Discussion Paper No. 559

Transcription

1 Dscusson Paper No. 559 VOLUNARY PARICIPAION IN HE DESIGN OF NON-EXCLUDABLE PUBLIC GOODS PROVISION MECHANISMS atsuyosh Sajo and akehko Yamato October 2001 he Insttute of Socal and Economc Research Osaka Unversty 6-1 Mhogaoka, Ibarak, Osaka , Japan

2 Voluntary Partcpaton n the Desgn of Non-excludable Publc Goods Provson Mechansms # by atsuyosh Sajo* and akehko Yamato** October 2001 # Research was partally supported by the Grant n Ad for Scentfc Research of the Mnstry of Educaton, Scence and Culture n Japan, the Nssan Foundaton and the Sumtomo Foundaton. * Insttute of Socal and Economc Research, Osaka Unversty, Ibarak, Osaka , Japan, and Research Insttute of Economy, rade and Industry, Kasumgasek, Chyoda, okyo , Japan. E-mal address: sajo@ser.osaka-u.ac.jp ** Department of Value and Decson Scence, Graduate School of Decson Scence and echnology, okyo Insttute of echnology, Ookayama, Meguro-ku, okyo , Japan. E-mal address: yamato@valdes.ttech.ac.jp

3 Abstract Groves-Ledyard (1977) constructed a mechansm attanng Pareto effcent allocatons n the presence of publc goods. After ths path-breakng paper, many mechansms have been proposed to attan desrable allocatons wth publc goods. hus, economsts have thought that the free-rder problem s solved, n theory. Our vew to ths problem s not so optmstc. Rather, we propose fundamental mpossblty theorems wth publc goods. In the prevous mechansm desgn, t was mplctly assumed that every agent must partcpate n the mechansm that the desgner provdes. hs approach neglects one of the basc features of publc goods: non-excludablty. We explctly ncorporate non-excludablty and then show that t s mpossble to construct a mechansm n whch every agent has an ncentve to partcpate. Correspondent: atsuyosh Sajo Insttute of Socal and Economc Research Osaka Unversty Ibarak, Osaka Japan Phone (country & area codes) (offce)/ (fax) E-mal: sajo@ser.osaka-u.ac.jp

4 Abstract Groves-Ledyard (1977) constructed a mechansm attanng Pareto effcent allocatons n the presence of publc goods. After ths path-breakng paper, many mechansms have been proposed to attan desrable allocatons wth publc goods. hus, economsts have thought that the free-rder problem s solved, n theory. Our vew to ths problem s not so optmstc. Rather, we propose fundamental mpossblty theorems wth publc goods. In the prevous mechansm desgn, t was mplctly assumed that every agent must partcpate n the mechansm that the desgner provdes. hs approach neglects one of the basc features of publc goods: non-excludablty. We explctly ncorporate non-excludablty and then show that t s mpossble to construct a mechansm n whch every agent has an ncentve to partcpate. Correspondent: atsuyosh Sajo Insttute of Socal and Economc Research Osaka Unversty Ibarak, Osaka Japan Phone (country & area codes) (offce)/ (fax) E-mal: sajo@ser.osaka-u.ac.jp

5 1. Introducton he provson of publc goods has an ncentve problem called the free-rder problem. As Samuelson (1964) ponted out, t s mpossble to attan a Pareto effcent allocaton through a decentralzed fashon, n partcular, a decentralzed prcng system. On the contrary, Groves and Ledyard (1977) proposed an explct procedure, called a mechansm, n whch the Nash equlbrum allocaton s Pareto effcent. Partcpants can pursue ther own self-nterest beng free rders f they choose n the mechansm, but the mechansm s free from these ncentves. In ths sense, as the subttle of ther paper shows, they found a soluton to the free-rder problem. Although the allocaton of the Groves-Ledyard mechansm s Pareto effcent, the mechansm s not ndvdual ratonal. hat s, the allocaton does not satsfy the condton where t s at least as good as each partcpant's ntal endowment. Followng the path-breakng paper by Groves and Ledyard (1977), Hurwcz (1979) and Walker (1981) fxed ths problem and succeeded n mplementng the Lndahl correspondence n Nash equlbra, whch satsfes both Pareto effcency and ndvdual ratonalty. Subsequently, numerous mechansms have been proposed that satsfy addtonal desrable propertes such as ndvdual feasblty and balancedness 1. Most mechansms developed thus far share one undesrable property, however: partcpants n the mechansms do not have freedom not to partcpate. As Olson (1965) notced, any non-partcpant can obtan beneft of a publc good that s provded by 1 See Groves and Ledyard (1987) and Hurwcz (1994). For the domnant strategy equlbrum concept, we have mpossblty results: Pareto effcency, ndvdual ratonalty, and ncentve compatblty (.e., truthtellng s a domnant strategy) are nconsstent. See Hurwcz (1972) wthout publc goods and Ledyard and Roberts (1974) wth publc goods. Sajo (1991) showed an mpossblty result wthout requrng Pareto effcency: a slghtly stronger ndvdually ratonal condton called an autarkcally ndvdual ratonalty and 1

