The Political Economy of

Voluntary Public Service

by

Arup Bose*, Debashis Pal**, and David E. M. Sappington***

Abstract

We characterize the voluntary public service policy that minimizes the expected cost ofdelivering a public service (e.g., jury or military service). We then examine whether amajority rule voting procedure will implement the voluntary public service policy (VPS)whenever it entails lower expected cost than mandatory public service (MPS). We �nd thatmajority rule often favors MPS in the sense that majority rule implements MPS when VPSwould secure the requisite public service at lower expected cost.

Keywords: voluntary public service, favoritism under majority rule, political economy.

October 2019

* Theoretical Statistics and Mathematics Unit, Indian Statistical Institute,Kolkata 700108, India (bosearu@gmail.com).

** Department of Economics, University of Cincinnati, Cincinnati, Ohio 45221 USA(debashis.pal@uc.edu).

*** Department of Economics, University of Florida, PO Box 117140, Gainesville, FL 32611USA (sapping@u�.edu).

We are particularly grateful to the associate editor and two anonymous referees for veryhelpful comments and suggestions. We also thank Felix Chang, Richard Romano, HuseyinYildirim, and seminar participants at the University of Florida, the University of Tokyo, andthe University of Zurich for helpful observations.

1 Introduction

Societies often require citizens to perform public service activities (e.g., jury service or

military service). In particular, citizens typically are not permitted to pay a fee to secure an

exemption from performing these activities. This is the case even though such an exemption

policy could admit Pareto gains. Speci�cally, an individual for whom the public service is

particularly onerous might be willing to pay for an exemption from the service more than

another individual would require in order to willingly perform the service.

Many reasons have been o¤ered for why an individual�s ability to purchase an exemption

from public service might be limited in practice. For instance, such ability might unfairly

favor relatively wealthy individuals. Furthermore, a public service like jury service might

play a valuable role in informing citizens about the merits of the jury system. Mandatory jury

service might also help to ensure a �trial by peers�or increase the likelihood that the most

capable jurors actually perform jury service (e.g., Bowles, 1980; Hans, 2008; Department of

Justice, Government of Western Australia, 2019).

We add to this list of potential explanations for the prevalence of mandated public service

another, seemingly more subtle, explanation. We demonstrate that a majority voting proce-

dure can systematically favor mandated public service over voluntary public service policies

that allow individuals to pay a fee to avoid performing the service. This favoritism can arise

even when none of the considerations identi�ed above are present, so the only costs associ-

ated with a public service policy are the administrative costs of implementing the policy and

the direct personal costs that individuals incur when they perform the service.

For concreteness, suppose the public service in question is jury service and a society

must choose between a voluntary jury service policy (VJS) and a mandatory jury service

policy (MJS). We specify conditions under which majority voting favors MJS in the sense

that a majority of citizens will vote for MJS even though expected social welfare would be

higher under VJS. We consider self-�nancing VJS policies that specify: (i) an exemption

fee that any individual can pay to avoid jury service; and (ii) a payment that is made to

1

any individual who voluntarily performs jury service. All jury service payments and policy

administrative and implementation costs are �nanced by the stipulated exemption fee under

self-�nancing VJS policies.

We demonstrate that if VJS and MJS entail the same administrative costs, an exemption

fee and jury service payment can be designed to ensure that all individuals prefer VJS to

MJS. Consequently, all individuals will vote for VJS rather than MJS, so a majority rule

voting procedure (�majority rule�) always implements the jury service policy (VJS) that

maximizes expected welfare. (Social welfare is the sum of the welfare levels achieved by each

member of society.)

In practice, VJS may be more costly to implement than MJS, in part because VJS

entails collecting and processing exemption fees and delivering jury service payments. We

show that VJS cannot secure Pareto gains when it is more costly to administer than MJS.

Consequently, as long as the additional administrative cost of VJS is not too pronounced,

some individuals will prefer an optimal (i.e., welfare-maximizing) VJS policy to MJS and

others will prefer MJS to VJS. The individuals who prefer VJS are those with the highest

and the lowest personal costs (c) of performing jury service. Individuals with the highest c�s

gain when they are permitted to avoid jury service by paying the exemption fee. Individuals

with the lowest c�s gain when they are paid to perform jury service. In contrast, individuals

with moderate c�s will prefer MJS to VJS because such c�s will both exceed the optimal jury

service payment and be exceeded by the optimal exemption fee.1

Even when VJS cannot secure Pareto gains, majority rule does not always favor MJS. To

illustrate, we show that majority rule always implements the welfare-maximizing form of jury

service when the distribution of c in the population is uniform. In this case, the expected

1Epple and Romano (1996) identify corresponding equilibrium preferences in a setting where majority ruledetermines the quality of a public good (e.g., public education) that will be implemented and �nanced witha proportional income tax. In their model, individuals with moderate income favor a relatively high levelof quality for the public good. In contrast, individuals with the lowest and the highest incomes prefer lowerlevels of quality. The preference of low-income individuals re�ects their limited wealth. The preference ofhigh-income individuals re�ects in part their equilibrium consumption of a substitute good (e.g., privateeducation).

2

personal gains from VJS exactly o¤set the corresponding losses both for individuals who elect

to perform jury service and for those who choose not to perform jury service. Consequently,

majority rule implements VJS if and only if VJS increases aggregate expected welfare in this

special case.

More generally, majority rule often favors MJS. It does so, for instance, when the number

of individuals eligible to perform jury service (N) is su¢ ciently large relative to the number of

trials (T ) and when individuals with the lowest c�s are relatively common in the population

(e.g., when the mean value of c exceeds the median value of c). In this case, nearly all

individuals choose not to perform jury service under VJS, so the personal gain from VJS

increases with c for nearly all c realizations. Under such circumstances, when the individuals

who prefer MJS (primarily those with the lowest c realizations) are relatively common in the

population, a majority will vote for MJS even when expected welfare would be higher under

VJS.

We show that majority rule often continues to favor MJS when T=N assumes larger

values and when c has standard symmetric, bell-shaped densities. This outcome re�ects in

part the aforementioned fact that the individuals who bene�t the most from VJS are those

with the highest and the lowest c realizations. Under many relevant conditions, majority

rule does not adequately capture the intensity of these individuals�preferences for VJS, so

a majority will vote for MJS even when expected welfare would be higher under VJS.2

We cannot claim that the identi�ed propensity for majority rule to favor mandatory

public service policies is the primary reason for the limited implementation of voluntary

public service policies in practice. Rather, the identi�ed propensity may best be viewed as

a heretofore under-appreciated additional potential explanation for the limited incidence of

voluntary public service policies in practice.

2Mulligan et al. (2004), among others, observe that majority voting often fails to accurately re�ect the inten-sities of voters�preferences, and so may systematically favor some policies (or groups of voters) over others.Other studies (e.g., Mulligan, 2015; Koch and Birchenall, 2016) demonstrate that voluntary service oftenis more e¢ cient than mandatory service. Our analysis complements these studies by idetifying conditionsunder which majority rule favors mandatory service even though voluntary service is more e¢ cient.

3

The ensuing discussion focuses on jury service, for concreteness. However, the analysis

is relevant in other settings where a jurisdiction decides whether to make the performance

of public service mandatory or voluntary. Many countries have wrestled with the question

of whether military service should be mandatory or voluntary, and some countries have

permitted their citizens to pay a fee to avoid military service (Ross 1994; Warner and Asch,

2001; Perri, 2013; Asal et al., 2017). Alternatively, the public service might entail the

provision of health care services in under-served regions of a country. In many countries,

newly-trained medical professionals are required to practice their specialty for a speci�ed

period of time in regions where relatively few established professionals choose to serve. In

some of these countries (e.g., Bolivia, Ethiopia, Ghana, India, Indonesia, Peru, South Africa,

and Thailand), a¤ected individuals can pay a fee to avoid this service. To illustrate, the fee

in the Meghalaya and Tamil Nadu States in India is 1 million rupees, which is approximately

21; 000 U.S. dollars (Frehywot et al., 2010).

Our analysis of the choice between mandatory and voluntary public service proceeds as

follows. Section 2 explains our formulation of MJS and VJS. Section 3 examines the gains

and losses that individuals experience when MJS is replaced by a welfare-maximizing VJS

policy. The �ndings in Section 3 are employed in Section 4 to identify conditions under

which a majority rule voting procedure favors MJS. Section 5 considers a modi�ed form of

VJS in which requests for exemption from jury service are not always granted. Section 6

concludes and suggests directions for future research. The proofs of all formal conclusions

are presented in the Appendix.

2 Mandatory and Voluntary Jury Service Policies

We consider a setting in which N individuals are eligible to serve as jurors on T � 1

trials. For simplicity, each trial is assumed to require a single juror. (If each trial requires r

jurors, then the ensuing analysis remains valid if T is replaced by eT = r T .) Each individualhas quasi-linear utility. The N > T individuals di¤er in their personal cost (c 2 [ c ; c ]) of

performing jury service. This cost might re�ect a �nancial opportunity cost of performing4

jury service, for example. c can be negative, re�ecting the possibility that some individuals

might enjoy serving as jurors or derive personal pleasure from performing what they regard

to be their civic duty. We model each individual�s personal cost of performing jury service

(c) as an independent draw of a random variable with distribution function G(c) and density

function g(c).

Under mandatory jury service (MJS), each of the N individuals must serve on a jury

when called upon to do so. Jurors are summoned randomly, so the probability that an

individual performs jury service under MJS is TN. The compensation paid to an individual

who performs jury service under MJS is normalized to 0. Consequently, the expected welfare

of an individual with personal cost c is WM(c) =TN[�c ] under MJS. The corresponding

expected social welfare is:

N

cZc

WM(c) dG(c) = � T ce, where ce �cZc

c dG(c) . (1)

Under voluntary jury service (VJS), each of the N potential jurors can either opt in or

opt out of jury service. An individual who opts in agrees to be called to perform jury service.

