Public Good Provision with Constitutional Constraint

Transcription

1 Public Good Provisio with Costitutioal Costrait Kag Rog School of Ecoomics, Shaghai Uiversity of Fiace ad Ecoomics, Shaghai , Chia Key Laboratory of Mathematical Ecoomics (SUFE), Miistry of Educatio, Shaghai , Chia Mar, 2017 Abstract: This paper studies the public good provisio problem, i which the pricipal faces a costitutioal costrait i the sese that i order for a public good provisio mechaism to be implemeted, it must first be approved by agets uder a prespecified votig rule. We fid that as log as the votig rule is ot the uaimity rule, the pricipal ca propose a mechaism such that first-best efficiecy of provisio of the public good is achieved. We also cosider various costraits, such as prohibitio of discrimiatory mechaisms or the existece of iterest groups ad vote buyig, which may prevet the proper fuctioig of the mechaism we propose. We discuss the optimal votig rule i such situatios. Keywords: Public goods; Costitutioal costrait; Votig; Efficiecy. JEL classificatio: D82, H41, D72 rog.kag@shufe.edu.c. Phoe: I thak Pak Hug Au, Da Berhardt, Sambuddha Ghosh, Xiag Su, Yu Wag, ad semiar participats at Wuha Uiversity, Xiame Uiversity, Shaghai Uiversity of Fiace ad Ecoomics, ad Nayag Techological Uiversity for helpful commets. I also thak the editor ad two referees for thoughtful commets that helped improve the paper sigificatly. All errors are mie. 1

2 1 Itroductio This paper cosiders the public good provisio problem. Imagie that a mayor of a city (or a pricipal) is plaig to provide a public good (e.g., build a public library) i the city. There is a large group of citizes (or agets) i the city. Each aget has a valuatio about the public good, which is ukow to the pricipal. The pricipal ca propose a mechaism, which is a mappig from agets reported valuatios to decisios about whether to provide the public good ad, if provided, how to allocate its cost. However, the pricipal faces a costitutioal costrait i the sese that the mechaism that he proposes ca be implemeted oly if the mechaism ca be approved by the agets uder a prespecified α- majority votig rule. This reflects the fact that i reality, implemetatio of a ew law i a city usually requires the approval of the city coucil (which ca be regarded as a committee that represets the agets i the city). Our mai research questio is: For ay prespecified costitutioal costrait (i.e., ay give votig rule), ca the pricipal propose a suitably desiged mechaism such that first-best efficiecy of provisio of the public good is achieved? Our mai model is a private value model, i which agets valuatios are i.i.d. The public good is a discrete good, which ca either be provided or ot provided. We assume that the expected value of a aget s valuatio exceeds the average cost of the public good, which implies that it is commo kowledge that the public good should be provided i a large ecoomy. Suppose that the votig rule is the α-majority rule (i.e., α fractio of yes votes is required to get a mechaism approved). We fid that as log as α is less tha oe, the the pricipal ca always desig a mechaism that will be approved by agets with probability oe i a large ecoomy. I additio, coditioal o approval of the mechaism, the probability that the public good will be provided goes to oe as the ecoomy becomes large. We thus obtai a efficiecy result for ay give α < 1. We ext briefly explai how we obtai the efficiecy result ad describe the difficulties. To achieve efficiecy, a prerequisite is that the mechaism proposed by the pricipal ca be 2

3 supported by more tha α fractio of agets. Note that a aget votes for a mechaism if ad oly if votig for the mechaism yields a higher payoff tha votig agaist the mechaism, coditioal o the evet that the aget s vote is pivotal. So, i order to iduce a aget to vote for a mechaism, we ca simply reward the aget by givig him a sufficietly large cash paymet if he votes yes ad his vote is pivotal. This is a key idea that we will use to costruct the mechaism that achieves efficiecy. However, certai difficulties arise. First, it appears that as log as the reward give to a pivotal aget who votes yes is sufficietly large, all agets will choose to vote for the mechaism. However, if all agets vote for the mechaism, the o aget is pivotal (except for the extreme case i which the votig rule is the uaimity rule). As a result, all agets votig for the mechaism is a weak equilibrium i the sese that every aget is idifferet betwee votig for the mechaism ad votig agaist it. Such a equilibrium may require a high level of coordiatio ad is quite fragile. I this paper, we will exclude such weak equilibrium. To obtai a strict equilibrium, we will eed to have some (maybe a very small fractio of) agets vote agaist the mechaism so that the evet that a aget is pivotal occurs with a o-zero probability. Secod, agets votig behavior is usually aoymous. So, the pricipal caot reward a aget directly based o the aget s votig behavior. However, the pricipal ca idirectly reward a aget based o the aget s reported valuatio by usig discrimiatory paymets whe the mechaism is approved. Thus, the mechaism proposed by the pricipal should be such that i equilibrium, a aget s votig behavior ad his reported valuatio are correlated so that the pricipal ca (idirectly) reward the aget who votes for the mechaism. We fid a very simple mechaism, called the Equal Cost Sharig with Pivotal Reward mechaism (or simply the ECSPR mechaism), that addresses the issues metioed above. The mechaism is as follows. Let be the miimum umber of agets required for a mechaism to be approved (i.e., is the smallest iteger that is greater tha or equal to α, where is the umber of agets i the ecoomy). The ECSPR mechaism, oce approved, 3

4 asks each aget to report either a high value h or a low value l. 1 If there are exactly agets who report high values, the the public good is provided with some probability, ad ay aget who reports the high value will be rewarded with payig less tha the average cost if the public good is provided (where the differece betwee the average cost ad the paymet made by a aget who reports the high value i this case is called the pivotal reward). 2 If more tha agets report high values, the public good is provided with probability oe ad its cost is equally distributed amog all agets. If exactly 1 agets report high values, the public good is ot provided, but the agets who report low values eed to make some paymets to the agets who report high values. I all other cases, the public good is ot provided ad o paymet is collected from ay aget. We fid that if the pricipal proposes the ECSPR mechaism (with suitably chose pivotal reward), the there is a strict Bayesia Nash equilibrium i the votig game. I this equilibrium, ay aget whose valuatio exceeds a threshold will vote for the mechaism ad report h, ad ay aget whose valuatio is less tha the threshold will vote agaist the mechaism ad report l. We fid that as the pivotal reward icreases, the threshold decreases. As a result, there exists a rage of pivotal rewards such that for ay pivotal reward i this rage, (i) the threshold is low eough such that the probability that a aget s valuatio exceeds the threshold is greater tha α, ad (ii) the threshold exceeds the lowest possible valuatio of a aget such that some very low-valuatio agets still prefer to vote agaist the mechaism. Iterestigly, this rage is idepedet of the umber of agets i the ecoomy. We ca use the law of large umbers to show that, as the ecoomy becomes large, as log as the pivotal reward remais withi the above rage, the mechaism will be approved with probability oe ad the provisio of the public good will approach its first-best efficiet level. 1 Obviously, the mechaism is a idirect mechaism. We ca trasform it to a truthful direct mechaism usig the revelatio priciple. However, it is more coveiet to state the mechaism as a idirect mechaism. 2 Correspodigly, ay aget who reports a low value i this case will make a paymet that is greater tha the average cost of the public good. 4