6 others. hs s due to the nature of a publc good called non-excludablty. In other words, Groves and Ledyard and ther followers found solutons to the free-rder problem once every partcpant decded to partcpate n the mechansms, but not solutons to the problem when agents have the ablty not to partcpate. hs partcpaton problem s mportant n many practcal stuatons, such as for nternatonal treates. he Kyoto Protocol to cope wth global warmng and clmate change s a specfc, recent example. It took years to agree on the basc framework, the Unted Natons Framework Conventon on Clmate Change (UNFCCC), to reduce the green house gases. UNFCCC was adopted n 1992 and entered nto force n he partes of UNFCCC adopted the Kyoto Protocol n he Protocol s a mechansm n our termnology to attan the goal of UNFCCC. he number of sgnatores ncludng the U.S.A. exceeded 186 n 2000, but n March 2001, Presdent Bush announced not to ratfy the Protocol snce t s harmful to the U.S. economy. herefore, the effectveness of the Protocol remans n doubt. In our framework, ratfcaton s equvalent to partcpaton. Another example s the League of Natons. Followng World War I Presdent Woodrow Wlson strongly supported the League, but the U.S. Congress never ratfed the reaty of Versalles 2. hus, our fundamental queston s: s there any mechansm satsfyng the condton that every agent always chooses partcpaton strategcally, called the voluntary partcpaton condton? In order to answer ths queston, we frst restrct our attenton to the Lndahl allocatons as a goal of our socety. hat s, our frst queston s whether or ncentve compatblty are nconsstent. However, our mpossblty results descrbed below do not depend on the choce of equlbrum concepts. 2

7 not any mechansm attanng Lndahl allocatons, called a Lndahl mechansm, can survve f we allow agents to choose partcpaton n the mechansm voluntarly. What we found s strkng. Each agent has an ncentve not to partcpate n the mechansm n a wde class of envronments. Based upon the prelmnary result, we return to the fundamental queston wth the two-agent economy. We fnd that no voluntary partcpaton mechansm exts under mld regularty condtons. Furthermore, ths result s ndependent of the choce of equlbrum concepts. he pcture s stll bleak even f the number of partcpants s at least three. Imposng Pareto effcency on a mechansm, we agan fnd a negatve result. he reason why we obtan the negatve result mght come from Pareto effcency on whch we mpose. We have a partal answer to ths queston. he voluntary contrbuton mechansm, whch cannot attan Pareto effcency, does not satsfy the voluntary partcpaton condton, ether. Mouln (1986), Palfrey and Rosenthal (1984), and Sajo and Yamato (1999) also analyzed the ssue of an ncentve to partcpate n a mechansm for the provson of a publc good. 3 Mouln and Palfrey-Rosenthal focused on specfc mechansms: Mouln studed the pvotal mechansm n dscrete publc goods economes wth quas-lnear preferences, whereas Palfrey and Rosenthal consdered a smple mechansm for the provson of a bnary publc good wth bnary contrbutons. Sajo and Yamato examned 2 Voluntary publc goods provson -- such as for publc broadcastng -- also faces the partcpaton problem. For example, part of publc broadcastng n Japan s supported by the publc broadcastng fee. Every famly must pay the fee by law, but many choose not to snce punshment s practcally non-exstent. 3 he partcpaton problem n an nsttuton has been examned manly n the context of votng and cartel formaton (e.g., see Brams and Fshburn (1983), Dxt and Olson (2000), Ledyard (1984), Okada (1996), Palfrey and Rosenthal (1983,1985), and Selten (1973)). 3

