Discussion Paper No. 559

VOLUNTARY PARTICIPATION
IN THE DESIGN OF NON-EXCLUDABLE
PUBLIC GOODS PROVISION MECHANISMS

Tatsuyoshi Saijo
and
Takehiko Yamato

October 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Voluntary Participation in the Design of Non-excludable
Public Goods Provision Mechanisms#

byTatsuyoshi Saijo*
andTakehiko Yamato**

October 2001

# Research was partially supported by the Grant in Aid for Scientific Research 1143002 of the Ministry of
Education, Science and Culture in Japan, the Nissan Foundation and the Sumitomo Foundation.
* Institute of Social and Economic Research, Osaka University, Ibaraki, Osaka 567-0047, Japan, and ResearchInstitute of Economy, Trade and Industry, 1-3-1 Kasumigaseki, Chiyoda, Tokyo 100-8901, Japan. E-mailaddress: saijo@iser.osaka-u.ac.jp** Department of Value and Decision Science, Graduate School of Decision Science and Technology,
Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan. E-mail address:
yamato@valdes.titech.ac.jp

Abstract

Groves-Ledyard (1977) constructed a mechanism attaining Pareto efficient
allocations in the presence of public goods. After this path-breaking paper, manymechanisms have been proposed to attain desirable allocations with public goods. Thus,
economists have thought that the free-rider problem is solved, in theory. Our view tothis problem is not so optimistic. Rather, we propose fundamental impossibility
theorems with public goods. In the previous mechanism design, it was implicitly
assumed that every agent must participate in the mechanism that the designer provides.
This approach neglects one of the basic features of public goods: non-excludability. We
explicitly incorporate non-excludability and then show that it is impossible to construct
a mechanism in which every agent has an incentive to participate.

Correspondent:
Tatsuyoshi Saijo
Institute of Social and Economic Research
Osaka University
Ibaraki, Osaka 567-0047
Japan
Phone 01181-6 (country & area codes)
6879-8582 (office)/6878-2766 (fax)E-mail: saijo@iser.osaka-u.ac.jp

Abstract

Groves-Ledyard (1977) constructed a mechanism attaining Pareto efficient
allocations in the presence of public goods. After this path-breaking paper, manymechanisms have been proposed to attain desirable allocations with public goods. Thus,
economists have thought that the free-rider problem is solved, in theory. Our view tothis problem is not so optimistic. Rather, we propose fundamental impossibility
theorems with public goods. In the previous mechanism design, it was implicitly
assumed that every agent must participate in the mechanism that the designer provides.
This approach neglects one of the basic features of public goods: non-excludability. We
explicitly incorporate non-excludability and then show that it is impossible to construct
a mechanism in which every agent has an incentive to participate.

Correspondent:
Tatsuyoshi Saijo
Institute of Social and Economic Research
Osaka University
Ibaraki, Osaka 567-0047
Japan
Phone 01181-6 (country & area codes)
6879-8582 (office)/6878-2766 (fax)E-mail: saijo@iser.osaka-u.ac.jp

