Role of Honesty in Full Implementation +

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1 1 Role of Honesty n Full Implementaton + Htosh Matsushma * Faculty of Economcs Unversty of Tokyo June (Frst Verson: March ) + Ths paper s a revsed verson of the manuscrpt enttled Non-Consequental Moral Preferences Detal-Free Implementaton Representatve Systems (Dscusson Paper CIRJE-F-304 Faculty of Economcs Unversty of Tokyo 2004) The research for ths paper was supported by a Grant-In-Ad for Scentfc Research (AENHI ) from the Japan Socety for the Promoton of Scence (JSPS) the Mnstry of Educaton Culture Sports Scence Technology (MEXT) of the Japanese government as well as a grant from the Center for Advanced Research n Fnance (CARF) at the Unversty of Tokyo I am grateful to the anonymous referee the assocate edtor for ther helpful comments All remanng errors are mne * Faculty of Economcs Unversty of Tokyo Hongo Bunkyo-ku Tokyo Japan

2 2 Abstract Ths paper ntroduces a new concept of full mplementaton that takes nto account agents preferences for understng how the process works We assume that the agents have ntrnsc preferences for honesty n the sense that they dslke the dea of lyng when t does not nfluence ther welfare but nstead goes aganst the ntenton of the central planner We show that the presence of such preferences functons effectvely n elmnatng unwanted equlbra from the practcal perspectve even f the degree of the preference for honesty s small The mechansms desgned are detal-free nvolve only small fnes eywords: Intrnsc Preferences for Honesty Detal-free Mechansms Full Implementaton Small Fnes Permssve Result JEL Classfcaton Numbers: C72 D71 D78 H41 1 Introducton Ths paper ntroduces a new concept of full mplementaton that takes nto account agents preferences for understng not just the consequence but also how the process works We nvestgate Bayesan envronments wheren a central planner s unaware of the desred alternatve to be chosen even f there exst multple agents they do receve ther prvate sgnal concernng ths alternatve The central planner delegates the alternatve

3 3 choce to these agents by desgnng a mechansm accordng to whch each agent makes announcements about ther prvate sgnals Full mplementaton requres that the values of the socal choce functon e the desred alternatves are nduced by the unque Bayesan Nash equlbrum The prevous works have constructed complcated mechansms whch are talored to the fner detal of specfcatons Ths complexty makes t dffcult to put the mplementaton theory nto practce 1 For nstance let us consder the mechansms provded by Abreu Matsushma (1992a 1992b 1994) n whch the agents are requred to make multple announcements about ther prvate sgnals at the same tme The central planner regards ther frst announcements as a reference fnes a small monetary amount to any agent who s the frst to devate from ths reference As long as the agents are honest durng ther frst announcements ths devce of small fnes functons to ncentvze the agents to keep ther all other announcements honest Incentvzng the agents to keep ther frst announcements honest at the outset mght be a more problematc ssue In order to solve ths ssue Abreu Matsushma ncorporated an addtonal ncentve scheme nto the devce of small fnes whch s however not detal-free e t depends heavly on the fner detal of specfcatons such as the probablty functon the utlty functons The falure to make the mechansms detal-free s the man drawback of the mplementaton theory from the practcal perspectve 2 1 See survey artcles such as Moore (1992) Palfrey (1992) Osborne Rubnsten (1994 Chapter 10) Maskn Sjöström (2002) 2 Another practcal problem concerns whether actual agents play teratvely undomnated strateges In expermental economcs t s a well known dea that even f subjects never enforce many teratons of removal at once they may learn to acheve a suffcently large number of teratons n the long run See

