A Solution to the Free-Rider Problem - CiteSeerX

Fundamental Difficulties in the Provision of Public Goods: 'A Solution to the Free-Rider Problem' Twenty Years After1

by

Tatsuyoshi Saijo* and Takehiko Yamato**

December, 1995/Revised May, 1997

1 We would like to thank Tim Cason, Jun Iritani, Ryo-ichi Nagahisa, Ken-ichi Shimomura, and Yoshikatsu Tatamitani for helpful comments and discussions. An earlier version of this paper was presented at the Conference on Axiomatic of Resource Allocation at Montreal, the Third International Meeting of the Society for Social Choice and Welfare at Maastricht, the 1996 Annual Meeting of the Japan Association of Economics and Econometrics at Osaka, the 1997 Public Choice Society Meeting, Kyoto University, Osaka University, University of Tokyo, and Fukuoka University. Research was partially supported by the Zengin Foundation for the Studies on Economics and Finance, Grant in Aid for Scientific Research D7730011 of the Ministry of Education, Science and Culture in Japan, and the Tokyo Center for Economic Research Grant. * Institute of Social and Economic Research, Osaka University, Ibaraki, Osaka 567, Japan. E-mail: [email protected] ** Faculty of Economics, Tokyo Metropolitan University, Hachioji, Tokyo 192-03, Japan. E-mail: [email protected] 1

Abstract

Groves-Ledyard (1977) constructed a mechanism attaining Pareto efficient allocations in the presence of public goods. After this path-breaking paper, many mechanisms 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 to this 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: Takehiko Yamato Faculty of Economics Tokyo Metropolitan University Hachioji, Tokyo 192-03 Japan Phone 01181-426 (country & area codes) 77-2326 (office)/77-2304(fax) E-mail: [email protected]

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1. Introduction It is well known that the provision of public goods has an incentive problem called the free-rider problem. Samuelson's (1964) view is pessimistic: it is impossible to attain a Pareto efficient allocation through a decentralized fashion, in particular, a decentralized pricing system. On the contrary, in a path-breaking paper Groves and Ledyard (1977), they proposed an explicit procedure, called a mechanism, that achieves a Pareto efficient allocation. The mechanism allows participants to pursue their own self-interest and to be free riders if they choose, but the mechanism eliminates these incentives. In this sense, they found a solution to the free-rider problem. Although their mechanism attains a Pareto efficient allocation, it does not satisfy individual rationality - a requirement that the allocation must be at least as good as each participant's initial endowment. Following Groves and Ledyard (1977), Hurwicz (1979) and Walker (1981) fixed this problem and succeeded in implementing the Lindahl correspondence which satisfies both Pareto efficiency and individual rationality. Subsequently, numerous mechanisms have been proposed that satisfy additional desirable properties such as individual feasibility and balancedness [See Groves and Ledyard (1987) and Hurwicz (1994)]. Most mechanisms developed to date 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 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 to not participate. This participation problem is important in many practical situations, such as for international treaties. For instance, it took 24 years to agree on the disposition of chemical weapons in the chemical weapons treaty, and the number of signatories was 3

more than 160 by the end of 1995. The treaty is a mechanism to produce a public good, i.e., world peace. In order to make the treaty effective, at least 65 countries must ratify it. Hungary was the 65th country to ratify the treaty - in October 1996 - and it will be effective in the middle of 1997. However, China and Russia have the most chemical weapons and have not ratified this treaty. Therefore, the effectiveness of the treaty remains in doubt. 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 Versailles 1. Our first question is whether or not any mechanism attaining Lindahl allocations, called a Lindahl mechanism, can survive 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 in a wide class of environments. Then what outcome would result in if we allow voluntary participation? In a simple two-agent economy, the Prisoner's dilemma game has been thought to be an abstraction of the free-rider problem. On the contrary, with voluntary participation we find that a Hawk-Dove game represents the free-rider problem properly. In this game, there are two equilibria: either one of two agents participates in the mechanism. That is, participation of all agents is not an equilibrium. Based upon these preliminary results, we ask a fundamental question: is there any mechanism satisfying the condition that every agent always chooses participation strategically, called the voluntary participation condition ? In a two-agent economy, the answer is negative: no voluntary participation mechanism exits under mild regularity conditions. Furthermore, this result is independent of the choice of equilibrium concepts. The picture is still bleak even if the number of participants is at least three. 1 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 family must pay the fee by law, but many choose not to since punishment is practically non-existent. 4

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. It is widely believed that a public good would be provided more easily for a small group than a large one as Olson (1965) argued. In symmetric Cobb-Douglas economies, we 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. In addition, in a sufficiently large replica of an economy, a participation incentive disappears. Finally, in a two-stage game with voluntary participation, the measure of the set of economies for which every agent chooses participation at equilibrium becomes smaller as the number of agents grows large. Moulin (1986) and Palfrey and Rosenthal (1984) also analyzed the issue of an incentive to participate in a mechanism for the provision of a public good.

