Uniform Price Mechanisms for Threshold Public Goods Provision with Private Value Information: Theory and Experiment

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1 Unform Pre Mehansms for Threshold Publ Goods Provson wth Prvate Value Informaton: Theory and Experment Zh L *, Chrstopher Anderson, and Stephen Swallow Abstrat Ths paper ompares two novel unform pre mehansms for provson pont publ goods to standard provson pont (PPM) and proportonal rebate (PR) mehansms wthn a Bayesan game wth prvate value nformaton. The unform pre auton mehansm (UPA) ollets an endogenously determned unform pre from everyone offerng at least that pre, whle the unform pre ap mehansm (UPC) ollets the unform pre from everyone offerng at least that pre, plus the full offer of everyone offerng less. By rebatng full amounts n exess of the pre, the unform pre mehansms reate regons where the expeted nrease n payment assoated wth a hgher offer s zero. We show that the unform pre mehansms support Bayesan Nash equlbra (BNE) wth hgher ontrbutons than BNE of PPM or PR, potentally nreasng effeny. We use laboratory experments to test whether these more effent BNE obtan, leadng to hgher ontrbutons or more frequent provson. Our mehansms outperform PR and PPM wth prvate values: UPC generates hgher aggregate ontrbutons and provson rates than PR and PPM; UPA attrats muh hgher ontrbutons, although t provdes less frequently. Ths rankng emerges beause hgh offers are more ommon (espeally among hghvalue people) n the unform pre mehansms, where t s low ost to venture hgh offers to potentally meet other hgh offers to support provson. Keywords: Unform pre auton, Unform pre ap, Proportonal rebate, Provson pont mehansm * Department of Eonoms, Unversty of Washngton Shool of Aquat and Fshery Senes, Unversty of Washngton Center for Envronmental Senes and Engneerng and Department of Agrultural and Resoure Eonoms, Unversty of Connetut.

2 . Introduton The provson pont mehansm (PPM) for publ goods provson s one where the good an be provded only f a threshold level of fundng ontrbutons s met. After Bagnol and Lpman (989) showed that the equlbrum outome s always effent n undomnated perfet equlbra wth omplete nformaton, t has been systematally studed, both theoretally and expermentally. PPM s popularty n the eonoms lterature an be attrbuted to the fat that many publ goods have an nherent threshold or dsrete nature for provson, suh as parks, publ rado broadastng, and envronmental onservaton projets, where a mnmum amount of fundng s needed to provde one unt. The PPM requres only a slght modfaton addton of the provson pont to the ommon open-ended donaton soltaton used by hartes, and hene many fundrasers use the PPM due to ts smplty and support for provson n equlbrum. 3 One pratal problem wth the PPM s how to deal wth the ontrbuton n exess of the provson ost, attrbutable to a ombnaton of nomplete value nformaton and mperfet oordnaton wthn the ontnuum of effent Nash equlbra. Wthout rebate, ontrbutors may vew the extra money as wasted, potentally reatng a dsnentve for ontrbuton n the frst plae, and espeally dsouragng hgh offers. However, rebatng extra money provdes an opportunty to shft the off-path nentves of the PPM, and perhaps attrat addtonal ontrbutons: ths has led to an addtonal lterature on whether and how dfferent methods for rebatng ontrbutons n exess of the provson ost affet ontrbutons. The most popular rebate rule s the proportonal rebate (PR), whh rebates the extra ontrbuton n proporton to the rato of an ndvdual s ontrbuton to the total ontrbuton. 4 The results n the expermental lterature are mxed: Marks and Croson (998) fnd no sgnfant dfferene between PPM and PR under omplete nformaton, whle Galmard and Palfrey (005) fnd PR (alled PCS n ther paper) generates sgnfantly hgher ontrbutons than PPM when value nformaton s prvate. Only a few of the possble fators affetng these mxed results have been explored (Rondeau et al.,999; Rondeau et al, 005; Spener et al., 009). Bagnol and Lpman (989) study PPM under omplete nformaton; Ntzan and Romano (990), MBrde (006), and Barber and Malueg (00a) dsuss threshold unertanty; Alboth et al. (00), Menezes et al. (00), Laussel and Palfrey (003), and Barber and Malueg (008, 00b) dsuss PPM wth prvate value nformaton. See Chen (008) for a reent revew of related expermental studes; for earler revews see Davs and Holt (993) and Ledyard (995). 3 See real world examples n Bagnol and Mkee (99) and Marks and Croson (998), or 4 Marks and Croson (998) and Galmard and Palfrey (005) provde omprehensve referenes on PR.

3 To provde a oherent framework for understandng how rebate rules affet ontrbutons and also to mprove upon PPM and PR, L et al. (04) ntrodue two novel unform pre mehansms, the unform pre auton mehansm (UPA) and the unform pre ap mehansm (UPC). In UPA, everyone who pays, pays the same pre: f there exsts a pre suh that the number of ontrbutons at or above that pre multpled by the pre equals the provson pont, then the good s provded, wth only those offerng at or above the unform pre payng the unform pre; the lowest suh pre wll be hosen f more than one unform pre s possble. UPC addresses an neffeny nherent n UPA, that ontrbutons an exeed the provson ost, but stll no unform pre meetng the provson rule exsts. In UPC, no one pays more than the unform pre: f the provson pont s exeeded, the lowest pre ap wll be alulated so whoever ontrbutes above the ap pays only the ap, and those ontrbutng less than the ap pay ther full offer, suh that the fnal olleted payments equal the provson pont. The objetve of the unform pre mehansms s to desgn a rebate rule that ndues hgher overall ontrbutons by allevatng partpants onern that ontrbutng more to support provson may lead to losng more of the (over)ontrbuton n the event of provson. Ths extent of onern s measured by the margnal penalty, the ost of ontrbutng an addtonal dollar ondtoned on provson (Marks and Croson, 998). In a true PPM, all overontrbuton s wasted so the margnal penalty s -, and n PR the margnal penalty s between - and 0. The two unform pre mehansms, on the other hand, have a wde range of aggregate ontrbutons where the margnal penalty s zero, and therefore they are expeted to ndue hgher ontrbutons. Through lab experments, L et al. (04) fnd that the unform pre mehansms do outperform PPM and PR under omplete nformaton. The nsght s that the lower margnal penalty faltates equlbrum seleton by makng t safer to tender hgher offers, and therefore the unform pre mehansms lead to hgher ontrbutons and more frequent provson. A key feature mssng n the applaton of these mehansms n a omplete nformaton game s that the rebate s rrelevant n equlbrum, sne exess ontrbutons never our n equlbrum, 5 so equlbrum theory s slent about how dfferent rebate rules may affet the ontrbuton behavor. In a Bayesan framework, however, exess ontrbutons an our when 5 Bagnol and Lpman (989) show n undomnated perfet equlbra, the provson pont s exatly met n PPM; and L et al. (04) verfy that PR and UPC have the same set of undomnated perfet equlbra as PPM.

