Measuring the Willingness to Pay to Avoid Guilt: Estimation using Equilibrium and Stated Belief Models

Transcription

1 Measurng the Wllngness to Pay to Avod Gult: Estmaton usng Equlbrum and Stated Belef Models Charles Bellemare Alexander Sebald Martn Strobel September 14, 2009 Abstract We estmate structural models of gult averson to measure the populaton level of wllngness to pay (WTP) to avod feelng gult by lettng down another player. We compare estmates of WTP under the assumpton that hgher-order belefs are n equlbrum (.e. consstent wth the choce dstrbuton) wth models estmated usng stated belefs whch relax the equlbrum requrement. We estmate WTP n the later case by allowng stated belefs to be correlated wth gult averson, thus provdng a drect test and control for a possble (false) consensus effect. All models are estmated usng data from an experment of proposal and response conducted wth a large and representatve sample of the Dutch populaton. We fnd that equlbrum and stated belef models both suggest that responders experence sgnfcant gult averson from lettng down proposers. Responders are on average wllng to pay up to 0.80 Euro to avod lettng down proposers by 1 Euro. Moreover, estmated WTP remans postve and sgnfcant n models usng stated belefs despte sgnfcant correlaton between gult averson and belefs. Fnally, we fnd no evdence that WTP s sgnfcantly related to the observable soco-economc characterstcs of players. JEL Codes: C93, D63, D84 Keywords: Gult averson, Wllngness to pay, Equlbrum and stated belefs models. We thank the team at CentERdata for support throughout the experment. The frst author thanks the Canada Char of Research n the Economcs of Socal Polces and Human Resources for support. We also thank Martn Dufwenberg, partcpants at the Workshop on Subjectve Expectatons n Econometrc Models n Québec, as well as partcpants at conferences n Copenhagen, Hafa, Granada, and Maastrcht for helpful comments and suggestons. Département d économque, Unversté Laval, CIRPÉE, {emal: cbellemare@ecn.ulaval.ca}. Department of Economcs, Unversty of Copenhagen, {emal: alexander.sebald@econ.ku.dk} Department of Economcs, Maastrcht Unversty, {emal: m.strobel@algec.unmaas.nl}

2 1 Introducton Persstent fndngs n expermental economcs suggest that n many strategc envronments people s preferences do not only depend upon the strateges played but also on the belefs they hold about other people s ntentons and expectatons [see e.g. Falk, Fehr, and Fschbacher, 2008;, Charness and Dufwenberg, 2006]. One specfc type of belef-dependent preferences whch has receved a lot of attenton recently s gult averson [Charness and Dufwenberg, 2006; Battgall and Dufwenberg, 2007; Vanberg, 2008; Ellngsen, Johannesson, Tjøtta, and Torsvk, 2009]. In that lterature an ndvdual s defned as gult averse f he values lvng up to hs expectatons of what other ndvduals expect of hm. Not dong so causes a feelng of gult whch negatvely affects the ndvdual s utlty and thus nfluences decson makng. The am of ths paper s to estmate structural models of gult averson to measure the populaton level of wllngness to pay (WTP) to avod feelng gulty. Exstng work test for the presence of gult averson by measurng the correlaton between players decsons and ther second-order belefs: ther expectatons of what others expect of them. The estmated correlatons typcally suggest sgnfcant gult averson n student populatons (e.g. Charness and Dufwenberg, 2006). Whle such tests provde ndcatons of the relevance of gult averson, they provde lttle nformaton concernng the quanttatve mportance of gult averson relatve to self-nterest. Measurng WTP thus has the potental to provde new nsghts on the quanttatve mportance of gult averson for players. To proceed, we conducted an experment wth a large and representatve sample of the Dutch populaton. The experment was based on a smple sequental two player game of proposal and response wth two addtonal nactve players. In the man treatment (henceforth treatment S) responders made ther decsons and were then asked to state ther second-order belefs: ther expectatons of the frst-order belefs or proposers. It has recently been argued that observng a sgnfcant correlaton between responders decsons and ther stated second-order belefs does not necessarly mply gult averson (see Charness and Dufwenberg, 2006; Vanberg, 2008; Ellngsen, Johannesson, Tjøtta, and 1

