Subjective Performance Evaluation and Inequality Aversion

DSCUSSON PAPER SERES ZA DP No. 338 Sut Poman Ealuaton and nqualty Ason Chstan Gund Judth Pzmk Mah 008 Foshungsnsttut zu Zukunt d At nsttut o th Study o ao

Sut Poman Ealuaton and nqualty Ason Chstan Gund Unsty o Wuzug and ZA Judth Pzmk Unsty o Bonn Dsusson Pa No. 338 Mah 008 ZA P.O. Box 740 5307 Bonn Gmany Phon: 49-8-3894-0 Fax: 49-8-3894-180 E-mal: za@za.og Any onons xssd h a thos o th authos and not thos o ZA. Rsah ulshd n ths ss may nlud ws on oly, ut th nsttut tsl taks no nsttutonal oly ostons. Th nsttut o th Study o ao ZA n Bonn s a loal and tual ntnatonal sah nt and a la o ommunaton twn sn, olts and usnss. ZA s an ndndnt nonot oganzaton suotd y Dutsh Post Wold Nt. Th nt s assoatd wth th Unsty o Bonn and os a stmulatng sah nonmnt though ts ntnatonal ntwok, wokshos and onns, data s, ot suot, sah sts and dotoal ogam. ZA ngags n ognal and ntnatonally omtt sah n all lds o lao onoms, dlomnt o oly onts, and dssmnaton o sah sults and onts to th ntstd ul. ZA Dsusson Pas otn snt lmnay wok and a ulatd to nouag dsusson. Ctaton o suh a a should aount o ts osonal haat. A sd son may aalal dtly om th autho.

ZA Dsusson Pa No. 338 Mah 008 ABSTRACT Sut Poman Ealuaton and nqualty Ason Many ms us sut oman aasal systms du to lak o ot oman masus. n ths ass, susos usually ha to at th oman o th suodnats. Usng suh systms, t s a wll stalshd at that many susos tnd to assss th mloys too good lnny as and that th aasals hadly ay aoss mloys o a tan suso ntalty as. W xlan ths two ass n a modl wth a suso, who has ns o th utlty o h nqualty as suodnats, and dsuss dtmnants o th sz o th ass. Extnsons o th as modl nlud th ol o suso s aotsm o on atula agnt and th ndognous ot ho o agnts. Whth nqualty as agnts xt hgh ots thn uly slontd ons, dnds on th sz o ot osts and nqualty ason. JE Classaton: M5, D63 Kywods: aasals, nqualty ason, oman aluaton, ntalty as, lnny as Cosondng autho: Chstan Gund Datmnt o Busnss and Eonoms Unsty o Wüzug Am Sandng D-97070 Wüzug Gmany E-mal: hstan.gund@un-wuzug.d Many thanks to Matthas Kakl and Not Shulz o hlul ommnts.

1. ntoduton Poman aasal s on o th most quntly dsussd tos o human sou managmnt sah. 1 Systmat oman aasal systms an mlmntd o sal uoss nludng omoton and tanng dsons, oman asd aal ay and sonnl lannng Clland t al. 1989. n many ass t s not ossl to otan ot oman masus on an nddual as. Bsds, xstng ot masus usually osond only to at o mloys tasks, whh may lad to dstotd nnts Mlgom & Rots 1988, olmstöm & Mlgom 1991. Tho, many ms mlmnt som knd o sut oman aluaton. Sut oman aluaton, how, s also aught wth som olms. On ssu s that th auay o atngs s not gn automatally. th at s a sdual lamant.. th own o a m and th aasal ats som knd o aal ay, sh may undot th oman o h suodnats n od to sa osts. ow, many susos a no sdual lamants ut thmsls mloyd woks n mult layd ms. Tho, susos an also nttd as agnts wth own utlty untons, whh may dat om th nal s ots. n ths stuaton ossl at ass nlud th ntalty as and th lnny as. n many ms th maoty o mloys gt atngs ao th aag mak. Tho, th aasals a y dnton skwd to th to nd o th sal and susos at som knd o lnnt.g. Jawaha & Wllams 1997. Bsds, sal studs nd that only a small aton o ossl oman lls s usd y th susos whn aluatng suodnats. Thy tnd to dntat only slghtly twn mloys.g. Muhy & Clland 1991. Ths ass may not only nnt o ms ut also lad to th olm that th oman aasal systm s not atd y sal mloys as Muhy 199 oss o th hama omany Mk & Co, n. Mo than 70 nt o mloys a loatd n only th o 13 oman atgos and only aout 5 nt ha got maks yond th aag on n ths as s Fgu 1. 1 Btz t al. 199 as wll as y & Wllams 004 od xtns suys o th ltatu. Oth ass that a dsussd n th ltatu nlud th halo, may and ny t s.g. Muhy & Clland 1995. 1

Fgu 1: Dstuton o sut oman aasals at Mk & Co., n. 0.3 0. 0.1 0 1 1-3- 3 3 4-4 4 5-5 Poman Ratng Not: 1 = unatal oman, 5 = xtonal oman. Sou: Muhy 199,. 40. Sal asons o ths ass a dsussd n th mal ltatu. susos a not wadd o auat atngs, thy may ha nsunt motaton to nst tm n gathng nomaton Fox t al. 1983. Futhmo, thy may also a ognt lmtatons and tnd to ous on som oman dmnsons ttn t al. 003 o ataly ao tan mloys.g. to nouag loyalty o to s th sl-ntst Fs & Judg 1991. Bsds, susos may ha a n o a lasant latonsh wth th suodnats Vama t al. 1996. Ngat dak may lad to undsal dsussons and s tho aodd ossl. 3 Th ontutons o Pndgast and Tol 1996 as wll as Pndgast 00 a two o th w thotal studs whh od xlanatons o th lnny as y assumng that th wll-ng o suodnats s at o th suso s utlty unton. n Pndgast and Tol 1996 a suso an dstot h ots ut s montod y th managmnt. Sn asd ots lad to nnt o assgnmnts, managmnt unshs th suso h ot dats om th managmnt s own osaton. Smlaly, Pndgast 00 assums that th suso as som ost whn datng om th tuth. Both ontutons 3 S also Bol 005 o a mo dtald ow o th ltatu.

