Group strategy-proof social choice functions

G oup s a egy-p oo social choice unc ions wi h bina y anges and a bi a y domains: cha - ac e iza ion esul s1 Sal ado Ba be ày Dolo s Be gaz and Be na do Mo enox Ap il 28 h, 2010 Abs ac : We de…ne di¤e en concep s o g oup s a egy-p oo ness o social choice unc ions. We discuss he connec ions be ween he de…ned concep s unde di¤e en assump ions on hei domains o de…ni ion. We cha ac e ize he social choice unc ions ha sa is y each one o hem and whose anges consis o wo al e na i es, in e ms o wo ypes o basic p ope ies. JEL Classi…ca ion Numbe : D71. Keywo ds: Social choice unc ions, Bina y anges, g oup s a egy-p oo ness, xy-mono onici y, xy-based ules. 1We would like o hank he commen s o Wal e Bosse and Luis Co chón. Sal ado Ba be à g a e ully acknowledges suppo om he Spanish Minis y o Science and Inno- a ion h ough g an "Consolida ed G oup-C" ECO2008-04756, and by he Gene ali a de Ca alunya, Depa amen d’Uni e si a s, Rece ca i Socie a de la In o mació h ough he Dis inció pe a la P omoció de la Rece ca Uni e si à ia, g an SGR2009-0419 and he Ba celona GSE Resea ch Ne wo k. Dolo s Be ga acknowledges he suppo o he Spanish Minis y o Science and Inno a ion h ough g an SEJ2007-60671 and o Gene ali a de Ca alunya, h ough g an SGR2009-0189. She also acknowledges he Resea ch Recogni- ion P og amme o he Ba celona GSE. Be na do Mo eno acknowledges …nancial suppo om he Spanish Minis y o Science and Inno a ion h ough g an ECO2008-03674. yMOVE, Uni e si a Au ònoma de Ba celona, and Ba celona GSE, Edi…ci B, 08193 Bella e a, Spain. E-mail: sal ado .ba b[email p o ec ed] zDepa amen d’Economia, Campus de Mon ili i, Uni e si a de Gi ona, 17071 Gi ona, Spain. E-mail: dolo s.b[email p o ec ed] xDepa amen o de Teo ía e His o ia Económica, Facul ad de Ciencias Económicas y Emp esa iales, Campus de El Ejido, 29071 Málaga, Spain. E-mail: b[email p o ec ed] 1 In oduc ion The Gibba d-Sa e hwai e Theo em es ablishes ha , when a social choice unc ion is de…ned on he uni e sal se o p e e ence p o…les o e kal e na- i es (k > 2), and i s ange con ains a leas h ee al e na i es, i can only be s a egy-p oo i i is dic a o ial. This esul is subjec o di¤e en quali…ca ions. One is ha , when he ule is de…ned on smalle se s o p o…les, he e may o may no exis o he ules ha a e s a egy-p oo , in addi ion o he dic a o ial ones. This is he case unde a a ie y o domains, ha include he ones o med by he Ca e- sian p oduc o single-peaked p e e ences, o o single-dipped p e e ences, o o sepa able p e e ences, among o he s. Ou s a emen s in his pape will be essen ially ue o unc ions de…ned on any domain, howe e small, asym- me ic o special i may be.1 A second quali…ca ion conce ns he ange o he social choice unc ion. In his pape we conside he subclass o unc ions ha ail o mee Gibba d and Sa e hwai e’s equi emen ha hei ange should con ain a leas h ee al e na i es. Speci…cally, we concen a e on ules ha a e no cons an and whose ange consis s o exac ly wo al e na i es, xand y. Because o his, i is known ha in ha case he e a e possibili ies o design non-dic a o ial s a egy-p oo ules. We wan o cha ac e ize hem all. This leads us o no ice ha he ange o a social choice unc ion may be bina y because he e a e only wo al e na i es in he ele an wo ld, bu i may also be bina y in he p esence o mo e han wo al e na i es, in which case his may be conside ed as one pa o he possible choices open o he mechanism designe . As we shall see, he cha ac e iza ion o bina y ules in his con ex equi es a numbe o p ecisions and ca e ul ea men ha can be a oided in wo lds whe e only wo al e na i es a e p esen o begin wi h. A hi d quali…ca ion e e s o he no ion o s a egy-p oo ness o be used. When we concen a e on ules wi h bina y anges, he e