3 Torsvk, 2009). The observed correlaton may nstead reflect a consensus effect whch occurs when ndvduals condton on ther behavor (and preferences) when statng ther belefs (Ross, Greene and House, 1977). 1 Ths effect has been thoroughly studed n psychology. For our smple game t means that responders stated second-order belefs are affected by ther ntended decsons rather than vce-versa. To address the possblty of a consensus effect we conducted an addtonal treatment, henceforth treatment X. In ths treatment responders where nformed of the true frst-order belefs of proposers before they made ther decsons. Hence, treatment X overcomes bases due to consensus effects by exogenously nducng second-order belefs ndependently of the preferences of responders. 2 We measure WTP n two dfferent ways. Frst, we estmate WTP combnng data from both treatments wth the second-order belefs stated n treatment S. We control for a possble bas n estmated WTP whch would result from consensus effects by allowng for correlaton between stated belefs and gult averson of players n treatment S. 3 Furthermore, combnng data from both treatments allows us to evaluate how much of the dfferences n measured gult averson across both treatments can be attrbuted to ths correlaton. Second, we estmate WTP assumng that belefs are consstent wth the relevant choce dstrbutons. Ths equlbrum approach s especally appealng for two reasons. Frst, t s frmly grounded n theory (see e.g. Harsany 1967, Battgall and Dufwenberg, 2007 and Battgall and Dufwenberg, 2009). 4 Second, the consstency requrement closes the 1 We wll call t a consensus effect although n the orgnal defnton Ross, Greene and House (1977) speak of a false consensus effect. Dawes (1989, 1990) argues that the label false s not justfed because the effect can be ratonalzed n a Bayesan framework. Engelmann and Strobel (2000) expermentally nvestgate ths ssue and found clear evdence aganst the falsty. For our purpose ths dstncton s however secondary. 2 Ellngson, Johannesson, Tjøtta, and Torsvk (2009) used a smlar method. 3 A smlar econometrc approach was followed by Bellemare, Kröger, and van Soest (2008). There, they estmate a structural model of choce under uncertanty usng ultmatum game data where belefs are allowed to be correlated wth nequty averse preferences. 4 Theoretcal models of gult averson do not necessary requre that belefs be n equlbrum to generate 2

4 model and thus crcumvents the need to collect data on (hgher-order) belefs. As a result, the equlbrum approach avods bases due to consensus effects whch arse when usng stated belefs. Obvously, one potental drawback of the equlbrum approach s that the consstency of decsons and belefs may be an overly restrctve assumpton n one shot games as players do not have any opportunty to learn about the expectatons of others. Our mans results are the followng. Frst, we fnd that WTP to avod lettng down player A s sgnfcantly hgher n treatment S than n treatment X when we do not allow for a correlaton between stated belefs and gult averson (e. no control for consensus effects). Interestngly, the measured WTP to avod lettng down player A s no longer sgnfcantly dfferent across both treatments once we allow stated belefs to be correlated wth gult averson. Ths s consstent wth a consensus effect. Quanttatvely, results from the stated belef model suggest that second movers are on average wllng to pay up to 0.80 Euro to avod lettng down player A by 1 Euro. Thrd, we fnd that the WTP to avod lettng down player A estmated usng the equlbrum model s smlar to the level of WTP predcted by the stated belef model once correlaton between gult averson and belefs s accounted for. Moreover, we do not fnd that WTP to avod lettng down any player vares sgnfcantly across varous soco-economc dmensons (age, educaton, ncome, etc.). 5 Fnally, we fnd no evdence that second movers are wllng to pay to avod lettng down nactve players. Ths result hold for both the stated and equlbrum belef models. The organzaton of the paper s as follows. In secton 2 we descrbe the game and expermental setup. In secton 3 we present our data. Secton 4 presents a model of smple gult. Secton 5 presents our econometrc model usng stated belefs whle secton 6 presents our econometrc model assumng equlbrum belefs. Secton 7 concludes. predctons about behavor. Battgall and Dufwenberg (2009) for example analyze strategc behavor n psychologcal games under the weaker requrement that belefs are ratonalzable. See ther secton 5.2 for a dscusson. 5 Recent expermental studes samplng the same populaton (Bellemare and Kröger (2007), Bellemare, Kröger, and van Soest (2008)) have on the other hand found that dstrbutonal preferences vary sgnfcantly across soco-economc dmensons. 3

5 2 The Game and the Expermental Setup The experment was done va the CentERpanel, an Internet survey panel managed by CentERdata at Tlburg Unversty. The panel conssts of about 4000 households, a representatve sample of the Dutch populaton. They are contacted weekly on Frdays and are requested to answer questons untl Sunday nght. Most of these questons are survey questons about household decsons but CentERdata also allows for smple nteractve experments. 6 Our experment s based on the followng game: Player A L R Player B x A (R) l r x B (R) y A (l) y A (r) x C (R) y B (l) y C (l) y D (l) y B (r) y C (r) y D (r) x D (R) In ths smple sequental game, there are four players: A, B, C and D. Player A can choose ether the outsde opton R or he can choose L to let player B decde. If player A chooses R then the game ends and the players receve ther payoffs x A (R), x B (R), x C (R) and x D (R), respectvely. If player A decdes to choose L then player B has to choose ether l or r. In both cases the game ends and the players receve ther correspondng payoffs, ether y A (l), y B (l), y C (l) and y D (l), respectvely or y A (r), y B (r), y C (r) and y D (r), respectvely. 6 For more detals and a descrpton of the recrutment, samplng methods, and past usages of the CentERpanel see: Computer screens from the orgnal experment (n Dutch) wth translatons are avalable upon request. 4