1. Introduction
The provision of public goods has an incentive problem called the free-rider
problem. As Samuelson (1964) pointed out, it is impossible to attain a Pareto efficient
allocation through a decentralized fashion, in particular, a decentralized pricing system.
On the contrary, Groves and Ledyard (1977) proposed an explicit procedure, called amechanism, in which the Nash equilibrium allocation is Pareto efficient. Participants canpursue their own self-interest being free riders if they choose in the mechanism, but themechanism is free from these incentives. In this sense, as the subtitle of their paper
shows, they found a solution to the free-rider problem.
Although the allocation of the Groves-Ledyard mechanism is Pareto efficient,the mechanism is not individual rational. That is, the allocation does not satisfy thecondition where it is at least as good as each participant's initial endowment. Following
the path-breaking paper by Groves and Ledyard (1977), Hurwicz (1979) and Walker(1981) fixed this problem and succeeded in implementing the Lindahl correspondence inNash equilibria, which satisfies both Pareto efficiency and individual rationality.
Subsequently, numerous mechanisms have been proposed that satisfy additionaldesirable properties such as individual feasibility and balancedness1.
Most mechanisms developed thus far share one undesirable property, however:participants in the mechanisms do not have freedom not to participate. As Olson (1965)
noticed, any non-participant can obtain benefit of a public good that is provided by
1 See Groves and Ledyard (1987) and Hurwicz (1994). For the dominant strategy equilibrium concept, wehave impossibility results: Pareto efficiency, individual rationality, and incentive compatibility (i.e., truthtelling is a dominant strategy) are inconsistent. See Hurwicz (1972) without public goods and Ledyard andRoberts (1974) with public goods. Saijo (1991) showed an impossibility result without requiring Pareto
efficiency: a slightly stronger individually rational condition called an autarkically individual rationality and1others. This is due to the nature of a public good called non-excludability. In other words,
Groves and Ledyard and their followers found solutions to the free-rider problem once
every participant decided to participate in the mechanisms, but not solutions to the
problem when agents have the ability not to participate.
This participation problem is important in many practical situations, such as forinternational treaties. The Kyoto Protocol to cope with global warming and climatechange is a specific, recent example. It took years to agree on the basic framework, theUnited Nations Framework Convention on Climate Change (UNFCCC), to reduce thegreen house gases. UNFCCC was adopted in 1992 and entered into force in 1994. The
parties of UNFCCC adopted the Kyoto Protocol in 1997. The Protocol is a mechanism in
our terminology to attain the goal of UNFCCC. The number of signatories including the
U.S.A. exceeded 186 in 2000, but in March 2001, President Bush announced not to ratify
the Protocol since it is harmful to the U.S. economy. Therefore, the effectiveness of theProtocol remains in doubt. In our framework, ratification is equivalent to participation.
Another example is the League of Nations. Following World War I President Woodrow
Wilson strongly supported the League, but the U.S. Congress never ratified the Treaty of
Versailles2.
Thus, our fundamental question is: is there any mechanism satisfying thecondition that every agent always chooses participation strategically, called the voluntary
participation condition? In order to answer this question, we first restrict our attention to
the Lindahl allocations as a goal of our society. That is, our first question is whether or

incentive compatibility are inconsistent. However, our impossibility results described below do not dependon the choice of equilibrium concepts.

2not any mechanism attaining Lindahl allocations, called a Lindahl mechanism, cansurvive if we allow agents to choose participation in the mechanism voluntarily. What
we found is striking. Each agent has an incentive not to participate in the mechanism ina wide class of environments.
Based upon the preliminary result, we return to the fundamental question withthe two-agent economy. We find that no voluntary participation mechanism exits under
mild regularity conditions. Furthermore, this result is independent of the choice ofequilibrium concepts.
The picture is still bleak even if the number of participants is at least three.
Imposing Pareto efficiency on a mechanism, we again find a negative result. The reason
why we obtain the negative result might come from Pareto efficiency on which we
impose. We have a partial answer to this question. The voluntary contribution
mechanism, which cannot attain Pareto efficiency, does not satisfy the voluntary
participation condition, either.
Moulin (1986), Palfrey and Rosenthal (1984), and Saijo and Yamato (1999) also
analyzed the issue of an incentive to participate in a mechanism for the provision of apublic good.3 Moulin and Palfrey-Rosenthal focused on specific mechanisms: Moulin
studied the pivotal mechanism in discrete public goods economies with quasi-linear
preferences, whereas Palfrey and Rosenthal considered a simple mechanism for theprovision of a binary public good with binary contributions. Saijo and Yamato examined

2 Voluntary public goods provision -- such as for public broadcasting -- also faces the participation problem.

For example, part of public broadcasting in Japan is supported by the public broadcasting fee. Every familymust pay the fee by law, but many choose not to since punishment is practically non-existent.
3 The participation problem in an institution has been examined mainly in the context of voting and cartel
formation (e.g., see Brams and Fishburn (1983), Dixit and Olson (2000), Ledyard (1984), Okada (1996), Palfrey
and Rosenthal (1983,1985), and Selten (1973)).

3

a specific two sage game: the first stage is a decision stage of participation in amechanism, and only the participants in the first stage play the second stage. On theother hand, we investigate participation incentive properties of a large class of
mechanisms in economic environments with a continuous public good.
The paper is organized as follows. In Section 2, we explain an example
illustrating our basic idea. In Section 3, we introduce notation and definitions. We
establish an impossibility result on participation incentives for the case of two agents in
Section 4 and that for the case of at least three agents in Section 5. In Section 6, weinvestigate the voluntary contribution mechanism. In the final section, we makeconcluding remarks.

2. An Example: Lindahl Mechanisms
We analyze the following symmetric two-agent economies with one private
good x and one pure public good y that is non-excludable and non-rival. The public
good can be produced from the private good by means of a constant return to scaletechnology, and let y = x be the production function of the public good. A consumption
bundle for agent i is denoted by ( xi , y) ∈ ℜ 2+ where xi ∈ ℜ + is the level of private goodshe consumes on her own, and y ∈ ℜ + is the level of public good. Two agents have thesame preferences that can be represented by a Cobb-Douglas utility function uαi ( xi , y ) =xαi y 1−α , where 0 < α < 1 and i = 1,2. Each agent's initial endowment is also the sameand given by ( ω i , 0) = (10, 0) for i = 1,2. We investigate situations in which the true
value of the preference parameter α is unknown to the mechanism designer, but theinitial endowment and the production technology are known.

