Working Paper. Exploring Voting Anomalies Using a Demand Revealing Random Price Voting Mechanism. WP October 2006

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1 WP October 2006 Workng Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York USA Explorng Votng Anomales Usng a Demand Revealng Random Prce Votng Mechansm Kent D. Messer, Gregory L. Poe, Cornell Unversty Danel Rondeau, Unversty of Vctora, Canada and UMR LAMETA INRA Montpeller, France Wllam D. Schulze, Cornell Unversty Chrstan Vossler, Unversty of Tennessee

2 It s the Polcy of Cornell Unversty actvely to support equalty of educatonal and employment opportunty. No person shall be dened admsson to any educatonal program or actvty or be dened employment on the bass of any legally prohbted dscrmnaton nvolvng, but not lmted to, such factors as race, color, creed, relgon, natonal or ethnc orgn, sex, age or handcap. The Unversty s commtted to the mantenance of affrmatve acton programs whch wll assure the contnuaton of such equalty of opportunty.

3 Explorng Votng Anomales Usng a Demand Revealng Random Prce Votng Mechansm Kent D. Messer, Gregory L. Poe Cornell Unversty Danel Rondeau Unversty of Vctora, Canada and UMR LAMETA INRA Montpeller, France Wllam D. Schulze Cornell Unversty Chrstan Vossler Unversty of Tennessee Abstract Recent papers show that n group decsons ndvduals have socal preferences for effcency and equty. However, the effect of socal preferences on votng, the predomnant fundng mechansm for publc goods, has not been thoroughly examned. Ths study nvestgates whether votng decsons are affected by the dstrbuton of net benefts assocated wth a proposed publc program usng a new Random Prce Votng Mechansm (RPVM). Theoretcal and econometrc analyss of expermental results presented n the paper suggest that observed dfferences from selfsh votng are caused by a concern for socal effcency, and that votng may be more effcent than prevously thought. JEL: C91, C92, D64, D72, H41 The research presented n ths paper was funded by Natonal Scence Foundaton Program n Decson, Rsk, and Management Scence. All opnons, errors, and conclusons are the sole responsblty of the authors. Messer, Poe, Rondeau, Schulze and Vossler 2006

4 1 Majorty-votng rules are used extensvely n modern democraces, by representatve legslatve bodes and n ballot ntatves (referenda), to determne the provson of publc goods. Such programs and ther fundng often mpose unequal costs and benefts on ndvduals. If voters have socal preferences, we should expect ther decsons to be nfluenced by the perceved or actual mpact of the votng outcome on others. For nstance, a strong supporter of a school bond may be worred that votng yes on the assocated tax may mpose costs on the elderly that exceed ther benefts. The elderly may worry that by votng no they hurt kds even though ther own chldren are grown. Mountng evdence from the felds of expermental and behavoral economcs suggests that, ndeed, at least a porton of ndvduals exhbts socal preferences n the form of socal welfare or equty concerns (see, for example, Fehr and Schmdt, 1999, Bolton and Ockenfels, 2000, Charness and Rabn, 2002, Engelmann and Strobel, 2004, and Fehr, Naef, and Schmdt, forthcomng). Ths study seeks to better understand the behavor of ndvduals n votng stuatons, ntroduces a new votng mechansm, detals ts theoretcal propertes and emprcal performance, and usng t, dscerns whch theory(es) of socal preferences best explan behavor. In the next secton, we ntroduce the ssues wth a set of experments that demonstrate that ndvdual decsons under a majorty-votng rule are nconsstent wth the predcton of the ratonal selfsh voter model. Two treatments are compared. In the frst, groups of three partcpants vote for or aganst a proposal that has a unform tax (cost) but provdes one of three levels of known, dfferent benefts (heterogeneous values) to each partcpant hgh, medum, and low. In the second treatment, all partcpants pay a unform tax and receve the same benefts (homogeneous values) f the program receves a majorty vote. Ths experment shows that a voter facng the prospect of a low beneft and hgh cost of provson s more lkely to vote for the

5 2 program f others stand to receve benefts that exceed the cost of the good. Smlarly, hgh beneft voters are more lkely to vote aganst a proposal that costs less than ther value f the proposal mposes a cost above value on others. In other words, there appears to be a systematc votng anomaly borne from an ndvdual s assessment of the mpact of the program on others. One way of explorng the orgns of ths observed votng behavor would be to collect a large number of yes/no (dchotomous) votes for a partcular program but at varyng mplementaton costs to trace out the demand functon and estmate the tax level at whch varous ndvduals swtch ther vote. The sample szes requred for ths approach are costprohbtve. As an alternatve, we propose n Secton 3 the Random Prce Votng Mechansm (RPVM), whch s best thought of as a generalzaton of the Becker-DeGroot-Marschak (BDM) mechansm (1964). The RPVM extends the prvate good BDM mechansm to publc goods. A proposed publc good s mplemented whenever a majorty of ndvduals ndcate a maxmum wllngness to pay (WTP) greater than or equal to a randomly selected prce. In that event, all subjects must pay the random prce. 1 Thus, n contrast to the dchotomous choce format, whch crudely bounds an ndvdual s value and only yelds a yes or no data pont, the RPVM elcts a pont estmate of the value of the good to subjects. Yet, ts coercve tax feature closely parallels real ballot ntatves. Furthermore, the RPVM s less complex than ncentve compatble publc goods fundng mechansms, such as the Smth (1979) Aucton and the Groves-Ledyard (1977) mechansm. In Secton 3, we show that the mechansm s ncentve compatble n expected utlty theory for selfsh ndvduals and develop theoretcal predctons of behavor for four alternatve types of socal preferences, and for both WTP and WTA wth both gans and losses. Votng