show that aotsm nass wth th wok s nnts. 4 Assssmnts a dstotd uwads th mo th suso lks a atula agnt and downwads th mo sh dslks a suodnat. n ths a w uld on ths agumnt and addtonally assum that mloys a nqualty as to ah oth. W an smultanously xlan oth th lnny as and th ntalty as wthn ou sml modl. agnts a nqualty as, t s oous that agnts ot om dstotd atngs to th man om an x ost st. ow, t s not oous how x ant nnts o dnt agnts a atd assumng that th s untanty n th oduton thnology. W show that th t o dstotd atngs on nnts dnds on th ty o th agnt uly sl-ntstd sus nqualty as. Whl uly slntstd agnts a not nlund y dstotd atngs, th hao o nqualty as agnts hangs. th suso dstots ots, th a stuatons, n whh y nqualty as agnts xt hgh ots than uly sl-ntstd agnts. t s now mo and mo atd that nqualty ason s an motant dng o o human hao n many stuatons. Basd on xmnts y.g. ownstn t al. 1989, Fh and Shmdt 1999 as wll as Bolton and Oknls 000 od suggstons o oatonalzatons o nqualty ason n onom modls. Rnt alatons o nqualty ason n nal-agnt-modls nlud nddual toh, 004, Nlson & Stow, 004, May & P, 004, Englma & Wamah, 005, Dmougn t al., 005 and tam asd nnt ontats toh 004, Dmougn & Flut 003a, Bl 004, ankod tounamnts Dmougn & Flut 003, Gund & Slwka 005 and ads slton olms on Smns 005, Dsau & Sangton 007. ow, nqualty ason has not n ald to oman aasal ssus so a. Smultanously and ndndntly om ths a Slwka 007 gs anoth xlanaton o th ntalty as y nooatng two not dntal sgnals o th agnts omans o a suso and a nal. Th ntalty as s du to th suso s nquty ason and a gsson to th man t o th suso s ot, aus sh wants to aod own dsadantags om datng om th nal s sgnal. 4 Slwka 007a ods a dnt modl on oman aasal wth on loss as agnt and shows n a two od modl that th lnny as oms mo lant o tm. Maod 003 nstgats otmal ontats wth sut oman aluatons whn th sgnals o a nal and an agnt aout th oman o th agnt a not olatd. n od to aod onlts wth th agnt th suso who s th nal n ths as omsss th atngs at th to whh s smla to th ntalty as. 3

n th nxt ston w od y odng a sml modl. A suso has to aas th oman o two mloys. utlty dnds on th m s ot and on th wll-ng o h nqualty as suodnats. W smultanously xlan oth th ntalty and lnny as wthn ths sml modl. W also dsuss dtmnants o th dg o oth ass and xtnt th modl to a stuaton wth aotsm, whn th suso has ns o th wll-ng o only on suodnat ston 3. Futhmo, agnts may antat th asd ots o th suso and ndognously hoos ots lls. W dsuss ths as n ston 4 and show that ots wth asd ots a hgh omad to a stuaton wth unasd ots th amount o nqualty ason s ath hgh. Ston 5 onluds.. A sml modl Suos that a suso S has to assss th oman o two agnts and and = 1, ; to dtmn th wags. al ot o agnt s haatzd y th tm 0. W assum that th suso s not a sdual lamant, ut mloyd n a m. Sh oss th tu oman 0 o oth agnts and thn has to stat a ot o ah agnt. Th wag W o agnt onssts o two omonnts. s a oman aasal ontngnt onus 0 n addton to hs xd wag w 0,.. W = w. Fo smlty w assum that oth agnts a qually nqualty as. To modl th utlts w us th utlty unton oosd y Fh and Shmdt 1999. th oman aasal o s low than th oman aasal o, agnt s alous o. Th sultng dsutlty nass n th dn o th oman atngs, n th ot dndnt wag omonnt, and n th dg o dsadantagous nqualty ason. nstad agnt s tt assssd, h may l som knd o omasson o gult wth agnt. Th xtnt o suh lngs s gn y th aamt. Thus, th utlty unton o an agnt who s nqualty as s gn y U, = W max {W W, 0} max {W W, 0} = w max {w w, 0} max {w w, 0} = w max {, 0} max {, 0}, and 0 < 1, 0. 5 5 Agnts a assumd to nqualty as wth st to outoms and do not tak nuts.g. ots nto aount. W thnk ths s a asonal assumton sn th s dn that th ast maoty o mloys 4

Th sal as o = = 0 tus th utlty o two uly sl-ntstd agnts who a only ntstd n th own wag. W assum < 1. Ths mls that agnt as mo o hs own utlty than o th oth agnt s utlty. n th smnal a Fh and Shmdt assum that. W do not nd ths assumton o ou modl. Dannng t al. 007 od xmntal dn that th oost may tu n som umstans. Futhmo, w assum that th suso s ntstd n th auay o h statmnt and n th utlts o th agnts. On th on hand, th suso sus om osts, h ot dats om h osaton. y, 0 masus th ntnsty o ths osts om dstotng aluatons. Ths osts may tu th otntal ost o ng d ound out to ha statd dstotd aluatons. 6 On th oth hand, S as o th utlts o th agnts. Fo xaml, sh may want to aod agumnts wth th agnts and a ngat wokng atmosh n gnal Bnadn & Bukly 1981. Th stngth o ths ato s sntd y th aamt. Th suso s utlty unton s tho gn y U S,, = w max {, 0} max{, 0} w max {, 0} max {, 0}, 0. Bo dng th otmal ots o th suso o nqualty as agnts, w dtmn th oman aasals o stuatons usually analyzd n mo tadtonal onom modls. Th sults an takn as nhmaks omad to th sults o ou modl: 1. th suso s mly sl-ontd and dos not a o h agnts utlts = 0, t s asy to show that sh wll ot th tu omans = = 1, y dng h utlty unton wth st to th ots, aus th would only osts ut no nts om datng om th osatons.. th suso as o agnts utlts 0 and th agnts a not nqualty as ut uly sl-ntstd = = 0, th suso maxmzs w w. Solng th st od ondton o lad to th onsd thmsls as to oms My 1975 so that dns n ots wll usually not d qually aoss mloys. 6 Pndgast 00 maks a smla assumton. 5