4Consider any mechanism implementing the Lindahl correspondence in Nashequilibria (see Hurwicz (1979), Walker (1981), Hurwicz, Maskin, and Postlewaite (1984),
Tian (1990), and so on).4 Suppose that each agent is able to choose whether she
participates in the mechanism. Then in order to achieve the desired Lindahl equilibrium
allocation by using the mechanism, every agent must choose participation. Therefore, we
ask a crucial question of whether each agent always have an incentive to participate in
the mechanism. Unfortunately, our answer to this question is negative.
To see why, let T ⊆ {1,2} be the set of agents who participate in the mechanism.5

An equilibrium allocation of the mechanism when the agents in T participate in it isdenoted by (( xTi )i∈T , y T ) .6 If two agents decide to participate in the mechanism, then( x1{ 1,2 } , x2{ 1,2 } , y { 1,2 } ) is a Lindahl allocation of the economy with two agents, since themechanism implements the Lindahl correspondence.7 It is straightforward to check that
there exists a unique Lindahl allocation given by ( x1{ 1,2 } , x2{ 1,2 } , y { 1,2 } ) = (10α, 10α, 20(1-α)).
Now suppose that some agent i does not participate in the mechanism, while{ j}

the other agent j ≠ i does, i.e., T = { j} . Then ( x j , y { j} ) is a unique Lindahl allocation of{ j}

the economy consisting of only one agent j . It is easy to see that ( x j , y { j} ) = (10α, 10(1-α)). Notice that non-participant i can enjoy her initial endowment, ω i , as well as the
4 The Lindahl correspondence is the same as the constrained Lindahl correspondence (Hurwicz, Maskin, andPostlewaite (1984)) under the present assumptions.
5 A "participant" stands for an agent who chooses participation in a mechanism, while an "agent" represents
any member who belongs to an economy.
6 A mechanism specifies a strategy set of each participant in T and an outcome function for each T ⊆ { 1 ,2 } .
This definition of a mechanism is more general than the usual one.

5

non-excludable public good produced by agent j ≠ i , y { j} . On the other hand, she is nolonger able to affect the decision on the provision of the public good. The following
condition should be satisfied if each agent has an incentive to participate in themechanism:(1)

uαi ( xi{ 1,2} , y { 1,2} ) ≥ uαi (ω i , y { j} ) for i, j = 1,2, j ≠ i ,where uαi is any Cobb-Douglas utility function. We call condition (1) the voluntary
participation condition.8 We show that no mechanism implementing the Lindahl
correspondence satisfies this condition. This fact can be illustrated for the case of α = 0.6
by using Kolm's triangle. See Figure 1. Point A denotes the Lindahl equilibrium
allocation when both agents participate in the mechanism: A = ( x1{1, 2} , x2{1, 2} , y {1, 2 } ) = (6,6, 8). Point B represents the allocation when agent 1 does not participate in themechanism, but agent 2 does: B = ( ω 1 , x2{ 2} , y { 2} ) = (10, 6, 4). Since u10.6 ( x1{1, 2} , y {1, 2 } ) ≈6.73 < u10.6 ( ω 1 , y { 2} ) ≈ 6.93, agent 1 would not participate in the mechanism when agent
2 does. The same thing holds for agent 2.

Figure 1 is around here.


3. Notation and Definitions

7 A mechanism implements the Lindahl correspondence if for each set of participants T ⊆ { 1 ,2 } and eacheconomy consisting of the participants in T , every equilibrium allocation is a Lindahl allocation and every
Lindahl allocation is an equilibrium allocation.
8 The voluntary participation condition is different from the individually rational condition which{ 1,2 } { 1,2 }
{ j}
α
,y
) ≥ uαi (ω i ,0) for i = 1,2. Since uαrequires that uαi ( xii (ω i , y ) ≥ ui (ω i ,0) , the voluntary

participation condition is stronger than the individually rational condition.

