ∈ [1/2, 2/3] is not covered inTheorem3. In this case, we can neither prove nor disprovewhetherefficiency or inefficiency obtains. Our conjecture is that whetherefficiency obtains depends on the distributions of the valuations ofthe agents.

5. Conclusions

This paper considers the public good provision problem inwhich the public good can be provided and payments can becollected from agents only if the proportion of agents whoobtain nonnegative interim expected utilities from themechanismweakly exceeds a required agreement rate α. We show that, if αis less than 1 − F(c), then there exists a sequence of α-referendasatisfying EXPIC, EXPBB and α-PIR such that as the size n ofthe economy becomes large, the probabilities of the public goodbeing provided in α-referenda approach 1 at a speed not slowerthan the exponential rate; and if α exceeds 1 − F(c), then theprobabilities that the public goodwill be provided in any sequenceof mechanisms that satisfy INTIC, EXABB and α-PIR approach 0 ata speed not slower than 1/n

13 . Hence, this paper not only obtains

asymptotic efficiency/inefficiency results for various requiredagreement rate settings, but also characterizes the speed atwhich the probability of provision reaches its efficient/inefficientlevel. In addition, as a methodological contribution, we proposethe standard deviation of an agent’s interim expected provisionas a measure of the agent’s influence in a mechanism. Wecharacterize the convergence speed of an agent’s influence in anysequence of anonymousmechanisms towards zero (Lemma 2) andthe convergence speed of any sequence of anonymous feasiblemechanisms towards a constant mechanism (Lemmas 3 and 4).

Our results imply that F(c) is the minimum fraction of theagents that need to be coerced into participation if we want toobtain efficiency asymptotically. An interesting question to ask iswhether we can design a mechanism to elicit payments and alsodetermine which agents need to be coerced in order to achieveefficiency. We do not have a clear answer to this question in thefinite economy, but we do have an answer when the economyis large. In particular, when the economy becomes large, ourLemmas 3 and 4 show that anymechanism satisfying INTIC, EXABBand α-PIR must converge to a mechanism where the interimexpected payment and provision functions are constant. By budgetbalance, this implies that if the public good is going to be provided,then the interim expected payment of an agent in any mechanismmust be roughly equal to the per capita cost c , independent of theagent’s valuation. This implies that if we want to obtain efficiency,then (roughly speaking), any agent with valuation less than cshould be coerced into participation no matter what mechanismwe are going to use.

Acknowledgments

I thank Steven Williams, Bart Taub and Dan Bernhardt for theirpatient guidance and helpful advice. I thank participants at theUIUC Research Seminar, Midwest Economics Association Meeting,Midwest Economic Theory Meeting, and Washington University’sEconomics Graduate Conference for helpful comments. I wouldalso like to thank David Martimort for a discussion of the initialidea of the paper. I gratefully acknowledge the financial supportfrom the UIUC Summer Research Fellowship. All errors are mine.

Appendix A

Proof of Theorem 1. (i) It can be easily verified that the mecha-nismMn

α = qn,α, tn,αi satisfies EXPIC, EXPBB and α-PIR (using thefact that Un,α

i (c) = 0).(ii) Let β = 1 − F(c).

K. Rong / Journal of Mathematical Economics 51 (2014) 187–196 195

Note 1vi≥cni=1 are independent variables and E(

ni=1 1vi≥c

n ) =

β , then we have:

P(qn,α = 1) = P

n

i=11vi≥c

n≥ α

≥ P

β −

ni=1

1vi≥c

n

≤ β − α

≥ 1 − P

β −

ni=1

1vi≥c

n

≥ β − α

≥ 1 − 2 exp−2n(β − α)2 (16)

where the last inequality follows from Hoeffding’s inequality,which says that P(|X − E(X)| ≥ t) ≤ 2 exp(−2nt2) where t > 0is a constant and X =

ni=1 Xin is the average of a sequence of

i.i.d. random variables Xini=1 with Pr(Xi ∈ [0, 1]) = 1 (see, e.g.,

Hoeffding, 1963).Thus, we have 1 − P(qn,α = 1) ≤ 2 exp−2n(β − α)2.

Proof of Lemma 2. 23

Let V = [v, v]n and Vi = [v, v] for any i. We have:

V

qn(v) − Eqn(v) −

i

(qni (vi) − Eqni (vi))

2

dF(v1) · · · F(vn)

=

V[qn(v) − Eqn(v)]2dF(v1) · · · F(vn)

− 2

i

V(qn(v) − Eqn(v))(qni (vi)

− Eqni (vi))dF(v1) · · · F(vn)

+

i

V[qni (vi) − Eqni (vi)]

2dF(v1) · · · F(vn)

=

V[qn(v) − Eqn(v)]2dF(v1) · · · F(vn)

− 2

i

Vi[qni (vi) − Eqni (vi)]

2dF(vi)

+

i

Vi[qni (vi) − Eqni (vi)]

2dF(vi)

= Var(qn(v)) −

i

Var(qni (vi))

= Var(qn(v)) − nVar(qni (vi)) (17)

where the first equality follows from the fact that qn1(v1), . . . ,qnn(vn) are independent, and the second equality follows from thefact that Eqn(v) = Eqni (vi) for any i.

