Clausthal University of Technology. Heterogeneous Social Preferences

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1 Clausthal Unversty of Technology Heterogeneous Socal Preferences Mathas Erle *, # Ths verson: June 2004 Abstract: Recent research has shown the usefulness of socal preferences for explanng behavor n laboratory experments. Ths paper demonstrates that models of socal preferences are partcularly powerful n explanng behavor f they are embedded n a settng of heterogeneous actors wth heterogeneous (socal) preferences. For ths purpose a smple model s ntroduced that combnes the basc deas of nequty averson, socal welfare preferences, recprocty and heterogenety. Ths model s appled to 43 games and t can be shown that ts predctve accuracy s clearly hgher than that of the solated approaches. Furthermore, t can explan most of the anomales (the contradctons ) that are dscussed n Goeree and Holt (2001). JEL-Classfcaton: C72, C92, D63 * Insttute of Busness Admnstraton and Economcs, Clausthal Unversty of Technology, Julus-Albert-Str. 2, Clausthal-Zellerfeld, Germany, m.erle@tu-clausthal.de. # For helpful comments I wsh to thank Heke Schenk-Mathes, J. Phlpp Semer and Jens-Peter Sprngmann.

2 2 1. Introducton When we meet people for the frst tme n our lfe we often ask ourselves what knd of person he or she s. And t s not ust a queston of curosty that we lke to know what type of person we are acquanted wth. Most often ths s the central queston for our decson to have further contact wth that person. Such a way of thnkng suggests that people are ndeed very dfferent from each other and that the type of the person we are occuped wth s of uttermost mportance. However, economc theory, by and large, gnores dfferences between people and usually assumes homogeneous preferences. Maybe economc theory msses too much of human behavor by dong so. The man purpose of ths paper s to demonstrate the usefulness of explctly modelng heterogenety n preferences for explanng the behavor of subects n 43 laboratory experments. Next to ths we try to corroborate the relevance of socal preferences for explanng human behavor. Expermental evdence makes clear that there are many games n whch Nash Equlbrum descrbes people s behavor qute well. However, there seem to be ust as many other games n whch laboratory behavor devates from the predctons of standard game theory by a wde margn. Obvously, there s a need for theoretcal nnovatons whch can explan the successes of game theory as well as ts falures. No doubt, theory has reacted to expermental evdence. There are several branches of new theoretcal approaches that can clam to have at least partal success n ntroducng superor concepts. Dynamc evolutonary approaches 1 (e.g. replcator dynamcs) often but not always converge to Nash Equlbra. Quantal Response Equlbra n general and the Logt Equlbrum n partcular (McKelvey and Palfrey 1995, 1998 and Goeree and Holt 2001) have been qute successful n explanng behavoral reactons due to parameter varatons n games wth dentcal Nash Equlbra. Fnally, there s a thrd strand of research whch was successful n explanng devatons from Nash Equlbrum. These are approaches of other regardng or socal preferences. The socal preferences approach can be dvded nto at least three mportant substrands: theores of ntentonal recprocty (Rabn 1993, Dufwenberg and Krchsteger 1998), the nequty averson approach (Bolton and Ockenfels 2000, Fehr and Schmdt 1999) and recently a theory of Socal Welfare Preferences 1 See Webull (1995) or Fudenberg and Levne (1998).

3 3 (Andreon and Mller 2002 and Charness and Rabn 2002). In ths paper we shall concentrate on the last two approaches. Bolton and Ockenfels (2000) as well as Fehr and Schmdt (1999) ntroduce concepts of nequty averson. It s assumed that there exst people who dslke nequalty and who actually sacrfce money to reduce t. Both concepts are partcularly successful n descrbng laboratory behavor when they assume heterogeneous actors. Bolton s and Ockenfels model s exclusvely defned for heterogeneous populatons of subects. Although Fehr s and Schmdt s model can be used for homogeneous populatons, all successful applcatons assume a mxture of nequty averse and strctly egostc subects, the latter beng ndvduals wth the standard utlty functons n game theory. The approaches dffer n the concrete defnton of nequty averson and Bolton and Ockenfels allow for more general preference dstrbutons of subects. However, the general verson of ther model s somewhat more complcated and ths makes t less sutable for drect applcaton. It thus cannot surprse very much that most further applcatons of the nequty averson approach use the smpler Fehr and Schmdt varant. In the meantme nequty averson has been challenged by numerous experments that have been carred out (e.g. Kagel and Wolfe 2000 and Charness and Rabn 2002). The most recent alternatve to nequty averson has been presented by Charness and Rabn (2002). They ntroduce a model of socal welfare preferences wth and wthout recprocty. Socal welfare preferences are characterzed by ndvduals who gve postve weght to aggregated surplus,.e. f other people are better off, c.p., utlty of ndvduals ncrease. The authors carred out 32 experments and compared the compatblty of several socal preference approaches wth the expermental data. Ther concluson s that socal welfare preferences show the best ft to the data. However, the comparson between socal welfare preferences and the nequty averson model s based because Charness and Rabn do not take nto account that the most frutful verson of the nequty averson model takes explctly nto account that there are dfferent types of actors,.e. that there s heterogenety of preferences. In fact, n Fehr, Kremhelmer and Schmdt (2002) as well as n Fehr, Klen and Schmdt (2001) nequty averse actors are only a mnorty n the populaton and the explanatory power of the model stems n partcular from the nterplay of strctly egostc and nequty averse subects. However, Charness and Rabn (2002) only consder the homogeneous populaton varant of the nequty averson theory. 2 The same crtque also apples to ther own model of socal wel- 2 Charness and Rabn (2002) are well aware of ths shortcomng as footnote 6 of ther paper shows.