6 otmal aasals = ~ = 1,. n ths as th suso ostats th agnts omans. Ths sult an nttd as th lnny as. Th sz o ths as s nasng n th dg o th suso s n o th agnts utlts and th montay nnts o th agnts. t s dasng n th suso s osts o gng datd ots. Eah mloy s ugadd to th sam dg so that th s no ntalty as n ths as. n ou modl th suso also has to hoos 1 and gn h osatons 1 and to maxmz h utlty y takng th nqualty ason o th agnts nto aount. Not that th suso s utlty unton dnds on h ots. Thus, w ha to dstngush th dnt ass, aus unn ots lad to nqualty twn th agnts. < = =. ] [ ] [ ] [ S w w w U Th as o qual ots an also ntgatd nto th oth ats o th suso s utlty unton so that w an wt ths as =. ] [ ] [ S w w U Assum that S oss. Sh wants to maxmz h utlty and has to dd whth t s otmal to hoos dnt o qual ots o th agnts. To dtmn th otmal ots and, w st look at th st at o h utlty unton. S dds to hoos a hgh ot o agnt o qual ots o oth agnts, sh maxmzs ths at o h utlty unton sut to th onstant whh lads to ] [ w = λ. Th sultng Kuhn-Tuk-ondtons a 0 1 = λ K1 0 1 = λ K

λ 0,, = 0 λ K3 Fom addng K1 and K w otan / =. K4 Possl solutons nlud oth dnt and qual ots. S hooss, t ollows om K3 that th agangan multlλ has to zo. Fom K1 and K w otan th aluatons o agnt and d d 1 / and = 1 / =. 7 ow, th st at o th suso s utlty unton s only ald o th ang d so that d 1 / has to hold. Sn th tm on th ght sd o nqualty 1 s ost, th dn o th osd omans has to suntly ost, too. Agnt has to outom hs ollagu to a sunt amount to omnsat o th aton o nqualty S announs d d. Not that th ght hand sd o 1 s nasng n th nqualty ason aamts and. Ths s ntut sn t atus an agnt s hgh dsutlty whn otanng a ot dnt om th on o hs ollagu. Futhmo, th ght hand sd s nasng n and n th ot dndnt wag omonnt whl dasng n th suso s osts o datng om th osd omans. S may hoos = n som ass, so that th agnts annot su om nqualty. n ths as w th ha λ = 0 o λ 0 s K3. λ = 0 and = th ots solng all Kuhn-Tuk-ondtons K1 to n / K3 a = 1 / and = 1 /, whh a qualnt to n n n n = = = /. 8 / 7 To atu th da o dnt ots w wll wt d 7 and d, = 1, and.

S sts = and w ha λ 0, w also otan n n n = = = / om K4. Sutatng K1 om K lads to th / lant ondton o statng ths ots: < /. ths ondton s satsd, th ots n and n sol th Kuhn-Tuk-ondtons K1 to K3. A smla asonng an ald to th maxmzaton o th sond at o th suso s utlty unton sut to th onstant 9. t s n otmal o th suso to g agnt a tt atng, whl sh oss that agnt s th to om s Andx A. S oss that agnt s oman s tt than th oman o agnt, th only soluton to th maxmzaton olm o th sond at o th utlty unton s to stat qual n n ots = / =. Sn th st at o th suso s utlty / unton osonds to th sond at o th utlty unton ots a qual, dnt ots a only otmal /. Othws S sts qual ots. Th daton o th otmal ots whn agnt s oman s at last as good as th oman o agnt ollows analogously. Ou sults a summazd n Pooston 1. Pooston 1: oth agnts a nqualty as and th oman dn twn th tt agnt and th wos agnt s suntly lag,.. /, th suso otmally ots = 1 / and d d = 1 /. th oman dn s ath small,.., / /, th suso announs qual ots n n = = /. / t s asy to s that th suso s ots tu oth th ntalty as and th lnny as. th a lag oman dns twn th agnts, th suso ts to dmnsh th wag ga twn th agnts. Ths t snts th ntalty as. Th hgh and,.. th mo nqualty as th agnts a, th lag s th ang o qual ots and th lag th ntalty as. 8 To atu th da o qual ots w wll wt o = 1,. 9 Th lant agang unton and th sultng Kuhn-Tuk-ondtons an ound n Andx A. n 8

Gn that < 1 holds, oth agnts oman atngs a hgh than th osd omans, whh snts th lnny as. Othws, only th ot o th agnt wth th low osd oman s adustd uwads. n ths as th sum o th uwad as o th wak mloy and th downwad as o th to om s always ost, so that th lnny as always xsts n sum. Ths dstoton nass n th n o th suso o th agnts utlts and n th ot dndnt wag omonnt. t s dasng n h osts o datng om th osd omans. nstng th as o qual ots n, t s also staghtowad that th suso gnally ats th agnt wth th low osd oman wth lnny. ow, smla to th as o dnt ots ths dos not always hold o th ot o th agnt wth th hgh osd oman. Agan, th sz o th lnny as dnds on th nqualty ason aamts and. Assum that, thn n < /. Pooston 1 stats that th suso announs n and n / /. Consquntly th ot o agnt s only adustd uwads < 1. n ontast th ot o agnt s low than hs osd oman, 1 and / <. ow, th uwad as o / th wak agnt always outwghs th ossl downwad as o th tt on. th agnts omans a xatly th sam = =, th suso ots = / o = 1,. Ths s th only as, n whh th oman aasals o n nqualty as agnts osond to th ots o sl-ntstd agnts smly aus th s no nqualty. Th lnny as s also snt n th as o uly sl-ntstd agnts, ut th ntalty as annot xland. 3. Faotsm n ou as modl o ston w assum that th suso aos all agnts. ow, t may wll th as that a suso has only ns o on o two suodnats h aot. Fo xaml, th suso knows on agnt wll and/o wants to oat ous loyalty. n ontast, th oth wok and th suso ha no sal latonsh. 9