6 3 behavor and the emprcal propertes of the mechansm tself are explored expermentally wth a desgn descrbed n Secton 4. Expermental results are presented n Secton 5. We report four key fndngs. Frst, the RPVM elcts preferences that are consstent wth observed dchotomous (yes/no) votes. Second, we show that behavor under the new mechansm s consstent wth theory, whch predcts truthful demand revelaton n groups of one or when a program results n equal dstrbuton of benefts to all members of a group (regardless of the type of socal preferences postulated). Thrd, no evdence of a WTP/WTA dscrepancy s found. Fnally, an econometrc analyss of votng patterns leads us to conclude that voters n our experments were motvated by the appeal of ther own potental gans as well as by a concern for the overall effcency of proposed programs (consstent wth altrusm as defned by Bergstrom, 2006). These fndngs suggest that votng may be more effcent than prevously thought. The paper contrbutes to the lterature n several ways. It demonstrates expermentally the presence of behavoral anomales n dchotomous votng. It ntroduces a new publc goods mechansm and analyzes ts theoretcal propertes. The applcaton of the mechansm demonstrates ts demand revealng propertes under all four Hcksan measures of welfare change and makes t possble to dentfy econometrcally whch type of socal preferences most powerfully explan observed votng behavor. The results and ther mplcatons for publc polcy, future research, and the overall effcency of votng are further dscussed n concluson. I. Evdence of Dstrbutonal Effects on Votng Decsons A thn emprcal lterature on the determnants of votng patterns n real ballots suggests the presence of non-selfsh consderatons n the voter s decson-makng process (Mueller, 1989;

7 4 Deacon and Shapro, 1975). In ths secton, we use smple dchotomous (yes/no) votng laboratory experments to show that votng anomales exst. These experments, as well as the RPVM experments dscussed later, were conducted at the Cornell Unversty s Laboratory for Expermental Economcs & Decson Research usng students drawn from undergraduate busness and economcs classes. The frst experment employed 174 subjects to explore the effects of heterogeneous versus homogeneous values on votng (see Appendx A for experment nstructons). All subjects were placed n groups of three voters and gven an ntal endowment of $10. In each decson task, the subject was assgned one of three possble nduced values: $2, $5, or $8, whch was the amount that the ndvdual would receve f the majorty of group members vote n favor. Each partcpant made sx votng decsons where her own value, the value of other group members, and the mplementaton cost vared across decsons. Exactly one partcpant receved each of the $2, $5, and $8 values n heterogeneous value decsons. To mnmze learnng effects, only one votng outcome was bndng and ths was randomly determned after all decsons were made. In one set of sessons partcpants faced heterogeneous dstrbutons of values (Table A1 n Appendx A). In the second set of sessons partcpants faced homogeneous and heterogeneous dstrbutons of values (Table A2 n Appendx A). The order of decsons was reversed n ndvdual sessons wthn each set (see Appendx A for detals). Varatons n the level of the unform tax were set to favor detecton of non-selfsh behavor. A $7.50 tax was used to examne the behavor of voters wth an nduced value of $8, taxes of $4.50 and $5.50 were mposed on partcpants wth a $5 nduced value, whle a $2.50 tax was utlzed to examne responses from those wth a $2 nduced value.

8 5 Fgure 1 provdes a comparson of votng n homogenous and heterogeneous condtons by subjects wth a $2 nduced value facng a $2.50 tax and those wth an $8 nduced value facng a tax of $7.50. When $2 subjects were n homogenous value groups ($2,$2,$2), 5.7% of subjects voted n favor of the program compared to 18.6% n the heterogeneous value case ($2,$5,$8). Usng a t-test, ths dfference s statstcally sgnfcant at the 1% level. For $8 value subjects facng a unform tax of $7.50, 86.4% vote n favor of the program when everyone stands to gan $8 whle only 73.3% vote yes when the dstrbuton of benefts s ($2,$5,$8). Ths dfference s sgnfcant at the 3% level. In contrast, the behavor of the $5 value ndvduals s qute smlar across homogeneous and heterogeneous settngs, yeldng no statstcal dfferences at tax rates of ether $4.50 or $5.50. In partcular, for the hgher cost of $5.50, 8.2% of subjects voted yes n the homogeneous treatment, whle 12.2% votes yes n the heterogeneous treatment (p=0.505). For the lower cost of $4.50, 92.3% votes yes n the homogeneous treatment, whle 84.6% votes yes n the heterogeneous treatment (p=0.288). The detecton of statstcal dfferences n the behavor of the $2 and $8 subjects across value settngs suggests that dstrbutonal consderatons mpact the utlty of voters and ther choces. In the heterogeneous value case, a tax of $7.50 would mpose a loss of $2.50 on subjects wth a $5 value and a loss of $5.50 on subjects wth a $2 value. The results suggest that subjects are concerned that they may mpose losses on others (ether all others, the average other subject or the worst of them). Such a votng pattern s consstent wth the conjecture made by Johannesson et al (1996) n a study on the value of statstcal lfe employng votes for hypothetcal safety programs. They argued that pure altrusm (consstent wth effcency n most studes) mples that hgh value voters wll alter ther votes n consderaton of other voters who

9 6 stand to gan less (or even lose). Smlarly, the results for the $2 subjects would be consstent wth those voters gettng postve utlty from the gans of others. As Bergstrom (2006) remnds us If we are to count the sympathetc gans each obtans from the other s enjoyment of the shared publc good, then we should not forget also to count the sympathetc losses each bears from the share of ts costs pad by the others (p. 399). II. The Random Prce Votng Mechansm and Behavoral Predctons In ths secton, we formally ntroduce the Random Prce Votng Mechansm and develop theoretcal predctons of bddng behavor for t. For ease of exposton, we focus the presentaton on the case where the game s played n the WTP for gans doman, and for the case where ndvduals have socal welfare motves (Charness and Rabn, 2002). These results are readly extended to the other three Hcksan measures (WTP to avod a loss and WTA a loss or forego a gan) and for three other forms of preferences consdered n ths paper. Once the results for the socal welfare functon have been establshed, we brefly dscuss these extensons. Behavoral predctons for all permutatons are summarzed n Table 1. The mechansm proposed here combnes the ncentve compatble propertes (under expected utlty maxmzaton) of the prvate goods BDM wth a majorty votng rule. In the BDM, a person s asked to provde a sgnal of ther WTP for an object. If a randomly drawn prce s no greater than the ndvdual s sgnal, the ndvdual buys the good at the random prce. If the prce s hgher, no transacton takes place. For expected utlty maxmzers, the second prce property of the BDM mechansm elmnates the ncentves for strategc bddng, makng truthful revelaton of one s value for the object a domnant strategy. Ths s confrmed by expermental tests (Irwn et al. 1998). The tradtonal BDM mechansm, however, cannot readly be used to elct the value of a publc