exis a numbe o a ac i e s a egy-p oo ules, and i becomes hen much mo e in e es ing o explo e he ex en o which some o hem may also be immune o ma- nipula ion by g oups. We analyze his ques ion ca e ully, unde a numbe o di¤e en possible no ions o g oup s a egy-p oo ness, and also by keeping in mind ha we wan ou s a emen s o hold o unc ions de…ned on any ype 1By "essen ially ue" we mean ha hey a e ei he ue wi hou quali…ca ion, o ue unde e y mino assump ions, o be discussed case by case. 1 o domains. One de…ni ion o g oup s a egy-p oo ness equi es ha i should no be possible o a g oup o agen s o de ia e om decla ing hei ue p e e ences and ge a s ic gain o each one o hem. Social choice unc ions a oiding his s ong ype o manipula ion a e called Weakly G oup S a egy-P oo . A second de…ni ion s a s om conside ing ha a g oup can p o… ably de i- a e i some o i s membe s de i e a s ic gain om doing so, while o he s simply emain indi¤e en while helping hei pa ne s. Rules ha a oid his weake o m o manipula ion a e called S ongly G oup S a egy-P oo . In an in e media e e sion o he p ope y, ha we simply call G oup S a egy- P oo ness, we allow ha only some agen s may gain om he de ia ion, bu we equi e ha all agen s in ol ed in ge ing he change should ac i ely pa icipa e in he manipula ion by ac ually de ia ing om hei u h ul p e - e ence. We p o ide cha ac e iza ions o he classes o social choice unc ions ha sa is y each one o hese h ee p ope ies, and also we elabo a e on why we single ou hese pa icula de…ni ions. Ou main cha ac e iza ion esul s iden i y wo ypes o basic p ope ies ha hese ules mus sa is y. These p ope ies mus be quali…ed in each case. Since we allow indi iduals o be indi¤e en be ween xand y, in some cases we will equi e ha hey a e sa is…ed "essen ially", and in o he cases no . By "essen ially" we mean ha he p ope ies will hold condi ional o he ac ha he p e e ences o indi iduals ha a e indi¤e en be ween he wo al e na i es in he ange emain cons an . Ou … s condi ion is ha o essen ial xy-mono onici y: i xob ains a a p o…le, and hen some people change hei p e e ences so ha he suppo o xinc eases, while he suppo o ydoes no , hen xmus s ill ob ain a he new p o…le. A mo e demanding equi emen in a simila spi i is ha o xy-s ong mono onici y. In ha case i xob ains a a p o…le, and p e e ence changes induce la ge suppo o x; hen xs ill be chosen a he new p o…le e en i suppo o ymay ha e also inc eased.2A second ype o equi emen e e s o he ype o in o ma ion on which ou ules may be based. We say ha hey a e xy-based i wha hey choose a each p e e ence p o…le only depends on he ela i e posi ion o x wi h espec o y o each indi idual. I is essen ially xy-based i he p ope y holds when we only compa e p o…les whe e indi iduals indi¤e en be ween 2In his second de…ni ion we d op he quali…ca ion o he p ope y being essen ial because he s a emen is no longe condi ioned o he p e e ences o indi¤e en indi iduals emaining cons an . 2 xand ykeep hei p e e ences unchanged. No ice also ha he equi emen will no apply in he case whe e all indi iduals a e indi¤e en be ween bo h al e na i es in he ange. We es ablish h ee cha ac e iza ion esul s in e ms o he abo e condi- ions, one o each o ou h ee ypes o g oup s a egy-p oo ness equi e- men s. A social choice unc ion is weakly g oup s a egy-p oo i and only i i is essen ially xy-based and essen ially xy-mono onic. I is s ongly g oup s a egy-p oo i and only i i is xy-based and xy-s ong mono onic. Finally, we show ha , when n3and unde a mild condi ion on he ichness o he domain, ules ha mee ou in e media e no ion o g oup s a egy-p oo ness a e also s ongly g oup s a egy-p oo , and hus sa is y he same p ope ies. The sophis ica ed eade will ealize ha ou condi ions a e pa o a la ge se o di¤e en