SERIE DOCUMENTOS BORRADORES

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1 ISSN E C O N O M Í A SERIE DOCUMENTOS No. 5, ulo de 00 Intenton-Based Economc Theores of Recprocty Darwn Cortés Cortés BORRADORES DE INVESTIGACIÓN

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3 DARWIN CORTÉS CORTÉS 3 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT DARWIN CORTES @p.unv-tlse1.fr ABSTRACT In recent years, several experments have shown ndvduals exhbt authentc recprocal behavour n anonymous one-shot nteractons. As recprocty has been shown to be relevant n several economc felds, there have also been several attempts to model recprocal behavour. I revew the ntenton-based models of recprocty and present an example n teachers management n the publc sector n whch government offers an ncentve scheme to mplement a program. The ncentve scheme has a prsoner s dlemma structure. In both smultaneous and sequental games, n equlbrum recprocal teachers may reach other equlbra dfferent from those predcted by the standard theory. Key words: Game theory, psychologcal games, Intenton-based models, recprocal behavour. JEL classfcaton: C700. RESUMEN Recentemente, varos expermentos han mostrado que los ndvduos exhben un comportamento auténtcamente recíproco en nteraccones anónmas que se dan una sola vez ( one-shot ). En tanto que se ha mostrado que la recprocdad es relevante en múltples campos de la economía, han exstdo varos ntentos por modelar el comportamento recíproco. Este documento revsa los modelos de recprocdad que se fundamentan en las ntencones y presenta un eemplo para el caso del maneo de los profesores en el sector públco, en el que el goberno ofrece un esquema de ncentvos para la mplementacón de un programa. Este esquema tene la estructura del dlema del prsonero. Tanto en los uegos smultáneos como secuencales, los resultados de equlbro pueden ser dstntos a los que predce la teoría convenconal. Palabras clave: teoría de uegos, uegos scológcos, modelos basados en ntencones, recprocdad. Clasfcacón JEL: C700. * Ths paper was presented as a DEA mémore to the MPSE - Ecole Doctorale de Scence Economque of the Unversté de Toulouse 1. I want to thank Paul Seabrght and Emmanuelle Aurol for ther comments. Julo de 00

4 4 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT 1. INTRODUCTION From several years ago, other socal scences dfferent from economcs lke psychology, socology and anthropology have ponted out human bengs tend to recprocate each other. Untl recent years t had not been clear whether ths behavour was only caused by some expectatons of future rewards or, at least n some cases, t was genune recprocal behavour. If the frst explanaton was true, the usual economc hypothess that ndvduals behave n a self-nterested manner could explan those behavours. Nevertheless, from two decades ago, several experments have shown ndvduals exhbt authentc recprocal behavour n anonymous one-shot nteractons. For example, n the ultmatum game a par of ndvduals has to dstrbute a fxed sum of money n a sequental move game. The proposer has to dvde the amount between hmself and the second subect. The responder can accept or reect the proposed dvson. If ndvduals were ratonal and self-nterested, the responder would accept any quantty of money and the proposer would gve the smallest possble quantty. However, evdence shows offers lower than 0% are atypcal and reected wth a hgh probablty, whle offers close to 50% are very common and rarely reected (Fehr and Fschbacher, 001). On the other hand, n the gft-exchange game the proposer (employer) offers a wage to the responder (worker). The worker can ether reect or accept t. If the worker reects both players gan nothng. If the worker accepts she has to exert a costly effort. The hgher the effort, both the lower the payoff she gets and the hgher the ncome the employer receves. Under the standard assumptons, the worker wll always choose the lowest effort and the employer wll only offer the lowest possble wage. Evdence suggests wages are clearly hgher than mnmum levels and wages and effort have a postve relaton (Fehr and Fschbacher, 001). Those and other experments have shown ndvduals actually recprocate each other. A recprocal ndvdual rewards knd behavour and punshes unknd behavour. The gft-gvng game llustrates the former, sometmes called postve recprocty, and the ultmatum game the latter (negatve recprocty). Addtonally, t has been shown recprocty can have an mportant role n some economc felds. In labour economcs, questonnare studes have shown managers are unwllng to cut wages because t can adversely affect work morale. Effectvely, wages cuts are consdered as an nsult by the workers (Bewley, 1995). Besdes, Akerlof (198) suggests recprocal behavour can explan why wages reman above the market clearng level. In fact, ths s supported by some experments that have shown recprocty contrbutes to the enforcement of contracts, as loyalty and trust are relevant n labour relatonshps. Further experments show ndvduals punsh free-rders n publc good provson games even f t reduces ther own payoffs; materal ncentves may crowd-out mplct ncentves that rely on recprocal behavour and recprocty can explan why n realty contracts are ncomplete, among other facts. 1 All these phenomena cannot be explaned assumng the self-nterest hypothess. There have been several attempts to model recprocal behavour. In ths document I revew the so-called ntenton-based models of recprocty, partcularly the models proposed by Rabn 1 Fehr and Gächter (000) survey expermental evdence. Frey (001) also surveys crcumstantal and econometrc evdence. Borradores de nvestgacón - No. 5