4 4 fare preferences n whch they assume a monomorphc populaton, agan. In ths paper we shall try to show that ths shortcomng serously lmts the explanatory power of ther model. Nevertheless, Charness and Rabn convncngly show that socal welfare preferences mght help explanng qute a lot of behavor n ther 32 games. Summarzng, Fehr and Schmdt have shown the usefulness of modelng heterogeneous populaton equlbra wth nequty averse and strctly egostc agents. Charness and Rabn have shown some evdence for socal welfare preferences and the relevance of recprocty. Ths paper tres to combne these approaches and analyzes whether ths ncreases explanatory power sgnfcantly. The focus n ths paper s on the applcaton of the basc dea. Therefore, the basc model has to be suffcently tractable for drect applcaton n a wde varety of games. In fact, we are gong to apply the model to 43 dfferent games and show that ts predctve accuracy s clearly greater than that of the solated models. However, the reader should be well aware that the model presented n ths paper s regarded ust as one sngle step to the development of operatonal models for explanng expermental and feld evdence. Its man purpose s to demonstrate the mportance of heterogenety of preferences. In secton 2 we shall ntroduce a very smple 2 players 3 types model of heterogeneous preferences wth explct modelng of recprocty. In secton 3 ths model s appled to all 2-player experments n Charness and Rabn (2002). Furthermore, ts predctve accuracy s compared wth that of the nequty averson and the socal welfare preference approach. In secton 4 the model s appled to eght games (each game s analyzed for 2 varants wth dentcal Nash Equlbra) from Goeree and Holt (2001). Fnally, a summary, some conclusons and some thoughts about future research are gven n secton Heterogeneous Socal Preferences: A Smple Model In ths secton I outlne a smple 2 players model wth lnear obectve functons. The man purpose of the model s to combne elements of the approaches of Fehr and Schmdt (1999) and of Charness and Rabn (2002) n a rather smple way that allows for drect applcaton to 2 person games. The man ngredents from Charness and Rabn (2002) are the concept of Welfare Preferences and negatve recprocty. The deas taken from Fehr and Schmdt (1999) are manly that there are dfferent types of actors,.e. there s heterogenety among the players, and the concept of nequty averse players. It s assumed that there are three knds of players: strctly egostc actors (SE actors), nequty averse actors (IA actors) and one type of

5 5 ndvduals who has Welfare Preferences (WP actors). It s further assumed that there s ncomplete nformaton n the sense that ndvduals only know ther own types but not the types of the actors they are playng wth. However, they do know the dstrbuton of types among all ndvduals n ther socety so that they have common prors about the dstrbuton of types. In accordance wth Charness and Rabn (2002) we do not explctly take nto account postve recprocty. We assume that all effects of postve recprocty are represented by IA and WP preferences. Charness and Rabn report that they fnd only lttle evdence for postve recprocty. Consequently, there s some hope that the neglect of postve recprocty does not lead to hgh naccuraces. In contrast to postve recprocty, negatve recprocty s modeled explctly. If other players msbehave parameters for socal preferences n IA and WP utlty functons are changed so that players become more envous and less generous, respectvely. The correspondng utlty functons are gven by equaton (1) U ( π π ) ( 1 σ t θt R) π + ( σ t + θ t R) ( ) ( ) 1 ρt θ t R π + ρt + θt R ( 1 θ R) π + θ Rπ,, = π, π > π, (1) t t π, π < π π = π = 1, 2,, t = SE, IA, WP. Here, π s the monetary payoff of player. ρ t represents player s concern for player s payoff f player s payoff s larger than player s. σ descrbes the weght that player puts on player s payoff f player gets the hgher payoff. Fnally, θ s the recprocty parameter and R s the recprocty varable whch s 1 f the other player has msbehaved and t s zero f no msbehavor has occurred. Strctly egostc players (SE) are characterzed by σ ρ = θ = 0,.e. they ust care SE = SE SE about themselves. Thus, SE players have standard game theoretc utlty functons. Inequty averse players (IA) parameters are characterzed by σ IA < 0 < ρ IA < 1. Ths means that they put negatve weght on the other player s payoff f ther own payoff s lower and that they put a postve weght on t f the own payoff s hgher than the other player s payoff: IA players dslke nequty. WP players always put a postve weght on the other player s payoff. It s assumed that 0 σ ρ 1. Furthermore, t s assumed that θ 0 for all types. < WP WP The recprocty varable R s dependent on msbehavor of the other player. Therefore, we have to defne msbehavor. Player regards player s acton as msbehavor f s acton volates s norm and f cannot make sure that hs fnal utlty s at least as large as the one he would have got f had acted accordng to s norms. Let us assume that dfferent types of

6 6 players have dfferent norms. Furthermore, t s mportant that each player s wllng to behave accordng to hs own norms f he knows that the other actor follows the same norms. Otherwse such norms would be eroded quckly. Consequently, norms have to be best responses n a game n whch the player s playng aganst another player who has dentcal norms. As ths has to be true for both players we can now defne norms. Defnton 1: The norm behavor for players of type t s defned by the (Subgame Perfect) Nash Equlbra of the correspondng complete nformaton game (wth IA or WP preferences, respectvely, and R = 0) n whch two players of the same type play aganst each other. The correspondng equlbra shall be called normatve reference equlbra. Ths means that players always follow the norms f they know that the other player belongs to the same type of actors as they themselves. However, f they do not know the other player s type they may devate from ther own norm. It s mportant to remember that devatng from one s norm s only a necessary (but not a suffcent) condton for msbehavor. Imagne a stuaton n whch actor has devated from s norm but n whch player can guarantee hmself an even hgher utlty than n the normatve reference equlbrum. In ths case he has no reason to be angry and to punsh. In such a case t s rather mplausble that devaton from the norm wll trgger any negatve recprocty. To exclude such cases from trggerng recprocty we add a maxmn condton: Negatve recprocty (R = 1) s only trggered f a devaton from the norm occurs and the maxmn utlty for player n the remanng game s smaller than hs utlty n the normatve reference equlbrum. Otherwse R = 0. The exstence of dfferent types of players wth dfferent utlty functons means that the games we are gong to analyze are games wth ncomplete nformaton. Therefore, t seems natural to apply standard tools of game theory. Let the materal game be the standard representaton of a game wth payoffs only n monetary unts. Then the heterogenous utlty game of the orgnal materal game s the extended verson of the latter nto a game wth ncomplete nformaton n whch three types of players, SE, IA and WP subects, wth known shares n the populaton exst and payoffs are gven n type specfc utltes. Defnton 2: Heterogeneous Socal Preferences (HSP) Equlbra are gven by the (perfect Bayesan) Equlbra of the heterogeneous utlty game. Note that we have defned IA and WP preferences only for 2-player games yet. Consequently, HSP Equlbra accordng to Defnton 2 are also only defned for 2-player games. Although extensons to n player defntons are not dffcult there are several possble ways to extend IA

