On the Incompatibility of Efficiency and Strategyproofness in Randomized Social Choice

Transcription

1 On the Incompatblty of Effcency and Strategyproofness n Randomzed Socal Choce Hars Azz NICTA and UNSW Sydney 2033, Australa Floran Brandl Technsche Unverstät München Garchng, Germany Felx Brandt Technsche Unverstät München Garchng, Germany Abstract Effcency no agent can be made better off wthout makng another one worse off and strategyproofness no agent can obtan a more preferred outcome by msrepresentng hs preferences are two cornerstones of economcs and ubqutous n mportant areas such as votng, auctons, or matchng markets. Wthn the context of random assgnment, Bogomolnaa and Mouln have shown that two partcular notons of effcency and strategyproofness based on stochastc domnance are ncompatble. However, there are varous other possbltes of lftng preferences over alternatves to preferences over lotteres apart from stochastc domnance. In ths paper, we gve an overvew of common preference extensons, propose two new ones, and show that the abovementoned ncompatblty can be extended to varous other notons of strategyproofness and effcency n randomzed socal choce. Introducton Effcency no agent can be made better off wthout makng another one worse off and strategyproofness no agent can obtan a more preferred outcome by msrepresentng hs preferences are two cornerstones of economcs and ubqutous n mportant areas such as votng, auctons, or matchng markets. The conflct between both notons s already apparent n Gbbard and Satterthwate s semnal theorem, whch states that the only sngle-valued socal choce functons that satsfy non-mposton a weakenng of effcency and strategyproofness are dctatorshps (Gbbard 973; Satterthwate 975). In ths paper, we study effcency and strategyproofness n the context of socal decson schemes (SDSs),.e., functons that map a preference profle to a probablty dstrbuton (or lottery) over a fxed set of alternatves (Gbbard 977; Barberà 979). Randomzed votng methods have a surprsngly long tradton gong back to ancent Greece and have recently ganed ncreased attenton n poltcal scence (Stone 20). Wthn computer scence, randomzaton s a very successful technque n algorthm desgn and s beng consdered more and more often n the context of votng (Contzer and Sandholm 2006; Procacca 200; Walsh and Xa 202; Servce and Adams Copyrght c 204, Assocaton for the Advancement of Artfcal Intellgence ( All rghts reserved. 202; Brrell and Pass 20; Azz, Brandt, and Brll 203b; Azz 203). There are varous ways of extendng preferences over alternatves to preferences over lotteres. We refer to these extensons as lottery extensons. Perhaps the most wde-spread lottery extenson s stochastc domnance (SD). Ths extenson s of partcular mportance because one lottery stochastcally domnates another one ff the former yelds at least as much expected utlty as the latter for any von-neumann- Morgenstern (vnm) utlty representaton consstent wth the ordnal preferences. However, settngs n whch the exstence of an underlyng vnm utlty functon cannot be assumed may call for other lottery extensons. For nstance, n ths paper, we put forward a partcularly natural new extenson called parwse comparson (PC ), whch arses as the specal case of skew-symmetrc blnear (SSB) utlty functons as proposed by Fshburn (982). Accordng to ths extenson lottery p s preferred to lottery q ff t s more lkely that p yelds a better alternatve than q. The PC extenson s more powerful than the SD extenson n the sense that, for the same preference relaton over alternatves, the SD preference relaton s contaned n the PC relaton. Apart from PC, we consder two other completons of SD due to Cho (202), namely the upward lexcographc (UL) and the downward lexcographc (DL) extenson. We furthermore consder a weakenng of SD that we call blnear domnance (BD) and whch s agan based on Fshburn s SSB utlty (Fshburn 984). Clearly, each of these lottery extensons gves to rse to dfferent varants or degrees of effcency and strategyproofness. Snce many lottery extensons are ncomplete,.e., some pars of lotteres are ncomparable, there are two fundamentally dfferent ways how to defne strategyproofness. The strong noton, frst advocated by Gbbard (977), requres that every msreported preference relaton of an agent wll result n a lottery that s comparable and weakly less preferred by that agent to the orgnal lottery. Accordng to the weaker noton, frst used by Postlewate and Schmedler (986) and then popularzed by Bogomolnaa and Mouln (200), no agent can msreport hs preferences to obtan another lottery that s strctly preferred to the org- SSB utlty functons are a generalzaton of vnm utlty functons.

