ERC. A Theory of Equity, Reciprocity and Competition. Gary E Bolton Smeal College of Business Penn State University, USA

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1 ERC A Theory of Equty, Recprocty and Competton Gary E Bolton Smeal College of Busness enn State Unversty, USA & Axel Ockenfels Faculty of Economcs and Management Unversty of Magdeburg, Germany September 1997 We demonstrate that a smple model, constructed on the premse that people are motvated by both ther pecunary payoff and ther relatve payoff standng, explans behavor n a wde varety of laboratory games. Included are games where equty s thought to be a factor, such as ultmatum, two-perod alternatng offer, and dctator games; games where recprocty s thought to play a role, such as the prsoner s dlemma and the gft exchange game; and games where compettve behavor s observed, such as Bertrand and Cournot markets, and the guessng game. Correspondence Bolton: 309 Beam, enn State Unversty, Unversty ark, A 16802, USA; (814) ; fax (814) ; geb3@psu.edu. Ockenfels: Unversty of Magdeburg, FWW, ostfach 4120, D Magdeburg, Germany; (+391) , fax (+391) ; axel.ockenfels@ww.un-magdeburg.de.

2 1. Introducton: The need for a unfyng explanaton The varous areas of nqury that consttute expermental economcs appear at tmes to be surveyng dstnct and solated regons of behavor. What we see n experments nvolvng market nsttutons s usually consstent wth standard notons of 'compettve' self-nterest. But other types of experments appear to foster sharply dfferent conduct. 'Equty' has emerged as an mportant factor n barganng games. 'Recprocty, of a type that dffers from the standard strategc concepton, s often cted to explan behavor n games such as the prsoner s dlemma. Many economsts wonder what-f-anythng connects these behavoral patterns. The ssue goes to the heart of what t s that expermental economcs can hope to accomplsh, f only because economsts have tradtonally placed a hgh value on generalty. If no connecton can be found, we are left wth a set of dsjont behavoral charts, each vald for no more than a lmted doman. But to the extent a common pattern can be establshed, laboratory research presents a broad, and a potentally powerful map of economc behavor. In ths paper, we descrbe a smple model we call ERC to denote the three mportant knds of behavor the theory captures: equty, recprocty and competton. We show that ERC s consstent wth a wde varety of expermental observatons gathered by many ndependent nvestgators. ERC s smple to apply n part, because t s not a radcal departure from standard modelng technques. The major nnovaton s the premse that, along wth the pecunary payoff, ndvduals are motvated by a 'relatve' payoff, a measure of how the pecunary payoff compares to that of the other players. Dfferent games present dfferent sets of tradeoffs between pecunary and relatve gans. What ERC demonstrates and the pont we wll stress s that a smple model of how pecunary and relatve motves nteract, organzes a large, and seemngly dsparate group of experments as one consstent pattern of behavor. Three experments provde a sense of the breadth of ERC. One experment, reported by Forsythe, Horowtz, Savn and Sefton (1994), nvolves two elementary strategc stuatons: the ultmatum game and the dctator game. In the ultmatum game, the proposer offers a dvson of $10, whch the responder can ether accept or reject; the latter acton leaves both players wth a payoff of zero. The dctator game dffers only n that the responder has no choce but to accept. The standard perfect equlbrum analyss of both games begns wth the assumpton that each player prefers more money to less. Consequently, the responder n the ultmatum game should accept all postve offers. Gven ths, the proposer should offer no more than the smallest - 1 -

3 monetary unt allowed. In the dctator game, the responder has no say, so the proposer should keep all the money. So, n both games, the proposer should end up wth vrtually the entre $10. Fgure 1 here. Fgure 1 dsplays the amounts proposers actually offered. Whle there s a great deal of heterogenety, average offers for both games are clearly larger than mnmal. Varous authors have gven these results an equty nterpretaton (Roth, 1995, provdes a survey). But equty s nsuffcent to explan everythng n Fgure 1. Offers are planly hgher n the ultmatum game. Ths has to do wth a fact well-known to those who do ultmatum experments: Responders regularly turn down proportonally small offers. So proposers adjust ther offers accordngly. roposers may care about equty they do gve money n the dctator game but t appears that t s responder concern for equty that drves the ultmatum game. Hence Fgure 1 llustrates a subtle nterplay between equty and strategc consderatons an nterplay that ERC captures. 1 The second experment, performed by Roth, rasnkar, Okuno-Fujwara and Zamr (1991), concerns a smple aucton market game. A sngle seller has one ndvsble unt of a good to offer nne buyers. Exchange creates a fxed surplus. Buyers smultaneously submt offers. The seller s then gven the opportunty to accept or reject the best offer. All subgame perfect equlbra have the seller recevng vrtually the entre surplus. Ten rounds of the aucton market experment were performed n each of four countres. In every case, by round 10, the transacton prce had converged to subgame perfect equlbrum. Hence the experment produces behavor that s remarkably consstent wth standard theory. The same study examned ultmatum game play. Whle there were some dfferences across countres, the qualtatve pattern was the same n all four places: offers were generally hgher than subgame perfecton predcts and a sgnfcant number of offers were rejected. Are the motves behnd market behavor fundamentally dfferent than those behnd the ultmatum game? ERC answers no, the same motvaton suffces to explan both games. 1 The results from dctator and ultmatum have been shown to be very stable when the experment s performed wth comparable nstructons. Forsythe et al. show that dctator gvng s stable wth respect to tme. Hoffman, McCabe, Shachat and Smth (1995) replcate the Forsythe et al. dstrbuton. Bolton, Katok and Zwck (forthcomng) demonstrate that the amount the dctator gves s stable wth respect to varous game manpulatons. Gvng behavor s not restrcted to people: capuchn monkeys gve food n what s an anmal verson of the dctator game; see de Waal (1996, p. 148). See Roth (1995) for a survey of the many ultmatum game experments