If Ni 2 [T;N ] individuals opt in under VJS, the probability that any particular one of these

individuals is called to perform jury service is TNi. An individual who performs jury service

under VJS receives the �xed payment w > 0. An individual who opts in but is not called to

perform jury service receives no payment (and makes no payment). Therefore, the welfare

of an individual with personal cost c who opts in under VJS is Wi(c) =TNi[w � c ].

An individual who opts out under VJS makes payment F > 0 and is thereby exempted

from jury service. The welfare of such an individual is Wo(c) = �F . An individual with

cost c prefers to opt in rather than opt out under VJS if:

Wi(c ) > Wo(c ) , T

Ni[w � c ] > �F . (2)

bc will denote the cost realization for which an individual is indi¤erent between opting in andopting out under VJS. (For expositional ease, we assume that an individual opts in when he

is indi¤erent between opting in and opting out.) Expression (2) implies that if bc 2 ( c ; c ),5

then bc is de�ned by:Wi(bc ) = Wo(bc ) , bc = w +

F NiT

. (3)

Expressions (2) and (3) imply that individuals with c 2 [ c ; bc ] opt in and individuals withc 2 (bc ; c ] opt out under VJS. Thus, as might be expected, the individuals who �nd juryservice to be relatively onerous are the ones who opt out.

A � 0 will denote the additional social cost of implementing VJS rather than MJS.

This cost includes the additional administrative cost associated with contacting citizens to

determine which ones wish to opt out, collecting and processing payments from those who

do opt out, and delivering payments to individuals who perform jury service. We consider

VJS policies that are self-�nancing in the sense that the sum of the incremental cost A and

payments to jurors (T w) is equal to the expected fees paid by individuals who are exempted

from jury service. These expected fees are the product of the fee (F ), the number of eligible

individuals (N), and the fraction of these individuals who opt out (1�G(bc ) ). Consequently,the �nancing constraint under VJS is:

T w + A = [ 1�G(bc ) ]N F . (4)

We also consider VJS policies that induce an adequate number of individuals to volunteer

to perform jury service. Speci�cally, we require a VJS policy to satisfy the following adequate

jury pool constraint, which states that at least T individuals are expected to opt in:

N G(bc ) � T . (5)

An optimal VJS policy is one that maximizes expected social welfare while satisfying the

�nancing and adequate jury pool constraints. Expected (social) welfare under VJS is:

N

bcZc

T

Ni[w � c ] dG(c)�N [ 1�G(bc ) ]F � A . (6)

6

3 Properties of an Optimal VJS

We now identify the properties of an optimal VJS policy. Doing so allows us to determine

which individuals bene�t and which individuals are harmed when VJS replaces MJS. We also

determine the rate at which an individual�s gain (or loss) from VJS varies with c. We employ

these �ndings in Section 4 to demonstrate that majority rule often favors MJS and to explain

why this is the case. Throughout the ensuing analysis, we assume A � 0 is su¢ ciently small

that w > 0 under VJS. Equation (4) implies this will be the case if A < [ 1�G(bc ) ]N F .Unless otherwise noted, the ensuing analysis considers the case where A > 0 is su¢ ciently

small that some individuals bene�t when an optimal VJS replaces MJS.

To begin, note that an optimal VJS policy allows as many high-cost individuals as possible

to opt out of jury service. Therefore, bc is set to ensure there are just enough jurors to coverthe number of trials, in expectation, i.e., G(bc ) = T

N. (Bose et al. (2019) provide a formal

proof of this conclusion.) Consequently, as the number of potential jurors increases (holding

constant the number of trials), a smaller fraction of potential jurors opt in and actually

perform jury service under an optimal VJS policy, i.e., @ bc@N

< 0 and bc ! c as T=N ! 0.

We now identify the individuals who gain and those who lose when an optimal VJS policy

replaces MJS. As explained further below, some individuals will experience a reduction

in welfare if VJS replaces MJS whenever VJS is relatively costly (i.e., whenever A > 0).

However, two groups of individuals bene�t in this case. First, individuals with the highest

personal costs of performing jury service gain when VJS allows them to pay F to avoid

jury service. Second, individuals with the lowest personal costs gain when they are paid w

to perform jury service, which is not particularly onerous for them. The individuals whose

welfare declines when VJS replaces MJS are those with intermediate c�s, as Lemma 1 reports.

Lemma 1. When an optimal VJS replaces MJS: (i) welfare increases for individuals with

the lowest c�s (c 2 [ c ; c1 ] where c1 � bc � AN �T ) and the highest c�s (c 2 [ c2; c ] where

c2 � bc + AT); whereas (ii) welfare declines for individuals with intermediate c�s (c 2 (c1; c2)).

7

Figure 1 illustrates how an individual�s net gain from VJS varies with his personal

cost of performing jury service (c). This net gain is WV (c) � WM(c), where WV (c) �

max fWi(c);Wo(c) g.[Figure 1 here ]

For individuals who opt in under VJS (i.e., those with c 2 [ c ;bc )), the net gain from VJSdeclines with c. This is the case because jury service becomes more onerous as c increases

and the probability of performing jury service after opting in under VJS is higher than under

MJS (i.e., 1 > TN). For individuals who opt out under VJS (i.e., those with c 2 (bc ; c ]), the

net gain from VJS increases with c because the personal cost that is avoided by opting out

of jury service increases with c. Lemma 2 records these conclusions and thereby identi�es

the slopes of the two linear segments in Figure 1.

Lemma 2. Under an optimal VJS, the rate at which an individual�s expected increase in

welfare from VJS (relative to MJS) varies with c is:

W 0�i(c) � W 0

i (c)�W 0M(c) = � N � T

N< 0 for c 2 [ c ;bc ) ; (7)

W 0�o(c) � W 0

o(c)�W 0M(c) =

T

N> 0 for c 2 (bc ; c ] . (8)

Corollary 1. jW 0�i(c) j R jW 0

�o(c) j , N � TN

R T

N, N R 2T . (9)

Corollary 1 indicates that the declining line segment in Figure 1 is more steeply sloped

than the increasing segment when N > 2T , in which case the probability of performing jury

service under MJS ( TN) is less than 1

2. Consequently, due to the relatively large increase in

the probability of performing jury service for an individual who opts in under VJS, his net

bene�t from VJS declines relatively rapidly with c.

Before proceeding to demonstrate that majority rule often favors MJS, we consider as a

benchmark the special case in which VJS is no more costly to implement than MJS.

8

Lemma 3. If A = 0, then VJS can be designed to ensure that every individual secures at

least the level of expected welfare he secures under MJS, and that nearly all individuals secure

strictly higher levels of expected welfare.

VJS can secure the welfare gains identi�ed in Lemma 3 by setting F = TNbc and w =�

N �TN

� bc, where G(bc ) = TN. This policy e¤ectively charges T

Nbc for an exemption from jury

service and sets the payment for performing jury service to satisfy the �nancing constraint (in

expression (4)). The identi�ed values of w and F leave the individual with c = bc indi¤erentbetween VJS and MJS. They also leave him indi¤erent between opting in and opting out

under VJS.3 Individuals with c > bc secure strict gains from VJS because these gains increasewith c. (Recall expression (8).) Individuals with c < bc also secure strict gains from VJS

because these gains increase as c declines (recall expression (7)) due to the higher probability

of performing jury service after opting in under VJS.

When A > 0, VJS can still enhance welfare by reducing the expected cost at which the

requisite jury service is performed. However, the limited set of policy instruments under

VJS (i.e., w and F ) does not allow compensation for performing jury service or payments

for avoiding jury service to �nely match the associated individual costs and bene�ts. Con-

sequently, as Lemma 4 reports, whenever VJS is more costly to implement than MJS, the

cost savings from the more e¢ cient delivery of jury services under VJS cannot be allocated

to generate Pareto gains.

Lemma 4. Suppose A > 0. Then a VJS policy that secures a strict increase in expected

welfare for some individuals (relative to MJS) necessarily reduces the expected welfare of

some other individuals.

3Because the adequate jury pool constraint binds under an optimal VJS policy, an individual who opts in iscertain to serve and an individual who opts out is certain not to serve. Therefore, when the individual withc = bc opts in under the identi�ed VJS policy, his expected welfare is w � bc = � N �T

N

� bc � bc = � TN bc.

This is precisely the expected welfare he secures if he opts out under VJS (�F = � TN bc ) and his expected

welfare under MJS.

9

4 Majority Voting for Jury Policy

Having determined how individual expected welfare changes when an optimal (i.e., welfare-

maximizing) VJS policy replaces MJS, we now examine when majority rule will favor MJS

by implementing MJS even though aggregate expected welfare would be higher under VJS.

We begin by considering the benchmark setting in which VJS and MJS are equally costly

to implement. Proposition 1 reports that majority rule always implements the welfare-

maximizing jury service policy in this setting because VJS increases the welfare of (nearly)

all individuals. (Recall Lemma 3.) Consequently, (nearly) all individuals will vote for the

policy that ensures the highest level of expected welfare (i.e., VJS).

Proposition 1. Suppose A = 0. Then all individuals (weakly) prefer an optimal VJS policy

to MJS, and majority rule always implements the optimal VJS policy.

When VJS is more costly to implement than MJS (so A > 0), VJS cannot be designed

to ensure that all individuals prefer VJS to MJS. (Recall Lemma 4.) When A is large, a

relatively large number of individuals prefer MJS to VJS because VJS entails a relatively

small w and a relatively large F to cover the high cost of implementing VJS. However, when

A is small, a relatively large number of individuals may prefer VJS to MJS because VJS

secures the requisite jury service at lower personal cost for jurors.