6

In the previous section, we saw that any Lindahl mechanism fail to satisfy thevoluntary participation condition in symmetric economies with two agents when Nashequilibrium is an equilibrium concept. We will show below that a similar negative result
holds for any equilibrium concept and any mechanism meeting mild conditions.
First of all, we introduce notation and definitions. In Section 2, we investigated
two-agent economies with one private good, one pure public good, and a constant returnto scale technology. We study the same situations with many agents. Let N = {1,2,...,n}be the set of agents, with generic element i. We assume that each agent i's preference
relation admits a numerical representation ui : ℜ 2+ → ℜ which is continuous, concave,and monotonic. Let U i be the class of utility functions admissible for agent i. Let P(N)be the collection of all no-empty subsets of N. For T ∈ P(N), letuT ≡ (ui )i∈T ∈UT ≡ × i∈T U i be a preference profile for the agents in T.
Agent i's initial endowment is denoted by ( ω i , 0). That is, there is no public
good initially. Let a distribution of initial endowments of the private good (ω i )i∈N be+1given. Given T ∈ P(N), a feasible allocation for T is a list ( xT , y) ≡ ( ( xi )i∈T , y) ∈ ℜ#T+

such that ∑ i∈T (ω i − xi ) = y . The set of feasible allocations for T is denoted by AT .
A mechanism is a function Γ that associates with each T ∈ P(N) a pair Γ(T ) =( ST , gT ), where ST = × i ∈T SiT and gT : ST → ℜ#T + 1 . Here SiT is the strategy space of agent i∈ T and gT is the outcome function when the agents in T play the mechanism. Given gT (s) =( xT , y), let gTi ( s) ≡ ( xi , y) for i ∈ T and gTy (s) = y .
An equilibrium correspondence is a correspondence µ which associates with eachmechanism Γ , each set of agents T ∈ P(N), and each preference profile uT ∈U T , a set of7strategy profiles µ( Γ , T , uT ) ⊆ ST , where ( ST , gT ) = Γ(T ) . We simply write µ( Γ , T , uT )as µ Γ (uT ) . Examples of equilibrium correspondences include dominant strategy
equilibrium correspondence, Nash equilibrium correspondence, and strong Nashequilibrium correspondence. The set of µ-equilibrium allocations of Γ for T at uT isdenoted by gT o µ Γ (uT ) ≡ {( xT , y) ∈ ℜ#T + 1 | there exists s ∈ ST such that s ∈ µ Γ (uT )and gT (s) = ( xT , y)}, where ( ST , gT ) = Γ(T ) .

4. The Case of Two Agents
Let an equilibrium correspondence µ be given. We introduce several
conditions on a mechanism.

Definition 1. The mechanism Γ satisfies non-emptiness under µ if for all T ∈ P(N) and alluT ∈ U T , gT o µ Γ (uT ) ≠ ∅.

Definition 2. The mechanism Γ satisfies feasibility under µ if for all T ∈ P(N) and all uT ∈U T , gT o µ Γ (uT ) ⊆ AT .

Non-emptiness says that there always exists an equilibrium. Feasibility
demands that an equilibrium allocation of the mechanism should always be feasible.
Note that we require feasibility only at equilibrium, but not out of equilibrium.
Moreover, a feasible mechanism does not necessarily satisfy individual feasibility (i.e.,8for all T ∈ P(N) and all s ∈ ST , gT (s) ∈ ℜ#+T + 1 ) nor balancedness (i.e., for all T ∈ P(N)and all s ∈ ST , gT (s) ∈ AT ).

Definition 3. The mechanism Γ satisfies the voluntary participation condition under µ if for

all uN ∈ U N , all ( x N , y N ) ∈ g N o µ Γ (uN ) , and all i ∈ N,
N −{i}),ui ( xiN , y N ) ≥ ui (ω i , y min
N −{ i }where y min∈

Arg miny N −{ i} ∈gyN −{ i} oµ Γ ( uN −{i} )ui (ω i , y N −{i} ) .

N −{ i }Since there is one public good and preferences satisfy monotonicity, y ministhe minimum equilibrium level of public good when all agents except i participate in themechanism. Consider an agent who decides not to participate in the mechanism. Thenshe can enjoy the non-excludable public good produced by the other agents without
providing any private good, while she cannot affect the decision on the provision of the
public good. The voluntary participation condition requires that no agent can benefit
from such a free-riding action. Note that when an agent chooses non-participation, she
has a pessimistic view on the outcome of her action: an equilibrium outcome that is mostunfavorable for her will occur.9 Moulin (1986) proposed a similar condition, called the

No Free Ride axiom, to characterize the pivotal mechanism when public goods are
discrete and costless, and preferences are quasi-linear.

9 A stronger condition on voluntary participation is conceivable for the case in which the non-participant has

a more optimistic view that a better equilibrium outcome will happen. However, we will derive
impossibility results regarding this weak condition on voluntary participation, and hence our results holdfor other stronger versions.