Fo smlty, w assum 0 and 0 to nstgat oman 1 = = n = aasals n suh a stuaton. 10 Tho, th suso only aos agnt 1. Now otmal ots o th suso dnd on th atual sz o th nqualty aamts o agnt 1, and, and on th sz o th osd oman dn. Agan, w ha to dstngush twn th ass. n Fst, agnt 1 s dong suntly tt than agnt,.. B = 1 /, 11 n th suso ots d d 1 / and n = / n =. Th aot, agnt 1, otans th hgh atng,.. d d n. Ths ots a smla to th ots n ston. Sn th suso now only as o agnt 1, th uwad as o th ots s nasng n. Although agnt s alous o agnt 1, th suso nglts ths at. Thus, n ontast to d 1 and d o ston, th ny aamt o agnt dos not nlun and algn ots. Not that B wll ngat < ½. Tho, th suso n ugads agnt 1 hs osd oman s slghtly wos than agnt s osd oman, agnt 1 dos not l muh omasson wth agnt whn ng a hgh ot. Both agnts atngs a hgh than th osd oman whh symolzs th lnny as. Although agnt 1 s th aot, h may lss ugadd than th oth agnt n som stuatons. Ths s a onsqun o agnt 1 s omasson o agnt. Tho, th suso wants to aod a lag dn n aluatons, whh lads to th ntalty as agan. Sond, agnts omans d only slghtly,.. 1 / = B B = 1 /, 1 th suso aluats oth n n n agnts qually and ots = = / 4 0. 5 n n. Ths xsson tus th ntalty as. Th ang o ths ntal s nasng n th nqualty ason aamts o agnt 1.. th hgh and, s nasng n th dg th suso as o hs aot th hgh, s nasng n th ot dndnt wag omonnt and s dasng n th osts o datng om th tuth. A omason o n wth n o ston als that th st tm / s hal o that o 4 n. Thus, th st tm s 10 Th ndx ndats that agnt 1 s th aot and th ndx n ndats that agnt s not th aot. 11 A daton o th thshold B an ound n Andx B. 1 W d B n Andx B. 10

nlund y th num o agnts th suso as o. < 1/ so that B s ngat, oth agnts th sam atngs although agnt 1 s oman s wos than agnt s oman. 1/ B s ost, t s ossl that ots a qual although agnt 1 s osd oman s slghtly tt than th osd oman o agnt. Ths s ntut sn a tt atng would mak agnt 1 l soy o agnt and das hs utlty. Not that n. Tho, agnt 1 s oman atng s atd y th lnny as. n ontast, agnt s atng s low than hs tu oman / n B. Only th osd omans a suntly los to ah oth, oth atngs a asd uwads. Othws, only agnt 1 ots om th aluaton. ow, th uwad as o agnt 1 always outwghs th ossl downwad as o agnt. d Thd, B, aasals a ˆ = 1 / and n < d ˆ = /. Now agnt 1 otans a low oman assssmnt than agnt. n n Nthlss, th assssmnt o agnt 1 s hgh than hs osd oman lnny as whl th oost s tu o agnt. Th as s a onsqun o agnt 1 ng alous o agnt s good oman. Whl th ot o agnt 1 s smla to that wthout aotsm o on atula agnt n ston, ths dos not hold o agnt s ot. n ston th a aamt angs o and so that th ot o agnt s asd uwads. agnt 1 s th aot and th suso dos not a o agnt, ths s not ossl anymo. Th onsdatons o ths ston an summazd n ooston. Pooston : th suso only as o on agnt, sh ots n n qual aasals = = / 4 0. 5 dnt ots n B n B and n d 1 = 1 / and n = / n n B as wll as d d d ˆ = 1 / and ˆ = / B. n n n < 11

4. Endognous Eot Cho o Agnts n th last two stons w analysd th suso s oman aasal dson whn th agnts omans and tho ots w xognously gn. Now w want to nstgat how th agnts nnts to xt ot and om wll a nlund y th asd ots o th suso thy antat h hao. 13 Thus, agnts ndognously hoos th ots whh dtmn th osd oman lls. W oma ots n suh a stuaton wth a stuaton n whh th suso dos not a o th agnts utlts,.. a stuaton o unasd ots. Th suso annot os th agnts ots and ot osts ut oss th omans 1 and. 14 On th ass o ths osatons sh stats h ots 1 and. Fo smlty w assum that th a only two oman lls: a low oman ll 0 and a hgh oman ll smultanously hoos th ots [ 0,1] wth oalty and W assum 0 wth wth oalty 1 /. 15 Both agnts. Whn xtng ot, an agnt alzs and sus om ot osts C = /. 1 0.. Ths ondton nsus that w only tak nto solutons o otmal ot lls nto onsdaton sn th ot unton s sttly ona. Moo, w sum that suso and agnts a mutually awa o th utlty untons. Bsds, all oth assumtons om ston aly to ths ston as wll. ntally, w wll xlo ot hos o uly sl-ntstd agnts as a nhmak. W wll oma th stuaton o an gost suso wth th as that th suso as o th two agnts. Thn, w wll od y dtmnng agnts ot hos o th as o agnts nqualty ason. 13 Ths s dnt om Pndgast 00. An agnt dos not xt th suso s aotsm and tho hs nnts a not atd n hs ontuton. 14 W assum that th ataton onstant dos not nd and thus oth agnts wok o th m. 15 Ths assumton nsus that th suso hooss dnt ots o and n th as o datng osatons o omans. 1