4 The thrd experment, by Fehr, Krchsteger and Redl (1993), nvolves what s sometmes referred to as the gft exchange game. Subjects assgned the role of frms offer a wage to those assgned the role of workers. The worker who accepts the wage then chooses an effort level. The hgher the level chosen, the hgher the frm's proft and the lower the worker's payoff. The game s essentally a sequental prsoner's dlemma, n whch the worker has a domnant strategy to choose the lowest possble effort. The only subgame perfect wage offer s the reservaton wage. Fgure 2 here. Fgure 2 compares the effort level actually provded wth the wage offered. Behavor s clearly nconsstent wth the horzontal lne that ndcates the equlbrum predcton. In fact, there s a strong postve correlaton between wage and effort. Ths s sometmes taken as evdence for recprocty (ths s the nterpretaton offered by Fehr et al.). The dctonary defnes recprocty as a "return for somethng done." Whle there s surely some relatonshp between ths concept and equty, the two are not equvalent. Dctator game gvng may nvolve an assessment of what s equtable, but t does not nvolve recprocty as defned by the dctonary. The postve correlaton evdent n Fgure 2 suggests to some that we need more than farness to explan behavor n the labor market game. Or do we? ERC mples that we do not. We begn by layng out the basc ERC model (secton 2). We then show that ERC can account for a varety of patterns reported for dctator, barganng, and related games, ncludng the Forsythe et al. experment (secton 3). Next we explan why the model predcts compettve behavor for a class of market games ncludng the Roth et al. experment, and the guessng game (secton 4). We then descrbe some basc results havng to do wth one-shot dlemmas. We can say more wth a parametrc model. We ft the smplest possble verson to the Fehr et al. data (secton 5). We show that the ft s robust by estmatng the Berg, Dckhaut and McCabe s (1995) nvestment game experment. We then make some observatons concernng repeated dlemmas (secton 6). We are not alone n our pursut. After layng out what ERC can do, we compare wth other approaches found n the lterature (secton 7). One model, Bolton (1991), does well explanng smple barganng games, but fals wth others. It turns out that the comparatve model s almost a specal case of ERC (secton 3.3)

5 2. The basc ERC model Snce the mmedate purpose s to explan lab data, our gudng crteron n constructng the model was that the mpled hypotheses should be both applcable to the lab envronment, and lab testable. Lab subjects can have no better than ncomplete nformaton about how ther game partners trade-off pecunary and relatve payoffs, and ths s what our propostons assume. On the other hand, n order to test the propostons, the nvestgator must be able to relably measure the underlyng trade-offs. We have found that much of what we need to know has to do wth the thresholds at whch behavor devates from the standard self-nterest assumpton. Ths nformaton s readly recovered from dctator and ultmatum game data, and we demonstrate throughout the paper that knowng the dstrbutons of these thresholds s suffcent to characterze many phenomena. 2.1 Formal statement of the model We concern ourselves wth n - player lab games, = 1,2, n, where players are randomly drawn from the populaton, and anonymously matched (face-to-face play s a known complcatng factor). All game payoffs are monetary and non-negatve, y 0 for all (ths s relaxed n secton 6). We assume that f a subject plays a game multple tmes, she never plays wth any partcular subject more than once. We can therefore analyze each game as one-shot. Each player acts to maxmze the expected value of hs or her motvaton functon, where and v = v y, σ ) (2.1) ( y / c, f c > 0 σ = σ ( c, y ) = s s relatve share of the payoff, 1/ n, f c = 0 n c = j= 1 y j s the total pecunary payout. Motvaton functons may be thought of as a specal class of expected utlty functons. We prefer motvaton functon because t emphaszes that (2.1) s a statement about the objectves that motvate behavor durng the experment. The weghts ndvduals gve these objectves may well change over the long-term, wth changes n age, educaton, poltcal or - 4 -

6 relgous belefs, etc. (Ockenfels and Wemann, 1996). It s, however, suffcent for our purposes that the trade-off be stable n the short term, for the duraton of the experment. 2 The formulaton n (2.1) s very smlar to that used by Bolton (1991). When extended to games wth more than two players, (2.1) specfes preferences over how the payoff s dstrbuted between self and others, but does not capture any preference there mght be over how the payoff s dstrbuted among the others. Whle ths level of abstracton s suffcent to explan much of what we see n varous games, there s evdence that (2.1) s more accurate than ths defense mght suggest. In secton 3.5, we dscuss several recent experments that fnd that subjects pay lttle attenton to how the payoff s dstrbuted across the rest of the group. The followng assumptons characterze (2.1): A0. v s contnuous and twce dfferentable on the doman of ( y, σ ). A1. Narrow self-nterest: v 1 0, v A2. Comparatve effect: v = 0 for σ = 1/n, and v 0. 2 < 22 A0 s for mathematcal convenence. A1 s a slghtly weakened verson of the standard assumpton made about preferences for money. A2 states that, holdng the pecunary argument fxed, the motvaton functon s concave down n the relatve argument, wth a maxmum around the allocaton at whch ones own share s equal to the average share. Ths assumpton mples that equal dvson has collectve sgnfcance hence we refer to equal dvson as the socal reference pont. 3 The lab data exhbts a great deal of heterogenety. The theory accounts for ths by postng a tenson, or trade-off, between adherng to the reference pont (the comparatve effect) and achevng personal gan (narrow self-nterest 4 ). Indvduals are dstngushed by how ths tenson s resolved. Much of what we need to know about ths tenson s captured by the 2 rasnkar (1997) examnes three large ultmatum game data sets and concludes that the trade-off s stable even wth repeated play. An objecton sometmes rased to the motvaton approach s that one can explan anythng by changng the utlty functons. Ths objecton mplctly assumes there s no way to nvaldate the functonal specfcaton. In the lab, however, we can, and often do, perform these types of valdaton tests. 3 A2 runs counter to the hypothess that people want to be frst n payoff rankng (Duesenberry, 1949). By ths hypothess, we would always see domnant strategy play n prsoner s dlemma and publc goods games, snce ths strategy s best from both a pecunary and relatve perspectve. Many people, however, fal to play domnant strategy n these games (see sectons 5 and 6). The equal splt behavor n dctator games also contradcts the hypothess