equi emen s ha ha e been used by di¤e en au ho s unde di¤e en names o he cha ac e iza ion o s a egy-p oo ules o e uni e sal domains. Names like Maskin mono onici y, s ong posi i e associa ion, and o he s ha e been used o deno e a ia ions o p ope ies ha one expec s o be sa is…ed by ules ha a e s a egy-p oo . And, indeed, many combina ions o p ope ies end up cha ac e izing he same ules when hese a e de…ned on ich enough domains. We eel ha ou choice o p ope ies is especially … , because hey allow us o cha ac e ize ules de…ned on all kinds o domains, possibly e y asymme ic and con aining ew p e e ences. The equi alence be ween ou s and o he p ope ies is no g an ed unde hese ci cums ances. Also no ice ha we do no insis on indi idual s a egy-p oo ness as a special case o cha ac e ize. This is because by a ecen esul o ou s, i is an es ab- lished ac ha indi idual and weak g oup s a egy-p oo ness a e equi alen when he ange o he social choice unc ion consis s o only wo (o h ee) elemen s (see Ba be à, Be ga, and Mo eno, 2010). A di¤e en ype o cha ac e iza ion esul s a e based on desc ip ions o how he ules would choose al e na i es a each p e e ence p o…le. The e exis wo ele an pape s ha ake his poin o iew. One is by La sson and S ensson (2006), who p o ide a cha ac e iza ion o s a egy-p oo ules: unde ou assump ion ha he ange is bina y, s a egy-p oo ness is equi - alen o weak g oup s a egy-p oo ness, as p o en in Ba be à, Be ga, and Mo eno (2010). Hence, hei cha ac e iza ion in e ms o he unc ional o m p o ides an al e na i e o he one we p esen he e. A second esul , his one due o Manjuna h (2009a), cha ac e izes he unc ional o m o s ong g oup s a egy-p oo ules when he e a e only wo al e na i es. We e-s a e he esul wi h some addi ional p ecisions and in o de o co e he case whe e 3 he ange is bina y bu p e e ences a e de…ned on a la ge se o al e na i es, and p o ide a no el p oo o i . The pape p oceeds as ollows. In Sec ion 2, we p o ide he amewo k, we p esen di¤e en e sions o g oup s a egy-p oo ness and discuss hei ela ionships unde di¤e en domain assump ions. In Sec ion 3 we p o ide he cha ac e iza ions in e ms o p ope ies. In Sec ion 4 we p o ide he announced addi ional cha ac e iza ion o s ongly g oup s a egy-p oo ules, he no el p oo , ha also allows us o comple e he p oo o one o he heo ems in he p eceding sec ion. Sec ion 5 concludes. 2 The se up and de…ni ions Le Abe a …ni e se o al e na i es A= x; y; z; w:::g:Le Nbe a …ni e se o agen s N= 1;2; :::; ng:Le Ube he se o all p eo de s on A(comple e, e‡exi e, and ansi i e bina y ela ions on A). Le Ri U be he se o admissible p e e ences o agen i2Nand le R  i2NRi. Fo any p e e ence ela ion Ri2 Ri, we deno e by Piand Ii he s ic and indi¤e ence pa o Ri, espec i ely. A p e e ence p o…le is deno ed by R= (R1; ::; Rn)2 R o also by R= (RC; RC)2 R when we wan o s ess he ole o a coali ion CN. Then RC2 RC i2CRiand RC2 RNnC deno e he p e e ences o agen s in Cand in NnC, espec i ely. Asocial choice unc ion (o ule)on a domain Ris a unc ion :R ! A. The ange o is deno ed by A . In his pape we concen a e on he amily o social choice unc ions wi h bina y ange, ha is, whose ange consis s o exac ly wo elemen s, ha we call xand y om now on. Le Rx ijRibe he subse o p e e ences such ha o any Rx i2 Rx i, xPx iy. Simila ly, de…ne Ry i. Le Rxy ijRibe he subse o p e e ences such ha o any Rxy i2 Rxy i,xIxy iy. We s a e ou esul s unde he ollowing minimal assump ion on he domain o admissible p e e ences: each indi idual has a leas one admissi- ble p e e ence whe e xis p e e ed o y, one whe e yis p e e ed o x; and one whe e he is indi¤e en be ween he wo. Tha is, o any i2Nand any 2 x; y; xyg,R i6=?.3 3Fo se e al o ou esul s, we could e en weaken his minimal condi ion on he domain and