4 a value profle wth hgher-than-expeted ndued values s realzed, and therefore the role of rebates an be explored by omparng the equlbra of eah mehansm. Ths paper frst eludates how the alternatve rebate rules affet the expeted payoff funton by examnng the expeted margnal penalty assoated wth hgher offers n eah mehansm. We then haraterze the Bayesan Nash equlbra (BNE) of the two unform pre mehansms, and ompare the equlbrum sets wth those of PPM and PR n an nomplete nformaton settng. Wth an almost always zero-expeted margnal penalty, UPA has a truthtellng BNE n a -player game, and dependng on parameters, may support BNE where the expeted group ontrbuton s lose to (more than 90% n our examples) the total expeted ndued value n a game wth 3 or more players. UPC has a BNE haraterzaton smlar to PPM and PR as shown n Galmard and Palfrey (005). By a numeral example, we fnd rebates ndue some BNE wth hgher ontrbutons from hgh-value types: PR and UPC generate ontrbutons omprsng equlbra more effent than PPM, and UPC ndues hgher ontrbutons from hgh-value people than PPM and PR. These rebates work by redung the expeted payment ost for hgh-value people more than low-value people, sne hgh-value people are more lkely to fae a value profle wth hgher-than-expeted ndued values and hene wll experene overontrbuton wth a hgher probablty and pay more of the exess ontrbuton n the absene of a rebate. Further, UPC has a margnal penalty struture that redues the overontrbuton ost more effetvely than PR by only protetng hgh ontrbutors: all exess ontrbutons n UPC are returned to those ontrbutng hgher than the pre-ap who are generally hgh-value people (as shown n Proposton below). These theoretal predtons are supported by our expermental results: UPC generates hgher aggregate ontrbutons and provson rates than PR and PPM; UPA attrats muh hgher ontrbutons, although t provdes less frequently. The rest of the paper s organzed as follows. Seton defnes presely the four mehansms to be ompared, and analyzes the effet of ther respetve margnal penalty strutures on ther expeted payments. Seton 3 haraterzes the BNE sets of UPA and UPC, demonstrates dfferenes n the mehansms BNE sets wth numeral examples, and explans the underlyng role of margnal penalty n dfferentatng the sets of equlbra. Seton 4 desrbes the expermental desgn and proedures. Seton 5 dsusses the observed group and ndvdual ontrbutons. Seton 6 syntheszes these results. 3

5 . The Mehansms and Ther Margnal-Penalty-Struture Effet on the Expeted Payment Suppose N agents eah have endowment I. Eah smultaneously hooses to make a ontrbuton to the provson of a threshold publ good wth a ost of PP. If the publ good s provded, eah agent reeves a prvate value of v ndependently drawn from a ommon knowledge value dstrbuton. If the publ good s not provded, all ontrbutons are refunded (money-bak guarantee).. Provson Pont Mehansm (PPM) The payoff funton for agent under PPM s I + () N v f = j PP j π = I otherwse Under PPM, f PP s met or exeeded, eah agent reeves the ntal endowment I mnus ther ontrbuton, plus ther prvate value, v, for the publ good; otherwse, they only get I. If ontrbutons exeed PP, PPM burns the exess. Alternatvely, exess ontrbutons an be returned through rebate mehansms, whh may affet ontrbuton strateges. PR, UPA, and UPC return exess ontrbutons n dfferent ways.. Proportonal Rebate (PR) Agent s payoff under PR s + + N N I v ( = j PP) f N j j= j PP () π = Under PR, f PP, the exess ontrbuton ( PP ) wll be rebated. The rebate to j j eah agent s proportonal to the rato of ther ndvdual ontrbuton to the total ontrbuton..3 Unform Pre Auton (UPA) Under UPA, a unform pre, UP, wll be alulated. UP s the lowest pre suh that the number of ontrbutons hgher than that pre tmes the pre s equal to PP. The payoff under UPA s (3) I j j= j otherwse j N I + v f,, < = j PP UP and UP j N π = I UP+ v f,, j= j PP UP and UP I otherwse. If an agent ontrbutes less than UP, she pays nothng and the full wll be rebated. If an agent ontrbutes UP or more, she wll pay only the 4

6 pre UP and the exess ontrbuton wll be rebated. To provde the good, UPA requres not only that the total ontrbuton meet or exeed PP, but also that the number of relatvely hgh ndvdual ontrbutons be suffent. More presely, PP and the group sze together determne a set of at most N possble pres, where PP s shared by n N agents offerng at least PP/n. If the ontrbutons n aggregate exeed PP, but annot satsfy np=pp for some n, the mehansm does not provde; wth suh an outome, UPA s not effent..4 Unform Pre Cap (UPC) UPC s a modfed verson of UPA that ensures the good an be provded whenever total ontrbutons exeed PP. The payoff under UPC s N I + v f = j PP and < UC j N (4) π = I UC + v f = j PP and UC j I otherwse where UC = mn{ p > 0 : + np = PP, n = { : p }}. Under UPC, f there are j { k : k < p} j exess ontrbutons, a unform pre ap (UC) wll be alulated. If an agent ontrbutes less than UC, she pays her full ontrbuton (under UPA she would have pad nothng). If an agent ontrbutes UC or more, she pays only the pre ap and the exess ontrbuton s rebated, just lke under UPA. UC s alulated as the lowest pre that ould ollet exatly PP. Sne ontrbutons lower than the pre wll not be rebated, the unform ap UC always exsts as long as ontrbutons at least meet PP; wth suh an outome, UPC s effent..5 The Effet of Margnal Penalty of Overontrbuton on the Expeted Payment Marks and Croson (998) use the onept of margnal penalty the prvate payoff loss assoated wth an addtonal unt of ontrbuton, ondtoned on provson to argue that rebate rules redue the ost of makng hgher prvate offers n the absene of oordnatng on a sngle equlbrum n ther omplete nformaton game, nreasng aggregate ontrbutons and the lkelhood of provson. It s nsghtful to translate ther onept to an ex ante expeted margnal penalty n our Bayesan framework, n order to better understand how rebate rules an nfluene BNE. Crually, n a Bayesan game, a profle wth hgh value-realzatons wll generate exess ontrbutons n equlbrum, meanng the effet of lower margnal penaltes an be dentfed wthn the equlbrum onept, as mehansms wth lower margnal penaltes may support strategy profles wth hgher ontrbutons as equlbra. 5