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They

focused on specific mechanisms: Moulin examined the pivotal mechanism in discrete public goods economies with quasi-linear preferences, whereas Palfrey and Rosenthal considered a simple mechanism for the provision of a binary public good with binary contributions. On the other 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 5, we also provide a formal model to show that a public good is less likely provided as the number 2 The problem of an incentive to participate in an institution has been examined mainly in the context of voting and cartel formation (see Brams and Fishburn (1983), Ledyard (1984), Okada (1996), Palfrey and Rosenthal (1983,1985), and Selten (1973) ) . 5

of agents grows large. In Section 6, we investigate the voluntary contribution mechanism. In the final section, we make concluding remarks.

2. An Example: Lindahl Mechanisms We consider the following symmetric economies with one private good x and one pure public good y. The public good can be produced from the private good by means of a constant return to scale technology, and let y = x be the production function of the public good. There are two agents . A consumption bundle for agent i is denoted by ( xi , y) ∈ ℜ 2+ where xi ∈ ℜ + is the level of private good she consumes on her own, and y ∈ ℜ + is the level of public good. We assume that two agents have the same preferences and they 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 same and given by ( ω i , 0) = (10, 0) for i = 1,2. We examine situations in which the true value of α is unknown, whereas the initial endowment and the production technology are known.

2.1 The Voluntary Participation Condition Take any mechanism implementing the Lindahl correspondence in Nash equilibria (e.g., see Hurwicz (1979), Nakayama (1980), Walker (1981), Hurwicz, Maskin, and Postlewaite (1984), and Tian (1990)). 3 Now imagine that each agent can decide whether she participates in the mechanism or not. Does each agent always have an incentive to participate in the mechanism? This question is essential since all agents must participate in the mechanism to achieve the desired Lindahl equilibrium allocation. Unfortunately, our answer to the above question is negative.

3 In the present context, the Lindahl correspondence is identical with the constrained Lindahl correspondence (Hurwicz, Maskin, and Postlewaite (1984)). In more general settings, however, the former is not Nash implementable, while the latter is. 6

To see why, let T ⊆ { 1,2} be the set of agents who participate in the mechanism.

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An equilibrium allocation of the mechanism when the agents in T participate in it is denoted by (( xTi )i∈T , y T ) .5 If two agents decide to participate in the mechanism, then ( x{11 ,2} , x{21,2 } , y{ 1,2 } ) is a Lindahl allocation of the economy with two agents, since the mechanism implements the Lindahl correspondence.

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It is easy to check that there exists

a unique Lindahl allocation given by ( x{11 ,2} , x{21,2 } , 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 the economy consisting of only one agent j . It is straightforward to see that { j}

( x j , y{ j } ) = (10α, 10(1-α)). Notice that the non-participant i can enjoy her initial endowment, ω i , as well as the non-excludable public good produced by agent j ≠ i , y{ j} . On the other hand, she is no longer 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 the mechanism: (1)

uαi ( x{i 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.7 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

4 An "agent" stands for any member who belongs to a society, whereas a "participant" stands for an agent who chooses participation in a mechanism. 5 A mechanism specifies strategy sets of participants in T and an outcome function for each T ⊆ { 1 ,2 } . This definition of a mechanism is more general than the usual one. 6 A mechanism implements the Lindahl correspondence if for each set of participants T ⊆ { 1 ,2 } and each economy consisting of the participants in T , every equilibrium allocation is a Lindahl allocation and every Lindahl allocation is an equilibrium allocation. 7 The voluntary participation condition is different from the individually rational condition which { 1 ,2 } { 1 ,2 } α { j} α requires that uα ,y ) ≥ uα i ( xi i (ω i ,0 ) for i = 1,2. Since ui (ω i , y ) ≥ ui (ω i ,0 ) , the voluntary participation condition is stronger than the individually rational condition.

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allocation when both agents participate in the mechanism: A = ( x{11,2 } , x{21,2 } , y{ 1 ,2 } ) = (6, 6, 8). Point B represents the allocation when agent 1 does not participate in the mechanism, but agent 2 does: B = ( ω 1 , x{22 } , y{ 2} ) = (10, 6, 4). Since u10.6 ( x{11,2 } , y{ 1 ,2 } ) ≈ 6.73 < u10.6 ( ω 1 , y{ 2} ) ≈ 6.93, agent 1 will not participate in the mechanism when agent 2 does. The same thing holds for agent 2.  Figure 1 is around here. 