8 a specfc two sage game: the frst stage s a decson stage of partcpaton n a mechansm, and only the partcpants n the frst stage play the second stage. On the other hand, we nvestgate partcpaton ncentve propertes of a large class of mechansms n economc envronments wth a contnuous publc good. he paper s organzed as follows. In Secton 2, we explan an example llustratng our basc dea. In Secton 3, we ntroduce notaton and defntons. We establsh an mpossblty result on partcpaton ncentves for the case of two agents n Secton 4 and that for the case of at least three agents n Secton 5. In Secton 6, we nvestgate the voluntary contrbuton mechansm. In the fnal secton, we make concludng remarks. 2. An Example: Lndahl Mechansms We analyze the followng symmetrc two-agent economes wth one prvate good x and one pure publc good y that s non-excludable and non-rval. he publc good can be produced from the prvate good by means of a constant return to scale technology, and let y = x be the producton functon of the publc good. A consumpton bundle for agent s denoted by ( x, y) R + 2 where x R + s the level of prvate good she consumes on her own, and y R + s the level of publc good. wo agents have the α same preferences that can be represented by a Cobb-Douglas utlty functon u ( x, y ) = α x y 1 α, where 0< α < 1 and = 1,2. Each agent's ntal endowment s also the same and gven by (ω, 0) = (10, 0) for = 1,2. We nvestgate stuatons n whch the true value of the preference parameter α s unknown to the mechansm desgner, but the ntal endowment and the producton technology are known. 4

9 Consder any mechansm mplementng the Lndahl correspondence n Nash equlbra (see Hurwcz (1979), Walker (1981), Hurwcz, Maskn, and Postlewate (1984), an (1990), and so on). 4 Suppose that each agent s able to choose whether she partcpates n the mechansm. hen n order to acheve the desred Lndahl equlbrum allocaton by usng the mechansm, every agent must choose partcpaton. herefore, we ask a crucal queston of whether each agent always have an ncentve to partcpate n the mechansm. Unfortunately, our answer to ths queston s negatve. o see why, let { 12, } be the set of agents who partcpate n the mechansm. 5 An equlbrum allocaton of the mechansm when the agents n partcpate n t s denoted by (( x ), y ). 6 If two agents decde to partcpate n the mechansm, then {, } {, } {, } ( x1 12, x2 12, y 12 ) s a Lndahl allocaton of the economy wth two agents, snce the mechansm mplements the Lndahl correspondence. 7 It s straghtforward to check that {, } {, } {, } there exsts a unque Lndahl allocaton gven by ( x1 12, x2 12, y 12 ) = (10α, 10α, 20(1- α)). Now suppose that some agent does not partcpate n the mechansm, whle {} j j {} j the other agent j does,.e., = { j}. hen ( x, y ) s a unque Lndahl allocaton of {} j {} j j the economy consstng of only one agent j. It s easy to see that ( x, y ) = (10α, 10(1- α)). Notce that non-partcpant can enjoy her ntal endowment, ω, as well as the 4 he Lndahl correspondence s the same as the constraned Lndahl correspondence (Hurwcz, Maskn, and Postlewate (1984)) under the present assumptons. 5 A "partcpant" stands for an agent who chooses partcpaton n a mechansm, whle an "agent" represents any member who belongs to an economy. 6 A mechansm specfes a strategy set of each partcpant n and an outcome functon for each { 12, }. hs defnton of a mechansm s more general than the usual one. 5

10 non-excludable publc good produced by agent j, y {} j. On the other hand, she s no longer able to affect the decson on the provson of the publc good. he followng condton should be satsfed f each agent has an ncentve to partcpate n the mechansm: { 12, } { 12, } {} j α (1) u ( x, y ) u α ( ω, y ) for, j = 1,2, j, where u α s any Cobb-Douglas utlty functon. We call condton (1) the voluntary partcpaton condton. 8 We show that no mechansm mplementng the Lndahl correspondence satsfes ths condton. hs fact can be llustrated for the case of α = 0.6 by usng Kolm's trangle. See Fgure 1. Pont A denotes the Lndahl equlbrum { 12, } allocaton when both agents partcpate n the mechansm: A = ( x 1, { 12, } x2, y { 12, } ) = (6, 6, 8). Pont B represents the allocaton when agent 1 does not partcpate n the { 2} mechansm, but agent 2 does: B = (ω 1, x 2, y { } 2 ) = (10, 6, 4). Snce 06. u1 { 12, } ( x 1, y { 12, } ) < u 1 (ω 1, y { 2 } ) 6.93, agent 1 would not partcpate n the mechansm when agent 2 does. he same thng holds for agent 2. Fgure 1 s around here. 3. Notaton and Defntons 7 A mechansm mplements the Lndahl correspondence f for each set of partcpants { 12, } and each economy consstng of the partcpants n, every equlbrum allocaton s a Lndahl allocaton and every Lndahl allocaton s an equlbrum allocaton. 8 he voluntary partcpaton condton s dfferent from the ndvdually ratonal condton whch α { 12, } { 12, } requres that u ( x, y ) u α α { j} α ( ω, 0) for = 1,2. Snce u ( ω, y ) u ( ω, 0 ), the voluntary partcpaton condton s stronger than the ndvdually ratonal condton. 6