5 1.1 Literature Review A cetral topic i the public good provisio literature cocers whether public goods ca be provided at a efficiet level. Various positive ad egative results have emerged. Mailath ad Postlewaite (1990) fid that a public good will ever be provided i a large ecoomy if the idividual ratioality (IR) costrait is required (see also Rob (1989)). I the case i which the IR costrait is ot a cocer, d Aspremot ad Gérard-Varet (1979) show that a public good ca be provided efficietly uder the so-called AGV mechaism. 3 Hellwig (2003) obtais a efficiecy result by cosiderig the situatio i which the IR costrait is required, but the cost of the public good is fixed ad idepedet of the umber of agets. Norma (2004) studies a public good provisio problem with use exclusios, ad obtais both efficiecy ad iefficiecy results, depedig o whether the average cost of the public good is less tha or exceeds a threshold. This paper is also related to Rog (2014), who proposes a so-called α idividual ratioality costrait, i which the IR costrait is oly required for α fractio of agets. Rog (2014) shows that the efficiecy result is obtaied if α is less tha a threshold, while the iefficiecy result is obtaied if α exceeds the threshold. A key differece betwee Rog (2014) ad this paper is that i the former, a mechaism ca be implemeted if ad oly if α fractio of agets obtai o-egative expected payoffs from the mechaism, while i this paper, a mechaism ca be implemeted if ad oly if α fractio of agets vote for the mechaism. Note that the fact that a aget votes for a mechaism does ot ecessarily mea that the aget will obtai a o-egative expected payoff from the mechaism; it oly meas that i the pivotal evet (i.e., i the case where the aget s vote is pivotal), the implemetatio of the mechaism yields a o-egative expected payoff for the aget. 4 I other words, the 3 The problems studied i these papers may be regarded as extreme cases of our model. I particular, the problems studied by Mailath ad Postlewaite (1990) ad Rob (1989) roughly correspod to the case of α = 1 i our model (see Corollary 1 (i) for a more detailed aalysis), while the problem studied by d Aspremot ad Gérard-Varet (1979) correspods to the case of α = 0 i our model (although the AGV mechaism is differet from the ECSPR mechaism proposed i this paper). 4 I additio, whether a aget s vote is pivotal depeds o the strategies used by other agets ad their valuatios. 5

6 fact that a aget s IR costrait is satisfied does ot ecessarily mea that the aget will vote for the mechaism (likewise, if a aget chooses to vote for a mechaism, it does ot ecessarily mea that the aget s IR costrait is satisfied). Fially, this paper is closely related to Dal Bó (2007), who studies the problem of how a outside party ca ifluece a committee s votig decisio o a issue (e.g., a project), which is assumed to be udesirable for all members of the committee. Dal Bó (2007) fids that if the outside party ca reward the decisive votes differetly, the it ca ifluece the committee s votig decisio by promisig each committee member a sufficietly large reward i the case i which the committee member s vote is pivotal ad the member votes yes (i equilibrium, all committee members will vote yes, so there is o cost for makig such a promise). Such rewards i the pivotal evet are similar to the pivotal reward proposed i this paper. However, a major differece betwee Dal Bó s model ad ours is that i his, the reward to a committee member is directly based o the member s votig decisio. This practice is usually illegal, ad thus ot feasible i reality. I cotrast, the reward to a aget i our model is based o the aget s reported valuatio. The paper is orgaized as follows. Sectio 2 cosiders the basic model, ad Sectio 3 cosiders a extesio i which there is a commo value compoet i agets valuatios. Sectio 4 discusses the choice of votig rule i situatios i which the ECSPR mechaism does ot work properly. Sectio 5 cocludes. 2 The Basic Model 2.1 Setup Suppose that a pricipal wats to build a public good (e.g., a public library) i a commuity, i which there are agets. The public good is a discrete commodity that will either be provided or ot provided. The cost of providig the public good is C(). We assume that C() is (weakly) icreasig i. This assumptio also reflects the fact that to serve a 6

7 larger commuity, the pricipal eeds to costruct a larger library, which icurs a greater cost. Give that is fixed for most parts of the paper, we write C() as C whe there is o cofusio. We deote aget i s valuatio about the public good by v i, which is kow oly to aget i. v 1,, v are idepedet ad idetically distributed i accordace with a commo distributio fuctio F, which is commo kowledge amog all agets ad the pricipal. The support of F is deoted by [v, v]. We assume that the desity fuctio of F is cotiuous ad strictly positive o [v, v]. To make the problem otrivial, we assume that the average cost of the public good exceeds the lowest possible valuatio of a aget, i.e., C() > v. Fially, we assume that C() < v e, where v e = Ev i. This assumptio implies that the public good should be provided i a large ecoomy, because v i v e as by the law of large umbers. Sice we maily focus o a large ecoomy, 5 first-best efficiecy meas that the public good should always be provided (i Sectio 3, we will cosider a exteded setup i with there is a commo value compoet i agets valuatios ad thus both provisio ad o-provisio of the public good may be optimal). The pricipal ca propose a public good provisio mechaism. A mechaism is a fuctio pair {q, {t i } i=1}, where q : [v, v] [0, 1] is the probability that the public good is provided ad t i : [v, v] R is the paymet collected from aget i. 6 Note that q ad t i are fuctios of the reported valuatios of all agets. Aget i obtais a (ex post) utility of v i q t i if the mechaism {q, {t i } i=1} is implemeted. We do ot directly impose ay idividual ratioality costrait o the mechaism. Istead, we assume that the pricipal faces a costitutioal costrait, i the sese that i order for a mechaism to be implemeted, the mechaism must first be approved by agets uder a prespecified α-majority votig rule. To make the problem otrivial, we assume that α > 1 F ( C ) (if α < 1 F ( C ), the the pricipal ca simply propose the equal cost sharig mechaism, which will be approved uder the α-majority votig rule because 5 I the public good provisio eviromet, it is atural to assume that is large. 6 We allow both idirect mechaisms ad direct mechaisms. However, for the sake of expositioal clarity, we use direct mechaisms whe describig mechaisms i geeral (also, ote that accordig to the revelatio priciple, ay idirect mechaism ca be equivaletly represeted as a truthful direct mechaism). 7