5 DARWIN CORTÉS CORTÉS 5 (1993) and Dufwenberg and Krchsteger (001). Ths approach emphaszes n the fact that recprocal ndvduals want to reward knd ntentons and to punsh unknd ntentons. To llustrate these theores I propose two examples n teachers management. The frst one conssts n a game that models teachers strategc behavour n the followng stuaton: government wants to mprove qualty of publc educaton for whch t ntends to mplement a program to make better teachers abltes. Government offers an ncentve scheme that has a prsoner s dlemma structure to enforce the program; n such a way that standard game theory wll predct both teachers are gong to partcpate. The second game slghtly modfes the materal payoffs of the frst one. I obtan that, n both smultaneous and sequental games, recprocal teachers may devate from partcpaton n equlbrum, as they consder partcpaton as an unknd behavour. Instead no partcpaton s regarded as a knd behavour. Of course, partcpaton of both teachers may also be an equlbrum when each teacher beleves the other s gong to partcpate. In that case, both teachers punsh the other s unknd ntenton. The text s organzed n three sectons. In the frst one, I provde an overvew of the economc theores about recprocty n order to gve a context to the ntenton-based theores. The second one s dvded n several subsectons n whch I present the examples and the theores mentoned. Wth expostve purposes I frst ntroduce the example and show the results obtaned usng the standard theory, and then I provde the model of recprocty and the new results. Last secton offers conclusons.. MODELLING RECIPROCIT In the standard theory self-nterest hypothess s formalzed by defnng ndvdual preferences solely on the materal resources the ndvdual has. One way to model recprocal behavour s enlargng the space n whch ndvdual preferences are defned to nclude others materal payoffs or welfare. When an ndvdual does not only care about the materal resources allocated to her but also cares about the relevant reference agents, we wll say she has socal preferences (Fehr and Fschbacher, 001 p. ). In fact, most of the theores that try to model recprocty ntroduce t as a socal preference. These theores have had nto account recprocty has two elements n nature: t s not only related to the consequences of others actons but also to the others ntentons. They have focused on one of those elements of recprocal behavour. Fehr and Schmdt (1999) and Bolton and Ockenfels (000) stress the fact that people desre to mantan equty and provde models of nequty averson. On the other hand, Rabn (1993) and Dufwenberg and Krchsteger (001) emphasze persons want to punsh nasty ntentons and to reward frendly ntentons. Levne (1998) bulds a model n whch ndvduals do not respond to ntentons but to the type of person they face. The type s determned by the degree of altrusm the ndvdual has. Charness and Rabn (000) and Falk and Fschbacher (000) develop theores that have elements from both ntenton-based recprocty and nequty averson models. Fnally, Segal and Sobel (1999) present an axomatc treatment of recprocty and altrusm whch s compatble wth some of the socal preferences models of recprocal behavour. It s worthy to pont out that some nequty averson models, whch only concern about payoffs dstrbuton, can mmc some predctons of ntenton-based recprocty models. However, Julo de 00

6 6 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT though ntenton-regardng models can be much more dffcult to handle than nequty averson models, expermental evdence suggests people punsh others even f punshment does not reduce nequty. 3 In the followng secton I present the pure ntenton-based economc models of recprocty. 3. MODELS OF INTENTION-BASED RECIPROCIT In these models n decdng what acton ndvduals are gong to follow they have nto account not only ther materal payoff but also ther belefs about others kndness. Specfcally, ndvdual utlty s composed by two parts: a materal payoff, whch s gven n terms of some measurable quantty, e.g. money; and a recprocty payoff that she obtans from assessng the others kndness. So, ndvduals wll do the acton that gves them the hghest utlty regardng both payoffs. For example, consder the game n Fgure 1. It presents a prsoner s dlemma. As usual when ndvduals only care about ther own materal payoff the Nash equlbrum s no cooperaton for both persons. However, notce that when an ndvdual chooses no cooperaton nstead of cooperaton she s reducng the other s materal payoff. So, when one of the agents decdes to cooperate, t can be nterpreted by the other as a knd acton, snce the former reduces hs payoff and ncreases the latter s at the same tme. If both players have hgh enough senstvty to recprocty concerns, cooperaton can be the best opton for them. FIGURE 1 It s worthy to pont out that belefs on kndness are formed assessng the other s ntentons. If player 1 s acton ncreases her payoff and the player s payoff smultaneously, player wll probably not consder that acton as knd. Further, t can happen that even f one player sacrfces hs materal payoff she s to be consdered as no knd. For nstance, n the game depcted n Fgure 1, assume player have no opton dfferent from cooperaton. Somehow ths player s forced to cooperate. So, we have a degenerate game composed by the left column of the game. In ths case, player 1 wll not beleve player s beng knd by cooperatng, as the latter has no choce. To llustrate the theores consdered n ths document, we are gong to analyze a qualtatve example from the teachers management. In the next secton s posed the basc problem. For a complete dscusson n ths regard, look at Falk and Fschbacher (000). Borradores de nvestgacón - No. 5

7 DARWIN CORTÉS CORTÉS A QUALITATIVE EXAMPLE OF TEACHERS MANAGEMENT Assume a government utlzes two teachers n offerng publc educaton. There s a teachers trade unon so that f both take the same decson wth respect to government polces, government cannot punsh them. Assume as well government wants to mprove the qualty of educaton offered and hence decde to mplement a program that rse teachers abltes. In order to exert that polcy, government brngs out an ncentve scheme as follows: If both teachers do not partcpate n the program, government cannot fre them and they contnue ganng the same payoff as before, say X. If both teachers enter n the program, they do a hgher effort and obtan the same payment X. 3 Payoffs cannot be lower than X because otherwse trade unon would mpede mplementaton of the governmental program. 4 Fnally, f teachers take dfferent decsons, trade unon s not workng anymore, so the teacher who does not partcpate s fred and obtans hs reservaton utlty and the teacher who partcpates receves a payoff X + d hgher than X. It s also assumed that teachers take ther decson smultaneously. The game s depcted n Fgure. It s easy to see that the ncentve scheme has a prsoner s dlemma structure. Government persuades teachers to partcpate offerng a contngent reward d to deter trade unon obstructons. Thus, players have an ncentve to partcpate n the program ndependently from the other s choce. In such a model, f teachers only care about ther materal payoff, the unque Nash Equlbrum n pure strateges s (partcpate, partcpate). FIGURE 3. INTRODUCING RECIPROCIT Suppose both teachers regard nceness, so they draw utlty from recprocty concerns. Notce that n ths example, as n the frst one, when a player attempts to maxmze her materal payoff reduces the other s payoff. As teachers are recprocal, they wll reward frendly actons and wll punsh hostle actons. Assume teacher has chosen to partcpate, so she can obtan ether X + d or X. If teacher 1 chooses to partcpate as well, he not only mnmzes teacher s payoff (she would obtan X nstead of X + d) but also maxmzes hs (he would get X nstead of 0). Thus, ths acton could be consdered unknd by teacher and hence she would not be wllng to devate from partcpaton because otherwse she would reward teacher 1. 3 Moral hazard s not an ssue here but t should be n a more realstc model. 4 In fact, the game structure s preserved even f partcpaton payoffs are hgher than X. It would be enough to assume the partcpaton premum to be lesser than d. Julo de 00

