Subjective Performance Evaluation and Inequality Aversion

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1 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

2 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 Bonn Gmany Phon: Fax: E-mal: [email protected] 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.

3 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 Wüzug Gmany E-mal: [email protected] Many thanks to Matthas Kakl and Not Shulz o hlul ommnts.

4 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 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 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 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 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

5 Fgu 1: Dstuton o sut oman aasals at Mk & Co., n 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 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 Bsds, susos may ha a n o a lasant latonsh wth th suodnats Vama t al 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.

6 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

7 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 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, 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

8 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 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

9 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

10 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.

11 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

12 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

13 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 = = / 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

14 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 = = / 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

15 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 = / 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

16 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 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

17 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 /. 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

18 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

19 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 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.

20 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 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

21 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

22 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

23 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 = 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

24 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

25 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

26 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 = = / n = n n n 3

27 λ 0 and S sts n n n =, w gt = = / 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

28 n η = 1 /. Equal ots = 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

29 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

30 7 Comason o,asd,as and,asd,go Pooston 3d:,, as asd,, go asd < 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.

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