7 7 and WP preferences to more general settngs. As ths s not necessary for the analyss n the remanng part of ths paper, such generalzatons reman obectves for future work. Let us now turn to the 2 players approach, agan. In the followng sectons we shall show that ths smple extenson of the establshed models leads to a surprsng ncrease n predctve accuracy. 3. HSP Equlbra n the Charness-Rabn-Games In ths secton and n secton 4 we wll apply the HSP Equlbrum concept to 43 dfferent games. The frst 27 games are the 2-players games that are presented n Charness and Rabn (2002). The remanng games are 8 treasure games whch confrm tradtonal Nash Equlbra and 8 contradcton games that contradct standard Nash predctons taken from Goeree and Holt (2001). To be able to derve concrete predctons, we have to make some further assumptons concernng the parameters n the IA and WP utlty functons. In ther recent papers Fehr and Schmdt successfully use σ = 2 and IA 0 ρ < 1 to stck to ther IA model rather closely we adopt < IA to explan laboratory behavor. As we try σ and assume that ρ = To keep the analyss as smple as possble and to gve recprocty a suffcent mportance assume that n case of negatve recprocty IA players always behave as f they had less money than the other player,.e. they put negatve weght on the other player s payoff: IA IA θ IA = θ IA ρ IA σ IA f R = 1 and π > π, = σ IA f R = 1 and π = π 0 otherwse ( π π, R) so that n case of R = 1 the sum of σ and θ as well as the sum of ρ and θ always equals 2. The correspondng utlty functon of player thus s 3π 2π f π < π or R = 1 IA U ( π, π, R) = 0.25 π π f π > π and R = 0. π f π = π and R = 0 Wth regard to the WP parameters Charness and Rabn (2002) reman rather vague. Although they do estmate these parameters n ther estmaton wth the best ft they get σ = WP and ρ WP = the results gve us only lmted gudance for our approach because they assume homogenous actors. However, as there s not a sngle estmaton wth σ WP, ρ WP > 0.5

8 8 we regard 0.5 as an upper boundary of these parameters. Furthermore, they assume σwp ρ WP. Fnally, to get a qualtatve dstncton between ρwp and σ WP we assume that ρ WP = 0.5 and σ = 0.3 whch works out qute well n the followng applcatons. Such a WP value of ρwp expresses the dea of Welfare Preferences most closely as t gves equal weghts to both players payoffs. Furthermore, t seems qute plausble that players gve less weght to the other player s payoff f they get less money than the other one. Wth regard to recprocty we assume that n case of negatve recprocty WP subects behave lke SE actors,.e. they are strctly egostc. Ths corresponds qute ncely to Charness and Rabn s dea of concern wthdrawal : they wthdraw ther wllngness to sacrfce to allocate the far share toward somebody who hmself s unwllng to sacrfce for the sake of farness. Consequently, θ WP = θ WP σ WP f π < π and R = 1 π = ρwp f π > π and R = 1 so that 0 f π = π or R = 0 (, π, R) U WP ( π, π, R) 0.7 π π = 0.5 π π, π, f f f π < π π > π π = π and R = 0 and R = 0. or R = 1 Of course, these parameters are rather crude and subectve frst estmates whch only serve as a frst step n developng a model of heterogeneous socal preferences. Fnally, we have to make assumptons about the dstrbuton of types. A rough estmate 3 of the Charness-Rabn games s that strctly egostc actors amount to approxmately 50% of subects, nequty averse players make up about 15% and the remanng 35% of the players are WP actors. For the purpose of better ntuton about the nature of HSP Equlbra let us frst dscuss the equlbrum n one of the 27 games, game Barc1 (whch s dentcal to Berk13), more explc- 3 The rough estmate was carred out the followng way: Frst, decsons n the frst seven games of Charness and Rabn (2002) whch are pure dctator decsons were determned and then an estmaton of populaton shares was carred out. Then equlbra of all 27 games were determned assumng ths dstrbuton (wth generously rounded values of the populaton shares). Next, another estmaton of populaton shares was carred out. Ths procedure was repeated twce. Fnally, the populaton shares were, agan, generously rounded so that we get promnent numbers as populaton shares.

9 9 tly. The game conssts of two stages. In stage 1 player A chooses monetary payoffs of 550 for both players, (550, 550), or he lets player B choose between (400, 400) and (750, 375). 4 If R = 0 then the correspondng utltes of the three types of players are gven by 5 U SE (R = 0) U IA (R = 0) U WP (R = 0) Out 550, , , 550 Enter 400, , , , , , Left Rght Left Rght Left Rght Table 1: Utltes n Barc1/ Berk13 Frst, we have to determne the normatve reference equlbra,.e.the subgame perfect equlbra n the complete nformaton game where an IA player plays wth an IA player (or a WP player playng wth a WP player). As can easly be seen, nequty averse B players prefer Left and, antcpatng ths, nequty averse A players thus choose Out. The normatve reference equlbrum of IA subects then s (Out, Left). Analogously, B players wth WP prefer Rght and WP-A players choose Enter because > 550. The normatve reference equlbrum of WP players thus s (Enter, Rght). The normatve reference ponts determne whether negatve recprocty wll be trggered. Because WP players regard Enter as part of ther norm and as Out mmedately fnshes the game WP subects cannot show negatve recprocty n ths game. However, nequty averse subects regard Out as the normatvely adequate behavor. Furthermore, the maxmn value of nequty averse B players after Enter equals 400 whch s strctly less than ther utlty n the normatve reference equlbrum. Consequently, f A chooses Enter ths wll trgger ther negatve recprocty reacton. However, n ths game utltes of nequty averse players do not change f R = 1, as can easly be checked. Knowng ths, we can determne equlbrum behavor of all three types n stage 2. SE and IA actors choose Left and WP subects choose Rght. Gven the dstrbuton of the types ths means that the probablty that player B plays Left s Next we can calculate expected utlty of player A choosng Enter. For SE players EU(Enter) = = The frst number n parentheses corresponds to the monetary payoff of player A, the second number to B s payoff. 5 One can read the followng table n the followng way, too: In stage 1 of the game: player A chooses a row (Out or Enter) and n stage 2 player B chooses a column (Left or Rght).