7 To understand the margnal penalty assoated wth eah of our mehansms, we examne the expeted payoff funton, ondtoned on others strateges, of a -player Bayesan game wth ontnuous types. Eah agent has an ndued value v, =,, ndependently drawn from [ vv, ] wth a pdf f( v) > 0, v [ vv, ], where v ( PP, PP ) to make a olletve ontrbuton neessary for provson. Wthout loss of generalty, we dsuss agent s expeted payoff, ondtoned on agent s ontrbuton funton () of v. For smplty, we assume () s strtly nreasng and dfferentable over [ vv, ] and we omt the endowment I n the expeted payoff funton. In PPM, agent s expeted payoff of ontrbutng ondtoned on v and () PPM (5) Eπ ( v, ( )) = v f ( v ) dv f ( v ) dv, v ( PP ) v ( PP ) where the ntegratons represent the probablty of provson, and the two terms on the RHS are the expeted beneft and ost, respetvely. Take the dervatve of (5) w.r.t., we have (6) Eπ ( v, ( )) ( ( ) ) ( ( )) f PP PPM = ( v ) ( ) ( ) f v dv v PP PP On the RHS of (6), the frst term represents the margnal net beneft due to an nreased provson probablty from a hgher ontrbuton. The seond term, whh s the one we are lookng for, s the expeted margnal penalty of overontrbuton: t says gven the same provson probablty (the lower bound of the ntegraton s unhanged), f nreases by, the ost also nreases by (the ntegrand) ondtoned on provson, refletng a --margnal penalty n PPM. Rebates and hene a lower margnal penalty redue ex ante ths expeted margnal penalty ost. In PR, gven the same (), we have PR PP (7) Eπ ( v, ( )) = v f ( v ) dv ( ) v ( PP ) f v dv v ( PP ) + ( v ) (8) ( ( ) ) ( ( )) Eπ ( v, ( )) f PP PP ( v) PR = ( v ) ( ) ( ) ( f v dv v PP PP + ( v)) Compared to (6), the only dfferene n (8) s the ntegrand on the RHS, whh s the margnal penalty n PR (exatly the same as defned n Marks and Croson, 998). Note that the margnal s 6

8 penalty s bounded between - and 0, almost always greater than -, and beomes loser to 0 as nreases. Gven and (), the expeted margnal penalty n PR s less than that n PPM, espeally for hgher ontrbutors, and hene hgher ontrbutons may be supported n equlbrum n PR, onsstent wth Marks and Croson. Our unform pre mehansms am to nrease ontrbutons by reatng regons of zeromargnal penalty, whh result n even lower expeted payments. In a -player game of UPA, the good s provded only f both agents ontrbute PP/ or above and eah pays PP/, so (9) 0 f < PP v ( ) ( ) f v dv f v dv f PP UPA Eπ ( v, ( )) = PP v ( PP ) v ( PP ) Compared to (5) and (7), the expeted payoff n UPA s ndependent of wthn [0, PP/) and [PP/, PP). Sne agent pays ether 0 or PP/, the margnal penalty s always 0 exept at PP/ aross whh the payment jumps from 0 to PP/. Thus, the expeted margnal penalty s always zero exept at PP/, where the expeted payment s (PP/) f ( v ) dv. More generally v PP ( ) n an N-player game (N>), as shown n L et al (04), f <UP( - ) where UP( - ) s the unform pre that provdes the good through payments of PP/m by m other agents, there exsts a utpont, (, UP( - )) at whh agent s ontrbuton s suffent to be nluded n payments of the next lowest unform pre, and the fnal payment jumps from 0 to the new pre, UP( - )= PP/(m+). Then, the margnal penalty of UPA s zero almost always, exept at the utpont wth a lump sum penalty, resultng n an expeted margnal penalty struture smlar to the -player ase. Gven the broad range of values wth no expeted margnal penalty, we onjeture hgher ontrbutons n UPA than the other mehansms. In UPC, the unform ap s PP when <PP/, and s max{pp ( ), PP/} when PP/, so we have (0) v f ( v ) dv f ( v ) dv f < PP v ( PP ) v ( PP ) UPC Eπ ( v, ( )) = v f v dv PP v f v dv ( PP ) ( ) [ ( )] ( ) v ( PP ) ( PP ) PP f( v ) dv f PP v ( PP ) 7

9 () ( ( ) ) ( ( )) f PP ( v ) ( ) < ( ) f v dv f PP PP v PP UPC Eπ ( v, ( )) = f ( ( PP ) ) ( v ) f > PP ( ( PP ) ) UPC has a hybrd expeted payoff struture of PPM (5) and UPA (9). Thus, when <PP/, UPC has the same expeted margnal penalty f ( v ) dv as PPM. When >PP/, v ( PP ) UPC has a zero-expeted margnal penalty, smlar to UPA: the expeted margnal ost of nreasng s only due to the nreased provson probablty whle not the margnal penalty of overontrbuton (.e., a zero-margnal penalty). At =PP/, the expeted margnal penalty s not defned. It s easy to verfy that, n an N-player game (N>), f s at or above UC( - ), the unform ap that provdes the good as a funton of gven -, then any nremental ontrbuton wll not hange the ap, reatng a margnal penalty of 0. If <UC( - ), there exsts a utpont,, at whh the margnal penalty hanges from - to 0 ( =PP/ n a - player game). Hene, UPC generally has a zero-expeted margnal penalty when the ndvdual ontrbuton s hgh enough, 6 and based on belefs about -, agents alulate an expeted margnal penalty between - and 0. Compared to PPM and PR, UPC may ndue even hgher ontrbutons bearng n mnd that the major exess offer would be from hgh ontrbutors. In ontrast to n a omplete nformaton game where PPM, PR, and UPC have the same Pareto effent equlbrum set, n Bayesan games these expeted margnal penaltes do dfferentate the mehansms equlbrum sets and provde theoretal benhmarks for further expermental omparsons, as shown n the next seton. 3. Bayesan Nash Equlbra To proeed, we fous on Bayesan games wth dsrete types. 7 We frst haraterze some bas propertes of the BNE sets of UPC and UPA as Galmard and Palfrey (005) do for PPM and PR. 6 Intutvely, ths s due to our desgn of UPC to fous on protetng hgh ontrbutors: the exess offer s only rebated to those ontrbutng above the ap, n ontrast to PR where all ontrbutors share the exess offer. 7 Although the role of rebates and the orrespondng margnal penalty struture an be learly revealed n a ontnuous-type analyss, these mehansms exept for UPA are muh more dffult to analyze wth prvate value nformaton and ontnuous types: there s only a small body of lterature on PPM (Alboth et al., 00; Menezes et al., 00; Laussel and Palfrey, 003), and even no analyss on PR. Solvng PR or UPC alone may need a full-length paper. On the other hand, t s easer to solve numerally a game wth dsrete types wthout losng the key nsghts 8