10 7 good. Anyone n a group of players mght alter hs sgnal (free-rde) f the object was a nonexcludable good and only one buyer s requred for all to enjoy full benefts. Whle the RVPM mantans the second prce property, t s necessary to add an allocaton rule that prevents strategc behavor. We do ths wth a majorty rule. In addton to beng a smple and famlar approach to collectve decson-makng, laboratory experments have shown that majorty votng can be ncentve compatble (Plott and Levne, 1978) lke more general bnary choce approaches (Farquharson, 1969). A. The RPVM The RVPM works as follows: N ndvduals are asked to sgnal the maxmum amount of money they would be prepared to pay for a program defned by a known vector ( π π π ) = 1, 2,..., N. In the WTP for gans doman, j π represents the ndvdual benefts to be receved by ndvdual j f the program s mplemented. The publc program that s submtted to a vote has two components: 1) the nduced values and 2) a transfer payment (C) from ndvduals to the mplementng authorty. Ths cost (unform tax) s randomly drawn from a dstrbuton wth probablty densty p(c) over the nterval [0,C max ], after ndvduals have sgnaled ther WTP. In what follows, we refer to ndvdual s sgnal as hs bd and denote t by B. For mplementaton of the program, a majorty of ndvduals (>50%) must have expressed a WTP that exceeds a per-person cost C. If a majorty of bds are greater than or equal to C, ndvdual receves a monetary payoff π C (the sum of whch could be negatve) to be added to an ntal endowment Y for a utlty level U = u ( Y + π C). If the majorty of bds are

11 8 below C, the program s not mplemented and subjects retan ther ntal endowment for utlty U = u ( Y). We assume that U s ncreasng. B. Value Revelaton as a Domnant Strategy of Selfsh Players It can readly be argued that a self-nterested ndvdual has a weakly domnant strategy to choose the truthful sgnal B = π. The demonstraton proceeds from a standard second prce argument. As Karn and Safra (1987) and Horowtz (Forthcomng) demonstrate, the smpler BDM s not always ncentve compatble outsde of the expected utlty framework. For ths reason, we lmt our analyss to expected utlty. We denote a vector of strateges chosen by the N-1 other players by B. 2 For our purposes, however, t wll be suffcent to characterze an admssble strategy profle smply by the par of numbers ( B, m B k ). m B s defned as the ( N + 1)/2 th largest bd n the vector. In other words, the nterval [B m, B k ] defnes the range over whch the bd of voter makes ths ndvdual the medan voter. For N=3, ths s the smallest of the two bds n B ; for N=5, t s the thrd largest of four, and so forth. Bk s defned as the ( N 1)/2 th largest bd n B (for N=3, B k s the largest of the other two bds, for N=5 t s the second largest, and so forth.). Note that for odd N, B < B. m k To establsh that B = π s a weakly domnant strategy we must establsh that EU ( B = π, B ) EU ( B π, B ) B n the ( N 1)( N 1) strategy space of the other players. We proceed by consderng dfferent subsets of ths space, and for each, demonstratng that no strategy exsts for player that provdes greater utlty than B = π ( π s fxed throughout).

12 9 Case 1: π Bm(< B k ), Decreasng B In the frst subset of B strategy profles, we group all profles for whch π B m (< B k ). Frst, we consder the possblty of decreasng a bd from B = π to a smaller amount. Snce Bm π, a majorty of players can be found among the other players to fund any project wth a random cost C Bm. Therefore, a reducton n B leaves the probablty that the program wll be mplemented unchanged at B m p( C) dc. Furthermore, for any B π, condtonal utlty levels 0 are u( Y + π C) f C Bm and u ( Y) otherwse. Thus, utlty levels are also unchanged by lowerng B. The concluson s that for all losses) to be realzed by reducng one s bd below π. Case 2 π Bm(< B k ), Increasng B B such that Bm π, there are no gans (nor any Contnung wth the same subset of others strategy profles, we now analyze the desrablty of ncreasng B beyond π. Frst, we note that snce π Bm, ncreasng the bd beyond π has no consequences on s expected utlty f the new bd B % B. Ths mmcs the argument m developed n Case 1. A sgnal B B m does not affect the probablty of fundng and does not modfy the condtonal utlty levels. Nether gans nor losses are therefore realzed by ncreasng B up to and ncludng B. m An ncrease beyond B m wll, however, decrease expected utlty. For Bm < B% Bk, B becomes the threshold bd n the sense that t defnes the largest realzaton of C that leads to the mplementaton of the program. Pushng B % nto ths range ncreases the probablty that the program wll be mplemented. However, any tme a program s mplemented where C > B, we m

13 10 know by the defnng condtons of Case 2 that π C < 0. Hence, u ( Y + π C) < u ( Y). These are programs that reduce player s utlty. It follows that choosng Bm < B% Bk produces lower expected utlty than the strategy B expected utlty as the strategy B = B whch, we have already establshed, yelds the same m = π. Ths establshes that π B = s strctly superor to Bm < B% Bkaganst profles characterzed by π Bm. Fnally, consder B % > B. For all such strateges, k B s the threshold value of C leadng k to mplementaton of the program. Increases of B beyond B k leave the probablty of mplementaton unchanged at B k p( C) dc. It follows that any strategy B % > B leads to the same k 0 expected utlty as the strategy B = Bk, whch we just establshed as beng nferor to B π =. Bm By vrtue of Cases 1 and 2, we have establshed that for any strategy profle π, EU ( B = π, B ) EU ( B π, B ). Case 3: ( ) Bm < Bk < π, Increasng B B such that We now turn to the subset of B strategy profles for whch ( ) B < B < π. We frst tackle the m k possblty of player bddng B % π. Snce B < π, t follows mmedately, as above, that the k probablty that the program wll be mplemented remans unchanged for all B > B k. Once agan, snce the condtonal payoffs are also unaffected by ncreases n B, t follows that for the category of strategy profles consdered here, EU ( B > π, B ) = EU ( π, B ). Case 4: ( ) Bm < Bk < π, Decreasng B