10 10 whch s less than 550, hs utlty when choosng Out. Consequently, SE players choose Out. As can easly be checked, the same s true for IA and WP players. Consequently, the unque HSP Equlbrum conssts of all A players optng for Out and 65 percent of B players choosng Left. By and large, ths equlbrum s confrmed by expermental behavor. In experment Barc1 (Berk13) 96 percent (86 percent) of A players have chosen Out and 93 percent (82 percent) of B players have chosen Left. In the same way all 2 player experments n Charness and Rabn (2002) have been analyzed. In Table 2 the structure of all 27 experments s summarzed. P(Enter) (P(Left)) represents the percentage of A players (B players) that have decded to let B choose (that have chosen Left). The correspondng equlbra are gven n Table 3. # Name Experment / Game P(Enter) P(Left) 1 Berk29 B chooses (400,400) vs. (750,400).31 2 Barc2 B chooses (400,400) vs. (750,375).52 3 Berk17 B chooses (400,400) vs. (750,375).50 4 Berk23 B chooses (800,200) vs. (0,0) Barc8 B chooses (300,600) vs. (700,500).67 6 Berk15 B chooses (200,700) vs. (600,600).27 7 Berk26 B chooses (0,800) vs. (400,400).78 8 Barc7 A chooses (750,0) or lets B choose (400,400) vs. (750,400) 9 Barc5 A chooses (550,550) or lets B choose (400,400) vs. (750,400) 10 Berk28 A chooses (100,1000) or lets B choose (75,125) vs. (125,125) 11 Berk32 A chooses (450,900) or lets B choose (200,400) vs. (400,400) 12 Barc3 A chooses (725,0) or lets B choose (400,400) vs. (750,375) 13 Barc4 A chooses (800,0) or lets B choose (400,400) vs. (750,375) 14 Berk21 A chooses (750,0) or lets B choose (400,400) vs. (750,375) 15 Barc6 A chooses (750,100) or lets B choose (300,600) vs. (700,500) 16 Barc9 A chooses (450,0) or lets B choose (350,450) vs. (450,350) 17 Berk25 A chooses (450,0) or lets B choose (350,450) vs. (450,350) 18 Berk19 A chooses (700,200) or lets B choose (200,700) vs. (600,600) 19 Berk14 A chooses (800,0) or lets B choose (0,800) vs. (400,400) 20 Barc1 A chooses (550,550) or lets B choose (400,400) vs. (750,375) 21 Berk13 A chooses (550,550) or lets B choose (400,400) vs. (750,375) 22 Berk18 A chooses (0,800) or lets B choose (0,800) vs. (400,400)

11 11 23 Barc11 A chooses (375,1000) or lets B choose (400,400) vs. (350,350) 24 Berk22 A chooses (375,1000) or lets B choose (400,400) vs. (250,350) 25 Berk27 A chooses (500,500) or lets B choose (800,200) vs. (0,0) 26 Berk31 A chooses (750,750) or lets B choose (800,200) vs. (0,0) 27 Berk30 A chooses (400,1200) or lets B choose (400,200) vs. (0,0) Table 2: The Charness-Rabn experments / games. P(Left): Percentage that subects choose Left ; P(Enter): Percentage that subects choose Enter. # Name Player A P(Enter) Player B P(Left) SE IA WP SE IA WP P(Enter) P(Left) 1 Berk29 [0,1] 1 0 [0.15,0.65] 2 Barc Berk Berk Barc Berk Berk [0,1] [0.50,0.85] 8 Barc [0,1] [0.15,0.65] 9 Barc [0,118/140] [0.15,0.57] [118/140,1] [0.57,0.65] [0,1] / [0,0.50] Berk [0,1] 0 [0,1] 0.65 [0,0.50] [0,1] 0 [0,1] 0.15 [0.50,0.85] [0,1] 1 0 [0,1] 0 [0,1] [0.15,0.65] Berk [0,1] 0 [0,1] 0.15 [0,0.85] 12 Barc Barc Berk Barc Barc Berk Berk Berk [0,1] 0 [0.50,0.85] 20 Barc Berk Berk [0,1] 1 [0.50,0.85] 23 Barc Berk Berk Berk Berk Table 3: HSP Equlbra n the Charness-Rabn games

12 12 How well do these equlbra explan laboratory behavor? To answer ths queston we frst have decde how to handle multple equlbra. Table 3 shows that n 12 of 27 games there are, ndeed, multple equlbra. Of course, some of these equlbra explan behavor better than other equlbra of the same game. We analyze three scenaros: (a) We only take nto account the best equlbra. By ths we mean those equlbra that have the smallest mean absolute error (MAE) over both the Entry-Out and the Left-Rght decson. (b) Accordng to the same standard we take nto account only the worst equlbra and (c) we separately analyze those games whch have a unque equlbrum. Fgure 1 shows the percentage of subects playng Left or Enter (P_Left) and the correspondng best case probabltes of Left (cases 1 to 27) and Enter (cases 28 to 47) n HSP equlbra (Prog_HSP) P_LEFT PROG_HSP Fgure 1: HSP predctons and laboratory behavor Fgure 1 seems to ndcate that HSP covers the man qualtatve features of the behavor n the Charness-Rabn games. Ths mpresson s further strengthened by some statstcal measures 2 ( ) 2 2 of predctve accuracy. Now let = 1 ( PPL PL) ( PL µ ) R be the quascoeffcent of determnaton wth PPL as the predcted probablty of choosng Left (or Enter), PL as the actual percentage of Left (or Enter) and µ as the mean percentage of Left (or Enter) over all strateges. Then R 2 = Furthermore, the mean absolute error of the HSP Equlbrum s MAE =