allow o some o he se s R i o be emp y. 4 The bes known nonmanipulabili y axiom is s a egy-p oo ness. I e- qui es he u h o be a dominan s a egy and i is a necessa y condi ion o implemen a ion in dominan s a egies (Gibba d, 1973 and Sa e hwai e, 1975). De…ni ion 1 An agen i2Ncan manipula e a social choice unc ion on Ra R2 R i he e exis s R0 i2 Risuch ha Ri6=R0 iand (R0 i; Ri)Pi (R). A social choice unc ion is s a egy-p oo on Ri no agen i2Ncan manipula e on R. Ano he o m o manipula ion is by means o coali ions. The ollowing de…ni ions e e o cases whe e agen s may gain om join changes o de- cla ed p e e ences. They di¤e on wo accoun s: he equi ed gains om manipula ion and he ac ions expec ed om coali ion membe s. Rega ding gains om manipula ion we may equi e ha each membe om de ia ing coali ions ob ains a s ic gain o else ha only some o hem do wi h he es no losing. Rega ding de ia ions we may ask ha all membe s o a coali- ion mis ep esen hei p e e ences o ha jus some o hem do. The h ee de…ni ions below will e‡ec hese modelling choices.4 De…ni ion 2 A coali ion Ccan s ongly manipula e a social choice unc ion on Ra R2 R i he e exis s R0 C2 RCsuch ha o all agen i2C, Ri6=R0 iand (R0 C; RC)Pi (R). A social choice unc ion is weakly g oup s a egy-p oo on Ri no coali ion CNcan s ongly manipula e on R. De…ni ion 3 A coali ion Ccan manipula e a social choice unc ion on R a R2 R i he e exis s R0 C2 RCsuch ha o all agen i2C,Ri6=R0 i and (R0 C; RC)Ri (R), and o some j2C, (R0 C; RC)Pj (R). A social choice unc ion is g oup s a egy-p oo on Ri no coali ion CNcan manipula e on R. De…ni ion 4 A coali ion Ccan weakly manipula e a social choice unc ion on Ra R2 R i he e exis s R0 C2 RCsuch ha o some agen l2C, 4We shall omi wha could ha e been a ou h e sion o g oup s a egy-p oo ness, one ha would equi e all agen s o gain bu would allow o some o hem no o change hei p e e ences. Tha would u n ou o be equi alen o weak g oup s a egy-p oo ness (see De…ni ion 2). 5 Rl6=R0 l, o all agen i2C, (R0 C; RC)Ri (R), and o some j2C, (R0 C; RC)Pj (R). A social choice unc ion is s ongly g oup s a egy- p oo on Ri no coali ion CNcan weakly manipula e on R. Rema ks (1) S a egy-p oo ness and weak g oup s a egy-p oo ness a e equi alen o social choice unc ions wi h bina y ange (see P oposi ion 1 and Theo em 1 in Ba be à, Be ga, and Mo eno, 2010). (2) When indi¤e ences a e no allowed, all h ee de…ni ions o g oup s a egy- p oo ness collapse in a single one. (3) S ong g oup s a egy-p oo ness implies g oup s a egy-p oo ness and he la e implies weak g oup s a egy-p oo ness. The con e se implica ions do no hold in gene al, as shown by he ollowing examples. Example 1 A ule ha is g oup s a egy-p oo bu no s ongly. Le n2 and #A2,x; y 2A. Then, o any R2 UN, de…ne he social choice unc ion as ollows: (R) = xi xPiy o any i2N, yo he wise. We show ha is no s ongly g oup s a egy-p oo . Le Rbe such ha each agen s ic ly p e e s x o yand le R0be such ha n1agen s s ic ly p e e xo e y, and he o he agen is indi¤e en be ween xand y. Obse e ha (R) = xand (R0) = y: Then, coali ion Ncould weakly manipula e a R0 ia R. The eade may check ha he ule sa is…es he wo weake s a egic condi ions. Example 2 A ule ha is weakly g oup s a egy-p oo bu no g oup. Le n2,#A2and agen s’p e e ences such ha o any i2N,R i6=? o any 2 x; y; xyg. Le kbe a dic a o on x; yg, ha is, (R) = xwhen Rk2 Rx k[ Rxy kand (R) = yo he wise. No e ha is (weakly g oup) s a egy-p oo . Howe e , coali ion C= k; jg j6=kcould manipula e a (Rxy k; Ry j; R j;kg) ia (Ry k; R0 j; R j;kg) o any R0 j2 Rx i[ Rxy iand any R j;kg2 RNn j;kg. Thus, is no g oup s a egy- p oo ( hus no s ongly). Be o e cha ac e izing he ules ha sa is y ou di¤e en equi emen s, le us ema k ha g oup s a egy-p oo ness and s ong g oup s a egy-p oo