11 In the prevous secton, we saw that any Lndahl mechansm fal to satsfy the voluntary partcpaton condton n symmetrc economes wth two agents when Nash equlbrum s an equlbrum concept. We wll show below that a smlar negatve result holds for any equlbrum concept and any mechansm meetng mld condtons. Frst of all, we ntroduce notaton and defntons. In Secton 2, we nvestgated two-agent economes wth one prvate good, one pure publc good, and a constant return to scale technology. We study the same stuatons wth many agents. Let N = {1,2,...,n} be the set of agents, wth generc element. We assume that each agent 's preference relaton admts a numercal representaton u : R + 2 R whch s contnuous, concave, and monotonc. Let U be the class of utlty functons admssble for agent. Let P(N) be the collecton of all no-empty subsets of N. For P(N), let u ( u ) U U be a preference profle for the agents n. Agent 's ntal endowment s denoted by (ω, 0). hat s, there s no publc good ntally. Let a dstrbuton of ntal endowments of the prvate good ( ) ω N # + 1 gven. Gven P(N), a feasble allocaton for s a lst ( x, y) (( x ), y) R + such that ( ω x ) = y. he set of feasble allocatons for s denoted by A. A mechansm s a functon Γ that assocates wth each P(N) a par Γ( ) = be ( S, g ), where S = S and g : S R # + 1. Here S s the strategy space of agent and g s the outcome functon when the agents n play the mechansm. Gven g (s) = ( x, y), let g s () ( x, y) for and g y (s) = y. An equlbrum correspondence s a correspondence µ whch assocates wth each mechansm Γ, each set of agents P(N), and each preference profle u U, a set of 7

12 strategy profles µ( Γ, u, ) S, where (S, g ) = Γ( ). We smply wrte µ( Γ, u, ) as µ Γ ( u ). Examples of equlbrum correspondences nclude domnant strategy equlbrum correspondence, Nash equlbrum correspondence, and strong Nash equlbrum correspondence. he set of µ-equlbrum allocatons of Γ for at u s denoted by g o µ Γ ( u ) {( x, y) R # + 1 there exsts s S such that s µ Γ ( ) and g (s) = ( x, y)}, where (S, g ) = Γ( ). u 4. he Case of wo Agents Let an equlbrum correspondence µ be gven. We ntroduce several condtons on a mechansm. Defnton 1. he mechansm Γ satsfes non-emptness under µ f for all P(N) and all u U, g o µ Γ ( u ). Defnton 2. he mechansm Γ satsfes feasblty under µ f for all P(N) and all u U, g o µ Γ ( u ) A. Non-emptness says that there always exsts an equlbrum. Feasblty demands that an equlbrum allocaton of the mechansm should always be feasble. Note that we requre feasblty only at equlbrum, but not out of equlbrum. Moreover, a feasble mechansm does not necessarly satsfy ndvdual feasblty (.e., 8

13 for all P(N) and all s S, g # + 1 (s) R + ) nor balancedness (.e., for all P(N) and all s S, g (s) A ). Defnton 3. he mechansm Γ satsfes the voluntary partcpaton condton under µ f for N N N all u N U N, all ( x, y ) g o µ Γ ( u ), and all N, u ( x, y ) u ( ω, y ), N N N mn {} N where y N Arg mn u y N {} yn {} ( ω, ) g N {} y oµ Γ ( {} ) mn {} u N. Snce there s one publc good and preferences satsfy monotoncty, ymn N {} s the mnmum equlbrum level of publc good when all agents except partcpate n the mechansm. Consder an agent who decdes not to partcpate n the mechansm. hen she can enjoy the non-excludable publc good produced by the other agents wthout provdng any prvate good, whle she cannot affect the decson on the provson of the publc good. he voluntary partcpaton condton requres that no agent can beneft from such a free-rdng acton. Note that when an agent chooses non-partcpaton, she has a pessmstc vew on the outcome of her acton: an equlbrum outcome that s most unfavorable for her wll occur. 9 Mouln (1986) proposed a smlar condton, called the No Free Rde axom, to characterze the pvotal mechansm when publc goods are dscrete and costless, and preferences are quas-lnear. 9 A stronger condton on voluntary partcpaton s concevable for the case n whch the non-partcpant has a more optmstc vew that a better equlbrum outcome wll happen. However, we wll derve mpossblty results regardng ths weak condton on voluntary partcpaton, and hence our results hold for other stronger versons. 9