4 4 The purpose of ths paper s to demonstrate the possblty of mplementng socal choce functons wthout harmng the detal-free concept of mechansm desgn The crucal assumpton s that each of these agents has an ntrnsc preference for honesty n the sense that she/he dslkes the dea of tellng whte les that do not nfluence her welfare but nstead go aganst the ntenton of the central planner Wth the ntrnsc preference for honesty n ths sense we do not need to ncorporate any addtonal ncentve scheme wth the devce of small fnes All we have to do s just to keep the agents frst few announcements rrelevant to the alternatve decson Hence by usng only detal-free mechansms we can fully exactly mplement any ncentve compatble socal choce functon n teratve domnance Apart from ncentve compatblty we do not requre any condton on socal choce functons These features are n contrast wth the prevous works n the mplementaton lterature where agents ntrnsc preferences for honesty were not generally taken nto account 3 Several expermental economcs researches such as Gneezy (2005) emphaszed that the role of ntrnsc preferences n ths manner s non-neglgble n economc decsons Charness Dufwenberg (2006) on other h rased the alarm that agents ntrnsc preferences to nfluence ther decsons are heavly dependent on contexts framngs For nstance agents ntrnsc costs of lyng may not be sgnfcant as long as they expect that the central planner beleves that they le Moreover snce each agent makes so many Camerer (2003) for nstance There s a dffculty n applyng ths dea to our stuaton n that each agent s own experences are severely lmted therefore she/he has to utlze the other agents experences Huck Jehel Rutter (2006) obtaned expermental results statng that learnng s affected by the framng of feedback nformaton about the other agents experences How to fx ths framng n the frst place s an nterestng queston but s beyond the purpose of ths paper

5 5 announcements at once t s nevtable that her/hs ntrnsc cost of lyng for each sngle announcement s severely lmted Despte the fraglty of the ntrnsc preferences n ths manner the result of ths paper should be regarded as beng qute permssve Overcomng ths fraglty only requres that the proporton of announcements that are rrelevant to the alternatve decson be suffcent Ths would show that even f each agent s ntrnsc cost of lyng for all of her/hs announcements s close to zero any ncentve compatble socal choce functon s fully exactly mplementable n teratve domnance 4 Ths paper s organzed as follows Secton 2 defnes the model Secton 3 specfes detal-free mechansms Secton 4 shows the man theorem 2 The Model Let A denote a fnte set of alternatves; the set of lotteres over the alternatves; N = {1 n} a fnte set of agents where n 2 Further let Ω denote a fnte set of prvate sgnals for agent N where we set ω Ω ; Ω = N Ω the set of prvate sgnal profles; p : Ω [01] a probablty functon over Ω accordng to whch the prvate 3 There are exceptons such as Glazer Rubnsten (1998) Elaz (2002) 4 We elmnate only strctly domnated messages by usng the same method used n the studes for vrtual mplementaton by Abreu Matsushma (1992a 1992b) Abreu Matsushma (1994) nvestgated exact mplementaton just lke ths paper does; however unlke ths paper they used teratvely weakly undomnated strateges where only weakly domnated strateges were elmnated

6 6 sgnal profle ω = ( ω ) N Ω s drawn romly A socal choce functon f : Ω A s defned as a mappng from prvate sgnal profles to alternatves The central planner wants to acheve the desrable alternatve f ( ω) A that depends on the prvate sgnal profle ω Ω whch s not known to her/hm She/he delegates the alternatve choce to the agents accordng to a mechansm G = ( M g t) where M = M N m M M s a fnte set of messages for each agent x : M t = ( t ) N t : M R When the agents announce a message profle m ( m ) M the central = N planner chooses any alternatve a A wth the probablty x( m)[ a ] makes a monetary transfer t( m ) to each agent wth certanty We focus on mechansms n whch each agent makes multple announcements about her/hs prvate sgnal; a postve nteger exsts such that for every N M = Ω M M 1 M = where m ( ) = m k k= 1 M M k =Ω m k M k for all k {1 } For every k {1 } we term mk Mk as the k th announcement of agent We defne a utlty functon for each agent N by u : A R M Ω R where there exst functons v : A Ω R c :[01] Ω R such that c (0 ω ) = 0 c ( r ω ) s contnuous ncreasng wth respect to r [01]