ness become equi alen unde he mild complemen a y domain condi ion e- qui ed in he ollowing p oposi ion. 6 P oposi ion 1 Le #A3and Rbe such ha each indi idual has a leas wo admissible p e e ences in Riwhe e xis p e e ed o yand wo whe e y is p e e ed o x: Then, any g oup s a egy-p oo social choice unc ion on Rwi h a bina y ange is also s ongly g oup s a egy-p oo . P oo . Le be a g oup s a egy-p oo social choice unc ion. Suppose ha is no s ongly g oup s a egy-p oo . Tha is, he e exis R2 R, a coali ion CN, and R0 C2 RCsuch ha o some agen l2C,Rl6=R0 l, o all agen s i2C; (R0 C; RC)Ri (R), and o some j2C, (R0 C; RC)Pj (R). I o any agen l2C,Rl6=R0 l, hen we ge a con adic ion o g oup s a egy- p oo ness. Thus, he e exis l2Csuch ha Rl=R0 l:De…ne CP= j2C:Rj=R0 jand (R0 C; RC)Pj (R)gand CI= k2C:Rk=R0 kand (R0 C; RC)Ik (R)g. By he complemen a y domain condi ion, o any j2CP, he e exis s R00 j2 RjnRjsuch ha (R0 C; RC)P00 j (R). I (R00 CP; R0 CnCP; RC) = (R) he e exis a coali ion CP, a p o…le (R00 CP; R0 CnCP; RC)2 R, and R0 CP=RCPsuch ha o any agen j2CPR00 j6=Rjand (R0 C; RC)Pj (R00 CP; R0 CnCP; RC) = (R)which is a con adic ion o g oup s a egy-p oo ness. Thus, (R00 CP; R0 CnCP; RC) = (R0 C; RC): I CI=? hen obse e ha he e exis R2 R, a coali ion CN, and R00 C(R00 CP; R0 CnCP)2 RCsuch ha o any agen i2C,Ri6=R00 iand (R00 CP; R0 CnCP; RC)Ri (R), and o some j2C, (R00 CP; R0 CnCP; RC)Pj (R). Then we ge a con adic ion o g oup s a egy-p oo ness. Thus, CI6=?. By he complemen a y domain condi ion, o any k2 CI, he e exis s R00 k2 RknRksuch ha (R00 CP; R0 CnCP; RC)P00 k (R). I (R00 CP[CI; R0 Cn(CP[CI); RC) = (R)coali ion CIcould manipula e ia RCI a (R00 CP[CI; R0 Cn(CP[CI); RC), which con adic s g oup s a egy-p oo ness. Thus, (R00 CP[CI; R0 Cn(CP[CI); RC) = (R0 C; RC). Then obse e ha he e exis R2 R, a coali ion CN, and R000 C(R00 CP[CI; R0 Cn(CP[CI))2 RCsuch ha o any agen i2C,Ri6=R000 iand (R00 CP[CI; R0 Cn(CP[CI); RC)Ri (R), and o some j2C, (R00 CP[CI; R0 Cn(CP[CI); RC)Pj (R), and hen we ge a con adic ion o g oup s a egy-p oo ness. Rema k 1 We ha e assumed in P oposi ion 1 ha #A3:This is because he complemen a y domain condi ion ha we assume in ou s a emen can only be sa is…ed in his case. When #A= 2, his condi ion canno be sa is…ed 7 and in ac he equi alence does no hold ( he ule in Example 1 when #A= 2 p o ides a coun e example). 3 Cha ac e iza ion esul s: p ope ies In his sec ion we p o ide ou … s se o cha ac e iza ion esul s. We p o e ha ou di¤e en e sions o he condi ion ha a ule should be xy-based and mono onic a e necessa y and su¢ cien o gua an ee ha hey sa is y ou di¤e en e sions o g oup s a egy-p oo ness.5 Fo each p e e ence p o…le R2 R;de…ne he se X(R) = i2N:xPiyg; Y(R) = j2N:yPjxg, and I(R) = k2N:yIkxg. We now de…ne he condi ions ha will cha ac e ize weak and s ong g oup s a egy-p oo ness. De…ni ion 5 A bina y social choice unc ion is essen ially xy-mono onic6 i and only i o any R; R02 R such ha Rh=R0 h o all h2I(R) I(R0); he ollowing holds: [X(R0)X(R); Y (R)Y(R0)(a leas one s ic inclusion), and (R) = x] ) (R0) = x; and [Y(R0)Y(R); X(R)X(R0)(a leas one s ic inclusion), and (R) = y] ) (R0) = y: De…ni ion 6 A bina y social choice unc ion is xy-s ongly mono onic i and only i o any R; R02 R he ollowing holds: [i ei he X(R0)X(R); Y (R)Y(R0)(a leas one s ic inclusion), o X(R0)X(R),?6=Y(R)$Y(R0)] and (R) = x) (R0) = x; [i ei he Y(R0)Y(R); X(R)X(R0)(a leas one s ic inclusion), o Y(R0)Y(R),?6=X(R)$X(R0)] and (R) = y) (R0) = y. 5Examples showing he ela ionship be ween he p ope ies de…ned in his sec ion a e a ailable upon eques . 