2.2 A Two-Stage Game with Voluntary Participation In the above, we find that each agent may not participate in the mechanism when the other agent does. Then a natural question arises: how many agents will participate in the mechanism if the participation decision is a strategic variable of each agent? In order to examine this question, we consider the following two-stage game with voluntary participation. In the first stage, each agent simultaneously decides whether she participates in the mechanism or not. In the second stage, knowing the other agents' participation decisions, the agents who selected participation in the first stage choose their strategies. Figure 2 illustrates the two-stage game for the case of α = 0.6. First of all, if both agents choose participation, then they play the mechanism whose equilibrium allocation is ( x{11,2 } , x{21,2 } , y{ 1 ,2 } ) = (6, 6, 8). The equilibrium payoff for each participant i is u10.6 ( x{i 1,2 } , y{ 1 ,2 } ) ≈ 6.73. Second, if one agent j chooses participation, while the other

agent i ≠ j selects non-participation, then only agent j chooses her strategy. The { j}

resulting equilibrium allocation is ( ω i , x j , y{ j} ) = (10, 6, 4). The payoff for non{ j}

participant i is u10.6 ( ω i , y{ j} ) ≈ 6.93, while that for participant j is u10.6 ( x j , y{ j} ) ≈ 5.10. Third, if both agents choose non-participation, then no public good is produced. In this case, the payoff for each agent i is u10.6 ( ω i , 0) = 0. 8

Given these equilibrium payoffs in the second stage, we can construct the payoff matrix for the first stage decision on participation as Table 1. It is a "Hawk-Dove" game. It has been generally thought that the free-rider problem in the provision of a public good can be represented by a "Prisoner's Dilemma" game. However, when voluntary participation is allowed, the problem is described by a Hawk-Dove game rather than a Prisoner's Dilemma game. We believe that a Hawk-Dove game represents the free-rider problem more appropriately than a Prisoner's Dilemma game does, since free-riding involves that people can benefit from not participating in a mechanism for the provision of a public good. There are two Nash equilibria of this Hawk-Dove game: one agent participates in the mechanism, whereas the other agent does not. The case in which both agents participate in it is not a subgame perfect equilibrium outcome of the two-stage game. The Lindahl allocation will not be achieved when voluntary participation is allowed.  Figure 2 and Table 1 are around here. 

3. Notation and Definitions In the above example, we find that Lindahl mechanisms fail to satisfy the voluntary participation condition in symmetric economies with two agents when Nash equilibrium is an equilibrium concept. In what follows, we show that similar negative results hold for any mechanism satisfying mild conditions in a large class of economies. Moreover, the results are independent of which equilibrium concept is used. First, we introduce notation and definitions to describe our general results. In Section 2, we examined two-agent economies with one private good, one public good, and a constant return to scale technology. We consider the same situations with many agents. Let N = {1,2,...,n} be the set of agents, with generic element i. We assume that

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each agent i's preference relation admits a numerical representation ui : ℜ 2+ → ℜ which is continuous, quasi-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), let uT ≡ (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). There is no public good initially. Let a distribution of initial endowments of the private good (ω i )i ∈N be given. +1 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 which associates with each T ∈ P(N) a pair Γ (T ) = ( ST , gT ), where ST = × i ∈T STi and gT : ST → ℜ#T + 1 . Here STi 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 each mechanism Γ , each set of agents T ∈ P(N), and each preference profile uT ∈U T , a set of strategy profiles µ( Γ , T , uT ) ⊆ S T , where ( ST , gT ) = Γ (T ) . We simply write µ(Γ , T , uT ) as µ Γ (uT ) . Examples of equilibrium correspondences are dominant strategy equilibrium correspondence, Nash equilibrium correspondence, or strong Nash equilibrium correspondence. The set of µ-equilibrium allocations of Γ for T at uT is denoted 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 some conditions on a mechanism. Definition 1. The mechanism Γ satisfies non-emptiness under µ if for all T ∈ P(N) and all uT ∈ U T , gT o µ Γ (uT ) ≠ ∅. 10

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

Non-emptiness means that there exists always an equilibrium. Feasibility says 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., for all T ∈ P(N) and all +1 ) nor balancedness (i.e., for all T ∈ P(N) and all s ∈ ST , gT (s) ∈ s ∈ ST , gT (s) ∈ ℜ #T +

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 ( x iN , y N ) ≥ ui (ω i , y min N −{ i } where y min ∈

Arg min

N −{ i } oµ Γ ( uN −{ i } ) y N −{ i } ∈g y

ui (ω i , y N −{ i} ) .

N −{ i } Since there is one public good and preferences satisfy monotonicity, y min is the

minimum equilibrium level of public good when all agents except i participate in the mechanism. Consider an agent who decides not to participate in the mechanism. Then she can enjoy the non-excludable public good produced by the other agents without providing any private good, while she cannot affect on 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 nonparticipation, she has a pessimistic view on the outcome of her action: an equilibrium

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outcome which is most unfavorable for her will occur. Moulin (1986) proposed a similar condition, called the No Free Ride axiom, when public goods are discrete and costless, and preferences are quasi-linear. Definition 4. The mechanism Γ satisfies the Robinson Crusoe condition under µ if for all i ∈ N and all ui ∈U i , if ( x{i i } , y{ i } ) ∈ g { i } o µ Γ (ui ) , then ( x{i i } , y{ i } ) ∈ Arg max ui ( x i , y ) . ( xi ,y )∈A{ i }

The Robinson Crusoe condition means that if only one agent participates in the mechanism, then she chooses an outcome which is best for her. We establish an impossibility result that three conditions mentioned above are incompatible in the case of two agents. Let U SCD = {(ui )i∈N ∀i ∈ N , ui ( x i , y ) = uαi ( xi , y ) = α ln xi + ( 1 − α ) ln y , α ∈(0 ,1) } be the class of symmetric Cobb-Douglas utility profiles 8.

