Dscusson Paper No. 07-034 Inequty Averson and Indvdual Behavor n Publc Good Games: An Expermental Investgaton Astrd Dannenberg, Thomas Rechmann, Bodo Sturm, and Carsten Vogt
Dscusson Paper No. 07-034 Inequty Averson and Indvdual Behavor n Publc Good Games: An Expermental Investgaton Astrd Dannenberg, Thomas Rechmann, Bodo Sturm, and Carsten Vogt Download ths ZEW Dscusson Paper from our ftp server: ftp://ftp.zew.de/pub/zew-docs/dp/dp07034.pdf De Dscusson Papers denen ener möglchst schnellen Verbretung von neueren Forschungsarbeten des ZEW. De Beträge legen n allenger Verantwortung der Autoren und stellen ncht notwendgerwese de Menung des ZEW dar. Dscusson Papers are ntended to make results of ZEW research promptly avalable to other economsts n order to encourage dscusson and suggestons for revsons. The authors are solely responsble for the contents whch do not necessarly represent the opnon of the ZEW.
Non-Techncal Summary There s a growng number of stylzed facts whch contradct the model of ratonal payoff maxmzng actors n economcs. Indvdual contrbutons to publc goods such as clmate protecton projects are promnent examples for behavor not n lne wth standard economc theory. In those cases, people cooperate although t seems not ratonal to do so. The contradcton between standard economc models of selfsh behavor and emprcal observatons has been a challenge for both theorsts and expermentalsts. In the last ten years a number of theores have been developed whch try to close ths gap n explanatory power. Most of these theores are based on the assumpton that people have some knd of otherregardng, or socal, preferences. These approaches seek to overcome the dsparty between standard game-theoretcal predctons and expermental observatons by alterng the underlyng utlty functon of subjects, but stck to the assumpton that subjects behave ratonally. Ths study ams to nvestgate the addtonal explanatory power of models wth otherregardng preferences. On the bass of a laboratory experment we present a smple two-steps procedure for a wthn-subject test of the nequty averson model of Fehr and Schmdt (1999). In the frst step, subjects play selected games n order to estmate ther ndvdual other-regardng preferences and are, thereupon, classfed accordng to ther behavor. In the second step, subjects wth specfc preferences as defned by the Fehr and Schmdt model are matched nto pars and nteract wth each other n a standard publc good game and a publc good game wth punshment possblty. Our results show that the specfc composton of pars sgnfcantly nfluences the subjects performance n the publc good games. We dentfy the averson aganst advantageous nequty and the nformaton about the co-player s type as the man nfluencng factors for the behavor of subjects.
Inequty Averson and Indvdual Behavor n Publc Good Games: An Expermental Investgaton Astrd Dannenberg a, Thomas Rechmann b, Bodo Sturm a, and Carsten Vogt c a Centre for European Economc Research (ZEW), Mannhem b Faculty of Economcs and Management, Unversty of Magdeburg c Department of Busness Admnstraton, Lepzg Unversty of Appled Scences E-mal: dannenberg@zew.de, thomas.rechmann@ww.un-magdeburg.de, sturm@zew.de, vogt@ww.htwk-lepzg.de July 2007 Abstract We present a smple two-steps procedure for a wthn-subject test of the nequty averson model of Fehr and Schmdt (1999). In the frst step, subjects played modfed ultmatum and dctator games and were classfed accordng to ther preferences. In the second step, subjects wth specfc preferences accordng to the Fehr and Schmdt model were matched nto pars and nteracted wth each other n a standard publc good game and a publc good game wth punshment possblty. Our results show that the specfc composton of groups sgnfcantly nfluences the subjects performance n the publc good games. We dentfy the averson aganst advantageous nequty and the nformaton about the coplayer s type as the man nfluencng factors for the behavor of subjects. JEL classfcaton: C91, C92, H41 Keywords: ndvdual preferences, nequty averson, expermental economcs, publc goods Acknowledgements: Fnancal support from the German Scence Foundaton s gratefully acknowledged. The authors thank Andreas Lange and Drk Engelmann for helpful comments. 2
1 Introducton Wthn expermental economcs there s a growng number of stylzed facts whch contradct the model of ratonal payoff maxmzng actors. People cooperate n socal dlemmas such as publc good games (Ledyard 1995), they reject hgh amounts of money n the ultmatum game (Güth et al. 1982, Camerer 2003) and last but not least they make postve contrbutons n the dctator game (Kahneman et al. 1986, Forsythe et al. 1994, Camerer 2003). The contradcton between the standard economc model of selfsh behavor and emprcal observatons has been a challenge for both theorsts and expermentalsts. In the last ten years a number of theores that try to close ths gap n explanatory power have been developed. Most of these theores are based on the assumpton that people have some knd of other-regardng, or socal, preferences. These approaches seek to overcome the dscrepances between standard gametheoretcal predcton and expermental observaton by alterng the underlyng utlty functon of the subjects, but stck to the assumpton that subjects behave ratonally. The models by Bolton and Ockenfels (2000) and Fehr and Schmdt (1999) are promnent examples for ths approach. They assume that people are wllng to pay money n order to avod unequal payoff dstrbutons. Besdes these nequty-averson theores, other approaches focus on ntentons of subjects,.e. the way a subject behaves affects whether a player cares postvely or negatvely about that subject. Rabn (1993) s the poneerng paper n ths drecton, whle Dufwenberg and Krchsteger (2004) and Falk and Fschbacher (2006) extended Rabn s approach to extensve form games. A common property of models wth other-regardng preferences s that subjects are heterogeneous n ther preferences. Ths mplcates that theoretcal prognoses about ndvdual behavor may dffer between subjects for the same decson problem. Ths study ams to nvestgate the addtonal explanatory power of models wth otherregardng preferences. Thereby, we focus on the model of nequty averson by Fehr and Schmdt (1999), n the followng F&S. There are two reasons for dong ths. Frstly, the F&S model s able to explan an mpressve amount of expermental evdence not n lne wth the standard model of selfsh behavor. Secondly, F&S use a model whch s from a theoretcal pont of vew qute parsmonous as only two addtonal parameters are added to the ndvdual utlty functon whch s stll solely based on monetary payoffs. Moreover, both parameters of the model can be estmated wth the help of smple laboratory technques. 3