6Lemma 7 in Manjuna h (2009b) shows ha when he se o admissible p e e ences is he se o all single-dipped p e e ences and a speci…c bina y ange es ic ion, some e sion o essen ially xy-mono onici y is a consequence o s a egy-p oo ness. 8 be ween xand y. Bu hen he second agen ob ains his bes ou come. This ends he p oo o S ep 1. S ep 2 Any s ongly g oup s a egy-p oo social choice unc ion wi h bina y ange is xy-based and xy-s ong mono onic. P oo o S ep 2: By Theo em 1, we know ha is essen ially xy-based and essen ially xy- mono onic. We now p o e by con adic ion ha is xy-based. Suppose no , hen he e exis R; R02 R such ha X(R)[Y(R)6=?,X(R) = X(R0); Y (R) = Y(R0), (R)6= (R0). Suppose … s ha X(R) = ?and hus Y(R)6=?(a simila a gumen applies i Y(R) = ?). By Lemma 1, (R) = (R0) = ywhich is a con adic ion. Thus, X(R)6=?and Y(R)6=?. By essen ially xy-based, (R0 X(R)[Y(R); RI(R)) = (R). I n= 2,R0= (R0 X(R)[Y(R); RI(R))and we ge he desi ed con adic ion since (R0)mus be di¤e en om (R). I n3, since (R)6= (R0) he e mus exis an agen i2Nsuch ha xIiy ha is xy-pi o al o (R0 X(R)[Y(R); RI(R)). By Lemma 2, is no s ongly g oup s a egy-p oo which is a con adic ion. We now p o e by con adic ion ha is xy-s ong mono onic. Suppose no , ha is he e exis R; R02 R such ha ei he (1) X(R0)X(R), Y(R)Y(R0)(a leas one inclusion s ic ), (R) = xbu (R0) = y; o else (2) X(R0)X(R),?6=Y(R)$Y(R0); (R) = xand (R0) = y. A simila a gumen holds o he o he possibili y whe e he oles o xand y a e exchanged. Fi s obse e ha by Lemma 1, X(R)6=?and Y(R)6=? (o he wise, i Y(R) = ?, hen Y(R0) = ?and X(R0)%X(R). By Lemma 1, (R0) = xwhich is he desi ed con adic ion. I X(R) = ?and Y(R)6=?, hen by Lemma 1 (R) = ywhich is he desi ed con adic ion). I case (1) holds, by essen ial xy-mono onici y and essen ial xy-basedness, (R0 X(R)[Y(R); RI(R)) = x(de…ne R00 = (R0 X(R)[Y(R); RI(R)), i ei he X(R00)% X(R)and Y(R00)$Y(R)o X(R00) = X(R)and Y(R00)$Y(R)we apply essen ial xy-mono onici y. I X(R00) = X(R)and Y(R00) = Y(R)we apply essen ial xy-basedness). I n= 2,R0= (R0 X(R)[Y(R); RI(R))and we ge he desi ed con adic ion since (R0)mus be di¤e en om (R). I n3, since (R)6= (R0) he e mus exis an agen i2Nsuch ha xIiy ha is xy-pi o al o (R0 X(R)[Y(R); RI(R)). By Lemma 2, is no s ongly 15 g oup s a egy-p oo which is a con adic ion. I case (2) holds, by essen ial xy-mono onici y and essen ial xy-basedness, (R0 X(R0); RNnX(R0)) = x(i X(R0)%X(R), ha is, X(R0)includes some agen s in I(R), we apply essen ial xy-mono onici y. I X(R0) = X(R)we apply essen ial xy-basedness). Then, (R0 X(R0); R0 Y(R); RNn X(R0)[Y(R)g) = x by essen ial xy-basedness. I n= 2,R0= (R0 X(R0); R0 Y(R); RNn X(R0)[Y(R)g)and we ge he desi ed con- adic ion since (R0)mus be di¤e en om (R). I n3, since (R)6= (R0) he e mus exis an agen i2N ha is xy- pi o al agen o (R0 X(R0); R0 Y(R); RNn X(R0)[Y(R)g)such ha xIiy. By Lemma 2, is no s ongly g oup s a egy-p oo which is a con adic ion. This ends he p oo o S ep 2. S ep 3 Any xy-based and xy-s ong mono onic social choice unc ion wi h bina y ange can be desc ibed as a e o ule when n3:When n= 2, is ei he a e o ule o a se ial dic a o . To show S ep 3, we use he ollowing claims. Obse e … s ha since is xy-based hen o any R i,R i2 R i, (R i; Ri) = (R i; Ri) o any Ri2 RNn igwhe e 2 x; y; xyg. In wha ollows, when we use R iwe e e o any R i2 R iwi hou loss o gene ali y. This is because all he s a emen s we make in his p oo om now on hold wha e e he ep esen a i e o he se R iis. Claim 1 Le n2. I is xy-based and xy-s ong mono onic hen is xy-Pa e ian. P oo o Claim 1 Le Rx2 i2NRx i, ha is, X(Rx) = N. Suppose o ge a con adic ion ha (Rx) = y. No e ha by xy-based and xy-s ong mono onici y, o any o he p o…le R2 R, (R) = ywhich con adic s ha has a bina y ange. Thus, (Rx) = x. Suppose ha he e is Rsuch ha xRiy o any i2Nand xPjy o some j2N; X(R)6=Nand (R) = y. Then X(R)6=?and Y(R) = ?. No e ha X(R)$X(Rx)and Y(R) = Y(Rx), o any Rx2 i2NRx i. By xy- s ong mono onici y, (Rx) = ywhich