13 13 If we take the best Nash Equlbrum as an estmator of laboratory behavor predctve accuracy s clearly lower (see Fgure 2). The quas-coeffcent of determnaton s even negatve (R 2 = 0.51) and the mean absolute error s much hgher (MAE = 0.263). Obvously, Nash Equlbra are a rather bad estmator of actual laboratory behavor P_LEFT PROG_NB Fgure 2: Subgame Perfect Nash Equlbrum predcton and laboratory behavor The Fehr-Schmdt model of nequty averson works a lttle better than Nash Equlbrum. Here we take parameters that the authors use n Fehr, Klen and Schmdt (2001) and Fehr Kremhelmer and Schmdt (2002),.e. σ IA = 2, ρ IA = 0.75 and assume that 60 percent of subects are SE actors and that 40 percent of subects are nequty averse. If, agan, we take only the best equlbra accordng to the Fehr et al. model we get R 2 = and MAE = The model clearly works worse than the concept of HSP Equlbra. To test the Charness-Rabn model of a populaton consstng only of (homogenous) WP ndvduals we assume that σ = 0.023, ρ = and θ = In addton, for ths test we accept Charness and Rabn s (2002, p. 840) defnton of msbehavor,.e. entry by A s characterzed as msbehavor n games 9, 11, 19, 20, 21, 23, 24, 25, 26, 27 of Table 2. Predctons of behavor are determned by the subgame perfect Nash Equlbrum gven the Charness-Rabn 6 These are the values of Charness and Rabn s best estmate. See Charness and Rabn (2002), p. 840.

14 14 utlty functon. It shows that the Charness-Rabn model does not work very well. R 2 = and MAE = 0.33 are both worse than the correspondng values of the Fehr-Schmdt approach. 7 However, comparng HSP Equlbra wth the other approaches seems to be a lttle bt unfar because the exstence of three types of actors leaves more room for ntermedate probabltes of choosng Left or Enter. Therefore, we can carry out another test that does not have such a compettve advantage n favor of HSP Equlbra. Ths s done by countng the number of correct predctons. By ths we mean that one of the two followng condtons holds: (a) Both the predcted probabltes of Enter (or Left) and the real percentage of Enter (or Left) are greater than or equal to 0.5; (b) Both the predcted probabltes of Enter (or Left) and the real percentage of Enter (or Left) are smaller than or equal to 0.5. It shows that HSP Equlbra are correct n 46 out of 47 cases, Fehr-Schmdt predctons are correct n 39 cases and Charness-Rabn predctons are correct n 37 cases. Agan, HSP Equlbra outperform the other concepts. Let us now turn to the case where we take the worst HSP Equlbrum n each game. Of course, n ths case R 2 and MAE decrease. However, R 2 = 0.19 and MAE = 0.19 are stll better than the correspondng values of best Fehr-Schmdt and Charness-Rabn equlbra. Fnally, f we only take nto account games wth unque HSP Equlbra then R 2 = and MAE = 0.13 whch are qute close to the case of the best equlbra. In sum, the concept of HSP Equlbra s able to organze the data for behavor n the experments of Charness and Rabn (2002) qute well and seems to be clearly superor to Nash Equlbra and the equlbra resultng from the approaches of Fehr-Schmdt and Charness-Rabn. It remans unclear, however, whether HSP Equlbra do as well n other games that have not been created for the specal purpose of analyzng socal preferences. So let us now look at some other games. 7 I also tested the concept of Logt Equlbra (Anderson, Goeree and Holt 1997). In ths case R 2 = 0.23 and MAE = Furthermore, I combned the Logt Equlbra concept wth the Charness-Rabn model. Ths lead to R 2 = and MAE = Consequently, t seems that the dea of stochastc game theory helps ncreasng predctve accuracy. However, n both cases the HSP equlbrum stll seems to be superor.

15 15 4. Treasures and Contradctons? An Applcaton to the Goeree-Holt-Games Recently, Jacob Goeree and Charles Holt (2001) analyzed ten pars of games. Each par conssted of two smlar games wth dentcal Nash Equlbra but dfferences n the absolute magntude of payoffs. Goeree and Holt showed that n each case n one of the two versons of the game Nash Equlbra descrbed laboratory behavor very well (these are called the treasures ) and that n the other verson (the contradctons ) Nash Equlbra made very poor predctons. In ths secton we shall analyze eght of the ten pars of games, agan. 8 We use the same parameters of ρ t, σ t and θ t and the same dstrbuton of types as n secton 3. It wll be shown that HSP Equlbra can (partally) solve the puzzle that Goeree and Holt have found The One-Shot Traveler s Dlemma Game In ths secton, let us consder the followng game: Two players ndependently choose an nteger number (N) between 180 and 300. If both players choose the same number they are pad ths amount n money. Otherwse each player gets the mnmum of both numbers plus (mnus) a transfer payment (T) from the player who chose the hgher number to the player wth the lower number. Let T > 1. The standard Nash Equlbrum of ths game s that both players choose 180. Otherwse each has an ncentve to underbd the other so that he can get the transfer payment. Ths Nash Equlbrum s unque and, furthermore, t s ndependent of T. Goeree and Holt (2001, ) carred out two laboratory treatments, one wth T = 5 and another wth T = 180. It showed that n the latter case laboratory behavor was close to Nash Equlbrum. About 80 percent chose numbers very close to 180. In contrast to ths, laboratory subects dd not at all behave accordng to the Nash predcton n the treatment wth T = 5. Here, even slghtly more than 80 percent of laboratory subects chose numbers that were close to the maxmum, 300, whch s not part of any Nash Equlbrum. Next, consder how HSP Equlbra correspond to laboratory behavor. Let us begn wth the T = 180 treatment. It shows that n ths case HSP works as well as Nash Equlbra because both concde: 8 The remanng two pars of games that deal wth ncomplete nformaton games are not too nterestng here because the concept of HSP Equlbra always has the character of ncomplete nformaton games anyway.