8 8 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT Now suppose teacher 1 chooses not to partcpate, so teacher gets X + d nstead of X. In ths case, teacher perceves teacher 1 s gvng up X for gvng her d and hence she could beleve teacher 1 acton to be knd. In ths stuaton, teacher would be unknd to player 1 f she remans partcpatng. So as teacher s recprocal she could change her decson (from partcpaton to no partcpaton) f she s better off dong so. Notce that one player s assessment of the other kndness depends not only on what the former beleves the latter s gong to do but also what the former beleves the latter beleves the former s gong to do. To form both belefs, farness of ntentons s determned assessng the equtablty of the fnal payoffs dstrbuton wth regard to the feasble set of payoffs. Dong so each player wll compare utlty she gets n both stuatons: partcpaton brngs her a hgher materal payoff than no partcpaton. Instead, no partcpaton brngs her a hgher recprocty payoff than partcpaton. So, f her recprocty senstvty s hgh enough, teacher wll decde to gve up d of her own payoff for gvng teacher 1 X. Dong the same analyss for the other teacher, we obtan that wth recprocal teachers we have two possble equlbra: 5 (not partcpate, not partcpate) and (partcpate, partcpate). But, when (not partcpate, not partcpate) wll be chosen? It depends on both the noton of farness and recprocty senstvty players have, and the amount of the materal payoffs. To see ths t s needed to ntroduce a formal model of recprocty RABIN (1993) S MODEL Rabn (1993) models recprocty based on psychologcal games proposed by Geanakoplos, Pearce and Stacchett (1989) (hereafter GPS). In such games, players payoffs depend not only on players actons but also on ther belefs. GPS show that many standard concepts have useful analogues n the framework they develop. Rabn s goal s to derve psychologcal games from materal games. Let us consder a normal form game wth two players, player 1 and player, who have mxed strategy sets and A 1, respectvely, obtaned from pure fnte strategy sets S 1 and S. Player s materal payoff s gven by the functon π : A1 A R. In order to construct the psychologcal game, let us assume that when a player chooses her strategy, her subectve utlty functon wll depend on three thngs: her strategy, her belef about the other s strategy and her belef about the other s belef about her strategy. 6 Let us call a1 A1and a Athe strateges of player 1 and player, respectvely; b 1 A 1 and b Aplayer s belef about player 1 s strategy and player 1 s belef about player s strategy, respectvely; and c 1 A 1 and c A player 1 s belef about player s belef about player 1 s strategy and player s belef about player 1 s belef about player s strategy. Observe that although a, b and c belong to the 5 In the next secton, we wll call them farness equlbra. 6 Hgher order belefs can be consdered but t s enough to take the frst two. Borradores de nvestgacón - No. 5

9 DARWIN CORTÉS CORTÉS 9 same set, they are dfferent n nature as a s a player s decson, b s player s belef ( ) and c s a player s belef. To ncorporate recprocty (farness n terms of Rabn) n the model we frst need to defne a kndness functon (, ) f a b whch measures how knd player s to player. If player beleves player chooses b, how knd s player by choosng a? When player chooses a, s selectng ( ) a payoff par ( a, b), ( b, a) chooses π π from the set of all the feasble payoffs to player when he {( ) } b. Let us call ths set ( ) π (, ), π (, ) Π b = a b b a a A. How knd player s beng depends on both the pont she chooses from Π ( b ) and the noton of kndness players have. To express ths noton n formal terms, we need to defne a functon for both player s kndness to player and player s belef about how knd player s beng to her. Rabn (1993) provdes some general propertes that sort of functons must have. h The followng payoffs are useful to do that: let π ( b) be player s hghest payoff n Π ( b ), l e π ( b) be player s lowest payoff among the Pareto- effcent ponts n Π ( b ), and π ( b) be an equtable payoff n Π ( b ). The followng propertes for kndness functons are suffcent condtons for the man result Rabn obtans: 7 Property 1: A kndness functon must be bounded and ncreasng. A kndness functon (, ) f a b s bounded and ncreasng f: a. There exsts a number N such that f ( a, b ) [ N, N] b. f( a, b) > f( a', b) f and only f π ( b, a) π ( b, a' ) for all a A and b A and; >. Ths property rules out the possblty of farness to generate nfntely postve or nfntely negatve utlty and brngs out a postve assocaton between the player s payoff and player kndness: gven b, the hgher player payoff s, the knder player s. Property : A kndness functon must be a Pareto splt. A kndness functon (, ) e Pareto splt f there exsts some π ( b) such that: f a b s a 7 They are presented as defntons n Appendx A n Rabn (1993), p For addtonal results s also needed to assume kndness functon to be affne, but that property s not relevant for our present purposes. Julo de 00

10 10 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT e e a. π ( b, a) > π ( b) mples that f( a, b ) > 0; π ( b, a) = π ( b) mples that ( ) e and π ( b, a) < π ( b) mples that f( a, b ) < 0; b. π h ( ) e ( ) l b π b π ( b) ; and h l c. f π ( b) > π ( b), then π h ( ) e ( ) l b > π b > π ( b) f a, b = 0; Ths property says that the far payoff to player s strctly between the best and the worst Pareto effcent payoffs n Π ( b ), provded that Pareto effcent set s not a sngleton. π e Among the class of functons defned by the prevous propertes, Rabn pcks the followng: Defnton 1: Player s kndness to player s gven by ( a, b ) e ( b, a ) π ( b ) mn ( b ) π ( b ) h π f π mn where π ( b) s the worst possble payoff for player n Π ( b ) h l π ( b) + π ( b) h mn ( b) =. If π ( b) π ( b) = 0 then ( ) f a, b = 0 It s easy to check ths functon has the general propertes presented above: Frst, f = 0 f h mn and only f player receves the equtable payoff. Ths s so because when π ( b) π ( b) and = player always gans the same payoff and there s no kndness ssue. Second, f < 0 when player s payoff s lesser than the equtable payoff. Ths happens when ether π ( b, a) pont smaller than the equtable payoff or ( b, a) s a Pareto-effcent π s not an effcent pont. Fnally, f > 0 only f player s payoff s greater hs equtable payoff and the Pareto set s not sngleton. Notce 1 also the functons take values n the nterval 1,.!. Player s belef about how nce player s to her can be expressed as a functon f ( b, c) Ths functon s formally equal to the prevous but t relates the two levels of belefs consdered n the model. Defnton : Player s belef about how knd player s to her s gven by Borradores de nvestgacón - No. 5