14 Defnton 4. he mechansm Γ satsfes the Robnson Crusoe condton under µ f for all {} {} {} N and all u U, f ( x, y ) g o µ Γ ( u ), then ( x {}, y {} ) Arg max ( x, y) A {} u ( x, y). he Robnson Crusoe condton means that f only one agent partcpates n the mechansm, then she chooses an outcome that s best for her. We establsh an mpossblty result that three condtons mentoned above are ncompatble n the case of two agents. Let U SCD = {( u ) N α N, u ( x, y) = u ( x, y ) = α ln x ( 1 α)ln y, α ( 0, 1 )} be the class of symmetrc Cobb-Douglas utlty profles10. + heorem 1. Let n = 2 and µ be an arbtrary equlbrum correspondence. Suppose that U and for all N, ω ω >0. If a mechansm satsfes non-emptness, feasblty, and U SCD = the Robnson Crusoe condton under µ, then t fals to satsfy the voluntary partcpaton condton under µ. he proof of heorem 1 s llustrated n Fgure 2. Consder the case n whch both agents have the same Cobb-Douglas utlty functon wth α = 0.6. By the Robnson Crusoe condton, a unque equlbrum allocaton of the mechansm when only agent 2 (resp. agent 1) partcpates n t s gven by Pont C (resp. Pont D) n Fgure 2. Moreover, f the mechansm satsfes the voluntary partcpaton condton, then at equlbrum, agent 1 (resp. agent 2) should receve a consumpton bundle n her weak upper contour 10 Here a Cobb-Douglas utlty functon s denoted by a natural logarthmc functon, whle t s an exponental functon n the example descrbed n Secton 2. he results n ths paper hold ndependent of whch functon s used. 10

15 set at C (resp. D) when both agents choose partcpaton. hese upper contour sets are denoted by the shaded areas n Fgure 2. However, snce they are dsjont, the feasblty condton s volated. A formal proof of heorem 1 s gven as follows: Fgure 2 s around here. Proof of heorem 1. Suppose by way of contradcton that the mechansm satsfes the α α voluntary partcpaton condton. Consder ( u, u ) U SCD wth α = It s easy to 1 2 check that by the Robnson Crusoe condton, a unque equlbrum allocaton of the {} {} mechansm for one agent economy s gven by ( x, y ) = ( 06. ω, 04. ω ), = 12,. Let V(( ω, y ), u ) {} j { j} 2 {( x, y) R + u ( x, y ) u ( ω, y )} be agent 's weak upper {} j 0.6 contour set at (, y ) for u, where (, y ) = ( ω, 04. ω) and j. Pck any ω {, } {, } {, } ω { 12, } ( x1 12, x2 12, y 12 ) g o µ Γ ( u, u ). By the voluntary partcpaton condton, {} j {, } {, } {} j 06. (2) ( x 12, y 12 ) V(( ω, y ), u ) (, j= 12;j, ). We clam that (3) {} j 06. ( x, y) V(( ω, y ), u ), 2x + y 2 (, j= 2 Suppose that (3) does not hold. hen for some and some ( x, y) R +, { } j u ( x, y) u ( ω, y ) and 2x + y = 2 ω. Let ( x, y ) be a maxmzer of the utlty 06. functon u ( x, y) = 06. ln x lny subject to the constrant 2 x y = 2ω. It s easy to + see that ( x, y ) = ( 06. ω, 08. ω ) and u x y u y j (, ) ( ω, ) = 06. ln ln2 < { } 11