con adic s wha we ha e jus p o ed. This ends he p oo o Claim 1. No e ha he coun e pa esul s o Claims 2 and 3 below exchanging he oles o xand ydo also hold. Claim 2 Le n2. I o some i2Nand some Ry i2 Ry i; (Ry i; Rx i) = y, 16 hen (Ry i; R0 i) = y o any R0 i2 RNn ig: P oo o Claim 2 Le R0 i2 RNn ig. Obse e ha (Ry i; R0 i) = yei he by xy-based i X(Ry i; R0 i) = Nn ig=X(Ry i; Rx i), o else by xy-s ong mono onici y i X(Ry i; R0 i)$Nn ig=X(Ry i; Rx i). This ends he p oo o Claim 2. Claim 3 Le n3. I o some i2Nand some Ry i2 Ry i; (Ry i; Rx i) = y, hen o any j2Nwe ha e ha (Ry j; Rx j) = y. P oo o Claim 3 By con adic ion, suppose ha (Ry i; Rx i) = yand (Ry j; Rx j) = x. I (Rxy i; Ry j; Rx  i;jg) = y hen (Ry j; Rx j) = yby xy-s ong mono onici y since ?6=X(Rxy i; Ry j; Rx  i;jg)$X(Ry j; Rx j)and Y(Rxy i; Ry j; Rx  i;jg) = Y(Ry j; Rx j). Thus, (Rxy i; Ry j; Rx  i;jg) = x. By xy-s ong mono onici y, (Ry i; Ry j; Rx  i;jg) = x, since X(Rxy i; Ry j; Rx  i;jg) = X(Ry i; Ry j; Rx  i;jg)and ?6=Y(Rxy i; Ry j; Rx  i;jg)$Y(Ry i; Ry j; Rx  i;jg). By Claim 2, since (Ry i; Rx i) = y hen (Ry i; Ry j; Rx  i;jg) = ywhich con adic s wha we ob ained abo e. This ends he p oo o Claim 3. Claim 4 Le n3. I o some i2Nand some Ry i2 Ry i, (Ry i; Rx i) = x hen (Ry C; Rx C) = x o any C,? $ C$N. P oo o Claim 4 Suppose, o ge a con adic ion, ha o some C,? $ C$N; (Ry C; Rx C) = y. Clea ly, C6= ig. No e also ha Ccan no be a single on (o he wise, i C= jg,j6=i, we would ge a con adic ion by Claim 3). Thus, #C > 1. No e also ha (Ry k; Rx k) = x o any k2N (o he wise, (Ry k; Rx k) = y, by Claim 4, (Ry i; Rx i) = ywhich is no he case). The e o e, wi hou loss o gene ali y, we can suppose ha i2C. We dis inguish wo subcases: Subcase 1 Le (Ry i; Rxy Cn ig; Rx C) = x. No e ha X(Ry i; Rxy Cn ig; Rx C) = X(Ry C; Rx C)and ?6=Y(Ry i; Rxy Cn ig; Rx C)$Y(Ry C; Rx C). The e o e, by xy- s ong mono onici y (Ry C; Rx C) = x, which is a con adic ion. Subcase 2 Le (Ry i; Rxy Cn ig; Rx C) = y. No e ha Y(Ry i; Rxy Cn ig; Rx C) = Y(Ry i; Rx i)and ?6=X(Ry i; Rxy Cn ig; Rx C)$X(Ry i; Rx i). The e o e, by xy- s ong mono onici y, (Ry i; Rx i) = y, which is a con adic ion. This ends he p oo o Claim 4. P oo o S ep 3: Fi s , by Claim 1, (R) = x o any Rsuch ha xRiy o any i2Nand xPjy o some j2Nand (R) = y o any Rsuch ha yRix o any i2N and yPjx o some j2N. Second, (R)can be any ou come o any Rwhe e 17 all agen s a e indi¤e en . Thi d, he a gumen di¤e s depending on nbeing wo o highe . I n= 2, suppose … s ha is such ha o some p o…le (Ry 1; Rx 2), whe e Ry 12 Ry 1and Rx 22 Rx 2, (Ry 1; Rx 2) = yand o some p o…le (Ry 2; Rx 1), whe e Ry 22 Ry 2and Rx 12 Rx 1, (Ry 2; Rx 1) = y. By Claim 2, o any Ry 12 Ry 1and Rx 22 Rx 2, (Ry 1; R2) = yand (Ry 2; R1) = y o any R22 R2and R12 R. Thus, can be ew i en as a e o ule o x. Second, suppose ha o some p o…le (Ry 2; Rx 1), whe e Ry 22 Ry 2and Rx 12 Rx 1, (Ry 2; Rx 1) = yand o any p o…le (Ry 1; Rx 2), whe e Ry 12 Ry 1and Rx 22 Rx 2, (Ry 1; Rx 2) = x. By Claim 2, o any Ry 22 Ry 2; (Ry 2; R1) = yand o any R12 R1. No e ha his ule can be ew i en as a se ial dic a o wi h o de 21. Thi d, suppose ha o some p o…le (Ry 1; Rx 2), whe e Ry 12 Ry 1and Rx 22 Rx 2, (Ry 1; Rx 2) = yand o any p o…le (Ry 2; Rx 1), whe e Ry 22 Ry 2and Rx 12 Rx 1, (Ry 2; Rx 1) = x. By Claim 2, o any Ry 12 Ry 1; (Ry 1; R2) = yand o any R22 R2. No e ha his ule can be ew i en as a se ial dic a o wi h o de 12. Finally, suppose ha o any p o…le (Ry 2; Rx 1), whe e Ry 22 Ry 2and Rx 12 Rx 1, (Ry 2; Rx 1) = xand o any p o…le (Ry 1; Rx 2), whe e Ry 12 Ry 1and Rx 22 Rx 2, (Ry 1; Rx 2) = x. By he coun e pa o Claim 2, o any Ry 22 Ry 2; (Ry 2; R1) = x o any R12 R1;and o any Ry 12 Ry 1; (Ry 1; R2) = x o any R22 R2. Thus, can be ew i en as a e o ule o y. I n3, suppose … s ha is such ha