To assess whether majority rule ensures the adoption of the jury service policy that

maximizes expected welfare, let AM denote the largest value of A for which a majority of

individuals prefer (an optimally designed) VJS to MJS. Also let AW denote the largest value

of A for which expected welfare is higher under VJS than under MJS. If AW > AM as

illustrated in Figure 2A, then for all A 2 (AM ; AW ), a majority will vote for MJS (because

A > AM) even though welfare is higher under VJS than under MJS (because A < AW ).

In this sense, majority voting favors MJS. If AW < AM as depicted in Figure 2B, then for

all A 2 (AW ; AM), a majority will vote for VJS (because A < AM) even though welfare is

higher under MJS than under VJS (because A > AW ). In this sense, majority rule favors

10

VJS.[Figure 2 here ]

Proposition 2 examines whether majority rule favors MJS or VJS as the number of

potential jurors becomes in�nitely large relative to the number of trials. The answer varies

with the relative magnitudes of the mean (ce) and median (cd) values of c.

Proposition 2. In the limit as T=N ! 0, majority rule favors MJS when ce > cd, favors

VJS when ce < cd, and favors neither MJS nor VJS when ce = cd (i.e., AM T AW ,

ce S cd).

Proposition 2 re�ects the fact that as T=N ! 0, nearly all individuals choose not to

perform jury service under an optimal VJS (i.e., bc ! c ). The primary di¤erence between

VJS and MJS in this case is that those who opt out make a small payment in return for the

slightly reduced probability of being called to perform jury service. Because the bene�t of a

reduced likelihood of performing jury service increases with c, individuals with the highest

c�s prefer VJS to MJS whereas nearly all individuals with the lower c�s prefer MJS to VJS.

Consequently, when individuals with the lowest c�s are relatively common in the population

(so cd < ce), majority rule may implement MJS when aggregate expected welfare would be

higher under VJS. Conversely, when individuals with the highest c�s are relatively common

in the population (so cd > ce), majority rule may implement VJS when aggregate expected

welfare would be higher under MJS.

A di¤erent conclusion emerges when the number of potential jurors approaches the num-

ber of trials.

Proposition 3. In the limit as T=N ! 1, majority rule favors VJS when ce > cd, favors

MJS when ce < cd, and favors neither MJS nor VJS when ce = cd (i.e., AM Q AW ,

ce S cd ).

Proposition 3 re�ects the fact that as T=N ! 1, nearly all individuals must opt in under

VJS to ensure that a juror is available for every trial. The primary di¤erence between VJS11

and MJS in this case is that those who opt in receive a small payment in return for the

slight increase in their probability of performing jury service. Because jury service becomes

more onerous as c increases, individuals with the smallest c�s prefer VJS to MJS whereas

nearly all individuals with the larger c�s prefer MJS to VJS. Consequently, when individuals

with the lowest c�s are relatively common in the population (so cd < ce), majority rule may

implement VJS when aggregate expected welfare would be higher under MJS. Conversely,

when individuals with the highest c�s are relatively common in the population (so cd > ce),

majority rule may implement MJS when aggregate expected welfare would be higher under

VJS.

Although majority rule can favor either MJS or VJS as T=N ! 1, the range of A values

for which the welfare-maximizing policy is not implemented is limited. This is the case

because each individual�s welfare changes very little when VJS replaces MJS. Consequently,

AW ! 0 and AM ! 0 as T=N ! 1.

In practice, T=N is often strictly positive and well below 1. Therefore, it is important to

determine whether majority rule favors MJS or VJS under these conditions. Propositions 2

and 3 suggest that majority rule may not favor either MJS or VJS when the distribution of

c is symmetric, so ce = cd. Proposition 4 con�rms that this outcome prevails when g(c) is

the special symmetric density for which all c realizations are equally likely.

Proposition 4. Majority rule favors neither MJS nor VJS (so AM = AW ) when g(c) is

the uniform density.

Majority rule favors neither MJS nor VJS when g(c) is the uniform density because when

A = AM in this case, the expected personal gains from VJS exactly o¤set the corresponding

losses, both for individuals who opt in and for those who opt out. Consequently, majority

rule implements VJS if and only if it increases aggregate expected welfare (so AM = AW ).

Two initial observations help to explain this conclusion. First, recall from Lemma 1

that under an optimal VJS, the individuals who prefer VJS to MJS are those with c�s in

12

[ c ; c1 ] [ [ c2; c ], whereas the individuals who prefer MJS to VJS are those with c�s in

(c1; c2). Therefore, when A = AM , the two regions [ c ; c1 ][ [ c2; c ] and (c1; c2) must contain

the same (expected) number of individuals. Second, for any bc 2 ( c ; c ), the two regions�c ; 1

2( c+ bc ) � [ � 1

2(bc + c ) ; c � and � 1

2( c+ bc ) ; 1

2(bc + c ) � contain the same (expected)

number of individuals when g(c) is the uniform density.4 These two observations imply that

c1 =12[ c+ bc ] and c2 = 1

2[bc+ c ] when A = AM and g(c) is the uniform density.

The personal gains from VJS decline with c at the constant rate W 0�i(c) = � N �T

Nfor

all individuals with c 2 [ c ; bc ] because all of these individuals opt in under VJS. (Recallexpression (7).) Consequently, gains and losses from VJS are symmetric around c1, which

(recall) is a cost realization at which the individual is indi¤erent between VJS and MJS. This

symmetry, the fact that c1 is the midpoint of [ c ; bc ], and the fact that g(c) is the uniformdensity imply that the increase in expected welfare associated with VJS on [ c ; c1 ] is equal

to the reduction in expected welfare associated with VJS on [ c1; bc ].Analogous observations imply that the personal gains and losses from VJS are symmetric

around c2. Consequently, when g(c) is the uniform density, the reduction in expected welfare

on [bc; c2 ] when VJS replaces MJS is equal to the corresponding increase in expected welfareon [ c2; c ]. Therefore, the expected welfare gains and losses from VJS (relative to MJS) are

identical when A = AM , which ensures AM = AW (and so majority rule favors neither MJS

nor VJS).5

The o¤setting gains and losses from VJS that underlie Proposition 4 re�ect the equal

likelihood of all c realizations under the uniform g(c) density. More generally, the welfare

gains and losses from VJS typically will not be exactly o¤setting even when the density of c

is linear. Bose et al. (2019) prove that majority rule favors MJS for any �nite T=N < 1 when

4This is the case because when g(c) is the uniform density, the intervals�c ; 12 (c+ bc ) � and � 12 (c+ bc ) ; bc �

contain the same (expected) number of individuals, as do the intervals� bc ; 12 (bc + c ) � and � 12 (bc + c ) ; c �.

5It can be shown that the expected welfare gain under VJS relative to MJS is proportional to AW �A wheng(c) is the uniform density. (See the Observation that follows the proof of Proposition 4 in the Appendix.)Therefore, given the symmetric gains and losses from VJS around c1 and c2 and the equal likelihood of allc realizations, a majority will prefer VJS to MJS if and only if A � AW when g(c) is the uniform density.

13

g(c) is a symmetric, piecewise-linear density function with an inverted-V shape. Proposition

5 provides a corresponding conclusion in a more general setting where g(c) is symmetric

about its mean, non-decreasing below its median, and (for analytic tractability) strictly log

concave.6

Proposition 5. Suppose g(c) is symmetric about its mean, non-decreasing below its median,

and strictly log concave. Then there exists a eN > 2T such that majority rule favors MJS

(i.e., AM < AW ) for all N 2 [ 2T; eN ).The eN identi�ed in Proposition 5 satis�es the following equality:

G(c2( eN))�G(ec1( eN)) = 1

2(10)

where c1(N) � bc � AWN � T , c2(N) � bc+ AW

N, and

ec1(N) � c1(2T )� [ c2(2T )� c2(N) ]. (11)

To understand this eN and the essence of the proof of Proposition 5, observe that majority

rule favors MJS (i.e., AW > AM) if G(c2) � G(c1) > 12when A = AW .7 The proof of

Proposition 5 establishes that under the speci�ed conditions: (i) G(c2) � G(c1) > 12when

A = AW and N = 2T ; (ii) c1 and c2 both decline as N increases (holding T constant);

and (iii) c1 declines more rapidly than c2 declines. Because AW > AM when N = 2T , AW

continues to exceed AM as N increases just above 2T . Equation (10) indicates that eN is

the largest N for which AW > AM for all N 2 [ 2T; eN ) if c1 and c2 both decline by thesame amount as N increases. Because c1 actually declines more rapidly than c2 declines as

N increases, eN understates the upper bound of the set of N�s for which AW > AM .

6g(c) is strictly log concave if log(g(c)) is a strictly concave function of c. If g(c) is strictly log concave, thenG(c) is strictly log concave, so d

dc (g(c)G(c) ) < 0 for all c 2 [ c ; c ]. Many common density functions are strictly

log concave, including the uniform, normal, exponential, and logistic densities. See Bagnoli and Bergstrom(2005) for additional discussion and analysis. Many studies in the public choice literature (e.g., Caplin andNalebu¤, 1988) focus on settings where relevant density functions are log concave.7Recall from Lemma 1 that individuals with c 2 (c1; c2) prefer MJS to VJS. Therefore, if G(c2)�G(c1) > 1

2when A = Aw, more than half of individuals prefer MJS to VJS when the two jury systems secure the sameexpected welfare (i.e., when A = Aw).