6 Players C and D are dummy players whose monetary payoffs are determned by the choces of player A and (possbly) B. 7 We ncluded C and D players to analyze how B s decson s affected by the presence of strategcally unnvolved players. The exstng lterature (e.g. Güth and Van Damme, 1998; Kagel and Wolfe, 2001) ndcates that the presence of one nactve player has a weak nfluence of behavor n smple games. Here, we use two nactve players n-order to make ther presence n the game more salent. Payoffs were systematcally vared across games wth the help of Optmal Desgn Theory (see Mueller and Ponce de Leon, 1996). Payoffs were presented n CentERponts - the currency that s usually used n experments conducted wth the CentERpanel. In total we nvted 3000 panel members to partcpate for both treatments. From all nvted partcpant 1962 responded and went through the whole experment. We next descrbe both treatments of our experment n detal. Treatment S Treatment S was conducted at the begnnng of We nvted 2000 CentERpanel members to partcpate n ths treatment out of the 2000 nvted panel members responded to the nvtaton by readng the openng screens of the experment. They were provded wth a descrpton of the game, the possble choces that players n the dfferent roles could make and ther assocated consequences. Before the revelaton of ther roles and monetary payoffs, members were gven the chance to resgn from the experment. 264 members resgned at ths stage, leavng us wth 1402 members who where then randomly assgned to a specfc game and to one of the four dfferent roles A, B, C and D. Followng the nformaton about ther role and ther game s payoffs, partcpants were asked to make ther choces. We used the strategy method (see Selten 1967). Ths means that A- and B-players made ther choces smultaneously whle B-players knew that ther decson was condtonal on A not choosng out. Ths helped us overcome the problems of 7 Our game s smlar to that analyzed by Charness and Rabn (2005) wth the dfference that we nclude the dummy players C and D. Furthermore, dfferent to them, we dd not ask players A to reveal ther expectatons about the possble choces of player B. 5

7 coordnatng nteractons n real tme va the panel. After makng ther decson, each A-player was asked to state ther frst-order belefs concernng the behavor of player B f they chose to let ths player decde the fnal allocaton. In partcular, A-players were presented the followng queston (Frst-order belefs of A-players) What do you thnk, how many B-Persons out of 100 wll choose l and how many r. Please ndcate ths number for each possble allocaton. 1. Number of B Persons out of 100 that wll choose B.1 : X A 2. Number of B Persons out of 100 that wll choose B.2 : Y A The computer program automatcally ensured that the numbers entered (X A +Y A ) added up to 100. To smplfy the task of partcpants, all belefs were elcted usng natural frequences. 8 After ther decsons (l or r), B-players were asked to state ther second-order belefs. In partcular, they were asked to answer the followng queston: (Second-order belefs of B-players) What do you thnk about Person A s belefs about the behavor of Persons B? Please ndcate ths number for each possble allocaton. 1. Person A beleves that X B B-Persons out of 100 choose B.1 2. Person A beleves that Y B B Persons out of 100 choose B.2 Agan, the computer program automatcally ensured that the numbers X B + Y B added up to 100. The decsons of A- and B-players were matched after the experment to determne the fnal payoff of players A, B, C and D. Before the experment partcpants were nformed that we expect at most 2000 persons to partcpate and that after the experment 50 8 Ths follows Hoffrage, Lndsey, Hertwg, and Ggerenzer (2000) who found that people are better at workng wth natural frequences than wth percent probabltes. 6

8 played games (50 players of each role) would be pad off. 9 In order to ncrease the number of B-player decsons whch were most nterestng for us, we put more persons nto the role of B than nto the other roles. More specfcally, we had prepared 1600 payoff-wse dfferent games for treatment S. Gven these 1600 games, we decded a-pror to randomly allocate each of our ntal 2000 nvted panel members to one of the four roles n the followng proportons: 1600 B-player roles (one for each game), 300 A-players, 50 C- players, and 50 D-players. We randomly pcked 50 out of the 300 games consstng wth A- and a B-players to whch we assgned C and D players. Ths means, we a-pror randomly pcked 50 payoff-wse dfferent games (out of 1600) wth A-, B-, C- and D- players whch were pad off after the experment. In the begnnng of the experment partcpants were then randomly allocated to a specfc role and a game ensurng that a-pror everybody had an equal chance to be n a game whch was pad off at the end (for detals see also the translated screens of the experment n the appendx). As announced before the experment, partcpants of the games that were pad out receved nformaton on the outcome of ther game and ther fnal payoffs a few weeks after the experment. Furthermore, the correspondng amounts were credted to ther bank accounts. Of the 1402 partcpants that completed the experment there were 1114 B-players, 214 A-players and 74 C- and D-players. 10 Treatment X Treatment X was conducted durng the summer of For ths treatment, we () selected all 214 games n treatment S wth decsons and stated frst-order belefs of A-players, () we re-contacted the A-, C- and D-players who had played these specfc games and asked 9 The experment was conducted usng CentERponts, the usual currency for CentERpanel members. For the sake of smplcty we state drectly the amounts n Euro. The exchange-rate was 100 CentERponts = 1 e. 10 Table 1 presents data from treatment S. As can be seen, the sample sze of treatment S s N= represents the number of B-players (out of the 1114) for whom we had a complete record of background characterstcs. 7