10 Then we solve for symmetr BNE of UPA n a 3-player game and dsuss the general BNE soluton struture and the value revelaton property of UPA n an N-player game. Lastly, we dfferentate the BNE sets of the four mehansms usng a numeral example. 3. Bas Propertes of the BNE sets of UPC and UPA Followng the model setup n the begnnng of Seton, agent s ndued value v s ndependently drawn from a fnte set V of real numbers wth a ommon knowledge probablty dstrbuton funton. V s the same for all agents. We assume, so, as n the nterestng (and hallengng real) ontext, provson s not optmal for any ndvdual. All above nformaton s ommon knowledge exept that the realzed v s only known to agent. Let v = ( v,, v ) N N V denote a profle of values, v ( ) = ( ( v),, N ( v N )) denote a ontrbuton strategy profle wth one ontrbuton funton for eah agent, and v = ( v,, v, v+, v N) and smlarly for ( v ). Further, let P( ) denote the provson probablty when agent ontrbutes gven the ontrbuton funtons of the others, let s ( v ( )) denote agent s fnal payment as a funton of v ( ), and let S( ) = E ( s( v ( )) ) denote s expeted payment gven ontrbuton. v Galmard and Palfrey (005) present three ommon propertes of the BNE sets of PPM and PR: no overbddng, payment monotonty and ontrbuton monotonty. We show that the BNE sets of UPC and UPA have three smlar propertes wth some regularty ondtons. Proposton (No overbddng). In UPC, any strategy n whh ( v) > v for some v s ex post weakly domnated; n UPA, any strategy n whh ( v ) s greater than the lowest PP / k v for k {,, N } and some v, s ex post weakly domnated. Proof: See Appendx. Lemma (Payment monotonty). In UPC, f P( ) > 0 and s less than PP and the utpont assoated wth some unform ap for some value profle v, S ( ) s strtly nreasng at ; n UPA, f P( ) > 0 and s less than PP and the utpont about the mehansms, as Galmard and Palfrey (005) do for PPM and PR. Therefore, we wll fous on dsrete types n the followng setons of the paper. 9

11 assoated wth some unform pre for some value profle v, S( ) s strtly nreasng at n the sense that when nreases to a hgher unform pre, S( ) nreases. Proof: See Appendx. Note that the utponts and are defned as n Seton.5. Proposton (Contrbuton monotonty). For both UPC and UPA, let * ( v ) be a symmetr Bayesan Nash equlbrum ontrbuton funton. 8 If P( * ( v) * ) 0 for all v > mn{ V } and > * ( v ) s less than the utpont assoated wth some unform ap/pre for some value profle v, then, for all v and v j, v > v ( v ) ( v ). * * j j Proof: See Appendx. The three propertes of UPC are exatly the same as those of PPM and PR, exept for the addtonal regularty ondton that or * ( v ) s less than the utpont assoated wth some unform ap for some value profle v. Ths regularty ondton elmnates the unnterestng and extremely rare ases that or * ( v ) s always greater than any possble unform ap for all value profles; n these rare ases S( ) s only weakly nreasng at. In addton to a smlar regularty ondton, the propertes of UPA n Proposton and Lemma are restated to reflet the step-funton form of the payment sheme, sne n UPA realzed payments an only be one of a fnte number of unform pres. Wth these adjustments, the proofs of the propertes of UPC and UPA are smlar to those for PR n Galmard and Palfrey (005). These regularty ondtons reflet some addtonal propertes of UPC and UPA due to ther margnal penalty struture. Next, we use symmetr BNE of UPA to show ts advantage n generatng hgher ontrbutons, and then demonstrate how UPC results n BNE sets dfferent from PPM and PR and dsuss the underlng nentves. 3. Symmetr BNE of UPA Sne there are only fnte possble pres n UPA, t s equvalent to a Bayesan game wth dsrete atons. It s known that a BNE soluton to ths knd of game has the form of a deson rule based on some rtal values. Therefore, we an solve the Bayesan game for UPA by 8 We abuse the notatons here and n the followng dsussons to emphasze the symmetry: v and are both salars, representng one agent s value and equlbrum ontrbuton strategy. 0

12 dentfyng the deson rules and the rtal values. We wll solve for symmetr BNE of UPA n a 3-player game frst, and dsuss the general BNE soluton struture n an N-player game. Wthout loss of generalty, we assume agents only ontrbute the possble unform pres. In a 3-player game, eah agent has three ontrbuton hoes {0, PP/3, PP/} and two rtal values wll be enough to haraterze the deson rule. Let v 3 and v denote the two rtal values for a symmetr pure-strategy weakly monoton BNE, assumng 0 v3 v v, where 0 = mn{ V } and v = max{ V }. The ontrbuton funton for eah agent s n the general form 0 f v v3 () ( v) = PP 3 f v3 < v v PP f v > v Then the BNE of UPA n the 3-player game are as follows. Proposton 3. In a 3-player Bayesan game, UPA has the followng four ategores of symmetr BNE: for =,, 3, BNE a) ( v ) = 0, wth = v = 3 v v. b) BNE ( v) = 0 f v PP 3 PP 3 f v > PP 3, wth v3 = PP 3 and v = v. ) d) BNE ( v) = 0 f v ˆ v PP f v > v ˆ, wth v > vˆ = v3 = v > PP and ˆv gven by ( vˆ PP 3) Pr( v > vˆ ) = ( vˆ PP ). 0 f v v3 BNE ( v) = PP 3 f v 3 < v PP, PP f v > PP wth v = PP > v3 > PP 3 and v3 gven by v 3 Pr( v > v ) = ( v3 PP 3) Pr( v > v3 ). Proof: See Appendx. Note that Pr( v > v ˆ) denotes the probablty of v > v. ˆ Smlarly for Pr( v > v ) and Pr( v > v 3 ). Remark. Gven our parameter assumpton that max{ } ( 3, ) V PP PP, solutons a and b always exst. The exstene of and d requres addtonal ondtons on PP, v, and the value dstrbuton. Frst, PP should be less than v, otherwse PP/ s not a feasble pre. Also, PP