14

The eN identi�ed in Proposition 5 often is much larger than 2T , so there is a substan-

tial range of TNvalues for which majority rule favors MJS when g(c) is as speci�ed in the

proposition. To illustrate, suppose g(c) is the Beta density with parameters � and �:

g(c) =c��1 [ 1� c ]��1

B(�; �)for c 2 [ 0; 1 ] , where B(�; �) �

Z 1

0

x��1 [ 1� x ]��1 dx.

The Beta density can assume many di¤erent shapes as � and � vary. The density has an

inverted-U shape (and so is concave) when � = � 2 (1; 2 ] and a bell shape when � > 2 and

� > 2. As Figure 3 illustrates, the Beta density is symmetric about its mean ( ��+�

) when

� = �, skewed to the right (with ce > cd) when � > � > 1, and skewed to the left (with

ce < cd) when � > � > 1. (Recall that ce denotes the mean c and cd denotes the median c.)

A symmetric Beta density becomes more concentrated about its mean as � = � increases.

The Beta density is log concave if � � 1 and � � 1 (Bagnoli and Bergstrom, 2005).

[Figure 3 here ]

Table 1 records the value of eN identi�ed in Proposition 5 (as a multiple of T ) for selected

values of � = �.

� = � eN =T2 12

5 64

10 310

20 1; 690

Table 1. Values of eN for Symmetric Beta Densities.

Table 1 indicates that the set of N�s for which majority rule is certain to favor MJS

can be particularly extensive when the extreme c realizations are relatively unlikely (i.e., for

symmetric Beta densities with relatively large values of �). In this case, c1 and c2 decline

relatively slowly as N increases, so G(c2) � G(c1) continues to exceed 12as N increases

substantially above 2T .15

Conditions more general than those speci�ed in Proposition 5 under which majority rule

favors MJS are di¢ cult to characterize analytically. This is the case because when N > T

is �nite, a non-trivial set of individuals with the smallest c realizations prefer VJS to MJS,

as do individuals with the highest c realizations. Consequently, the properties of the entire

distribution of c, rather than just the mean and median of the distribution, can determine

whether majority rule favors MJS or OJS.

Although general analytic conclusions are elusive, numerical solutions reveal that: (i)

majority rule tends to favor MJS systematically when g(c) is a symmetric, log concave

function; and (ii) majority rule often favors MJS when g(c) is not symmetric. To illustrate

the �rst of these two conclusions, consider the symmetric Beta density with � = � = 2, so

ce = cd = 12. (This density is illustrated in Figure 3.) Table 2 identi�es for the setting where

A = AM : (i) the cost realization, bc, at which the individual is indi¤erent between opting inand opting out under OJS; and (ii) the boundaries of the regions of c realizations in which

individuals prefer OJS to MJS ( [ c ; c1 ][ [ c2; c ] ) and the region in which individuals prefer

MJS to OJS ((c1; c2) ). The Table also identi�es the values of AM and AW for a range of

values of T=N 2 (0; 1). Table 2 reports that majority rule systematically favors MJS in the

identi�ed setting (i.e., AM < AW for all T=N 2 (0; 1)).8

8Mize et al. (2007) report there were 148; 558 state jury trials and 5; 463 federal jury trials in the U.S. in2006. Table 2 takes the number of trials to be 1; 000, for expositional ease. If the values of N and T inTable 2 were to be multiplied by k > 0, the values of Am and Aw would also be multiplied by k. The entriesin the other columns in the table would not change.

16

N bc (AM) c1 (AM) c2 (AM) AM AWAMAW

1; 001 0:982 0:499 0:982 0:482 0:488 0:9891; 010 0:941 0:494 0:946 4:470 4:611 0:9691; 100 0:814 0:459 0:849 35:486 37:820 0:9382; 000 0:500 0:326 0:674 173:648 187:500 0:9262; 500 0:433 0:293 0:642 209:364 226:017 0:9265; 000 0:287 0:214 0:579 292:122 314:238 0:93010; 000 0:196 0:157 0:544 348:495 371:916 0:93750; 000 0:084 0:075 0:511 426:743 444:391 0:960100; 000 0:059 0:054 0:506 446:799 460:932 0:969

1; 000; 000 0:018 0:018 0:501 482:262 487:772 0:989

Table 2. Outcomes for the Beta Density with � = � = 2. [T = 1; 000 ]

Corresponding conclusions arise for other symmetric Beta densities. Numerical solutions

reveal that AM < AW for all T=N < 1, regardless of the value of � = �.9

Majority voting also often favors MJS when g(c) is an asymmetric density with ce > cd.

Although majority rule favors VJS as T=N ! 1 (recall Proposition 3), Table 3 indicates

that majority rule often favors MJS whenever T=N is even slightly less than 1. The table

records outcomes for the setting in which g(c) is the Beta density with parameters � = 2 and

� = 3. This density has a bell shape and is skewed to the right (with ce > cd), as illustrated

in Figure 3. (The mean of this density is 0:4. The median of this density is approximately

0:38573.)

Table 3 indicates that majority rule favors MJS (so AM < AW ) in this setting whenever

T=N < 0:9992 (� 1;0001:000:757

). Numerical solutions demonstrate that similar conclusions arise

for many other Beta densities with ce > cd. (See Bose et al. (2019) for details.)

9This conclusion re�ects numerical solutions that consider all values of � = � 2 ( 1; 25 ] in increments of 1,all values of N=T 2 (1; 2 ] in increments of 0:001, all values of N=T 2 [ 2; 10 ] in increments of 0:01, all valuesof N=T 2 [ 10; 1000 ] in increments of 1, and all values of N=T 2 [ 1; 000; 10; 000 ] in increments of 10.

17

N bc (AM) c1 (AM) c2 (AM) AM AWAMAW

1; 000:01 0:986 0:386 0:986 0:006 0:00600 1:0181; 000:757 0:942 0:385 0:942 0:421 0:42095 1:000

1; 010 0:860 0:381 0:864 4:790 85:962 0:9671; 100 0:691 0:348 0:725 34:243 37:269 0:9192; 000 0:386 0:236 0:535 149:601 165:533 0:9042; 500 0:329 0:211 0:507 177:589 196:258 0:9055; 000 0:212 0:152 0:452 239:872 264:366 0:90710; 000 0:143 0:111 0:423 280:295 307:478 0:91250; 000 0:060 0:053 0:395 334:699 360:327 0:929100; 000 0:042 0:038 0:391 348:566 372:203 0:937

1; 000; 000 0:013 0:013 0:386 373:245 391:337 0:954

Table 3. Outcomes for the Beta Density with � = 2 and � = 3. [T = 1; 000 ]

To understand why majority rule often favors MJS when ce � cd and T=N 2 (0; 1),

recall from Proposition 2 that this favoritism prevails when T=N ! 0. In this case, nearly

all individuals opt out of VJS, so bc ! c and the private gain from VJS increases with c

at a constant rate for nearly all individuals. Now consider the changes that arise as T=N

increases. A non-negligible fraction of individuals now opt in under VJS, so bc increasesabove c and a nontrivial region [ c ; c1 ] arises in which individuals prefer VJS to MJS.

To assess the impact of these changes, suppose A = AM , so one-half of the population

prefers VJS to MJS. When [ c ; c1 ] expands as T=N increases, [ c2; c ] contracts su¢ ciently to

ensure that exactly half of the population continues to prefer VJS to MJS. Let SO denote the

set of individuals who prefer VJS to MJS. The expansion of [ c ; c1 ] adds to SO individuals

whose net gain from VJS is relatively sensitive to c. The contraction of [ c2; c ] removes

from SO individuals whose net gain from VJS is relatively insensitive to c. (Recall from

Corollary 1 that jW 0�i(c) j > jW 0

�o(c) j for all T=N < 12.) When g(c) has an inverted-U

shape or a bell shape, the lowest c realizations are relatively unlikely. Consequently, to

ensure SO includes exactly half of the population as T=N increases, [ c ; c1 ] expands by more

than [ c2; c ] contracts. The net e¤ect is to introduce greater variation in the net gain from

VJS (i.e., greater variation in the intensity of preference for VJS) within SO.

The outcome of majority rule depends only on the number of individuals who prefer VJS

18

to MJS, not the intensity of the preference for VJS. Therefore, majority rule fails to account

fully for the increased intensity of preference for VJS as T=N increases toward 12, which

causes majority rule to favor MJS (i.e., AM < AW ).

Majority rule can favor VJS when ce < cd and T=N < 1. (Recall Proposition 2.) However,

numerical solutions indicate that majority rule favors VJS only when T=N is very close to 0

unless cd is substantially greater than ce. (See Bose et al. (2019) for details.)

Before proceeding to consider modi�ed VJS policies, we brie�y illustrate when majority

rule might implement MJS even though expected welfare would be higher if VJS were im-

plemented. To do so, suppose a summons to perform jury service requires an individual to

miss one day of work, and the corresponding loss of income constitutes the individual�s per-

sonal cost (c) of performing jury service.10 Further suppose the annual income of potential

jurors has a Gamma distribution with parameters � = 2 and � = 310;000

, re�ecting actual

approximations of the U.S. income distribution.11 Social Security Online (2017) reports that

the average individual compensation subject to federal income tax in the U.S. in 2017 was

approximately $48,250. Re�ecting this statistic, we scale income levels to ensure a mean

annual income of $50; 000.12 We further assume there are 250 working days annually, so

the mean daily income is ce = $50;000250

= $200. In addition, we assume the probability of

being summoned to perform jury service under MJS is 0:15. This assumption re�ects Mize

et al. (2007, p. 8)�s �nding that �To secure enough jurors to hear cases, state courts mail an

estimated 31.8 million jury summonses annually to approximately 15 percent of the adult

10Mize et al. (2007, p. 8) observe that even though many individuals are summoned to perform jury serviceeach year, �less than 1 percent of the adult American population�is actually empaneled on a jury. Conse-quently, it may be reasonable to assume that, on average, individuals who are summoned to perform juryservice spend one day in this capacity. The assumption that an individual who is summoned to performjury service forfeits one day�s income may be most relevant for self-employed individuals.