2 nal one. In other words, the strong verson always nterprets ncomparabltes n the worst possble manner (such that they volate strategyproofness) whle the weak verson nterprets them as actual ncomparabltes that cannot be resolved. Usually, the strong noton s much more demandng than the weak one. Whenever a lottery extenson s complete, however, both notons concde. One of the best known results about SDSs s a consequence of a characterzaton by Gbbard (977), who attrbutes t to Hugo Sonnenschen: when ndvdual preferences are lnear, every Pareto optmal and strongly SDstrategyproof SDS s a random dctatorshp,.e., one of the agents s chosen at random and then pcks hs most preferred alternatve. 2 Gbbard s proof requres the unversal doman of lnear preferences and cannot be extended to arbtrary subdomans (Chatterj, Sen, and Zeng 204). Moreover, n many mportant subdomans of socal choce such as house allocaton, matchng, and coalton formaton, tes are unavodable snce agents are ndfferent among all outcomes n whch ther allocaton, match, or coalton s the same (Sönmez and Ünver 20; Azz, Brandt, and Seedg 203; Bouveret and Lang 2008; Elknd and Wooldrdge 2009; Azz, Brandt, and Seedg 203). Wthn the specal doman of random assgnment, Bogomolnaa and Mouln (200) have been able to show that there s no anonymous, SD-effcent, and strongly SDstrategyproof SDS. As a consequence, all generalzatons of random dctatorshp to weak preferences, volate SDeffcency or strong SD-strategyproofness. 3 Azz, Brandt, and Brll (203b) recently conjectured that the mpossblty by Bogomolnaa and Mouln even holds when only requrng weak SD-strategyproofness and proved ths for the rather lmted class of majortaran SDSs. 4 We complement and strengthen these results by provng the followng theorems (always assumng anonymty):. PC -strategyproofness s ncompatble wth PC - effcency n the context of neutral SDSs. 2. UL-strategyproofness s ncompatble wth ULeffcency. 3. BD-strategyproofness s ncompatble wth Pareto optmalty n the context of parwse SDSs. 4. BD-group-strategyproofness s ncompatble wth Paretooptmalty n the context of neutral SDSs. The frst result s a proof of a partcularly natural weakenng of the above mentoned conjecture by Azz, Brandt, and Brll (203b). The second result mght be surprsng because the correspondng statement for the DL-extenson does 2 Whle random dctatorshp s strongly SD-strategyproof, t only satsfes weak SD-group-strategyproofness. 3 Random seral dctatorshp, for nstance, somewhat surprsngly volates SD-effcency (Bogomolnaa and Mouln 200; Bogomolnaa, Mouln, and Stong 2005; Azz, Brandt, and Brll 203b). 4 Wthn the doman of random assgnment wth unt-demand and the doman of dchotomous preferences, respectvely, the condtons of the conjecture are compatble (Bogomolnaa and Mouln 200; Bogomolnaa, Mouln, and Stong 2005). not hold (random dctatorshp satsfes both DL-effcency and DL-strategyproofness). The thrd and the fourth result sgnfcantly strengthen theorems by Azz, Brandt, and Brll (203b) (Theorem ) and Bogomolnaa, Mouln, and Stong (2005) (Proposton 3), respectvely. The assumpton of anonymty s crucal as all our mpossbltes fal to hold when omttng anonymty. Seral dctatorshp, an extreme example of a non-anonymous SDS, s defned for a fxed sequence of the agents and lets each agent narrow down the set of alternatves by pckng hs most preferred of the alternatves selected by the prevous agents. Seral dctatorshp trvally satsfes all reasonable notons of effcency and strategyproofness. Snce lotteres can guarantee ex ante farness va randomzaton, anonymty and neutralty are typcally two mnmal condtons that far SDSs are expected to satsfy. 2 Related Work Apart from some early precursors (Zeckhauser 969; Fshburn 972), the frst formal study of strategyproof randomzed socal choce was conducted by Gbbard (977). A recent survey of randomzed socal choce s contaned n a book chapter by Barberà (200). Usng stochastc domnance for strategyproofness, effcency, and farness condtons was popularzed by Bogomolnaa and Mouln (200). They focussed on a subdoman of randomzed socal choce called random assgnment, n whch each outcome s a one-to-one assgnment of objects to agents. Recently, Cho (202) extended the approach of Bogomolnaa and Mouln (200) by ntroducng new lottery extensons such as ones based on lexcographc preferences. Azz, Brandt, and Brll (203b) examned the tradeoff between effcency and strategyproofness for socal decson schemes and ntated the analyss of strct maxmal lotteres, a lttle known SDS due to Kreweras and Fshburn. Recently, Azz (203) proposed a new SDS that compromses between RSD and effcent but manpulable SDSs. A lne of nqury that has been especally popular n AI and mult-agent systems s to check how well strategyproof SDSs approxmate common determnstc votng rules such as Borda s rule (Contzer and Sandholm 2006; Procacca 200; Brrell and Pass 20; Servce and Adams 202). 3 Prelmnares Let N = {,..., n} be a set of agents wth preferences over a fnte set A wth A = m. The preferences of agent N are represented by a complete and transtve preference relaton R A A. The set of all preference relatons wll be denoted by R. In accordance wth conventonal notaton, we wrte P for the strct part of R,.e., a P b f a R b but not b R a and I for the ndfference part of R,.e., a I b f a R b and b R a. A preference profle R = (R,..., R n ) s an n-tuple contanng a preference relaton R for each agent N. The set of all preference profles s thus gven by R n. We wll compactly represent a preference relaton as a comma-separated lst wth all alternatves among whch an agent s ndfferent placed n a set.