16 { j} < 0. hus, u ( x, y) u ( ω, y ) > u ( x, y ), whch contradcts the fact that ( x, y ) s the maxmzer of u ( x, y ) subject to 2x + y = 2 ω. 06. However, by (2) and (3), x + x + y ω. hs contradcts the { 1 12, } { 2 12, } { 12, } 2 feasblty condton on the mechansm. Q.E.D. 5. Pareto Effcent Mechansms In ths subsecton, we show an mpossblty result on the voluntary partcpaton condton n the case of at least three agents. We propose the followng two condtons on a mechansm. Let an equlbrum correspondence µ be gven. Defnton 5. he mechansm Γ satsfes symmetry under µ f for all P(N) and all u U, f u, j. = u and ω = ω for all, j and ( x, y) g o µ Γ ( u ), then x = x for all j j j Defnton 6. he mechansm Γ satsfes Pareto effcency only for partcpants under µ f for all P(N) and all u U, g o µ Γ ( u ) P ( u ), where P ( u ) {( x, y) A there does not exst ( x, y ) A such that u ( x, y ) u ( x, y ) for all and u ( x, y ) > u ( x, y) for some }. 11 By usng an argument smlar to the below, t s not hard to check that the four condtons mentoned n heorem 1 are ncompatble for any α ( 051., ). 12

17 Symmetry requres that f all partcpants have the same preferences and endowments, then they receve the same consumpton bundle at equlbrum. herefore, every partcpant pays the same amount of the prvate good for the provson of the publc good. Pareto effcency only for partcpants means that every equlbrum allocaton of the mechansm should be Pareto effcent for partcpants, but not necessarly effcent wth respect to all agents. heorem 2. Let n 3 and µ be an arbtrary equlbrum correspondence. Suppose that U and for all N, ω ω >0. If a mechansm satsfes non-emptness, feasblty, U SCD = symmetry, and Pareto effcency only for partcpants under µ, then t fals to satsfy the voluntary partcpaton condton under µ. Proof. ake ( u α SCD ) U. ake any equlbrum allocaton (( x ), y N ) g N N N N o µ Γ ( u ). By symmetry, x = x for all, j N. By feasblty and Pareto effcency N N j N only for partcpants, ( x N, y N ) s a maxmzer of the utlty functon α ln x+ ( 1 α)ln y, N subject to nx + y = nω. It s easy to check that ( x, y N ) = (ωα, nω( 1 α) ). In a smlar N { } way, we can show that ( x, y N { } ) = (ωα, ( n 1) ω( 1 α )) for N. herefore, the dfference between the utlty level when all agents partcpate n the mechansm and that when all agents except partcpate n t s gven by α N (4) u x y u α {} (, ) ( ω, y ) N N = α ln α + ( 1 α )[lnn ln( n 1)] f ( α, n) 13

18 for N. We prove that the sgn of f ( α, n) s negatve when α = 06. and n Note that the functon lnn ln( n 1 ) s decreasng n n. herefore, for n 3, f (., 06 n) f (.,) 06 3 = 06. ln [ln3 ln 2] < < 0. hs mples that the voluntary partcpaton condton s volated. Q.E.D. Remark: By usng an argument smlar to the proofs of heorems 1 and 2, we can show that the condton of Pareto effcency only for partcpants can be replaced by a weaker condton n heorem 2: f a mechansm satsfes non-emptness, feasblty, symmetry, and Pareto effcency only wth respect to n-1 partcpants (.e., for all P(N) wth # = n 1and all u U, g o µ Γ ( u ) P ( u )), then t fals to satsfy the voluntary partcpaton condton. hs result holds when there are at least two agents and hence heorem 1 on the two-agent case s a corollary of t. Although the result s logcally better than heorem 2, the condton of Pareto effcency only wth respect to n-1 partcpants would not have a meanngful economc nterpretaton, except the case of two agents n whch the condton s equvalent to the Robnson Crusoe condton. 6. he Voluntary Contrbuton Mechansm In the prevous secton, we found negatve results on voluntary partcpaton for any mechansm satsfyng non-emptness, feasblty, symmetry, and Pareto effcency only for partcpants. However, Pareto effcency only for partcpants s not necessary to obtan such results. In ths secton, we study the voluntary contrbuton mechansm that does not satsfy Pareto effcency only for partcpants when the equlbrum concept s 12 By usng an argument smlar to the below, t s not dffcult to check that the sgn of f( α, n) s negatve 14