11Salem and Mount (1974, Table II) report that the U.S. income distribution in the 1960s was reasonablywell captured by a Gamma density with parameters of this magnitude. (The Gamma density is g(c) =1

�(�)

���e��cc��1

�for c > 0, where �(�) �

1R0

e�xx��1dx.) Singh and Maddala (1976) report that the U.S.

income distribution generally is better approximated by a Gamma density than by the two other densitiesmost commonly employed for this purpose �the Pareto density and the log normal density.

12The mean of the identi�ed Gamma distribution is �� = 20;000

3 . This mean is multiplied by 7:5(= 50; 000=[ 20; 000=3 ] ) to generate a mean annual income of $50; 000.

19

population.�

In this setting, AMN� $18:91 and AW

N� $23:57, so majority rule will implement MJS

even though VJS would secure a higher level of expected welfare when AN2 ($18:91; $23:57).

Furthermore, under VJS, (only) individuals with annual incomes below $17; 081 volunteer to

perform jury service. When A = AM , individuals who perform jury service are paid $39:17

whereas those who opt out of jury service pay $29:16 to do so.13

The opportunity cost of performing jury service may re�ect many factors other than

daily income, so g(c) may not be readily captured by the Gamma density. To illustrate

the outcomes that arise under more symmetric variation in c, suppose g(c) is a normal

density with mean $200 and variance $100. If N = 200 million and T = 30 million, as

above, majority rule will implement MJS even though VJS would secure a higher level of

expected welfare when AN2 ($1:95; $2:33). Furthermore, individuals with annual incomes

below $47; 408 volunteer to perform jury service under VJS. When A = AM , individuals who

perform jury service are paid $159:24 whereas those who opt out of jury service under VJS

pay $30:39.14

5 More Limited Exemption from Jury Service

We have shown that the individuals with the highest c�s choose not to perform jury service

under a welfare-maximizing VJS policy. Some might view this outcome to be �unfair�if, for

example, high c realizations are correlated with high income levels. This outcome might also

reduce the performance of the jury system if individuals with the highest c�s are the best

jurors. In addition, if individuals with similar c�s are deemed to be the closest peers, then

13If the average opportunity cost declines from $200 to $100 in this setting, majority rule implements MJSwhen VJS would secure a higher level of expected welfare if AN 2 ($9:45; $11:79). Under VJS, individualswith annual incomes below $8; 540 volunteer to perform jury service. When A = Am, individuals whoperform jury service are paid $19:58 whereas those who opt out of jury service pay $14:58 to do so. IfA = 1

2 Am, then individuals who perform jury service are paid $24:31 whereas those who opt out of juryservice pay $9:85.

14If the variance declines from 100 to 50 in this setting, majority rule implements MJS when VJS wouldsecure a higher level of expected welfare if AN 2 ($1:38; $1:65). Under VJS, individuals with annual incomesbelow $48; 167 volunteer to perform jury service. When A = Am, individuals who perform jury service arepaid $162:40 whereas those who opt out of jury service pay $30:28.

20

the identi�ed outcome might be viewed as limiting the likelihood of a �trial by peers� for

individuals with high c�s.

To begin to address these potential concerns, consider a modi�ed VJS policy in which

an individual�s request for exemption from jury service is approved with probability p 2

(0; p ], where p < 1.15 The individual pays the fee F > 0 if (and only if) his request is

approved. If the request is denied, the individual joins those who agree to perform jury

service in the pool of eligible jurors. The expected number of individuals in this pool is

Ni = N [ 1� p ( 1�G(bc ) ) ], where bc 2 [ c ; c ] is the c realization for which the individualis indi¤erent between opting in (i.e., agreeing to perform jury service when called upon to

do so) and opting out (i.e., requesting to be exempted from performing jury service). An

individual in the pool of eligible jurors is paid w > 0 if and only if he is called upon to

perform jury service. Each individual in the pool is summoned to perform jury service with

probability TNi2 (0; 1 ].

When p is su¢ ciently small that the adequate jury pool requirement, Ni � T , is not

constraining,16 p = p under the welfare-maximizing VJS policy. (This conclusion and all

subsequent conclusions in this section are proved in Bose et al. (2019).) Thus, requests

for exemption from jury service are granted with the highest permissible probability. This

outcome ensures that, to the greatest extent possible, jury service is performed by those for

whom it is least costly to deliver. Under this optimal VJS policy, the individuals who prefer

VJS to MJS are those for whom c 2 [ c ; c1 ] [ [ c2; c ], where c1 = bc � AT

Nip [ 1�G(bc ) ]N and

c2 = bc + AT

NipG(bc )N . Individuals with c 2 (c1; c2) prefer MJS to VJS. Furthermore, @ bc@ p < 0

under this policy, so as a request for exemption from jury service becomes less likely to be

granted, fewer individuals request an exemption.

The primary qualitative conclusions drawn above persist under this modi�ed VJS policy.

15A more complete treatment of these concerns would quantify and account explicitly for the social value offairness, jury accuracy, and trial by peers.

16The adequate jury pool requirement is not constraining when p � 1 � TN . This setting is of particular

interest because it allows the welfare-maximizing VJS policy to be determined by factors other than thosethat underlie the welfare-maximizing VJS policy analyzed in Sections 3 and 4.

21

Speci�cally, majority rule favors neither MJS nor this modi�ed VJS policy (so AW = AM)

when g(c) is the uniform density. Furthermore, numerical solutions reveal that majority

rule systematically favors MJS (so AM < AW ) when g(c) is a bell-shaped Beta density with

ce � cd.17

6 Conclusions

We have characterized the voluntary public service policy that maximizes the expected

welfare of individuals who are eligible to perform the service, where an individual�s welfare

is the di¤erence between the payment he receives and the personal cost (c ) he incurs in

performing the service. We have also demonstrated that majority rule often favors mandatory

public service in the sense that mandatory service is implemented under majority rule even

when expected welfare would be higher if voluntary service were implemented. This is the

case, for instance, when the most extreme realizations of c are relatively unlikely, so the

density for c has the common bell shape.

The failure of majority rule to systematically implement the welfare-maximizing public

service policy re�ects in part the limited set of policy instruments under consideration.

Individuals in our model were only permitted to opt in or opt out of the public service in

return for a single associated �xed payment. In principle, voluntary service policies could

admit varying probabilities of being exempted from service, p, in return for payments that

increase with p. Although such policies likely would be more costly to design, implement,

and administer, they have the potential to more closely align private and social bene�ts

under a voluntary service policy, and therefore could increase the propensity for majority

rule to systematically implement the welfare-maximizing public service policy.17This conclusion re�ects numerical solutions that consider all values of � 2 [ 1; 15 ] in increments of 1, allvalues of � 2 [�; 25 ] in increments of 1 (except � = � = 1), and all values of p 2 [ 0:1; 0:9 ] in incrementsof 0:01. Bose et al. (2019) prove analytically that when g(c) is a symmetric, piecewise linear density withan inverted-V shape, majority rule always favors MJS over the modi�ed VJS policy (so Am < Aw). Boseet al. (2019) also show that the extent to which majority rule favors MJS when g(c) is V -shaped canbe mitigated to some extent when p is small. This is the case because as p declines, the welfare gainsand losses from VJS become less sensitive to c (because the individuals with the highest c�s who requestexemption from jury service are not ensured of exemption). Consequently, any de�ciency of majorityvoting in fully re�ecting the intensity of preferences for VJS is ameliorated.

22

A complete assessment of the likelihood that majority rule will implement the welfare-

maximizing public service policy must account for potential divergence between the private

and social bene�ts of performing public service. In practice, the personal cost an indi-

vidual incurs when he performs public service (e.g., his foregone earnings) can di¤er from

the corresponding reduction in social value caused by his diminished participation in other

activities.18

Such a complete assessment must also account for potential variation in individual e¢ cacy

of performing public service. If individuals who �nd public service to be particularly costly

are particularly adept at performing public service, then the welfare-maximizing voluntary

service policy may induce fewer individuals to opt out of performing public service. As the

discussion in Section 5 suggests, majority rule may continue to favor mandatory service in

this setting if increased participation by the individuals best suited to perform public service

is secured by denying some of their requests to opt out of public service.

Additional bene�ts of mandatory and voluntary public service must also be considered

when assessing whether common voting procedures will implement welfare-maximizing poli-

cies. As noted in the Introduction, the additional potential bene�ts of mandatory public

service include its ability to promote equity and fairness and to disseminate valuable in-

formation about the role and merits of public service. Voluntary programs may provide

additional bene�ts if they can: (i) reduce wasteful, costly lobbying to avoid public service;

or (ii) induce governments to manage productive resources more e¢ ciently (Martin, 1972).

Explicit consideration of these extensions of our analysis await further research.

18Bowles (1980, p. 371) notes that the costs of jury service �lie mainly in the value of production lost to theeconomy as a result of the absence of jurors from work.�Koch and Birchenall (2016) identify conditionsunder which social welfare increases when a voluntary military enlistment policy replaces a draft, therebyallowing the most productive individuals to continue to earn high wages and pay high taxes.