Theorem 1. Let n = 2 and µ be an arbitrary equilibrium correspondence. Suppose that U ⊇ U SCD and for all i ∈ N, ω i = ω > 0 . If a mechanism satisfies non-emptiness, feasibility, and the 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 3. 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 3. Moreover, if the mechanism satisfies the voluntary participation condition, then at 8 Here a Cobb-Douglas utility function is denoted by a natural logarithmic function, while it is a exponential function in the example described in Section 2. The results in this paper hold independent of which function is used. 12

equilibrium, agent 1 (resp. agent 2) should receive a consumption bundle in her weak upper contour set at C (resp. D) when both agents choose participation. These upper contour sets are denoted by the shaded areas in Figure 3. However, since they are disjoint, the feasibility condition is violated. A formal proof of Theorem 1 is given as follows:  Figure 3 is around here.  Proof of Theorem 1. Suppose by way of contradiction that the mechanism satisfies the voluntary participation condition. Consider ( uα1 , uα2 ) ∈U SCD with α = 0.6. It is easy to check that by the Robinson Crusoe condition, a unique equilibrium allocation of the mechanism for one agent economy is given by ( x{i 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 ui , where (ω i , y{ j } ) = (ω , 0.4ω ) and j ≠ i . Pick any ( x{11 ,2} , x{21,2 } , y{ 1,2 } ) ∈ g{ 1,2 } o µ Γ (u10.6 , u 20.6 ) . By the voluntary participation condition, (2)

( x{i 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 ) ∈ℜ2+ , 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
ui0.6 ( xi∗ , y ∗ ) , which contradicts the fact that ( x∗i , y ∗ ) is the maximizer of ui0.6 ( xi , y ) subject to 2 xi + y = 2ω . However, by (2) and (3), x{11 ,2 } + x{21,2 } + y{ 1 ,2 } ≠ 2ω . This contradicts the feasibility condition on the mechanism.

5. Pareto Efficient Mechanisms 5.1. An Impossibility Result on the Voluntary Participation Condition 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 ) ∈ g T o µ Γ (uT ) , then xi = x j for all i, 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 , g T o µ Γ (uT ) ⊆ (uT ) , where

(uT ) ≡ { ( xT , y ) ∈ AT

there does not exist ( xT′ , y ′) ∈ AT such that ui ( xT′ , y ′ ) ≥ ui ( xT , y ) for all i ∈T and ui ( xT′ , y ′ ) > ui ( xT , y ) for some i ∈T }.

Symmetry requires that if all participants have the same preferences and endowments, 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 not necessarily efficient with respect to all agents. The following lemma is useful below. 14

Lemma 1. Let n ≥ 2 and µ be an arbitrary equilibrium correspondence. Suppose that (i) for all i ∈ N, ω i = ω > 0 ; and (ii) a mechanism satisfies non-emptiness, feasibility, symmetry, and Pareto efficiency only for participants under µ . Then for each uαN ≡ ( uαi ) i∈N ∈U SCD and each T ∈ P(N ) , there exists a unique µ-equilibrium allocation of the mechanism for T at uTα ≡ ( uαi ) i∈T given by

( xTi , y T ) = ( ωα , # Tω (1 − α ) ), i ∈T , where #T is the cardinality of T.

Proof. Take any equilibrium allocation (( xTi )i∈T , y T ) ∈ g T o µ Γ (uT ) . By symmetry, xiT = xTj for all i, j ∈ T. By feasibility and Pareto efficiency only for participants, ( xiT , y T ) is a maximizer of the utility function α ln x + ( 1 − α ) ln y , subject to # Tx + y =# Tω . It is easy to check that ( xTi , y T ) = ( ωα , # Tω ( 1 − α ) ).

Theorem 2. Let n ≥ 3 and µ be an arbitrary equilibrium correspondence. Suppose that U ⊇ 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 the voluntary participation condition under µ .

Proof. Take ( uαi ) i∈N ∈U SCD . By Lemma 1, the difference between the utility level when all agents participate in the mechanism and that 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)

for i ∈ N . We prove that the sign of f (α , n) is negative when α = 0.6 and n ≥ 3 . Note that the function ln n − ln(n − 1) is decreasing in n. Therefore, for n ≥ 3 , f ( 0.6 , n) ≤

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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.

Remark: By using an argument similar to the proofs of Theorems 1 and 2, we can show that the condition of Pareto efficiency only for participants can be replaced by a weaker condition 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 , g T o µ Γ (uT ) ⊆ (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.