19 Nash equlbrum. o our surprse, ths mechansm does not satsfy the voluntary partcpaton condton, even though the name of the mechansm contans the term "voluntary". Defnton 7. he voluntary contrbuton mechansm s a mechansm such that for all P(N) and, S = [ 0, ω ] and g () s = ( ω s, s) for s S. he above defnton of the voluntary contrbuton mechansm s a generalzaton of the usual one, n whch all agents are supposed to partcpate, to the case n whch voluntary partcpaton s allowed. When the equlbrum concept s Nash equlbrum, each agent selects her contrbuton out of her endowment to the provson of the publc good, s, to maxmze her utlty u ( ω s, j sj ), gven contrbutons of the other agents n, ( s j ) j {} the voluntary contrbuton mechansm. n heorem 3. Let n 3. Suppose that ()U ; () for all N, ω ω >0 ; and () µ s a U SCD = Nash equlbrum correspondence. hen the voluntary contrbuton mechansm fals to satsfy the voluntary partcpaton condton. α SCD N Proof. ake ( u ) U. Let ( x, y N ) be the consumpton bundle that each agent N receves at the unque symmetrc Nash equlbrum f all agents n N decde to partcpate for any α ( , 1 ) and any n 3. 15

20 N n the mechansm. It s easy to see that ( x, y N ) = (ωαn/( 1+ α( n 1)), ω( 1 α) n/( 1+ α( n 1)) ). Also, let y N { } be the publc good level at the unque Nash equlbrum allocaton of the mechansm played among n-1 partcpants n N straghtforward to check that y N { } = ω(1-α)( n 1/(1+α(n-2)). ) hus, for N, α (5) u x N y N u α N {} (, ) ( ω, y ) { }. It s = α ln α + ( 1 α)[lnn ln( n 1)] +α lnn + ( 1 α)ln[ 1+ α( n 2)] ln[ 1+ α( n 1)] h( α, n). We show that the sgn of h( α, n) s negatve when α = 06. and n By partally dfferentatng h( α, n) wth respect to n, we have h( α, n) ( α)[ α αn( α αn)] = n nn ( 1)[ 1+ α( n 2)][ 1+ α( n 1 )]. If α = 06. and n 3, then h ( α, n ) 049. [ nn ( 3 ) + 3n+ 5 = ] > 0. n n( n 1)( 3n 1)( 3n+ 2) Moreover, lm h( α, n) = 0. Hence, for any fnte number n 3, h( 0. 6, n) < 0. hs n mples that the voluntary partcpaton condton s volated. Q.E.D. 7. Concludng Remarks We see that the solutons to the free-rder problem, whch have been proposed n mechansm desgn theory, are not necessary solutons to the free-rder problem when 13 By usng an argument smlar to the below, t s not dffcult to check that the sgn of h( α, n) s negatve for any α ( 0. 25, 1 ) and any n 3. 16

21 partcpaton n mechansms s voluntary. Furthermore, we show that t s qute dffcult or mpossble to desgn a mechansm wth voluntary partcpaton. As Olson (1965) argued, a publc good would be less lkely provded as the number of agents becomes large. Sajo and Yamato (1999) confrmed ths conjecture by provng that n a two-stage game wth voluntary partcpaton, the measure of the set of symmetrc Cobb-Douglas economes for whch every agent chooses partcpaton at equlbrum becomes smaller as the number of agents grows large. In a smlar way, we can show that the measure of the set of economes for whch the voluntary partcpaton condton s satsfed s strctly decreasng as the number of agents ncreases. hs would be another result supportng Olson s conjecture. In the voluntary partcpaton condton defned above, t s mplctly assumed that each agent has the most optmstc conjecture on the number of other agents who wll not partcpate n the mechansm f she does not, that s, she expects no agent other than her to choose non-partcpaton n the mechansm. On the other hand, n the ndvdually ratonal condton usually dscussed n the lterature on mechansm desgn, t s assumed that each agent has the most pessmstc conjecture on that number, that s, she expects all other n-1 agents to select non-partcpaton, too. However, an agent mght have an ntermedate conjecture: her conjecture on the number of other non- partcpants can take on a whole range of values from 0 to n-1. An open queston s to examne other condtons on voluntary partcpaton takng account of these possble conjectures. Sajo, Yamato, Yokotan, and Cason (1998), and Cason, Sajo, and Yamato (2001) observed that cooperaton has emerged though spteful behavor n ther experments on the voluntary contrbuton mechansm wth voluntary partcpaton. Our theory n ths 17

