Discussion Paper No. 948

PRE-NEGOTIATION COMMITMENT
AND INTERNALIZATION
IN PUBLIC GOOD PROVISION
THROUGH BILATERAL NEGOTIATIONS

Noriaki Matsushima
Ryusuke Shinohara

September 2015
Revised August 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanPre-negotiation commitment and internalization in public
good provision through bilateral negotiations∗
Noriaki Matsushima†

Ryusuke Shinohara‡

August 31, 2017

Abstract
We investigate public good provision through bilateral negotiations between apublic-good supplier and the beneficiaries of the good. We find that although a prenegotiation commitment on the production level of the public good by the supplierenhances the internalization of beneficiaries’ preferences, the good is not necessarily
provided eﬃciently. We show that with the commitment, the supplier produces the
public good at an eﬃcient level in equilibrium if and only if its bargaining power is
suﬃciently weak. In addition, the public good may be produced excessively as a resultof the commitment when the supplier’s bargaining power is suﬃciently strong.
Keywords: Public good; Simultaneous bilateral bargaining; Supplier bargaining
power; Nash bargaining solution.
JEL Classification: C78, D42, D62, H41, H44.

∗

This paper extends a previously circulated unpublished paper “The eﬃciency of monopolistic provisionof public goods through simultaneous bilateral bargaining.” We are grateful to Naoki Watanabe, Wataru
Kobayashi, Emmanuelle Taugourdeau, Takuro Yamashita, and Shintaro Miura for their useful discussions
and suggestions. In addition, we thank the participants of the 12th Meeting of the Society for Social Choiceand Welfare, the 68th European Meeting of the Econometric Society, PET 15, 71st Annual Congress of
the International Institute of Public Finance, the meeting of the Japanese Economic Association, andthe meeting of the Japan Institution of Public Finance. We gratefully acknowledge the financial support
from the KAKENHI Grant-in-Aid for Scientific Research (S) No.15H05728, (A) No. 17H00984, (B) No.
15H03349, (C) Nos. 24530248, 15K03361, and Young Scientists (B) No. 24730165, RISTEX (JST), andthe program of the Joint Usage/Research Center for “Behavioral Economics” at ISER, Osaka University.
The usual disclaimer applies.
†
Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka
6-1, Ibaraki, Osaka 567-0047, Japan.
Tel: (81)-6-6879-8571.
Fax: (81)-6-6879-8583.
E-mail:nmatsush@iser.osaka-u.ac.jp‡

Ryusuke Shinohara, Department of Economics, Hosei University, 4342 Aihara-machi, Machida, Tokyo
194-0298, Japan. Tel: (81)-42-783-2534. Fax: (81)-42-783-2611. E-mail: ryusukes@hosei.ac.jp1

1

Introduction

We examine the role of commitments prior to negotiations in the provision of public
goods. In bargaining theory, many studies have examined how a pre-commitment aﬀects
negotiation outcomes.1 However, few studies have investigated this topic in public goods
theory. Here, we examine how a pre-commitment on the production level of a public good
aﬀects the provision of the good, as negotiated between the supplier and the beneficiariesof the good.
In our case, the supplier has the technology to produce the public good, but does
not benefit from the public good itself. On the other hand, the beneficiaries enjoy the
public good, but cannot produce it on their own. Hence, the provision of the public
good via a negotiation, including a monetary transfer, is beneficial for the supplier andbeneficiaries. This situation is worth analyzing in the context of environmental economics
because it includes a bargaining problem over pollution reduction: a private firm in an
upstream area produces a good by polluting a river, and residents in the downstream
area suﬀer from this pollution. While the firm must pay costs to control the pollution
level, the residents benefit from the pollution reduction. For mutual benefit, the polluterand pollutees need to negotiate a pollution reduction in exchange of compensation. The
pollution reduction can be considered a public good for the residents because they benefit
non-exclusively from it, irrespective of whether they pay compensation for it. In summary,
our study examines two important topics in public economics: public good provision andenvironmental externalities.
With regard to environmental externalities between polluters and pollutees, as in theexample above, the eﬀectiveness of the negotiation between them to achieve Pareto efficiency has been investigated numerous times since the seminal work of Coase (1960).
However, many subsequent papers report a limitation of the Coase theorem (or collective
negotiation) in various environments, including two-player models (e.g., Porter, 1988; Huber and Wirl, 1998; Robson and Skaperdas, 2008; MacKenzie and Ohndorf, 2013, 2016),
1

See, for example, Muthoo (1999, Ch. 8).

2

multiple-player models (Dixit and Olson, 2000; Ellingsen and Paltseva, 2012, 2016), andunilateral actions before negotiation (e.g., Hoel, 1991; Konrad and Thum, 2014). In addition, although the eﬃciency in public good provision has been investigated theoreticallyand experimentally (e.g. Ray and Vohra, 1997, 2001; Dixit and Olson, 2000; Thoron etal., 2009; McEvoy, 2010), some studies, such as Ray and Vohra (1997, 2001) and Dixit
and Olson (2000), show that negotiations over the provision of a public good rarely overcome the well-known free-rider problem. From the viewpoint of these lines of research,
it is natural to evaluate whether some tactics prior to negotiations facilitate allocativeeﬃciency.
To address our question, we build a bargaining model under complete information,
based on simultaneous bilateral negotiations. In this model, there is a supplier of a public
good and n consumers of the good. The supplier negotiates with each consumer, simultaneously and bilaterally. Thus, the supplier has n simultaneous bilateral bargaining
sessions. The supplier and each consumer in the session anticipate the outcome of theother bilateral bargaining sessions because all sessions are simultaneous. The anticipationin one session of the results of the other sessions aﬀects the surplus of the session. In equilibrium, each session correctly anticipates the outcome of the other sessions. We assume
that, through a Nash bargaining solution, each session shares the correctly anticipatedsurplus. The supplier and a consumer in each session share the surplus in proportion totheir relative levels of bargaining power, which are given exogenously.2
From the viewpoint of real-world situations, it is reasonable to examine the provisionof a public good through bilateral negotiations. In particular, in resolving environmental
externalities, polluters and pollutees often negotiate individually, rather than collectively.
According to Depres et al. (2008), who study the case of Vittel, a drinking water bottler
in France, the company negotiated and contracted with farmers to solve the problem inwhich farmers’ practices were significantly polluting Vittel’s water catchment area. Seem2

In the context of industrial organization, several studies employ the simultaneous bilateral bargaining
model (Chipty and Snyder, 1999; Raskovich, 2001, 2003; Marshall and Merlo, 2004; Matsushima andShinohara, 2014; Collard-Wexler et al., 2017).

3

ingly, the Vittel case is diﬀerent from ours because Vittel is the only pollutee, and there
are numerous polluters (farmers). However, Depres et al. (2008) report some important
aspects, which seem to be widely applicable to negotiations in many environmental problems: a bilateral negotiation is preferable to a collective negotiation for the pollutee andfor some of the polluters. In the case of Vittel, collective bargaining would most likely
have increased the farmers’ (polluters’) bargaining power. Hence, negotiating with eachfarmer individually provided the company with a stronger relative bargaining position.
On the other hand, some farmers who trusted their bargaining skills preferred individual
negotiations to exploit the advantages of bargaining.3
First, we examine a case in which the supplier does not commit to a level of the
public good prior to negotiations with consumers. In this case, the supplier and eachconsumer negotiate the joint production level of the public good, as well as transfers fromthe consumer to the supplier. We prove that the realized public good level through thenegotiation is far from the eﬃcient level (Proposition 1), owing to a kind of free-rider
problem on public goods. When the supplier and consumers negotiate bilaterally, there isan externality between the bilateral negotiations. As a result of this externality, a consumer
in bilateral bargaining can free ride a public good that is provided by compensation fromother consumers.
Second, we examine a case in which the supplier commits to the level of the public
good prior to the negotiations. In this case, the level of the public good is separated
from the negotiation, and the supplier and each consumer negotiate the transfers only.
Formally, we model this case as follows: in the first stage, the supplier commits to thelevel of the public good; in the second stage, observing the level, each consumer bilaterally and simultaneously negotiates with the supplier over his/her monetary compensation
3
Note that some aspect of the Vittel case might be captured by our situation. According to Depres etal. (2008, p. 423),

the quality of Vittel water had several positive spillovers (employment at the Vittel company,
as well as in related activities, such as thermalism and tourism).
Those spillovers were achieved by the farmers contracted by Vittel protecting the water quality. On
the other hand, non-contracted farmers were able to enjoy the spillovers, even if they did not engage inprotecting the water quality. (Indeed, some farmers did not enter into a contract with Vittel.)4for the provision of the public good. We prove that under mild conditions, there is a
threshold value of the supplier’s bargaining power below which the public good is produced eﬃciently in equilibrium (see Theorem 1). This result shows that the public goodis produced eﬃciently if and only if the supplier’s bargaining power is “suﬃciently weak.”We evaluate to the extent to which the pre-commitment leads to the eﬃcient provisionof the public good under specific functional forms (see Section 6). We show that as thenumber of consumers increases, the eﬃcient provision of the public good is less likely tobe observed in equilibrium. However, even if the number of consumers is very large, theeﬃcient provision of the public good in equilibrium may be observed with a certain highlikelihood (see Proposition 3, and the discussion thereafter). In addition, we explore whatlevel of the public good is provided when the supplier’s bargaining power is beyond thethreshold. Here, we provide a numerical example in which the supplier with “suﬃciently
strong” bargaining power produces excessive amounts of the public good. That is, thepre-commitment may lead to an excessive provision of the public good (see Section 6.2).
We note several interesting points with regard to our results. While the pre-commitment
on the production level of the public good facilitates the internalization of consumers’ preferences, the public good is not necessarily provided eﬃciently. Then, as shown in Theorem
1 and Section 6.2, when the supplier can pre-commit to the level of the public good, it has
an incentive to set the level “suﬃciently high” so that it receives transfers from all consumers.4 In doing so, the supplier takes into account consumers’ preferences for the public
good. In this sense, the internalization of the consumers’ preferences succeeds throughthe pre-commitment. However, depending on the level of per-capita transfers from theconsumers, which is governed by the supplier’s bargaining power, the supplier may set the
public good level above the eﬃcient level prior to the negotiations. Therefore, although
the internalization of the preferences are completed via the pre-commitment, the supplier
may not provide the public good eﬃciently (see also Remark 2).
The remainder of this paper is organized as follows. Section 2 reviews the related
4

Refer to the discussion immediately after Theorem 1.

5literature. Section 3 presents the preliminaries, and Section 4 analyzes the case of no
pre-commitment. Section 5 analyzes the case of a pre-commitment. Section 6 presents
an analysis under parametric functions, and Section 7 discusses the extension of the basicmodel. Section 8 concludes the paper.