8 1 F ( C ) fractio of agets will support the mechaism i a large ecoomy). We impose two additioal requiremets o the mechaism {q, {t i } i=1}. First, we require that the mechaism be aoymous. That is, q ad t i do ot deped o the idetities of agets. This assumptio is eeded to avoid the uiterestig case i which the pricipal simply asks a specific aget to pay all the costs of the public good, ad such a mechaism will be approved by agets uder ay α-majority rule as log as α ( 1)/. Secod, we require that the mechaism satisfy the ex post budget balace costrait. 7 That is, i=1 t i(ˆv 1,..., ˆv ) Cq(ˆv 1,..., ˆv ) = 0 for ay reported valuatios (ˆv 1,..., ˆv ). The game cosidered i this paper is called the votig game. Its timig is as follows. 1. The pricipal proposes a public good provisio mechaism {q, {t i } i=1}. 2. Each aget kows his ow valuatio about the public good, without kowig other agets valuatios. Each aget chooses whether to vote for the mechaism or vote agaist it, ad reports his valuatio. If at least α fractio of agets vote for the mechaism, the game proceeds to the ext step. Otherwise, the game eds immediately (i this case, the public good is ot provided ad there is o paymet collected from ay aget) The mechaism is implemeted based o the reported valuatios of all agets. We ca also modify the above game such that agets report their valuatios after approval of the mechaism, ad this modificatio will ot chage agets equilibrium behavior. This is because i the game i which a aget makes his votig decisio ad reports his valuatio at the same time, the aget kows that his reported valuatio will be used oly after the mechaism is approved. 7 Note that ex post budget balace (EXPBB) is a stroger requiremet tha ex ate budget balace (EXABB). However, it is also well kow i the literature that i Bayesia mechaism desig, if there are two or more agets ad agets types are idepedet, the EXABB ad EXPBB are equivalet i the sese that for ay mechaism that satisfies EXABB, we ca always costruct side paymets to esure that EXPBB is satisfied ad that ay aget obtais the same iterim expected payoff as before (see, e.g., Börgers ad Norma (2009)). 8 Grüer ad Koriyama (2012) cosider a model where if a public good provisio mechaism is vetoed, the the situatio of a majority votig with equal cost sharig will be eforced. I cotrast, our model assumes that if a mechaism is vetoed, there is o reegotiatio ad all agets obtai a payoff of zero. 8

9 Notice that our defiitio of the mechaism {q, {t i } i=1} requires that the mechaism oly depeds o the agets reported valuatios ad does ot deped o the agets votig behavior. This assumptio is atural, because votig is usually aoymous ad thus the pricipal does ot kow which aget votes yes ad which aget votes o (eve whe votig is ot aoymous, the pricipal is usually prohibited from discrimiatig betwee agets based o their votig behavior). Our mai research questio is, for ay give α-majority rule, ca the pricipal propose a mechaism that will be approved by agets with a probability close to oe ad (coditioal o approval of the mechaism) the public good will be provided with a probability close to oe? 2.2 Equilibrium otio As is typical i ay game that ivolves votig, our votig game may have some uiterestig equilibria. For example, assumig that α > 1/, for ay give mechaism proposed by the pricipal, a trivial equilibrium is that all agets vote agaist the mechaism (regardless of their valuatios), ad thus the public good will ever be provided. 9 Such a equilibrium is uiterestig, because it requires a very high level of coordiatio betwee agets. More precisely, i such a equilibrium, ay aget is actually idifferet betwee votig for the mechaism ad votig agaist it. To avoid such a equilibrium, the equilibrium cocept we will use is the strict Bayesia Nash equilibrium (or simply the strict BNE). The strict BNE is a BNE with the additioal requiremet that for almost ay type of ay aget, if the aget deviates from his equilibrium actio to ay other actio, the aget will be strictly worse off (rather tha weakly worse off, as i the stadard BNE). More precisely, if agets strategy profile (σ 1,..., σ ) is a strict BNE, the for ay aget i, there is some zero-measure set Ṽi such that for ay v i i [v, v]\ṽi, followig the equilibrium actio σ i (v i ) yields a strictly higher 9 Similarly, assumig that α 1, aother trivial equilibrium is that all agets vote for the mechaism (regardless of their valuatios), ad (uder some suitably desiged mechaism) the public good will always be provided. 9

10 payoff tha followig ay other actio. I the ext subsectio, we propose a mechaism, called the Equal Cost Sharig with Pivotal Reward mechaism (or simply the ECSPR mechaism), which approximately attais first-best efficiecy uder a very atural strict Bayesia Nash equilibrium. 2.3 ECSPR mechaism Let be the smallest iteger greater tha or equal to α. That is, if there are or more tha yes votes, the mechaism will be approved. The ECSPR mechaism, deoted by {q E, {t E i } i=1}, is a idirect mechaism. It asks each aget to report either a high value, deoted by h, or a low value, deoted by l. That is, lettig M = {h, l}, q E is a mappig from M to [0, 1], ad t E i is a mappig from M to R. Sice we require that the mechaism be aoymous, what matters for the value of q E is the umber of agets who report h ad the umber of agets who report l. I additio, what matters for the value of t E i is the umber of agets who report h, the umber of agets who report l, ad whether aget i reports h or l. Lettig x be the umber of agets whose reported values are h (which implies that the umber of agets whose reported values are l is x), Table 1 describes the value of q E for every possible x, ad describes the value of t E i for every possible x ad every possible reported value of aget i (otig that the first colum i Table 1 deotes the umber of h reports, rather tha the umber of yes votes). The parameters η 1 > 0, η 2 > 0, ad p (0, 1) are give. We call η 1 the pivotal reward. The ECSPR mechaism has two basic features. First, whe the umber of agets reportig high values is equal to or greater tha + 1, the public good is provided with probability oe ad the cost of the public good is shared equally by all agets. Secod, whe the umber of agets who report high values is exactly, the public good is provided with probability p. I additio, i this case, if the public good is provided, the ay aget who reports h makes a paymet less tha the average cost, while ay aget who reports l makes a paymet more tha the average cost. 10