22 paper suggests that no cooperaton wll emerge. Reconclng theoretcal results to expermental results s an open area of our future research. References BRAMS, S. J., AND P. C. FISHBURN (1983): "Paradoxes of Preferental Votng," Mathematcs Magazne, 56, CASON,.,. SAIJO, AND. YAMAO (2001): Voluntary Partcpaton and Spte n Publc Good Provson Experments: An Internatonal Comparson, Purdue Unversty Center for Internatonal Busness Educaton and Research (CIBER) workng paper. DIXI, A. AND M. OLSON (2000): "Does Voluntary Partcpaton Undermne the Coase heorem? Journal of Publc Economcs, 76, GROVES,., AND J. LEDYARD (1977): "Optmal Allocaton of Publc Goods: A Soluton to the 'Free Rder' Problem," Econometrca, 45, GROVES,., AND J. LEDYARD, "Incentve Compatblty Snce 1972," n Informaton, Incentves, and Economc Mechansms: Essays n Honor of Leond Hurwcz, eds.,. Groves, R. Radner, and S. Reter, (Mnneapols, Unversty of Mnnesota Press, 1987), HURWICZ, L. (1972): "On Informatonally Decentralzed Systems," n Decson and Organzaton: A Volume n Honor of Jacob Marschak, eds., R. Radner and C. B. McGure, (Amsterdam, North-Holland), HURWICZ, L. (1979): "Outcome Functons Yeldng Walrasan and Lndahl Allocatons at Nash Equlbrum Ponts," Revew of Economc Studes, 46, HURWICZ, L. (1994): "Economc Desgn, Adjustment Process, Mechansms, and Insttutons," Economc Desgn, 1, HURWICZ, L., E. MASKIN, AND A. POSLEWAIE (1984): "Feasble Implementaton of Socal Choce Correspondences by Nash Equlbra," mmeo. LEDYARD, J. (1984): "he Pure heory of Large wo-canddate Electons," Publc Choce, 44, LEDYARD, J. AND J. ROBERS (1974): "On the Incentve Problem wth Publc Goods," mmeo, Northwestern Unversty. 18

23 MOULIN, H. (1986): "Characterzatons of the Pvotal Mechansm," Journal of Publc Economcs, 31, OKADA, A. (1996): "he Organzaton of Socal Cooperaton: A Noncooperatve Approach," KEIR Dscusson Paper, Kyoto Unversty. OLSON, M. (1965): "he Logc of Collectve Acton: Publc Goods and the heory of Groups," Cambrdge: Harvard Unversty Press. PALFREY,., AND H. ROSENHAL (1983): "A Strategc Calculus of Votng," Publc Choce, 41, PALFREY,., AND H. ROSENHAL (1984): "Partcpaton and the Provson of Dscrete Publc Goods: A Strategc Analyss," Journal of Publc Economcs, 24, PALFREY,., AND H. ROSENHAL (1985): "Voter Partcpaton and Strategc Uncertanty," Amercan Poltcal Scence Revew, 79, SAIJO,. (1991): Incentve Compatblty and Indvdual Ratonalty n Publc Good Economes, Journal of Economc heory, 55, SAIJO,., AND. YAMAO (1999): A Voluntary Partcpaton Game wth a Nonexcludable Publc Good, Journal of Economc heory, 84, SAIJO,,. YAMAO, K. YOKOANI, AND. N. CASON (1998): "Voluntary Partcpaton n Publc Good Provson Experments: Is Sptefulness a Source of Cooperaton?" mmeo, Osaka Unversty. SAMUELSON, P. A. (1964): "he Pure heory of Publc Expendture," he Revew of Economcs and Statstcs, 36, SELEN, R. (1973): "A Smple Model of Imperfect Competton, where 4 Are Few and 6 Are Many," Internatonal Journal of Game heory, 2, IAN, G. (1990): "Completely Feasble Contnuous Implementaton of the Lndahl Correspondence wth a Message Space of Mnmal Dmensons," Journal of Economc heory, 51, WALKER, M. (1981): "A Smple Incentve Compatble Scheme for Attanng Lndahl Allocatons," Econometrca, 49,

24 A 8 0 ω B 6 Fgure 1. A Lndahl mechansm does not satsfy the voluntary partcpaton condton.

25 2 1 D C 0 ω Fgure 2. No mechansm satsfes the the voluntary partcpaton condton for the two-agent case.