One nterestng mplcaton of models wth other-regardng preferences such as F&S s that they allow wthn-subject tests,.e. controlled experments wth the same subject but dfferent decson problems. Due to the fact that these theores predct gven dfferent preferences dfferent behavor of subjects, one may test hypotheses at the ndvdual level wth the followng two-steps procedure. In a frst step, ndvdual other-regardng preferences are measured by means of approprately desgned games. In a second step, the same subjects nteract wth each other n a controlled envronment under specfc rules for whch hypotheses regardng the ndvdual behavor have been derved n advance. Under the assumpton that preferences are stable at least wthn a short tme perod, ths approach allows a robust test of such models n the laboratory. Remarkably, ths approach has already been mentoned by Fehr and Schmdt (1999), p. 847, One of the most nterestng tests of our theory would be to do several dfferent experments wth the same group of subjects. Our model predcts a cross-stuaton correlaton n behavor. For example, the observatons from one experment could be used to estmate the parameters of the utlty functon of each ndvdual. It would then be possble to test whether ths ndvdual s behavor n other games s consstent wth hs estmated utlty functon. Our study mplements such a two-steps procedure. In the frst step, we measure the ndvdual F&S preferences by means of two smple experments, a modfed ultmatum game and a modfed dctator game. In the second step, subjects wth specfc preferences are matched together nto pars and nteract wth each other n a standard publc good game and a publc good game wth a punshment possblty. We dstngush between three groups of pars. In partcular, we form a group of far subjects where both players of the par are hghly nequty averse, a group of egostc subjects where both players are very lttle nequty averse, and a mxed group where one player s far and the other one s egostc. Due to the composton of treatments wth subjects wth specfc preferences we are able to derve and test hypotheses accordng to the F&S model. Furthermore, we control the nformaton subjects receve about the type of ther co-player. Blanco et al. (2006) were the frst who employed the two games that help to elct the ndvdual weghts of F&S nequty averson. Our approach s based on the method ntroduced by Blanco et al., but dffers from ther approach n some aspects (see secton 3 for a dscusson). 4
Our results dffer from the results descrbed n comparable papers n respect of two aspects. Frstly, the weght of averson aganst dsadvantageous nequty vares very lttle throughout our subject pool and has a medan of zero. Secondly, our results show that the specfc composton of groups sgnfcantly nfluences the subjects' performance n the publc good games: As long as subjects are nformed about the type of ther co-player, far groups contrbute more to the publc good than egostc or mxed groups. Moreover, far groups are more lkely to cooperate n the fnal perod of the publc good games than other groups. It turns out, furthermore, that explct nformaton s a key factor for ths dfference n behavor: As long as far subjects are not nformed on the fact that ther co-player s far, too, they act lke egostc subjects. Only the explct nformaton that they are playng wth a far co-player sgnfcantly enhances ther contrbutons. The remander of the paper s organzed as follows. Secton 2 sets the stage by descrbng the F&S model whch underles our experment. Secton 3 descrbes the desgn of our experment ncludng treatments and hypotheses. Secton 4 presents the expermental results. Secton 5 summarzes and dscusses our results. Secton 6 concludes and gves a bref outlook on further research. 2 Theoretcal background: The model of Fehr and Schmdt (1999) 2.1 Preferences Accordng to Fehr and Schmdt (1999) ndvduals are not exclusvely motvated by the absolute payoff they can earn but also value allocatons due to ther dstrbutonal consequences. Partcularly, assumng that ndvduals suffer from nequalty F&S ntroduce the followng utlty functon for subject : n n 1 1 (, π j ) = π α max{ π j π,0} β max{ π π j, } U π n 1 n 1 0 (1) j j where π and π j denote the absolute payoffs to subjects and j, respectvely, n denotes the total number of players nvolved n some decson problem, α 0 measures the mpact of s dsutlty from dsadvantageous nequalty whle β 0 measures the correspondng 5
mpact of advantageous nequalty. In the two player case whch s partcularly relevant for our expermental settng, (1) reduces to U ( π π ) π α max{ π π,0} β max{ π π,0}, j j j =. (2) F&S assume β < 1,.e. players are not wllng to burn ther money to elmnate advantageous nequalty. In addton, they assume that players put a stronger weght on dsadvantageous nequalty,.e. α β. In our experment, we wll obtan the weghts α and β from modfed ultmatum and dctator games (see secton 3.1). 2.2 Voluntary contrbuton games 2.2.1 The standard voluntary contrbuton game The assumpton of such preferences may have a strong mpact on the theoretcal predctons on the outcomes n several classes of games. In a publc good (PG) game for example, preferences of the F&S-type may lead to much hgher cooperaton rates compared to the predctons derved by standard economc theory. To see ths, look at the followng voluntary contrbuton game. Each player = 1,..., n s gven some ntal endowment y whch can be devoted to the producton of some publc good. Player s contrbuton to the publc good s denoted by g, the producton functon for the publc good s smply gven by the sum over n j all contrbutons = 1 g j. Let us assume that the margnal per capta return of an nvestment n the publc project s gven as some constant 1 n < a < 1. Then the monetary payoff for player s gven by ( g1 g n ) = y g + a = 1 n j π,..., g. Obvously, ths game consttutes a socal dlemma. The margnal return to an nvestment n the publc good s a whle the margnal costs for such an nvestment amount to 1. Thus, for player t s a domnant strategy to choose g = 0. Snce ths holds for all players dentcally, the unque equlbrum of ths j game s characterzed by contrbutons g j = 0 j and the publc good wll not be provded at all. However, the provson would be benefcal snce the collectve margnal return s na whch s clearly above the margnal costs of provson. Hence, the socal optmum s acheved 6