16 16 Result 1: The one shot traveler s dlemma game wth T = 180 has a unque HSP Equlbrum n whch all types, SE, IA and WP actors, choose 180 regardless of ther roles as player 1 or player 2. Sketch of the Proof: (a) One can easly verfy that no player and no type has an ncentve to devate from the Equlbrum. Therefore t consttutes an equlbrum. (b) Assume that there exsts another HSP Equlbrum n whch N > 180 s played wth postve probablty. Let N max be the hghest number of the equlbrum canddate that s played wth postve probablty. Frst, we can easly rule out that t could be an equlbrum that all types play the same number N > 180 wth probablty one as t s always advantageous for SE players to underbd the others by one unt. Second, t can be shown that regardless of the strateges of IA and WP players underbddng ncentves of SE players are so strong that only N = 180 remans an equlbrum canddate for SE actors. Thrd, gven that SE players always play 180 and that ther share of the populaton s 0.5, t can be shown that no IA player would choose to play N max wth postve probablty because they always prefer playng 180 to N max. Furthermore, t can be shown that expected utlty of IA subects playng N > 180 s always negatve whereas they can realze a strctly postve expected utlty f they choose 180. Consequently, N = 180 remans the only canddate for equlbrum behavor of IA players. Fourth, gven that SE and IA players play 180, the best remanng scenaro for WP subects playng N > 180 gves them an expected utlty of whch s strctly less than 180, the expected utlty of playng N = 180. Thus, there cannot exst another equlbrum n whch N > 180 s played wth strctly postve probablty. Let us now turn to the case T = 5. Here we have multple HSP Equlbra: Result 2: In the one shot traveller s dlemma game wth T = 5 there exsts more than one HSP Equlbrum. (1) One Equlbrum s that all types choose N = 180. (2) Another HSP Equlbrum contans mxed strateges. Here, WP players choose N WP = 300 and IA players choose N IA = 288. In both cases players choose ther numbers wth probablty one. In contrast to ths, SE subects play a mxed strategy wth (rounded) probabltes: P(299) = 0.06; P(298) = 0.092; P(297) = ; P(296) = ; P(295) = 0.1; P(294) = ; P(293) = ; P(292) = ; P(291) = The second equlbrum corresponds well to the behavor of subects n the laboratory. The ntuton s that WP and IA players play some knd of a coordnaton game. Wth T = 5 t s

17 17 suffcent for WP players that the other WP players choose 300, regardless of how the other types behave. In ths case they prefer N WP = 300. IA subects have a very strong averson of gettng less than other players. In general they could coordnate ther behavor on any number that s smaller than (or equal to) SE and WP players choces. However, only N IA = 288 makes SE subects that play ther mxed strateges (gven above) ndfferent to all numbers between 291 and 299. Fnally, SE players experence a tradeoff between effcently underbddng WP players (299) and between beng underbdden by IA subects. However, because 288 s suffcently below N WP = 300 and because WP players have a much larger share n the populaton t pays for SE subects to rsk beng exploted by IA subects. Note that ths equlbrum works only f T = 5,.e. t s mportant for ths equlbrum that beng underbdden by others s not too costly. Ths s why WP subects can coordnate on 300 and SE players rsk beng underbdden by IA subects. Summarzng, HSP Equlbra explan laboratory behavor n the traveller s dlemma n both treatments wth low and hgh transfer payments qute well Matchng Pennes Games Three varatons of a Matchng Pennes game are the subect of ths secton. Table 4 gves the basc structure of the games n whch the players have to move smultaneously. Player 2 Left Rght Player 1 Top A,40 40,80 Bottom 40,80 80,40 Table 4: Matchng Pennes Games In the symmetrc game A = 80, n the asymmetrc game A = 320 and n the reverse asymmetrc game A = 44. In all three varatons there does not exst a Nash Equlbrum n pure strateges. In partcular, n the symmetrc game the equlbrum n mxed strateges conssts of both subects playng each strategy wth equal probablty. Note that ths behavor remans the Nash Equlbrum strategy for player 1 n the other varatons, too. Nash Equlbrum descrbes laboratory behavor n the symmetrc case very well. Here, player 1 plays Top n 48 percent of all subects and player 2 chooses Left n 48 percent, too. In contrast to ths, Nash Equlbrum fals to explan player 1 s behavor n the other cases. In the asymmetrc game 96 percent of subects chose Top and 84 percent of players 2 selected Rght. In the reversed asymmetrc game 92 percent of the row players took Bottom and 80 percent of

18 18 the column players decded to take Left. Nash Equlbrum thus only explans behavor n the symmetrc game whch leads Goeree and Holt (2001, 1407) to summarze: the Nash mxed-strategy predcton seems to work only by concdence... Let us now turn to HSP Equlbra. In the symmetrc game we get the followng equlbra: Result 3: In the symmetrc Matchng Pennes Game all strategy combnatons are HSP Equlbra whch fulfll the followng condtons: [ Left] p [ Left] p[ Left] p = SE IA [ Top] p [ Top] [ Top] p p = SE IA. For example, f all types mx Left-Rght or Top-Bottom wth probabltes (0.5, 0.5) ths consttutes a HSP Equlbrum. The same s true for SE subects choosng Top or Left and the other types choosng Bottom of Rght. In any case, the aggregate probablty of choosng Left or Top must be 0.5. Obvously, ths explans behavor exactly as good as Nash Equlbrum does. Consder now the asymmetrc game. In ths case column players of type WP have a domnant strategy (Left) so that one can derve the next HSP Equlbrum that dffers fundamentally from the Nash Equlbrum. Result 4: In the asymmetrc Matchng Pennes Game there exsts a unque HSP Equlbrum wth ( p [ Top], p [ Top], p [ Top], p [ Left], p [ Left], p [ Left] ) = ( 1,1,1,0,0, 1) and SE IA WP SE IA WP,.e. all row players choose Top, SE and IA column players choose Rght and WP column players choose Left. Ths means that the equlbrum aggregate probablty of playng Top equals 1 (expermental behavor: 96 percent) and the aggregate probablty of playng Left s 0.35 (expermental behavor: 16 percent). In contrast to Nash Equlbrum the concept of HSP Equlbra reacts to the payment varaton! A smlar pcture can be drawn n the reversed asymmetrc game: Result 5: In the reversed asymmetrc Matchng Pennes Game there exsts a unque HSP Equlbrum wth ( p [ Top], p [ Top], p [ Top], p [ Left], p [ Left], p [ Left] ) =,1,0,1,1, SE IA WP SE IA WP. Consequently, the aggregate probablty that row players play Top equals 0.30 (compared wth 8 percent n the laboratory) and the aggregate probablty of Left s 0.91 (expermental 57 77