14 11 As Case 4, we consder the possblty of reducng s bd from B = π > B k. Such a move has ether no effect on s utlty (f B % B ) or strctly reduces t (f k B % < B ). k By the same logc presented n Case 3, choosng B k B% < π has no mpact on the expected utlty of player. Ths s not the case f the bd s further reduced to B m B% < B k. Now, B % becomes the threshold that defnes the lowest level of C leadng to mplementaton of the program. As the bd decreases below B k, t lowers the probablty that the program wll be funded. Ths decrease n probablty s entrely assocated wth realzatons of C n the nterval ( B %, Bk). Snce B < π, k mplementaton of those programs would have benefted player. It follows that bddng % B (and gettng the same expected utlty as wth k B π B strategy profle characterzed by ( ) B < B < π. m k = ) domnates B m B% < B k for all Further reductons of the sgnal below B m do not reduce the probablty of fundng any further, nor does t change the condtonal utlty n the two states of the world. Thus the expected utlty for B % < B s the same as that for m B B% < B whch s n turn lower than the m k utlty for B = π. Collectng the results from Cases 3 and 4 establshes that EU( B = π, B ) EU( π, B ) for all B such that ( ) Case 5: Bm π Bk : Increasng or Decreasng B B < B < π. m k Ths last case ncludes all remanng strategy profles B not yet consdered. As we can nfer from the analyss of prevous cases, these strategy profles put player s strategy B = π n the crtcal zone where t s the determnant of the probablty that the program wll be

15 12 mplemented. Increasng or decreasng B leads to a change n the probablty of fundng the publc program. As B ncreases up to B k, the ncreased probablty of fundng s assocated wth cases where C > π, necessarly lowerng expected utlty. As before, further ncreases of B % beyond Bk do not result n further decreases n expected utlty, but do not offer any ncrease ether. It follows that the strategy B = π strctly domnates B > π % when B s such that Bm π Bk. By smlar reasonng, B = π also strctly domnates B < % π. The reducton n bd nto the regon B % B decreases the probablty that benefcal programs (wth m C < π ) wll be mplemented but offers no offsettng gan. Further decreases n B % have no addtonal effect on expected utlty. For Case 5, Bm π Bk,we conclude that EU( B = π, B ) > EU( B π, B ) for all B such that Bm π Bk. Wth cases 1 to 5, we have explored the entre strategy space of the N-1 other players, and consdered all possble devatons from the strategy B = π. Yet, under no crcumstances can departng from the strategy B = π yeld an ncrease n player s expected utlty. B π = s therefore a weakly domnant strategy for Player. Wth all players postulated to be selfsh and wth an ncreasng utlty functon, all other players also have a domnant strategy to play Bj = π j. Ths establshes that truthful revelaton by all players s a Bayesan Nash Equlbrum of the RPVM game. Whle devatng from B = π wll, on occason, not change the expected utlty of player, ths s only true when B s known and gven. Wth any uncertanty surroundng the other player s choce of strategy, all strateges B π wll necessarly have a

16 13 lower level of expected utlty snce they wll be played wth some probablty aganst a B for whch B = π domnates. In ths context, B π = s a weakly domnant strategy of the RPVM. C. Behavoral Predctons for Players wth Socal Preferences In ths secton, we look more closely at the theoretcal predctons emanatng from more explct models of ndvduals wth socal preferences. The correspondng utlty functons wll contan new arguments (the potental payoffs to others) and exhbt partculars that requre a less general soluton concept than shown n the prevous secton. We present wth some detal the soluton for ndvduals wth socal effcency preferences expressng ther WTP for a program conferrng gans. We then summarze the predctons for four alternatve models and for the remanng three welfare settngs. An ndvdual wth socal effcency preferences s postulated to have utlty that s ncreasng (decreasng) n the gans (losses) of others and of the form ( ) U = u Y + π C + α (π j C), where α 0 parameterzes the ntensty of ndvdual j altrusm (for pure selfshness, α = 0 ). Ths s a purely altrustc ndvdual who weghts equally the gans and losses to others. It can be consdered a specal case of Charness and Rabn s (2002) socal welfare functon. To compute the Bayesan Nash Equlbrum we once agan rely on the crtcal values B m and B k, the nterval defnng the range over whch the bd of voter makes ths ndvdual the medan voter. Thus, s expected utlty can be expressed as

17 14 (1) Bm EU( B, B ) = p( C) U Y + π C+ α ( π j C) dc 0 j B + p( CU ) Y+ π C+ α ( π j C) dc B j m Bk Cmax ( ) ( ) + pcu ( ) Y dc+ pcu ( ) Y dc B Bk. The frst term s the expected utlty condtonal on the randomly drawn cost beng below B m. Here, s bd s rrelevant snce there s already a majorty of voters wllng to pay more than the cost of mplementng the program. The second and thrd terms cover the nterval over whch the bd of ndvdual wll have a margnal effect on the probablty that the program s mplemented. Condtonal on C fallng n that range, B s effectvely the medan bd. The last term s the nterval over whch has no effect on the outcome snce no matter how large B s, too few ndvduals have bd hgh enough to mplement the program. In searchng for an equlbrum, we focus on affne bddng strateges. Let ndvdual conjecture that ndvduals m and k choose bds of the form (2) Bm = γ m πm + αmπ j j m and (3) Bk = γ k πk + αk π j j k where. γ k and γ m are postve (stll unknown) constants. Substtutng these expressons n Equaton 1 and maxmzng wth respect to B yelds the frst order condton: p( B) U Y + B + ( j B) = p( B) U( Y). j (4) π ( α π )