2

Literature review

We review several studies on the eﬀectiveness of pre-play actions before (i) public good
provision (e.g., Hoel, 1991; Varian, 1994; Ray and Vohra, 1997, 2001; Dixit and Olson,
2000; Konrad and Thum, 2014) or (ii) Coasian bargaining (e.g., MacKenzie and Ohndorf,
2013; Ellingsen and Paltseva, 2016), which are important factors in our study.
As in our study, Ray and Vohra (1997, 2001) and Dixit and Olson (2000) examine
public good provision through bargaining, under complete information. Ray and Vohra(1997, 2001) introduce two models in which a coalition with several agents can force
each of the agents to provide a negotiated level of a public good. Dixit and Olson (2000)
introduce a two-stage model in which agents determine whether to participate in a Coasianbargaining process, which is itself eﬃcient, given the number of participants determined in
the first stage. However, in these studies, the internalization of beneficiaries’ preferences
fails, and the provision of public goods is ineﬃcient, except in specific cases.5 Comparedwith these studies, our Proposition 1 shares these negative results. Fortunately, in ourmodel, the pre-commitment on the level of the public good can improve the eﬃciency ofproviding the good. In addition, Proposition 3 contributes to the literature of the groupsize eﬀects of public good provision, following the seminal work of Olson (1965) (see thediscussion in Section 6.1).6 In a diﬀerent context, incorporating incomplete information
on agents’ costs when providing public goods into the model of Hoel (1991), Konrad
and Thum (2014) show that a unilateral pre-negotiation commitment by an agent on the5

Ellingsen and Paltseva (2012) show that even when providing excludable public goods, players do notnecessarily participate in the provision of the goods when they are allowed to negotiate access to the goods,
ex post.
6
For a comprehensive survey on this topic, see Pecorino (2015).

6

level of the public good reduces the likelihood of achieving eﬃcient outcomes, which is anegative result.7 In contrast to their result, our study shows certain positive aspects ofthe pre-commitment, although our model of public good provision diﬀers to theirs.8
Several researchers have examined how the restriction of feasible bargaining outcomes
aﬀects the eﬃciency of negotiations. MacKenzie and Ohndorf (2013) show that the eﬃciency of two-person Coasian bargaining can be enhanced by a restriction imposed by anoutside authority on agents’ property rights, which diminishes the feasible set of pollution
abatement levels. Recently, Ellingsen and Paltseva (2016) analyzed the validity of the
Coase theorem with N players by introducing a pre-play stage, in which players voluntarily negotiate contracts with regard to how to play the game and how much money istransferred among them. They show significantly diﬀerent results between the two-playercase and the N -player case. They prove that an eﬃcient outcome is achievable in the twoplayer case, but not in the general N -player case, owing to strategic nonparticipation in
the negotiation stage.9 Our results are similar, in that the negotiation outcome is eﬃcientif there are two players (a supplier and one consumer), and never eﬃcient otherwise (seeProposition 1). Our result implies that restrictions on the strategies employed prior to negotiations, via the commitment of a player, helps to negotiate a better outcome. Similarly,
Ellingsen and Paltseva (2016) show that an appropriate property right, which is related to
the restriction of players’ strategies, achieves an eﬃcient allocation via contracting amongplayers.
7

Hoﬀmann et al. (2015) test Konrad and Thum’s (2014) result experimentally.
In the context of a voluntary contribution to public goods (e.g., Bergstrom et al., 1986), Varian (1994)
considers a game of sequential contribution games, in which the player moving first can credibly commit to
its contribution. He shows that such a unilateral commitment does not increase the total level of a public
good.
9
It is also shown in the context of a common agency game that the contract that achieves eﬃciency
suﬀers from strategic nonparticipation. Bernheim and Whinston (1986) and Laussel and Le Breton (2001)
show that, in common agency games, the eﬃcient provision of a public good is achieved through transfer
schemes that are completely contingent on the level of the good. Furusawa and Konishi (2011) show that
this eﬃciency in equilibrium is vulnerable to voluntary participation behavior in the contracting process.
8

7

3

The model

Consider an economy with a pure public good and a private good (money), with a supplierof the public good and n ≥ 2 consumers of the public good.10 Only the supplier has thetechnology to provide the public good, but without any compensation from the consumers,
it has no incentive to provide it. On the other hand, consumers enjoy the public good, butcannot produce the public good by themselves. Hence, each consumer needs to delegate
the provision of the public good to the supplier in exchange for monetary compensation.
How much money is compensated from each consumer to the supplier is decided throughbargaining.
Formally, the set of players is denoted by {s, 1, . . . , n}, where s is the supplier and i(i = 1, . . . , n) is a consumer. Let N be the set of consumers. The level of the public good
is typically denoted by g ≥ 0, and consumer i’s transfer to the supplier is denoted by Ti .
The cost function of the public good is denoted by c : R+ → R+ , such that c(0) = 0, and cis an increasing, convex (sometimes, weakly convex), and twice continuously diﬀerentiable
function. When the supplier provides g units of the public good and receives payment∑
Ti from each i ∈ N , its payoﬀ is i∈N Ti − c(g) (a profit). Each consumer i receives
payoﬀ v(g) − Ti , where v : R+ → R+ is a benefit function from the public good, suchthat v(0) = 0 and v is an increasing, concave (sometimes, weakly concave), and twice
continuously diﬀerentiable function. Note that in this model, each consumer has the same
benefit function, v.
Definition 1 As a reference level for the public good, we defineg(m) ∈ arg max mv(g) − c(g),g≥0for each m ∈ {1, . . . , n}.
Furthermore, we impose the following conditions on v(g) and c(g).
10See Remark 1 for the case of n = 1.

8

Assumption 1 We impose the following assumption on c(·) and v(·):(1.1) limg→0 v ′ (g) = ∞, limg→∞ v ′ (g) = 0, limg→0 c′ (g) = 0, and limg→∞ c′ (g) = ∞.
(1.2) For all g > 0, c′ (g)/c(g) > v ′ (g)/v(g).
Condition (1.1) guarantees interior solutions of the model: that is, g(m) > 0 for eachm ∈ {1, . . . , n}.11 Condition (1.2) describes the relation between the ratio of the marginal
benefit of the public good to the gross benefit and the ratio of the marginal cost to theoverall cost. Condition (1.2) implies that c(g)/v(g) is increasing in g, that is, the cost pervalue is increasing in the level of the public good, which sounds reasonable.12
We model a bargaining process between the supplier and consumers in Sections 4 and 5,
based on simultaneous and bilateral bargaining, which is often used in the context of buyer–
supplier relationships (e.g., Horn and Wolinsky, 1988; Marshall and Merlo, 2004).13 Thisis based on observations of real-world negotiations concerning environmental externalities,
as discussed in the Introduction. The question we address is how the commitment on
the production level of the public good prior to negotiations aﬀects the eﬃciency of theprovision of the good. To answer this question, we consider a case in which the supplierdoes not pre-commit the level of the public good (Section 4), and then a case in which the
supplier unilaterally commits to the level of the public good prior to negotiations (Section
5).

4

No pre-commitment on the level of the public good

In this section, the supplier does not commit to the production level of the public goodprior to negotiations. Hence, the supplier negotiates the production level and the transfers
11
Note that our main results hold whenever g(m) is interior for each m ∈ {1, . . . , n}. Thus, we can obtainthese results if we assume limg→0 v ′ (g) = ∞, limg→∞ v ′ (g) = 0, and c′ (g) is finite, instead of assuming
Condition (1.1).
12
The conditions in Assumption 1 are satisfied by reasonable parametric benefit and cost functions (seealso Proposition 2). By l’Hôpital’s rule and (1.1), we obtain

lim

g→0

13

c(g)
= 0.
v(g)

See also the references in Milliou and Petrakis (2007).

9

(1.3)

with consumers. We provide a two-stage model of negotiation, based on simultaneous andbilateral bargaining. In the first stage, the supplier and each consumer i ∈ N bilaterally
negotiate the joint production level of the public good gi ≥ 0 and the transfer to the
supplier from consumer i Ti (≥ 0). Hence, in this stage, the supplier faces n independent
bilateral negotiations. We assume that every bilateral negotiation is simultaneous andNash bargaining. We further assume that the outcome of each negotiation is given by
the (asymmetric) Nash bargaining solution, in the belief that the bargaining outcomeswith the other parties are determined in the same way. Hence, the joint surplus of one
bilateral bargaining process is maximized through the choice of the public good level,
and the maximized joint surplus is divided between the consumer and the supplier in the
proportion of 1 − β to β, where β ∈ [0, 1] represents the supplier’s bargaining power. We
assume that, in equilibrium, the negotiators in each bilateral negotiation have consistentbeliefs about the other negotiations: the outcome of each negotiation is predicted correctlyby the others. Let (gj , Tj )j∈N be the equilibrium outcome of the simultaneous bilateralbargaining. The second stage is an approval stage by the supplier, in which the supplierdecides whether to execute the first-stage outcome (gj , Tj )j∈N . If the supplier approves∑∑(gj , Tj )j∈N , it provides j∈N gj units of the public good in exchange for transfers j∈N Tj ,∑∑∑and the supplier’s and consumer i’s payoﬀs are j∈N Tj −c( j∈N gj ) and v( j∈N gj )−Ti ,respectively. If the supplier disapproves, it annuls (gj , Tj )j∈N , and the payoﬀ to every
player is zero. We establish this stage to guarantee the supplier’s participation constraint.
In our situation, the supplier reaps no benefit from producing the public good in theabsence of transfers from consumers, although it has the technology to do so. Hence, it isnatural for the supplier to check whether it will benefit from a negotiation.14
The method of simultaneously applying the Nash bargaining solution follows Chipty
and Snyder (1999) and Raskovich (2003). However, note that in their models, free riding
is impossible. See, for example, the condition “vi (0, q−i ) = 0” in Raskovich (2003, p. 410,
14

We do not explicitly model consumers’ approval of (gj , Tj )j∈N , because the participation constraint
always holds for all consumers since the Nash bargaining solution guarantees them non-negative payoﬀs.
Thus, consumers will approve the second-stage bargaining outcome.

10

12th line from the bottom).
A non-cooperative foundation for the asymmetric Nash bargaining solution has beenpresented by several studies. As Binmore et al. (1986) show, the subgame perfect equilibrium outcome in Rubinstein’s bargaining model, with alternating oﬀers and a risk of
breakdown, approximates the Nash bargaining solution. In addition, they discuss the relationship between the Nash bargaining solution and asymmetric bargaining power. Hence,
we can approximately interpret that in the first-stage game of our model, each pair of
the supplier and a consumer play the alternating oﬀer bargaining game with a risk of
breakdown, anticipating the other sessions’ outcomes. Therefore, by backward induction,
our solution is consistent with a subgame perfect Nash equilibrium.
We solve this model in a backward manner. In the second stage, given (gj , Tj )j∈N ,∑∑the supplier approves it if j∈N Ti > c( j∈N gj ), is indiﬀerent between approval and∑∑disapproval if j∈N Tj = c( j∈N gj ), and disapproves otherwise. Henceforth, we assume∑∑
that when j∈N Tj = c( j∈N gj ), the supplier disapproves.15
In the first stage, the supplier negotiates with each consumer bilaterally and simultaneously. In the equilibrium of this stage, the bargaining participants (the supplier and a
consumer) correctly anticipate the other bargaining outcomes and conduct a Nash bargaining process. That is, for each i ∈ N , the supplier and consumer i negotiate over (gi , Ti )to maximize the Nash product function, given (gj , Tj )j̸=i .
Players’ surpluses in each bilateral negotiation take diﬀerent forms, depending on theoutcomes of the other bilateral negotiations.
First, consider bilateral bargaining between the supplier and consumer i, given (gj , Tj )j̸=i∑∑such that j̸=i Tj ≤ c( j̸=i gj ). Then, based on the second-stage equilibrium, if this bargaining breaks down, the supplier does not approve (gj , Tj )j̸=i in the next stage. Hence,the payoﬀs to the supplier and consumer i are zero. If this bargaining reaches an agreement (gi , Ti ) and the supplier approves (gj , Tj )j∈N in the next stage, then the supplier and∑∑∑consumer i receive the payoﬀs Ti + j̸=i Tj − c(gi + j̸=i gj ) and v(gi + j̸=i gj ) − Ti ,15

The assumption does not change the subsequent result.