11 DARWIN CORTÉS CORTÉS 11 f! ( b, c ) π e ( c, b ) π ( c ) h π mn ( c ) π ( c ) mn e h mn where π ( c ) and π ( c ) have analogue defntons. If π ( c ) π ( c ) ( ) f! b, c = 0. Usng both functons f( a, b) and f ( b, c) = 0 then! we can defne a utlty functon for player. Dong so, we are assumng players have a shared noton of farness. Ths utlty functon ntegrates the materal payoff and the recprocty payoff: 8 (,, ) (, ) +! (, ) 1 + (, ) U a b c π a b f b c f a b The frst term s the materal payoff and the second the recprocty payoff. The constant reflects how senstve player s to recprocty matters regardng player and we wll assume t s postve. Ths utlty functon gathers the man feature about recprocal behavour. If player ( )!, she wll want to punsh hm beng beleves player s treatng her unkndly f ( b, c ) < 0 unknd, that s choosng a such that (, ) player s beng nce f! ( b, c ) > 0, she wll be nce. Furthermore, the hgher f ( b, c) ( ) f a b to be low. On the contrary, f player thnks! s, the more materal payoff player s wllng to gve up to reward player. Fnally, ths utlty functon has the property that whenever player s hostle to player, player s utlty s lesser than her materal payoff. That s, an ndvdual s not able to completely recover her welfare takng revenge once other has treated her badly. These preferences together to the elements already defned for the materal game form a psychologcal game. Usng the concept of psychologcal Nash Equlbrum defned by GPS, Rabn (1993) proposes the followng defnton, Defnton 3: The par of strateges ( a, a ) ( A, A ), a. a arg max a A U( a, b, c) s a farness equlbrum f, for = 1,, 1 1 b. c = b = a Ths noton of equlbrum s analogous to Nash Equlbrum, but appled to psychologcal games. Condton b. of the defnton requres all hgh-level belefs to correspond actual behavour. 8 Ths utlty functon s slghtly dfferent from whch Rabn uses. We have added the term n the recprocty payoff. Julo de 00

12 1 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT Consderng agan our example, we can calculate the teachers utlty functons regardng recprocty. Though theory s posed for mxed strateges we only analyze equlbrums n pure strateges. In Fgure 3, we can see the utlty values once condton b. of farness equlbrum s satsfed. FIGURE 3 Frst, note that when player s beng unknd to player, player s recprocty payoff s negatve, whch reduces hs overall utlty and ntroduces ncentves to devate. However, the profle of strateges (partcpate, partcpate) s a farness equlbrum for all values of X and because response for unkndness s unkndness. Consder now, what happens f player devates to no partcpaton. Ths acton ncreases player s recprocty payoff because he consders player s beng knd. In fact, player wll devate to no partcpaton strategy f the loss n materal payoff, δ, s less than the gan n recprocty payoff, 1. The profle (not partcpate, not partcpate) wll be a farness equlbrum whenever 1 δ < for = 1,. Ths condton s satsfed when ether δ s low enough or s hgh enough. If the government gves a reward too lttle when one teacher partcpates and the other does not or f both teachers have a strong feelng to recprocate the other, devatng from partcpaton wll be an equlbrum. Ths model has been extended to nclude sequental actons. In prncple, such a model would be more adequate to realty because recprocal actons have an mplct delay. One s knd wth somebody that has been knd. Besdes, extendng the model to sequental games s also essental for appled research (Rabn (1993), p. 196), as ndvduals can change ther motvaton due to nformaton provded by past decsons. 3.4 SEQUENTIAL GAMES Consder now a slghtly modfed sequental verson of our game of teachers, depcted n Fgure 4. Assume there s no trade unon anymore, so government can offer a lesser materal payoff f both teachers partcpate n the program, X-ε, 0 < ε < X. In the frst step, teacher 1 decdes whether to partcpate or not n the program offered and once he has decded, teacher has to take her decson. Assumng further no recprocty (and perfect and complete nformaton), t can be seen, solvng by backward nducton, the profle (Partcpate, partcpate) wll be the Borradores de nvestgacón - No. 5

13 DARWIN CORTÉS CORTÉS 13 unque Subgame Perfect Nash Equlbrum. Government s strategy to mplement the program s completely successful as teachers wll always partcpate. Let us ntroduce recprocty. Suppose 1 chooses do not partcpate (NP) n the program. Player can choose ether X or X+δ (or mx). Her choce wll depend on both her kndness and the belef she has about the 1 s ntenton to choose NP. When teacher 1 chooses NP gves teacher a payoff at least X and at most X+δ. Instead, when teacher 1 chooses P gves teacher a payoff at least 0 and at most X-ε. So wll beleve 1 s beng knd when he chooses NP and f recprocty payoff s hgh enough she wll choose X nstead of X+δ (or mx). To establsh f (NP, np) wll be an equlbrum, we have to evaluate what teacher 1 beleves when chooses np. FIGURE 4 It s convenent to pont out one dfference n the analyss of recprocty n normal games and extensve games. In normal games teacher wll always choose do not partcpate, provded recprocty payoff supersedes materal payoff. Ths does not happen n a sequental model because, for nstance, once teacher knows teacher 1 has chosen to partcpate, there s no reason to mantan the decson of do not partcpate uncondtonally. In that case teacher would consder teacher 1 s beng hostle and thus she would partcpate n the program as well. Unlke normal games, n sequental games uncondtonal np does not occur because player s optmzng n each subgame. On the other hand, n modellng recprocty n sequental games t s not plausble to assume players are gong to keep ther ntal belefs along the game. Player s belef about how knd player 1 s beng once the latter has decded do not partcpate s dfferent from the former s belef once the latter has decded to partcpate. It means t s necessary to analyze changes n belefs n each node of the game n order to establsh equlbrum condtons. Furthermore, t s not possble to consder each subgame separately. Player belef about how nce s player 1, gven he has already decded to not partcpate, depends on whch payoffs she would had had f player 1 had decded to partcpate. Therefore, backward nducton cannot be used to obtan the equlbrum. Dufwenberg and Krchsteger (001) (henceforth DK (001)) provde a concept of Julo de 00