13 should be hgh enough relatve to v to support and d (espeally for d). We use a unform value dstrbuton over [0, ] to demonstrate these addtonal ondtons n the appendx. 9 Remark. A ategory of symmetr BNE n UPA nludes all the ontrbuton strateges that are equvalent to a symmetr BNE where only the possble unform pres are used as ontrbuton hoes, as one of the lsted BNE n the proposton. For example, any strategy wth ndvdual ontrbutons always below PP/3 s a symmetr BNE n ategory a, sne ontrbutng below PP/3 s equvalent to ontrbutng 0. Beause these small ontrbutons annot affet payments, refletng the zero-margnal penalty struture, they annot be exluded as equlbra. Ths feature advantages UPA n value revelaton: n d, agents an reveal ther true values wthout affetng ther payments over the entre value range exept for the nterval [PP/3, v 3 ] where agents need to ontrbute less than PP/3 to follow the equlbrum strategy. Generally, a BNE nludng more of the feasble unform pres (d wth two postve pres vs. wth only one) supports a larger truth-tellng value range: b to d all support larger truth-tellng value ranges than a (.e., the non-truth-tellng ranges of [PP/, v ], [PP/, ˆv ] and [PP/3, v 3 ] are all smaller than [PP/3, v ]); the omparsons among b to d depend on parameters. We show n the appendx that d, f t exts, has the largest truth-tellng range. In the speal ase of a two-player game, truth-tellng over the entre value range s supported n UPA as agents ontrbute PP/ when and 0 otherwse (See Appendx). In an N-player game, UPA has a soluton struture smlar to that n the 3-player game, whh s summarzed n the followng proposton. Proposton 4. In an N-player Bayesan game, assumng max{ } (, ) V PP N PP, UPA always has the followng two ategores of symmetr BNE: for =,, N, BNE e) ( v ) = 0, for all v V. f) BNE ( v) = 0 f v PP N PP N f v > PP N BNE wth unform pres hgher than PP/N and/or wth more than one unform pre may or may not exst, dependng on whether the orrespondng rtal values an be dentfed from 9 These BNE also apply to a game wth ontnuous types.

14 a system of polynomal equatons gven the parameters of PP, V and the value dstrbuton. The BNE wth the most numerous dfferent pres, f t exts, s g), wth v = PP > v > PP k, for k = 3,, N, and the rtal values are gven k by a system of N- polynomal equatons. Proof: See Appendx. The general proedure to solve the N-player game and the general form of the expeted payoff funton at eah possble unform pre used to onstrut the system of polynomal equatons are gven n the appendx. Remark 3. Although t s not nentve ompatble n general (f. Borgers et al., 05), UPA may stll support relatvely hgh value revelatons for pure publ goods due to the same reason dsussed n Remark, n ontrast to the dret seral ost sharng mehansm (Mouln, 994) where nentve ompatblty s obtaned only for exludable publ goods. 0 Therefore, we would expet relatvely hgh ontrbutons n UPA even n BNE. Note that ths value revelaton property s prmarly due to the step-funton style of payment sheme, whh s fully aptured by the almost always zero-expeted margnal penalty struture. Also, the equlbrum f of UPA results n the same equlbrum outome as the onservatve equal-osts mehansm for pure publ goods (Mouln, 994), where everyone needs to pay PP/N and the good s not 0 If we have n equlbrum v = PP > v > PP 3 > > v > ( ) > > 3 N PP N v PP N, the truth-tellng value N ranges supported by the symmetr BNE n UPA are as follows: let PP/k denote the hghest unform pre n the set of BNE that exst gven the model parameters, then the non-truth-tellng value range s [PP/m, v ] or [PP/(k-), ], where m {k, k+,, N}, v s a rtal value defned n the same fashon as n the equlbrum g; m alternatvely, the truth-tellng value range s [, ] and [0, PP/N]; and f k=n and N s large, the non-truth-tellng value ranges wll be relatvely small. 3 m

15 provded otherwse,.e., the onservatve equal-osts mehansm just mplements a partular BNE ategory f of UPA. 3.3 BNE-Set Comparsons by a Numeral Example To further llumnate how UPC and UPA dffer from PPM and PR, we adopt the approah used n Galmard and Palfrey (005). In Galmard and Palfrey s (005) envronment, eah agent s value s ndependently drawn from a unform dstrbuton over a set of three values, Through a well-onstruted 3-player, 3-value numeral example, they fnd that, although they share some bas BNE propertes, PPM and PR have dfferent equlbrum sets, and they use those examples to motvate hypotheses for ther subsequent experment. We replate ther numeral results and show that our unform pre mehansms have equlbrum sets dfferent from those of PPM and PR usng the same parameters. Further, there are systemat dfferenes n the equlbrum sets that are explaned by expeted margnal penaltes, whh we use to predt how the four mehansms wll dffer ontrbutons: a lower expeted margnal ost of ontrbutng more (exessvely) supports hgher ontrbutons n equlbrum. v {9, 45, 90}, =,, 3. The provson pont s 0. Let ={ L, M, H } be the ontrbuton funton, where the supersrpts L, M and H denote the ontrbuton from v = 9, 45, and 90 respetvely, and only nteger-valued ontrbutons are allowed. They fnd PPM has 35 nontrval pure-strategy, symmetr, and weakly monoton Bayesan Nash equlbra, and PR has All the nontrval equlbra are ategorzed nto four effeny groups (f. Galmard and Palfrey s (005) Table ), measured by expeted total net beneft (total ndued values mnus the ost). PR has four equlbra n the most effent group, and PPM has none; PPM and PR respetvely have 4 and 7 equlbra n Wth the neffeny of provdng the good only when everyone has a value hgher than the equal-ost share PP/N, the sngle unform pre guarantees that truth-tellng s a domnant strategy n the onservatve equal-osts mehansm, whh s onsstent wth the haraterzaton for domnant strategy mplementaton (Shwartz and Wen, 03). Ths also explans why truth-tellng s a domnant strategy n UPA n a -player game where PP/ s the only postve unform pre. A losed-form soluton for UPC s BNE set s muh more hallengng to obtan, beause UPC has a hybrd expeted payoff struture between PPM and UPA (see Eq. (0) n Seton.5). Nevertheless, we keep UPC n ths paper beause ts rebate struture provdes an mportant pont of omparson. As we wll see below, the ombnaton of the expeted-payoff analyss and omparson of the numerally alulated BNE-set provdes suffent nsghts to understand the key role of rebates n attratng hgher ontrbutons n equlbrum, whh may not be theoretally demonstrable even wth a losed-form soluton gven the multplty of equlbra. 3 Nontrval means the provson probablty s greater than zero. 4