11

respectively. Therefore, the supplier and consumer i decide (gi , Ti ) to maximize the Nash∑∑∑product function β ln(Ti + j̸=i Tj −c(gi + j̸=i gj ))+(1−β) ln(v(gi + j̸=i gj )−Ti ). Thus,
given (gj , Tj )j̸=i , the bargaining between the supplier and consumer i achieves (gi∗ , Ti∗ ) suchthat{gi∗ } = arg max v(gi +gi ≥0

Ti∗ = v(gi∗ +∑gj ) − c(gi +

j̸=i

∑

∑

gj ) and

(1)

j̸=i



gj ) − (1 − β) v(gi∗ +

j̸=i

∑

gj ) − c(gi∗ +∑gj ) +∑

Tj  .

j̸=i

j̸=i

j̸=i

∑

∑

∑

(2)

From (2),

Ti∗ +

∑

Tj − c(gi∗ +j̸=iThus, Ti∗ +∑gj ) = β v(gi∗ +

j̸=i

∑

∗
j̸=i Tj > c(gi +v(gi∗ +gj ) − c(gi∗ +

j̸=i

j̸=i

gj ) +

Tj  .

j̸=i∑

j̸=i gj ) if and only if∑∑gj ) − c(gi∗ +gj ) +

j̸=i

j̸=i

∑

Tj > 0.

(3)

j̸=i

The left-hand side of (3) is the joint surplus of the supplier and consumer i. From (3), thesupplier approves ((gi∗ , Ti∗ ), (gj , Tj )j̸=i ) if and only if the joint surplus is positive. Note that∑
if the supplier approves ((gi∗ , Ti∗ ), (gj , Tj )j̸=i ), consumer i’s payoﬀ v(gi∗ + j̸=i gj ) − Ti∗ =∑∑∑(1−β)(v(gi∗ + j̸=i gj )−c(gi∗ + j̸=i gj )+ j̸=i Tj ) is non-negative. In contrast, if the joint
surplus is negative (i.e., (3) does not hold), the bargaining does not reach an agreement;that is, (gi∗ , Ti∗ ) = (0, 0).
Second, consider the case of∑∑j̸=i Tj > c(j̸=i gj ).

Based on the second-stage equi-

librium, even if this bilateral bargaining breaks down, the supplier approves the out∑
come. Hence, in the case of disagreement, the supplier and consumer i receive j̸=i Tj −∑∑c( j̸=i gj ) and v( j̸=i gj ), respectively. If this bargaining reaches agreement (gi , Ti ) andthe supplier approves (gj , Tj )j∈N in the next stage, then the supplier and consumer i re∑∑∑ceive Ti + j̸=i Tj − c(gi + j̸=i gj ) and v(gi + j̸=i gj ) − Ti , respectively. As in the12previous case, (gi∗ , Ti∗ ) maximizes the Nash product function β ln(Ti − c(gi +∑∑∑c( j̸=i gj )) + (1 − β) ln(v(gi + j̸=i gj ) − Ti − v( j̸=i gj )): that is,{gi∗ } = arg max v(gi +

∑

gi ≥0

Ti∗ = v(gi∗ +∑gj ) − c(gi +j̸=i∑gj ) − v(
gj )

j̸=i

∑

∑

− (1 − β) v(gi∗ +j̸=i gj ) +gj ) and

(4)

j̸=i

j̸=i



∑

gj ) − c(gi∗ +

j̸=i

∑




∑
∑
gj ) − v(gj ) − c(gj ) .

j̸=i

j̸=i

(5)

j̸=i

∑
∑
The maximized joint surplus of this negotiation is v(gi∗ + j̸=i gj ) − c(gi∗ + j̸=i gj ) −∑∑(v( j̸=i gj ) − c( j̸=i gj )), which is non-negative, from (4). From (5),

Ti∗ +

∑

Tj − c(j̸=i∑gj ) = β v(gi∗ +j∈N∑gj ) − c(gi∗ +

j̸=i

+

∑
j̸=i

∑

∑
Ti − c(gj ) > 0,

j̸=i

j̸=i




∑
∑
gj ) − v(gj ) − c(gj )
j̸=i

j̸=i

implying that the supplier’s payoﬀ is positive and so it always approves ((gi∗ , Ti∗ ), (gj , Tj )j̸=i ).
From (1) and (4), in every bargaining in which the supplier and a consumer reach an∑∑
agreement, the supplier and consumer i maximize v(gi + j̸=i gj∗ ) − c(gi + j̸=i gj∗ ), given(gj∗ )j̸=i through the choice of gi . Thus, if the tuple of the public goods at equilibrium isdenoted by (gj∗ )j∈N , then{gi∗ } = arg max v(gi +
gi ≥0

∑

gj∗ ) − c(gi +

j̸=i

∑

gj∗ ) for each i ∈ N .

j̸=i

By this property of the equilibrium, we obtain the next proposition.
Proposition 1 In any equilibrium, the equilibrium level of the public good∑∗j∈N gj isexactly g(1).
Proof. Suppose that in an equilibrium, (gj∗ , Tj∗ )j∈N is the first-stage outcome, which the
supplier approves in the second stage. Recall that by Assumption 1, g(1) is the unique13

maximizer of v(g) − c(g) and g(1) > 0. Therefore, gi∗ +∑∗j̸=i gj = g(1) for each i ∈ N . ■There always exists an asymmetric equilibrium such that a consumer is the one andonly contributor, and the others free ride; that is, (gj , Tj )j∈N , such that (gi , Ti ) = (g(1), βv(g(1))+(1 − β)c(g(1))) for some i ∈ N , and (gj , Tj ) = (0, 0) for each j ∈ N \{i}. In addition to thisasymmetric equilibrium, if β is suﬃciently low, then there exists a symmetric equilibrium
at which gi∗ = g(1)/n, for each i ∈ N .16 Note that the asymmetric equilibrium always
exists in any case, but the symmetric equilibrium does not always exist. Proposition 1
shows that at any equilibrium, the level of the public good is unique.
By Proposition 1, in each bilateral negotiation, the total surplus nv(g) − c(g) is nevermaximized, which induces the ineﬃcient provision of the public good (recall g(1) < g(n)).
This is because no bilateral negotiation internalizes consumers’ benefits from other bilateral negotiations. Proposition 1 resembles the well-known result of under-provision of the
public good through voluntary contribution (e.g., Bergstrom et al., 1986) in that no player
takes other players’ benefits into account when deciding on a contribution to a public good.
In our model, owing to externalities among bilateral negotiations, some consumers may
be able to enjoy the public good at no cost. In the asymmetric equilibrium, the levelof the public good provided by some consumer is g(1), which is suﬃciently high so as
not to generate a positive surplus from negotiations between the supplier and the otherconsumers. Since there is no gain from negotiating, some consumers send no transfers andbecome free riders.

5Pre-commitment on the production level of the publicgood

In this section, the supplier commits to the level of the public good prior to negotiations,that is, the supplier decides the level of the public good before it negotiates with eachconsumer on transfers. The modified model in this section is as follows: in the first stage,
16

The analysis is available upon request.

14

the supplier decides the commitment level of the public good g ≥ 0. In this stage, thesupplier does not provide the public good at this level or incur the cost. The public good isprovided after the supplier’s approval in the third stage. In the second stage, the supplierand each consumer i ∈ N bilaterally and simultaneously negotiate over the division of
the joint surplus, thus, determining Ti ≥ 0. The equilibrium outcome is determined in asimilar way to the model in the previous section. In the third stage, the supplier decides
whether to approve a project (g, (Tj )j∈N ), which is a tuple of the level of the public good
and the transfers. If the supplier approves, then it provides g units of the public good∑and receives Ti from each consumer i. As a result, the supplier’s payoﬀ is i∈N Ti − c(g)and each consumer i’s payoﬀ is v(g) − Ti . Otherwise, no public good is provided and nomoney is transferred. Then, the supplier’s and each consumer’s payoﬀs are zero.

5.1

The third stage

The analysis of this stage is the same as before. Given (g, (Ti )i∈N ), the supplier approves∑
it if and only if i∈N Ti > c(g).

5.2

The second stage

In the second stage, who contributes to the project and how much money each contributortransfers to the supplier are determined. Who contributes to the project depends on who
is pivotal to the project, as defined below.
Definition 2 Let g ≥ 0 be a level of the public good. Let i ∈ N . Let Tj be a transferfrom consumer j ∈ N \{i}. Given (Tj )j̸=i , consumer i is pivotal to the approval of a project∑∑(g, (Tj )j∈N ) if j̸=i Tj ≤ c(g) < v(g) + j̸=i Tj .17
The pivotal consumers are defined based on the third-stage equilibrium. We consider
bargaining between the supplier and consumer i ∈ N when i is pivotal to the approvalof a project. If this bargaining breaks down, then the supplier disapproves in the third17

The definition of a pivotal consumer is the same as that of Raskovich (2003).