Using equality (17) and the fact thatV [qn(v) − Eqn(v) −

i(qni (vi) − Eqni (vi))]

2dF(v1) · · · F(vn) ≥ 0, we have:

Var(qni (vi)) ≤1nVar(qn(v)). (18)

23 The technique used in the proof also appeared in Becker (2012). In particular,inequality (18) belowmay be regarded as a special case of Proposition 2.1 in Becker(2012).

Notice that 0 ≤ qn(v) ≤ 1 for any n and v ∈ V , so Var(qn(v)) ≤

1. Thus, Var(qni (vi)) = O(1/n).

Proof of Lemma 3. First, by Lemma 1, ifqn, tni

ni=1

satisfies

INTIC, then qni (vi) is nondecreasing on [v, v].For a given ε > 0, using Chebyshev’s inequality, Lemma 2

and the fact that Var(qn(v)) ≤14 (note qn(v) follows a Bernoulli

distribution that can take two values: 0 and 1), we have:

Pqni (vi) − Eqni (vi)

≥ ε

≤Var(qni (vi))

ε2≤

14nε2

.

Fix v ∈ (v, v). If n ≥1

4m(v)ε2, then

Pqni (vi) − Eqni (vi)

> ε + ε0

≤1

4n(ε + ε0)2< m(v) for any ε0 > 0. (19)

Now ifqni (v) − Eqni (vi)

> ε + ε0, then we have either qni (v)−

Eqni (vi) > ε + ε0 or qni (v) − Eqni (vi) < −(ε + ε0). Since qni (vi) is anondecreasing function, we have either qni (vi)−Eqni (vi) > (ε+ε0)

for all vi ≥ v or qni (vi) − Eqni (vi) < −(ε + ε0) for all vi ≤ v. Thus,P(|qni (vi) − Eqni (vi)| > ε + ε0) ≥ min(P(vi ≥ v), P(vi ≤ v)) =

m(v) which is a contradiction with (19). As a result, we must haveqni (v) − Eqni (vi) ≤ ε + ε0 for all n ≥

14m(v)ε2

. Since ε0 > 0 is

arbitrary, thenqni (v) − Eqni (vi)

≤ ε for all n ≥1

4m(v)ε2. Hence, for

any given N , we haveqni (v) − Eqni (vi)

≤1

2√

m(v)Nfor all n ≥ N .

This implies thatqni (v) − Eqni (vi)

≤1

2√

m(v)nfor any given n.

Proof of Lemma 4. We first show that there is some vi ∈ [v, v]

such that Un(vi) ≥ 0. Suppose this is not true, then we must haveUn(vi) < 0 for all vi ∈ [v, v]. This implies that rn(v) = 0 for anyv ∈ [v, v]

n. Since rn(v) = 0 < α, we must have qn(v) = 0 andtni (v) = 0 for all i and all v ∈ [v, v]

n by α-PIR. This implies thatUn(vi) = 0 for all vi ∈ [v, v]. Contradiction!

We thus have shown that there is some vi ∈ [v, v] such thatUn(vi) ≥ 0. Now, by the definition of vn, we must have vn

∈ [v, v]

and Un(vn) ≥ 0.We next show that tni (vi) is bounded. On the one hand, we

have tni (vi) = viqni (vi) − vivn q

ni (s)ds − Un(vn) ≤ v, where the

equality follows from the fact thatqn, tni

ni=1

satisfies INTIC and

the inequality follows from the facts that Un(vn) ≥ 0 and qni (vi) ≥

0. On the other hand, by EXABB, we have Etni (vi) ≥ cEqni (vi). Thisimplies that E[viqni (vi)] − E[

vivn q

ni (s)ds] − Un(vn) ≥ cEqni (vi), i.e.,

Un(vn) ≤ E[viqni (vi)] − E[ vivn q

ni (s)ds] − cEqni (vi) ≤ v. This latter

inequality and the fact that vivn q

ni (s)ds ≤

vivn q

ni (vi)ds (because qni is

a nondecreasing function by INTIC) imply that tni (vi) = viqni (vi) − vivn q

ni (s)ds−Un(vn) ≥ viqni (vi)−

vivn q

ni (vi)ds−Un(vn) = viqni (vi)−

(vi − vn)qni (vi) − Un(vn) = vnqni (vi) − Un(vn) ≥ −v (the lastinequality is because vn

∈ [v, v] ⊂ R+). We thus have shown thattni (vi) ≤ v for any vi ∈ [v, v].