9 them whether we could use ther decsons and belefs (f any) for a follow-up experment and () nvted 1000 new members of the CentERpanel to partcpate n the experment. 719 out of the 1000 nvted panel members responded to the nvtaton by readng the openng screens of the experment. As n treatment S, they were gven the chance to resgn from the experment after the structure of the game was explaned but before they learned ther role and the detaled payoffs. 159 members resgned at ths stage, leavng us wth 560 members who where then all assgned to the role of player B and confronted wth ther specfc game. 11 In contrast to treatment S, the B-players n treatment X were not asked for ther second-order belefs but were presented the frst-order belefs of ther matched A-player (taken from treatment S) before makng ther decsons. All other features of the treatment are otherwse dentcal to treatment S. Smlar to treatment S we nformed partcpants before the game that 25 games played were gong to be randomly selected and pad out. As before the subjects receved nformaton about the decsons a few weeks later and for the players of the selected games ncludng A-, C- and D-players the correspondng amounts were credted to ther bank account. 3 Data Table 1 presents the sample means and standard devatons of the allocatons to A-, B-, C-, and D-players at the three end knots of the game. [Insert Table 1 here] The average allocaton ranges between 20 and 25 Euros per player dependng on the role and the termnal node. Frst-order belefs of A players were elcted n treatment S and are provded to B- players n treatment X. We analyze the frst-order belefs of A players n treatment S by 11 Hence the 214 games were used on average more than twce. Table 1 presents data from treatment X. The sample sze of treatment X s N=540. Analogous to treatment S, 540 represents the number of B-players (out of the 560) for whom we had a complete record of background characterstcs. 8

10 estmatng the followng lnear regresson b A = α 0 + α 1 y A + α 2 y B + α 3 y C + α 4 y D + ɛ (1) where b A denotes the probablty placed by player A on player B playng r (frst-order belefs of player A), and where y k = y k (r) y k (l) denotes the payoff dfference when player B chooses r relatve to l for player k {A, B, C, D}. The estmated equaton s the followng (wth standard errors n parenthess) ba = (0.019) y A (0.001) y B (0.001) (0.001) yc (0.000) y D We fnd that A-players expect that B-players are more lkely to chose r when B-player payoffs from dong so ncrease relatve to payoffs from choosng l. Interestngly, frst-order belefs do not vary sgnfcantly wth payoffs of A-, C-, and D-players. Ths suggests that A-players do not expect that B-players wll take nto account the well beng of other players when makng ther decsons. 4 A model of smple gult averson In ths secton, we specfy a structural econometrc model of gult verson. Our startng pont s the model of smple gult proposed by Battgall and Dufwenberg (2007). 12 We start by assumng that a B-player s utlty of playng r s gven by U (r) = y B (r) + φ A G A (r) + φ CD G CD (r) (2) where y B (r) denotes hs payoff, G A (r) denotes gult towards player A (condtonal on player A s belefs), and where G CD (r) denotes gult towards players (C, D) (condtonal on players C and D s belefs). Player B s utlty of choosng l s defned analogously by replacng r for l and s omtted for brevty. 12 Note, Battgall and Dufwenberg (2007) also present an extended model of gult from blame whch assumes that a player cares about others nferences regardng the extent to whch he s wllng to let down. 9

11 φ CD The parameter φ A controls player B s senstvty to gult towards player A. Smlarly, controls player B s senstvty to gult towards players (C, D). Note, as margnal utlty of own ncome y B s normalzed to 1, the (absolute) values of φ A and φ CD represent player B s wllngness to pay to avod respectvely lettng down A-players and C, D-players by 1 CentERpont. The gult varables from choosng r are defned as where E ( Y A also G A (r) = [ E ( ) Y A y A (r) ] 1 [ y A (r) < y A (l) ] (3) G CD (r) = [ E ( ) Y CD y CD (r) ] 1 [ y CD (r) < y CD (l) ] (4) ) denotes the expected payoff of player A, where y CD (n) y C (n)+y D (n) for ) denotes the expectaton of the sum of payoffs of players n {l, r}, and where E ( Y CD C and D. 13 These expectatons are gven by E ( ) Y A = b A y A (r) + (1 b A )y A (l) (5) = b A [ y A (r) y A (l) ] + y A (l) where b A E ( Y CD ) = b CD = b CD y CD [ y CD (r) + (1 b CD )y CD (l) (6) (r) y CD (l) ] + y CD (l) denotes player A s subjectve belef that player B wll play r, whle b CD denotes players C and D s subjectve belef that player B wll play r. Player B lets down player A by choosng r f ths provdes player A wth a fnal payoff y A (r) below hs expectaton. Smlarly, player B lets down players C and D by choosng r f ths provdes these players wth a fnal payoff y CD (r) below ther expectaton. Hence, we assume that a player cares about the extent to whch he lets other players down, where G A (r) and G CD (r) measure the amount of let down from choosng r. From (2), (3), and (4) t also follows that player can only let down player A (or players CD) by choosng the alternatve provdng A (or players CD) wth hs lowest payoff We also estmated a model allowng separate gult from lettng players C and D. The results are essentally dentcal to those obtaned by groupng players C and D together and led to no sgnfcant ncrease n the log-lkelhood functon. 14 For example, f y A(r) < ya (l), then GA (r) > 0 and GA (l) = 0. 10