14 14 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT sequental recprocty whch allows them to propose a new soluton concept, Sequental Recprocty Equlbrum (SRE). 3.5 DUFWENBERG AND KIRCHSTEIGER (001) S MODEL As we have sad, when recprocty s ncorporated n sequental games t s necessary to dstngush between a player ntal and subsequent belef. Once a subgame has been attaned, a player s belef can change and, as kndness depends on belef, kndness may therefore change as well. DK (001) deal wth ths by keepng track of how belefs change when a new subgame s reached and by assumng players choces take nto account the belefs they hold n the most recently reached subgame. To do that, as Rabn they adopt the psychologcal games framework; but unlke GPS (1989), who only regards to games where solely ntal belefs can affect player s assessments, DK (001) propose a noton of recprocty n whch player s belefs change n each subgame. Formally, they pose a t-player extensve form game wthout nature and wth perfect recall. Any such a game Γ s descrbed by a fnte set of nodes organzed n a tree, a collecton of nformaton sets, a set of choces avalable at each decson node, a functon assgnng each nformaton set to the player who moves at the decson nodes n that set, and a collecton of payoff functons assgned to each endnode (Mas-Colell, Whnston and Green (1995)). Let { 1,..., t} T = be the set of players where t. It s convenent to add new notaton to that used n secton 3.3, as there are now several players. Let A be the set of player s behavour strateges, a ; B be the set of possble player s belefs about player s strateges, b ; and Ck be the set of player s belef about player s belef about player k s strateges, c k. As n Rabn s model, belefs are mxed strateges, so we have B = A and Ck = Bk = Ak. Besdes, player s materal payoff s now gven by the functon π : A Rwhere A=Π TA. Now, let us proceed to formalze how the player s belefs change when new subgames are reached. To keep track of how each player s behavour, nceness and percepton of other s nceness dffer across subgames, let R be the set of nodes that are startng nodes of all possble subgames n Γ, and let Γ r be the subgame whch startng pont s r R. Let us defne the r- part of Γ r as the set of nodes n Γ r that do not belong to some proper subgame of Γ r. For a strategy a A, let a () r be the strategy that has the same choces as a but assgnng a probablty equal to 1 to the choces that drve to node r. In an analogous way, defne b ( ) () b B c C c r for and k k k players are playng ( b ) and belevng ( c ) k r and, respectvely. Thus, player decdes to play a belevng other, whereas n the r-part of the subgame Γr, player s playng a ( r) A and belevng other players to play b ( r) ( ck () r ) k k ( ) and to beleve. Ths means that even f players ntally beleve that others mx ther choces, the Borradores de nvestgacón - No. 5

15 DARWIN CORTÉS CORTÉS 15 subsequent percepton of kndness s trggered by the actual choce (DK (001), p. 8). In terms of our example, consder the proper subgame startng n the player s rght sde node and call that node r. Player beleves player 1 s choosng hs strategy as b1 = p' NP+ ( 1 p' ) P. Before 1 plays, at node r f s bg (1 or near 1) player wll thnk player 1 s knd. However, once r s reached, player does not consder player 1 to be knd anymore. At r, player s belef s b1 ( r) = P. DK (001) also change the noton of effcency suggested by Rabn (1993), whch says that the lowest effcent strategy s chosen from Π ( b ). They argue that, n a sequental game framework, the set of Pareto-effcent strateges relevant to establsh the equtable payoff cannot depend on belefs, as ths can drve us to no exstence of equlbrum. 9 DK (001) s effcency noton can be formulated as { E = a A there exsts no a ' Asuch that for all r ( a ) A, k T t holds that π ( ( )) () ( ) π ( ) ( ) ( ) ( ) wth strct nequalty for some ( ) a ' r, a r a r, a r, k k ( r )}, a, k The concept of effcency has a central role n the ntenton-based theores. To llustrate ths pont consder the game depcted n Fgure 5. We have the same game of Fgure 4 but now player 1 can do an acton Z n whch both players obtan a payoff -X. Let us suppose player 1 beleves wth probablty one player s playng the strategy np, p. It can be seen 1 beleves he selects the materal payoff π ( Pnpp) = x ε from the feasble set { xx, ε, x},. In the game of Fgure 4, player 1 would be consdered unknd, now are we wllng to accept player 1 s beng knd due to the mere possblty of Z to be chosen? To rule out ths unreasonable consderaton we restrct our attenton to effcent payoffs n order to determne the equtable payoff. DK (001) propose the noton of effcency above to do that. 10 e We can defne the equtable payoff as π ( b ) h l ( ) = ½( π ( a ( b ) ) + π ( a,( b ) ), whch s essentally the same defned n secton 3.3. Unlke that one, a subndex has been added to e to ndcate ( ) l that ths s the equtable payoff for and π a( b) s now the lowest payoff n E. In turn, kndness, kndness belef and utlty functons can also be defned n a smlar way as before. 9 Look at DK (001) p. 9 for an example of no exstence of equlbra due to a belef dependent concept of effcency. 10 However, ths dstncton does not make any dfference wth respect to our example, because all the strateges are effcent under both concepts. Julo de 00