3 For example a P b I c s represented by R : a, {b, c}. A preference relaton R s lnear f x P y or y P x for all dstnct alternatves x, y A. A preference relaton R s dchotomous f x R y R z mples x I y or y I z. Our central object of study are socal decson schemes,.e., functons that map the ndvdual preferences of the agents to a lottery (or probablty dstrbuton) over alternatves. A socal decson scheme (SDS) s a functon f : R n (A). A mnmal farness condton for SDSs s anonymty, whch requres that f(r) = f(r ) for all R, R R n and permutatons π : N N such that R = R π() for all N. Another farness requrement s neutralty. For a permutaton π of A and a preference relaton R, π(x) R π π(y) f and only f x R y. Then, an SDS f s neutral f for all R R n, f(r)(x) = f(r π )(π(x)) for all x A. An SDS f s parwse (or a neutral C2 functon) f t s neutral and for all preference profles R and R, f(r) = f(r ) whenever for all alternatves x, y, { N x R y} { N y R x} = { N x R y} { N y R x}. In other words, the outcome of a parwse SDS only depends on the anonymzed comparsons between pars of alternatves (Young 974; Zwcker 99). An SDS f s majortaran (or a neutral C functon) f t s neutral and for all preference profles R and R, f(r) = f(r ) whenever for all alternatves x, y, { N x R y} { N y R x} ff { N x R y} { N y R x}. It s easy to see that the three classes of SDSs form a herarchy: every majortaran SDS s parwse and every parwse SDS s anonymous. Two anonymous SDSs that have been recently analyzed n a framework smlar to ths paper are random seral dctatorshp (RSD) and strct maxmal lotteres (SML) (Azz, Brandt, and Brll 203b). RSD s the canoncal generalzaton of random dctatorshp to weak preferences. It s defned by pckng a sequence of the agents unformly at random and then nvokng seral dctatorshp (.e., each agent narrows down the set of alternatves by pckng hs most preferred of the alternatves selected by the prevous agents). SML s a lttle known class of parwse SDSs due to Kreweras and Fshburn that return a mxed quasstrct Nash equlbrum of the pluralty game. Computng RSD was recently shown to be #P-complete (Azz, Brandt, and Brll 203a) whle SML can be computed effcently usng lnear programmng. 4 Lottery Extensons In order to reason about the outcomes of SDSs, we need to make assumptons on how agents compare lotteres. A lottery extenson maps preferences over alternatves to (possbly ncomplete) preferences over lotteres. We wll now defne the lottery extensons consdered n ths paper. For a more detaled account of the lottery extensons SD, DL, and UL, we refer to Cho (202). R DL R PC R SD R BD R UL Fgure : Incluson relatonshps between lottery extensons. An arrow denotes set ncluson between two relatons, e.g., R BD R SD. DL, PC, and UL are extensons that yeld complete preference relatons over sets. Throughout ths secton, let R R be a preference relaton and p, q (A). The frst extenson we propose s called blnear domnance (BD) and requres that for every par of alternatves the probablty that p yelds the more preferred alternatve and q the less preferred alternatve s at least as large as the other way round. Formally, p R BD q ff x, y, x P y : p(x)q(y) p(y)q(x). (BD) Apart from ts ntutve appeal, the man motvaton for BD s that p blnearly domnates q ff p s preferable to q for every SSB utlty functon consstent wth R (Fshburn 984). Stochastc domnance (SD) prescrbes that for each alternatve x A, the probablty that p selects an alternatve that s at least as good as x s greater or equal to the probablty that q selects such an alternatve. Formally, p R SD q ff x: p(y) q(y). (SD) y : yr x y : yr x It s well-known that p R SD q ff the expected utlty for p s at least as large as that for q for every von-neumann- Morgenstern utlty functon compatble wth R. A novel strengthenng of SD, consdered n ths paper for the frst tme, s the parwse comparson (PC) extenson. The reasonng behnd PC s to prefer p to q f the probablty that p yelds an alternatve preferred to the alternatve returned by q s at least as large than the other way round. 5 Formally, p R PC q ff q(x)p(y). (PC ) xr y p(x)q(y) xr y Fnally, we defne the downward lexcographc (DL) extenson and the upward lexcographc (UL) extenson ntroduced by Cho (202). Accordng to DL the lottery wth hgher probablty on the top ranked alternatve s preferred, n case of equalty, the one wth hgher probablty on the second ranked alternatve, and so on. Formally, p R DL q f p = q or x: (p(x) > q(x) and y, y R x: p(y) = q(y)). (DL) 5 Interestngly, ths extenson may lead to ntranstve preferences over lotteres, even when the preferences over alternatves are transtve (Blyth 972; Fshburn 988).