23

Appendix. Proofs of Formal Conclusions

Proof of Lemmas 1 and 2. (2) and (3) imply that individuals with c 2 [ c ; bc ] opt inwhereas individuals with c 2 (bc ; c ] opt out under VJS. Also, Wi(c) = w � c (becauseG(bc ) = T

N) and Wo(c) = �F . Therefore:

W�i(c) = Wi(c)�WM(c) = w � c��� T

Nc

�= w �

�N � TN

�c

) W 0�i(c) = � N � T

N< 0 for c 2 [ c ;bc ) ; and

W�o(c) = Wo(c)�WM(c) = �F ��� T

Nc

�= �F + T

Nc

) W 0�o(c) =

T

N> 0 for c 2 (bc ; c ] , (12)

so Lemma 2 holds.

Because G(bc ) = TN, the de�nition of bc implies:

w � bc = �F , F = bc� w . (13)

c1 is the largest realization of c 2 (c ;bc ) for which Wi(c) � WM(c). Therefore, (12)implies:

w � c1 = � T

Nc1 ) c1

�N � TN

�= w1 ) c1 =

�N

N � T

�w . (14)

Because G(bc ) = TNand the �nancing constraint holds:

T w + A = [N � T ]F ) T w + A = [N � T ] [bc� w ]) T w + A = [N � T ]bc�N w + T w ) w =

1

N[ (N � T ) bc� A ] . (15)

The second equality in the �rst line of (15) re�ects (13). (14) and (15) imply:

c1 =

�N

N � T

�1

N[ (N � T ) bc � A ] = bc � A

N � T . (16)

c2 is the smallest realization of c 2 (bc ; c ) for which W0(c) � WM(c). Therefore, (12)implies:

�F = � T

Nc2 ) c2 =

N F

T. (17)

(13) and (15) imply:

F = bc� 1

N[ (N � T )bc � A ] = T

Nbc + A

N. (18)

24

(17) and (18) imply:

c2 =N

T

�T

Nbc + A

N

�= bc + A

T. (19)

Lemma 1 follows from (12), (16), and (19). �

Proof of Lemma 3. As demonstrated in the text, the �nancing and adequate jury poolconstraints are satis�ed as equalities when F = T

Nbc and w = � N �T

N

� bc. Furthermore, (3)implies Wi(bc ) =Wo(bc ) because:

w + FNiT

=

�N � TN

�bc + T

Nbc � TT

�= bc :

Therefore, because c1 = c2 = bc when A = 0, Lemma 1 implies the proof is complete ifWV (bc ) = WM(bc ). This equality holds because:

WV (bc ) = WM(bc ) , w � bc = � T

Nbc

,�N � TN

�bc � bc = � T

Nbc , � T

Nbc = � T

Nbc . �

Proof of Lemma 4. Under an optimal VJS that ensures WV (c) > WM(c) for some c 2[ c ; c ], there exists a bc 2 ( c ; c ) de�ned by:

Wi(bc ) = Wo(bc ) , w � bc = �F , w = bc � F . (20)

Suppose the VJS policy can be designed to ensure WV (c) � WM(c) for all c 2 [ c ; c ].Then it must be the case that:

WV (bc ) � WM(bc ) , w � bc � � T

Nbc , w �

�N � TN

�bc (21)

, bc � F ��N � TN

�bc , T

Nbc � F , N F � T bc . (22)

The �rst equivalence in (22) re�ects (20). Because G(bc ) = TN, the last inequality in (21)

implies:

[ 1�G(bc ) ]N F � w T � [ 1�G(bc ) ]N F � � N � TN

�bc T=

�N � TN

�N F �

�N � TN

�bc T =

�N � TN

�[N F � T bc ] � 0 . (23)

The inequality in (23) re�ects (22). (23) implies that (4) cannot hold for any A > 0.Therefore, it cannot be the case that the VJS policy ensures WV (c) � WM(c) for all c 2[ c ; c ]. �

25

Proof of Proposition 1. The proof follows immediately from the associated discussion inthe text. �

Proof of Proposition 2. The average expected cost that individuals incur under VJS isTN

R bcc c dG(c)

G(bc) +AN. The corresponding average expected cost under MJS is T

Nce. Therefore, the

average expected net gain from VJS is:

T

Nce � T

N

R bccc dG(c)

G(bc) � A

N. (24)

(24) and the de�nition of AW imply:

T

Nce � T

N

R bccc dG(c)

G(bc) � AWN

= 0 ) AWT

= ce �R bccc dG(c)

G(bc ) . (25)

De�ne: c1 (A) = bc� A

N � T and c2 (A) = bc+ AT. (26)

Lemma 1 and the de�nitions of AM and AW imply:

AM T AW , G (c2(AW ))�G (c1(AW )) S 1

2, and (27)

AM Q AW ,Z c2(AW )

c1(AW )

dG(c) R 1

2. (28)

From (25): bc+ AWT

= bc+ ce � R bcc c dG(c)G(bc ) . (29)

L�Hôpital�s Rule implies:

limbc! c

" R bccc dG(c)

G(bc )#= limbc! c

� bc g(bc )g(bc )

�= limbc! c

[ bc ] = c . (30)

Because G(bc ) = TN, (29) and (30) imply that as T=N ! 0:

bc+ AWT

! limbc! c

�bc+ AWT

�= limbc! c

"bc+ ce � R bcc c dG(c)G(bc )

#

= c+ ce � c = ce . (31)

From (25):

bc � AWN � T = bc� � T

N � T

�AWT

= bc� T

N � T

"ce �

R bccc dG(c)

G(bc )#. (32)

26

Because G(bc ) = TN, (30) and (32) imply that as T=N ! 0:

bc � AWN � T ! lim

T=N! 0

�bc� AWN � T

�

= limT=N! 0

"bc� � T

N � T

� ce �

R bccc dG(c)

G(bc )!#

= c� [ 0 ] [ ce � c ] = c . (33)

(27), (31), and (33) imply that when T=N is su¢ ciently small:

AM T AW , G (ce)�G (c ) S 1

2, G (ce) S G

�cd�, ce S cd . �

Proof of Proposition 3. As N ! T , nearly all individuals must opt in under VJS toensure that every trial has a juror. Consequently:bc ! c as N ! T . (34)

(25) and (34) imply that as N ! T :

AWT

= ce �R bccc dG(c)

G (bc) ! ce �R ccc dG(c)

G (c)= ce � c

e

1= 0

) AW ! 0 as N ! T . (35)

(26) and the de�nition of AM imply:

G

�bc+ AMT

��G

�bc� AMN � T

�=1

2. (36)

(34) and (35) imply:

G

�bc+ AMT

�! 1 as N ! T . (37)

(26), (34), and (35) imply:

c2 (AW ) = bc+ AWT

! c as N ! T . (38)

(26), (36), and (37) imply:

G

�bc� AMN � T

�= G (c1(AM)) ! 1

2) c1(AM) ! cd as N ! T . (39)

(26) and (34) imply:

limN!T

c1 (AW ) = c � limN!T

�AWN � T

�. (40)

27

From Finding 4 in the proof of Proposition 5 below:

@AW@N

=

Z bcc

G(c) dc : (41)

(34), (35), (40), (41), and L�Hôpital�s Rule imply:

limN!T

�AWN � T

�= lim

N!T

Z bcc

G(c) dc = limN!T

�G(c) c jbcc �

Z bcc

c g(c) dc

�

= G(c) c jcc �Z c

c

c g(c) dc = c� ce. (42)

(40) and (42) imply:limN!T

c1(AW ) = ce . (43)

From (27):AW R AM , G (c2 (AW ))�G (c1 (AW )) R

1

2. (44)

(38) and (43) imply that as N ! T :

G (c2 (AW ))�G (c1 (AW )) ! 1�G (ce) . (45)

(44) and (45) imply that as N ! T :

AW R AM , 1�G (ce) R 1

2, G (ce) Q 1

2= G

�cd�, ce Q cd . �

Proof of Proposition 4. For expositional ease, suppose c = 0, so g(c) = 1c. (Bose et al.

(2019) prove this normalization is without loss of generality.) Because G (bc ) = TN:bc

c=

T

N) bc = T

Nc : (46)

From (25):

AWT

= ce �R bc0c dG (c)

G (bc) =c

2�

(bc)22bc =

1

2[ c� bc ] . (47)

(26) and (47) imply:

c2 (AW ) = bc+ AWT

= bc+ 12[ c� bc ] = 1

2[ c+ bc ] . (48)

(26) and (47) also imply:

c1 (AW ) = bc� AWN � T = bc� AW

T

�T

N � T

�= bc� 1

2[ c� bc ] T

N � T

28

= bc � 1 + 12

�T

N � T

��� c T

2 [N � T ] =bc [ 2N � T ]� c T

2 [N � T ] . (49)

(48) and (49) imply:Z c2(AW )

c1(AW )

dG (c) =1

c[ c2 (AW )� c1 (AW ) ] =

1

c

�[N � T ] [ c+ bc ]� bc [ 2N � T ] + c T

2 [N � T ]

�

=bc [N � T � 2N + T ] + c N

2 c [N � T ] =N [ c� bc ]2 c [N � T ] . (50)

(27) and (50) imply:

AM Q AW , N [ c� bc ]2 c [N � T ] R

1

2, N [ c� bc ] R c [N � T ]

, N bc Q T c , bc Q T

Nc . (51)

(46) and (51) imply AM = AW . �

Observation. The expected welfare gain from VJS is proportional to AW �A when g(c) isthe uniform density.