f each player contrbutes hs entre ntal endowment to the publc good leadng to payoff SO π = any whch s above the payoff players receve n the Nash equlbrum ( π NE = y ). F&S have shown that ths result s fundamentally altered f players are endowed wth nequalty averson accordng to (1). They prove the followng results: 1. If a β < 1, then t s a domnant strategy for player to choose g = 0. + 2. Let k, 0 k n, denote the number of players wth a β < 1. Then, f ( n 1) a 2 k, there exsts a unque equlbrum wth = 0 { 1,..., n}. 3. If for all players j { 1,...,n} ( n 1) < ( a + β 1) ( α β ) j wth a β > 1 the condton j j + j k + (3) g + holds, then equlbra wth postve contrbutons to the publc good exst. All k players wth a β < 1 choose = 0 + g whle all other players contrbute g j g [ 0, y] =. The ntuton behnd these results s not too dffcult. Frstly, f a player wth a β < 1 nvests one monetary unt n the publc good hs monetary return s a whle he gans a maxmum non-monetary utlty of β. Now, f the sum of both returns s less than one t s obvously the best strategy not to nvest nto the publc good, rrespectvely of what other players do. Secondly, f there are suffcently many players wth a β < 1, then player wll not be wllng to contrbute even f he shows stronger nequalty averson,.e. for hm a β > 1 holds. The reason s that relatvely few far players are not able to suffcently reduce dsadvantageous nequalty. Thrdly, f there are suffcently many players wth a β > 1, they can sustan cooperaton amongst themselves, even f the other players do not contrbute. However, ths requres that the contrbutors are not too upset about the dsadvantageous nequalty toward the free rders. (Fehr and Schmdt 1999, p. 840). + j + + + j 2.2.2 The voluntary contrbuton game wth punshment The dea that punshment of defectve players may ncrease contrbuton rates to the publc good s straghtforward. In a settng wth standard preferences, however, punshment s a non credble threat. Imagne a two-stage game: Stage one s the voluntary contrbuton game as 7
descrbed n the secton above. Stage two of the game ncorporates the possblty for players to enact some punshment on ther opponents. Snce punshment s costly t wll not be carred out by ratonal players nterested only n ther absolute materal payoff on the second stage. Snce players antcpate the outcome on the second stage they wll defect n the frst stage of the game. Ths outcome s substantally altered f preferences of the F&S-type are nvolved. F&S show that the exstence of a group of so called condtonally cooperatve enforcers may enhance the prospects for cooperaton. These ndvduals [ 1,...,n' ] must show suffcently strong averson aganst advantageous nequalty,.e. ther preferences must obey β 1 a. In addton, punshment must not be too costly. Let c denote the margnal costs of punshment. If c < ( n 1)( 1+ α ) ( n' 1)( α + β ) α [ 1,..., n' ] (4) and all other players [ n'+1,..., n] do not care about nequalty,.e. for them α β = 0, then the followng strateges form a subgame perfect equlbrum: In the frst stage each player contrbutes g g [ 0,..., y] = =. If each player does so, there are no punshments on the second stage of the game. If, however one of the players [ n'+1,..., n] devates and chooses g < g, then each enforcer j [ 1,..., n' ] carres out some punshment on player p j ( g g ) ( n c) = ' whle all other players do not punsh. 2.2.3 Introducng uncertanty The analyss n F&S s based upon the assumpton that players know ther opponents type,.e. they know k, the number of players wth preferences β < 1 a. For ths reason, n most of our expermental treatments, we nformed the partcpants, prevous to the publc good games (see secton 3.1), on how ther opponent had behaved n the modfed ultmatum and dctator game played before. Thus, these subjects were prncpally able to derve the correspondng type of ther co-player. In one treatment, we dd not nform the subjects about ther opponent. These subjects were only able to predct ther opponent s type wth some probablty. 1 In the 1 In the followng we assume rsk neutral behavor. 8
case of two players, followng condton (3), a far player wth preference β j > 1 a wll choose to contrbute to the provson of the publc good f the followng condton s met: 2 E ( k) where ( k) a + β j < (5) α j + β 1 j E denotes the expected value of k. Obvously, for n = 2, k can only take on the values one or zero,.e. E ( k) s the probablty a far player attaches to the possblty that hs co-player s an egostc type. Note that as a consequence of ntroducng uncertanty the parameter α matters. Ths s dfferent n the case of perfect knowledge whch can be easly seen by settng ( k) = 0 If ( k) > 0 E. Then, (5) reduces to > 1 a. β j E, however, for ncreasng values of α j t becomes more dffcult to fulfl (5),.e. to ensure that the condton s stll met, players must have hgher values of β j. 3 The ntuton behnd ths s as follows. On the one hand, f a far but unformed subject contrbutes to the publc good he runs the rsk of havng an egostc opponent and, therefore, beng exploted. On the other hand, f he does not contrbute, he runs the rsk of havng a far opponent and possbly explotng her. In other words, postve contrbutons to the publc good would ncrease the rsk of dsadvantageous nequty and decrease the rsk of advantageous nequty. Hence, the subject s only wllng to contrbute, f hs averson aganst dsadvantageous nequty s suffcently low and hs averson aganst advantageous nequty s suffcently large. Condton (5) can also be used to llustrate the effect of nformaton. If far players are nformed about the behavour of ther opponents n former modfed ultmatum and dctator games ths ncreases ther confdence about ther opponent s type. Partcularly, n ths case they should expect ther opponent to be a far type wth hgher probablty f he or she behaved accordngly n the former games. Techncally, ths means because E( k ) decreases, condton (5) s more easly met for nformed subjects. Hence, we should observe a hgher 2 If one skps F&S s assumpton that α β, whch mght be approprate for our subject pool (see secton 4.1), then the addtonal condton E ( k) a 2 must hold. See Fehr and Schmdt (1999), p. 862, for detals. 3 Ths can be easly seen when (5) s solved for β j. In ths case, condton (5) reads ( k) 1 a α j + E( k) 1 E( k) E β j >. 1 9