16 16 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT FIGURE 5 Defnton 4: Kndness of player to another player n the r-part of a subgame Γ r s gven by the functon f : A x Π B R defned by ( ( ),( ( )) ) ( ( ),( ( )) ) (( ) ) e f a r b r = π a r b r π b Apart from dfferences already mentoned, defnton 4 s analogous to defnton 1. f dffers formally from f n that f s not normalzed and thus, n prncple, t may take values extremely hgh or low. However, due to we are analysng central ponts (as we subtract an average from the payoff chosen), t s not expectable to obtan an extreme number, so property 1a n secton 3.3 can be hold wthout large nconvenences. On the other hand, t s straghtforward to check defnton 4 holds propertes and 1b. Defnton 5: Player s belefs about how knd player s to n the r-part of a subgame Γ r s gven by the functon: ~ f ~ f : B x Π k C k R defned by ( b () r,( c () r ) ) π b () r, c () r k k e ( ( k ) ) ( ck () r ) = π k ( ) Ths defnton s formally equal to the prevous. The same comments for f f can be done for : f! n relaton to f!. Defnton 6: Player s utlty n the r-part of a subgame Γ r s a functon ( B x Π C ) R U : A x Π defned by k k U a, ( ) = () r c () r k k ( ( ) ( ) ) ( ) ( ) k wth respect to ( ( )! + /{}( ) ( ( ) ( ( )) )) π,,, a r b r f a r b r f b r ck r T k Utlty functon n Rabn s model has the term ( ) 1 f 1 + nstead of f. For comparson purposes, t has been preferred to keep the functons as alke as those the authors propose. As senstvty Borradores de nvestgacón - No. 5

17 DARWIN CORTÉS CORTÉS 17 for recprocty s a nonnegatve number, recprocty payoff ncreases utlty f player beleves ~ ~ player s knd ( f > 0 ), and reduces utlty f player beleves player s unknd ( < 0 ) f. Appendng ths knd of utlty functons to an extensve game, we get the tuple Γº = ( Γ, ( U ) T ). DK (001) call Γºa psychologcal game wth recprocty ncentves. There s a noton of equlbrum assocated to these games that can be formulated as Defnton 7: The profle â = (â ) T s a Sequental Recprocty Equlbrum (SRE) f for all T and for all r R t holds that a. aˆ () ( ) () () ( ) r arg max a A r, aˆ U a, b r, ck r b. b = aˆ for all c. ck = aˆ k for all, k a A r, aˆ ( ) k where A (r,a) s the set of strateges player can use f she behaves accordng to a (r) at nformaton sets outsde the r-part of Γ r, but s free to choose any alternatve n the r-part of Γ r. Condton a. says player maxmzes hs utlty at node r gven hs belefs and gven that he follows hs equlbrum strategy outsde the r-part of Γ r. Ths entals belefs to assgn a probablty one to the sequence of choces that allow r to be reached. Condtons b. and c. says n the equlbrum belefs are correct and correspond to the actual strategy. DK show every psychologcal game wth recprocty ncentves has at least one SRE. To do that, they frst defne the sze of a subgame as the number of ts subgames, then they smultaneously determne equlbrum choces of the subgames wth the same sze, startng from the smallest (sze equal one) untl arrvng to the complete game. 11 In the game that appears n secton 3.4, frst teacher 1 decdes whether to partcpate or not n the program offered by the government and then teacher does so. We showed there, no recprocty mples profle (partcpate, partcpate) to be the sole Subgame Perfect Nash Equlbrum. How the analyss s affected when teachers are recprocal? We can fnd t out usng the theory developed n ths secton. When there s recprocty between agents, the game becomes a psychologcal game wth recprocty ncentves. Examnng SRE for dfferent levels of recprocty senstvty we can say: 1 1. If teacher s senstvty to recprocty,, s low enough, profle (partcpate, partcpate) s an δ equlbrum behavour. Specfcally ths occurs when < x ( δ + ε ). In ths case, each player wll beleve the other s gong to partcpate, whch wll n turn be consdered as unknd. Those belefs render a negatve recprocty payoff to both players and therefore each teacher prefers to partcpate n the government program. From the prevous nequalty t can also be seen 11 Demonstraton appears n DK (001) p Detaled calculatons are ncluded n the Appendx. As the game has two players, we smplfy notaton, so, = 1,, s agent s senstvty to recprocty. Julo de 00

18 18 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT that gven a senstvty to recprocate level for both teachers, 1 and, the hgher δ relatve to x and ε s, the more lkely both teachers to partcpate. Government should take ths nto account n order to make teachers to partcpate n the program he proposes.. If teacher s nclnaton to recprocate,, s hgh enough, profle (do Not Partcpate, do not partcpate) holds n all SRE. Regardless 1, when teacher has a strong nclnaton to recprocate, she wll obtan a hgh recprocty payoff f teacher 1 decdes to not partcpate, so she wll play np (nstead of p) when teacher 1 does so. Notce player would also get a hgher materal payoff dong so than that she had obtaned f teacher 1 plays P (nstead of NP). Teacher 1 knows all ths, and thus he wll choose to play NP to get a hgher materal payoff than that he would get f he had played P. Ths equlbrum behavour cannot be predcted when we assume no recprocty. The scheme proposed by government does not work n the way government expects due to recprocty between teachers. 3. Gven a hgh s leanng to recprocate, t also happens (Partcpate, partcpate) to be an equlbrum behavour when teacher 1 also has a strong tendency to recprocate. Ths arses when each player thnks the other s gong to play p, as each one expects an unknd acton from the other. There are self-fulfllng expectatons. 4. For ntermedate values of 1 and, equlbrum behavours are mxed strateges. In equlbrum, for player probablty of no partcpaton, p, s gven by ε p = δ x. As t can be nferred from prevous analyss, ths probablty ncreases when ncreases. In addton, p reduces f the rato between ε and δ decreases and ncreases f x ncreases. δ ε can be vewed as the nverse of the ncentve government provdes to player to partcpate. Player tres to gan δ (she gans d f (NP, p) s chosen) but she loses ε f (P, p) s chosen. She evaluates how much she can obtan and lose from partcpaton. Ths evaluaton affects p n the way descrbed. On the other hand, an ncrease n x ncreases p because ceters parbus t makes less attractve to partcpate. For player 1, t s not possble to do the same knd of analyss due to parameters affect hs probablty of no partcpaton, q, n a complex way. In fact, for a gven, s equlbrum behavour s unque whereas, n general, 1 s equlbrum behavour s not unque for a gven Fnally, from the results obtaned for ths game we can analyze a sequental verson of the teacher s game wth trade unon. In that game ε = 0, so payoffs n profles (NP, np) and (p, p) are equal to X. The most nterestng result n ths case s no partcpaton to be an equlbrum behavour only n mxed strateges. Analyss s as follows. We know teacher wll always play p when teacher 1 plays P, 14 so teacher would get X n ths profle. On the other hand, f teacher 1 plays NP, teacher can get ether X or X+δ. For (NP, p) to be possble n equlbrum, player has to beleve wth probablty one that player 1 beleves player wll choose p. But n ths 13 Look at Remarks 4 and 5 n the Appendx. 14 Look at Remark 1 n the Appendx Borradores de nvestgacón - No. 5