5.2 The Number of Agents and a Participation Incentive: Olson's Conjecture Olson (1968) claimed that a public good is less likely provided as the number of agents becomes larger. In this section, we present a formal model to confirm this conjecture from the viewpoint of a participation incentive in a mechanism for the provision of a public good: as the number of agents increases, each agent is less likely to have an incentive to participate in a Pareto efficient mechanism in symmetric CobbDouglas economies. The participation incentive disappears with a large number of agents. Suppose that U = U SCD and for all i ∈ N, ω i = ω > 0 . Let the initial endowment of the private good ω be fixed. Then an economy is specified by a list uαN ∈U SCD . Since we assume that all agents have the same Cobb-Douglas utility function in an 16

economy, the set of all economies is represented by the interval (0, 1) endowed with Lebesgue measure λ. Given a mechanism Γ and an equilibrium correspondence µ , let U vp (n ) be the set of economies for which Γ satisfies the voluntary participation

condition under µ when the number of agents is #N = n , i.e., U vp (n ) ≡ { α ∈(0 ,1)  N −{ i } ∀( x N , y N ) ∈ g N o µ Γ ((uαi )i∈N ) , ∀i ∈ N, uαi ( xiN , y N ) ≥ uαi (ω i , y min ) }. We find the

following properties of the measure of this set.

Theorem 3. Let n ≥ 2 and µ be an arbitrary equilibrium correspondence. Suppose that (i) U = U SCD ; (ii) for all i ∈ N, ω i = ω > 0 ; and (iii) a mechanism satisfies non-emptiness,

feasibility, symmetry, and Pareto efficiency only for participants under µ . Then (a) the measure of the set of economies for which the mechanism satisfies the voluntary participation condition under µ strictly decreases as the number of agents increases; (b) the measure of the set of economies for which the mechanism satisfies the voluntary participation condition under µ vanishes as the number of agents grows large; and (c) for each α ∈(0 ,1) , there exists an integer number n P (α ) ≥ 1 such that if n ≤ n P (α ) , then the mechanism satisfies the voluntary participation condition under µ ; and if n > n P (α ) , then the mechanism fails to satisfy the voluntary participation condition under µ .9

Proof. (a) Take ( uαi ) i∈N ∈U SCD . As we see in the proof of Theorem 2, it follows from Lemma 1 that there exists a unique equilibrium allocation of the mechanism for T = N and T = N − { i} . For each i ∈ N , uαi ( xiN , y N ) − uαi (ω i , y N −{ i } ) = f (α , n) as we defined in (4). For each n ≥ 2, let α P (n ) be a value α ∈(0 ,1) satisfying f (α , n) = 0 . Figure 4 depicts the function α P .10 For α P (n) < α < 1 , f (α , n) < 0 , i.e., uαi ( xiN , y N ) < uαi (ω i , y N − i ) ; for α

9 Note that the result stated in Theorem 3-(a) implies neither the one in Theorem 3-(b) nor the one in Theorem 3-(c). 10 For example, α P (2) = 0.5, α P (3) ≈ 0.203, α P (4) ≈ 0.120, and α P (5) ≈ 0.082. 17

= α P (n ) , f (α , n) = 0 , i.e., uαi ( xiN , y N ) = uαi (ω i , y N − i ) ; and for 0 < α < α P (n) , f (α , n) > 0 , i.e., uαi ( xiN , y N ) > uαi (ω i , y N − i ) . Therefore, U vp (n ) = ( 0 , α P (n)] . Next we prove that α P (n ) is strictly decreasing in n. Note that ∂f (1 − α ) dα P = − ∂n = . ∂ f n(n − 1)[ 1 + ln α /(1 − α )] dn ∂α We show that

dα P < 0 . Since (1 − α ) > 0 and n(n-1) > 0, it remains to prove that dn

1 + ln α /( 1 − α ) < 0. It is easy to check that the function 1 + ln α /( 1 − α ) is increasing in $ α. Further, by using L'Hopital's rule, lim[1 + ln α /(1 − α )] = 0 . These imply that α →1

1 + ln α /( 1 − α ) < 0. Hence, the measure of U vp (n ) , λ(U vp (n)) , is strictly decreasing in n. (b) We show that lim λ (U vp (n )) = 0 . It is sufficient to prove that lim α P (n) = 0 . n→∞

n→∞

Notice that lim f (α , n) = α ln α . For all α ∈(0 ,1) , α ln α < 0. Also, for α = 1, α ln α = 0. n→∞

$ Moreover, by L'Hopital's rule,

F GG GH

1 ln α lim α ln α = lim = lim − α 1 α →0 α →0 1 α →0 α α2

I JJ = lim (−α ) = 0. JK α →0

By these facts, lim α P (n) = 0 or 1. However, α P (n ) is strictly decreasing in n. Hence, n→∞

lim α P (n) = 0 .

n→∞

(c) Take any α ∈(0 ,1) . Let n∗ (α ) be a value satisfying the equation f (α , n∗ (α )) = 0 . We can rewrite this equation as ln[n∗ (α ) /(n∗ (α ) − 1)] = k(α ) ≡ α ln α /(α − 1) > 0 . Hence, n∗ (α ) = e k(α ) /( e k(α ) − 1) > 1 . Let n P (α ) be the largest integer

18

less than or equal to n∗ (α ) . Since the function ln[n /(n − 1)] is strictly decreasing in n, it follows that if n ≤ n P (α ) , then f (α , n) ≥ 0 ; and if n > n P (α ) , then f (α , n) < 0 .