level of cooperaton n groups consstng of nformed far players than n groups consstng of unnformed far players. 3 Expermental desgn 3.1 Games We used four dfferent games (games A, B, C, and D) n our expermental desgn. Thereby, the only purpose of games A and B whch were orgnally ntroduced by Blanco et al. (2006) was to measure each subject s preferences accordng to the F&S model. After these games, some of the subjects wth certan preferences played games C and D n varous treatments (see secton 3.2). The desgn of the games s presented n the followng. 4 Game A s desgned to measure the subjects averson aganst dsadvantageous nequty. The game resembles the responder s basc decson stuaton n the ultmatum game but abstracts from strategc nteracton, such that we can rule out ndvdual behavor caused by strategc consderatons such as ntentons or recprocty. 5 In ths game, each subject has to decde n 22 cases (numbered from #1 to #22) n the role of player 1 between two pars of payoffs (par I and par II) each wth an amount of money for hmself or herself and another subject n the role of player 2. Payoffs (see the left part of Table 1) are chosen n a way that except for #1 subjects always have to choose between par I, a dsadvantageously unequal dvson of 10.00, and par II, an equal dstrbuton wth 2.00 for both players. All cases were arranged n a descendng order by the amount of money subjects could earn n par I. In ths game, a purely selfsh subject should choose par I from #1 to #20 and par II for #21 and #22. 6 A subject strongly dslkng dsadvantageous nequty, n contrast, would choose par I n #1 and par II from #2 to #22. Subjects wth other-regardng preferences accordng to F&S between these two extremes would be expected to swtch from choosng par I to par II after #2 but pror to #21. We descrbe ndvdual behavor n game A as consstent f (1.) a subject has a unque swtchng pont from par I to par II and (2.) the swtchng pont s between #2 and #21. 4 See the appendx for the nstructons that we dstrbuted to our partcpants. 5 The dfference to the payoffs of the orgnal ultmatum game s the fact that the conflct pont payoffs (n ) are changed to (2, 2) nstead of the orgnal (0, 0). 6 In the followng we assume ratonal behavor of all subjects. 10
Regardng the frst condton, a subject wth averson aganst dsadvantageous nequty consstent wth the F&S model who swtches for a specfc case from par I to par II should choose for all subsequent cases par II. As the payoffs for player 1 n par I are arranged n descendng order, a swtch back to par I n any of the subsequent decsons s not consstent. Ths would lead to a lower own payment and to hgher dsadvantageous nequty than n the case that was rejected before. In relaton to the second condton, t s useful to consder the decson cases outsde of the consstent area between #2 and #21. A subject who chooses par II n #1 already s not regarded as consstent because he or she could attan an equal allocaton wth hgher own payoff by choosng par I. A subject who chooses par II n #22 only or never swtches to par II at all has a negatve value for the weght of averson aganst dsadvantageous nequty ( α < 0 ),.e. lkes dsadvantageous nequty, and s therefore not consstent wth the F&S model. Wth the subject s swtchng pont we can determne the upper and lower bounds of the ndvdual α. We approxmate the ndvdual value for α by choosng the mean of the correspondng nterval (see Table 1). 7 Game B whch resembles the decson problem n the dctator game s desgned to measure the subjects averson aganst advantageous nequty. 8 Agan, each subject had to decde between two pars of payoffs (par I and par II) each wth an amount of money for hmself or herself n the role of player 1 and another subject n the role of player 2 n 22 cases (from #1 to #22; see the rght part of Table 1). Payoffs are chosen n a way that subjects had to choose between par I, an extremely unequal but advantageous dstrbuton of 10.00, and par II, an equal dstrbuton of dfferent amounts from 0.00 to 21.00. All cases were arranged n an ascendng order by the amount of money subjects could earn n par II. In ths game, a purely selfsh subject would choose par I from #1 through #20 and par II for #22. In the case of #21, ths subject would be ndfferent between par I and par II. A subject strongly dslkng advantageous nequty would always choose par II. Subjects wth farness preferences accordng to F&S would be expected to swtch from choosng par I to par II after #1 but before #21. 7 There are two exceptons to ths rule. Frstly, we cannot determne an upper bound for α of a subject who swtches from par I to par II n #2. Therefore, we assgn to those subjects the value of the lower bound, α = 2. 18. Secondly, we assgn the value α = 0 to a subject who swtches from par I to par II n #21, although the correspondng nterval for ths case s 0. 08 α 0. 04. 8 Strctly speakng, game B s equvalent to the dctator game only for decson #11. However, smlar to the dctator game, game B creates a trade-off between own monetary payoff whch creates advantageous nequty and a lower but equally dstrbuted payoff. 11
Table 1: Payoffs n game A and game B swtchng pont from par I to par II game A game B par I par II par I par II payoffs (n ) for player payoffs (n ) for player # 1 2 1 2 α 1 2 1 2 β 1 5.00 5.00 2.00 2.00-10.00 0.00 0.00 0.00 1.00 2 4.44 5.56 2.00 2.00 2.18 10.00 0.00 0.50 0.50 0.98 3 4.42 5.58 2.00 2.00 2.13 10.00 0.00 1.00 1.00 0.93 4 4.39 5.61 2.00 2.00 2.02 10.00 0.00 1.50 1.50 0.88 5 4.36 5.64 2.00 2.00 1.90 10.00 0.00 2.00 2.00 0.83 6 4.32 5.68 2.00 2.00 1.77 10.00 0.00 2.50 2.50 0.78 7 4.29 5.71 2.00 2.00 1.66 10.00 0.00 3.00 3.00 0.73 8 4.24 5.76 2.00 2.00 1.54 10.00 0.00 3.50 3.50 0.68 9 4.19 5.81 2.00 2.00 1.41 10.00 0.00 4.00 4.00 0.63 10 4.14 5.86 2.00 2.00 1.30 10.00 0.00 4.50 4.50 0.58 11 4.07 5.93 2.00 2.00 1.18 10.00 0.00 5.00 5.00 0.53 12 3.92 6.08 2.00 2.00 1.00 10.00 0.00 5.50 5.50 0.48 13 3.86 6.14 2.00 2.00 0.85 10.00 0.00 6.00 6.00 0.43 14 3.81 6.19 2.00 2.00 0.79 10.00 0.00 6.50 6.50 0.38 15 3.68 6.32 2.00 2.00 0.70 10.00 0.00 7.00 