18 15 Ths equaton has a degenerate soluton at pb ( ) = 0 that can safely be gnored. The nteror soluton equates expected utlty under the two states of the world (the program s funded or not). Solvng for B, the optmal bd s gven by: (5) B π + απ j j =. 1 + ( N 1) α The optmal strategy has a form that matches the prors of ndvdual regardng the bddng strateges of m (Equaton 2) and k (Equaton 3) for γ 1 ( 1 ( N 1) α ) = +. Thus, f all N players adopt the lnear conjecture and bd ther optmum, all conjectures are smultaneously proven correct. No one has any ncentve to devate, establshng that (Equaton 5) s a Bayesan Nash Equlbrum. 3 It s useful to note that the prvate BDM s nestled n the RPVM. Settng N=1 yelds the famlar BDM result: B = π. A number of testable behavoral predctons emerge from ths soluton. 1) If π j = π j, B = π. Truthful revelaton s optmal wth equal payoffs because bddng above nduced value ncreases the probablty that the program wll be funded n the range where costs exceed everyone s benefts. Bddng below value s also sub-optmal snce t reduces the probablty that the program wll be mplemented, n the range where everyone could beneft. 2) The optmal bd s ncreasng n one s nduced value snce (6) B 1 = > 0. π 1 + ( N 1) α 3) For socal welfare preferences, an ncrease (decrease) n the sum of gans of others ncreases (decreases) s optmal bd: (7) B α = > 0. π 1 + ( N 1) α j

19 16 4) By drect extenson of Equaton 7, ndvdual wll ncrease (decrease) hs bd when movng from a homogenous dstrbuton where π = π j = π j, to a heterogeneous dstrbuton where all payoffs other than hs own are ncreased (decreased). For example, whle everyone n a ($2,$2,$2) dstrbuton s predcted to bd $2, the $2 ndvdual n a ($2,$5,$8) dstrbuton would set B > $2. The ndvdual values the hgher benefts to others and s thus prepared to ncur (n expectaton) a personal cost to ncrease the probablty that the program wll be mplemented. On the other hand, an $8 type would bd less n the ($2,$5,$8) dstrbuton than for a ($8,$8,$8) program regardless of the scenaro presented. These results are summarzed n the frst row of Table 1. WTA and Losses for A The theory can be renterpreted to descrbe the optmal bddng strategy for an ndvdual expressng hs mnmum WTA compensaton to forego gans. In ths case, C represents the randomly determned compensaton to be pad n exchange for not recevng a payoff defned by. B then denotes the smallest amount that ndvdual would accept. If a majorty of bds are less than or equal to C, compensaton C s pad but the gans are not, for a utlty level U Y + C+ α C. Otherwse, s pad and utlty s U Y + π + α π j. j j Re-dervng the optmal bddng strategy yelds exactly Equaton 5 and the same theoretcal predctons, although the vector now represents ndvdual opportunty costs of mplementng the compensaton program. An ncrease n the opportunty cost of any player mples a decrease n the socal value of the compensaton offer, and therefore ncreases the mnmum acceptable level of compensaton requred by voters. In Table 1, the smallest π s therefore the smallest absolute nduced value, be t a gan or a loss. 4

20 17 The optmal strateges for the WTA compensaton for a program that mposes a loss and for the WTP for a program that elmnates a loss also replcate Equaton 5. Alternatve Forms of Other-Regardng Preferences for B Smlar approaches can be followed to analyze the optmal bddng strategy of voters who have dfferent forms of socal preferences. Of nterest are three other utlty specfcatons that can be dentfed by the expermental data we collected. They are the Maxmn utlty (Charness and Rabn, 2002) (MM), a verson of Bolton and Ockenfels (2000) (ERC) theory of equty, and Fehr and Schmdt (1999) (FS) nequty averson preferences. For emprcal reasons dscussed later, we also produce behavoral predctons for a model where players care smultaneously about socal effcency of the program and the welfare of the poorest player (.e. Maxmn preferences). Perhaps the most ntutve approach to understandng the results of Table 1 s to focus mmedately on the predctons n the last column of Column 4. Frst, no matter what type of preferences, gans or losses, WTA or WTP, the predcted bd when all payoffs are equal (homogenous dstrbuton) s B = π for all players. In contrast, heterogeneous dstrbutons wll have vared effects under alternatve preferences. To revew these effects, consder the vector Π =(2, 5, 8) as an example. B.1 Maxmn Wth Maxmn preferences, utlty depends on one s own payoff as well as on the potental gans (losses) of the ndvdual who stands to gan the least (lose the most) from mplementng the program. Denotng the payoff for ths worst off player by π w,we wrte ths socal component nto the utlty functon by addng the term α ( π ) + to player s own w C earnngs. However, t s now necessary to dentfy the worst off player. In WTP cases, the person

21 18 who potentally gans the least (loses the most), s the one wth the smallest nduced value ( π w = $2 n our example). Ths s true regardless of whether the WTP s for a gan or to avod a loss. The predcton s that ths person wll bd exactly $2, whle others wll bd less than nduced value n order to reduce the probablty that a net loss wll be mposed on the $2 ndvdual. In WTA scenaros, the ndvdual wth a hgh absolute nduced value s the worst off and the predcton s that those wth lower absolute nduced values wll set B > π to reduce the probablty of mposng costs on the $8 ndvdual. B.2 ERC Preferences Wth ERC preferences, ndvdual nequty averson manfests tself as dsutlty when the ndvdual s payoff dffers from the mean group payoff. The socal component of utlty assumed 1 N here s α ( π C) ( π j C) N j= 1 (wrtten for a WTP gans context). Those wth a payoff exactly equal to the mean wll contnue to optmally bd B = π. However, others wll behave dfferently. In the gans doman, ndvduals wll be wllng to pay less than ther prvate value for a program that provdes addtonal ncome, and wll requre a smaller amount of compensaton to forego such gans. Ths happens because the program creates nequtes that offset part of the ndvdual s own payoff. The opposte behavor should be observed n the losses doman. Indvduals wll requre more than ther prvate value n compensaton to accept a loss and be wllng to pay more to avod a loss. B.3 FS Preferences FS preferences dffer from ERC preferences n two aspects. Frst, the averson to nequty comes from a drect comparson of one s payoff wth that of other ndvduals (rather than wth the mean). Second, FS preferences allow for dfferent valuatons of postve and negatve