16 the seond most effent group; 3 equlbra of PR and all the remanng 3 equlbra of PPM fall nto the fourth group; the remanng equlbra of PR are n the thrd group. In our unform pre mehansms, UPC has 8 nontrval symmetr, pure-strategy monoton BNE, half n the most effent group and half n the fourth group; UPA has two ategores of BNE, both n the fourth group. Table lsts all the most effent equlbra of PPM, PR and UPC and two equlbra for eah ategory n UPA (see the appendx for a omplete lst of the nontrval BNE of PPM, PR, and UPC). Table The most effent Bayesan Nash equlbra under eah mehansm* PPM All n the nd effent group PR All n the st effent group UPC All n the st effent group UPA All n the 4 th effent group {3, 34, 45} {,, 60} {0, 0, 6} {9, 45, 50} Category {4, 34, 44} {,, 58} {,, 60} {0, 34, 34} {5, 35, 4} {3, 3, 56} {,, 58} {9, 33, 90} Category {6, 35, 4} {4, 4, 54} {3, 3, 56} {0, 0, 5} * {X, Y, Z} denotes the ontrbuton from value 9, 45, and 90, respetvely. UPC ndues muh hgher ontrbutons from the hghest value type than PPM: the lowest ontrbuton from the hgh type n UPC (56) s muh more than the hgh type s maxmum ontrbuton n PPM (45) n ther most effent equlbrum sets. Together wth a smlar pattern between PR and PPM, ths omparson sheds lght on how the rebate and ts orrespondng lower expeted margnal penalty nfluene ontrbutons of hgh types: the rebate n UPC and PR redues the expeted payment ost more sgnfantly for hgh-value agents sne they wll experene overontrbuton wth the hghest probablty, and they ould pay the most n the absene of a rebate. For example, {,, 60} s a BNE n both PR and UPC, but s not an equlbrum n PPM sne a hgh-value agent would be better off to devate from 60 to. 4 nsght reflets the expeted margnal penalty: PR and UPC both have lower expeted margnal penaltes than PPM, espeally for hgh ontrbutors who are generally hgh types based on Proposton. Compared to PR wth a rebate orrespondng to a smaller but stll not zero-expeted margnal penalty, UPC as desgned further redues the expeted ost wth a zero-expeted margnal penalty on hgh (enough) ontrbutons, and hene ndues even hgher ontrbutons Ths 4 Galmard and Palfrey (005) show that f agent wth a value v has any nterm proftable devaton, then one of these devatons s proftable: 0 or 0-x-y, where x, y are n {,, 60}. Ths works for UPC as well, a fat used n the followng paragraphs. 5

17 from the hgh type than does PR. 5 Spefally, {0, 0, 6} s not a BNE n PR beause an agent wth value 90 has an nentve to devate from 6 to 0: the agent would rather redue her ontrbuton than pay the expeted proportonal share. The realzed value profle that drves the dfferene between mehansms s when the other two agents are (9, 90), ths agent s ontrbuton of 6 wll reeve only a proportonal rebate (8.) n PR, whle a full rebate above the unform ap n UPC (=6 the unform ap 4),.e., the margnal penalty n UPC s low enough to elmnate the devaton nentve but that n PR s not. Wth a muh broader range of zero-expeted margnal penalty, UPA generates the hghest ontrbutons n equlbrum, as demonstrated by the lsted four BNE: n Category, {9, 45, 50} results n the same equlbrum outome and payoff as {0, 34, 34} but supports truthtellng for low and medum types, and also supports hgher ontrbutons from hgh types sne ontrbutng 45 or 50 leads to the same payments as ontrbutng 34 beause the next hghest possble unform pre s 5; n Category, {9, 33, 90} and {0, 0, 5} are smlarly equvalent, and the former even supports truth-tellng for the hgh type and on average results n group ontrbutons qute lose to the expeted total ndued values (93%), whh wthout the ost of exluson s not that muh worse than the dret seral ost sharng mehansm n terms of value revelaton. Also, note that ths BNE set s onsstent wth Proposton 3: {0, 34, 34} and {0, 0, 5} orrespond to the equlbrum b and respetvely. The observatons above support the role of the expeted margnal penalty n dfferentatng the equlbrum sets and orrespondng aggregate ontrbuton levels. In UPC and PR, t s only when the ontrbuton s hgh enough that the expeted margnal penalty approahes zero, and hene we fnd the effet s the most sgnfant for the hgh-value type n the numeral example. Beause more effent equlbra omprse a larger porton of UPC s set of nontrval BNE, we hypothesze that t wll generate hgher ontrbutons than PR and PPM. Smlarly, UPA has equlbra wth muh hgher value revelaton, but wth lower provson rate due to the number of less effent equlbra and the addtonal onstrant on ontrbuton profle as n the 5 Note that f we take an average of all the nontrval BNE wthn eah mehansm, the two observatons above about UPC stll hold: the average ontrbuton funtons are {8, 5, 4}, {7, 7, 5}, and {,, 54} for PPM, PR, and UPC, respetvely. And the evdene beomes even stronger f we nrease the ontrbuton preson to 0.: {.3, 9., 40.8}, {8.6, 30.0, 5.6}, and {3.4, 3.4, 55.4} for PPM, PR, and UPC, respetvely, and UPC has the hghest equlbrum ontrbuton from the hgh value type, 63.8, among all nontrval BNE of the three mehansms. 6

18 omplete nformaton ase. For PPM and PR, based on the results from Galmard and Palfrey (005), we hypothesze PR s better than PPM n both ontrbuton and provson rate. 5 Expermental Desgn and Proedures In eah perod, 0 subjets eah learn ther random, prvate v, whh was ndependently drawn from a unform dstrbuton on over a set of nne values, v {4, 5, 6, 7, 8, 9, 0,, }, =,, 0. The provson pont PP s 36, 45% the total expeted ndued value (80) n a group of sze 0. Feasble postve UPA pres are {3.6, 4, 4.5, 5., 6, 7., 9, }. These expermental parameters are hosen n a way that provdng the good s always soally optmal, and a wde range of unform pres are possble. All the nformaton above s ommon knowledge. Fgure presents the average equlbrum ontrbuton at eah ndued value based on the nontrval BNE set of eah mehansm wth nteger-valued ontrbutons n ths 0-player, 9-value game, and shows that the observatons from the numeral example n Seton 3.3 stll hold. 6 0 Average Equlbrum Contrbuton PPM PR UPC UPA Indued Value Fgure Average Equlbrum Contrbutons by Indued Value 6 Note that the average equlbrum ontrbutons n UPA are based on the nontrval BNE that only unform pres are hosen as ontrbuton hoes and hene are the lower bound. Also, although the ontrbuton preson s 0. n the experment, there are two justfatons for nteger preson we used here. Frst, as mentoned n the prevous footnote, nreasng preson wll only make the dfferenes among the mehansms more sgnfant, espeally n favor of UPC, and hene wll not hange our man hypotheses. Seond, a searh wth a 0. preson s mpossble n terms of the omputer runtme gven our relatvely large group sze (0) and many value types (9). Even wth an nteger preson, t took a full day to searh for UPC s equlbra on 4 nodes of a hgh-speed lmate modelng luster onsstng of 00 nodes (560 ores, 4.5 TB memory, 49.3 peak Tflops). Wth a preson of 0., the strategy spae for eah value type wll be 0 tmes larger, whh mples mpossblty gven that we have 9 dfferent values. For UPA, however, we dd obtan the omplete BNE set sne there are only 8 postve unform pre hoes. 7