 Figure 4 is around here.  In Figure 4, the region below (resp. above) the curve α = α P (n), which is defined in the proof of Theorem 3-(a), represents the set of pairs of n and α in which the voluntary participation condition is satisfied (resp. not satisfied). As Theorems 3-(a) and 3-(b) show, the measure of the set of α for which agents have (resp. do not have) participation incentives becomes smaller (resp. larger) and converges to zero (resp. one) as the number of agents n grows large. We can interpret the result stated in Theorem 3-(c) as follows. Consider an economy consisting of one agent whose preferences are represented by a Cobb-Douglas utility function with parameter α ∈(0 ,1) . For each type of agent α ∈(0 ,1) , there exists some replication n P (α ) such that for all n with n P (α ) < n , the voluntary participation condition is violated in the n replica of the one-agent economy with type α . That is, in a sufficiently large replica of any one-agent economy, the participation incentive disappears.

5.3 Equilibria of a Two-Stage Game with Voluntary Participation We turn to the question of how many agents will participate in the mechanism when the participation decision is a strategic variable of each agent in symmetric CobbDouglas economies. We consider the two-stage game with voluntary participation described in Section 2.2: in the first stage, each agent simultaneously chooses whether or not she participates in the mechanism, and in the second stage, the agents who decided to participate in the mechanism choose their strategies. In Section 2.2, we examine a 19

Lindahl mechanism when n = 2, α = 0.6, and ω = 10, as illustrated in Figure 2 and Table 1. Here, we investigate any mechanism satisfying non-emptiness, feasibility, symmetry, and Pareto efficiency only for participants in a class of symmetric Cobb-Douglas economies with n agents. Let

be the collection of equilibrium sets of agents who choose to participate in

the mechanism in the first stage and α P (n) be the function defined in the previous section. Figure 4 illustrates how many agents participate in the mechanism at equilibrium, depending on the value of preference parameter α when n = 5. As the value of α increases, i.e., as the value of the public good relative to the private good decreases, the equilibrium number of participants in the mechanism tends to be smaller. First, when 0 < α < α P ( 5 ) ,

= {N}, i.e., all five agents participate in the mechanism.

Second, when α = α P ( 5) , T ∈

if and only if #T = 4 or 5, i.e., four or five agents

participate in the mechanism. Third, when α P ( 5) < α < α P ( 4 ) , T ∈

if and only if #T =

4, i.e., four agents participate in the mechanism, and so on. Finally, when α P ( 2) < α < 1 , T∈

if and only if #T = 1, i.e., only one agent participates in the mechanism. In general, we have the following characterization of

:

Theorem 4. Let n ≥ 2 and µ be an arbitrary equilibrium correspondence. Suppose that (i) U = U SCD ; (ii) for all i ∈ N, ω i = ω > 0 ; and (iii) a mechanism satisfies non-emptiness,

feasibility, symmetry, and Pareto efficiency only for participants under µ . Case 1: 0 < α < α P (n) . In this case,

= {N}, i.e., all agents participate in the

mechanism. Case 2: α = α P (n − b) , b = 0,1,2,..., n-2. In this case, for all T ∈ and #

=

, #T = n-b or n-b-1

F n I + F n I , i.e., n-b or n-b-1 agents participate in the mechanism. GH n −bJK GH n−b −1JK

20

Case 3: α P (n − b ) < α < α P (n − b − 1) , b = 0,1,2, ..., n-3. In this case, for all T ∈ = n-b-1 and #

=

, #T

F n I , i.e., n-b-1 agents participate in the mechanism. GH n−b −1JK

Case 4: α P ( 2) < α < 1 . In this case, for all T ∈

, #T = 1 and #

= n, i.e., one agent

participates in the mechanism.

Remark: Since the function α P (n ) is strictly decreasing in n and lim α P (n) = 0 as we n→∞

show in the previous subsection, Theorem 4 implies that the measure of the set of economies for which participation of all agents is an equilibrium outcome of the twostage game strictly decreases and vanishes as the number of agents grows large. This would be another result supporting Olson's (1968) conjecture that a public good is less likely provided as the number of agents increases.