11 Table 1: Descriptio of the ECSPR mechaism (coditioal o approval of the mechaism). x E t i (if aget i reports h) E t i (if aget i reports l ) E q 0 N/A C p C ( ) p ( 1) p C C C C C 1 1 N/A 1 Notice that for the ECSPR mechaism to work properly, we eed to assume that α ( 1)/. This is because if α > ( 1)/, the we have =, which implies that the budget caot be balaced for the case i which there are reported high values, because i this case, all agets will receive pivotal rewards, while o aget cotributes to the rewards. 2.4 Equilibrium aalysis For ay give η 1 > 0 ad p, defie z(η 1, p) as the uique z [0, 1] that satisfies z = 1 F ( C p z η 1 p 1 z Let t(η z +1 p).10 1, p) = C p p z(η 1,p) η 1 1 z(η 1,p) z(η 1,p) +1 p (so, 1 F (t(η 1, p)) = 10 I particular, the uiqueess of the solutio z i the equatio follows from the followig facts: (i) the left-had side of the equatio is strictly icreasig i z; (ii) the right-had side of the equatio is decreasig i z because 1 F ( C p z η1 p 1 z +1 p) = 1 F ( C p z η1 ) = 1 F ( C z p 1 z p +1 p p z η1 ) = p 1 z +1 p 1 F ( C pη 1 ) ad 1 p p +(1 p )z 0 (which is due to the fact that p ); (iii) whe z p z η1 approaches zero, the right-had side 1 F ( C p 1 z +1 p) = 1 F ( C pη 1 ) approaches z p +(1 p )z 1 F ( C η 1), which is greater tha the left-had side (i.e., 0); ad (iv) whe z approaches 1, the right-had side approaches 1 F ( C ), which is less tha the left-had side (i.e., 1). pη 1 1 p 11

12 z(η 1, p)). If we set t(η 1, p) = v, the η 1 = ( C 1 F (t(η 1, p)) = 1 F (v) = 1). We let η 1 (p) = ( C Lemma 1. Assume that α 1, p 1 p v) (usig the fact that z(η p 1, p) = 1 p v). p ad 0 < η 1 < η 1 (p). I the votig game i which the pricipal proposes the ECSPR mechaism, there is a strict Bayesia Nash equilibrium i which a aget votes for the mechaism ad reports h if the aget s valuatio is i [t(η 1, p), v], ad a aget votes agaist the mechaism ad reports l if the aget s valuatio is i [v, t(η 1, p)). Proof of Lemma 1: We use yes to deote a aget s actio of votig for the ECSPR mechaism ad o to deote a aget s actio of votig agaist the ECSPR mechaism. Sice agets choose their votig decisios ad report their values simultaeously, a aget has the followig four actios to choose from: (yes, h), (yes, l), (o, h), (o, l). Cosider aget i, whose valuatio is v i. Suppose that all agets other tha i follow the t(η 1, p)-threshold strategy. We will show that it is optimal for aget i to follow such a strategy. Give that all agets other tha i follow the t(η 1, p)-threshold strategy, there are four possibilities to cosider: (a) 2 or fewer others vote yes ad report h. I this case, there is othig that i ca do to get the mechaism approved, ad thus aget i is idifferet betwee the four actios. (b) Exactly 1 others vote yes ad report h. I this case, (o, h) ad (o, l) both give a payoff of zero, while (yes, h) ad (yes, l) give payoffs of p(v i ( C η 1)) ad 1 +1 η 2, respectively. (c) Exactly others vote yes ad report h. The (o, h) ad (yes, h) give payoff v i C, while (o, l) ad (yes, l) give payoff p(v i ( C + η 1 )). (d) If + 1 or more vote yes ad report h, the aoucemet is irrelevat. It is obvious that (yes, l) is strictly worse tha (o, l) i expectatio sice the two actios give the same payoff except whe 1 others vote yes ad report h (i which case (yes, l) is strictly worse tha (o, l) as 1 +1 η 2 < 0). 12

13 (yes, h) ad (o, h) oly chages the outcome whe 1 others vote yes ad report h, so (yes, h) is weakly worse tha (o, h) if p(v i ( C η 1)) 0 v i C + η 1 0. (1) Also, (o, l) is weakly worse tha (o, h) if sice p(v i ( C + η 1)) v i C. (2) Hece, if (o, h) is a best reply, it follows from (1) that v i C η 1. But the, if p <, v i ( C + η 1) < v i C < 0 (where the last iequality follows from the fact that v i C η 1 < C ), it follows that p(v i ( C + η 1)) > (v i ( C + η 1)) = (v i C ) η 1 =(v i C ) (v i C + η 1) v i C (3) so (2) caot hold. We thus have show that (o, h) is suboptimal for ay v i [v, v] if p <. If p =, it ca be verified that (i) if v i < C η 1, the the last iequality i (3) will be strict (although the first iequality i (3) will become a equality) ad thus (2) caot hold (which implies that (o, l) is strictly better tha (o, h)); (ii) if v i > C η 1, the (1) caot hold (which implies that (yes, h) is strictly better tha (o, h)); ad (iii) if v i = C η 1, the (1) ad (2) will be equalities (which implies that i is idifferet betwee (o, h), (yes, h) ad (o, l), all of which are actually optimal actios for i). So (o, h) is suboptimal for almost ay type of aget i (i particular, for ay v i C η 1). Hece, (yes, h) ad (o, l) are the oly possibilities whe all others follow the t(η 1, p)- 13