19 DARWIN CORTÉS CORTÉS 19 stuaton, player would obtan X from both (NP, np) and (p, p) and hence there would be no recprocty ssue (recprocty payoff equal to zero). Therefore, player would prefer to play another strategy, as profle (NP, p) offers player a hgher materal payoff. A smlar analyss can be done for player CONCLUSIONS Evdence has shown that sometmes people behave n dfferent ways from whch s predcted by assumng ndvduals are self-nterested. Furthermore, when persons devate from self-nterested behavour they do not always try to ncrease the well-beng of others. On the contrary, t has been found ndvduals usually respond n a knd manner to knd actons and n an unknd manner to unknd actons. In response to these fndngs, several economc theores have attempted to model recprocty behavour. In ths document, we have revewed the so-called ntenton-based theores of recprocty, specfcally the models made by Rabn (1993) and Dufwenberg and Krchsteger (001). These theores have receved ths name because they emphasze people want to punsh hostle ntentons and to reward nce ntentons. To do that, they adopt the psychologcal games framework developed by Geanakoplos, Pearce and Stacchett (1989). In ths framework ndvdual utlty depend not only on strateges but also on belefs. Rabn (1993) develops a theory for -players normal form games and ntroduces a new equlbrum noton called farness equlbrum. Dufwenberg and Krchsteger (001) n turn extend Rabn s theory dealng wth t-players sequental games and present the noton of sequental recprocty equlbrum. The man nnovaton they do s to keep track of belefs about ntentons as the game evolves. Players maxmze ther behavour n each subgame takng nto account belefs about ntentons formed n the prevous stages. In a partcular subgame players use belefs that comes from the most recently reached subgame. There are other dfferences between these models. Rabn (1993) uses a kndness functon neutral to unts of measure of the stakes, so that kndness cannot nfntely ncrease or decrease utltes. Ths also allows ndvdual kndness to reduce as long as payoffs become larger. Instead Dufwenberg and Krchsteger (001) measure kndness n the same unts of materal payoffs (.e. money), whch has the advantage kndness does not dsappear when payoffs rse but the dsadvantage t also makes utlty to be senstve to lnear transformatons as recprocty payoff s measured n money squared. Moreover, they dffer n the effcency noton used to defne the equtable payoff. Rabn (1993) s noton depends on belefs and then t only consders strateges on the equlbrum path; whereas DK (001) defnes neffcent strateges as those that yeld a weakly lower payoff for all player (strctly lower for some) than other alternatves n all the subgames. Fnally, Rabn (1993) specfes kndness n the utlty functon n such a way to capture the dea that whenever a player s treated unkndly, her overall utlty wll be lower than her materal payoff (her ablty to take payback s not perfect). DK (001) s specfcaton does not capture that. We have also llustrated the theores studed wth a smple example n teachers management. We have proposed an mplementaton mechansm for a governmental polcy when there s a teachers trade unon. Both teachers have to decde to partcpate (p) or not (np) n a Julo de 00

20 0 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT governmental program. In order to mplement the polcy, government proposes a game wth a prsoner s dlemma structure. Wthout recprocal teachers, n both games (normal and sequental forms) there s a unque equlbrum: teachers partcpate n the governmental program. Wth recprocal teachers, we obtan addtonal results. In the normal form game, there are two farness equlbra: one n whch each teacher s knd to the other and other n whch both teachers are unknd. If n equlbrum both teachers are knd to each other, government cannot mplement the program. In the sequental game n turn we have multple equlbra. We consdered two games: a sequental verson of the prevous one and a game n whch there s no trade unon and hence government can gve a lesser payoff f both teachers partcpate n the program. Now teacher does not choose np uncondtonally as n the normal form, as teacher behaves optmally off the equlbrum path. In both games, condtonal cooperaton can be part of a SRE. However, under trade unon np s an equlbrum behavour only n mxed strateges. One lmtaton of the ntenton-based approach s that one ndvdual only has recprocal behavour when other ndvduals have shown to have knd ntentons. Suppose n our example player s constraned to choose do not partcpate. Player 1 wll not consder ths acton as knd because player has no opton. In fact, although nowadays there s almost consensus about the exstence of recprocal behavour, there s stll dsagreement about the foundatons of that behavour. For nstance, other theoretcal approaches focus on nequty averson (Bolton and Ockenfels, 000) or the type of persons one faces (Levne, 1998). Hence, n an nequty model player 1 wll behave knd when there s an nequty ssue even f player s forced to choose do not partcpate. Dscusson s opened regardng ths pont. 15 Another lmtaton of ths approach s that equlbrum analyss s rather complex and there are multple equlbra due to self-fulfllng belefs. In the normal form game suggested, for example, both equlbra emerge for ths reason, so t s not possble to establsh whch one s gong to occurs. On the other hand, even though treatment of belefs n the sequental model s very nnovatve t makes dffcult to buld tractable models. Fnally, despte smplcty of our examples, they suggest t wll be worthy to take nto account recprocty n theores that try to model government-teachers relatonshps. On one hand, a sgnfcant part of lterature on recprocty has shown recprocal behavour s relevant n the analyss of employer-employee relatonshps. It has been documented employers are reluctant to decrease wages n crss tmes because they do not want to reduce employees morale to work. In partcular, t would be nterestng to fnd out how recprocty affects the man results of multagent settngs. 16 On the other hand, some emprcal research has shown teachers trade unons can affect negatvely student performances (qualty of educaton) (Hoxby, 1996). 15 Falk and Fschbacher (000) show evdence that supports ntentons matter. Fehr and Schmdt (001) survey exstng models on farness and recprocty 16 One of the man results n these settngs s that under moral hazard, prncpal can use relatve performance of agents to elct a hgher effort (yardstck competton). Cf. Laffont and Martmort (00). Borradores de nvestgacón - No. 5