4.1 Sl-ntstd Agnts Fst, w look at th hao o uly sl-ntstd agnts and onsd th as o a suso, who dos not a o agnts utlts. Sh, tho, always ots th agnts tu oman lls = so that ots a unasd. An agnt s xtd utlty EU s gn y hs xd wag lus th xtd onus mnus th ot osts: EU,unasd,go = w 1 /. 16 Solng th st od ondton EU / = = 0 lads to th otmal ot o a sl-ntstd gost agnt: = /., unasd, go / = snts th magnal tun o a uly sl-ontd agnt om ahng th hgh oman n th as o unasd ots. t s oous that, unasd, go s dasng n th ot osts aamt. Th mo an agnt s om xtng ot th hgh th oman dndnt wag omonnt and th dn twn th two oman lls th hgh s hs ot ll. th suso as o th agnts utlts and tho dstots h ots n th sns w ha dd n ston, th xtd utlty o an agnt s gn y EU,asd,go = w 1. 17 Th st od ondton EU / = = 0 lads to,, =. asd go / n, ots a ndndnt o th suso s ns o gost agnts. th suso stats asd ots, sh ugads h statmnts y alzd oman ll. W, tho, an omulat / ndndnt o th 16 Th ndx unasd ndats that th suso stats th tu oman whl th ndx go ndats th as o uly sl-ontd agnts. 17 Th ndx asd ndats that th suso dos not ot th tu osd omans. 13

Pooston 3a: Th s no nnt t o ot dstotons o uly sl-ntstd agnts. 4. nqualty As Agnts and Unasd Rots W now look at th hao o nqualty as agnts. Fst, w onsd th as o unasd ots wth th suso otng th agnts tu omans =. Tho, an agnt s xtd utlty s gn y 18 EU,unasd,as = w 1 1 1 1 /. oth agnts alz th sam low o hgh oman ll, th a no nqualty osts. ow, agnts su om nqualty thy a dnt sults. Th st od ondton EU / = = 0 gs us th otmal ot o agnt gn ot : 1 = /. ntutly, amguous ts a nlunng th otmal ot h. Suos that only agnt xts ot whh, o ous, annot an qulum. Thn agnt s wllng to xt th hgh ot th mo h an n th good stat whn s alzd. ang alzd th good stat whl th oth agnt s wos o, maks agnt l omasson wth hs ollagu and so hs ot dass n. t us all ths ngat t th ng ahad t. agnt dd not xt ot, agnt ould n alous o th oth agnt. Thus, hs magnal tun om ahng th hgh oman ll s low than th magnal tun o a uly sl-ntstd agnt. agnt xts ot, too, t s ossl that agnt has a low omnsaton than agnt x ost and s nous. n, agnt wants to aod ths stuaton and hs ot nass n. Futhmo, th ngat nlun o on agnt s ot s dmnshd ut not anlld out as long as th s stll a ost oalty that agnt alzs. W all ths t th atn ot t. Th hgh ths t s th hgh a th nnts to xt ot and to ah th hgh oman ll. 18 Th ndx as ndats that oth agnts a nqualty as. 14

n th symmt Nash qulum 1, unasd, as, unasd, as = oth agnts xt ot, unasd, as 1 =. Comaat stats show that, s nasng n th oman dndnt wag, unasd as omonnt, n th oman dn and n th dsadantagous nqualty aamt. 19 An agnt wth a hgh xts hgh ot to das th oalty o an unaoual stuaton.. a stuaton wh h has only alzd has alzd. Moo, ot dass n., whl th oth agnt A omason o nqualty as agnts ots,, unasd as wth th ots o uly sl-ntstd agnts ndats that ots a hgh o nqualty as agnts 1, whh an wttn as 0 < / o and lads to 3, Pooston 3: th suso dos not dstot ots, nqualty as agnts xt hgh ots n a symmt Nash qulum than uly sl-ntstd agnts and only < /. Condton 3 shows that nqualty as agnts xt hgh ots ot osts a suntly low o th ny aamt s suntly hgh. Futhmo, ondton 3 holds th oman dndnt wag omonnt and th oman dn a suntly hgh. t llustats that th ost ts on nnts whh sult om a ost ot o th oth agnt ha to qut hgh. th atn ot t domnats th ngat ng ahad t, nqualty as agnts xt hgh ots than uly sl-ntstd agnts, whn ots a not dstotd. 19 S Andx C. 0 Thus, th xsts an ntal [ 1, / ] o that guaants nto solutons and hgh ots o nqualty as agnts. 15

4.3 nqualty As Agnts and Basd Rots Now, lt us onsd th as o asd ots, whn th suso as o th two nqualty as agnts. Followng Pooston 1, th suso wll ot d d 1 / and = 1 / = sh oss = and =. sh oss o oth agnts, sh stats n n = = / n n = = /. Thus, agnt s xtd utlty s gn y EU = w 1 1 1 1 1 1 1 1 1 1 To dtmn th otmal ots, w dntat th agnt s xtd utlty wth st to :. EU = 1 = 0. Solng o gs us th otmal ot o agnt gn ot 1 = / o 1/ 1 = Now th ot ho dnds on th osts o statng a dstotd oman aluaton. Agan, th ot o agnt nass n th ot o agnt whh s haatzd y th atn ot t. 1 ow, although ths t s stll ost t s low than th t ots a not dstotd. Gn that agnt alzs and agnt only, agnt sus lss om ng hnd sn th suso adusts th oman aasals and so dus th dn n ayos. Th mtgaton o th ngat nlun o though a ost. 1 Ths holds aus w assum a suntly lag oman dn / 16.