7.00 0.33 16 3.53 6.47 2.00 2.00 0.58 10.00 0.00 7.50 7.50 0.28 17 3.33 6.67 2.00 2.00 0.46 10.00 0.00 8.00 8.00 0.23 18 2.85 7.15 2.00 2.00 0.30 10.00 0.00 8.50 8.50 0.18 19 2.72 7.28 2.00 2.00 0.18 10.00 0.00 9.00 9.00 0.13 20 2.22 7.78 2.00 2.00 0.10 10.00 0.00 9.50 9.50 0.08 21 1.43 8.57 2.00 2.00 0.00 10.00 0.00 10.00 10.00 0.03 22 0.10 9.90 2.00 2.00-0.14 10.00 0.00 10.50 10.50 0.00 We label ndvdual behavor n game B as consstent f (1.) a subject has a unque swtchng pont from par I to par II and (2.) ths swtchng pont s between #2 and #22.,.e. f the ndvdual weght of averson aganst advantageous nequty meets β < 1. Relatng to the 0 frst condton, a subject wth averson aganst advantageous nequty consstent wth the F&S model swtchng from par I to par II at one pont should also choose par II n all cases after the swtchng pont. As the payoffs for player 1 n par II are arranged n an ascendng order, a swtch back to par I n any of the subsequent cases s not consstent. Ths would lead to the same advantageous nequty than was rejected before but now wth hgher opportunty costs n terms of equal payoffs for both players. For the second condton, we consder agan the decson cases outsde of the consstent area between #2 and #22. A subject choosng par II already n #1 has β 1,.e. s wllng to burn money n order to produce equal payoffs. A subject who does not swtch at all dsplays affecton for advantageous nequty. Both behavoral patterns are not consstent wth F&S. Smlar to game A, we can determne the upper and lower bounds for the ndvdual s β by means of a subject s swtchng pont. We 12
approxmate the ndvdual value of β by choosng the mean of the correspondng nterval (see Table 1). 9 In a recent paper, Blanco et al. (2006) present an expermental test of the F&S model whch s smlar to our approach. In ther experment, one and the same cohort of subjects plays, among others, an ultmatum game to derve α, and a modfed dctator game to derve β. Whle the modfed dctator game they used s practcally dentcal to game B n ths study, our game A dffers from the ultmatum game wth the strategy method n Blanco et al. n an mportant aspect. 10 In the ultmatum game, each subject reacts to a specfc proposal of hs or her co-player,.e. there s a dstnct element of strategc nteracton n ths game. In our game A, however, each subject hmself or herself decdes only between dfferent dstrbutons of 10.00 and the equal payoff wth 2.00 for both, whch means that there s no drect nteracton between both subjects and no room for strategc consderatons. In other words, contrary to Blanco et al. we use only non-strategc games 11 for the elctaton of both F&S parameters. Whle ths dfference does not matter from a theoretcal pont of vew nether wth regard to the F&S model nor the standard game theory t stll may nfluence the behavor of subjects (see secton 4.1). Fehr and Schmdt (2005) recommend usng strategc games n order to elct preference parameters that capture not only trats of nequty averson but, moreover, strategc consderatons lke ntentons or recprocty. As our man focus s on the effect of nequty averson, we do not follow ths recommendaton. Our games are explctly of non-strategc nature. One more remark on the methodology to elct ndvdual F&S preference parameters should be n order. A basc assumpton underlyng our desgn of the games A and B s that ndvduals are only drven by preferences of the F&S-type. Ths means n partcular that ndvduals do not hold any specfc preferences wth regard to effcency. An example wll help to llustrate ths pont. In our game B, the sum of both payoffs n par II rses from 0 n #1 to 21 n #22. An ndvdual carng for effcency only wll swtch from par I to par II after #10 or after #11. Gven an ndvdual cares for equty and effcency, we underestmate 9 As before, there s an excepton to ths rule. We assgn the value β = 0 to a subject who swtches from par I to par II n #22, although the correspondng nterval n ths case s 0. 05 β 0. 10 Game B n the present study dffers n a small but mportant aspect of the correspondng game n the Blanco et al. study. In our game B, selfsh subjects have to swtch from par I to par II n the last decson row (#22). The same holds for game A (#21 and #22). Ths feature enables us to select subjects who do not behave consstently nether accordng to F&S nor to the standard theory of selfsh behavor (as a specal case of F&S). 11 Followng Fehr and Schmdt (2005, p. 47), we regard such games as strategc where each player has an nfluence on each other player s materal payoff. 13
hs or her averson aganst advantageous nequty for β >. 53 and overestmate t for β <.53. Game C s a standard two-player PG game wth a voluntary contrbuton mechansm. 12 Two players get a fxed balance of 3.00 for show-up and are endowed wth 10.00 each. They decde smultaneously how much (f any) money from the endowment to contrbute to a publc good. Each monetary unt that the subject keeps for hmself or herself rases the ndvdual payoff by exactly that amount. Both subjects receve 0.70 for each 1.00 contrbuted to the publc good,.e. the margnal per capta return s constant and equal to 0.7. The game was played usng a partner desgn over 10 rounds wth the number of rounds as common knowledge. After each round subjects were nformed about ther own contrbuton and the contrbuton of the co-player as well as the payoffs of both players. Game D conssts of two stages. Stage 1 s equvalent to game C,.e. subjects play the same two-player PG game as descrbed above. Stage 1 n game D s followed by a stage 2. In ths stage subjects have the possblty to assgn hs or her co-player negatve ponts,.e. a punshment mechansm (e.g. Fehr and Gächter 2000, 2002) s ntroduced. Each negatve pont reduces the payoff of the co-player by 1.00. However, the assgnment of negatve ponts s costly. Each negatve pont assgned reduces the punsher s own payoff by ether 0.17 or 0.50 (dependng on the treatment). Agan, the game was played usng a partner desgn over 10 rounds wth the number of rounds as common knowledge. 13 Subjects were pad separately for games A and B and games C and D. The payments from games A and B were computed as follows: All subjects wthn a sesson were randomly matched nto pars of subjects. After ths, t was determned (agan by chance) whether game A or B would be relevant for the payoffs. After the selecton of the relevant game, a random draw selected whch number of the payoff lst (between #1 and #22) would be relevant. Fnally, a random draw decded whch person wthn a par determned the payoffs,.e. whose decson as player 1 was realzed. Accordng to ths rule, each of the 22 decson cases n game A and B had the same chance to be relevant for the payment. Subjects were nformed about ths payoff rule n advance and we checked the comprehenson of ths desgn feature n 12 Blanco et al. (2006), too, played a standard two-player PG game (n a one-shot verson) after elctng the ndvdual nequty averson weghts. 13 Theoretcally, due to possble punshment and costs of punshment n game D, t s possble to encounter cases wth negatve payoffs. We were prepared to handle these cases (payments would have been set to zero). Luckly, though, a case lke ths dd not occur durng the experments. 14