21 DARWIN CORTÉS CORTÉS 1 5. REFERENCES Akerlof, George (198), Labor contracts as a partal gft exchange, Quarterly Journal of Economcs 97: Benabou, Roland and Jean Trole (00), Intrnsc and extrnsc motvaton, mmeo. Bewley, Truman (1995), A depressed labor market as explaned by partcpants, Amercan Economc Revew 85, Papers and Proceedngs: Bolton, Gary and Axel Ockenfels (000), ERC - A theory of Equty, Recprocty and Competton, Amercan Economc Revew 90: Charness, Gary and Matthew Rabn (000), Socal preferences: some smple tests and a new model, Unversty of Calforna at Berkeley, mmeo. Dufwenberg, Martn and Georg Krchsteger (001), A theory of sequental recprocty, Dscusson paper. Center, Tlburg Unversty. Falk, Armn and Urs Fschbacher (000), A theory of recprocty, Insttute for Emprcal Research n Economcs, Unversty of Zurch, Workng Paper, 6. Fehr, Ernst and Armn Falk (001), Psychologcal Foundatons of Incentves, Unversty of Zurch, mmeo. Fehr, Ernst and Urs Fschbacher (001), Why socal preferences matter - The mpact of non-selfsh motves on competton, cooperaton and ncentves, Unversty of Zurch, mmeo. Fehr, Ernst and Smon Gächter (000), Farness and retalaton: The economcs of recprocty, Journal of Economc Perspectves 14: Fehr, Ernst and Klaus Schmdt (001), Theores of farness and recprocty - Evdence and economcs applcatons, Unversty of Zurch, mmeo. Frey, Bruno and Reto Jegen (001), Motvaton crowdng-out theory, Journal of Economc Surveys 15 (5): Geanakoplos, John, Davd Pearce and Enno Stacchett (1989), Psychologcal games and sequental ratonalty, Games and Economc Behavour 1: Hoxby, Carolne (1996), How teachers unons affect educaton producton, The Quarterly Journal of Economcs: Laffont, Jean-Jacques and Davd Martmort (00), The theory of Incentves: The prncpal agent model, Harvard Unversty Press. Levne, Davd (1998), Modellng altrusm and sptefulness n experments, Revew of Economc Dynamcs 1: Mas-Colell, Whnston and Green (1995), Mcroeconomc theory, Oxford Unversty Press. Rabn, Matthew (1993), Incorporatng farness nto game theory and economcs, Amercan Economc Revew, 83 (5): Seabrght, Paul (00), Blood, Brbes, and the crowdng-out of altrusm by fnancal ncentves Unversty of Toulouse, mmeo. Segal, Uz and Joel Sobel (1999), Tt for tat: Foundatons of preferences for recprocty n strategc settngs, Unversty of Calforna at San Dego, mmeo. Julo de 00

22 INTENTION-BASED ECONOMIC THEORIES OF RECIPROCIT APPENDIX Equlbrum analyss of the Sequental Game Remark 1: If teacher 1 partcpates, teacher also partcpates n every SRE Note that only the recprocty payoff can make choose np, as the materal payoff per se dctates a choce of p for. However, for any possble strategy of, teacher gets less when 1 chooses P than when he chooses NP. Whatever 1 beleves about s strategy, 1 s choce of P s unknd, and hence must beleve that 1 s unknd. Thus the recprocty payoff as well as the materal payoff makes teacher to choose p. Remark : If teacher 1 does not partcpate, the followng holds n all SRE: a. If b. If δ εx >, teacher does not partcpate δ < x ( δ + ε ), teacher partcpates c. If δ δ < < x ( ), teacher does not partcpate wth a probablty of δ + ε εx ε p = 1+ δ x Notce that f 1 does not partcpate, can gve 1 a materal payoff of at least 0 and at most x so the equtable payoff of 1 s x/. If chooses no partcpaton, 1 receves x. Therefore, s kndness of no partcpaton s x/. Smlarly, s kndness of partcpaton s -x/. In order to calculate how knd beleves 1 s after choosng NP we have to specfy s belef of 1 s belef about s choce after NP. 17 Denote ths by p. Then s belef about how much payoff 1 ntends to gve to by choosng NP s p x+(1-p ) (x+δ), and snce s payoff resultng from 1 s choce of P would be x, 18 s belef about 1 s kndness from choosng NP s p x + (1 - p )(x+δ) (p x + (1 - p ) (x + δ) + δ) + x - ε) = 0.5 ((1 - p ) δ + ε). Ths mples that when 1 does not partcpate and the second order belef s p, s utlty of no partcpaton s gven by x (x/) ((1 - p ) δ + ε), whereas s utlty of partcpaton s (x + δ) -0.5 (x/)((1 - p ) δ + ε). The former s larger than the latter f (x/) ((1 - p ) δ + ε) > δ. In equlbrum, the second order belef must be correct. Hence, f n equlbrum does not partcpate, the condton must hold for p = 1. Ths s the case f δ εx >. On the other hand, f n equlbrum partcpates, that condton must not hold for p = 0; Ths mples that δ < x ( δ + ε ) For ntermedate values of δ δ < < x( δ + ε ), nether no partcpaton nor partcpaton can be of an equlbrum. In εx 17 In prncple we also need s belef about 1 s behavor. However, after 1 has already chosen NP, already knows what 1 has done, and s belef has to be n accordance wth her knowledge. 18 In any SRE player partcpates after a partcpaton of 1 (Remark 1) Borradores de nvestgacón - No. 5