12 So far, the analyss has assumed that player B knows b A B forms expectatons (hs second-order belefs) b A = E(b A ) and b CD and b CD. In realty, player = E(b CD ) over the possble values of the frst-order belefs of the other players. Player B s expected utlty E(U (r)) (condtonal on the game) can be derved by replacng b A n (5) wth E(b A ) and b CD n (5) wth E(b CD ). The expectaton E(U (l)) s derved analogously. 5 Estmaton usng stated belefs In ths secton we estmate the model of the prevous secton usng stated second-order belefs. Our estmaton framework explctly deals wth the possble correlaton between stated belefs and gult averson whch would arse n the presence of a consensus effect. In our model, the exstence of a consensus effect mples that B-players wth gult averson (.e. hgher values of φ A ) state second-order belefs b A (r) resultng n hgher mpled levels of G A ( ) of the relevant alternatve. We estmate our stated belef model combnng data from both treatments. Ths allows us to asses how much of the dfferences n estmated φ A across both treatments s attrbutable to the possble correlaton between stated belefs and gult averson n treatment S. To proceed, we assume that the senstvty to gult towards player A s gven by φ A = φ A + γd + u φa (7) where u φa s a normally dstrbuted dosyncratc component of gult averson wth mean zero and varance σ 2 φ. D denotes a dummy varable takng a value of 1 for players n treatment X, and 0 otherwse. Ths varable captures dfferences of φ across both treatments whch are not accounted for by the model. 15 We next model stated second-order belefs b A n treatment S. Snce reported probabltes may well be zero or one, we allow for censorng at 0 and 1, as n a two-lmt tobt model. In partcular, we model the stated second-order belefs as: 15 We also estmated a model where we allowed φ A to depend on observable characterstcs of players (age, gender, educaton, and ncome). We faled to fnd any sgnfcant ncrease n the model log-lkelhood. Results are avalable upon request. 11

13 b A (r) = x δ ρu φa 1[y A (r) < y A (l)] + ρu φa 1[y A (r) > y A (l)] + u b b A = 0 f b A < 0 = b A f 0 < b A < 1 = 1 f b A > 1 where u b denotes a mean zero normally dstrbuted random varable wth varance σ 2 b, and x denotes a vector of payoffs characterzng the game. Note, the model above allows the unobserved part of gult averson u φa to affect the stated belefs n a manner whch s consstent wth the consensus hypothess when ρ > 0. To see ths, consder frst games where playng rght provdes gult to player B, that s games such that y A (r) < y A (l). Recall that there s no gult from playng left n ths case. Then t follows from (5) that B-players wth relatvely hgher gult averson (hgher values of u φa ) are more lkely to thnk that player A expects that a lower proporton of B players wll choose r. Hence, lower values of b A wll be stated whch (from (3) and (5) ) results n hgher gult G A (r) from choosng r. Next consder games where playng left provdes gult to player B, that s games such that y A (r) > y A (l). Recall that there s no gult from playng rght n ths case. Then t follows from (5) that B players wth relatvely hgher gult averson (hgher values of u φa ) are more lkely to thnk that player A expects that a hgher proporton of B players wll choose r. Hence, hgher values of b A gult G A (l) from choosng l. wll be stated whch results n hgher The prevous dscusson mples that any postve correlaton between second-order belefs and gult averson may lead to an overstatement of the mportance of gult averson. A formal test of the correlaton between gult averson and belefs can be performed by testng the null hypothess ρ = 0 aganst the alternatve ρ > 0. In the event that ρ > 0, a value of γ sgnfcantly dfferent from zero would suggest that accountng for correlaton between stated belefs and gult averson s not suffcent to explan the behavoral dfferences across both treatments. As second-order belefs of B-players concernng C- and D-players were not elcted, t wll not be possble to estmate φ CD. However, t s possble to control for the effect of 12

14 gult towards nactve players when estmatng φ A. To do so, we replace (6) nto (4) and (4) nto (2). Takng expectatons over b A we get an expresson of the expected utlty of player B from choosng r E(U (r)) = y B (r) + φ A G A (r) (8) +φ CD (1 b CD )(y CD (l) y CD (r))1 [ y CD (r) < y CD (l) ] where G A (r) s now evaluated at b A. Note from (8) that gult towards nactve players s a functon of a known varable (y CD (l) y CD parameter φ CD (1 b CD ) whch can be estmated. 16 (r))1 [ y CD (r) < y CD (l) ] and an unknown Fnally, we assume that player B has prvate nformaton about a part of hs utlty of choosng left and of choosng rght. We model ths by addng λε r to E(U (r)) n (8) and λε l to E(U (l)) (not presented), where λ denotes a scale parameter. We assume that the unobserved prvate utltes ε n for n {l, r} are..d across players and choces and follow a type 1 extreme value dstrbuton. The model s estmated usng full nformaton maxmum smulated lkelhood. 17 We estmated a restrcted and unrestrcted verson of the model wth stated belefs. The restrcted model was estmated settng ρ = 0, thus mposng ndependence between stated belefs and gult averson. Our unrestrcted verson of the model conssted of estmatng all parameters ncludng ρ, thus allowng for a correlaton between gult averson and stated belefs. [Insert Table 2 here] Table 2 presents the results of the restrcted and unrestrcted versons of the model usng stated belefs. We dscuss frst the results of the restrcted model. We fnd that the 16 Estmatng φ CD (1 b CD ) as a sngle parameter mplctly assumes that φ CD across. We also expermented wth a random coeffcent specfcaton allowng φ CD (1 b CD ) does not vary (1 b CD ) to vary across. Ths dd not lead to a sgnfcaton ncrease n the log-lkelhood functon value. We thus report pont estmates of φ CD (1 b CD ). 17 Detals concernng the log-lkelhood functon and computaton can be found n the appendx of the paper. 13