22 19 dfferences between ndvdual payoffs. The functon we employ s α β Max ( π C) ( π C),0 Max ( π C) ( π C),0 j j N 1 N 1. Fehr and Schmdt postulate j j that ndvduals are less affected by dfferences n ther favor than by stuatons where they are the poor party n the comparson (a stuaton that would be characterzed by α β ). Practcally, ths mples that all ndvduals (even one wth a payoff equal to the mean) get dsutlty from a heterogeneous dstrbuton. It follows that all ndvduals n any game wth heterogeneous dstrbutons wll be wllng to pay less than ther nduced value for gans and wllng to accept less than nduced value to forego a gan. By the same logc, all ndvduals wll be wllng to pay more than nduced value to avod unequal group losses and requre greater compensaton to accept them. B.4 Combnng Socal Effcency (Pure Altrusm) and Maxmn Preferences As Charness and Rabn (2002) postulated n ther work, t s actually possble that ndvduals may have preferences that smultaneously reflect a concern for both socal effcency and care for those who stand to gan least or lose most from the mplementaton of a program. Because we wll be estmatng ths model n the emprcal secton of the paper, we ntroduce t here and n Table 1. The combned components of socal concerns are gven by ( C) + α ( π C) + β π j w j. Whle we stll obtan that ndvduals n groups of one or facng homogeneous dstrbutons optmally bd ther nduced values, combnng these two types of preferences modfes the behavoral predctons n nterestng ways. The worst off player n heterogeneous dstrbutons (the lowest absolute nduced value n WTP and largest n WTA) (qualtatvely) abandon ther Maxmn because they now care about

23 20 the fate of others whose welfare could be mproved (more) by the program. Thus, the worst off player n WTP games ncrease ther bd aboveπ w whle those n WTA games decrease ther bd. A player wth a payoff equal to the mean of the dstrbuton on the other hand, would now defntely behave lke a Maxmn. The pure altrusm model left ths player ambvalent between favorng the best off or worst off ndvduals n under the socal welfare functon assumpton. Ths ambvalence s shattered n favor of the worst off ndvdual who now has greater weght on the average player s utlty. For ndvduals at or above average payoff, the concerns for effcency and the worst off ndvdual both push the bd n the drecton of π w. However, for ndvduals wth nduced values between π w and the average payoff of the dstrbuton, the two sources of utlty are actually n conflct and weght n opposte drecton. On the one hand, concern for the worst off pulls the optmal bd toward π w to mnmze potental losses for that player. Effcency, on the other hand, calls for movng one s bd away from own payoff and n the drecton of the average payoff. Ths mples a fxed pont between π w and the average payoff defnng a value where bd s equal to nduced value. III. Expermental Desgn To test the theores outlned above, an addtonal 276 partcpants were recruted from a varety of undergraduate busness and economcs courses. Each sesson conssted of ether two WTP experments: WTP-Gans and WTP-Losses (n=138) or two WTA experments: WTA- Gans and WTP-Losses (n=138), representng all four welfare settngs. All sessons conssted of four parts; an example sesson s as follows: Part A: WTP-Losses, low-ncentve practce rounds usng the RPVM n a prvate settng where the cost and payoffs were determned for each round.

24 21 Part B: WTP-Losses, hgh-ncentve prvate and publc RPVM treatments where the treatment and cost whch resulted n earnngs were determned for one randomly selected treatment at the end of the experment. Part C: WTP-Gans, low-ncentve practce rounds usng the RPVM n a prvate settng where the cost and payoffs were determned for each round. Part D: WTP-Gans, hgh-ncentve prvate and publc RPVM treatments where the treatment and cost whch resulted n earnngs were determned for one randomly selected treatment at the end of the experment. To control for potental order effects, the order of parts was vared across sessons swtchng between ABCD and CDAB as descrbed above. Further, Part B and Part D vared the order of the treatments wth respect to the amount of the nduced values, votng group sze, and the dstrbuton of values among group members. In publc RPVM treatments, subjects were provded complete nformaton about the payoff amounts of the other subjects. To prevent order effects from potentally deteroratng socal preference behavor as s common n repeated voluntary contrbuton games (Davs and Holt, 1993), subjects submtted bds for the treatments n Part B and Part D wthout feedback. At the end of the experment one of the nne RPVM programs was mplemented from both Part B and Part D by havng the subjects draw from a bag of marked poker chps. The exchange rate for Part A and Part C was ffteen expermental dollars for one US dollar, whle the exchange rate for Part B and Part D was one expermental dollar for one US dollar. The experment lasted approxmately one and one-half hours and the average payoff was $35. Subjects receved wrtten nstructons (see the example for WTP-Gans n Appendx B) and were permtted to ask questons at the begnnng of each part of the experment. The

25 22 nstructons used language parallel to that found n publc referenda. The WTP nstructons drected each subject to vote whether to fund a program by submttng a bd that represented the hghest amount that you would pay and stll vote for the program. The WTA nstructons drected each subject to vote whether to mplement a program by submttng an offer that represents the lowest amount of compensaton that you would accept and stll vote aganst the program. Each subject was seated at an ndvdual computer equpped wth a prvacy sheld. Subjects were assgned nto votng groups of varyng sze of ether one or three. For the groups of three, the admnstrators announced the groups and asked each group member to rase ther hand so that they could be dentfed by other members of ther group. Ths ensured that subjects were aware of who was n ther votng group for all treatments. No communcaton was allowed. For smplcty, consder the WTP-Gans experment. In each treatment, subjects started wth an ntal balance of $10 and were assgned an nduced value ($1, $2, $4, $5, $6, $8 or $9). Subjects then decded how much to bd rangng from zero to the entre ntal balance. After the subjects submtted ther bds, the cost for the program was determned by usng a random numbers table wth values from zero to nne. The frst random number from the table represented the dollars amount, the second number the dmes amount, and the thrd number the pennes amount. For example, f the frst random number was a four, the second was a nne, and the thrd was a four, the determned cost would have been $4.94. Consequently, the cost was unformly dstrbuted between $0.00 and $9.99 wth dscrete ntervals of $0.01. Treatments conssted of groups of three or one partcpant where treatments wth group sze of one were dentcal to the prvate good BDM as each subject s bd consttutes a majorty. In WTP-Gan treatments, f the majorty of the bds were greater than or equal to the randomly determned cost, then the program was funded. In ths case, all of the subjects n the votng group