a quz before the experment started. Subjects who behaved consstently n games A and B (whch had been played at frst) were nvted to play games C and D. 14 The payments from games C and D were determned n a smlar way: After both games, a random draw determned whch game (C or D) would be relevant. Followng ths decson, one of the 10 perods was selected randomly and the payments were realzed accordng to the decsons n ths round. As before, the payoff rule was common knowledge to all partcpants. 3.2 Subject pool and treatments We ran 25 sessons wth 18 to 20 subjects n each sesson. All n all, 492 subjects partcpated n games A and B and 160 of these subjects were nvted to play games C and D. Sessons lasted about 90 mnutes and the average earnng was 5.90 for games A and B and 15.20 for games C and D. The sessons were conducted n November 2006 and May 2007 at the Magdeburg Expermental Laboratory (MaXLab). The experment was fully computerzed and anonymous. 15 In the laboratory, subjects were randomly allocated to separate cabns and had no mutual contact durng or after the experment. The man characterstcs of our subject pool are dsplayed n Table 2. Though the majorty of our subjects are students of economcs, the fracton of non-economsts (41%) s qute hgh. Almost two thrds of our subjects already had experence n experments,.e. have partcpated n at least one expermental sesson before. Table 2: Subject pool descrptve statstcs absolute frequency relatve frequency n percent total 492 100.0 consstent choces 371 75.4 study Management Scence (MS) 173 35.2 Economcs (Ec) 39 7.9 other Economcs (othec) 79 16.1 MS or Ec or othec (Econ) 291 59.1 other than Econ (NonEcon) 201 40.9 experenced (at least one experment before) 339 68.9 male subjects 270 54.9 female subjects 222 45.1 14 Those subjects who were not nvted to take part n games C and D played a one-shot prsoner s dlemma game. The only purpose of ths game was to avod a zero payment for subjects who ddn t play games C and D (there was no show-up-fee). After ths game these subjects were pad and left the laboratory. The mean payment ( 3.60) was accordngly low. In the followng we do not refer to the results of ths game. 15 We used Z-tree for programmng. See Fschbacher (2007). 15
In order to check the robustness of our desgn, we mplemented two modfcatons durng the experment. In two of our sessons, we modfed the order of play. Whle n most sessons, subjects frst played game A followed by game B, n two sessons (for 40 of 492 subjects), we had subjects play game B frst, then A. Moreover, n seven of our sessons (for 136 of 492 subjects), payoffs n game A were slghtly dfferent: The payoff n par II of game A was 0.00 for both subjects nstead of 2.00. Both changes do not lead to sgnfcantly dfferent dstrbutons of the values for α and β at the 5% level (K-S test, two-taled). 16 Therefore, we felt free to pool the data of all 25 sessons of games A and B conducted. In order to analyze the predctve power of the F&S model as well as some comparatve statc effects we defned several treatments for games C and D, whch are descrbed below. 17 Treatments dffer wth respect to (1.) the preference parameter β of the subjects that form a par n games C and D, (2.) the state of nformaton about the co-player s behavor n games A and B, and (3.) the value of parameter c, whch defnes the margnal costs of punshment n game D. Table 3: Treatments n game C Treatment Parameter β, = 1, 2 Informaton Observatons EGO β < 0. 3 yes 35 MIX β 1 < 0. 3 β2 > 0. 3 yes 13 FAIR β > 0. 3 yes 17 FAIR(n) β > 0. 3 no 15 Σ 80 Notes: (1.) All subjects n FAIR and FAIR(n) have α = 0. (2.) No one n our subject pool has β = 0. 3. (3.) There s no dfference n the dstrbutons of β between FAIR and FAIR(n) at the 5% level (K-S test). Our four treatments n game C dffer wth respect to the composton of the (two-person)- groups playng the game. Detals are dsplayed n Table 3. Thereby, one group of two players ( = 1, 2) makes one statstcally ndependent observaton. In treatment EGO, two subjects wth β < 0. 3 are matched together to form a common group. Treatments MIX, FAIR, and FAIR(n) are defned correspondngly. In all treatments except for FAIR(n), all subjects are 16 Unless t s explctly noted, n the followng all tests are two-taled. 17 The defnton of treatments s determned by the structure of the dstrbuton of α and β wthn our subject pool. Due to the fact that there s vrtually no dsperson for α (see secton 4.1) we had to focus the defnton of treatments on β. 16
nformed about ther respectve co-player s former behavor n games A and B. In FAIR(n), players dd not receve any nformaton about each other s former behavor. All subjects who played game C also completed game D, the PG game wth punshment possbltes. The dfferent treatments n game D are dsplayed n Table 4. Wthn the EGO treatment, we dstngushed between hgh and low costs of punshment. Therefore, pars of subjects from the EGO treatment n game C were allocated nto two separate EGO treatments n game D, EGO(h) and EGO(l). As n game C before, n most of the treatments, subjects were nformed about ther co-player s behavor n games A and B, agan wth the notable excepton of treatment FAIR(h, n). Table 4: Treatments n game D Treatment Parameter β, = 1, 2 Informaton Costs of Punshment Observatons EGO(l) β < 0. 3 yes low 9 EGO(h) β < 0. 3 yes hgh 26 MIX(h) β 1 < 0. 3 β2 > 0. 3 yes hgh 10 FAIR(h) β > 0. 3 yes hgh 17 FAIR(h, n) β > 0. 