7 7 #{ k {1 } mk ω} u( a t m ω) = v( a ω) + t c( ω) Note that c #{ k {1 } m ω } ω k ( ) mples agent s ' ntrnsc cost of lyng when she/he receves the prvate sgnal ω Ω announces the message m M Ths ntrnsc cost depends on the proporton of her/hs dshonest announcements #{ k {1 } m k ω } Ths cost does not depend on the absolute number of dshonest announcements Hence the ntrnsc cost of lyng for each sngle announcement s severely lmted whenever the number of announcements that each agent s requred to make s very large Further note that v ( a ω ) mples the utlty of agent for her/hs materal nterest Moreover we assume quas-lnearty rsk-neutralty n terms of monetary transfers We shall confne our attenton to socal choce functons f that satsfy ncentve compatblty n terms of the agents materal nterests n that for every N ω Ω ω Ω /{ ω } (1) Ev [ ( f( ω) ω) ω ] Ev [ ( f( ω ω ) ω) ω ] where E[ ω ] s the expectaton operator gven ω Incentve compatblty mples that truth-tellng s a Bayesan Nash equlbrum n the drect mechansm rrespectve of whether or not the agents have ntrnsc preferences for honesty Let u = ( ) denote a utlty functon profle A combnaton ( Gu ) defnes a u N Bayesan game A strategy for each agent N s defned as a functon s : Ω M We denote s ( ) = s k k= 1 s ( ) ( ( )) ω = s k ω k = 1 where sk : Ω Ω sk ( ω) Ω

8 8 denotes the k th announcement of agent Let S denote the set of strateges for agent A strategy profle s denoted by ( s N Let S N s = ) S s( ω ) = ( s ( ω )) N s ( ω ) = ( s j ( ω j )) j N /{ } The soluton concept used n ths paper s teratve domnance whch s defned as follows Let S = ( 0) S (0) (0) S = S N Recursvely for every h = 12 let S denote ( h ) the set of strateges s for each agent that are undomnated wth respect to ( h 1) S S = S ; n other words there exst no m M no ω Ω such that for ( h 1) ( h 1) j j N /{ } every s ( h 1) S E[ u ( x( m s ( ω )) t ( m s ( ω ))( m s ( ω )) ω) ω ] > Eu [ ( xs ( ( ω)) t( s( ω)) s( ω) ω) ω ] where we denote u( α r m ω) = u( a r m ω) α( a) for each α Let S ( h) = S ( h) a A N S ( ) ( h) = S A strategy profle s S s sad to be teratvely undomnated n ( Gu ) h= 0 f ( ) s S We defne the honest strategy s * S for agent by * sk ( ω) = ω for all k { 1 } all ω Ω The honest strategy profle s * = ( s * ) S nduces the value of the socal choce functon N f ( ω ) for every ω Ω wth no monetary transfers; n other words for every ω Ω * ( ( ))[ f( )] 1 xs ω ω = * ( ( )) 0 t s ω = for all N

9 9 3 Mechansm Desgn We fx a postve real number ε > 0 such that (2) ε < (1 ω ) for all N c whch mples that ε s selected less than the ntrnsc dsutlty for each agent when she/he les durng all her/hs announcements Note that there exsts such an ε because c( r ω ) s contnuous ncreasng wth respect to r [01] c(0 ω ) = 0 Moreover we fx two postve ntegers ˆ such that > ˆ (3) ˆ ε < c( r+ ω) c( r ω) for all ˆ r [01 ] (4) ( ˆ ) ε > max v ( a ω) v ( a ω) 2 ( aa ω ) A Ω N From nequalty (2) the contnuty ncrease of c note that there exst such a ˆ In fact by fxng as suffcently large we can choose ˆ to satsfy the followng two propertes: ˆ s suffcently close to unty to satsfy nequalty (3) ˆ s suffcently large to satsfy nequalty (4) Based on ( ε ˆ ) defned above we specfy the mechansm denoted by G( ε ˆ ) = ( M g t) as follows; for every m M

10 10 #{ k { ˆ + 1 } f(( mk ) N) = a} xm ( )[ a] = ˆ for all a A for every N t ( m) = ε f there exst k {2 } such that mk m1 ( m ) = ( m ) for all h { 1 k 1} j h j N j1 j N t ( m ) = 0 f there exsts no such k The central planner requres each agent to announce number of tmes the type of prvate sgnal that was observed She/he romly selects one announcement profle ( m ) M from the last ˆ profles chooses the alternatve j k j N j k N f (( mjk ) j N) A where k { ˆ + 1 } She/he mposes a fne of ε > 0 f only f the agent s the frst to devate from her own frst announcement Note that the early ˆ announcement profles e ( m j k) for ˆ j N k {1 } are rrelevant to the alternatve decson x( m ) 5 The mechansm small fnes gven by ε > 0 G( ε ˆ ) nvolves only Note that the mechansm G( ε ˆ ) s detal-free n the followng sense Let us select ε > 0 as close to zero a postve real number λ (01) as beng close to unty a postve real number Q > 0 such that t s suffcently large Moreover let us select 5 Note that these profles are relevant to the monetary transfers ( t ( m)) N