7 thresholds at whch behavor dverges from the narrow self-nterest assumpton. Each player has two thresholds, r (c) and s (c), defned as follows (note that y = cσ ): r ( c) : = arg max v ( cσ, σ ), c > 0 and s (c): v ( cs, s ) = v (0,1/ n), c > 0, s 1/n σ As we demonstrate n secton 3, r corresponds to the dvson that fxes n the dctator game, and s corresponds to s rejecton threshold n the ultmatum game. ostulates A0 to A2 guarantee there s a unque r [1/n, 1] and a unque s (0, 1/n] for each c. ostulate A3 provdes an explct characterzaton of the heterogenety that exsts among players. Let r f and s f be densty functons. A3. Heterogenety: For all c > 0: f r ( r c) > 0, r [ 1/ n, 1] and f s ( s c) > 0 s (0, 1/n]. Hence we assume that the full range of thresholds s represented n the player populaton. 2.2 A useful two-player game example It wll be useful to have an example motvaton functon to llustrate some key ponts as we go along, although we emphasze that we wll not use the example to prove any propostons. Consder the addtvely separable motvaton functon for player, nvolved n a two-player game (we contnue to wrte y as cσ ), b v ( c 2 2 σ, σ ) = acσ ( σ 1/ 2) ; 0, b > 0 a. The component n front of the frst mnus sgn s smply an expresson of standard (rsk neutral) preferences for the pecunary payoff. The component after the frst mnus sgn delneates the nfluence of the comparatve effect. In essence, the further the allocaton moves from equal dvson, the hgher the loss from the comparatve effect. Fgure 3 dsplays a partcular parameterzaton of (2.2). Fgure 3 here. 4 The reason we nsst on the narrow qualfer s that we are not at all convnced that any of the behavor mpled by ERC s altrustc. See remarks n secton

8 The functonal form (2.2) allows us to express the range of heterogenety posted by A3 n a very succnct form. A player s type s characterzed by the margnal rate of substtuton between pecunary and relatve argument, and s equal to the value a/b. Strct relatvsm s represented by settng a = 0. Strct narrow self-nterest s a lmtng case (b 0). 2.3 ERC-equlbra As players gan experence wth game rules and the subject populaton, play tends to settle down to a stable pattern (see Roth and Erev, 1995). ERC makes equlbrum predctons ntended to characterze the stable patterns. The basc framework s an ncomplete nformaton game n whch each player s r and s are prvate nformaton, but the denstes r f and s f are common knowledge. That s, we assume that, n the stable state, players have learned the dstrbuton from whch ther playng partners are drawn. But consstent wth our assumpton that playng partners are randomly and anonymously assgned, ndvdual motvaton functons are treated as prvate nformaton. Defne an ERC-Nash equlbrum as a Bayesan Nash equlbrum solved wth respect to player motvaton functons. Defne an ERC-subgame perfect equlbrum as a Bayesan subgame perfect equlbrum wth respect to player motvaton functons. (Except where noted, the ERC-subgame perfect equlbra we derve are sequental equlbra, because the games present no opportuntes for updatng strategc nformaton.) ERC predctons about ndvdual optmalty that are ndependent of nformaton consderatons apply startng from the frst round. The dctator game and second mover behavor n the gft exchange game are examples, as we show below. ERC does not attempt to capture learnng or framng effects (at least not ths verson). Secton 7 compares what ERC explans to what present learnng theores explan. Framng effects refer to the nfluence on behavor that expermenters observe from changes n how the game s posed to subjects. When usng ERC to make comparatve predctons across games, we assume that the frame s held constant, n the sense that the drectons gven to subjects are parallel across games. 5 We compare the model to populaton patterns found n the data. It s possble for an naccurate theory of ndvdual behavor to match populaton behavor out of serendpty. But n 5 The experments of Forsythe et al. (1995) and Roth et al. (1991) are good examples of parallel presentaton of dfferent games

9 ths case, the sheer volume of populaton data organzed by ERC makes serendpty hghly mprobable. 3. Games of farness: dctator, shrnkng pe barganng, best shot, and mpunty These games, when played n the lab, are always fnte (a fnte number of possble actons). For smplcty, we derve many of the results n ths secton assumng a contnuous strategy space. Unless otherwse stated, all propostons characterze ERC-subgame perfect equlbra (recall the nformaton condtons descrbed n secton 2.3). 3.1 Dctator and ultmatum games, and the relatonshp between them Frst consder a dctator game n whch the [D]ctator dstrbutes a pe of maxmum sze k > 0 between self and a recpent. We represent the dctator s dvson as the par (c, σ D ). So the dctator s payoff s cσ and the recpent payoff s c cσ. D D roposton 3.1: For all dctator allocatons, c = k, and σ r D (c) [½, 1]. roof: Follows drectly from A1 and the defnton of r (c) gven n secton 2.1. D = The dctator game has been the subject of several studes (e.g., Forsythe et al., 1994; Hoffman et al., 1995; Bolton et al., forthcomng). Whle the precse dstrbuton of dctator gvng vares wth framng effects, proposton 3.1 appears equally vald for all studes: Dctators dstrbute all the money and (almost) always gve themselves at least half. (Those takng less than half, lke the one dctator n Fgure 1, account for less than 1 percent of the data n the studes lsted.) As an llustratve example, consder the addtvely separable motvaton functon gven n (2.2), and suppose that k = 1. A straghtforward calculaton shows that 1 a = + r mn, 1. 2 b Hence the dctator s decson s a drect reflecton of the margnal rate of substtuton between pecunary and relatve payoff. Now consder an ultmatum game between a []roposer and a [R]esponder. For the moment, we assume the cake sze, k > 0, s common knowledge. We represent the proposal by - 8 -

10 (c, σ ), nterpreted analogously to the dctator notaton. To keep the statements of the propostons as smple as possble, we assume that f a responder s ndfferent between acceptng and rejectng, that s, f 1 σ = s R ( c), then the responder always accepts proposal (c, σ ). We assume s (c) s dfferentable. roposton 3.2 characterzes the responder s ERC-subgame perfect equlbrum strategy, and proposton 3.3 characterzes the proposer s. roposton 3.2: The probablty a randomly selected responder wll reject, p c, σ ), satsfes the followng: () p has the value 0 when ncreasng n σ over the nterval (½, 1); () fxng a σ = ½ and the value 1 when ( σ = 1; () p s strctly σ (½, 1), p s strctly decreasng n c. roof: () For all responder, v c / 2,1/ 2) v (0,1/ 2). Hence, equal dvson s never R ( R rejected. The defnton of s (c) mples that the responder rejects the offer f 1 σ < s R (c), s (0, 1/n]. Therefore, the densty ( s c) σ = 1 offers are turned down. () Ths follows from ntegratng over f s. () s (c) s mplctly defned by v ( cs, s ) = v (0,1/ 2) for s ½. Dfferentatng yelds v cs, s )[ s cs ] v ( cs, s ) s ( + 2 =. Hence, Ths completes the proof. sv 1 ( cs, s ) s ( c) = < 0. cv ( cs, s ) + v ( cs, s ) 1 2 roposton 3.3: For all ultmatum proposals, c = k and roof: For any fxed c > 0, all proposers prefer σ ½. σ = ½ to any never turned down. It follows that any equlbrum proposal has p( c, σ ) s strctly decreasng n c when σ > ½ and constant for propose dvdng all of k. σ < ½, and σ = ½ s σ ½. By proposton 3.2, σ = ½, so the proposer wll Many studes, begnnng wth Güth et al. (1982), confrm propostons 3.2() and 3.3. The experment of Bolton and Zwck (1995) vvdly llustrates that lower offers tend to have a hgher probablty of rejecton. Slonn and Roth (forthcomng) present evdence that the probablty of rejecton tends to decrease as c ncreases