Proof. Take ( uαi ) i∈N ∈U SCD . Suppose that in the first stage, every agent i ∈ T ⊆ N chooses to participate in the mechanism, while every agent i ∉ T chooses not to participate in it. By Lemma 1, the second stage equilibrium allocation is uniquely determined: for each i ∈ T, ( xTi , yT ) = ( ωα , ω (1 − α )# T ) and for each i ∉ T, ( ω i , yT −{i} ) = ( ω , ω (1 − α )(# T − 1) ). The set T is an equilibrium set of participants if and only if (i) no agent in i ∈ T will choose non-participation; and (ii) no agent in i ∉ T will select participation. In other word, T ∈

if and only if

(i) for all i ∈ T, uαi ( xiT , y T ) ≥ uαi (ω i , y T −{ i} ) , i.e., f (α ,# T ) ≡ α ln α + (1 − α )[ln# T − ln(# T − 1)] ≥ 0; and (ii) for all i ∈ N-T, uαi ( xiT ∪{ i } , y T ) ≤ uαi (ω i , y T ) , i.e., f (α ,# T + 1) ≤ 0. Case 1: 0 < α < α P (n) . By the definition, f (α , n) > 0 . Moreover, since the function ln# T − ln(# T − 1) is strictly decreasing in # T, f (α ,# T + 1) ≥ f (α , n) > 0 for 0 ≤#T ≤ n − 2 . First, since f (α , n) > 0 and N − T = ∅ , it follows from (i) and (ii) that N ∈

21

. Second,

since f (α , n) > 0 and f (α ,# T + 1) > 0 for 0 ≤#T ≤ n − 2 , it follows from (ii) that T ∉ 0 ≤#T ≤ n − 1 . Hence, we conclude that

for

= {N}.

Case 2: α = α P (n − b) , b = 0,1,2,..., n-2. In this case, f (α , n − b ) = 0 , f (α ,# T + 1) > 0 for 0 ≤#T ≤ n − b − 2 , and f (α ,# T ) < 0 for n − b + 1 ≤# T ≤ n . First, since f (α , n − b ) = 0 and f (α , n − b + 1) < 0 , it follows from (i) and (ii) that T ∈

for #T = n − b . Second, since

f (α , n − b − 1) > 0 and f (α , n − b ) = 0 , it follows from (i) and (ii) that T ∈

for

#T = n − b − 1 . Third, for 0 ≤#T ≤ n − b − 2 , since f (α ,# T + 1) > 0 , it follows from (ii) that T∉

. Fourth, for n − b + 1 ≤# T ≤ n , since f (α ,# T ) < 0 , it follows from (i) that T ∉

Hence, we conclude that T ∈

.

if and only if #T = n-b or n-b-1.

Case 3: α P (n − b ) < α < α P (n − b − 1) , b = 0,1,2, ..., n-3. In this case, f (α ,# T + 1) > 0 for 0 ≤#T ≤ n − b − 2 and f (α ,# T ) < 0 for n − b ≤# T ≤ n . First, since f (α , n − b − 1) > 0 and f (α , n − b) < 0 , it follows from (i) and (ii) that T ∈

for #T = n − b − 1 . Second, for

0 ≤#T ≤ n − b − 2 , since f (α ,# T + 1) > 0 , it follows from (ii) that T ∉ n − b ≤# T ≤ n , since f (α ,# T ) < 0 , it follows from (i) that T ∉ that T ∈

. Third, for

. Hence, we conclude

if and only if #T = n-b-1 .

Case 4: α P ( 2) < α < 1 . In this case, f (α ,1) > 0 and f (α ,# T ) < 0 for all 2 ≤#T ≤ n . First, since f (α ,1) > 0 and f (α ,2 ) < 0 , it follows from (i) and (ii) that T ∈ Second, for 2 ≤#T ≤ n , since f (α ,# T ) < 0 , it follows from (i) that T ∉ conclude that T ∈

for #T = 1 .

. Hence, we

if and only if #T = 1.

6. The Voluntary Contribution Mechanism 6.1 The Voluntary Participation Condition In the previous section, we see that the voluntary participation condition is not satisfied by any mechanism satisfying non-emptiness, feasibility, symmetry, and Pareto efficiency only for participants. However, Pareto efficiency only for participants is not necessary to obtain our negative result. In this section, we study the voluntary contribution mechanism which does not satisfy Pareto efficiency only for participants 22

when the equilibrium concept is 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 g Ti ( s) = (ω i − si , ∑ i∈T si ) for s ∈S T .

This definition of the voluntary contribution mechanism is a generalization of the usual one to the case in which voluntary participation is allowed. In most literature on the voluntary contribution mechanism, all agents are supposed to participate in it. Let µ be a Nash equilibrium correspondence. Then in the voluntary contribution mechanism, each agent i chooses 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} .

Theorem 5. Let n ≥ 3. Suppose that (i) U ⊇ U SCD ; (ii) for all i ∈ N, ω i = ω > 0 ; and (iii) µ is a Nash equilibrium correspondence. Then the voluntary contribution mechanism fails to satisfy the voluntary participation condition.

Proof. Take ( uαi ) i∈N ∈U SCD . Let ( x iN , y N ) be the consumption bundle that each agent i receives at the unique symmetric Nash equilibrium if all agents in N decide to participate in the mechanism. It is easy to see that ( x iN , 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 Nash equilibrium 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)

uαi ( xiN , y N ) − uαi (ω i , y N −{ i } ) 23

= 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 . By partially differentiating h(α , n) with respect to n, we have

∂h(α , n) −(1 − α )[1 − 2α + αn(1 + α − αn)] = . ∂n n(n − 1)[1 + α (n − 2)][1 + α (n − 1)] If α = 0.6 and n ≥ 3 , then ∂h(α , n ) 0.4[9n(n − 3) + 3n + 5] = >0. ∂n n(n − 1)( 3n − 1)( 3n + 2) Moreover, lim h(α , n) = 0 . Hence, for any finite number n ≥ 3 , h(0.6 , n) < 0. This n→∞

implies that the voluntary participation condition is violated.