14 threshold strategy (except whe p = ad v i = C η 1, i which case i is idifferet betwee (o, h), (yes, h) ad (o, l)). Let z(η 1, p) = P rob(v j t(η 1, p)) be the probability that aget j i chooses (yes, h) ad ρ(k) = ( 1)! k!( 1 k)! zk (1 z) 1 k (4) be the probability that k others select (yes, h). Now, (yes, h) ad (o, l) give differet payoffs i two cases: (a) Whe 1 others select (yes, h), i which case (yes, h) gives payoff p(v i ( C η 1)) ad (o, l) gives 0. (b) Whe others select (yes, h), i which case (yes, h) gives payoff v i C ad (o, l) gives p(v i ( C + Sice η 1 )). Hece the differece is = ρ( 1)p(v i C + η 1) + ρ( )[v i C p(v i ( C + ρ( ( 1)! 1) = ( 1)!( )! z(η 1, p) 1 (1 z(η 1, p)) η 1))]. ( 1)! = ( 1)!( )! z(η 1, p) 1 (1 z(η 1, p)) = 1 z(η 1, p) ( 1)! z(η 1, p)!( 1 )! z(η 1, p) (1 z(η 1, p)) 1 = 1 z(η 1, p) ρ( ) z(η 1, p) it follows that > (resp. <) 0 if ad oly if 1 z(η 1, p) z(η 1, p) p(v i C + η 1) + v i C p(v i ( C + η 1)) > (resp. <) 0, 11 i.e., (v i C )(1 + p 1 z(η 1, p) p) > (resp. <) p z(η 1, p) z(η 1, p) η 1, 11 The if ad oly if argumet holds if ρ( ) > 0, which is true because it ca be show that 0 < z(η 1, p) < 1 (uder the assumptio that 0 < η 1 < η 1 (p)). I particular, if η 1 = 0, the t(η 1, p) = C, ad if η 1 = η 1 (p), the t(η 1, p) = v. Usig the fact that t(η 1, p) is strictly decreasig i η 1 (see Lemma 3), we have t(η 1, p) (v, C ) whe 0 < η 1 < η 1 (p), which implies that z(η 1, p) (0, 1). 14

15 i.e., v i > (resp. <) C p z(η 1,p) η 1 p 1 z(η 1,p) z(η 1,p) + 1 p. We have thus show that it is optimal for aget i to adopt the t(η 1, p)-threshold strategy, give that all other agets use the t(η 1, p)-threshold strategy. Thus, all agets usig the t(η 1, p)-threshold strategy is a Bayesia Nash equilibrium. Furthermore, from the aalysis above, it is also a strict Bayesia Nash equilibrium. Notice that η 2 does ot appear i the threshold t(η 1, p). This is because η 2 is used by the pricipal as a puishmet for icosistet strategies (i particular, it ca prevet a aget from deviatig to (yes, l) i the t(η 1, p)-threshold strategy equilibrium). 12 As log as η 2 is positive, this puishmet will be effective, ad the exact size of η 2 will ot affect the agets equilibrium behavior. The t(η 1, p)-threshold strategy equilibrium may ot be the uique strict Bayesia Nash equilibrium of the votig game (we will discuss other strict BNE i Sectio 2.7). However, if we restrict the equilibria of the game to the set of symmetric cosistet-strategy equilibria, the the t(η 1, p)-threshold strategy is ideed uique. A equilibrium is symmetric if all agets use the same strategy. A aget s strategy is cosistet if the aget votes for the mechaism if ad oly if the aget reports h (or equivaletly, the aget votes agaist the mechaism if ad oly if the aget reports l). A cosistet-strategy equilibrium does ot restrict a aget s strategy space to the set of cosistet strategies; it istead requires that whe all other agets use cosistet strategies, a aget s best respose is a cosistet strategy. Lemma 2. Assume that α 1, p ad 0 < η 1 < η 1 (p). I the votig game i which the pricipal proposes the ECSPR mechaism, the t(η 1, p)-threshold strategy equilibrium is the uique symmetric cosistet-strategy strict Bayesia Nash equilibrium. I the remaider of the paper (except Sectio 2.7), we will focus o the t(η 1, p)-threshold 12 O the other had, the feature that the public good is provided with a probability less tha oe i the case where the umber of h reports is ca prevet a aget from deviatig to (o, h) i the t(η 1, p)- threshold strategy equilibrium. 15

16 strategy equilibrium. I such a cosistet-strategy equilibrium, there is a clear lik betwee agets votig behavior ad their reported values. This makes the equilibrium easy to aalyze. We ext aalyze the relatioship betwee t(η 1, p) ad η 1 ad p. The fact that the pivotal reward η 1 > 0 does ot automatically mea that i equilibrium, all agets will vote yes for the mechaism. This is because whe a aget s vote is pivotal, votig yes (ad reportig h) will cause the public good to be provided with some probability, while votig o will result i o provisio of the public good. So, if a aget has a very low valuatio, the he may choose to vote o. However, we ca show that as η 1 icreases, the fractio of agets who choose to vote o will decrease. This result is illustrated i the ext lemma. I additio, the lemma shows that as p icreases, the threshold t(η 1, p) decreases. Lemma 3. Assume that p. The t(η 1, p) is strictly decreasig i η 1 ad p. Recall that if we set t(η 1, p) to be v, the we have v = C i.e., η 1 = ( C 1 p v) p p p 1 F (v) η 1 F (v) 1 F (v) +1 p = C = η 1 (p). This implies that for ay give p (0, pη 1, 1 p ], we ca set η 1 to be sufficietly close to η 1 (p) so that t(η 1, p) is sufficietly close to v ad thus z(η 1, p) is sufficietly close to oe. Therefore, the probability that a aget will vote for the mechaism ad report high value is sufficietly close to oe, ad the probability that the public good will be provided is sufficietly close to oe. 13 We thus have the followig result. Propositio 1. Assume that α 1 ad p. Suppose that the pricipal proposes the ECSPR mechaism ad that agets use the t(η 1, p)-threshold strategy i the votig game. The for ay ζ > 0, there exists a pivotal reward η 1 > 0 such that the probability of provisio of the public good i equilibrium is greater tha 1 ζ. Accordig to Lemma 3, t(η 1, p) is strictly decreasig i p. So, if we set p to its maximum possible value,, the the ECSPR mechaism will be supported by the maximum fractio 13 Notice that we caot let η 1 = η 1 (p) (or η 1 > η 1 (p)), because if this is the case, the the t(η 1, p)- threshold strategy equilibrium is such that all agets vote for the mechaism. But such a equilibrium is ot a strict Bayesia Nash equilibrium, because ay aget is idifferet betwee votig for the mechaism ad votig agaist it (as log as α ( 1)/). 16