11 The addtvely separable motvaton functon of (2.2) mples a negatve relatonshp between s and r ; specfcally, s = r r 1/ 4 2. As far as we know, there s no data on whether a relatonshp exsts (let alone ths one), although a relatonshp of some sort s plausble. An experment clarfyng ths ssue would help us towards a more precse verson of the model. Forsythe et al. (1994) found that, on average, offers are hgher n the ultmatum game than n the dctator game. ERC predcts ths relatonshp. By propostons 3.1 and 3.3, we may assume that all proposals dvde all of k, whch we normalze to sze 1. roposton 3.4: On average, offers n the ultmatum game wll be hgher than offers n the dctator game. In fact, no one offers more n the dctator game, and the only players who offer the same amount are those for whom r (1) = ½. roof: That proposers who have r (1) = ½ offer the same n both games s obvous. Suppose nstead that r (1) = 1. Snce a demand of (c, ultmatum game, t s clear that the proposal wll be we wrte out the frst order condtons (normalze v ( 0,1/ 2) = 0 ): σ ) = (1,1) s always turned down n the σ < 1. For all other proposers, r (½, 1), FOC for the dctator game: v σ, σ ) v ( σ, σ ) 0 D1 ( D D + D 2 D D = p (1, σ ) v p( σ, σ ) FOC for the ultmatum game: v1 ( σ, σ ) + v 2 ( σ, σ ) = > 0. 1 p(1, σ ) By nspecton, σ > σ D. Ths completes the proof. 3.2 Unknown pe sze games Suppose now that the responder must decde whether to accept or reject an offer of y monetary unts wthout knowng the pe sze, but knowng that the pe was drawn from some dstrbuton, f(k), wth support [ k, k ]. Suppose y < k /2. Mtzkewtz and Nagel (1993), Kagel et al. (1996), and Rapoport et al. (forthcomng) have all shown that responders are more lkely to reject y under these crcumstances than f they know for certan that the pe s k, and less lkely to reject than f they know t s k. The same s true n ERC. Let p u (y) denote the probablty that y wll be rejected by a randomly selected responder. For smplcty, we assume that the sze of the

12 offer does not convey any nformaton about the pe sze (hence for ths proposton, ERCsubgame perfecton does not mply sequental equlbrum). roposton 3.5: For all y < k /2, k y k y p k, < p u (y) < p k,. k k roof: Suppose y < k /2. Then there exsts a responder who, f he knew the pe sze was k, s just ndfferent between y and rejectng. Then, keepng n mnd postulate A2, 1 y k y v ( 0, ) = v ( y, ) < v ( y, ) f ( k) dk, 2 k k k whch ndcates that and players wth smlar rejecton thresholds are less lkely to reject when they do not know the sze of the pe. The rest of the proof s analogous. 3.3 Two perod alternatng offer games Each round of a two-perod alternatng offer game s played lke an ultmatum game, wth players swtchng roles from frst to second perods. If the frst perod offer s rejected, the pe s dscounted pror to the second perod counterproposal. Bolton (1991) descrbes a comparatve model of two perod alternatng offer barganng, and shows that the comparatve statcs ft the data well. The comparatve model s almost a specal case of ERC. The comparatve model assumes that v ( cσ, σ ) s strctly ncreasng n σ for all. The proof of proposton 3.4 mples that, for players wth r (c) > ½, v s an ncreasng functon n some neghborhood around the proposer s demand, σ. (Ths s also true for the two-perod game.) So the comparatve statcs of the two models are, for these players, n local agreement. ERC predcts that barganers wth r (c) = ½, wll offer half the pe even f an offer of somewhat less s very unlkely to be turned down. Expermenters make menton of such people (e.g., Kagel, Km and Moser, 1996). The comparatve model makes no room for them, whch s why t s almost a specal case ERC s a more accurate model, even for barganng games. 3.4 Impunty and best shot We concern ourselves wth the mn versons of mpunty and best shot games, and compare these to the mn-ultmatum game. In all three games, a proposer moves ether left or

13 rght. The responder observes the proposer s move and then ether accepts or rejects. The games dffer only n the payoffs, whch are lsted n Table 1. Table 1 here. Note that the standard subgame perfect equlbrum s the same for all three games: the proposer plays rght, and the responder plays accept. Applyng ERC to the mn-ultmatum game s straghtforward, and yelds results qualtatvely equvalent to those for the full verson. Applcaton of ERC to the other games leads to markedly dfferent predctons: roposton 3.6: For the mpunty game: () The only outcomes wth a postve probablty of occurrng are (2,2) and (3,1). () The proporton of (3,1) outcomes s equal to the proporton of the populaton for whom v (3, ¾) > v (2, ½). () The probablty of the (3,1) outcome s hgher for mpunty than for the mn-ultmatum game. roof: () For all responders, v R (2, ½) v R (0, ½) and v R (1, ¼) > v R (0,0). () Gven responders behavor, the proposer s choce s effectvely between (2,2) and (3,1). () In ultmatum, all proposers who choose rght prefer (3,1) to (2,2). But not all who choose left prefer (2,2) to (3,1). By (), mpunty proposers choose rght ff they prefer (3,1) to (2,2), and by (), an offer of (3,1) s never rejected. Experments by Güth and Huck (1997) and Bolton and Zwck (1995) furnsh evdence for 3.6() and 3.6(). Bolton, Katok and Zwck (forthcomng) provde evdence for 3.6(). roposton 3.7: The probablty of the (3,1) outcome s greater n best shot than n the mnultmatum game. The proporton of (3,1) offers rejected n best shot s the same as n mnultmatum. roof: For the proposer, the expected value of playng rght s the same n both games. The expected value of playng left n the best shot game s strctly smaller than n the ultmatum game: Let p be the probablty a randomly chosen best shot responder prefers (1,3) to (1,1). Then p v (1, ¼) + (1 p) v (1, ½) < v (2, ½) for all p (0,1]. For the second half of the proposton, note that, after an offer of (3,1), responders n mn-best shot and mn-ultmatum have dentcal choces avalable to them