3 no hgh 15 Σ 77 Notes (see also Table 3): (1.) As we have only three ndependent observatons n the MIX(l) treatment t s omtted from hypotheses testng n game D. (2.) None of the subjects wth β > 0. 3 n MIX(h) fulflls condton (4). 3.3 Hypotheses Based on the defnton of treatments and the theoretcal consderatons n secton 2, we are able to derve specfc hypotheses whch follow from the F&S model for our subject pool. Thereby we focus at least for the non-parametrc tests n secton 4 on the last perod of both games C and D. In dong so we can exclude any repeated game effects, whch may come nto play f there s a repeated nteracton between only two subjects. In the followng we assume that, whenever F&S predct the exstence of multple equlbra, subjects play each of them wth the same probablty. As there s no dfference n the prognoss regardng the contrbuton behavor between games C and D we are able to formulate jont hypotheses for both games. Note that there s no treatment n game D where punshment s part of an equlbrum strategy: In MIX(h) none of the subjects wth β > 0. 3 fulflls condton (4),.e. there are no subjects who may enforce cooperaton. In FAIR(h) cooperaton wthout punshment s an equlbrum because n the 17
case of mutual cooperaton there are no ncentves for subjects wth β > 0. 3 to punsh each other. In ths case, punshment would reduce the own payoff and create advantageous nequalty. Therefore, we can derve the followng hypotheses regardng the contrbutons to the publc good n dfferent treatments for games C and D: H1: In the EGO and MIX treatments of both games, zero contrbutons of all subjects should be observed. H2: In the FAIR treatments, postve contrbutons to the publc good should be observed more frequently than n the correspondng EGO and MIX treatments. H3: In the FAIR treatments wth nformaton, postve contrbutons to the publc good should be observed more frequently than n the correspondng FAIR(n) treatment wthout nformaton. 4. Results The results secton conssts of two man parts. The frst part analyzes the subjects behavor n games A and B. The second part focuses on games C and D and refers to the treatments and hypotheses descrbed above. 4.1 Behavor n games A and B As dscussed n the prevous secton, we are able to select subjects wth consstent preferences whch are n lne wth the assumptons α 0 and β < 1 n the F&S model. All n all, out 0 of 492 subjects who partcpated n the experment 371 (75%) behaved consstently n games A and B. Thereby, we do not detect any sgnfcant correlatons between the soco-economc characterstcs n Table 2 and the consstency of choces at the 5% level (Spearman s ρ ). Fgure 1 presents the dstrbuton of the F&S parameters, α and β. A bref look at the dstrbuton of the values for α, the weght for the averson aganst dsadvantageous nequty, shows that about 86% of all subjects behave selfshly n game A (lower rght n Fgure 1). There s a second peak n the range of 0 8 < α 1. 0. These are subjects who choose the. payoff n par II ( 2.00 for both) nstead of the payoff n par I ( 3.92 for themselves and 18
6.08 for the other subjects) n #12 of game A. The mean value for α s 0. 102. The medan value s 0 ndcatng that selfsh behavor s the domnant pattern n our subject pool. The dstrbuton of the values for β, the weght for the averson aganst advantageous nequty, looks qute dfferent (upper left n Fgure 1). We observe two peaks n the dstrbuton. Frstly, for 0 < β 0. 1,.e. for subjects who behave rather selfshly and swtch from par I to par II n #21 or #22 of game B. Secondly, there s a peak for the range 0 4 < β 0. 6,.e. for. subjects who swtch from par I to par II between #10 and #13 and who prefer a rather equal allocaton of the 10 vs-à-vs an advantageous but very unequal allocaton. The mean of the β values s 0.356, the medan s equal to 0.375. The scatter plot n Fgure 1 (upper rght) shows the jont dstrbuton of the two parameters. Remarkably, n our subject pool we found very few subjects (45 of 371,.e. 12% of consstent choces) meetng the F&S condton α β. The correspondng data ponts le below the 45-degree lne n the scatter plot. It s apparent that α and β are not sgnfcantly correlated and a test for correlaton confrms ths (Spearman s ρ = 0. 345, p = 0.299). Gven the qute heterogeneous subject pool n our experment, we are able to test whether there are any correlatons between the soco-economc characterstcs n Table 2 and the ndvdual values for α and β. We observe a small but sgnfcant negatve correlaton between subjects who study Management Scence, Economcs or a related feld (e.g. busness nformaton systems ) and the value for β (Spearman s ρ = 0. 154, p = 0.003). Ths observaton s n lne wth the well-known fact that economcs students behave n general more selfshly than other students (e.g. Marwell and Ames 1981, Frank et al. 1996, and Carter and Irons 1991). However, ths s the only sgnfcant correlaton between α or β and the soco-economc characterstcs at the 5% level. 19
Fgure 1: Dstrbuton of F&S parameters for consstent choces (.9;1.0) (.8;.9] (.7;.8] (.6;.7] (.5;.6] (.4;.5] (.3;.4] (.2;.3] (.1;.2] (0;.1] 0 percent 0 20 40 60 80 alfa 0.5 1 1.5 2 2.5 0.2.4.6.8 1 beta Note: A random nose (5% of the graphcal area) s ntroduced to the data n the scatter plot before plottng. 0 20 40 60 80 percent 0 (0;.2] (.2;.4] (.4;.6] (.6;.8] (.8;1.0] (1.0;1.2] (1.2;1.4] (1.4;1.6] (1.6;1.8] (1.8;2.0] (2.0;2.2] (2.2;2.4] (2.4;2.6] Because of the smlarty between the approaches of revealng ndvdual F&S preferences by Blanco et al. (2006) and our expermental technque, t s nterestng to compare the ndvdual values for α and β. Fgure 2 presents the cumulated denstes for α and β for economsts ( Econ ), non economsts ( NonEcon ), and all subjects ( All ) n the present study (these data are the same as the ones n Fgure 1). Addtonally the correspondng cumulated denstes of the above mentoned Blanco et al. study are depcted. 18 The dfference n the cumulated denstes for α between our subjects and the partcpants n Blanco et al. s remarkable and hghly sgnfcant (K-S test, p = 0.000). Whle n Blanco et al. only 15% have α = 0, n our study the fracton of subjects wthout any averson aganst dsadvantageous nequty s more than fve tmes hgher. Regardng the weght of advantageous nequty, β, the dfferences between Blanco et al. and our data are not sgnfcant at the 5% level (K-S test). 18 We thank Hans Normann and hs colleagues for provdng the data of ther experment. All of the 72 subjects n Blanco et al. (2006) were non-economsts. There were 13 subjects who dd not behave consstently accordng to our defnton. Therefore, the parameters of only 59 subjects are dsplayed for Blanco et al. n Fgure 2. 20