11 11 ˆ to be suffcently large such that ˆ s greater than λ ˆ ( ) ε s greater than Q Note that nequaltes (3) (4) hold whenever c ( r+ λ ω ) c ( r ω ) ε for all r [01 λ] max v ( a ω ) v ( a ω ) Q 2 ( aa ω ) A Ω N whch are very weak restrctons because ε s selected such that t s close to zero Q s selected such that t s suffcently large Hence we can say that G( ε ˆ ) does not depend on the fner detals of specfcatons such as the probablty functon the utlty functons 6 4 Man Theorem The followng theorem shows that wth ncentve compatblty truth-tellng s the unque teratvely undomnated strategy profle n ( G( ε ˆ ) u) whch mples that any ncentve compatble socal choce functon s fully mplementable n teratve domnance In contrast to the prevous works we do not need any condtons such as Bayesan monotoncty (Jackson (1991)) no consstent decepton (Matsushma (1993)) measurablty (Abreu Matsushma (1992b)) n addton to ncentve compatblty 6 The constructon of G( ε ˆ ) depends on the socal choce functon f Needless to say G( ε ˆ ) functons rely crucally on the ncentve compatblty of the socal choce functon

12 12 Inequalty (3) guarantees that each agent s wllng to keep her/hs early ˆ announcements honest because her/hs ntrnsc cost of lyng for all of these announcements s greater than the small monetary fne ε Gven that the early ˆ announcements are honest nequalty (4) along wth ncentve compatblty guarantees that the devce of small fnes functons n ncentvzng each agent to keep her/hs latter ˆ announcements honest n the same manner as n Abreu Matsushma (1992a 1992b 1994) The Theorem: The honest strategy profle s * ( G( ε ˆ ) u) S s unquely teratvely undomnated n Proof: Fx s S N arbtrarly Further fx ω Ω arbtrarly Suppose that s ( ω ) s ( ω ) for some j some k { 2 ˆ } j k j j k 1 j Then agent s never fned at the tme of announcng m k = ω for all k { 1 ˆ } Next suppose that s ω ) = s ( ω ) for all k { 2 ˆ } all j j k ( j j k 1 j If sk ( ω) ω for all k { 1 ˆ } then by announcng m k = ω for all k { 1 ˆ } nstead agent can save the dsutlty for lyng c #{ k {1 } s ( ω ) ω } #{ k {1 } s ˆ ω k ( ω) ω} c( ω) k ( )

13 13 whch s greater than ε due to (3) If s ω ) s ( ω ) for some k { 2 ˆ } then agent k ( k 1 s fned an amount ε Snce the early ˆ announcements of agent do not nfluence the alternatve decson t follows that agent s wllng to replace the early ˆ announcements ˆ ( s k ( )) k= 1 ω wth * ˆ k k 1 ( s ( ω )) = Fx k { ˆ + 1 } arbtrarly Suppose that s = s for all j N all k { 1 k 1} * j k j k Further fx ω Ω arbtrarly Suppose that s ( ω ) ω k Let m M denote the message for agent such that mk = ω for all k { 1 k} mk = sk ( ω) for all k { k + 1 } Frst suppose that Then t(( s ω)) s ( ω ) j ω j for some j jk = ε t( m s ( ω )) = 0 whch along wth (4) mply that agent prefers m to s ω ) Next suppose that ( Then t(( s ω)) s ( ω ) j = ω j for all j jk = ε t( m s ( ω )) ε whch along wth the ntrnsc preferences for honesty ncentve compatblty gven by nequalty (1) mply that agent strctly prefers m to s ω ) Hence we have proved that s * s the unque teratvely undomnated ( strategy profles QED

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