9Definition 4. The mechanism Γ satisfies the Robinson Crusoe condition under µ if for all i∈ N and all ui ∈U i , if ( xi{ i} , y { i} ) ∈ g { i} o µ Γ (ui ) , then ( xi{ i} , y { i} ) ∈ Arg max ui ( xi , y ) .
( xi , y )∈A{ i }The Robinson Crusoe condition means that if only one agent participates in themechanism, then she chooses an outcome that is best for her.
We establish an impossibility result that three conditions mentioned above areincompatible in the case of two agents. Let U SCD = {(ui )i∈N ∀i ∈ N , ui ( xi , y ) = uαi ( xi , y )= α ln xi + ( 1 − α )ln y , α ∈(0 ,1) } be the class of symmetric Cobb-Douglas utility profiles10.

Theorem 1. Let n = 2 and µ be an arbitrary equilibrium correspondence. Suppose thatU ⊇ U SCD and for all i ∈ N, ω i = ω > 0 . If a mechanism satisfies non-emptiness, feasibility, andthe Robinson Crusoe condition under µ , then it fails to satisfy the voluntary participation
condition under µ .

The proof of Theorem 1 is illustrated in Figure 2. Consider the case in which
both agents have the same Cobb-Douglas utility function with α = 0.6. By the Robinson
Crusoe condition, a unique equilibrium allocation of the mechanism when only agent 2(resp. agent 1) participates in it is given by Point C (resp. Point D) in Figure 2. Moreover,
if the mechanism satisfies the voluntary participation condition, then at equilibrium,
agent 1 (resp. agent 2) should receive a consumption bundle in her weak upper contour

10 Here a Cobb-Douglas utility function is denoted by a natural logarithmic function, while it is anexponential function in the example described in Section 2. The results in this paper hold independent of
which function is used.

10

set at C (resp. D) when both agents choose participation. These upper contour sets aredenoted by the shaded areas in Figure 2. However, since they are disjoint, the feasibility
condition is violated. A formal proof of Theorem 1 is given as follows:
Figure 2 is around here.

Proof of Theorem 1. Suppose by way of contradiction that the mechanism satisfies thevoluntary participation condition. Consider (uα1 , uα2 ) ∈U SCD with α = 0.6.11 It is easy tocheck that by the Robinson Crusoe condition, a unique equilibrium allocation of themechanism for one agent economy is given by ( xi{ i} , y { i} ) = (0.6ω , 0.4ω ) , i = 1,2 . Let
V ((ω i , y { j} ), ui0.6 ) ≡ {( xi , y ) ∈ℜ 2+ ui0.6 ( xi , y ) ≥ ui0.6 (ω i , y { j} ) } be agent i's weak upper
contour set at (ω i , y { j} ) for ui0.6 , where (ω i , y { j} ) = (ω , 0.4ω ) and j ≠ i . Pick any( x1{ 1,2 } , x2{ 1,2 } , y { 1,2 } ) ∈ g {1, 2 } o µ Γ (u10.6 , u20.6 ) . By the voluntary participation condition,(2)( xi{ 1,2 } , y { 1,2 } ) ∈ V ((ω i , y { j} ), ui0.6 )( i , j = 1,2 ; j ≠ i ).

We claim that(3)∀( xi , y ) ∈ V ((ω i , y { j} ), ui0.6 ) ,2 xi + y ≠ 2ω

( i , j = 1,2 ; j ≠ i ).

Suppose that (3) does not hold. Then for some i and some ( xi , y ) ∈ℜ 2+ ,ui0.6 ( xi , y ) ≥ ui0.6 (ω i , y { j} ) and 2 xi + y = 2ω . Let ( x∗i , y ∗ ) be a maximizer of the utility
function ui0.6 ( xi , y ) = 0.6 ln xi + 0.4 ln y subject to the constraint 2 xi + y = 2ω . It is easy to
see that ( x∗i , y ∗ ) = (0.6ω ,0.8ω ) and ui0.6 ( x∗i , y ∗ ) − ui0.6 (ω i , y { j} ) = 0.6 ln 0.6 + 0.4 ln 2 <11

- 0.30 + 0.28 < 0. Thus, ui0.6 ( xi , y ) ≥ ui0.6 (ω i , y { j} ) > ui0.6 ( x∗i , y ∗ ) , which contradicts the factthat ( x∗i , y ∗ ) is the maximizer of ui0.6 ( xi , y ) subject to 2 xi + y = 2ω .
However, by (2) and (3), x1{ 1,2 } + x2{ 1,2 } + y { 1,2 } ≠ 2ω . This contradicts thefeasibility condition on the mechanism. Q.E.D.