15 estmate of φ A s and sgnfcant, ndcatng sgnfcant gult averson n treatment S. The estmated magntude of φ A s surprsngly large. It suggests that B players are on average wllng to pay up to Euros to avod lettng down A players by 1 Euro. As argued before the estmated value of φ A n the restrcted model could be based downward by the presence of a consensus effect. Evdence of such a bas s provded by the postve and sgnfcant estmate of γ. The later result suggests that estmated gult averson n treatment X s sgnfcantly weaker than that of treatment S. Nonetheless, the estmated level of gult averson n treatment X s sgnfcant. 18 The estmated value of φ CD (1 b CD ) s negatve and nsgnfcant, suggestng weak gult averson from lettng down nactve players. The estmated varance of u φa parameter s not well dentfed n the restrcted model. s small and nsgnfcant, ndcatng that ths Concernng the parameters n the belef equatons, we fnd that B-players payoffs have a sgnfcant effect on stated belefs and are of the predcted sgn: B-players state hgher probabltes of choosng r when ther payoffs of playng rght y B (r) s hgher, and lower probabltes when ther payoffs of playng left y B (l) s hgher. We also fnd that B-players state sgnfcantly hgher probabltes b A choosng r ncreases. of choosng r when the payoff of player A when We next dscuss results of the unrestrcted model. Frst, note that the estmate of ρ s postve and sgnfcant, ndcatng a sgnfcant postve correlaton between gult averson and stated belefs. As we dscussed above, a postve and sgnfcant estmate of ρ s consstent wth the consensus hypothess. Allowng for ths correlaton has an mportant mpact on our man model estmates. In partcular, the estmated value of φ A remans negatve and sgnfcant. Interestngly, the estmated level of gult averson n treatment S s now , almost half the estmated magntude n the restrcted model. Ths suggests that B-players are now on average wllng to pay up to Euros to avod lettng down A players by 1 Euro. Furthermore, the estmated value of γ s no longer sgnfcantly dfferent from zero once correlaton between gult averson and belefs 18 A ch-square test of the null hypothess that φ A + γ = 0 aganst the alternatve φ A + γ < 0 s rejected at conventonal levels (p-value = 0.033). 14

16 s accounted for. Ths ndcates that the correlaton between gult averson and belefs accounts for most of the dfferences n measured WTP across both treatments. Together these results ndcate that gnorng the correlaton between the senstvty to gult and stated belefs n treatment S leads to a substantal bas of the estmated level of gult averson. Concernng gult towards the nactve players, the estmated value of φ CD (1 b CD ) remans negatve and nsgnfcant, suggestng agan weak gult averson from lettng down players C and D. Fnally, the estmated parameters of the belef equaton n the unrestrcted model are smlar to those of the restrcted model. In partcular, B-players state hgher probabltes of choosng r when ther payoffs of playng rght y B (r) s hgher, and lower probabltes when ther payoffs of playng left y B (l) s hgher. We also fnd that B-players state sgnfcantly hgher probabltes b A of choosng r when the payoff of player A when choosng r ncreases. Hence, t seems that B-players thnk that A players wll expect them to take nto account ther well beng when makng ther decsons. 6 Estmaton assumng equlbrum belefs In ths secton we estmate WTP to avod gult under the assumpton that second-order belefs are n equlbrum. We do so usng only data from treatment S. Estmaton of an equlbrum model usng data from treatment S s reasonable gven that B-players made ther decsons n that treatment before knowng that they later had to state ther second-order belefs. As a result, decsons n treatment S could not have been nfluenced by the belefs elctaton procedure. We exclude data from treatment X at ths pont snce each B player n that treatment was provded the frst-order belefs of player A before makng hs decson. As these frst-order belefs were not restrcted to be consstent wth the choce dstrbutons, mposng consstency for estmaton of the model parameters n treatment X would almost surely result n a msspecfed model. To estmate the equlbrum model, we use the followng specfcatons of φ A and φ CD 15