o some p o…le (Ry i; Rx i), whe e Ry i2 Ry iand Rx j2 Rx j o any j2Nn ig, (Ry i; Rx i) = y. Then, by Claims 2 and 3, (Ry k; Rk) = y o any k; any Rk2 Ry k;and any Rj2 Rj; j2Nn kg. Tha is, he ou come will be y o any p o…le whe e he e is one agen ha s ic ly suppo s yo e x. Thus, is a e o ule o x. Le now suppose ha is such ha o all p o…les (Ry i; Rx i), whe e Ry i2 Ry i and Rx j2 Rx j o any j2Nn ig, (Ry i; Rx i) = x. Then, by Claim 4 and he coun e pa s o Claims 2 and 3, he ou come will be x o any p o…le whe e he e is one agen ha s ic ly suppo s xo e y. Thus, is a e o ule o y. This ends p oo o S ep 3, and hence he p oo o Theo ems 2, 3, and 4. 18 5 Final Rema ks In his pape we ha e p o ided di¤e en de…ni ions o s a egy-p oo ness in on o possible manipula ions by g oups, and se e al cha ac e iza ions o ules sa is ying hese p ope ies when hei ange is es ic ed o co e wo al e na i es. We eel ha , when a ainable, non-manipulabili y by g oups (in i s di¤e - en o ms) is an a ac i e p ope y, since in many con ex s di¤e en agen s can be expec ed o explo e he possibili y o bene… ing om join ac ions, in addi ion o indi idual ones. Ea ly au ho s on he issue o s a egy-p oo ness did indeed e e o he in e es o a oiding such join s a egic beha io (Pa - anaik, 1978, Dasgup a, Hammond, and Maskin, 1979, Peleg, 1984 and 2002). T ue, in many domains, and o unc ions wi h non-bina y anges, i may be excessi e o ask o hese p ope ies. Bu no always! Fo in e es ing cases when hey may be ul…lled because o domain es ic ions, see Moulin (1999), Pápai (2000), Ba be à and Jackson (1995). In ac , ou pape con empla es ano he case whe e join manipula ions can be a oided, his ime because he anges o ou unc ions a e es ic ed. We ha e allowed o agen s o ha e p e e ences o e o he al e na i es ha a e no in he ange, and been ca e ul in ollowing up he implica ions o ha ex ension in he domains o he ules. This is in con as wi h he wo k o au ho s who assume ha only wo al e na i es a e a ailable when he ange consis s o wo o hem. We insis in he di¤e ence, because we wan o emphasize ha he choice o es ic he ange is indeed a possible ool o he mechanism designe , e en when mo e han wo choices a e in p inciple socially a ailable. We ha e also looked o cha ac e iza ions ha a e essen ially independen o he cha ac e is ics o he domains o de…ni ion o he ules. This is because he se s o ules sa is ying ou di¤e en e sions o non-manipulabili y by g oups could in p inciple be a ying as he domains o de…ni ion change om one applica ion o ano he . By selec ing p ope ies ha a e necessa y and su¢ cien o ou condi ions o be sa is…ed, we go o he essen ials o he ques ion. And, when needed, ou quali…ca ions on he minimal equi emen s on domains o ou esul s o hold a e made explici a each poin . We ha e also insis ed in examining he ole o indi iduals who a e in- di¤e en be ween he al e na i es in he ange (bu no iden ical in o he espec s). The p esence o indi¤e ences is always a sou ce o p oblems in social choice, and i also complica es and en iches ou analysis he e. 19 We lea e i o he in e es ed eade o examine how ou analysis would be simpli…ed (and some imes educed o p e iously exis ing esul s) when only wo al e na i es a e p esen a all, and/o when indi¤e ences among al e na i es a e uled ou . Le us also men ion ha we ha e concen a ed on he no ions o weak and s ong g oup s a egy-p oo ness, The in e media e no ion o g oup s a egy- p oo ness has been p o en o be equi alen o he s ong e sion unde mild domain assump ions, bu no o he pa icula case o wo al e na i es only. Cha ac e iza ions o ules sa is ying he in e media e p ope y in his pa ic- ula case a e le as an open p oblem. Re e ences [1] S. Ba be à, D. Be ga, and B. 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