15

stage, and the supplier’s and consumer i’s payoﬀs are zero. If an agreement is reached∑in the bargaining, then the supplier’s payoﬀ is j∈N Tj − c(g) and consumer i’s payoﬀ isv(g) − Ti . Then, the joint surplus of this bargaining isv(g) − Ti + Ti +∑Tj − c(g) = v(g) +

j̸=i

∑

Tj − c(g) > 0.

j̸=i

In Nash bargaining, this joint surplus is divided between the supplier and consumer i in
the proportion of β and 1 − β. Hence,Ti = v(g) − (1 − β) v(g) +∑Tj − c(g) = βv(g) + (1 − β)c(g) − (1 − β)j̸=iFrom (6), we confirm that∑

Tj . (6)j̸=i∑j∈N Tj > c(g). Thus, the transfer from the pivotal consumer

is necessary for the supplier to provide the public good.
Second, we consider the case in which consumer i is not pivotal to the project g, that∑∑is, the case in which either (i) v(g) + j̸=i Tj ≤ c(g) or (ii) j̸=i Tj > c(g) is satisfied.
In case (i), the joint surplus of the bargaining is zero because the supplier disapproves.
Case (ii) means that the supplier approves the project at the third stage, irrespective ofwhether the bargaining with consumer i succeeds. Hence, the supplier’s net surplus from∑∑this bargaining is Ti + j̸=i Tj − c(g) − ( j̸=i Tj − c(g)) = Ti and the consumer i’s isv(g) − Ti − v(g) = −Ti ; the joint surplus of this bargaining is zero. In any case, the joint
surplus of the bargaining with the non-pivotal consumer i is zero, which implies Ti = 0;
thus, consumer i free rides the public good.
Finally, we examine how many consumers become pivotal and how much money thepivotal consumers transfer to the supplier. Suppose that m pivotal consumers (1 ≤ m ≤ n)exist. Let M ⊆ N be the set of pivotal consumers. Since condition (6) holds for all m
consumers, solving the system of equations yields (Tjm )j∈M such that for each j ∈ M ,Tjm =βv(g) + (1 − β)c(g)(> 0 if g > 0).
β + (1 − β)m

16

(7)

The payoﬀ to the supplier is
πSm (g) ≡∑Tjm − c(g) =j∈M

β(mv(g) − c(g)),β + (1 − β)m(8)

the payoﬀ to the pivotal consumer i ∈ M is
πim (g) ≡ v(g) − Tim =(1 − β)(mv(g) − c(g)),β + (1 − β)mand the payoﬀ to the non-pivotal consumer i ∈ N \M is v(g) (the superscript “m” of Tjm
and πSm refers to the number of pivotal consumers). Henceforth, we call pivotal consumers
contributors and non-pivotal consumers free riders.
The sum of the joint surplus of the bargaining sessions with m pivotal consumers is
mv(g) − c(g), and the supplier’s share of the surplus and the pivotal consumer’s share areβ/(β + (1 − β)m) and (1 − β)/(β + (1 − β)m), respectively. Since β + (1 − β)m is thesum of the bargaining powers over the supplier and m pivotal consumers, the surplus isdistributed in proportion to the bargaining power.18
Lemma 1 shows that the equilibrium number of contributors is determined according
to the level of the public good in the first stage.
Lemma 1(1.a) For any given g ≥ 0, such that nv(g) ≤ c(g), the equilibrium number of contributors
at g is zero.
(1.b) For any given g > 0, such that nv(g) > c(g), m is the equilibrium number ofcontributors at g if and only if
c(g)
c(g)
<m≤
+ 1.
v(g)
βv(g)

(9)

At least one integer m exists that satisfies (9).
Proof. (1.a) If g satisfies nv(g) ≤ c(g), there is no surplus in any bilateral bargaining.
18

Note that (Tjm )j∈M is supported by other bargaining models. See Section 7.1.

17

Hence, no consumer pays a positive fee; the number of contributors is zero.
∑(1.b) m is the number of contributors if and only if j∈M \{i} Tjm ≤ c(g) < v(g) +∑mj∈M \{i} Tj for each i ∈ M . From (7), these inequalities hold if and only if (m−1)βv(g) ≤c(g) < mv(g), implying (9). Clearly, an integer m exists that satisfies c(g)/v(g) < m ≤(c(g)/v(g)) + 1. Hence, at least one integer m exists that satisfies (9) because β ≤ 1.

■

Note that if nv(g) > c(g), then the pivotal condition restricts the number of contributors, which means that the supplier does not necessarily receive positive transfers fromall consumers. We summarize the second-stage equilibria as follows:

The second-stage equilibria. After the level of the public good is decided at g in
the first stage, the equilibrium outcome of the second-stage subgame is (M, (Tjm )j∈N ),where M is the set of contributors, the equilibrium number of contributors m = |M | isdetermined according to Lemma 1, and the equilibrium transfer isβv(g) + (1 − β)c(g)
β + (1 − β)m
Tjm =



0

if j ∈ M,(10)
if j ∈ N \M.

Note that for some level of the public good in the first stage, there may be multiple
second-stage equilibria that support diﬀerent equilibrium numbers of contributors, because(9) may include multiple integers. For the backward induction analysis, we focus on theequilibrium that maximizes the payoﬀ to the supplier among the second-stage equilibria
in every second-stage subgame.
To investigate which second-stage equilibrium maximizes the supplier’s payoﬀ, we startby restating (9), which is based on the level of the public good. Define ḡ m for eachm ∈ {1, . . . , n}, and g m for each m ∈ {2, . . . , n}, such thatm=c(g m )c(ḡ m )
and
m
=
+ 1.
v(ḡ m )βv(g m )18(11)

Note that c(g)/v(g) is increasing in g, by (1.2) in Assumption 1. Thus, by (1.2) inAssumption 1 and (1.3), ḡ m can be defined uniquely for each m ∈ {1, . . . , n}, and g m
can be defined uniquely for each m ∈ {2, . . . , n}. We adopt the convention that g 1 ≡ 0,
by (1.3).
Lemma 2(2.a) For any given g > 0, such that nv(g) > c(g), m is the equilibrium number ofcontributors at g if and only if
g m ≤ g < ḡ m .

(9’)(2.b) For each m ∈ {2, . . . , n}, g m < ḡ m−1 if β < 1, and g m = ḡ m−1 if β = 1.
Proof. (2.a) is the restatement of (1.b) in Lemma 1. We now show (2.b). By definition,c(ḡ m−1 )/v(ḡ m−1 ) = m−1 and c(g m )/(βv(g m )) = m−1. Under these conditions, we obtainc(g m )/v(g m ) = β(m − 1) ≤ m − 1 = c(ḡ m−1 )/v(ḡ m−1 ). Since c(g)/v(g) is increasing in g,
we obtain g m ≤ ḡ m−1 .

■Lemma 2 shows that g m and ḡ m make the lower and upper bounds of the public goodlevel, where m is the equilibrium number of contributors in the second stage. In addition,
Lemma 2 shows the condition under which the second-stage subgame has multiple equilibria that support diﬀerent numbers of contributors. By (2.b), for each m ∈ {2, . . . , n},g m−1 ≤ g < ḡ m−1 (the range of m − 1 contributors) and g m ≤ g < ḡ m (the range of m contributors) overlap if and only if β < 1. Thus, in the case of β < 1, if the supplier chooses g
between g m and ḡ m−1 in the first stage, multiple second-stage equilibria exist that support
the existence of m − 1 contributors and that of m contributors in the subsequent second
stage.
When there are multiple numbers of contributors in the equilibria of a second-stage
subgame, we focus on the equilibrium that maximizes the payoﬀ to the supplier from theset of second-stage equilibria. Lemma 3 is helpful to clarify which second-stage equilibrium
maximizes the supplier’s payoﬀ.

19

Lemma 3 For each g > 0 and each m ∈ {1, . . . , n − 1},
πSm (g) ≤ πSm+1 (g),(12)

which is satisfied with equality if and only if β = 0.
Proof. From (8), we obtain
πSm+1 (g) − πSm (g) =β(βv(g) + (1 − β)c(g))≥ 0,(β + (1 − β)(m + 1))(β + (1 − β)m)■

which is satisfied with equality if and only if β = 0.

Lemma 3 shows that for each g > 0, πSm (g) is non-decreasing in m. By Lemma 3, in thesecond stage, immediately after the supplier chooses g such that nv(g) > c(g), the secondstage equilibrium that maximizes the supplier’s payoﬀ within the set of the second-stage
equilibria is the equilibrium at which the number of contributors is the maximal integer
among m that satisfies (9).

The refined second-stage equilibrium.

For each g ≥ 0, denote the equilibrium

number of contributors by m(g) ∈ {0, . . . , n}. After the level of the public good g is decidedm(g)in the first stage, the equilibrium outcome of the second-stage subgame is (M, (Tj)j∈N ),

where M is the set of contributors, the equilibrium number of contributors m(g) = |M | is
0 if g = g 1 (= 0),








1 if g ∈ (g 1 , g 2 ),
m(g) = k if g ∈ [g k , g k+1 ) (k ∈ {2, . . . , n − 1}),







n if g ∈ [g n , ḡ n ),







0 if g ≥ ḡ n ,m(g)and the equilibrium transfer (Tj)j∈N conforms to (10).
20

(13)

We learn from (13) that the equilibrium number of contributors increases as the level ofthe public good increases. Hence, the supplier sets the level of the public good “suﬃciently
high” if and only if it receives transfers from all consumers.

5.3

The first stage

As a preparation for Theorem 1, we introduce some notation. Using (11), we defineβ(g, m) ∈ R+ for each g ≥ 0 and m ∈ (1, n], such thatβ(g, m) ≡

c(g)
1
·
.
v(g) m − 1

For each g and m, β(g, m) is the value of bargaining power at which g is the lower boundover which m contributors exist: that is, g = g m if β = β(g, m). By this,β (g(m), m) =c(g(m))1c(g(m)) 1·
and β (g(m), m + 1) =· .
v(g(m)) m − 1v(g(m)) m(14)

Thus, g(m) = g m if β = β (g(m), m) and g(m) = g m+1 if β = β (g(m), m + 1).
Imposing an additional condition of the monotonicity of πSm (Condition 1 below), we
establish a necessary and suﬃcient condition under which the supplier produces the public
good eﬃciently.
Condition 1 πSm (g m ) < πSn (g n ) for each m ∈ {1, . . . , n − 1}.
Theorem 1 Under Assumption 1 and Condition 1, there is an equilibrium at which the
supplier produces the eﬃcient level of the public good (g(n)) if and only if β satisfies
0 ≤ β ≤ β (g(n), n).
The proof of Theorem 1 is provided in the Appendix. As Theorem 1 shows, β(g(n), n)marks the threshold value of the supplier’s bargaining power, below which the supplierproduces the public good eﬃciently. The supplier provides the public good eﬃciently at
an equilibrium if and only if its bargaining power is “suﬃciently weak,” under several
assumptions and conditions. At this equilibrium, the supplier receives transfers from all21

consumers, as we see in the proof of Theorem 1, and its payoﬀ is “suﬃciently close” tozero because β is suﬃciently small.
Condition 1 can be interpreted as a condition with regard to the change rate of a joint
surplus of the supplier and contributors when the number of contributors increases. Aftersome calculation, we obtain
πSn (g n ) − πSm (g m ) > 0 if and only ifnv(g n ) − c(g n )β + (1 − β)n,>mmmv(g ) − c(g )β + (1 − β)mwhere nv(g n ) − c(g n ) is the surplus of the supplier and of the contributors when all consumers are contributors, and mv(g m ) − c(g m ) is that when m consumers are contributors.
Those surpluses are evaluated at minimum levels of the public good when the contributors
are m and n. Since (β + (1 − β)n)/(β + (1 − β)m) ≥ 1, Condition 1 requires that the
surplus increases at a greater rate than a certain number when the number of the contributors increases from m to n. Remarks with regard to Condition 1 are in order. First,
this condition is not necessary to prove the suﬃciency of Theorem 1. Thus, even without
Condition 1, if the supplier’s bargaining power is suﬃciently weak, then it provides the
public good at the eﬃcient level. Second, Condition 1 is satisfied in many applications
because it is satisfied by reasonable parametric benefit and cost functions, as we see in
Proposition 2.
Intuitively, we can confirm from Lemma 3 that for any level of the public good, themore consumers from which the supplier receives transfers, the higher is the supplier’s
payoﬀ; hence, the supplier desires to receive transfers from all consumers. Since the public
good is pure, the supplier must set the level of the public good “suﬃciently high” in orderto receive transfers from all consumers; if the level of the public good is low, so that thecost of the public good is compensated by the transfers from fewer than n consumers,then free riders exist. However, there is another problem: the supplier may choose an
excessively higher level of the public good. If the supplier’s bargaining power is strong,
the transfers from contributors are large. Then, there may be a case in which the cost ofg(n) is compensated by fewer than n consumers. In this case, even if the supplier chooses
22g(n), there are free riders. Then, the supplier sets the level of the public good over g(n)to eliminate the free riders because it wants to receive transfers from all consumers. This
leads to excessive provision of the public good.19 Therefore, the supplier provides the
public good eﬃciently if and only if its bargaining power is suﬃciently weak, such thatthe cost of g(n) is compensated by all consumers, but not by fewer than n consumers.
Remark 1 We have not examined the case of n = 1. In this case of the models in
Sections 4 and 5, the public good is obviously always produced eﬃciently for any value ofthe supplier’s bargaining power in equilibrium. This is very diﬀerent from the result in
the case of n ≥ 2. The diﬀerence between the cases of n ≥ 2 and n = 1 is the existenceof externalities between bilateral negotiations. Thus, we conclude that the externalities
between bilateral bargaining sessions cause the Pareto ineﬃciency.