17 of agets. I additio, if we set p =, the t(η 1, p) = C η 1. I the remaider of the paper, we will set p = t(η 1, p ) as t(η 1 ) ad z(η 1, p ) as z(η 1 ). := p. With some slight abuse of otatio, we will simply write 2.5 Large ecoomy This subsectio cosiders a large ecoomy. For simplicity, throughout this subsectio, we assume that C = c for some costat c for ay. To get the ECSPR mechaism approved, we do ot eed to have almost all agets vote for the mechaism; it is sufficiet that more tha α fractio of agets choose to vote for the mechaism. To accomplish this i a large ecoomy, the pivotal reward η 1 should be strictly greater tha η m 1, where η m 1 is such that 1 F (t(η m 1 )) = α. That is, η 1 should be strictly greater tha c F 1 (1 α) (because t(η 1 ) = C η 1 = c η 1 ). We call the term c F 1 (1 α) the lower boud of pivotal reward. O the other had, the term C v = c v measures the amout of the reward the pricipal eeds to give a pivotal aget to iduce the lowest valuatio aget to vote for the mechaism. We call c v the upper boud of pivotal reward. If the pivotal reward is strictly betwee the lower boud ad the upper boud of pivotal reward, the there is a positive probability that the ECSPR mechaism will be rejected by agets whe the ecoomy is small. However, as the ecoomy becomes large, the probability that the mechaism will be approved goes to oe. Ituitively, this is due to the law of large umbers, which esures that the approval rate of the mechaism coverges to z(η 1 ), a value that is greater tha α. We thus obtai the followig result. Theorem 1. Assume that α 1 ad C = c for ay. Suppose that the mechaism proposed by the pricipal is the ECSPR mechaism with p = p, ad that agets use the t(η 1 )-threshold strategy i the votig game. If η 1 (c F 1 (1 α), c v), the the probability of provisio of the public good i equilibrium goes to oe as goes to ifiity. We call ay pivotal reward that is strictly betwee the lower boud ad the upper boud of pivotal reward the effective pivotal reward. A careful aalysis of the effective pivotal 17

18 reward will lead to several iterestig observatios. First, the rage of the effective pivotal reward is idepedet of, because the lower boud ad the upper boud of pivotal reward are idepedet of. This also implies that the puishmet for a aget who votes o i the evet that the mechaism is just barely approved is also idepedet of (more precisely, the rage of the puishmet per aget, η 1, is ( α (c F 1 α (1 α)), (c v))). This 1 α 1 α property is ice, because it meas that the puishmet is bouded eve i a large ecoomy, ad this makes our mechaism a practical oe. Secod, as α chages from 1 F (c) to 1, 14 the rage of the effective pivotal reward becomes smaller ad smaller. This reflects the fact that as α icreases, it becomes harder to desig a mechaism to achieve first-best efficiecy. We ow compare our result with that of Mailath ad Postlewaite (1990). Mailath ad Postlewaite (1990) show that if the idividual ratioality costrait is required for all agets, the as the ecoomy becomes large, the probability of provisio of the public good goes to zero for ay sequece of mechaisms. This implies that i our votig game, if the votig rule is a uaimity rule, the the probability of provisio of the public good goes to zero as the ecoomy becomes large (see the proof of Corollary 1 (i) for a detailed aalysis). O the other had, our result (Theorem 1) shows that as log as the votig rule is ot the uaimity rule (oticig that ( 1)/ approaches 1 as becomes large), the first-best efficiecy ca be achieved approximately. We use p(m) to deote the maximum probability (uder all BNEs) that the public good will be provided i the votig game whe the mechaism proposed by the pricipal is M. We thus have: Corollary 1. (i) If α = 1, the for ay sequece of mechaisms {M } =1 where M deotes a mechaism for the -aget ecoomy, we have that p(m ) 0 as. (ii) If α < 1, the there exists a sequece of mechaisms {M } =1 such that p(m ) 1 as. 14 If α < 1 F (c), the the pricipal ca simply propose the equal cost sharig mechaism, which will be approved by agets uder the α-majority rule i a large ecoomy. 18

19 It is worth emphasizig the differece ad the lik betwee settig α = 1 ad the requiremet of (IR) (as i Mailath ad Postlewaite (1990)). I particular, settig α = 1 requires that i order for a mechaism to be approved, ay aget must obtai a oegative payoff by votig for the mechaism, coditioal o the evet that his vote is pivotal (i.e., coditioal o the evet that all other agets vote for the mechaism), while (IR) requires that ay aget obtai a o-egative payoff from implemetatio of the mechaism. So, although both requiremets require a aget s payoff to be o-egative, the payoff i the former case is a coditioal expected payoff, while the payoff i the latter case is a ucoditioal expected payoff. Despite this differece, there is a very close lik betwee settig α = 1 ad (IR). I particular, i the proof of Corollary 1, we show that for ay budget-balaced mechaism M 1 proposed by the pricipal ad ay BNE of our votig game (with α = 1), we ca always fid a slightly differet mechaism M 2 such that M 2 satisfies (IC), (BB), ad (IR). 15 I additio, M 2 ad M 1 are equivalet i the sese that they yield the same probability of provisio i equilibrium. So, Corollary 1 (i) immediately follows from the impossibility result of Mailath ad Postlewaite (1990). 2.6 A example Example 1. Assume that the agets valuatios are i.i.d. with a uiform distributio o [0, 1]. Assume that C = c = 0.4, ad α = 2. The the lower boud of pivotal reward per 3 aget is c F 1 (1 α) = 0.067, the upper boud of pivotal reward per aget is c v = 0.4, ad the threshold valuatio t(η 1 ) = 0.4 η 1. Figure 1 depicts the probability of provisio i the votig game as a fuctio of the pivotal reward for = 100, 500, 1000, The figure makes it clear that as the ecoomy becomes large, the probability of provisio goes to oe whe the pivotal reward is strictly betwee the lower boud ad upper boud of pivotal reward. I additio, for ay give 15 I particular, M 2 is costructed as follows. If the valuatio profile v = (v 1, v 2,..., v ) is such that some aget chooses to vote agaist the mechaism M 1 i equilibrium, the M 2 is such that the public good is ot provided ad o paymet is collected from ay aget. Otherwise (i.e., if v is such that all agets choose to vote for M 1 ), the M 2 yields the same probability of provisio ad paymet scheme as M 1. 19