19 Table shows the mehansm treatments presented n eah sesson: the frst treatment s always PPM (5 perods), to famlarze subjets wth the baselne game. The followng treatments (5 perods eah) apply the other mehansms n a Latn Square to ontrol for order effets. Table Treatment Arrangement of Expermental Sessons Treatment Order st nd 3rd (5 perods) (5 perods) (5 perods) Sesson PPM PR UPC Sesson PPM PR UPA Sesson 3 PPM UPC PR Sesson 4 PPM UPC UPA Sesson 5 PPM UPA PR Sesson 6 PPM UPA UPC The expermental software was developed n z-tree (Fshbaher, 007). At the start of eah treatment, the expermenter read the nstrutons aloud as subjets read along. Subjets were then gven an ntal budget of 5 expermental dollars. Subjets then smultaneously hoose a ontrbuton, [0,5] (wth a preson of 0.) towards the projet. At the end of eah perod, subjets were nformed whether the projet s provded, and ther earnngs, payment and rebates. At the end of a sesson, earnngs were totaled aross all perods. Subjets were reruted through unversty-wde daly dgest emal server (manly for undergraduates), and from an emal lst of students nterested n partpatng n experments. A total of 60 subjets partpated n the sx omplete sessons, produng 4500 ndvdual level observatons. 8

20 Group Contrbuton UPA UPC Perod PR PPM Fve-perod Provson Rate Fgure Group Contrbutons n Eah Perod and 5-Perod Provson Rate by Mehansm 9

21 6 Results We measure the performane of the mehansms by two ndators: aggregate group ontrbuton and the provson rate. The provson rate reflets the effeny of the mehansm, as provson s always effent gven our parameter values; therefore, t s a dret test of our hypotheszed dfferenes n mehansm effeny based on the BNE sets derved above. In addton, group ontrbuton s a measure of the extent of value revelaton. Although none of these mehansms are nentve ompatble, revelaton propertes may be of nterest n applatons where smallsale, real money, real good plot programs are used to provde estmates of publ value for nonmarket goods that are then appled over a broader populaton (e.g., Champ et al, 00; Swallow et al., 008; Haskell et al., 00; Bush et al., 03; Swallow, 03). Fgure gves an overvew of group ontrbutons n eah perod, and fve-perod provson rates, by mehansm. Grey lnes represent sesson-spef group ontrbutons, dark lnes represent averages over sessons, and dark dots represent average fve-perod provson rates. 6. Group Contrbutons Table 3 Two-fator Random Effets Models of Group Contrbuton Group Contrbuton () () PR (0.946) (0.9) UPC.770*.74*** (0.946) (0.943) UPA 4.46***.4*** (0.946) (.06) Provson Rate -8.68*** (.83) Constant (PPM) 36.4*** 4.09*** (0.7) (.86) Log-lkelhood Ch-square (df) 7.4 (3) (4) R overall Number of observatons Number of perods (treatment-spef) 0 0 Standard errors n parentheses; *** p<0.0, ** p<0.05, * p<0.; : Provson rate over prevous 5 perods, whh yelds the largest log-lkelhood among to 5-perod lags. In Fgure, UPA generates muh hgher group ontrbutons than the others, and UPC looks to be slghtly hgher than PR and PPM. Table 3 presents results from two-fator random effets regressons of group ontrbuton on mehansm dummes (group- and perod-spef, f. Marks and Croson, 998), fousng on the observatons from the last 0 perods. 0 0 We exlude the observatons from the frst fve perods to solate potental mehansm-learnng or order effets n the early perods. We use the same rule for all the followng regressons and statstal tests unless stated otherwse. 0

22 Model provdes a baselne that nludes only mehansm dummes, usng PPM as the base. Average predted group ontrbutons are not dstngushable from the PP value of 36 n PPM (36.4, p=0.733) and PR (36.50, p=0.557), but are sgnfantly hgher than PP n UPC (38.0, p=0.07) and UPA (50.70, p<0.00). Model ontrols for the prevous fve perods provson rate, sne ndvduals efforts to redue ther payments may nfluene the equlbrum seleton proess as they try to ontrbute just enough to obtan regular provson as a group. It s sgnfantly negatve (p<0.00), whh s evdene of heap rdng (f. Issa et al. 989) where ndvduals reveal less of ther value when provson has been ourrng. A lkelhood rato test advses usng Model for nterpretaton. Model reflets an orderng of group ontrbutons by mehansm that s broadly onsstent wth hgher ontrbutons ourrng where the expeted margnal penalty s lower, espeally for the margnal penalty strutures of our new mehansms. UPA wth an almosteverywhere zero-expeted margnal penalty and a BNE set supportng group ontrbutons lose to the expeted total ndued values s sgnfantly hgher than the others all wth p<0.00 (lkelhood rato test). Smlarly, the lower expeted margnal penalty from UPC leads to sgnfantly hgher aggregate ontrbutons than PPM (p=0.004) and PR (p=0.009). Inreasng ontrbutons for a hgher probablty of provson wll not result n losng money due to overontrbuton n a broad ontrbuton range n these mehansms. However, we annot rejet the hypothess that PR and PPM generate the same group ontrbutons, onsstent wth Marks and Croson (998), but dfferent from Galmard and Palfrey (005). The latter dfferene ould be attrbutable to the lower ost-beneft rato (the provson pont dvded by the total expeted ndued value; our 45% vs. Galmard and Palfrey's 6%) or the larger group sze (0 vs. 3) n our experment. 6. Provson Rate The group ontrbutons lead to a smlar orderng of provson rates among the effent mehansms, as shown n Fgure (dots). Wth the same provson ondton among UPC, PR and PPM, UPC desgned wth margnal penalty n mnd performs better than both PPM and PR. Spefally, UPC has a provson rate 68.8%, whh s sgnfantly hgher than those for Sne ndued values are randomly assgned n eah perod, provson rate over several prevous perods reflets how far away the expeted group ontrbuton s from the provson pont. Marks and Croson (998) has a ost-beneft rato of 50% and a group of sze 5.