5. Pareto Efficient Mechanisms
In this subsection, we show an impossibility result on the voluntary
participation condition in the case of at least three agents. We propose the following two
conditions on a mechanism. Let an equilibrium correspondence µ be given.

Definition 5. The mechanism Γ satisfies symmetry under µ if for all T ∈ P(N) and all uT∈ U T , if ui = u j and ω i = ω j for all i, j ∈ T and ( xT , y ) ∈ gT o µ Γ (uT ) , then xi = x j for alli, j ∈ T.

Definition 6. The mechanism Γ satisfies Pareto efficiency only for participants under µ if for
all T ∈ P(N) and all uT ∈ U T , gT o µ Γ (uT ) ⊆ P (uT ) , where P (uT ) ≡ { ( xT , y ) ∈ AT there
does not exist ( xT′ , y ′ ) ∈ AT such that ui ( xT′ , y ′ ) ≥ ui ( xT , y ) for all i ∈T andui ( xT′ , y ′ ) > ui ( xT , y ) for some i ∈T }.

11 By using an argument similar to the below, it is not hard to check that the four conditions mentioned in

Theorem 1 are incompatible for any α ∈( 0.5,1) .

12

Symmetry requires that if all participants have the same preferences andendowments, then they receive the same consumption bundle at equilibrium. Therefore,
every participant pays the same amount of the private good for the provision of the
public good. Pareto efficiency only for participants means that every equilibrium
allocation of the mechanism should be Pareto efficient for participants, but notnecessarily efficient with respect to all agents.

Theorem 2. Let n ≥ 3 and µ be an arbitrary equilibrium correspondence. Suppose thatU ⊇ U SCD and for all i ∈ N, ω i = ω > 0 . If a mechanism satisfies non-emptiness, feasibility,
symmetry, and Pareto efficiency only for participants under µ , then it fails to satisfy thevoluntary participation condition under µ .

SCD. Take any equilibrium allocation (( xiN )i∈N , y N ) ∈Proof. Take (uαi )i∈N ∈Ug N o µ Γ (uN ) . By symmetry, xiN = x N
j for all i, j ∈ N. By feasibility and Pareto efficiency
only for participants, ( xiN , y N ) is a maximizer of the utility function α ln x + ( 1 − α )ln y ,subject to nx + y = nω . It is easy to check that ( xiN , y N ) = ( ωα , nω ( 1 − α ) ). In a similarway, we can show that ( xiN −{ i} , y N −{ i} ) = ( ωα , (n − 1)ω ( 1 − α ) ) for i ∈ N . Therefore, thedifference between the utility level when all agents participate in the mechanism andthat when all agents except i participate in it is given by(4)uαi ( xiN , y N ) − uαi (ω i , y N −{ i} ) = α ln α + ( 1 − α )[ln n − ln(n − 1)] ≡ f (α , n)13for i ∈ N . We prove that the sign of f (α , n) is negative when α = 0.6 and n ≥ 3 .12 Notethat the function ln n − ln( n − 1) is decreasing in n. Therefore, for n ≥ 3 , f (0.6 , n) ≤f (0.6 ,3) = 0.6 ln 0.6 + 0.4[ln 3 − ln 2 ] < −0.3 + 0.2 < 0. This implies that the voluntary
participation condition is violated. Q.E.D.

Remark: By using an argument similar to the proofs of Theorems 1 and 2, we can showthat the condition of Pareto efficiency only for participants can be replaced by a weakercondition in Theorem 2: if a mechanism satisfies non-emptiness, feasibility, symmetry,and Pareto efficiency only with respect to n-1 participants (i.e., for all T ∈ P(N) with
#T = n − 1 and all uT ∈ U T , gT o µ Γ (uT ) ⊆ P (uT ) ), then it fails to satisfy the voluntary
participation condition. This result holds when there are at least two agents and hence
Theorem 1 on the two-agent case is a corollary of it. Although the result is logically
better than Theorem 2, the condition of Pareto efficiency only with respect to n-1
participants would not have a meaningful economic interpretation, except the case of
two agents in which the condition is equivalent to the Robinson Crusoe condition.

6. The Voluntary Contribution Mechanism
In the previous section, we found negative results on voluntary participation forany mechanism satisfying non-emptiness, feasibility, symmetry, and Pareto efficiency
only for participants. However, Pareto efficiency only for participants is not necessary toobtain such results. In this section, we study the voluntary contribution mechanism that
does not satisfy Pareto efficiency only for participants when the equilibrium concept is
12 By using an argument similar to the below, it is not difficult to check that the sign of f (α , n) is negative14

Nash equilibrium. To our surprise, this mechanism does not satisfy the voluntary
participation condition, even though the name of the mechanism contains the term
"voluntary".