ot o th oth agnt s also dasd. Tho, th atn ot t s dntly dmnshd. Futhmo, th suso adusts ots th ngat ng ahad t s dmnshd ½. agnt has only alzd, hs oman aasal s nasd y th suso and ayo dns a dud. Sn th dstoton tsl dnds on th nqualty aamts and, th magntud o th duton s also nlund y th sz o. ow, also dass th ot o agnt, whh dass nnts. Tho, ng ahad t s only dud ½. agnt ls not muh symathy o hs ollagu.. < ½, t n nass. Ths susng sult s ndndnt o th sum o adantagous and dsadantagous nqualty ason. Although oth agnts a uatd < 1 and although th suso dass nqualty x ost, th hang n ots s not stong nough to dmnsh th ng ahad t and x ant n nass t s small. Not that th suso nass h oman statmnt y / n ah ossl stuaton ndndnt o th alzd oman ll o th oth agnt. Ex ant nnts to xt ot a tho not atd. n a symmt Nash qulum 1, asd, as, asd, as = oth agnts xt ot, asd, as 1 1 =. 3 By omang th ots o nqualty as agnts n th stuatons wth and wthout asd ots, w an show that, asd, as and only, unasd, as 1 4. 4 n ths as th duton o th ng ahad t domnats th duton o th atn ot t. Condton 4 dmonstats that nnts to xt ot wth unasd ots a always hgh omad to a stuaton wth asd ots ½. agnts do not show omasson wth ah oth = 0 ut a alous, asd ots always das nnts. nstad ½ and th s a duton o th ngat ng ahad t, nnts an hgh whn ots a skwd. Ths lads to Not that w susum ths nnt t und th ng ahad t aus t s lant agnt has alzd a hgh oman ll than agnt. 3 Otmal ots a small than on aus o ondton., asd, as 4 A daton an ound n Andx C. Not that ondton 4 s mo stt than ondton. 17

Pooston 3: oth agnts a nqualty as and ½, nnts to xt ot n a symmt qulum a hgh th suso dos not a o agnts and ots a unasd. oth agnts a nqualty as wth ½ and ot osts a ath hgh.. / 1, nnts to xt ot n a symmt qulum a hgh wth asd ots. ntutly, th t o dstotng ots on nnts s amguous. On th on hand, nnts to alz a hgh th mo an agnt sus om hang a low oman ll than th oth agnt th hgh n a stuaton wth unasd ots. Ths ost t wokng a th atn ot t s dmnshd sn th suso s ots du th wag ga twn oth agnts dnt oman lls a alzd. On th oth hand, ots das n th amount o omasson wth th oth agnt,.. th hgh n a stuaton wth unasd ots. Ths ngat t may also dmnshd y th suso s ots. Fnally, w oma th ots o nqualty as agnts wth th ots o uly slsh agnts whn th suso as o th agnts utlts. Sn w onntat on nto solutons, an ntal K K = 1, wth = 1 and has to xst, whh s dd n Andx C. Th ot ost aamt has to ao, whh dss th lant ondton o ots to small than on o ths as s ondton. Thn a omason o ots o nqualty as agnts wth th ots o uly slsh agnts shows that ots o nqualty as agnts a hgh < K. ow, th lant ntal, K dos not always xst. Takng ondton 1 / nto aount, t s oously that w nd a suntly lag dn twn th osd oman lls and. Futhmo, th nqualty aamt has to suntly hgh o a gn to guaant th xstn o th ntal. Ths lads to th ollowng ooston: 18

Pooston 3d: th oman dn and a ath hgh o a gn, an ntal, K o th ot ost aamt xsts, so that dstotng ots lad to hgh ots o nqualty as agnts omad to ots o uly sl-ntstd agnts n a symmt Nash qulum. Two ts ha to takn nto aount. Fst, although th ost atn ot t s dmnshd, t s stll ost and tho nass ots o nqualty as agnts omad to uly sl-ntstd ons. Sond, uly sl-ntstd agnts only tak th own oman ay nto aount, whn ddng aout th amount o ot. n ontast, nqualty as agnts also onsd th nom o th oth agnt and sally th stuaton, n whh thy thmsls ha alzd whl th oth agnt has only alzd. Wthout dstotd ots, a hgh dass nnts, whh haatzs th ngat ng ahad t. Ths t also xsts wth dstotd ots. ow, a hgh and lads to a hgh dstoton o ots and tho to lss nqualty, whn ots a sut to th ntalty and lnny as. s suntly hgh.. ½, th ngat ng ahad t s dmnshd. Tho, Pooston 3d dmonstats th ondtons o th ost t domnatng th ngat nnt t. Fgu summazs ou sults o ½. t tus and omas th ots o agnts wth dnt ns uly sl-ntstd sus nqualty as wokng wth a suso who th stats asd ots o th tu oman o oth agnts. Fst, th ots o sl-ontd gost agnts a low than ots o nqualty as agnts wthout dstotd ots th ost t o nqualty as agnts.. th atn ot t domnats th ngat ng ahad t. Ths holds o th st two stons o Fgu. Sond, lookng at th as that th suso as o th agnts, ots o slontd agnts a low than thos o nqualty as ons th manng atn ot t domnats th manng ng ahad t. 5 Thd, w an oma th aasals o nqualty as agnts y oth tys o susos: Dstotng ots nass nnts th duton o th atn ot t s domnatd y th duton o th ngat ng ahad t. 5 Not that th atn ot t s hgh ots a unasd. 19

Fgu : Comason o ots o agnts wth dnt ns dndnt on th suso s ns o ½, unasd, as, asd, as, go, unasd, as, go, asd, as, go, unasd, as, asd, as, go, asd, as, unasd, as K = 1 1 5. Conluson Many mal studs ha shown that sut oman atngs o susos a sut to th ntalty and lnny as: Susos tnd to dntat only slghtly twn th suodnats so that atngs a omssd. Moo, atngs a otn skwd towads th to nd o th atng sal. Ou analyss os a sml xlanaton o oth ass: susos a o th utlty o th nqualty as suodnats, atngs may dstotd n oth ways. Th xtnt o th ass s nlund y th sz o agnts nqualty ason and th dn n osd omans, o nstan. t s motant to not that w a takng an x ant st whn analyzng nnts. nnts a nstgatd o ot ontngnt wags a ad. Ths s th usual st o nal agnt modls. Pous modls wth nqualty as agnts ha also hosn ths aoah. t s tho ngltd n ths modls that th outom o oman aasal may also at utu hao o mloys. A low atng, o nstan, may dsouag mloys n th utu and may tho wakn nnts to xt ot. n ontast, t may also ossl that low atd mloys show som knd o now mo than hao. n atula, gttng a low atng mans ng atd wos than th aag, on ason o th xta motaton may to suad oths o on s alty. Bol 006 oss that asd oman aasal outoms nlun utu ots. study s asd on data o a nanal s m. Sh dntats twn th lnny and ntalty as and shows that lnnt atngs ostly at oman momnt. n ontast, th ntalty as has a ngat t on utu oman. 0