14 roposton 3.7 mples that, relatve to the ultmatum game, best shot behavor moves towards, but s not dentcal to the standard subgame perfect equlbrum. rasnkar and Roth s (1992) best shot experment comes close to convergence. 6 On the other hand, Duffy and Feltovch s (forthcomng) best shot experment clearly does not converge, even after 40 teratons, although as predcted, best shot s closer to perfect equlbrum than a correspondng ultmatum game. In sum, the expermental evdence s consstent wth proposton Three-way ultmatum and the soldarty game We conclude ths secton wth a dscusson of experments that bear on the queston of whether motvaton s adequately captured by motvaton functon preferences for dstrbuton between self and the group, or whether they are better captured by altrustc preferences, where a person cares about the dstrbuton across all ndvduals. Güth and van Damme (forthcomng) report on a three-way ultmatum game experment n whch the proposer proposes a three-way splt of the pe, and one responder can accept or reject. The thrd player, a recpent, does nothng save collect any payoff the other two agree to gve hm. The experment fnds that nformaton about the recpent s share has no drect nfluence on the responder s decson to accept or reject. ERC predcts the same, because the dstrbuton among the other players does not enter nto the motvaton functon (see secton 2.1). Selten and Ockenfels (forthcomng) observe a smlar phenomenon studyng the soldarty game. In ths game, each player n a three-person group ndependently rolls a de to determne whether they (ndvdually) wn a fxed monetary sum. Before the de s rolled, each announces how much she wshes to compensate the losers, for both the case where there s one loser, and for the case where there are two. Most subjects gve the same total amount ndependent of the number of losers. In addton, gfts for one loser are postvely correlated wth the expectaton about the gfts of others. Selten and Ockenfels demonstrate that nether the behavoral pattern nor the relaton between decsons and expectatons are easy to justfy f subjects have standard altrustc preferences. They conclude that most subjects, even though they are wllng to sacrfce money for soldarty, are unnterested n the welfare of recpents, and only care about ther own share of ther wnnngs. 6 So does Harrson and Hrschlefer (1989), but the ncomplete nformaton aspect of the game renders the result ncomparable to the theory

15 Bolton et al. (forthcomng) fnd that the total gft dctators leave multple recpents s stable, but how dctators dstrbute gfts across recpents appears, n most cases, to be arbtrary. Wemann (1994) analyzes a publc goods experment drected at the queston of whether ndvdual behavor of others, or just aggregate group behavor nfluences the decson to contrbute. He concludes that, "Whether or not the ndvdual contrbutons [to a publc good] are common knowledge has no mpact on subject's behavor" (p.192). 4. Compettve behavor In the last secton, we showed that f a game creates a trade-off between absolute and relatve motvatons, we can observe behavor whch sharply contradcts standard theoretcal predctons. But people do not always play far. Many market nsttutons apparently nduce 'compettve,' self-nterested behavor. In ths secton we show that typcal market envronments nteract wth ERC-motvatons n a way that algns absolute and relatve motves. As a consequence, tradtonal Nash equlbra are ERC-Nash equlbra. Some well known expermental results come from games wth symmetrc equlbrum payoffs, so we begn wth the symmetrc case. It turns out that ERC mples an nterestng dfference between Bertrand and Cournot games wth respect to symmetry, and we turn to ths ssue at the end of the secton. Bertrand and Cournot games are the standard textbook examples of (olgopolstc) markets: Suppose demand s exogenously gven by M = p + q, where M s a constant, p denotes the prce and q the quantty. Suppose n 1 dentcal frms produce at constant margnal cost θ (< M). In Cournot games, frms choose quanttes q [, M θ] y θ, where q [, M θ] 2 ( q) = ( M q ) q q 0 yeldng profts gven by 0. In Bertrand games, frms choose prces p θ yeldng profts equal to y ( p) = ( p θ )( M p ) / n~ f sets the lowest prce along wth n ~ 1 other frms, or equal to zero f there exsts a frm j whch sets a lower prce. All pure strategy spaces are fnte. For smplcty, we assume that the nterval between admssble prce offers,, s small; specfcally, (p θ)(m p ) > (1/n)(p θ)(m p) for all p > θ, and for all n (so there s a pecunary ncentve to undercut p, when all others bd p). The nformatonal assumptons lad out n secton 2.3 contnue to apply

16 roposton 4.1: For n 1, and for ether prce (Bertrand) or quantty (Cournot) competton, all Nash equlbra are ERC-Nash equlbra. roof: For n = 1, σ 1so that the ERC-monopolst smply maxmzes hs profts. For n > 1, observe that all Nash equlbra n both the prce and the quantty game, yeld equal equlbrum profts for all frms (see Bnmore, 1992). Hence, a frm that devates from hs Nash equlbrum strategy can nether gan wth respect to absolute nor relatve payoffs. Ths completes the proof. The remanng propostons n ths secton provde a stronger characterzaton of ERCequlbra. In propostons 4.2 to 4.2b, we wll suppose that v (0,0) s the worst possble outcome for all. Ths s a mld assumpton, mplyng that the worst thng that can happen to s to have to watch others receve a postve payoff whle recevng none hmself. We further suppose that, gven any (y, σ), the value of ( y, σ) s bounded wth respect to both and n. Ths assumpton v has a purely techncal purpose. Bounded wth respect to (fxng n) would follow mmedately f we made the realstc, but less mathematcally convenent, assumpton that the populaton were fnte; we smply mpose boundedness on the nfnte populaton (see A3). Wth respect to n, the mportant mplcaton s that v ( y,1/ n) s bounded for fxed and y. That s, for a fxed pecunary payoff, the value to of achevng the socal reference proporton s bounded wth respect to the number of players n the game. We thnk assumng the value of v ( y,1/ n) s fxed wth respect to n would be reasonable, but boundedness wll be suffcent. We frst show that the compettve outcome s the unque ERC-Nash equlbrum for the Bertrand game. The ntuton s qute smple: For large n, there s a hgh probablty that at least one player cares suffcently about hs pecunary payoff to undercut hgh bds n pursut of pecunary gan. In equlbrum, everyone knows that the probablty of such a person s hgh, and so, n equlbrum, everyone undercuts because ths s what s necessary to preserve relatve as well as pecunary postons. roposton 4.2: For prce competton and for n large enough, the market prce n all ERC-Nash equlbra s equal to cost θ or to θ +, the standard Nash equlbrum prces for n > 1 frms. roof: Let γbe the probablty that the composton of players n the game s suffcently narrowly self-nterested n the sense that, for each admssble p,