23 PPM (57.5%, p=0.054 one-taled z-test) and PR (53.8%, p=0.05 two-taled), wth the latter two not statstally dstngushable (p=0.60). 3 Ths orderng emphaszes the advantage of the zeromargnal penalty n UPC n ndung not only hgher ontrbutons, but also a larger proporton of hgher group ontrbutons. The smlarty between PR and PPM n provson rate s stll onsstent wth Marks and Croson (998), but ontrasts wth Galmard and Palfrey (005) who found PR s sgnfantly better than PPM. One may argue that the provson rate n PR s drven down by the low provson rate n the 5-perod nterval 6 to 0. However, PR and PPM are stll not statstally dfferent (p=0.638) when fousng on the last 5 perods data, although PR has a nomnally hgher provson rate than PPM (58.3% vs. 54.4%). Beause the profle of offers, n addton to the total amount, affets the provson deson n UPA, t has a provson rate (35.0%) sgnfantly lower than the other mehansms, all wth p<0.0. To understand how the margnal penalty struture of our unform mehansms ndues nentves that result n hgher group ontrbutons and provson rates, we examne ndvdual level ontrbutons at dfferent ndued values, where BNE has dfferent predtons aross mehansms. 6.3 Indvdual Contrbutons Fgure 3 shows average ndvdual ontrbutons at eah ndued value aross mehansms. Average observed unform pres from UPA and UPC are also nluded to show how beng lose to where the margnal penalty hanges sharply dfferentates ontrbutons aross mehansms. Average ontrbutons nrease wth ndued value n all mehansms, but they are yet hgher n the unform pre mehansms (Fgure 3). Consstent wth group ontrbuton, UPA stands out as generatng muh hgher ontrbutons at all value levels; UPC has generally slghtly hgher ontrbutons, espeally at hgh values (0 to ). Comparng Fgure and 3, we fnd the observed average ontrbutons are followng farly losely the average ontrbutons of the equlbra shown n Fgure. Sne Fgure only shows the lower bound of equlbrum ontrbuton, onsstent-wth-equlbrum hgher offers n UPA may make t more promnent n Fgure Wloxon rank sum test gves smlar results: UPC vs. PPM (p=0.0545, one-taled), UPC vs PR (p=0.05, twotaled), PPM vs. PR (p=0.60, two-taled) 4 The man dfferene between Fgure and 3 s that we do not observe zero average ontrbutons from low value types n the lab data, whh s qute normal n the publ good experments lterature sne usually subjets just do not ontrbute zero (Ledyard, 995).

24 0 Indvdual Contrbuton PPM PR UPC UPA Observed Ave. Pre Cap of UPC Observed Ave. Pre of UPA Indued Value Fgure 3 Mean Contrbutons by Indued Value To nvestgate statstally how ndvdual ontrbuton vares wth ndued value and mehansm, we run a seres of subjet-treatment random effets tobt models of dollar amount ontrbuted. Table 3 shows the results, usng PPM as an exluded base mehansm. Table 3. Random Effets Tobt Models of Indvdual Contrbuton Contrbuton () () (3) (4) PR (0.348) (0.45) (0.348) (0.45) UPC (0.348) (0.47) (0.349) (0.47) UPA.546*** 0.766*.383*** 0.65 (0.348) (0.44) (0.350) (0.45) Value 0.55*** 0.490*** 0.53*** 0.489*** (0.003) (0.079) (0.003) (0.079) PR Value (0.080) (0.079) UPC Value ** 0.064** (0.085) (0.084) UPA Value *** *** (0.080) (0.080) Provson Rate *** *** (0.6) (0.60) Constant (PPM) *** (0.36) (0.63) (0.55) (0.8) Log-lkelhood Ch-square (df) 609 (4) 644 (7) 638 (5) 673 (8) R overall Number of observatons Number of groups Number of perods Standard errors n parentheses; *** p<0.0, ** p<0.05, * p<0. : Provson rate over prevous 5 perods, whh yelds the largest log-lkelhood among to 5-perod lags. 3

25 Model s a baselne model whh estmates mehansm-spef nterepts wth mehansm dummes; varaton n slope s aptured by ndued value. Provson rate and nteraton terms among mehansms and ndued value, are added n Models to 4, of whh Model 4 s hosen for nterpretaton based on lkelhood rato tests. An ndvdual s value has a large and sgnfantly postve (p<0.00) effet on her ontrbuton, and there are no sgnfant offsettng negatve oeffents from the varous mehansms. Ths result provdes strong statstal evdene of a postve relatonshp between ndvdual ontrbuton and ndued value, whh s onsstent wth Proposton, but has not been wdely doumented aross provson pont mehansms, though Rondeau et al. (005) and Spener et al (009) fnd smlar effets n one-shot PR games, and Galmard and Palfrey (005) show a smlar result based on medan bd funtons for PPM and PR. For PPM, the result s onsstent wth related theoretal predtons by Alboth et al. (00) and Laussel and Palfrey (003). UPA has a larger nterept than the others and a sgnfantly steeper slope than PPM and PR all wth p<0.00. UPA s slope s also nomnally steeper (p=0.93) than UPC. Combned, these results ndate that UPA generates hgher ontrbutons throughout the value range. Ths result reflets the prevalene of hgh value revelaton BNE wthn UPA s equlbrum set. Wth an nterept slghtly, but not sgnfantly, lower than those for PPM and PR, UPC has a sgnfantly steeper slope than PPM (p=0.04) and PR (p=0.06), whh mples UPC elts hgher ontrbutons from hgher valued people than do PPM and PR, onsstent wth our observaton from the numeral example n Seton 3.3. We annot rejet the hypothess that PR and PPM have the same nterept and slope. These results ndeed reflet the dfferenes n the BNE sets (Fgure ): UPC, PR and PPM dffer the most at value, where UPC ndues a ontrbuton of 0, ompared to 6 n PR, and only 4 n PPM. By Wloxon rank sum test, UPC does generate hgher ontrbutons at (6.7) than PPM (5.4) and PR (5.46), both wth borderlne sgnfane (p=0.093 and p=0.09, one-taled), and no sgnfant dfferene between PPM and PR (p=0.380, one-taled). Gven the multplty of equlbra, the borderlne sgnfane suggests we refne the analyss to ompare mehansms where margnal adjustments to ontrbutons are most onsequental for subjets, n the range of the unform ap. Spefally, we look for the range of ndued values whose observed ontrbutons are wthn one standard devaton of the average 4