Fom ou x ant ont o w w show that lnnt atngs do not at nnts o uly sl-ntstd agnts n ontast to nqualty as agnts. nooatng ths x ost st nto thotal modls may on ntstng to o utu sah. Takng addtonal wag osts o a nal nto aount, whh ould sult om suh atngs, ould anoth omsng dton o sah. Th addtonal xtd wag osts may outwgh ost nnt ts. Thn th nal may not ot om ngagng nqualty as agnts omad to uly sl-ntstd agnts. 1

Andx A and S would dd to hoos a low ot o agnt o qual ots o oth agnts, sh maxmzs th sond at o h utlty unton sut to. Th lant agang unton s = [ w ] η. Th sultng Kuhn-Tuk-ondtons a 1 η = 0 K1 1 η = 0 K η 0,, = 0 Fom addng K1 K w otan η K3 / =. K4 W show y ontadton that annot a soluton to ths maxmzaton olm. holds, η has to zo and w otan = 1 / and = 1 / om K1 and K. ow, only holds o 1 / 1 / o <. Ths lads to a / ontadton sn w assum. Smlaly, = and η = 0 annot a soluton. Fo η = 0 w otan = 1 / and = 1 / om K1 and K. But = only holds o = whh agan lads to a ontadton. Th last / ossl soluton s = and η 0. Sutatng K om K1 lads to = η 0. Wth = w gt η = and th lant onstant s. Th n sultng ot o oth agnts s / / = /. Sn w assum only soluton to th maxmzaton olm o th sond at o th suso s utlty unton whh satss all Kuhn-Tuk-ondtons s to stat qual ots n o 0., th

Andx B Daton o th thsholds B and B and th otmal ots Th suso s utlty unton an dsd y U S [ w = [ w n n ] ] n n n n n n. Assum 0. To dtmn th otmal ots 0 and 0, S dds n whth to hoos dnt o qual ots o oth agnts and thn ks thos ots that maxmz h utlty und ths onstant. Agan, w st look at th st at o h utlty unton. Th lant agang unton s = [ w n ] n n λ Th sultng Kuhn-Tuk-ondtons a n 1 λ = 0 K5 λ = 0 K6 n λ 0, n n, = 0 By addng K1 and K w gt λ K7 n n / =. K8 n n. Th a two ossl solutons: n o n =. S sts n, t ollows om K7 that λ has to zo. K5 and K6 lad to th ots d = 1 / and d n = / n. ow, th st at o S s utlty unton s only ald o th ang. Thus 1 / B, all ondtons a satsd and n n d and n d sol th maxmzaton olm. n som ass, S may st =. n ths as w th ha λ = 0 o λ 0 s K7. n n λ = 0, = 1 / and n = / n wll th ots o th aot and th oth agnt stly s K5 and K6. Ths ots sol all Kuhn- Tuk-ondtons and tho a th soluton to th maxmzaton olm n n B. Ths ots a qualnt to = = / 4 0. 5. n = n n n 3

λ 0 and S sts n n n =, w gt = = / 4 0. 5 n om K8. Sutatng K5 om K6 lads to th lant ondton o statng ths ots: < 1 / B. Sn w assumd 0 n = n, qual ots n a th soluton to th maxmzaton olm o th st at o th utlty unton th oman n dn s ath low.. B. n n B and dnt ots a th soluton S dds to hoos a low ot o h aot o qual ots o oth agnts, sh maxmzs th sond at o h utlty unton sut to th onstant. Th lant agang unton s = [ w n ] n n η n Th sultng Kuhn-Tuk-ondtons a 1 η = 0 K5 η = 0 K6 η 0, n n, n = 0 Addng K5 and K6 lad to n η K7 n / =. K8 n. n Agan, w show y ontadton that annot a soluton to ths maxmzaton n olm. d holds, η has to zo whh lads to ˆ = 1 / and n d ˆ n = n / s K5 and K6. ow, n only holds o n < 1 / B. Ths lads to a ontadton sn w assum n. Smlaly = and η = 0 annot a soluton. Wth η = 0 w otan n n n = 1 / and n = n / om K5 and K6. But n = only holds o B whh lads to a ontadton sn B < 0 and w n = assum. n Th only ossl soluton s = and η 0. Sutatng K6 om K5 lads to n 1 n n = η 0. Takng n = nto aount w gt 4

n η = 1 /. Equal ots = 4 0.5 sol all n / n Kuhn-Tuk-ondtons n 1 / B. Sn w assum n, th only soluton to th maxmzaton olm o th sond at o th suso s utlty unton whh satss all Kuhn-Tuk-ondtons s to stat qual ots. Sn th st at o th suso s utlty unton osonds to th sond at o th utlty unton ots a qual, dnt ots a only otmal n ots. B. Othws S sts qual Th daton o th otmal ots ollows analogously, th aot s oman s at most as good as th oman o th oth agnt.. 0. Ou sults a summazd n Pooston. n 5

6 Andx C Comaat stats: as unasd 1,, =,, 1 as unasd = 0 0 1,, < = as unasd aus o assumton as unasd,, 1 = 0 Comason o,asd,as and,unasd,as:,, as asd,, as unasd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

7 Comason o,asd,as and,asd,go Pooston 3d:,, as asd,, go asd 1 1 1 1 1 1 1 1 1 < Not, that dnomnato and numato a oth ngat: 0 < and 0 1 <, aus w assum. Thus, has to n th ntal K, 1, 1 = o K, 1, 1 = sn w onntat on nto solutons. Not that ths ondton only holds o suntly lag oman dns and o a gn.