6

Analysis under parametric functions

6.1

Number of consumers and likelihood of allocative eﬃciency

We consider an example in which v(g) = θg α and c(g) = γg δ , where α, θ, δ, and γ are
positive constants such that 0 < α < 1 < δ. Note that v(g) and c(g) satisfy Assumption
1. From (11) and (14), we obtain
(

) 1
(
) 1
(
) 1
βθ(m − 1) δ−αmθ δ−αmαθ δ−αm, ḡ =, g(m) =
,
(15)
g =
γ
γ
δγ
(
)
γ
α
1
α
δ−α
β(g, m) =g, β(g(m), m) =+ 1 , and β(g(m), m + 1) = .
θ(m − 1)
δ m−1
δ
m

19

In Section 6.2, we provide an example in which the public good is provided excessively in equilibrium.

23

Moreover, we obtain thatπSm (g m )

(
) α (
)
β
βθ(m − 1) δ−αβθ(m − 1)=mθ − γ ·β + (1 − β)m
γ
γ
1
)
(
α
(βθ)δ δ−α=(m − 1) δ−α .
αγ

Since α/(δ − α) > 0 and m ≥ 1, πSm (g m ) is increasing in m. Thus, Condition 1 holds.
Proposition 2 Assumption 1 and Condition 1 hold if v(g) = θg α and c(g) = γg δ , whereα, θ, δ, and γ are positive constants, such that 0 < α < 1 < δ.
By Proposition 2, Theorem 1 applies to this example.
Corollary 1 Suppose that v(g) = θg α and c(g) = γg δ , where α, θ, δ, and γ are positive
constants such that 0 < α < 1 < δ. The supplier provides the public good eﬃciently inequilibrium if and only if β ∈ [0, β(g(n), n)], where β(g(n), n) = αn/(δ(n − 1)).
Proposition 3 follows directly from β(g(n), n) being decreasing in n.
Proposition 3 The set of the supplier’s bargaining powers in which the supplier provides
the public good eﬃciently at an equilibrium, [0, β(g(n), n)], shrinks and converges to [0, α/δ]
as n becomes large, if v(g) = θg α and c(g) = γg δ , where α, θ, δ, and γ are positive
constants such that 0 < α < 1 < δ.
From Proposition 3, if we treat β(g(n), n), the upper bound of the set in Proposition 3,
as the measure of the likelihood for the eﬃcient provision of public good, we conclude thatthe equilibrium likelihood of the eﬃcient provision of the public good becomes lower as thenumber of consumers increases. However, note that this likelihood does not necessarily
vanish, even if the number of consumers approaches infinity (α/δ is close to one if α and δ
are close to one). This implies that there is a possibility that a suﬃciently weak supplierprovides the public good eﬃciently, even if it negotiates with many consumers.
A comparison of our results with those of Dixit and Olson (2000) might be important,because whether beneficiaries are pivotal to public good provision plays a role in both sets24

of results. Under the two-stage game of public good provision, Dixit and Olson (2000)
examine mixed-strategy Nash equilibria, and show that each beneficiary’s equilibrium
probability of participation diminishes to zero as the number of the players increases.
Thus, the likelihood of the eﬃcient provision of the public good is extremely low when thenumber of beneficiaries is large.20 Intuitively, in their game, the probability of being pivotal
to the public good provision aﬀects the participation probability of each beneficiary.21
Note that in our analysis, the probability of being pivotal is irrelevant. As discussed
in Section 5, in our model, the supplier has an incentive to set the level of the public
good such that all consumers are pivotal to the provision of the public good. This is thesame, even when the number of consumers is very large. Hence, Proposition 3 is derived
from a diﬀerent reason to that of Dixit and Olson (2000). Although our model and themeasurement of the likelihood of eﬃciency are diﬀerent to those of Dixit and Olson (2000),
we share a similar implication: as the number of consumers (beneficiaries) increases, thelikelihood of achieving eﬃciency in equilibrium decreases.

6.2

Equilibrium level of the public good when the supplier’s bargaining
power is suﬃciently strong

Theorem 1 proves that if β > β(g(n), n), there is no equilibrium that supports the eﬃcient
provision of the public good. We discuss which of underprovision and overprovision of the
public good causes ineﬃciency in public good provision.
√
Consider an example in which v(g) = g and c(g) = g.22 Then,g(m) =( m )2
2, g m = (β(m − 1))2 , β(g(m), m) =m1
, and β(g(m), m + 1) = .
2(m − 1)
2Note that β(g(n), n) = n/(2(n−1)). In the Appendix, we show that there is an equilibrium
20

A similar implication can be obtained from the results of the voluntary participation game for a public
good mechanism (e.g., Saijo and Yamato, 1999; Shinohara, 2009; Furusawa and Konishi, 2011; Konishiand Shinohara, 2014).
21
The meaning of pivotal here is the same as that in our analysis: without a pivotal beneficiary, thepublic good is not provided.
22
The linear cost function violates Assumption (1.1). However, note that Theorem 1 holds wheneversolutions of the model are interior. See footnote 11.

25

at which g n units of the public good are supplied. Since g(n) < g n , the supplier produces
the public good over the eﬃcient level at this equilibrium. Recall that in the paragraphs
after Theorem 1, we mentioned that the supplier has an incentive to increase the level ofthe public good to receive transfers from all consumers. This example presents a case inwhich this incentive leads to the overprovision of the public good in equilibrium when the
supplier’s bargaining power is strong.
Finally, we note that the possibility cannot be denied that the supplier with strong
bargaining power underprovides the public good under other benefit and cost functions.23
Thus, the ineﬃciency in public good provision is due to the overprovision of the public
good in some cases and the underprovision of the public good in other cases.
Remark 2 We discuss the relation between the pre-commitment and the internalizationof consumers’ preferences as a result of simultaneous bilateral bargaining. The supplier
receives transfers from all consumers if its bargaining power is suﬃciently weak (see theproof of Theorem 1). It may also receive transfers from all consumers if its bargaining
power is strong, as shown in the example in this section. In general, when the supplier
receives transfers from all consumers, consumers’ benefits from the public good are embedded in the supplier’s payoﬀ (see (8)). In this sense, the internalization succeeds when
the supplier can pre-commit to the production level of the public good. This property
may induce the supplier to provide the public good eﬃciently. However, the supplier with
suﬃciently strong bargaining power provides the public good ineﬃciently. Thus, while thepre-commitment enhances the internalization of consumers’ preferences through the bargaining, this does not necessarily lead to the eﬃcient provision of a public good. Whether
the opportunity of the pre-commitment successfully yields eﬃcient provision depends on
the supplier’s bargaining power.
23

The formal analysis of the case in which the supplier has strong bargaining power is available uponrequest.

26

7

Extensions

7.1

Other bargaining models

We briefly discuss how the main result changes if we consider other kinds of bargaining inthe second stage of the model in Section 5.24

Sequential bargaining

We consider a model in which the supplier bilaterally, but

sequentially negotiates with each consumer.25 Formally, we replace the second-stage bargaining game of the model in Section 5 with the following sequential bilateral bargaining
game: the supplier first negotiates with consumer 1, then negotiates with consumer 2 after
the bilateral bargaining with consumer 1, ..., and finally, bilaterally negotiates with consumer n after the bilateral bargaining sessions with the other consumers. Each bilateralbargaining session is assumed to be Nash bargaining. The supplier has only one bilateralnegotiation with each of the n consumers. We can show that the equilibrium transfer
attained through this sequential bargaining is the same as that in (10). Hence, a result
similar to Theorem 1 can be obtained under sequential bilateral bargaining.

Multilateral bargaining based on cooperative games

Some solutions of the co-operative games support the outcome through simultaneous bilateral bargaining in themodel in Section 5, while others do not. We consider a cooperative game ({s} ∪ N, wg ),where wg : 2{s}∪N → R+ is the characteristic function, such that for any nonempty subset
C ⊆ N , wg ({∅}) = wg ({s}) = wg (C) = 0; wg ({s} ∪ C) = max{|C|v(g) − c(g), 0}, where |C|
represents the cardinality of C. First, we confirm that for each level of the public goodin the first stage g ≥ 0, the payoﬀs attained at the simultaneous bilateral bargaining aresupported as one of the core elements. Thus, if we focus on this core element, then we
obtain the same result as that in Theorem 1.
Second, we consider other solutions of the cooperative game. Since several famous
24

The formal analysis in this section is available upon request.
A sequential bargaining protocol based on the Nash bargaining solution is analyzed by Valencia-Toledo
and Vidal-Puga (2017).
25

27

solutions, such as the Shapley value, kernel, and nucleolus, satisfy the equal-treatment
property (ETP), we focus on the solution with the ETP.26 Given the solution, it is always
supported in an equilibrium that the supplier chooses g(n) in the first stage, which is
diﬀerent from the main result of this study. Which solution is appropriate in the analysis
of pure public good provision seems to depend on real-world situations, which we would liketo analyze. As discussed in the Introduction, in some cases of environmental externalities,
negotiations are conducted bilaterally. Those cases should not be captured by solutions
based on multilateral negotiations.

7.2

Heterogeneous consumers

We prove that it is partially correct that the supplier produces the public good ineﬃciently
if its bargaining power is suﬃciently strong, even when consumers are heterogeneous. We
consider the case in which there are two consumers (N = {1, 2}) and they have diﬀerent
benefit functions and bargaining power to the supplier. The payoﬀ to consumer i ∈ N isvi (g) − Ti , where vi : R+ → R+ is consumer i’s benefit function from the public good thatsatisfies vi (0) = 0, vi′ > 0, vi′′ ≤ 0, and is twice continuously diﬀerentiable. Let βi ∈ [0, 1]
be the supplier’s bargaining power with consumer i ∈ N . The cost function c(g) satisfiesc(0) = 0, c′ > 0, c′′ > 0, and is twice continuously diﬀerentiable. The timing of thegame is the same as that for the basic model. In this extended model, the Pareto-eﬃcient
allocation is not achieved in equilibrium if the supplier’s bargaining power over some (notevery) consumer is suﬃciently strong. In this sense, the main result in the case of identicalconsumers remains partially true.