17 v (( p θ )( M p ),1) > v ((1/ n)( p θ)( M p),1/ n) for at least one. By assumpton A3, as n ncreases, γncreases monotoncally to 1. Choose n large enough, so that γsatsfes where bd s above [ (1 γ ) v ((1/ n)( p θ)( M p ),1/ n) + γ v (0,0) v (0,1/ n) ] 0 max < M M p M s the monopoly prce. A maxmum exsts because of the boundedness assumpton. Now suppose there s an ERC-Nash equlbrum n whch the maxmum possble wnnng p H > θ +. Snce transactons are never made at a prce of greater than p H s strctly domnated by offerng a prce of H, p H, bddng p (recall that we assume that v (0,0) s the worst possble outcome for all ). Therefore, n equlbrum, all prces bd wth postve probablty by any player must be p H or lower. Hence play t. It follows that the expected value to frm of bddng p H s the wnnng bd only f all n frms p H s β v ( 1/ n)( p θ)( M p ),1/ n) + (1 β) v (0,0) ( H H where β s the probablty that all frms other than bd value of frm bddng p H s (4.1) (( p θ)( M p ),1) + (1 β)k [ ] p H. On the other hand, the expected β v (4.2) H H For suffcently narrowly self-nterested agents, (4.2) > (4.1). Therefore, suffcently selfnterested players always bd lower than any player s p H. Gven ths, the expected value of bddng p H for (1 γ ) v ((1/ n)( p H θ)( M p H (1 ),1/ n) + γ v (0,0) γ ) v ((1/ n)( p M θ)( M p M ),1/ n) + γ v (0,0) < v (0,1/ n) whch contradcts the assumpton that p H s a best response for at least some player (any player can guarantee hmself v (0,1/n) by playng θ ). Snce a constructon lke (4.2) s always possble f p H > θ +, t follows that p H = θ or θ + for suffcently large n

18 In the guessng game, n > 1 players smultaneously choose a number z from an nterval [0, k]. For smplcty, we assume that the number of choces s fnte, and that the nterval between any two consecutve choces s. The wnner s the player whose number s closest to γ z, γ < 1. The wnner receves a fxed prze; f there s a te, wnners share the prze equally. The guessng game s very smlar to a Bertrand game, save that the cake to be dstrbuted s fxed. Nagel s (1995) experment shows that play converges to the unque standard Nash equlbrum, z 0. roposton 4.2a: For n large enough, the unque Nash equlbrum n the guessng game s equvalent to the (unque) ERC-Nash equlbrum. roof: Showng that z 0 s an ERC-Nash equlbrum s straghtforward. For the proof n the other drecton: Note that any outcome n whch wns has a payoff greater than v (0,1). Fx a strategy profle for the other n 1 players, and let x be the modal average mpled by the dstrbuton. If n s large enough, player s nfluence on the average s neglgble (and so we can gnore t). So when n s large enough, by guessng k ( k γ x, player can guarantee herself greater than v 0,1) + v (0,0). Substtutng ths value for ( 0,1/ n), the rest of the proof closely k parallels proposton 4.2. v How large must n be? By proofs of propostons 4.2 and 4.2a, the answer depends on the prevalence of suffcently narrowly self-nterested subjects n the populaton. A player for whom r(c) 1 certanly satsfes the requrement. Hoffman et al. (1994) performed a dctator game n a buyer-seller frame smlar to Bertrand games (wth players beng randomly assgned buyer and seller postons). The proporton gvng zero was about 45 percent. 7 Then the probablty of at least one subject wth r = 1 n a group of n subjects s n. Assumng that r s not too senstve to the sze of the pe, a lower bound on the probablty of at least one suffcently selfnterested player n a group of 3 s over 83 percent. It appears then that n need not be very large for ERC-Nash equlbrum market prces to shrnk to the standard Nash prce. Holt (1995) 7 We refer to Hoffman et al. s buyer-seller dctator game wth contest selecton of roles. They also ran a buyerseller dctator game wth random selecton of roles. The proporton gvng zero was lower, but the proporton almost gvng zero (10 percent or less) was about 40 percent, and n ths sense our calculaton s approprate for both treatments. We refer to these partcular dctator treatments because they are roughly framed (buyer-seller) n the same way as Bertrand experments. We nevertheless thnk of the resultng calculatons as llustratons. A

19 reports some evdence that outcomes of olgopoly games are less compettve wth two players than wth three or more, but no partcular effect for numbers greater than two. Interestngly, ERC mples that the aucton market game studed by Roth et al. (1991) (dscussed n secton 1) s suffcently dfferent from the Bertrand game to obtan compettve results ndependent of the number of buyers. Recall that, n ths game, buyers smultaneously bd on an object owned by a sngle seller. The lowest bd s submtted to the seller who ether accepts or rejects; f the latter, all players receve a zero monetary payoff. We prove that obtanng the (compettve) subgame perfect equlbrum does not depend on the number of buyers, so long as there are at least two. We normalze the surplus that can be shared from the transacton to 1, and we represent a bd by the proporton of the surplus that the buyer proposes keepng (defned ths way, the relaton to proposton 4.2 wll be clear). A bd wns f t s both the lowest submtted and large enough to be acceptable to the seller. Analogous to the Bertrand game, we suppose that the nterval between permssble bds,, s small. roposton 4.2b: Consder an aucton market game havng at least two buyers. Under the assumpton that the seller accepts, all ERC-subgame perfect equlbra for the market game have a wnnng buyer bd of 0 or. roof: Suppose, contrary to proposton, that there s an equlbrum n whch z H > s the hghest bd that wns wth postve probablty. The proof that, n equlbrum, no one ever bds hgher s analogous to the proof of proposton 4.2 f one substtutes "prce (p)" for "bd (z)" and "frm" for "buyer". However, n contrast to the Bertrand game, n ths market one buyer wth the smallest bd s chosen randomly, and dvdes the surplus wth the seller, who s an actual subject n the experment. Consequently, equatons (4.1) and (4.2) of proposton 4.2 become β [ (1/n) v ( z H, z H ) + (1 1/n) v (0, 0) ] + (1 β ) v (0,0) β v ( z H, z H ) + (1 β ) [ ]. (4.1a) (4.2a) The nequalty (4.2a) > (4.1a) holds for all players, regardless of type. Ths contradcts the assumpton that bddng z H s a best response. Ths completes the proof. careful, meanngful calculaton requres runnng dctator and Bertrand games n closely parallel frames (parallel drectons)