Next, we prove the lemma. Let ϵ > 0 be such that P(vi ≤

v + ϵ) = P(vi ≥ v + ϵ). For any ϵ ∈ (0, ϵ), define µ(ϵ) asthe unique value such that P(vi ≥ v − µ(ϵ)) = P(vi ≤ v + ϵ).Define h(ϵ) = P(vi ≤ v + ϵ). Notice that as ϵ → 0, we haveµ(ϵ) → 0 and h(ϵ) → 0. Fix v ∈ (v + ϵ, v − µ(ϵ)). For anyvi ∈ (v + ϵ, v − µ(ϵ)), we have tni (vi) ≤ vi(qni (vi) − qni (v)) +

tni (v) ≤ v 1√h(ϵ)n

+ tni (v), where the first inequality follows fromthe incentive compatibility constraint and the second inequalityfollows from Lemma 3. Thus, Etni (vi) =

(v+ϵ,v−µ(ϵ))

tni (vi)dFi(vi)+

196 K. Rong / Journal of Mathematical Economics 51 (2014) 187–196

[v,v+ϵ]

tni (vi)dFi(vi) +[v−µ(ϵ),v]

tni (vi)dFi(vi) ≤ v 1√h(ϵ)n

+ tni (v) +

2vh(ϵ). That is, tni (v) − Etni (vi) ≥ −v 1√h(ϵ)n

− 2vh(ϵ). Similarly,

it can be shown that tni (v) − Etni (vi) ≤ v 1√h(ϵ)n

+ 2vh(ϵ). Thus,tni (v) − Etni (vi) ≤ v 1

√h(ϵ)n

+2vh(ϵ) for any v ∈ (v+ϵ, v−µ(ϵ)).This implies that for any v ∈ (v, v) and for any 0 < η < m(v) =

minP(vi ≤ v), P(vi ≥ v), we havetni (v) − Etni (vi)

≤ v 1√

ηn +

2vη.

Appendix B

On convergence rates of expected provision level and per capitaexpected payment towards zero in Al-Najjar and Smorodinsky(2000):

In Al-Najjar and Smorodinsky (2000), the public good provisionmechanism is denoted by (δ, c), where c = (c1, . . . , cN) and N isthe number of agents. Note that δ : Ω → 0, 1 is the provisionfunction and ci : Ω → R is the payment function for agent i. Ω isthe underlying type space of all agents.

Al-Najjar and Smorodinsky (2000) assume that the distributionof any agent’s valuation is continuous on (0, t], and is discrete at 0with the probability at 0 being greater than ϵ for some ϵ > 0 for allagents.24 The proposition on p. 331 in Al-Najjar and Smorodinsky(2000) says that supδ,c Eδ ≤

t+β

[Rη,N +η] for any 0 < η ≤ ϵ, whereRη,N =

1√

ηπ1

√N

and β is a constant. For simplicity, we drop theconstraint η ≤ ϵ. It should be noted that imposing the constraintη ≤ ϵ will not change the result below. Now, define:

η∗(N) = argminη>0

t+

β[Rη,N + η].

Then, the best bound for supδ,c Eδ is t+β

[Rη∗(N),N + η∗(N)].The first-order condition of the above minimization problem canbe written as −

12η

−32 1

√π

√N

+ 1 = 0 (it can be verified thatthe second-order condition for the problem is satisfied). That is,

24 Al-Najjar and Smorodinsky (2000) also considered the case where thedistribution of agents’ valuations is fully discrete and in that case, the convergencerate of expectedprovision towards zero is indeedN−

12 (see the corollary inAl-Najjar

and Smorodinsky, 2000).

η∗(N) =1

(4πN)13. Thus,wehave supδ,c Eδ ≤

t+β

[Rη∗(N),N+η∗(N)] =

t+β

(416 + 4−

12 )π−

13 1

N13

= O(N−13 ).

Thus, the convergence rate of supδ,c Eδ towards zero is on

the order of N−13 , not N−

12 as stated on p. 331 of Al-Najjar and

Smorodinsky (2000). That is, the probability of the public goodbeing provided converges to zero at a rate of N−

13 .

Now for the bound on expected payments, from p. 342of Al-Najjar and Smorodinsky (2000), we have that E(cn) ≤

t+[Vn(δ, η) + η] for any 0 < η ≤ ϵ. Thus,

EcnN ≤ t+[Rη,N + η] ≤

t+[1

√ηπ

1√N

+η] and

Ecn ≤ t+[1

√ηπ

√N+ηN] for any 0 < η ≤ ϵ.

By a similar proof, we have

EcnN ≤ O(N−

13 ). That is, the per capita

expected payment converges to zero at a speed not slower thanN−

13 . As a result, the expected aggregated payment is O(N

23 ).

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