8

Concluding remarks

We examine public good provision through simultaneous bilateral negotiations between
a supplier of a public good and the consumers of the good. Our question is whether apre-negotiation commitment for the public good plays an eﬀective role in achieving the26

See Peleg and Sudhölter (2007) for the definitions of those solutions.

28

eﬃcient provision of the public good. First, we show that under simultaneous bilateralnegotiations, the eﬃcient provision of the public good is never accomplished without the
pre-commitment (Proposition 1). Second, we show that if the supplier has an opportunity
to pre-commit to the production level of the public good, then under some conditions, thesupplier produces the public good eﬃciently in equilibrium if and only if its bargaining
power is suﬃciently weak (Theorem 1). In addition, we investigate the likelihood of theeﬃcient public good provision in equilibrium under reasonable parametric benefit andcost functions, and show that the equilibrium likelihood of the eﬃcient provision of the
public good diminishes as the number of consumers increases. However, in some case,the equilibrium likelihood is suﬃciently high, even if the number of consumers is verylarge (Section 6.1). A supplier with strong bargaining power, and who obtains a large
surplus, provides the public good ineﬃciently. In some cases, this ineﬃciency stems fromthe excessive provision of the public good (Section 6.2).
Our contribution is to clarify the relation between the eﬀectiveness of the pre-commitment
and the supplier’s strength in bargaining. This provides some positive and negative aspects of the commitment prior to negotiations: while the pre-commitment on the level ofthe public good plays a role in enhancing the internalization of beneficiaries’ preferences,the public good is not necessarily provided eﬃciently, even under this enhanced internalization. Our results seem to have some implication to real-world cases of the resolutionof environmental externalities. In real-world bargaining, the bargaining power would bedetermined by the relative positions of a polluter and pollutees. If the polluter is a big
company, then it may naturally be stronger than the pollutees. In this case, the polluter
abates pollution with a greater pre-commitment than without it, but the reduction levelof the pollution with pre-commitment may not be an eﬃcient level. Note that the increasein the production level of the public good through commitment is not necessarily desirable
for the whole economy.

29

Appendix
Preliminaries
Before proving Theorem 1, we provide some preliminaries.
By {g(m)} = arg maxg≥0 mv(g) − c(g), we obtain several observations, as follows.
1. By (8), if the number of contributors is fixed at m, then πSm (g) is maximized at
g = g(m). This implies that πSm (g) is increasing in g ∈ [0, g(m)) and decreasing ing ∈ [g(m), ∞).
2. g(m) < g(m + 1) for each m ∈ {1, . . . , n − 1}.
3. g(n) is the (Pareto) eﬃcient level of the public good because g(n) maximizes nv(g)−c(g).
Corollary 2 shows that πSm (g(m)) is increasing in m.
Corollary 2 For each m ∈ {1, . . . , n − 1}, πSm (g(m)) < πSm+1 (g(m + 1)).
Proof. By Lemma 3, πSm (g(m)) ≤ πSm+1 (g(m)). From the above-mentioned first and■second observations, πSm+1 (g(m)) < πSm+1 (g(m + 1)).
Lemma 4 shows the relative relationship between β(g(m), m) and β(g(m), m + 1).
Lemma 4(4.a) lim β (g(m), m) = ∞ andm→1

lim β (g(m), m + 1) =m→1c(g(1)).
v(g(1))(4.b) For each m > 1, β (g(m), m + 1) < β (g(m), m).
Proof. (4.a) By (14), as m → 1,

β (g(m), m) =c(g(m))c(g(m))c(g(1))→ ∞ and β (g(m), m + 1) =→.
v(g(m))(m − 1)v(g(m))mv(g(1))■(4.b) follows immediately from (14).

30Note that while g(m) is dependent on m, but independent of β, g m and g m+1 dependon m and β. Lemma 5 shows how g(m) and g m (or g m+1 ) are related according to thevalues of m and β.
Lemma 5 Let β ∈ (0, 1] and m ∈ (1, n]. Then,g m+1 ≤ g(m) if β ≤ β (g(m), m + 1) ,g m ≤ g(m) < g m+1 if β (g(m), m + 1) < β ≤ β (g(m), m) , andg(m) < g m if β > β (g(m), m) .

(16)
(17)
(18)

Proof. g m satisfiesc(g m )β=.
v(g m )(m − 1)
From (14), if β ≤ β (g(m), m), thenc(g m )c(g(m))= β ≤ β (g(m), m) =.
v(g m )(m − 1)v(g(m))(m − 1)
We obtain g m ≤ g(m) because c(g)/v(g) is increasing in g. Conversely, if g m ≤ g(m), thenc(g m )c(g(m))≤v(g m )(m − 1)v(g(m))(m − 1)
because c(g)/v(g) is increasing in g.

Then, by (14), β ≤ β (g(m), m).

Thus, β ≤

β (g(m), m) if and only if g m ≤ g(m). Similarly, we obtain β ≤ β (g(m), m + 1) if andonly if g m+1 ≤ g(m). Finally, we obtain (16)–(18).

■

Lemma 6 shows what level of the public good g maximizes πSm (g) under the constraintg ∈ [g m , g m+1 ), for each m and β.
Lemma 6 Let m ∈ {1, . . . , n} and β ∈ (0, 1].
(6.a) If β ≤ β (g(m), m + 1), then g m+1 ≤ g(m) (see (16)); hence, πSm (g) is increasing
within the interval [g m , g m+1 ). Thus, there is no maximizer when restricting to this31interval.
(6.b) If β (g(m), m + 1) < β ≤ β (g(m), m), then g(m) ∈ [g m , g m+1 ) (see (17)); hence,
within the interval [g m , g m+1 ), πSm (g) is maximized at g = g(m).
(6.c) If β (g(m), m) < β, then g(m) < g m (see (18)); hence, within the interval [g m , g m+1 ),πSm (g) is maximized at g = g m .

Proof of Theorem 1(⇐) Suppose that 0 ≤ β ≤ β (g(n), n). If β = 0, the supplier’s payoﬀ is zero, irrespectiveof its choice of g. Thus, g(n) is one of the optimal levels for the supplier.
Hereafter, we restrict our focus to the case in which 0 < β ≤ β (g(n), n). Given
that the number of contributors is m(g) in (13), we investigate the public good level thatmaximizes the supplier’s payoﬀ. From 0 < β ≤ β (g(n), n) and (17), we obtaing n ≤ g(n) < g n .

(19)

We partition the set {1, . . . , n − 1} (the number of contributors less than n) into threesubsets as follows:{}
N1 ≡ m ∈ {1, . . . , n − 1}| g m+1 ≤ g(m) ,}{N2 ≡ m ∈ {1, . . . , n − 1}| g m ≤ g(m) < g m+1 , and{}N3 ≡ m ∈ {1, . . . , n − 1}| g(m) < g m .

We examine which number of the contributors maximizes the supplier’s payoﬀ for eachof the three subsets. First, consider the numbers of contributors in N1 . Let m ∈ N1 . Notethat m is the number of contributors if and only if g ∈ [g m , g m+1 ). Because g m+1 ≤ g(m),πSm (g) is increasing in [g m , g m+1 ). Since πSm (g) is continuous at every g, we can define32πSm (g m+1 ) andlim πSm (g) = πSm (g m+1 ) =g↑g m+1()
β mv(g m+1 ) − c(g m+1 ).
β + (1 − β)mThus, supg∈[gm , gm+1 ) πSm (g) = πSm (g m+1 ). By Lemma 3, πSm (g m+1 ) < πSn (g m+1 ). Because
πSn (g) is increasing if g < g(n) and g m+1 ≤ g n ≤ g(n), we further obtain πSn (g m+1 ) ≤πSn (g(n)). Therefore,
πSm (g m+1 ) < πSn (g(n)) for each m ∈ N1 .

(20)

Second, we consider the numbers of contributors in N2 . Let m ∈ N2 . Since g m ≤g(m) < g m+1 , πSm (g) is maximized at g = g(m). Then, by Corollary 2,
πSm (g(m)) < πSn (g(n)) for each m ∈ N2 .

(21)

Third, consider the numbers of contributors in N3 . Let m ∈ N3 . Given that the
number of contributors is m, the supplier’s payoﬀ πSm (g) is maximized at g = g m becauseg(m) < g m . By Lemma 3, πSm (g m ) < πSn (g m ). Since πSn (g) is increasing if g < g(n) andg m < g n ≤ g(n), we further obtain πSn (g m ) ≤ πSn (g(n)). Therefore,
πSm (g m ) < πSn (g(n)) for each m ∈ N3 .

(22)

Finally, we prove that the supplier’s payoﬀ is maximized at g = g(n). Since g(n) ∈[g n , g n ) from (19), then given that the number of contributors is n, the suppliers’ payoﬀ
is maximized at g = g(n). Thus, (20) and (22) imply that the supplier’s payoﬀ when thenumber of contributors belongs to N1 ∪ N3 is less than πSn (g(n)). (21) implies that the
supplier’s payoﬀ when the number of contributors belongs to N2 is less than πSn (g(n)). In
conclusion, the supplier chooses g(n) in equilibrium.
(⇒) To the contrary, suppose that β > β (g(n), n) and an equilibrium exists at which
the supplier chooses g(n). Suppose that on the path of this equilibrium, the supplier33

chooses g(n) in the first stage, the number of contributors is µ ∈ {1, . . . , n} in the second
stage, and the project is approved in the third stage. The supplier obtains πSµ (g(n)) inthis equilibrium.
Note that the supplier chooses a level of the public good in the set [g µ , g µ ] when µ
is the number of contributors. At the same time, by (18), we obtain g(n) < g n fromβ > β(g(n), n). Thus, in this equilibrium, the number of contributors µ is less than n:µ < n.
We need to consider two possibilities: g µ < g(n) and g µ = g(n). Suppose first thatg µ < g(n). We obtain g(µ) < g(n) because µ < n. Then, because πSµ (g) is decreasingin g if g > g(µ), if the supplier sets a level of the public good a bit lower than g(n) in
the interval [g µ , g µ ], the supplier’s payoﬀ increases (the supplier can choose such a level
because of g µ < g(n)). Second, suppose that g µ = g(n). By Condition 1 and µ < n,πSµ (g µ ) = πSµ (g(n)) < πSn (g n ). In any case, the supplier does not choose g(n) in the■equilibrium, which is a contradiction.