19 19 probablty: 80 percent). Agan, HSP Equlbra explan the man qualtatve features of behavoral change due to the varatons of payoffs. Once more, HSP Equlbrum turns a contradcton nto a treasure A Coordnaton Game wth a secure outsde opton In ths secton we analyze coordnaton games wth a secure outsde opton. However, the outsde opton s domnated by a mxed strategy of Left and Rght so that t should never be part of a Nash Equlbrum. Table 5 gves the structure of the games: Player 2 Left Rght Secure Player 1 Top 90,90 0,0 A,40 Bottom 0,0 180,180 0,40 Table 5: Coordnaton Games wth outsde opton There are two versons of the game. In the frst A = 0 and n the second A = 400. The games have three equlbra: (1) (Bottom,Rght), (2) (Top,Left) and an equlbrum n mxed strateges (3) ( p p ), p ) =,, Left, Rght Top Note that the Nash Equlbra are ndependent of A. In the A = 0 treatment 96 percent (84 percent) of the row players (column players) have chosen Bottom (Rght),.e. the strateges for the pareto domnant Nash Equlbrum. 80 percent of the subects managed to coordnate on ths equlbrum. Ths was dfferent n the A = 400 treatment. Here only 64 percent (76 percent) of the row (column) players have chosen Bottom (Rght) and only 32 percent of the pars coordnated on ths equlbrum. More than 50 percent of the outcomes were uncoordnated non-nash outcomes. Agan, Nash Equlbrum was a bad predctor for one of the versons (A = 400). It turns out that HSP Equlbra n the A = 0 treatment concde wth Nash Equlbra,.e. n the treasure verson HSP Equlbra do equally well. However, HSP Equlbra n the A = 400 verson dffer from Nash Equlbra. Result 6: There are three HSP Equlbra n the extended coordnaton game wth A = 400: (1) SE, IA and WP players play (Bottom, Rght). (2) All types of row players play Top. SE and IA column players choose Left and WP column players choose Secure. (3) Row players: SE and IA subects play Top and WP subects play Top wth probablty

20 20 1/21. Column players: SE and IA subects play Left wth probablty 442/1755 and Rght wth probablty 1313/1755. WP subects choose Secure. The thrd HSP Equlbrum gves an aggregate probablty that players choose Top of 2/3. The aggregate probablty of Left s about 0.16, the probablty of Rght (Secure) s 0.49 (0.35). Consequently, only about 27 percent of the pars can be expected to coordnate on one of the two Nash Equlbra. Obvously, ths estmate looks more pessmstc than the expermental experence. Nevertheless t better fts the data than Nash Equlbra A Mnmum-Effort Coordnaton Game Ths game s a specal knd of a team problem. Each of the two players smultaneously mn e e ce,, = 1, chooses hs effort e and payoffs are determned by the formula = {, } 2 and. Here c represents a cost parameter that s assumed to be smaller than 1 and ce gves ndvdual s costs. Each ndvdual gets the output mn{e, e }. It s straghtforward that there are multple Nash Equlbra. In fact, every feasble effort s part of an equlbrum f all actors coordnate on the correspondng value. Ths s true for any c < 1 and s ndependent from the magntude of c. In the expermental desgn by Goeree and Holt (2001) efforts could be any nteger number between 110 and 170. They carred out two treatments, one wth c = 0.1 and one wth c = 0.9. It shows that behavor n these two treatments clearly dffers. Wth c = 0.1 choces of effort concentrate near the upper boundary, 170. In contrast to ths, most subects have chosen efforts near 110, the lower boundary, f c = 0.9. Wthout doubt, both treatments are n accordance wth Nash Equlbrum. However, the concept of Nash Equlbrum gves no hnts why there s so much dvergence between the treatments. Unfortunately, there are also multple HSP Equlbra. In fact, f all types coordnate on any feasble effort ths represents a HSP Equlbrum, too. Consequently, as wth Nash Equlbrum, HSP Equlbra are n accordance wth expermental behavor. Ths, of course, s not surprsng because any effort choce s part of one of the many Nash Equlbra of the game. Even worse, the behavoral dfferences between the two treatments cannot be explaned, ether. One mght argue, however, that wth c = 0.1 the coordnaton problem s weakened. For example, even f all SE and IA players choose e = 110 t s an optmal behavor for WP players to coordnate on e = 170,.e. t s suffcent for WP subects that coordnaton only between them s arranged properly. Gven that all WP players coordnate on e = 170 t s optmal for all SE players to coordnate on 170, too, even f all IA subects stck to e = 110. Fnally, as y

21 21 all types playng 170 s a HSP Equlbrum, IA players would follow the other types of players. Nevertheless, such a knd of reasonng s not part of the concept of HSP Equlbrum so that we have an unresolved equlbrum selecton problem The Kreps Game In the prevous secton we dealt wth a game that had multple equlbra. Every feasble effort could be part of a Nash Equlbrum and there was no relable way to dscrmnate between the Nash Equlbra. In ths secton Nash even works worse. There are, agan, multple Nash Equlbra but for one player the only pure strategy whch s not part of any Nash Equlbrum s chosen most of the tmes n one of the treatments. The structure of the Kreps Game s gven n Table 6. Left Mddle Non-Nash Rght Top 200,50 0,45 10,30 20, 250 Bottom 0, , ,30 I: 50,40 (II: 350,400) Table 6: Two varants of the Kreps Game The only dfference between the varants can be found n the cell (Bottom,Rght). In varant I payoffs are rather small and n varant II they are much larger. However, both varants of the Kreps Game have two Nash Equlbra n pure strateges, (Top,Left) and (Bottom,Rght). Furthermore there exsts one Nash Equlbrum n mxed strateges n whch the column player randomzes between Left (wth probablty 1/21) and Mddle (wth probablty 20/21) and the row player randomzes between Top (wth probablty 150/155) and Bottom (wth probablty 5/155). Consequently, only the pure column strategy Non-Nash s not part of a Nash Equlbrum n both varants of the game. Here, varant II represents the treasure treatment n whch 96 percent of the subects n the laboratory have chosen Bottom and 84 percent have chosen Rght. So behavor s n accordance wth the pareto optmal Nash Equlbrum. However, n treatment I wth relatvely low payoffs behavor of subects n the laboratory was completely at odds wth the concept of Nash Equlbrum. 68 percent of the column players have chosen Non-Nash, the only strategy that s not part of any Nash Equlbrum. In contrast to Nash Equlbrum HSP Equlbrum does not rule out the pure strategy Non- Nash:

22 22 Result 7: Varant I of the Kreps Game has multple HSP Equlbra. One of them s characterzed by the followng behavor: SE and IA column players choose Non-Nash and WP column players choose Non-Nash wth probablty 57/77 and Left wth probablty 20/77. SE row players choose Top wth probablty 107/140 and Bottom wth probablty 33/140. IA row players choose Bottom and WP row players choose Top. Accordng to ths HSP Equlbrum, Non-Nash s played wth an aggregated probablty of approxmately 91 percent (compared to 68 percent n the laboratory) and the probablty of Left s about 9 percent (26 percent). The aggregate probablty of choosng Top s 73.2 percent (compared to 68 percent n the laboratory). HSP Equlbrum thus gves a clearly better predcton than Nash Equlbrum. It should be mentoned, however, that the HSP Equlbrum from Result 7 s also vald for treatment II where t s hardly played at all. In addton, note that only n varant II there exsts a HSP Equlbrum n whch all types of row players choose Bottom and all types of column players choose Rght Should you trust others to be ratonal? Let us now turn to games n the extensve form. What s of partcular nterest here s whether the logc of backward nducton holds relably. Ths means that we have to analyze whether players who move frst should trust ther followers to behave ratonal and whether they should beleve threats that are not credble. In ths secton we concentrate on the frst queston. Look at the game n Fgure 3 n whch the frst player has to decde whether to stop the game and choose a safe payoff or whether he should let the second player choose between two other payoff combnatons. Player 1 S R (80,50) P Player 2 N I: (20,10) II: (20,68) (90,70) Fgure 3: An extensve form game

23 23 Agan, there are two varants of the game. The dfference between the varants conssts of the payoff of player 2 n case of an R-P play. In varant I player 2 looses much money f he chooses P nstead of N. In varant II payoff dfferences for player 2 are rather small f he has to choose between P and N. In any case there s the same unque subgame perfect Nash Equlbrum, namely (R,N). Goeree and Holt (2001) show that laboratory behavor fts well to the Nash predcton n varant I: 84 percent of the frst movers have chosen R and 100 percent of the second movers selected N. However, thngs are qute dfferent n varant II. Here only 48 percent of the frst movers decded to take R and 75 percent of the second movers have chosen N. Although most second movers behaved ratonally, there are suffcently many subects who devate from the ratonal second move so that t pad on average for the frst movers to choose the nonequlbrum strategy S. Or to put t another way: In varant II the frst movers have good reason not to trust the other players to behave ratonally. Agan, Nash Equlbrum does well n one treatment but fals n the other one. In varant I, the treasure treatment, the HSP Equlbrum concdes wth the Nash Equlbrum. Consequently, HSP Equlbra are exactly as successful n ths varant as Nash Equlbra. However, varant II has a dfferent HSP Equlbrum. Result 8: HSP Equlbrum play n varant II of the extensve form game s characterzed by the followng behavor: SE and IA types of the frst movers choose S and WP subects take R. The second movers choose N f they are SE or WP types. They prefer P f they are IA actors. The ntuton for ths result s that nequty averse players prefer P to N. The reason for ths s that they dslke beng n the dsadvantaged poston much more than they dslke nequty n ther own favor. 9 Because IA second movers devate from the subgame perfect Nash behavor SE and IA subects prefer S as frst movers. Consequently, HSP Equlbrum predcts that 65 percent of the frst movers choose S (compared to 52 percent n expermental behavor) and only 35 percent choose R. Furthermore, accordng to HSP Equlbrum 15 percent of the second movers opt for P (compared to 25 percent n the experments). Agan, HSP Equlbrum s a better estmator of actual behavor than subgame perfect Nash equlbrum. 9 Ths, of course, s due to the parameters for nequty averson taken from Fehr, Klen and Schmdt (2001) and Fehr, Kremhelmer and Schmdt (2002).

24 24 Fnally, note that the secton ttle should you trust others to be ratonal that has been adopted from Goeree and Holt (2001) does not really ft to the game when looked upon from a HSP perspectve. In equlbrum all players behave ratonally, so that one can trust n the others ratonalty. However, what the frst movers do not know s the motvaton of the second movers. Consequently, from a HSP pont of vew the problem consdered here s better descrbed by the queston should you trust others not to be envous? 4.7. Should you beleve a threat that s not credble? In ths secton we deal wth the problem of credble vs. ncredble threats. Many Nash Equlbra nvolve ncredble threats that make them rather mplausble. For ths reason Selten (1965) ntroduced the crteron of subgame perfectness to rule them out. However, t has often been shown that subgame perfectness does not always ft to actual behavor very well. Let us now look at two games wth ncredble threats that are represented n Fgure 4. Player 1 S R (70,60) P Player 2 N I: (60,10) II: (60,48) (90,50) Fgure 4: Two extensve form games wth ncredble threats The structure of the games are very smlar to those n the prevous secton. However, n the games consdered here player 2 dslkes player 1 to play R. The Nash Equlbrum (S,P) works only wth the use of the ncredble threat of player 2 to play P. Subgame perfectness rules ths out so that (R,N) remans the unque subgame perfect Nash Equlbrum. Agan, the only dfference between the two varants of the game conssts of dfferent payoffs n case of (R,P) play. In varant I playng P s very costly to player 2 because he looses 80 percent of hs profts. In contrast to ths varant II s a game n whch playng P only costs 4