20 About the assumpton concernng seller behavor: From the pont of vew of ERC, ts valdty s an emprcal queston. In fact, Roth et al. report that no best bd was ever rejected n a non-practce round (p. 1075). The assumpton s bascally equvalent to postng that ( 1/,1] v ( σ, σ ) > v (0,1/ n) σ n, 8 whch mples an asymmetry wth respect to farness: I reject offers that are very unfar to me but accept offers that are very unfar to you. Asymmetry of ths sort s suggested by Loewensten et al. (1989), and by Fehr and Schmdt (1997). Whle ERC has no problem accommodatng ths assumpton, we have avoded t to hghlght the fact that t s not relevant to any proof n ths paper save proposton 4.2b where t has but a very mnor role. In partcular, the assumpton s not necessary to explan the compettve behavor of buyers n the Roth et al. game. Is there a restrcton we could place on the motvaton functon to guarantee the compettve results n ropostons 4.2 and 4.2a for any szed group (greater than 1, of course)? The only one we can thnk of s a stronger asymmetry assumpton: v ( c,1) > v ( c / n,1/ n) for all and c. But ths s falsfed by dctator game experments. roposton 4.1 shows that the standard Cournot-Nash equlbrum s an ERC-Nash equlbrum. We do not know f ths s the unque ERC-Nash equlbrum for the type of ncomplete nformaton game played n the lab (the type we have been studyng). If we assume complete nformaton, however, and restrct to pure strateges, we can prove unqueness. roposton 4.3 extends the classc textbook graph proof of duopoly Cournot equlbrum n pure strateges (e.g. Bnmore, 1992, p. 290) to ERC motvatons. roposton 4.3: Consder a Cournot duopoly n whch both players know one another s motvaton functon. If r (c) > 1/2 for all c for at least one player, then the unque ERC-Nash equlbra n pure strateges s the standard Nash equlbrum. roof: Fgures 4 (a) and 4 (b) here. In fgures 4 (a) and (b), the x axs shows the quantty of frm j and the y axs the quantty of frm. The thck lnes show the standard Nash-reacton curves of player (BE) and player j 8 Strctly speakng, proposton 4.2b requres that the seller accepts all bds, not just those greater than 1/n. The proof, however, s easly extended: Suppose that the z H n the proof gves the seller less than 1/n. Revse both (4.1a) and (4.2a) to reflect the fact that undercuttng ncreases the probablty the seller wll accept

21 (CF). Two thngs need to be proved. Frst, observe that for all quantty combnatons lyng on the dagonal AD, the margnal utlty wth respect to relatve payoffs s zero, because payoffs are equal (assumpton A2). Snce the margnal utlty wth respect to absolute payoffs s strctly ncreasng for at least one player (note that r (c) > 1/2 for all c mples v > 1 0 for σ = 1/ 2 ), the only locaton on AD whch can be an ERC-Nash equlbrum s pont X, the Cournot equlbrum. Second, note that: (1) on the Nash-reacton curves, y ( q ) = 0 and y ( q ) = 0, respectvely; j (2) y ( q ) > 0 ff ( q, q j ) s wthn ABE, y j ( q ) > 0 ff ( q, q j ) s wthn ACF; (3) y < y j ff q, q ) s wthn ADE, y > y j ff ( q, q j ) s wthn ACD; and (4) σ ( q ) > 0, k =, j, everywhere ( j n the nteror of ACE. Wth these propertes, t s easy to see that ERC-reacton curves are bounded by the Nash-reacton curves and the dagonal: j's ERC-reacton curve must le somewhere n the darkly shaded areas and 's ERC reacton curve must le somewhere n the brghtly shaded areas. (The areas nclude the Nash-reacton curves for both players and exclude AD wth the excepton of pont X for at least one player.) The only possble pont of ntersecton of ERC reacton curves s X. Ths completes the proof. k k j j The proof requres a suffcently self-nterested player n a weaker sense than do the Bertrand propostons, specfcally r (c) > 1/2 for one player. From dctator games, we estmate the proporton of r(c) > 1/2-players to be 80 percent. Ths s a conservatve estmate most dctator studes fnd a lower proporton than ths. Then we estmate the probablty of a standard Nash equlbrum (under complete nformaton) to be at least 96 percent. The calculaton gnores the pure strategy requrement. Evdence for the standard Cournot-Nash equlbrum s less than conclusve. Holt (1985) conducted sngle-perod duopoly experments of the type we study here. Whle n the begnnng some subjects try to cooperate, quantty choces tend ultmately to Cournot level. Holt (1995) surveys a number of studes, and reports some support for Nash equlbrum, but also expresses reservatons. Huck, Normann and Oechssler (1997) report rough convergence to Nash equlbrum n the four-person case. Fnally, ERC mples symmetrc payoffs are mportant to Cournot outcomes n a way that they are not to Bertrand games. Consder a Cournot duopoly n whch frm has a cost advantage: θ < θ. The standard Nash equlbrum proft of frm s greater than the proft of j