Definition 7. The voluntary contribution mechanism is a mechanism such that for all T ∈P(N) and i ∈ T, SiT = [0 , ω i ] and gTi ( s) = (ω i − si , ∑ i∈T si ) for s ∈ST .

The above definition of the voluntary contribution mechanism is a
generalization of the usual one, in which all agents are supposed to participate, to thecase in which voluntary participation is allowed.
When the equilibrium concept is Nash equilibrium, each agent i selects her
contribution out of her endowment to the provision of the public good, si , to maximize
her utility ui (ω i − si , ∑ j∈T s j ) , given contributions of the other agents in T, ( s j ) j∈T −{ i} inthe voluntary contribution mechanism.

Theorem 3. Let n ≥ 3. Suppose that (i)U ⊇ U SCD ; (ii) for all i ∈ N, ω i = ω > 0 ; and (iii) µ is aNash equilibrium correspondence. Then the voluntary contribution mechanism fails to satisfy thevoluntary participation condition.

SCD
Proof. Take (uα. Let ( xiN , y N ) be the consumption bundle that each agent ii ) i∈N ∈U

receives at the unique symmetric Nash equilibrium if all agents in N decide to participate

for any α ∈(0.203,1) and any n ≥ 3 .

15

in the mechanism. It is easy to see that ( xiN , y N ) = ( ωαn /(1 + α (n − 1)) ,ω (1 − α )n /(1 + α (n − 1)) ). Also, let y N −{ i} be the public good level at the unique Nashequilibrium allocation of the mechanism played among n-1 participants in N − {i} . It is
straightforward to check that y N −{ i} = ω(1-α) (n − 1) /(1+α(n-2)). Thus, for i ∈ N ,(5)

N N
N −{ i }α)uα
i ( xi , y ) − u i (ω i , y= α ln α + (1 − α )[ln n − ln(n − 1)] + α ln n + (1 − α ) ln[1 + α (n − 2)] − ln[1 + α (n − 1)]≡ h(α , n) .
We show that the sign of h(α , n) is negative when α = 0.6 and n ≥ 3 .13 By partiallydifferentiating h(α , n) with respect to n, we have∂h(α , n ) −(1 − α )[1 − 2α + αn(1 + α − αn)].
=n(n − 1)[1 + α (n − 2)][1 + α (n − 1)]∂n
If α = 0.6 and n ≥ 3 , then∂h(α , n) 0.4[9n(n − 3) + 3n + 5]>0.
=n(n − 1)( 3n − 1)( 3n + 2)∂nMoreover, lim h(α , n) = 0 . Hence, for any finite number n ≥ 3 , h(0.6 , n) < 0. Thisn→∞implies that the voluntary participation condition is violated. Q.E.D.

7. Concluding Remarks
We see that the solutions to the free-rider problem, which have been proposedin mechanism design theory, are not necessary solutions to the free-rider problem when

13 By using an argument similar to the below, it is not difficult to check that the sign of h(α , n) is negative

for any α ∈(0.25 ,1) and any n ≥ 3 .

16

participation in mechanisms is voluntary. Furthermore, we show that it is quite difficult
or impossible to design a mechanism with voluntary participation.
As Olson (1965) argued, a public good would be less likely provided as thenumber of agents becomes large. Saijo and Yamato (1999) confirmed this conjecture byproving that in a two-stage game with voluntary participation, the measure of the set ofsymmetric Cobb-Douglas economies for which every agent chooses participation atequilibrium becomes smaller as the number of agents grows large. In a similar way, we
can show that the measure of the set of economies for which the voluntary participation
condition is satisfied is strictly decreasing as the number of agents increases. This would
be another result supporting Olson’s conjecture.
In the voluntary participation condition defined above, it is implicitly assumed
that each agent has the most optimistic conjecture on the number of other agents whowill not participate in the mechanism if she does not, that is, she expects no agent otherthan her to choose non-participation in the mechanism. On the other hand, in theindividually rational condition usually discussed in the literature on mechanism design,
it is assumed that each agent has the most pessimistic conjecture on that number, that is,she expects all other n-1 agents to select non-participation, too. However, an agent might
have an intermediate conjecture: her conjecture on the number of other non- participants
can take on a whole range of values from 0 to n-1. An open question is to examine otherconditions on voluntary participation taking account of these possible conjectures.
Saijo, Yamato, Yokotani, and Cason (1998), and Cason, Saijo, and Yamato (2001)
observed that cooperation has emerged though spiteful behavior in their experiments on
the voluntary contribution mechanism with voluntary participation. Our theory in this17

paper suggests that no cooperation will emerge. Reconciling theoretical results toexperimental results is an open area of our future research.