Proof . As in the proof of Proposition 4, assume c = 0 without loss of generality. ThenG (bc ) = T

Nand bc = T

Nc when g(c) is the uniform density. The expected welfare gain from

VJS (relative to MJS) given c is:

WV (c)�WM(c) = w � c� T

N[� c ] = w �

�N � TN

�c for c 2 [ 0;bc ]; and

WV (c)�WM(c) = �F � T

N[� c ] = �F +

�T

N

�c for c 2 [ bc ; c ]. (52)

(52) implies:Z TNc

0

�w �

�N � TN

�c

�dc =

T

Nc

�w � c

2

T

N

�N � TN

��, and

Z c

TNc

��F +

�T

N

�c

�dc =

�N � TN

�c

��F + c

2

T

N

�N + T

N

��. (53)

(53) implies:Z c

0

(WV (c)�WM(c) ) dc = c

�T

Nw �

�N � TN

�F +

c

2

T

N

�N � TN

��N + T

N� T

N

��

29

= c

�T

Nw �

�N � TN

�F +

c

2

T

N

�N � TN

��

=c

N

�T w �

�N � TN

�N F +

c

2T

�N � TN

��

=c

N

�T w � (1�G (bc ))N F + c

2T (1�G (bc )) �

=c

N

��A+ c

2T

�c� bcc

��=

c

N[AW � A ] . (54)

(54) re�ects (4) and (47). �

Proof of Proposition 5. Without loss of generality, assume c = 0 and c = 1, so ce = 12.

G(c) is strictly log concave when g(c) is strictly log concave (Bagnoli and Bergstrom, 2005).Therefore:

@

@c

�g(c)

G(c)

�< 0 , @

@c

�G(c)

g(c)

�> 0 for all c 2 [ 0; 1 ] . (55)

Suppose A = AW . Then Lemma 1 implies:

c1 = bc� AWN � T and c2 = bc+ AW

T. (56)

The de�nitions of AW and AM imply:

AW > AM , G (c2)�G (c1) >1

2. (57)

Findings 1 �12 below demonstrate that G (c2)�G (c1) > 12for all N 2 [ 2T; eN ) under the

speci�ed conditions, where eN is de�ned in (10) above.

Finding 1. bc � 12:

Proof. G(bc ) = TN� 1

2when N � 2T . Therefore, bc � 1

2because g (c) is symmetric and

strictly increasing. �

Finding 2. @ bc@N

= � TN2g(bc ) = � 1

T[G(bc ) ]2g(bc ) < 0 for all bc > 0.

Proof.G(bc ) = T

N) g(bc ) � @bc

@N

�= � T

N2

) @bc@N

= � T

N2 g(bc ) = � 1

T

�T 2

N2 g(bc )�= � 1

T

[G(bc ) ]2g(bc ) . �

30

Finding 3. AWT= 1

2�bc + N

T

R bc0G(c) dc .

Proof. From (25):AWT

= ce �R bc0c g(c) dc

G(bc ) =1

2�R bc0c g(c) dc

G(bc ) . (58)

Integration by parts provides:Z bc0

c g(c) dc = [ c G(c) ]bc0 �Z bc0

G(c) dc = bc G(bc )� Z bc0

G(c) dc . (59)

(58) and (59) imply:

AWT

=1

2�R bc0c g(c) dc

G(bc ) =1

2�bc G(bc )� R bc

0G(c) dc

G(bc )=1

2� bc+ R bc0 G(c) dc

G(bc ) =1

2� bc+ N

T

Z bc0

G(c) dc . �

Finding 4. @AW@N

=R bc0G(c) dc .

Proof. Finding 3 implies:

@AW@N

=@

@N

�T

�1

2� bc�+N Z bc

0

G(c) dc

�

= � T@bc@N

+

Z bc0

G(c) dc+N

�G(bc ) @bc

@N

�

= T@bc@N

�� 1 +

N

TG(bc ) �+ Z bc

0

G(c) dc =

Z bc0

G(c) dc . � (60)

Finding 5. @@c

�G(c)g(c)

�> 0 for all c ) g(ec )G(c)� g(c)G(ec ) < 0 for c < ec .

Proof. For c < ec :@

@c

�G (c)

g (c)

�> 0 ) G(c)

g(c)<G(ec )g(ec ) ) g(ec )G(c)� g(c)G(ec ) < 0 . �

Finding 6. @c2@N< 0 .

Proof. (56) and Findings 2 and 3 imply:

31

c2 = bc+ AWT

= bc+ 12� bc+ N

T

Z bc0

G(c) dc =1

2+N

T

Z bc0

G(c) dc

) @c2@N

=1

T

Z bc0

G(c) dc+N

TG (bc) � @bc

@N

�=

1

T

Z bc0

G(c) dc+@bc@N

=1

T

Z bc0

G(c) dc� 1

T

[G(bc ) ]2g(bc ) =

1

T g(bc )�g(bc )Z bc

0

G(c) dc� (G(bc ))2 �

=1

T g(bc )� Z bc

0

[ g(bc )G(c)� g(c)G(bc ) ] dc � < 0 . (61)

The inequality in (61) re�ects Finding 5. �

Finding 7. @c1@N< 0 .

Proof. Finding 3 implies:

AW = T

�1

2� bc �+N Z bc

0

G(c) dc :

Therefore, (56) and Findings 2 �4 imply:

@c1@N

=@

@N

�bc� AWN � T

�=

@bc@N

+AW

[N � T ]2��

1

N � T

�@AW@N

= � 1

T

[G(bc ) ]2g(bc ) +

AW

[N � T ]2� 1

N � T

Z bc0

G(c) dc

= � 1

T

[G(bc ) ]2g(bc ) � 1

N � T

Z bc0

G(c) dc+1

[N � T ]2�T

�1

2� bc�+N Z bc

0

G(c) dc

�

= � 1

T

[G(bc ) ]2g(bc ) +

T

[N � T ]2�1

2� bc �+ Z bc

0

G(c) dc

�N

(N � T )2� 1

N � T

�

= � 1

T

[G(bc ) ]2g(bc ) +

T

[N � T ]2�1

2� bc �+ T

[N � T ]2Z bc0

G(c) dc

=1

T [N � T ]2 g(bc )�� (N � T )2 (G(bc ))2 + T 2 g(bc )�1

2� bc�+ T 2 g(bc )Z bc

0

G(c) dc

�

=N2

T [N � T ]2 g(bc )"��1� T

N

�2(G(bc ))2 + � T

N

�2g(bc )�1

2� bc�+ � T

N

�2g(bc )Z bc

0

G(c) dc

#

32

=N2

T [N � T ]2 g(bc )�� (1�G(bc ))2 (G(bc ))2 + (G(bc ))2 g(bc )�1

2� bc�+ (G(bc ))2 g(bc )Z bc

0

G(c) dc

�

=N2 [G(bc ) ]2

T [N � T ]2 g(bc )�� (1�G(bc ))2 + g(bc )�1

2� bc�+ g(bc )Z bc

0

G(c) dc

�

=N2 [G(bc ) ]2

T [N � T ]2 g (bc)�� 1 + 2G(bc ) + g(bc )�1

2� bc�� (G(bc ))2 + g(bc )Z bc

0

G(c) dc

�

=N2 [G(bc ) ]2

T [N � T ]2 g(bc )�� 1 + 2G(bc ) + g(bc )�1

2� bc�� Z bc

0

G(bc ) g (c) dc+ Z bc0

g(bc )G(c) dc �

=[G(bc ) ]2

T�1� T

N

�2g(bc )

�� 1 + 2G(bc ) + g(bc )�1

2� bc�� Z bc

0

[G(bc ) g (c)� g(bc )G (c) ] dc �

=[G(bc ) ]2

T [ 1�G(bc ) ]2 g(bc )�� 1 + 2G(bc ) + g(bc )�1

2� bc�� Z bc

0

[G(bc ) g (c)� g(bc )G (c) ] dc � .(62)

Because @@c

�G(c)g(c)

�> 0, Finding 5 implies that G(bc ) g(c) � g(bc )G (c) > 0 for c < bc .

Therefore, the expression in (62) is negative if � 1 + 2G(bc ) + g(bc ) � 12� bc �< 0.

De�nef(x) = � 1 + 2G (x) + g (x)

�1

2� x

�for x 2

�0;1

2

�.

Observe that f(0) = � 1 + g(0)2< 0. This inequality holds because g is unimodal and

symmetric around 12. Also:

f

�1

2

�= � 1 + 2G

�1

2

�+ g

�1

2

��1

2� 12

�= 0 , and

f 0(x) = 2 g(x) + g0(x)

�1

2� x

�� g(x) = g(x) + g0(x)

�1

2� x

�. (63)

(63) implies that f 0(x) > 0 for all x 2�0; 1

2

�because g(x) is increasing on

�0; 1

2

�. Therefore,

f(0) < 0; f�12

�= 0; and f 0(x) > 0 for all x 2

�0; 1

2

�; which implies that f(x) < 0 for all

x 2�0; 1

2

�: Hence, @c1

@N< 0, from (62). �

Finding 8.�12�G (c)

� R c0G (y) dy � 1

2[G (c) ]2

�12� c

�for all c 2 [ 0; 1

2].