26 23 receved ther personal payoff amount n addton to the ntal balance, but also had to pay the determned cost. If the majorty of bds were less than the randomly determned cost, then the program was not funded. In ths case, all of the subjects n the votng group nether receved ther personal payoff amount nor pad the cost, and thus, the subjects receved only ther ntal balance. For all welfare settngs, the majorty of the publc good treatments wth heterogeneous values were conducted wth a symmetrc dstrbuton,.e. ($2,$5,$8) (93 subjects for WTP; 93 for WTA). In addton, to help dentfy the parameters of the alternatve socal welfare bd functons, sessons were conducted that had heterogeneous values wth asymmetrc dstrbutons,.e. ($4,$5,$9) and ($1,$5,$6) (45 subjects for WTP; 45 subjects for WTA). 5 For the WTP experments, n the prvate good treatments, a subject s optmal strategy was to ether submt a bd equal to her nduced value or one penny less, due to dscrete costs. For the votng groups of three, the majorty rule ntroduced a coercve tax element, because f a majorty of the group submtted bds greater than or equal to the randomly determned prce, then everyone had to pay the prce regardless of ther ndvdual bds. For the WTP-Losses experments f a majorty of the bds was less than the random cost, the program was not funded. Consequently, all group members have ther personal loss amount deducted from ther ntal balance of $10. If the majorty of bds were greater than or equal to the determned cost, the program was funded and all votng group members had to pay the determned cost from ther ntal balance of $10 but dd not have the personal loss amount deduced. For WTP-Losses, the same logc holds as the majorty rule could force a low value subject to pay a hgher cost then ther nduced value and the hgh value subject may be dened the opportunty of payng a cost lower than ther nduced value. The logc of how the vote creates

27 24 a coercve tax element for both the nduced gans and nduced losses treatments s dentcal n the WTA-Gans and WTA-Losses experments. For the WTA experments, subjects submtted offers that represented the lowest amount of compensaton they would accept where the optmal offers were ether the nduced value or one penny above t. The nduced gans and losses were the same as the WTP settng and the possble compensaton agan ranged from $0.00 to $9.99. To avod ncome effects, the ntal balance was $5 whch made the expected earnngs n the WTA settng equvalent to the WTP settng. In WTA-Gans, an offer was the lowest amount a subject would accept to vote aganst the program whch otherwse would provded the subject a gan. If the majorty of the offers were less than or equal to the random compensaton, then the program was not mplemented and all votng group members receved the compensaton n addton to ther ntal balance. If the majorty of the offers were greater than the random compensaton, the program was mplemented and the group members receved ther personal payoff amount n addton to ther ntal balance. In contrast, n WTA-Losses, an offer represented the lowest amount a subject would accept to vote n favor of the program, whch forced the subject to pay the nduced loss f funded. Therefore, f the majorty of offers were less than or equal to the random compensaton, the program was mplemented and all group members receved the compensaton and the ntal balance but had to pay ther nduced losses. If the majorty of the offers were greater than the random compensaton, the program was not mplemented and all group members kept ther ntal balance.

28 25 IV. Results Smlar to other studes usng the BDM mechansm (Boyce et al. 1992; Irwn et al. 1998), the goal of the ntal low-ncentve practce rounds was to gve subjects an opportunty to gan experence wth the mechansm before ntroducng addtonal complextes to the decson envronment. Repeated low ncentve prvate rounds provded subjects an opportunty to receve feedback on how ther bds and offers affected ther payoff. Over ten practce rounds, subjects bds/offers converged towards nduced value, startng at $0.69 above nduced value n the frst round and declnng by 70% to only $0.21 above nduced value n the tenth round. By the last practce round subjects offers/bds were statstcally ndstngushable from ther nduced values n all four welfare settngs (One Sample T-test). A. Comparng RPVM wth Dchotomous Choce Votng Drect comparsons can be made between the dchotomous choce data presented earler n Secton 2 and selected treatments from the RPVM experment, n partcular, the ($2,$2,$2), ($5,$5,$5), ($8,$8,$8) and ($2,$5,$8) value dstrbutons for the WTP-Gans settng. Frst, we fnd a close correspondence between dchotomous choce votng at a partcular cost and the number of RPVM subjects who bd at or above that same cost (and are thus ndcatng they would vote yes at ths cost). For example, 23.7% of RPVM subjects wth a $2 value ndcated that they would pay at least $2.50 for a program that had benefts that were dstrbuted heterogeneously ($2,$5,$8). Ths percentage s statstcally ndstngushable from the 18.6% of subjects who voted yes n the smlar dchotomous choce votng settng (p = 0.410). As shown n Table 2, n fact, none of the dchotomous choce votng treatments yelded results that were statstcally dfferent than the results of the RPVM.