20 size of the ecoomy, the figure shows that as the pivotal reward icreases, the probability of provisio icreases. So, there is a mootoic relatioship betwee the pivotal reward ad the probability of provisio of the public good. =10000 =100 =500 =1000 Figure 1: Probability of provisio vs pivotal reward i Example 1. Fially, we wat to emphasize that the fact that a mechaism is supported by a aget does ot ecessarily mea that the aget obtais a payoff greater tha zero from implemetatio of the mechaism. I Example 1, the average cost of the public good is 0.4, ad accordig to our mechaism, the cost is equally distributed amog agets whe the public good is provided (except for the case where there are some pivotal agets, which occurs with probability zero i a large ecoomy). This implies that aroud 40% of the agets are hurt if the mechaism is implemeted. I other words, the idividual ratioality costraits of aroud 40% of the agets are violated uder the mechaism, although more tha 2/3 of agets will choose to vote for the mechaism (provided that the pivotal reward is i betwee the lower boud ad the upper boud of pivotal reward). 2.7 Other equilibria ad equilibrium selectio The previous sectios focus o the t(η 1 )-threshold strategy BNE. I this subsectio, we will discuss the properties of other strict BNE ad explai why the t(η 1 )-threshold strategy 20

21 BNE is a atural choice of equilibrium. Sice t(η 1 )-threshold strategy BNE is the uique symmetric cosistet-strategy strict BNE (Lemma 2), ay symmetric strict BNE that is differet from the t(η 1 )-threshold strategy BNE must ivolve the use of icosistet strategies. We say that a strict BNE is a icosistet-strategy strict BNE if it is ot a cosistetstrategy strict BNE. We have the followig result. Lemma 4. Assume that α 1 ad p = p. I ay symmetric icosistet-strategy strict BNE of the votig game i which the pricipal proposes the ECSPR mechaism, we must have: (i) aget i votes agaist the mechaism ad reports l if v i C η 1; (ii) aget i votes for the mechaism ad reports h if v i C ; (iii) there exists a set ˆV ( C η 1, C ) (where ˆV has a o-zero measure) such that if v i ˆV, the aget i votes agaist the mechaism ad reports h, ad if v i ( C η 1, C )\ ˆV, the aget i votes for the mechaism ad reports h. Note that Lemma 4 shows that i ay symmetric icosistet-strategy strict BNE, a aget will ever use the icosistet actio (yes, l). The other icosistet actio (o, h) may be used oly whe C η 1 < v i < C. A implicatio of Lemma 4 is that whe agets use a symmetric icosistet-strategy strict BNE, the ECSPR mechaism will be approved with a lower probability ad thus the public good will be provided with a lower probability tha the case i which agets use the t(η 1 )-threshold strategy BNE (otig that t(η 1 ) = C η 1). Aother implicatio of Lemma 4 is that i ay symmetric icosistet-strategy strict BNE, the strategy used by a aget must have at least two thresholds, because as v i varies from v to v, three equilibrium actios (i.e., (o, l), (o, h), (yes, h)) occur. Thus, the t(η 1 )-threshold strategy BNE which ivolves oly oe threshold i its equilibrium strategy is the most simple/saliet BNE amog all symmetric strict BNE. Fially, ote that the t(η 1 )-threshold strategy BNE achieves efficiecy i a large ecoomy, which implies that the total expected surplus of all agets reaches its maximum. This i tur 21

22 implies that the t(η 1 )-threshold strategy BNE (weakly) Pareto domiates all other symmetric icosistet-strategy strict BNE i a large ecoomy i the ex ate sese, i.e., a aget s ex ate payoff uder the t(η 1 )-threshold strategy BNE is (weakly) higher tha the aget s ex ate payoff uder ay symmetric icosistet-strategy strict BNE (otig that comparig agets ex ate payoffs across equilibira is relevat if agets decide which strategies to use before they kow their (private) valuatios). 3 Extesio: Commo Value Compoet 3.1 Setup I the previous sectio, we assume that the pricipal kows the distributio of agets valuatios. This implies that i a large ecoomy, eve without kowig agets private valuatios, the pricipal kows whether the public good should be provided or ot. This makes the votig stage of our model less iterestig, sice the votig stage oly serves as a device for gettig the public good provided (if the public good should be provided), rather tha a device that the pricipal uses to collect useful valuatio iformatio from agets to help the pricipal to make the decisio about provisio of the public good. I this sectio, we will relax the assumptio that the pricipal kows the exact distributio of agets valuatios. Followig the commo value compoet model of Ledyard ad Palfrey (2002), we assume that the agets valuatios have a commo value compoet. More specifically, a aget s (adjusted) valuatio is r i = v i + m, where v i is aget i s private valuatio ad m is the commo value compoet. We assume that v 1,..., v are i.i.d. with a cotiuous distributio fuctio F, ad m takes a value i the rage of [m, m]. All agets kow the exact value of m, but the pricipal oly kows the distributio of m. This implies that a aget kows the exact distributio of ay other aget s valuatio, but the pricipal does ot kow the exact distributio of the valuatio of ay aget. We assume throughout this sectio that the total cost of the public good is C() = c for ay, where c is a 22

23 costat. 3.2 Aalysis Defie m = c v e, where v e is the expected value of v i. Obviously, if m > m, the v e +m > c ad thus the public good should be provided i a large ecoomy. If m < m, the v e + m < c ad thus the public good should ot be provided i a large ecoomy. We assume that m < m < m, which implies that both provisio ad o-provisio of the public good may be optimal, depedig o the realized value of m. Suppose that the priciple proposes the ECSPR mechaism with p = p =. Note that from the poit of view of the agets playig the game iduced by the mechaism, it is the same as with idepedet private values (except that there is a commo shift of the support of agets valuatios). Let t(η 1 m) be the threshold t(η 1 ) that we defie i Sectio 2.4 whe the realizatio of the commo value compoet is m, ad z(η 1 m) be the probability that a aget s valuatio is greater tha t(η 1 m). Let η 1 be such that z(η 1 m ) = α. That is, η 1 is the pivotal reward that iduces just α fractio of agets to vote for the mechaism i a large ecoomy i the t(η 1 m)-threshold strategy equilibrium whe m = m. We have the followig result regardig the relatioship betwee z(η 1 m) ad m. Lemma 5. z(η 1 m) is icreasig i m. I additio, z(η 1 m) is strictly icreasig i m at ay m where z(η 1 m) (0, 1). Note that z(η 1 m ) = α (0, 1). Accordig to Lemma 5, if m > m, the z(η 1 m) > α, ad if m < m, the z(η 1 m) < α. So, if the pricipal proposes the ECSPR mechaism with η 1 = η 1 ad agets use the t(η 1 m)-threshold strategy equilibrium, the (i) if m > m, the mechaism will be approved with probability oe (ad the public good will be provided with probability oe) i a large ecoomy; ad (ii) if m < m, the mechaism will ot be approved (ad the public good will ot be provided) i a large ecoomy. That is, the first-best efficiet level of provisio of the public good ca always be achieved i a large ecoomy. This result is summarized i the followig theorem. 23