Proof. The intermediate value theorem ensures there exists � 2 ( c; 12) such that

1

2�G(c) = G(

1

2)�G (c) =

�1

2� c

�G0(�) =

�1

2� c

�g(�) . (64)

33

g(�) � g(c) because g(c) is increasing in c and � 2 ( c; 12). Therefore, (64) implies:

1

2�G(c) �

�1

2� c

�g(c)

)�1

2�G(c)

� Z c

0

G(y) dy ��1

2� c

�g(c)

Z c

0

G(y) dy . (65)

(65) implies the Finding holds if:�1

2� c

�g(c)

Z c

0

G(y) dy � 1

2[G(c) ]2

�1

2� c

�

, R (c) � g(c)

Z c

0

G(y) dy � 12[G(c) ]2 � 0 . (66)

Di¤erentiating R(c) provides:

R0 (c) = g(c)G(c) + g0(c)

Z c

0

G(y) dy �G(c) g(c) = g0(c)

Z c

0

G(y) dy � 0 . (67)

The inequality in (67) holds because g(c) is increasing for c < 12. Because R (0) = 0 from

(66), (67) implies R (c) � 0 for all c 2�0; 1

2

�. �

Finding 9.�� @c1@N

�� � �� @c2@N

��Proof. From (56):

c2 � c1 = bc+ AWT��bc� AW

N � T

�=AWT

�1 +

T

N � T

�=AWT

�N

N � T

�

) @ (c2 � c1)@N

=

�N

N � T

�@

@N

�AWT

�� AW

T

�T

(N � T )2�

=1

N � T

�N

T

@

@N(AW )�

AWN � T

�

) @ (c2 � c1)@N

� 0 , N

T

@

@N(AW ) � AW

N � T . (68)

(68) and Findings 3 and 4 imply:

@ (c2 � c1)@N

� 0 , N

T

Z bc0

G(c) dc � T

N � T

�1

2� bc+ N

T

Z bc0

G(c) dc

�

, N

T

�1� T

N � T

� Z bc0

G(c) dc � T

N � T

�1

2� bc �

34

, N

T[N � 2T ]

Z bc0

G(c) dc � T

�1

2� bc �

,�N � 2TN

� Z bc0

G(c) dc � T 2

N2

�1

2� bc �

,�1� 2 T

N

� Z bc0

G(c) dc ��T

N

�2 �1

2� bc �

, [ 1� 2G(bc ) ] Z bc0

G(c) dc � [G(bc ) ]2 � 12� bc �

,�1

2�G(bc ) � Z bc

0

G(c) dc � 1

2[G(bc ) ]2 � 1

2� bc � .

Finding 8 implies that this inequality holds. �

Finding 10. AW > AM if N = 2T:

Proof. bc = G( TN) = G(1

2) = 1

2when N = 2T: Furthermore, from (56), when N = 2T :

c1 = bc� AWN � T =

1

2� AW

T:

Therefore, (56) and (57) imply:

AW > AM , G

�1

2+AWT

��G

�1

2� AW

T

�>1

2: (69)

From (58):

AWT

=1

2�R bc0c g (c) dc

G(bc ) ) 1

2+AWT

= 1�R 1

2

0c g(c) dc

G�12

� = 1� 2Z 1

2

0

c g(c) dc

) 1

2� AW

T=1

2� 12+

R bc0c g(c) dc

G(bc ) =

R 12

0c g(c) dc

G�12

� = 2

Z 12

0

c g(c) dc . (70)

G(c) = 1�G(1� c) because g (c) is symmetric. Therefore:

G

1� 2

Z 12

0

c g(c) dc

!= 1�G

2

Z 12

0

c g(c) dc

!. (71)

(69) �(71) imply:

AW > AM , G

�1

2+AWT

��G

�1

2� AW

T

�>1

2

35

, 1�G 2

Z 12

0

c g(c) dc

!�G

2

Z 12

0

c g(c) dc

!>1

2

, 1� 2G 2

Z 12

0

c g(c) dc

!>1

2, 2G

2

Z 12

0

c g(c) dc

!<1

2: (72)

De�ne H(c) = 2G(c) for 0 � c � 1

2. Then:

h(c) � H 0(c) = 2G0(c) = 2 g(c) for 0 � c � 1

2. (73)

(73) implies that H(c) is a distribution function on�0; 1

2

�with corresponding density func-

tion h (c) :

Let Y be a random variable on�0; 1

2

�with density function h(c): Then:

E (Y ) =

Z 12

0

c h(c) dc = 2

Z 12

0

c g(c) dc . (74)

(73) and (74) imply that (72) holds if and only if

H (E (Y )) <1

2. (75)

Let M (Y ) denote the median of Y , so H (M(Y )) = 12: Then:

H (E (Y )) <1

2, H (E (Y )) < H (M(Y )) , E(Y ) < M(Y ) . (76)

E(Y ) < M(Y ) because h(c) is strictly increasing in c for c 2�0; 1

2

�(e.g., van Zwet 1979;

Dharmadhikari and Joag-Dev, 1983). Therefore, AW > AM . �

We have shown that c1 and c2 both decline as N increases. We now determine howG(c2)�G(c1) changes when c1 and c2 decline by the same amount (x).

Finding 11.@

@x[G (c2 � x)�G (c1 � x) ] < 0 for x 2 ( 0; c2 �

1

2) . (77)

Proof.@

@x[G(c2 � x)�G(c1 � x) ] = � g(c2 � x) + g(c1 � x) < 0 :

The inequality holds here because (56) implies that when N = 2T :

c2 =1

2+AWT

>1

2and c1 =

1

2� AW

T<1

2.

Therefore, g(c2) = g(c1) when N = 2T because g is symmetric around 12. Furthermore, g(c)

is strictly increasing for c � 12and strictly decreasing for c � 1

2: Hence, if both c2 and c1

36

decrease by x, g(c2� x) > g(c2); and g(c1) > g(c1� x): Consequently, g(c2� x) > g(c1� x).�

Finding 12. There exists a eN > 2T such that G(c2)�G(c1) > 12for all N 2 [ 2T; eN ).

Proof. Express c1 and c2 as functions of N , as in (11). (57) and Finding 10 imply:

G(c2(2T ))�G(c1(2T )) >1

2. (78)

Furthermore, Findings 6 and 7 imply c2(N) < c2(2T ) and c1(N) < c1(2T ) for N > 2T .

De�ne: ec1(N) � c1(2T )� [ c2(2T )� c2(N) ] for N � 2T . (79)

(79) implies ec1(2T ) = c1(2T ). For N > 2T , c2(N) is less than c2(2T ) by the amountc2(2T )�c2(N). (79) implies that ec1(N) is less than c1(2T ) by the identical amount becausec1(2T ) � ec1(N) = c2(2T ) � c2(N). Because ec1(N) and c2(N) decline by the same amountas N increases above 2T , G(c2(N))�G(ec1(N)) is a decreasing function of N , from Finding11.

Finding 9 implies that c1(N) declines more rapidly than c2(N) declines as N increasesabove 2T . Therefore, ec1(N) > c1(N) for N > 2T . Consequently, because G(c) is strictlyincreasing in c:

G(c2(N))�G(c1(N)) >1

2if G(c2(N))�G(ec1(N)) � 1

2for N > 2T . (80)

We next prove:

There exists a �nite eN such that G(c2(N))�G(ec1(N)) = 1

2. (81)

To prove (81), observe initially that because ec1(2T ) = c1(2T ), (78) implies:

G(c2(2T ))�G(ec1(2T )) = G(c2(2T ))�G(c1(2T )) >1

2. (82)

Because G(c2(N))�G(ec1(N)) is a decreasing function of N , (82) implies that (81) holds if:limN!1

fG(c2(N))�G(ec1(N)) g � 1

2. (83)

To show that (83) holds, �rst observe that when N = 2T , G(bc ) = TN= 1

2) bc = 1

2.

Therefore, Finding 2 implies:

c2(N) �1

2for all N � 2T and lim

N!1c2(N) =

1

2. (84)

Next observe that Finding 3 implies:

AW (2T )

T= 2

Z 12

0

G(c) dc � 1

4: (85)

37

The inequality in (85) holds because:

G(c) � c for all c 2 [ 0; 12] (86)

)Z 1

2

0

G(c) dc �Z 1

2

0

c dc =1

2c2���� 120

=1

8:

To prove that (86) holds, de�ne �(c) � G(c)� c. Observe that:

�(0) = �(1

2) = 0 . (87)

In addition:�0(c) = g(c)� 1 ) �00(c) = g0(c) � 0 , and

�0(0) = g(0)� 1 � 0 . (88)

(87) and (88) imply that �(c) � 0 for all c 2�0; 1

2

�, so (86) holds.

(56) and (85) imply:

c2(2T ) = bc (2T ) + AW (2T )T

� 1

2+1

4=3

4, and

c1(2T ) = bc (2T )� AW (2T )T

� 1

2� 14=1

4

) c1(2T ) � c2(2T )�1

2. (89)

Because limN! 1

c2(N) =12, (79) and (89) imply:

limN! 1

ec1 (N) = c1(2T )�hc2(2T )� lim

N! 1c2(N)

i= c1(2T )� [ c2(2T )�

1

2] � 0 . (90)

(84) and (90) imply that (83) holds. � �

38

𝑊𝑉(𝑐) −𝑊𝑀(𝑐)

0 𝑐

𝑐 𝑐1 �̂� 𝑐2 𝑐

Figure 𝟏. Net Benefits from OJS.

VJS Maximizes Welfare | MJS Maximizes Welfare

Majority Prefer VJS | Majority Prefer MJS

Majority Choice Efficient | Majority Choice | Majority Choice Efficient

Inefficient

________________________|______________|_______________________ 𝐴

𝐴𝑀 𝐴𝑊

Figure 2A. Majority Rule Favors MJS for 𝑨 ∈ (𝑨𝑴, 𝑨𝑾).

VJS Maximizes Welfare | MJS Maximizes Welfare

Majority Prefer VJS | Majority Prefer MJS

Majority Choice Efficient | Majority Choice | Majority Choice Efficient

Inefficient

________________________|______________|_______________________ 𝐴

𝐴𝑊 𝐴𝑀

Figure 2B. Majority Rule Favors VJS for 𝑨 ∈ (𝑨𝑾, 𝑨𝑴).

0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

Figure 3. Beta Density with Parameters (α, β)

α = 2, β = 2

α = 2, β = 3

α = 3, β = 2

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