6.2 The Number of Agents and a Participation Incentive In the following two subsections, we again focus on symmetric Cobb-Douglas economies. First, we show that as the number of agents grows large, each agent is less likely to have an incentive to participate in the voluntary contribution mechanism.

Theorem 6. Let n ≥ 2. Suppose that (i) U = U SCD ; (ii) for all i ∈ N, ω i = ω > 0 ; and (iii) µ is a Nash equilibrium correspondence. Then (a) the measure of the set of economies for which the voluntary contribution mechanism satisfies the voluntary participation condition strictly decreases as the number of agents increases; and (b) for each α ∈(0 ,1) , there exists an integer number nV (α ) ≥ 1 such that if n ≤ nV (α ) , then the voluntary contribution mechanism satisfies the voluntary participation condition; and if

24

n > nV (α ) , then the voluntary contribution mechanism fails to satisfy the voluntary

participation condition.

Proof. Take ( uαi ) i∈N ∈U SCD . As we show in the proof of Theorem 5, there exist a unique equilibrium allocation of the mechanism under the present assumptions. For each i ∈ N , uαi ( xiN , y N ) − uαi (ω i , y N −{ i } ) = h(α , n) as we defined in (5). For each n ≥ 2, let α V (n) be a value α ∈(0 ,1) satisfying h(α , n) = 0 . For α V (n) < α < 1 , h(α , n) < 0 , i.e., uαi ( xiN , y N ) < uαi (ω i , y N − i ) ; for α = α V (n) , h(α , n) = 0 , i.e., uαi ( xiN , y N ) = uαi (ω i , y N − i ) ; and for α < α V (n ) , h(α , n) > 0 , i.e., uαi ( xiN , y N ) > uαi (ω i , y N − i ) . Hence, U vp (n ) = ( 0 , α V (n)] . Moreover, the function α V is strictly decreasing in n.11 These imply the

desired results.

Remark: Since lim [ uαi ( xiN , y N ) − uαi (ω i , y N − i ) ] = lim h(α , n) = 0 for all α ∈(0 ,1) , n→∞

n→∞

lim U vp (n) = (0,1). In other words, the measure of the set of economies for which

n→∞

the voluntary participation condition is satisfied is the set of all economies when the number of agents is infinite. But this is because the difference between participating in the voluntary contribution mechanism and withdrawing from it vanishes as the number of agents grows large. It is easily checked that for all i ∈ N , lim xiN = ω and n→∞

lim ( y N − y N −{ i} ) = 0 , that is, as the number of participants increases, the individual

n→∞

contribution of the private good for the public good as well as the difference between the public good level for n participants and that for n-1 participants converge to zero, although the level of the public good produced remains to be strictly positive.

6.3 Equilibria of a Two-Stage Game with Voluntary Participation 11 For example, α V (2) ≈ 0.39, α V (3) ≈ 0.25, α V (4) ≈ 0.18, and α V (5) ≈ 0.14. 25

Next we study a two-stage game with voluntary participation for the voluntary contribution mechanism in symmetric Cobb-Douglas economies, as analyzed in Section 5.3. Figure 2 illustrates the two-stage game for the case of n = 2, α = 0.6, and ω = 10. It is straightforward to derive the payoff matrix for the first stage decision, similar to Table 1. There are two pure Nash equilibria: one agent participates in the mechanism, while the other agent does not. The case in which both agents choose participation is not a Nash equilibrium. In general, how many agents participate in the mechanism at equilibrium depends on the value of preference parameter α. We have a characterization of equilibrium sets of participants,

, which is similar to that for a Pareto efficient

mechanism (Theorem 5), by using the function α V defined in the previous section instead of α P .

7. Concluding Remarks We find that solutions to the free-rider problem thus far are not necessary solutions to the free-rider problem with voluntary participation. Furthermore, we show that it is quite difficult or impossible to design a mechanism with voluntary participation. We implicitly assume that every agent has an access to the technology of the public good production. If only a government has an access to the technology, then it would be much simple to force agents to participate in a public good provision mechanism: the government uses a mechanism satisfying strict individual rationality only if every agent votes "yes" to participation in the mechanism. Even though some agents have an access to the technology, this unanimity mechanism works providing that there exits a body that have enough power to force the agents not to produce the public good. Of course, analyzing this problem is still an open area of our future research. 26

In an experiment conducted by Saijo, Yamato, Yokotani, and Cason (1997), they find that cooperation has emerged though spiteful behavior in a repeated Hawk-Dove game with randomly matched pairs. Our theory in this paper predicts that no cooperation will emerge. An open question is to reconcile theory results to experimental results.

27

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