22 frm j. But frm may choose a smaller quantty n the ERC-Nash equlbrum n order to boost the relatve payoff. On the other hand, consder cost heterogenety n Bertrand games;.e., each 1 2 k frm s randomly assgned to costs { θ, θ,, θ } k < θ K. Then, the compettve prce s, the lowest θ n the market, and t s also a standard Nash equlbrum. 9 It contnues to be an ERC equlbrum f the market s large enough; the proof s analogous to that of proposton Dlemma games: a smple quanttatve ERC model All dlemma games share two defnng characterstcs. Frst, f players are purely narrowly self-nterested, then ther set of choces ncludes a domnant strategy that yelds the hghest payoff regardless of what others do. Second, devaton from the domnant strategy contrbutes to a hgher jont payoff for the group, and enough contrbutons produce an outcome areto superor to the domnant strategy outcome. Domnant strategy s not a good descrpton of the behavor we typcally see. In ths secton, we show that ERC s consstent wth many of the patterns we do observe n the prsoner s dlemma (D) and assocated one-shot dlemma games (we dscuss repeated dlemma games n secton 6). We can say more wth a ftted parametrc model. In sectons 5.2 and 5.3 we ft the gft exchange and nvestment games, both essentally (sequental) Ds What s necessary to nduce cooperaton n smultaneous and sequental D s? We wll demonstrate that, n ERC, the extent of cooperaton can depend on the nteracton between () heterogenety wth respect to how players trade-off pecunary for relatve gans, and () the sze of payoffs, especally the sze of the effcency gans that can be acheved through cooperaton. These factors are mportant n both smultaneous and sequental Ds, although the factors nteract n somewhat dfferent ways across the two games. Table 2 here. 9 Ths holds f there s more than one frm wth mnmum cost. If there s only one frm wth mnmum cost, there s a Nash equlbrum n whch the prce s the second lowest cost and the frm wth mnmum cost gets all the surplus. 10 Roughly speakng, for n large enough there s one frm among the frms wth mnmum cost whch s suffcently self-nterested so that t undercuts any prce greater than mnmum cost. 11 Not every player n ether the gft exchange or the nvestment game has a domnant strategy, so techncally speakng nether game s a dlemma game. But, as wll become clear, they are both very close off-shoots

23 Consder the D payoff matrx n Table 2. To llustrate how trade-offs between pecunary and relatve payoffs matter to ERC predctons, we wll suppose that ndvduals can be descrbed b 1 by the motvaton functon gven n (2.2), v ( cσ, σ ) = acσ σ. Then the margnal 2 2 rate of substtuton between pecunary and relatve payoffs, a/b, fully characterzes a subject s type. The populaton dstrbuton of types wll be denoted by F(a/b). 12 To see what nfluences cooperaton n a one-shot smultaneous D, examne the optmal decson rule for a subject wth type a/b: 2 C f D a b p 1/ 2 < 4( 1 m)( 1+ 2m) 2 = : g( m, p). Here, p s the probablty that the opponent cooperates. Thus, cooperaton s nfluenced by the extent to whch subjects are motvated by relatve payoffs, the magntude of the mpcr, m, and the proporton of cooperatng subjects n the populaton. There s always an equlbrum n whch nobody cooperates, but dependng on the shape of F(a/b), there may also be equlbra n whch a proporton of subjects cooperate whle others defect. We next consder the sequental D, n whch the second mover decdes after beng nformed of the frst mover s acton. We obtan an nterestng result: Cooperaton requres both subjects who are wllng to sacrfce pecunary for relatve gans, and subjects who are mostly nterested n absolute payoffs. To see ths, examne the optmal decson rules (the nformaton assumptons lad out n secton 2.3 contnue to apply): second mover: C f D 1. frst mover plays C 2. a b < g( m, 1) frst mover: Cf D m( 1+ p$) > 0 2. a b 1 p$ > 8( mp$ + m 1)( 1+ 2m) 2 12 Of course, the results we derve wll be specal to ths class of motvaton functons. But keep n mnd that our goal here s to demonstrate that partcular factors can play an mportant role n what ERC predcts

24 Here, p$ = p$( m) = F( g( m, 1 )) s the probablty that the second mover responds cooperatvely f the frst mover cooperates. The second mover s optmal decson rule corresponds to the one appled n the smultaneous D wth p = 0 or 1 respectvely. The second mover cooperates ff she s suffcently motvated by the relatve payoff, and the frst mover cooperated. The frst mover cooperates ff she s suffcently motvated by pecunary payoffs, and the expected monetary net return of cooperaton (= 1+ m( 1+ p$) ) s postve. The reasonng behnd the requred frst mover motvaton s smple: A frst mover who s nterested n relatve payoff can guarantee equal payoffs by defectng, snce n ths case, the second mover defects for sure. Only f a frst mover s suffcently nterested n hs absolute payoff, wll he take the chance of beng exploted n an attempt to trgger second mover cooperaton. Heterogenety guarantees that the proporton of both frst and second movers who 1 p$( m) cooperate ncreases wth the mpcr ( p$ ( m) > 0 and / m 2 8( mp$ + m 1)( 1+ 2m) $p (m) s very small, a suffcently hgh mpcr may nduce the frst mover to cooperate. 13 < 0). Even f Several studes support the vew that potental effcency gans and the propensty of others to cooperate (measured n ERC by the margnal rate of substtuton between absolute and relatve payoffs) are major determnants of cooperaton n both smultaneous and sequental Ds. In a well-known survey, Rapoport and Chammah (1965) demonstrate that cooperaton rates n Ds ncrease when the gans from cooperaton ncrease, or when the sucker payoff decreases. 14 Ledyard (1995) surveys the lterature on publc good games, and concludes that, besdes communcaton, the mpcr s the only control varable that has a strong postve effect on cooperaton rates. Many experments show a strong relaton between own and opponent decsons. Cooper, DeJong, Forsythe, and Ross (1996) found two behavoral types n one-shot Ds, whch are perfectly n lne wth the ERC-decson rules n Ds derved above: "egosts", who always defect, and "best response altrusts", for whom C (D) s a best response to C (D). 15 Smlarly, rutt (1970) and Rapoport and Chammah (1965, p ) found strong postve nteractons between cooperatve choces of players. Several studes have manpulated the 13 The specfc class of motvaton functons also mples an ncome effect. Suppose we multply all payoffs n Table 2 by a fxed postve number. Then there s a stronger tendency for all players to behave accordng to standard game theoretc predctons. Rabn s (1993) model makes a smlar predcton. 14 Rapoport and Chammah (1965), p. 39, Fgure 1. Lave (1965) ncludes smlar results