29 26 Second, the dfferences between RPVM homogeneous and heterogeneous treatments mrror that found for dchotomous choce votng. For $2 value subjects, a statstcally dfferent and hgher percentage of subjects n heterogeneous value treatments bd at or above $2.50 (p = 0.004). Ths dfference across treatments, -11.9%, s qute smlar to the -12.9% dfference n the dchotomous choce experment. For $8 value subjects, a statstcally dfferent and lower percentage of subjects n heterogeneous value treatments bd at or above $7.50 (p = 0.003). Ths dfference across RPVM treatments, 10.8%, s qute smlar to 13.1% dfference n the dchotomous choce experment. Smlar to dchotomous choce, no dfference across RPVM treatments s found for $5 subjects at costs of $4.50 (3.2% dfference, smlar to 4.0% n dchotomous choce) (p = 0.320) or $5.50 (-4.3% dfference, smlar to -7.7% n dchotomous choce) (p = 0.320). Overall, there s a very close correspondence between dchotomous choce votng and RPVM bddng both n levels and n terms of homogeneous versus heterogeneous treatment dfferences. B. RPVM Bddng Behavor and the Nature of Socal Preferences The RPVM experments yeld 76 unque (hgh ncentve) treatments, where a treatment s defned by a specfc welfare settng (e.g. WTP-gans), the subject s nduced value, and the dstrbuton of other players values (f any). To facltate comparsons between bddng behavor and nduced values, we pool the data from all treatments and regress ndvdual bds on 76 ndcator varables to produce estmates of the average bd n each treatment. As each ndvdual produces multple observatons, we estmate robust standard errors adjusted for clusterng at the ndvdual level. Gven all decsons from the ndvdual are made wthout feedback, there are no controls for learnng behavor. Tables 3 and 4 present the treatment-specfc mean bds for the gans and loss settngs, respectvely, for both WTP and WTA. Estmates that are statstcally

30 27 dfferent than nduced value at the 5% level are talczed. Inspecton of these results suggests that behavor does not appear to exhbt WTP/WTA dscrepances. Bddng behavor for the heterogeneous value treatments suggests that socal preferences do play a role. Whereas mean bds are statstcally equal to value n 39 of the 40 treatments nvolvng prvate good or homogeneous value settngs, there are many nstances n heterogeneous treatments where mean bds are statstcally dfferent than nduced value. Overall, as llustrated n Fgure 2, low-value subjects tend to bd above value and hgh-value subjects to bd below value. As can be seen n the cumulatve dstrbutons n Fgure 3, subjects bds/offers n the heterogeneous value treatments systemcally devated from ther bds/offers n ether the prvate treatments or the publc homogeneous value treatments. As can be seen by nspectng Tables 3 and 4, n seven out of the eght treatments where the lowest-value subject has an nduced value more than a dollar less than the mddle-value subject (subjects wth nduced gans and losses of $1 and $2), subjects sgnfcantly rase ther WTP/WTA relatve to the nduced value. Lkewse, when the hghest-value subjects had an nduced value that was more than a dollar hgher than the mddle-value subject ($8 and $9 values), subjects sgnfcantly lowered ther WTP/WTA n seven of the eght treatments. When the low-value (hgh-value) subject has a value close to the mddle-value subject, statstcal dfferences between bds and nduced values are not generally observed. There s not a systematc dvergence from nduced values for mddle-value ($5) subjects. Symmetrc dstrbutons produce bds that are roughly equal wth value, although n one of four cases there s a statstcal dfference. In asymmetrc dstrbuton treatments, there s a weak tendency for mddle-value subjects to bd below value when ther value s above average (.e. the

31 28 $1, $5, $6 dstrbuton) and a weak tendency to bd average value when ther value s below average (.e. the $4, $5, $9 dstrbuton). Fnally, we nvestgate the extent to whch the socal welfare theores dscussed n Secton 3 are consstent wth observed bddng behavor usng data from publc good treatments. In partcular, we estmate the unknown parameters (.e. α and β) of the theory-specfc optmal bd functons. Estmated parameters that are statstcally dfferent than zero, wth the correct sgn, provde evdence that a partcular theory has the ablty to organze the data. Further, estmated parameters shed lght on the relatve mportance of socal versus selfsh preferences on bddng behavor. Consstent wth our prevous framework, we use a lnear regresson approach to estmate unknown parameters; to allow for heteroscedastcty and the correlaton of ndvdual-level responses, we estmate robust standard errors adjusted for clusterng at the subject level. The bd functon parameters for the two equty models are drectly estmable (mposng the constrant that the coeffcent on π equals one). However, the bd functons for the Socal Effcency, Maxmn, and the combned Effcency-Maxmn theory are nonlnear n the unknown parameter(s). Ths does not preclude lnear regresson as the bd functons can be re-wrtten as lnear n unknown parameters and our estmates of nterest recovered from these n a straghtforward fashon. For example, we can express the Maxmn bd functon as: B = π (8) δ1 π + δ 2 w δ = 1 where 1 ( 1+ ) and ( 1+ α ) α α δ 2 =. The parameter α s overdentfed. It can be easly shown that δ 2 = 1 - δ 1, and we can mpose ths restrcton drectly nto the model to resolve the dentfcaton ssue. The restrcted model s:

32 29 B π = δ ( π π ) (9) w 1 w Wth an estmate of δ 1 n hand, an estmate of α and ts standard error can be obtaned usng the delta-method. In a smlar ven, exactly dentfed specfcatons that correspond to the Effcency and Effcency-Maxmn theores can be constructed. Unfortunately, t s not possble to estmate ndvdual-specfc coeffcents from our desgn. We nstead constran the unknown parameters to be equal across ndvduals, and what we estmate are best thought of as bd functons for the representatve ndvdual n the sample. Further, for estmaton purposes we nclude an error term and an overall model constant. Although the theoretcal bd functons do not mply a constant term, whether or not one should be ncluded s essentally an emprcal queston. If the mean of the error term s not zero, for nstance, omttng the constant term would serve to dstort coeffcent estmates. Table 5 presents bd functons, estmated by poolng the entre sample as well as estmated separately for the WTP and WTA treatments. Poolng WTP (WTA) gans and loss data s justfed by statstcal tests, and data from all welfare settngs can be justfably pooled for all but the Maxmn specfcaton. The two equty-based specfcatons are not supported by the data. The parameter of the ERC model s not statstcally dfferent than zero and has the ncorrect sgn. The two parameters of the FS model are statstcally dfferent than zero. However, the result α < 0 s nconsstent wth the theory. In partcular, t suggests that ndvduals bd to ncrease dsadvantageous nequalty (.e. reduce equalty). Consstent wth the respectve theores, the estmated parameter for both the Socal Effcency and Maxmn model s postve and statstcally dfferent from zero at the 1% level. The estmate of α = n the WTP Effcency model mples that the weghts on selfnterest, 1 ( 1 + 2α ), and effcency, ( 1+ 2α ) α, are equal to 0.90 and 0.05, respectvely. Ths