17 φ A = φ A + u φa (9) φ CD = φ CD + u φcd (10) where the elements of (9) have been defned prevously n (7), φ CD denotes the mean of φ CD, and where u φcd s a normally dstrbuted dosyncratc component wth mean zero and varance σ 2 φ.19 Contrary to (7), (9) and (10) do not nclude the treatment dummy D as data from treatment X s not used n the estmaton. Under these assumptons, the probablty p (r) that player B wll play r n a gven game gven belefs (b A, b CD ) s gven by p (r) = exp (E(U (r))/λ) exp(e(u (r))/λ) + exp(e(u (l))/λ) ha (u φa where the ntegraton s taken over the dstrbutons of u φa s gven n (8). )h CD (u φcd and u φcd )du φa du φcd (11) and where E(U (r)) To close the model, we assume that belefs of B-players are consstent wth the choce dstrbuton. Ths restrcton mplctly suggests the followng assumptons on the nformaton sets of the players n the game. Frst, we assume that A, C, and D players know the dstrbutons of φ A φ CD and φ CD. They do not know however the exact values of φ A and of the B-player they are matched wth. Second, A, C, and D-players do not know the prvate component ε (n) of the B-player they are matched wth, but they know ther populaton dstrbutons. All other elements of the utlty functon are assumed to be known. Hence, A, C and D players can use ths nformaton to derve ther frst-order belefs concernng the behavor of player B. These frst-order belefs have two characterstcs. Frst, they are dentcal across players (b A = b CD ) gven all players share the same nformaton set. Second, frst-order belefs wll concde wth the observed dstrbuton p (r) gven n (11). Fnally, B-players are assumed to know all ths,.e. they know what 19 Hence we assume that the varances of u φa and u φcd are dentcal. Allowng these varances to dffer does not produce sgnfcant ncreases n the log-lkelhood functon value (p-value = 0.912). 16

18 A, C, and D-players can nfer. Hence, they algn ther second-order belefs wth the frstorder belefs of other players. Ths mples that the followng equlbrum restrctons are assumed to hold b A = b CD = p (r) for all = 1, 2,..., N (12) Note that the equlbrum restrctons mply that φ CD the stated belef approach where only the product φ CD can be dentfed. Ths dffers from (1 b CD ) s dentfed. Identfcaton of φ CD follows from (8) and the equlbrum restrctons (12) whch provde dentfcaton of b CD. To estmate our equlbrum model, let d (r) denote a bnary decson varable takng a value of 1 when player {1, 2,..., N} chooses r, and 0 otherwse. The model loglkelhood s gven by Q N (θ) = 1 N N log [d (r) p (r) + (1 d (r)) (1 p (r))] (13) =1 where θ denotes the vector of model parameters. Estmaton of θ s done teratvely. In partcular, for a gven value of θ, t s smple to solve for the fxed pont p (r) for each player. Gven these fxed ponts, we then update θ to maxmze (13) gven the games { (y A (l), y A (r), y B (l), y B (r), y CD (l), y CD (r)) : = 1, 2,..., N } As a result, the fxed ponts are updated teratvely wth each new value of θ untl equaton (13) s maxmzed. Estmates of the equlbrum model are gven n the last column of Table 2. We fnd that the estmated value of φ A s and sgnfcantly dfferent from zero. Interestngly, the estmated value of φ A n the equlbrum model s smlar to the correspondng value estmated n the unrestrcted verson of the stated belef model. Furthermore, the estmated gult averson towards the nactve players φ CD s small and nsgnfcant. Ths parallels our fndngs usng the stated belef model and ndcates that we do not loose much by excludng gult towards nactve players. Ths result s n lne wth earler expermental research documentng the nsenstvty towards nactve players (see e.g. Güth 17

19 and van Damme (1998), Kagel and Wolfe (2001)). Fnally, we fnd that σφ 2 s postve but mprecsely measured suggestng that gult averson does not vary sgnfcantly across the populaton. 7 Concluson Ths paper has focused on estmatng the populaton level of WTP to avod gult usng equlbrum and stated belef models of gult averson. Our applcaton focused on a smple game of proposal and response played by a large and representatve sample of the Dutch populaton. Results from both equlbrum and stated belefs models provde the same nsght: responders have a sgnfcant WTP to avod gult. In lne wth the consensus hypothess, we found a sgnfcant correlaton between stated belefs and gult averson n the stated belef model. We also found that ths correlaton had an mportant mpact on the measured level of WTP. In partcular, our estmates ndcate that the estmated WTP n the stated belef model can be exaggerated by a factor close to 2 f consensus effects are not taken nto account. Interestngly, the estmated WTP n the equlbrum model s close to the estmated WTP n the stated belef model. We nterpret ths fndng as an ndcaton that the equlbrum model provdes a good frst approxmaton of the level of WTP n the populaton even n one shot games. Future research s needed to nvestgate whether ths result apples to more general models ncorporatng second-order belefs (see Dufwenberg and Krchsteger, 2004). Overall, our estmates suggest that B-players are on average wllng to pay up to 0.80 Euros to avod lettng down A players by 1 Euro. On the other hand, we fal to fnd that players are wllng to pay to avod lettng down nactve players. Ths result holds both for the equlbrum and stated belef models. Fnally, our expermental desgn shares mportant smlartes wth the one used by Ellngsen, Johannesson, Torsvk and Tjøtta (2009). Nevertheless, our results ndcate that sgnfcant gult averson remans after controllng for consensus effects. An nterestng 18

20 drecton for future research s to examne the factors whch can explan ths dfference. Soco-economc and cultural dfferences across subject pools are n prncple possble explanatons. Yet, we found no evdence that gult averson vares sgnfcantly across socoeconomc dmensons (e.g. age, educaton, ncome) whch dstngush our representatve subject pool from student subject pools. Ths suggests that cultural (or other unobservable) characterstcs can possbly account for the dfferences n measured gult averson across both populatons. 19