Fgure 2: Cumulated densty for F&S parameters comparson percent 100 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 alfa percent 100 90 80 70 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 beta Econ All NonEcon Blanco et al. 4.2 Behavor n games C and D The results secton for games C and D conssts of fve parts. The frst and second parts analyze the subjects contrbutons n the standard PG game (game C) and the PG game wth punshment possblty (game D). Here, we are able to test whether the prognoses from the F&S model apply to our subject pool. The thrd part nvestgates the effect of nformaton and the fourth part the effect of punshment. Fnally, we present Tobt estmates for the contrbuton to the publc good n game C and the punshment behavor n game D. 4.2.1 Hypotheses testng of the F&S model n game C A total of 160 subjects played the PG game C. The mean contrbuton per perod over all subjects and all perods s 5.90. In order to test our hypotheses derved from the F&S model, we analyze the behavor of subjects n the treatments EGO, MIX, and FAIR (see Table 3). These treatments dffer only n ther composton of ndvdual types, namely egostc, far or both types. The attrbutes egostc and far arse only from the value of β, as n the absence of uncertanty about the co-player s type, the subjects contrbutons n the standard PG game solely depend on β. 19 19 If we control for α and apply the non-parametrc tests to all subjects wth α = 0 (86% of our subjects n games C and D) we fnd the same results as n the tests ncludng all subjects. Seeng ths, α seems ndeed to have no sgnfcant effect on subjects contrbuton behavor. We wll return to ths pont later. 21
The prognoss of the F&S model s the same for EGO and MIX. Contrbutons of zero are the only equlbrum. Ths s the same allocaton that standard economc theory predcts for both games. In contrast, accordng to F&S n the FAIR treatment exst the standard noncontrbuton equlbrum as well as equlbra wth postve contrbutons. Fgure 3 shows the mean contrbutons n game C for the three treatments wth nformaton, namely FAIR, EGO, and MIX. The mean contrbuton per perod s 7.10 for FAIR, 6.00 for EGO, and 4.30 for MIX. A Mann-Whtney U test (MW U test) shows that the dfferences between treatments are not sgnfcant at the 5% level except for FAIR and MIX (p = 0.013). In each perod of all treatments the share of subjects contrbutng a postve amount of money to the publc good s sgnfcantly greater than zero (Bnomal Sgn test, one-taled, p = 0.000). Therefore, we can sum up these observatons nto the followng result. Result 1: The contrbutons of the subjects n the EGO and the MIX treatment to the publc good are sgnfcantly hgher than the levels predcted by F&S and by the standard model of pure selfshness. The contrbutons n FAIR are n lne wth F&S but not wth the standard model. 10 9 8 7 6 5 4 3 2 1 0 Fgure 3: Contrbutons n game C FAIR EGO MIX 1 2 3 4 5 6 7 8 9 10 Perods As already explaned above (secton 3.3), our analyss focuses partcularly on the behavor n the fnal perod. The mean contrbuton n the fnal perod s 6.30 n FAIR, 4.20 n EGO, and 2.40 n MIX. Thereby, the dfferences n fnal perod contrbutons are sgnfcant between MIX and FAIR (MW U test, p = 0.007) and weakly sgnfcant between EGO and FAIR (p = 0.090). There are, n contrast, no sgnfcant dfferences between EGO and MIX. In consderaton of our hypothess H2, ths s what we expected. Accordng to ths, the F&S 22
model seems to have some explanatory power for the behavor n the fnal perod of the standard PG game. In order to check whether ths result s robust, we also consder the share of subjects who defect n the fnal perod by contrbutng nothng to the publc good. 54% of the subjects n EGO, 58% of the MIX subjects but only 32% of the FAIR subjects defect n the fnal perod. Thus, the shares of defectng subjects are relatvely hgh n EGO and MIX compared to the FAIR treatment. Applyng a 2 χ test, we fnd that these dfferences are not sgnfcant at the 5% level. One has to take nto consderaton, however, that the 2 χ test refers to ndependent observatons,.e. mean contrbuton of both players n a par, and not to ndvduals. As a consequence, defecton s only counted f both players n a par contrbute nothng. If one of the two players contrbutes a small amount to the publc good, mean contrbuton s postve and, therefore, represents cooperaton n the test. For ths reason, we repeat the test wth the crtcal value for defecton equal to 3. Thus, we defne defecton when the mean contrbuton of both players n a par s below 3 and cooperaton when t s equal or above 3. In ths case, the dfferences n shares of defectng subjects s sgnfcant between MIX and FAIR (p = 0.018) and weakly sgnfcant between EGO and FAIR (p = 0.063). Agan, there are no sgnfcant dfferences between EGO and MIX. Furthermore, we can reject the hypothess that cooperaton and defecton are equprobable for FAIR (p = 0.008) whereas we cannot reject ths hypothess for EGO and MIX at the 5% level. Ths holds regardless of whether we take zero contrbutons or contrbutons below 3 as defecton. Result 2: The F&S model has some explanatory power for the behavor n the fnal perod of the PG game: Subjects n the FAIR treatment contrbute more to the publc good than subjects n EGO and MIX. Furthermore, the share of subjects who defect n the fnal perod s lower n FAIR than n EGO and MIX. Fgure 3 llustrates the relatvely poor performance of the subjects n the MIX treatment. Even though the dfferences between MIX and EGO are not sgnfcant, t s remarkable that the contrbutons n MIX are always lower. The MIX treatment s the only heterogeneous treatment where two dfferent types of subjects make up a par, namely a far subject and an egostc subject. It s nterestng to fnd out whether both types choose low contrbutons or whether one of the types s responsble for ths development. The more detaled analyss ndcates that the frst perod plays an mportant role where the only nformaton subjects have about ther co-player s the behavor n games A and B. The egostc subjects n MIX contrbute on average 6.30 n the frst perod whereas the far subjects contrbute only 23