References
BRAMS, S. J., AND P. C. FISHBURN (1983): "Paradoxes of Preferential Voting," Mathematics
Magazine, 56, 207-214.
CASON, T., T. SAIJO, AND T. YAMATO (2001): “Voluntary Participation and Spite in Public
Good Provision Experiments: An International Comparison,” Purdue University
Center for International Business Education and Research (CIBER) working
paper.
DIXIT, A. AND M. OLSON (2000): "Does Voluntary Participation Undermine the Coase
Theorem?” Journal of Public Economics, 76, 307-335.
GROVES, T., AND J. LEDYARD (1977): "Optimal Allocation of Public Goods: A Solution to
the 'Free Rider' Problem," Econometrica, 45, 783-811.
GROVES, T., AND J. LEDYARD, "Incentive Compatibility Since 1972," in Information,
Incentives, and Economic Mechanisms: Essays in Honor of Leonid Hurwicz, eds., T.
Groves, R. Radner, and S. Reiter, (Minneapolis, University of Minnesota Press,
1987), 48-111.
HURWICZ, L. (1972): "On Informationally Decentralized Systems," in Decision andOrganization: A Volume in Honor of Jacob Marschak, eds., R. Radner and C. B.
McGuire, (Amsterdam, North-Holland), 297-336.
HURWICZ, L. (1979): "Outcome Functions Yielding Walrasian and Lindahl Allocations at
Nash Equilibrium Points," Review of Economic Studies, 46, 217-225.
HURWICZ, L. (1994): "Economic Design, Adjustment Process, Mechanisms, andInstitutions," Economic Design, 1, 1-14.
HURWICZ, L., E. MASKIN, AND A. POSTLEWAITE (1984): "Feasible Implementation of Social
Choice Correspondences by Nash Equilibria," mimeo.
LEDYARD, J. (1984): "The Pure Theory of Large Two-Candidate Elections," Public Choice,44, 7-41.
LEDYARD, J. AND J. ROBERTS (1974): "On the Incentive Problem with Public Goods,"
mimeo, Northwestern University.

18

MOULIN, H. (1986): "Characterizations of the Pivotal Mechanism," Journal of Public
Economics, 31, 53-78.

OKADA, A. (1996): "The Organization of Social Cooperation: A Noncooperative
Approach," KEIR Discussion Paper, Kyoto University.
OLSON, M. (1965): "The Logic of Collective Action: Public Goods and the Theory ofGroups," Cambridge: Harvard University Press.
PALFREY, T., AND H. ROSENTHAL (1983): "A Strategic Calculus of Voting," Public Choice,41, 7-53.
PALFREY, T., AND H. ROSENTHAL (1984): "Participation and the Provision of Discrete
Public Goods: A Strategic Analysis," Journal of Public Economics, 24, 171-193.
PALFREY, T., AND H. ROSENTHAL (1985): "Voter Participation and Strategic Uncertainty,"
American Political Science Review, 79, 62-78.
SAIJO, T. (1991): “Incentive Compatibility and Individual Rationality in Public Good
Economies,” Journal of Economic Theory, 55, 203-212.
SAIJO, T., AND T. YAMATO (1999): “A Voluntary Participation Game with a Nonexcludable Public Good,” Journal of Economic Theory, 84, 227-242.
SAIJO, T, T. YAMATO, K. YOKOTANI, AND T. N. CASON (1998): "Voluntary Participation in
Public Good Provision Experiments: Is Spitefulness a Source of Cooperation?"
mimeo, Osaka University.
SAMUELSON, P. A. (1964): "The Pure Theory of Public Expenditure," The Review of
Economics and Statistics, 36, 387-389.
SELTEN, R. (1973): "A Simple Model of Imperfect Competition, where 4 Are Few and 6
Are Many," International Journal of Game Theory, 2, 141-201.
TIAN, G. (1990): "Completely Feasible Continuous Implementation of the Lindahl
Correspondence with a Message Space of Minimal Dimensions," Journal of
Economic Theory, 51, 443-452.
WALKER, M. (1981): "A Simple Incentive Compatible Scheme for Attaining Lindahl
Allocations," Econometrica, 49, 65-71.

19

1

6

6
10
2

01

A
8

ω

B 6
4

02

Figure 1. A Lindahl mechanism does notsatisfy the voluntary participation condition.

2
1

C

D
01

ω

02

Figure 2. No mechanism satisfies the the voluntary
participation condition for the two-agent case.