24 Theorem 2. Assume that α 1. Suppose that there is a commo value compoet m i a aget s valuatio. Suppose that the mechaism proposed by the pricipal is the ECSPR mechaism with p = p ad η 1 = η1, ad that agets use the t(η1 m)-threshold strategy i the votig game. We have: (i) if m > m, the the probability of provisio of the public good i equilibrium goes to oe as goes to ifiity; (ii) if m < m, the the probability of provisio of the public good i equilibrium goes to zero as goes to ifiity. 4 Choice of Votig Rule A implicatio of the results i the previous sectios is that the choice of votig rule is irrelevat with respect to the efficiecy of provisio of the public good, as log as the votig rule is ot the uaimity rule. This is because as log as the proposed mechaism is the ECSPR mechaism (with a properly set η 1 ), the it ca rectify ay predetermied (but usually too striget or too weak) votig rule, so that first-best efficiecy of provisio of the public good is always achieved. I this sectio, we cosider the situatios or costraits that may prevet the proper fuctioig of the ECSPR mechaism. I such situatios, the iitial choice of the votig rule becomes importat. I this sectio, the votig rule is o loger exogeously give; istead, there is a costitutioal desig stage prior to the votig game. Our questio is: Which votig rule should the pricipal choose at the costitutioal desig stage? I particular, whe will the simple-majority rule be optimal, ad whe will a super-majority rule be optimal? The aalysis i this sectio is based o the model developed i Sectio 3. I additio, we assume that 1 2 α 1 with α < 1 2 i the case of α = 1). (we require α 1 2 i reality, ad we require α 1 because we seldom observe a votig rule because the efficiecy result will ever hold 24

25 4.1 No discrimiatory mechaism We first cosider the situatio i which the pricipal caot use ay discrimiatory mechaism, i.e., the pricipal caot charge differet agets with differet paymets. This situatio is likely to arise if the pricipal has fairess cocers, or it is required by law that the pricipal caot use ay discrimiatory mechaism, or it is simply too costly to implemet a discrimiatory mechaism. I such situatios, the oly mechaism the pricipal ca use is the Equal Cost Sharig mechaism (or simply the ECS mechaism). The ECS mechaism, {q ECS, {t ECS i } i=1}, is such that q ECS (v) = 1 ad t ECS i (v) = C() = c for ay v [v, v]. If the pricipal proposes the ECS mechaism, the a aget will vote for the mechaism if ad oly if the aget s valuatio is above the average cost. 16 This is because (i) the ECS mechaism, oce approved, will provide the public good with probability oe ad require each aget to pay the average cost of the public good, regardless of agets reported valuatios, ad (ii) for ay aget, votig for the ECS mechaism will iduce it to be approved with a higher probability (ad thus the public good will be provided with a higher probability) tha votig agaist it. Notice that as the commo value compoet m icreases, the approval rate uder the ECS mechaism also icreases, because the approval rate uder the ECS mechaism P (r i c) = 1 F m (c) = 1 F (c m) is a icreasig fuctio of m (where F m is the cdf of r i ad F is the cdf of v i ). If α is too large (so that α > 1 F (c m )), the the public good will be uder-provided (i particular, for m that is greater tha but close to m, the public good should be provided but it will ot be provided, as the ECS mechaism caot be approved). If α is too small (so that α < 1 F (c m )), the the public good will be over-provided (i particular, for m that is less tha but close to m, the public good should 16 Note that lettig agets vote o the ECS mechaism is equivalet to the followig two-step procedure: First, let the agets vote o whether the public good should be provided, ad secod, if the votig result is such that the public good is provided (i.e., there are eough yes votes i the first step), the the cost will be equally distributed amog all agets. This two-step procedure ad its welfare property are also studied by Ledyard ad Palfrey (1994) ad Ledyard ad Palfrey (2002), i which it is called the referedum with equal cost shares. 25

26 ot be provided but it will be provided, as the ECS mechaism will be approved). Oly if α = 1 F (c m ) := α, will the first-best provisio level of the public good be achieved for ay possible realizatio of m. So, the α -majority votig rule is the optimal votig rule. Notice that α = 1 F (c m ) = 1 F (v e ). If the distributio F is such that its mea equals to its media, the α = 1 F (v e ) = 1 2. If the distributio F is such that the mea is greater tha the media, the α < 1 2. Sice we require α 1, the costraied-optimal votig rule will be 2 α = 1. Notice that most 2 widely used distributios are such that the mea is equal to the media or greater tha the media. Examples of the former case iclude uiform distributio, ormal distributio, t distributio (whe the mea is well defied), etc. Examples of the latter case iclude logormal distributio, Chi-squared distributio, expoetial distributio, etc. 17 I summary, we have the followig result, which provides a ratioale for the popularity of the simplemajority rule i practice. Propositio 2. Suppose that the pricipal ca oly use the ECS mechaism. Suppose that there is a commo value compoet i a aget s valuatio, ad that a aget s private valuatio follows a distributio whose mea is equal to or greater tha the media. The the simple-majority rule is optimal (i the sese that the maximum efficiecy of provisio of the public good is achieved). If F is such that the mea of v i is less tha the media of v i, the α = 1 F (v e ) > 1/2, i.e., a super-majority rule is optimal. Such a distributio usually has a log left tail, which implies that most agets have high valuatios ad very few agets have low valuatios. I reality, such distributios are quite rare. Propositio 3. Suppose that the pricipal ca oly use the ECS mechaism. Suppose that there is a commo value compoet i a aget s valuatio, ad that a aget s private 17 The distributios i the former case are usually symmetric, which implies that the fractio of agets who have high valuatios ad the fractio of agets who have low valuatios are equal. The distributios i the latter case usually have a log right tail, which implies that most agets have relatively low valuatios, but a few agets may have very high valuatios. 26