Remark 3 Proof of the necessity above does not depend on the equilibrium selectionin the second and third stages presented in Sections 5.1 and 5.2. In addition to theequilibrium presented in Section 5.1, in the third stage, there is an equilibrium at which∑
the supplier chooses execution if and only if j∈N Tj ≥ c(g). Depending on which thirdstage equilibrium we consider, the condition under which µ consumers transfer to the
supplier is g(n) ∈ [g µ , g µ ) or g(n) ∈ (g µ , g µ ]; that is, in any equilibrium, if µ consumers
transfer to the supplier, then g(n) ∈ [g µ , g µ ]. In the proof, we consider this interval.
Analysis in Section 6.2
Suppose that the second and third-stage equilibria are the same as those in Sections 5.1
and 5.2. We now prove that the supplier’s payoﬀ is maximized when it chooses g n . Supposethat β > β(g(n), n) = 1/(2(n − 1)) + 1/2.
Note that β(g(m), m) = 1/(2(m − 1)) + 1/2 is decreasing in m. Since β(g(m), m + 1) =1/2 < β(g(m), m) = 1/(2(m − 1)) + 1/2, it follows that g(m) < g m+1 for each m (see34Lemma 5). We obtain that β(g(m), m) = β at m ≡ 2β/(2β − 1). Let m̃ ≡ 2β/(2β − 1).
Then, g m ≤ g(m) if m ≤ m̃ and g(m) ≤ g m if m ≥ m̃ (see Lemma 5). In summary,g m ≤ g(m) < g m+1 if m ≤ m̃ and g(m) ≤ g m < g m+1 if m ≥ m̃.
We also obtain

πSm (g(m)) =βm2
and πSm (g m ) = β 2 (m − 1).
4(1 + (1 − β)(m − 1))

Since πSm (g(m)) is increasing in m, we obtain πSm (g(m)) < πSm̃ (g(m̃)) for each m < m̃.
By (6.b) of Lemma 6, in [0, g m̃ ), the supplier’s payoﬀ is maximized at g = g(m̃), and the
maximized payoﬀ is
πSm̃ (g(m̃)) =β2.
2β − 1Since πSm (g m ) is increasing in m, we obtain πSm (g m ) < πSn (g n ) for each m ∈ [m̃, n). From
(6.c) of Lemma 6, in [g m̃ , ∞), it is maximized at g = g n and the maximized payoﬀ isπSn (g n ) = β 2 (n − 1). The diﬀerence between those payoﬀs is
πSn (g n ) − πSm̃ (g(m̃)) =β 2 ((2n − 1)β − n)n
> 0 since β >.
2β − 1
2(n − 1)Since πSm (g(m)) is increasing in m, we also obtain πSn (g n )−πSm (g(m)) > 0 for all integers
m, such that m ≤ m̃. Therefore, in this numerical example, the supplier oversupplies the
public good when its bargaining power is suﬃciently strong.

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39

Online Appendix for“Pre-negotiation commitment and internalization in public good
provision through bilateral negotiations”
Noriaki Matsushima and Ryusuke Shinohara
Symmetric equilibria in Section 4
We briefly discuss the possibility of the (non)existence of symmetric equilibria in the modelin Section 4. We focus on an equilibrium in which gi∗ = g(1)/n for each i ∈ N , and every∑∑∗∗
consumer i ∈ N is pivotal:j̸=i Tj ≤ c( j̸=i gj ), for each i ∈ N . In this case, for eachi ∈ N,
Ti∗ = v(∑
gj∗ ) − (1 − β) v(j∈N∑gj∗ ) − c(

j∈N

∑

j∈N

gj∗ ) +∑Tj∗  .

j̸=iSolving these simultaneous equations yields
Ti∗ =

βv(

∑

∑

∗
j∈N gj ) + (1 − β)c(

β + (1 − β)n∗j∈N gj )for each i ∈ N.

∑∑
Using this transfer value, i∈N gi∗ = g(1), and c( j̸=i gj∗ ) = (n − 1)g(1)/n, we obtain that∑∑∗∗j̸=i Tj ≤ c( j̸=i gj ) if and only if)(g(1)
c(g(1))(1 − n) + nc n−1
n
[
(
(
))] .
β≤
(n − 1) v(g(1)) − (n − 1) c(g(1)) − c n−1g(1)nNote that the right-hand side of the previous inequalities is not necessarily positive. If it
is positive, then the symmetric equilibrium exists only if the supplier’s bargaining poweris suﬃciently small. Otherwise, the equilibrium does not exists for any β.

Equilibrium level of the public good when β is suﬃciently highWe focus on the analysis of the first stage, given the second- and third-stage equilibrium in
Subsections 5.1 and 5.2. Assuming Condition 1, we examine the case in which the supplieridetermines the level of the public good in the first stage.
Suppose that β > β (g(n), n). Then, by Lemma 5, g(n) < g n . Similarly to the proofof Theorem 1, we make a partition of the number of contributors less than n, such that
N1 = {m ∈ {1, . . . , n − 1}| g m+1 ≤ g(m)},N2 = {m ∈ {1, . . . , n − 1}| g m ≤ g(m) < g m+1 }, andN3 = {m ∈ {1, . . . , n − 1}| g(m) < g m }.
By the same calculation in the proof of Theorem 1,27 we obtain that for each m ∈ N1 andeach g ∈ [g m , g m+1 ),πSm (g) < πSm (g m+1 ) < πSm+1 (g m+1 ) ≤ πSn (g n ).
For each m ∈ N2 and each g ∈ [g m , g m+1 ),∗πSm (g) ≤ πSm (g(m)) ≤ πSm (g(m∗ )),in which m∗ is the maximal integer in N2 . For each m ∈ N3 and each g ∈ [g m , g m+1 ),πSm (g) ≤ πSm (g m ) < πSn (g n ).
In summary, for each m ∈ N1 ∪ N3 and each g ∈ [g m , g m+1 ),πSm (g) < πSm (g n ).
For each m ∈ N2 and each g ∈ [g m , g m+1 ),∗πSm (g) ≤ πSm (g(m∗ )).
27

See also Lemma 6.

ii

We obtain that g(m∗ ) < g(n) by m∗ < n and that g(n) < g n . However, under our∗

general setting, we cannot determine the sign of πSm (g(m∗ )) − πSn (g n ). Therefore, the∗equilibrium level of the public good is g(m∗ ) if πSm (g(m∗ )) > πSn (g n ) and is g n otherwise.
In conclusion, in any subcase, the supplier chooses an ineﬃcient level of the public goods,that is, a level higher or lower than the eﬃcient level.

Simultaneous bilateral bargaining and the core in Section 7.1
We show that the payoﬀs attained with simultaneous bilateral bargaining belong to thecore of some appropriate cooperative game.
Let N = {s} ∪ N , where s is the supplier and N = {1, . . . , n} is the set of consumers,
with n ≥ 2. For each g > 0, such that nv(g) > c(g), define the characteristic function wg :2N → R+ as follows: for any nonempty subset C ⊆ N , wg ({∅}) = wg ({s}) = wg (C) = 0;
wg ({s} ∪ C) = max{|C|v(g) − c(g), 0}.28 By this definition, the cooperation of the supplierand consumers is necessary for a positive surplus. If C is suﬃciently large, |C|v(g) > c(g)holds; hence, the supplier provides g (decided in the first stage) and the consumers in
C pay the fee to the supplier.The supplier is assumed to commit the level of the public
good decided in the first stage. In the negotiation in coalitions, only the surplus division is∑considered. A payoﬀ profile u = (ui )i∈N belongs to the core of (N , wg ) if i∈C ui ≥ wg (C)∑for each C ⊆ N and i∈N ui = wg (N ).
Let β ∈ [0, 1] be the supplier’s bargaining power. Let g > 0 be a level of the public
good such that nv(g) − c(g) > 0. Let m be a number of pivotal consumers. Note thenthat mv(g) > c(g) holds by the pivotal condition. Let M ⊆ N be a set of contributors.
Let us , uc , and uf denote the payoﬀs to the supplier, the contributor, and the free rider
28|C| represents the cardinality of C.

iii

through the simultaneous bilateral bargaining in the second stage, respectively. Then,β(mv(g) − c(g))≥ 0,
β + (1 − β)m(1 − β)(mv(g) − c(g))= v(g) − Tim =≥ 0, andβ + (1 − β)m

us = πSm (g) =uc
uf= v(g) > 0.

Proposition A1 A payoﬀ profile (ui )i∈N , such that us = us , ui = uc for any i ∈ M , andui = uf for any i ∈ N \M belongs to the core of (N , wg ).
∑Proof. Trivially,ggi∈C ui ≥ w (C) for each C ⊆ N , such that w (C) = 0.

We need toconsider the coalition {s}∪C, such that C ⊆ N and wg ({s}∪C) > 0; that is, |C|v(g) > c(g).
Let C c ⊆ C be the set of contributors in C and let C f ⊆ C be the set of free riders in C.
We obtain∑ui =i∈{s}∪Cβ + (1 − β)|C c |(mv(g) − c(g)) + |C f |v(g).
β + (1 − β)mThen, we obtain
β + (1 − β)|C c |(m − |C c |)(mv(g)−c(g))−(|C c |v(g)−c(g)) =(βv(g) + (1 − β)c(g)) ≥ 0.
β + (1 − β)m
β + (1 − β)m
Thus,

∑

i∈{s}∪C ui≥ |C c |v(g) − c(g) + |C f |v(g) = wg ({s} ∪ C). In conclusion, (ui )i∈N■

belongs to the core of (N , wg ).

Remark A1 We can prove that the core of (N , wg ) coincides with the set of payoﬀs∑{(ui )i∈N | ui ≥ 0 for each i ∈ N andi∈N ui = nv(g) − c(g)}.

Sequential bilateral bargaining in Section 7.1
We replace the second-stage bargaining of the model in Section 5 with the following sequential bilateral bargaining model. The supplier first negotiates with consumer 1, second
negotiates with consumer 2 after the bilateral bargaining with consumer 1, ..., and finally,iv

negotiates with consumer n bilaterally. The supplier has only one bilateral negotiation
with each consumer.
We focus on the third-stage equilibrium at which for each project (g, (Tj )j∈N ), the∑supplier executes the project if and only if j∈N Tj > c(g).
Given this third-stage equilibrium, we solve the second-stage sequential bargaining by
the backward application of the Nash bargaining solution. Given the outcome (g, (Tj )j̸=n ),the supplier negotiates with consumer n.
• If∑

j∈N \{n} Tj > c(g) or if∑j∈N \{n} Tj ≤ c(g) and v(g) ≤ c(g) −∑j∈N \{n} Tj , thenthe surplus of the bargaining session of the supplier and consumer n is zero; hence,
Tn = 0.
• If∑

j∈N \{n} Tj ≤ c(g) and v(g) > c(g)−∑j∈N \{n} Tj , the surplus of this bargaining is

positive and consumer n pays to the supplier Tn = v(g)−(1−β)(v(g)+∑c(g)) = βv(g) − (1 − β)( j∈N \{n} Tj − c(g)).