Rns Bnadn, J.., Bukly, R. M. 1981: Statgs n Rat Tanng. Th Aadmy o Managmnt Rw 6, 05-1. Bl, P. R. 004: nquty Ason and Tam nnts. ESE Dsusson Pa, Unsty Collg ondon. Bol, J. C. 005: Sut Poman Ealuaton. ESE Busnss Shool. Unsty o Naaa. Bol, J. C. 006: Th Dtmnants and Poman Ets o Suso Bas. ESE Busnss Shool. Unsty o Naaa. Bolton, G. E., Oknls, A. 000: A Thoy o Equty, Roty, and Comtton. Aman Eonom Rw 100, 166-193. Btz, R. D., Mlkoh G. T., Rad, W. 199: Th Cunt Stat o Poman Aasal Rsah and Pat: Conns, Dtons, and mlatons. Jounal o Managmnt 18, 31-35. Clland, J. N., Muhy, K. R., Wllams, R. E. 1989: Multl Uss o Poman Aasal: Paln and Colats. Jounal o Ald Psyhology 74, 130-135. Dannng, A., Rhmann, T., Stum, B., Vogt, C. 007: nquty Ason and nddual Bhao n Pul Good Gams: An Exmntal nstgaton. ZEW Dsusson Pa No. 07-034. Dmougn, D., Flut, C. 003a: Gou s. nddual Poman Pay Whn Woks A Enous. Cah d hh/wokng Pa 03-18. Dmougn, D., Flut, C. 003: nquty Ason n Tounamnts. Snt Ss 003s- 18. Dmougn, D., Flut, C., lm, C. 005: Outut and Wags wth nqualty As Agnts. umoldt Unsty Bln. Mmo. Dsau, R., Sangton, D. E. M. 007: Equty and Ads Slton. Jounal o Eonoms and Managmnt Statgy 16, 85-318. Englma, F., Wamah, A. 005: Otmal nnt Contats Und nquty Ason. ZA Dsusson Pa No. 1643. Fh, E., Shmdt, K. M. 1999: A Thoy o Fanss, Comtton, and Cooaton. Th Quatly Jounal o Eonoms 114, 817-868. Fs, G. R., Judg, T. A. 1991: Psonnl/human Rsous Managmnt: A Poltal nlun Pst. Jounal o Managmnt 17, 447-488. Fox, S., Bzman, A., man, E. 1983: Th alo Et: s t a Untay Cont? Jounal o Ouatonal Psyhology 56, 89-96. Gund, C., Slwka, D. 005: Eny and Comasson n Tounamnts. Jounal o Eonoms and Managmnt Statgy 14, 187-07. olmstöm, B., Mlgom, P.R. 1991: Multtask Pnal-Agnt Analyss: nnt Contats, Asst Ownsh, and Jo Dsgn. Jounal o aw, Eonoms, and Oganzaton 7, 4-5. toh,. 004: Moal azad and Oth-Rgadng Pns. Jaans Eonom Rw 55, 18-45. 8

ttn, C. D., ak, D. F., My, M. W. 003: Sutty and th Wghtng o Poman Masus: Edn om a Baland Soad. Th Aountng Rw 78, 75-758. Jawaha,. M., Wllams, C. R. 1997: Wh All Th Chldn a Ao Aag: Th Poman Aasal Puos Et. Psonnl Psyhology 50, 905-96. y, P. E., Wllams, J. R. 004: Th Soal Contxt o Poman Aasal: A Rw and Famwok o th Futu. Jounal o Managmnt 30, 881-905. ownstn, G. F., Thomson,., Bazman, M.. 1989: Soal Utlty and Dson Makng n ntsonal Contxts. Jounal o Psonalty and Soal Psyhology 57, 46-441. Maod, W. B. 003: Otmal Contatng wth Sut Ealuaton. Th Aman Eonom Rw 931, 16-40. May, B., P, T. 004: Pnzn d Anzgstaltung Rskoason und sozaln Pänzn. Ztsht ü Btswtshat 74, 1047-1075. My,. 1975: Th Pay o Poman Dlmma. Oganzatonal Dynams 3, 39-50. Mlgom, P.R., Rots, J. 1988: An Eonom Aoah to nlun Atts n Oganzatons. Aman Jounal o Soology, Sulmnt, 94, S-154-S-179. Muhy, K. J. 199: Poman Masumnt and Aasal: Motatng Manags to dnty and Rwad Poman. n: Buns, W. J. d.: Poman Masumnt, Ealuaton, and nnts, Boston, 37-6. Muhy, K. R., Clland, J. N. 1991: Poman Aasal: An Oganzatonal Pst, Bostn: Allyn and Baon. Muhy, K. R., Clland, J. N. 1995: Undstandng Poman Aasal. Thousand Oaks: Sag. Nlson, W. S., Stow, J. 004: nnt Pay o Oth-Rgadng Woks. Mmo, Duk Unsty. Pndgast, C. 00: Untanty and nnts. Jounal o ao Eonoms 0, S115- S137. Pndgast, C., Tol, R. 1996: Faotsm n Oganzatons. Jounal o Poltal Eonomy 104, 958-978. Slwka, D. 007a: oss Ason and Bas n Sut Poman Ealuatons. Mmo Unsty o Cologn. Slwka, D. 007: Auay, Soal Pns and Bas n Sut Poman Ealuaton. Mmo Unsty o Cologn. Vama, A., Dns, A. S., Pts,.. 1996: ntsonal At and Poman Aasals: A Fld Study. Psonnl Psyhology 49, 341-360. on Smns, F. 005: Fanss, Ads Slton, and Emloymnt Contats. Unsty o Munh. Mmo. 9