21 A Techncal appendx We present here the log-lkelhood functon of the model wth stated belefs. We observe for each player n treatment S a choce and a stated belef. Let c {l, r} denote the choce of player, and let b A denote hs stated second-order belef concernng the choce of playng r. Fnally, defne x = {(y j (r), yj (l)) : j {A, B, CD}} as the relevant payoff vector for player. Gven our model assumptons, t follows that condtonal on u φa, the lkelhood of ( ) observng c, b A s the product of the condtonal choce and belef lkelhoods ( ) ( ) L(c, b A x, u φa ) = 1 [c = l] Pr c = l x, u φa F b A x, u φa ( ) ( ) +1 [c = r] Pr c = r x, u φa F b A x, u φa where ( ) Pr c = r x, u φa ( ) Pr c = l x, u φa exp (E(U (r))/λ) = exp (E(U (r))/λ) + exp (E(U (l))/λ) ( ) = 1 Pr c = r x, u φa and ( ) F b A x, u φa = Φ = f = Φ ( ) x δ+ρuφa 1[y A(r)<yA (l) ] ρu φa 1[y A(r)>yA (l) ] σ b ( b A x δ+ρuφa 1[y A(r)<yA (l) ] ρu φa 1[y A(r)>yA (l) ], f b A = 0 σ b ) /σ b, f 0 < b A < 1 ( ) 1 x δ+ρuφa 1[y A(r)<yA (l) ] ρu φa 1[y A(r)>yA (l) ] σ b, f b A = 1, where Φ ( ) and f ( ) denote respectvely the standard normal cumulatve and densty functons. The lkelhood contrbuton of player s obtaned by ntegratng out over the dstrbuton of u φa L(c, b A x ) = ( L(c, b A x, u φa )h u φa ) du φa (14) 20

22 where h ( ) denotes the normal densty functon wth mean zero and varance σ 2 φ. For players n the treatment X, belefs are assumed exogenous. Hence, ther lkelhood contrbuton s smply ther condtonal choce probablty L(c x ) = = ( L(c x, u φa )h [ ( 1 [c = l] Pr u φa ) du φa c = l x, u φa (15) ) ( )] ( ) + 1 [c = r] Pr c = r x, u φa h u φa du φa The sample log-lkelhood s gven by 1 N N =1 ( ( ) ) log L(c, b A x ) T + log (L(c x )) [1 T ] where T s a dummy varable takng the value of 1 when player took part n treatment X, and 0 otherwse. Gven no closed form soluton exsts to ths ntegrals n (14) and (15), a numercal approxmaton must be performed. In the paper, we approxmate the lkelhood contrbuton by smulaton. In partcular, we approxmate (14) and (15) usng the followng smulators { where ( h u φa L(c, b A x ) = 1 R L(c x ) = 1 R R r=1 R r=1 L(c, b A x, u φa,r ) L(c x, u φa,r ) } u φa,r : r = 1,..., R denotes a sequence of R draws taken from the dstrbuton ). Sequences are randomly drawn for each of the N players n the experment. We use Halton draws to lower the smulaton nose of the estmator (see Tran (2003) for detals). 21

23 Stated belefs - Treatment S Exogenous belefs - Treatment X x y l y r x y l y r Player A (9.978) (16.750) (16.416) (9.900) (16.778) (16.491) Player B (7.806) (17.703) (17.138) (8.022) (17.574) (16.964) Player C (2.194) (16.393) (16.120) (2.039) (16.722) (16.780) Player D (2.194) (16.683) (16.768) (2.039) (15.826) (16.717) Table 1: Sample mean and standard devatons of the allocatons across players n treatments S (N = 1078) and X (N = 540). Entres are measured n Euros. 22

24 Stated belefs Equlbrum belefs Restrcted (ρ = 0) Unrestrcted ( ρ = ) Preference parameters φ A ** ** *** (0.217) (0.312) (0.167) φ CD (see note) (0.078) (0.080) (0.205) γ 0.870*** (0.288) (0.411) λ 3.360*** 3.022*** 3.138*** (0.258) (0.238) (0.087) σφ ** (0.111) (2.351) (1.613) Belef parameters y A (r) 0.012** 0.013** (0.005) (0.005) y A (l) *** (0.005) (0.005) y B (r) 0.071*** 0.067*** (0.005) (0.005) y B (l) *** *** (0.005) (0.005) x A (0.001) (0.001) σb *** 0.054*** (0.003) (0.004) Constant 0.491*** 0.484*** (0.038) (0.035) Log-lkelhood Table 2: Estmated parameters of the stated belef model usng data from treatments S and X. Asymptotc standard errors are n parenthess. Estmates for the stated belef model presented under the headng φ CD correspond to estmates of φ CD (1 b CD ). See secton 5 for detals. *, **, *** denote sgnfcance at the 10%, 5%, and 1% level respectvely. Estmates are based on 1078 and 540 B-players n treatments S and X. 23

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