∑

j∈N \{n} Tj −Given this outcome of the Nash bargaining solution, we next investigate bilateral bargaining with consumer n − 1.
We investigate the bilateral bargaining with consumer k ∈ {1, . . . , n − 1}, assumingthat we have investigated the bilateral bargaining with consumer n to consumer k + 1,
by the backward application of the Nash bargaining solution. Let (Tj )nj=k+1 be the Nashbargaining outcome of the bargaining with consumers k + 1 to n. Before reaching thebargaining with consumer k, the supplier negotiates with consumer 1 to k − 1. Let (Tj )k−1
j=1
be the outcome of the bargaining before the bargaining with consumer k. Given (Tj )k−1
j=1 ,the supplier and consumer k negotiate, anticipating that (Tj )nj=k+1 is obtained in thesubsequent negotiations.
• If

∑k−1

j=1 Tj +∑n

j=k+1 Tj

> c(g) or if∑k−1

j=1 Tj +∑n

j=k+1 Tj

≤ c(g) and v(g) ≤∑∑nc(g) − ( k−1
j=1 Tj +j=k+1 Tj ), then the surplus of the bargaining session of the

supplier and consumer k is zero; hence, Tk = 0.

v

• If

∑k−1

j=1 Tj +∑n

j=k+1 Tj ≤ c(g) and v(g) > c(g) −∑k−1

j=1 Tj −∑n

j=k+1 Tj , the surplusof this bargaining is positive and consumer k pays to the supplier Tk = v(g) − (1 −∑∑β)(v(g) + j̸=k Tj − c(g)) = βv(g) − (1 − β)( j̸=k Tj − c(g)).
Given this Nash bargaining outcome, we return to the bilateral bargaining with consumer k −1. Repeating a similar procedure until reaching the bargaining with consumer 1,∑
we finally obtain the following results. For each k ∈ N , Tk = βv(g)−(1−β)( j̸=k Tj −c(g))
if∑Tj ≤ c(g) and v(g) > c(g) −j∈N \{k}

∑

Tj

(23)

j∈N \{k}and Tk = 0 if∑j∈N \{k}Tj > c(g) or ∑

Tj ≤ c(g) and v(g) ≤ c(g) −j∈N \{k}∑Tj  .

(24)

j∈N \{k}Conditions (23) and (24) are equivalent to the pivotal condition and the non-pivotal
condition, respectively. For each k ∈ N , if (23), k is pivotal to the supplier’s approval of a∑
project and Tk = βv(g)−(1−β)( j̸=k Tj −c(g)); if (24), then k is non-pivotal and Tk = 0.
This is the same as the outcome of the simultaneous bilateral bargaining. Therefore, evenunder the sequential bilateral negotiations, Theorem 1 still holds.

Multilateral bargaining based on cooperative games in Section 7.1
Since several famous solutions, such as the Shapley value, kernel, and nucleolus, satisfy
the equal treatment property (ETP), we focus on the solution with the ETP. Since theconsumers have identical benefit functions in the basic model, all distinct consumers iand j are interchangeable in the sense that, for each coalition D ⊆ {s} ∪ N that contains
neither i nor j, wg (D ∪ {i}) − wg (D) = wg (D ∪ {j}) − wg (D). If the solution satisfies the
ETP, then it assigns the same payoﬀ to all interchangeable players (i.e., all consumers in
our model). Formally, let (ui )i∈{s}∪N ∈ Rn+1
be payoﬀs that are assigned by the solution+∑
with the ETP and that satisfy us + i∈N ui = wg ({s} ∪ N ). Then, a surplus-sharingvi

ratio exists that is common to all consumers, rc ∈ [0, 1/n], such that ui = rc wg ({s} ∪ N )for each i ∈ N and us = (1 − nrc )wg ({s} ∪ N ). If g is the level of the public goodin the first stage, then the supplier’s payoﬀ, πSETP (g), and consumer i’s, πiETP (g), are
πSETP (g) = max{(1 − nrc )(nv(g) − c(g)), 0} and πiETP (g) = max{rc (nv(g) − c(g)), 0},
respectively. Given the solution, it is always supported at an equilibrium that the supplierchooses g(n) in the first stage, Therefore, under the solution satisfying the ETP, the public
good is always provided eﬃciently, which is diﬀerent from the main result of this study.
This diﬀerence comes from the property that all consumers obtain the same payoﬀ under
the solution with the ETP. This means that they all transfer the same amount of money
to the supplier and they cannot free ride. On the other hand, the simultaneous bilateralbargaining allows asymmetry of consumers’ payoﬀs and free riding.
What solution is appropriate in the analysis of pure public good provision seems todepend on real-world situations which we would like to analyze. As discussed in Introduction, in some cases with regard to environmental externalities, negotiations are conductedbilaterally. Those cases should not be captured by solutions based on multilateral negotiations. Theoretically, it would be a problem whether the ETP solution is appropriate forthe analysis of public good provision. Under the ETP solution, consumers cannot free ridea public good because they pay the same compensation for the provision of a pure public
good. However, by the non-excludability of the public good, we may naturally consider
that the consumer has an option to free ride. We can easily confirm that, with this option,
each consumer is made better oﬀ by free riding the public good provided by the others,
even under the ETP solution.29
Suppose that (n−1)v(g) > c(g) and i ∈ N is a consumer whose payoﬀ is uETP
≡ rc (nv(g)−c(g)) underithe solution with the ETP. Since (n − 1)v(g) > c(g), then wg ({s} ∪ N \{i}) > 0, which means that thecoalition {s} ∪ N \{i} produces g units of the public good even if consumer i opts out from {s} ∪ N . Thus,
since the public good is pure, consumer i can enjoy the free-riding payoﬀ v(g) even if he/she opts out of N .
The free-riding payoﬀ v(g) clearly outperforms uETP
because rc ≤ 1/n and uETP
≤ v(g) − (c(g)/n) < v(g).
i
i
29

vii

Heterogeneous consumers in Section 7.2
The analysis is based on the assumption that v1 (g) < v2 (g) for all g > 0 and β1 ≤ β2 .
Under this assumption, for any level of the public good, consumer 2’s benefit from thepublic good is greater than consumer 1’s; the supplier’s bargaining power with consumer
2 is not lower than that with consumer 1.
The analysis of the third stage is the same as that of the case of identical consumers.
Based on the third-stage equilibrium, consumer i ∈ N is said to be pivotal to (g, Tj ) (i ̸= j)
if vi (g) + Tj > c(g) ≥ Tj .
In the second stage, if consumer i is non-pivotal to (g, Tj ), then Ti = 0. If consumer i
is pivotal, thenTi = vi (g) − (1 − βi )(vi (g) + Tj − c(g)) = βi vi (g) − (1 − βi )(Tj − c(g)).

(25)

If consumer i is pivotal and consumer j is not, then Ti = βi vi (g)+(1−βi )c(g) and Tj ={i}

0. The supplier’s payoﬀ is πS (g) ≡ βi (vi (g)−c(g)) (The upper-script letter {i} means thatthe supplier receives transfers from consumer i). Consumer i’s payoﬀ is (1−βi )(vi (g)−c(g))
and consumer j’s payoﬀ is vj (g).
If consumers 1 and 2 are pivotal, then (25) is satisfied for each i ∈ N ; hence,Ti =βi vi (g) − (1 − βi )βj (vj (g) − c(g)),βi + βj − βi βj(26)where j ∈ N \{i}.30 The supplier’s payoﬀ is
πSN (g) ≡ T1 + T2 − c(g) =β1 β2(v1 (g) + v2 (g) − c(g)),β1 + β2 − β1 β2in which the upper-script letter N means that the supplier receives transfers from consumers 1 and 2.
Consumer i is pivotal to (g, Tj ) if and only if vi (g) + Tj > c(g) ≥ Tj (j ∈ N \{i}). By
30

We assume in the subsequent analysis that Ti > 0 for each i ∈ N when both consumers are pivotal.

viii

(26), vi (g) + Tj > c(g) if and only if vi (g) + vj (g) > c(g). From (26), c(g) ≥ Tj if and onlyif0≥1
(βj (vj (g) − c(g)) − (1 − βj )βi vi (g))βi + βj − βi βj

if and only if
vj (g) −(1 − βj )βivi (g) ≤ c(g).
βjTherefore, consumer i is pivotal to (g, Tj ) if and only if g satisfies
vj (g) −(1 − βj )βivi (g) ≤ c(g) < vi (g) + vj (g) (j ∈ N \{i}).
βj

(27)

Since v1 (g) < v2 (g) for each g > 0 and β1 ≤ β2 ,(1 − β2 )β1(1 − β1 )β2
v2 (g) < v2 (g) −v1 (g).
β1
β2

v1 (g) −Thus, from (27), consumers 1 and 2 are pivotal if and only if g satisfies
v2 (g) −(1 − β2 )β1
v1 (g) ≤ c(g) < v1 (g) + v2 (g).
β2

(28)

Clearly, (28) does not hold at g = 0.
To simplify the discussion on (28), we further assume that for each i ∈ N , vi (g) = λi g,where 0 < λ1 < λ2 , and c(g) = g 2 /2. Under those functions, consumers 1 and 2 are pivotal
if and only if g satisfies()(1 − β2 )β1
2 λ2 −λ1 ≤ g < 2(λ1 + λ2 ).
β2Let g(N ) ≡ arg maxg≥0

(29)

∑

j∈N vj (g) − c(g) and g({i}) ≡ arg maxg≥0 vi (g) − c(g) for eachi ∈ N . Then, g(N ) = λ1 + λ2 and g({i}) = λi for each i ∈ N . Note that whetherg(N ) = λ1 + λ2 satisfies (29) depends on the values of λi and βi (i ∈ N ).
In Claim 1, we show that the eﬃcient provision of the public good is achieved at
an equilibrium of the game only if consumers 1 and 2 are both pivotal to the eﬃcientix

provision.
Claim 1 There is an equilibrium of the game that supports the provision of g(N ) unitsof the public good only if)((1 − β2 )β1
λ1 .
g(N ) ≥ 2 λ2 −
β2

(30)

Proof. Suppose, to the contrary, that (30) does not hold, but an equilibrium exists at
which g(N ) units of the public good are provided.
Since (30) does not hold, it is impossible that both consumers 1 and 2 are pivotal tothe provision of g(N ) units of the public good. Hence, one of the consumers is pivotal
to the provision. For each g > 0, if consumer 1 is solely pivotal to the provision of gunits of the public good, then 0 < g ≤ 2λ1 .31 This, together with 2λ1 < g(N ), implies
that v1 (g(N )) − c(g(N )) < 0; hence, consumer 1 cannot be solely pivotal to the provisionof g(N ) units of the public good. Therefore, if g(N ) is chosen in the first stage, thenconsumer 2 is solely pivotal to the provision of it.
Consider g̃, such that max{2λ1 , λ2 } < g̃ < g(N ). Note that at g̃, consumer 1 cannot
be solely pivotal because 2λ1 < g̃, and consumer 2 can be pivotal because g̃ < 2λ2 . The{2}

supplier’s payoﬀ when it chooses g̃ is πS (g̃) = β2 (v2 (g̃) − c(g̃)) and its payoﬀ when it{2}chooses g(N ) is πS (g(N )) = β2 (v2 (λ1 + λ2 ) − c(λ1 + λ2 )). Since λ2 < g̃ < g(N ) and
{2}

{2}

v2 (g) − c(g) is maximized at g = g({2}) = λ2 , we obtain πS (g(N )) < πS (g̃), which
contradicts that g(N ) units of the public good are provided in equilibrium.

31

Here, 2λ1 is the maximal level of the public good that guarantees v1 (g) − c(g) ≥ 0.

x■