4 Upward lexcographc orderng s dual to the former,.e., p R UL q f p = q or x: (p(x) < q(x) and y, x R y : p(y) = q(y)). (UL) The followng example llustrates the defntons of the extensons. Consder the preference relaton R : a, b, c and lotteres Then, p P P C p]; [p R BD p = 2/3a + 0b + /3c q = 0a + b + 0c. q; p P DL q]; and [q R BD q; q P UL p]. and p; [p R SD q]; [q R SD The ncluson relatonshps between the lottery extensons are depcted n Fgure. 5 Effcency and Strategyproofness In ths secton, we present general defntons of effcency and strategyproofness whch gve rse to varyng levels of effcency and strategyproofness dependng on whch lottery extenson s used to defne them. The relatonshps between these concepts are depcted n Fgure 2. Effcency prescrbes that there s no lottery that all agents prefer to the one returned by the SDS. Each lottery extenson yelds a correspondng noton of effcency. Let E be a lottery extenson. Gven a preference profle R, a lottery p E-domnates another lottery q f p R E q for all N and p P E q for some N. An SDS f s E-effcent f, for every R R n, there does not exst a lottery that E-domnates f(r). An alternatve s Pareto domnated f there exsts another alternatve such that all agents strctly prefer the latter to the former. An SDS s Pareto optmal (or ex post effcent) f t puts probablty zero on all Pareto domnated alternatves. It s well-known that SD-effcency mples Pareto optmalty. Our frst theorem, the proof of whch s omtted due to space restrctons, shows that Pareto optmalty s stronger than BD-effcency. Theorem. Pareto optmalty mples BD-effcency but the converse does not hold. Strategyproofness prescrbes that no agent can obtan a more preferred outcome by msrepresentng hs preferences. Agan, we obtan varyng degrees of ths property dependng on the underlyng lottery extenson. Let E be a lottery extenson. An SDS f s E-manpulable f there exst preference profles R and R wth R j = R j for all j such that f(r ) P E f(r). An SDS s E-strategyproof f t s not E-manpulable. An SDS s strongly E-strategyproof f for all R and R wth R j = R j for all j such that f(r) R E f(r ). 6 For complete lottery extensons (DL, PC, and UL), the weak and the strong notons of strategyproofness concde. An SDS f s E-group-manpulable f there exsts an S N and preference profles R and R wth R j = R j for all j N \ S such that f(r ) P E f(r) for all S. An SDS s E-group-strategyproof f t s not E-group-manpulable. 6 Please note that n some papers (e.g., Bogomolnaa and Mouln 200) the term strategyproofness refers to the strong noton of strategyproofness. 6 Results and Dscusson Recent research has shown that there exsts an nterestng tradeoff between effcency and strategyproofness n randomzed socal choce (Azz, Brandt, and Brll 203b). For example, RSD satsfes strong SD-strategyproofness and Pareto optmalty, but volates SD-effcency. SML, on the other hand, satsfes PC -effcency and ST - strategyproofness, where ST s a weakenng of BD, but volates SD-strategyproofness. Ths secton contans four mpossblty results that mprove our understandng of the nterplay between effcency and strategyproofness and have nontrval consequences on concrete SDSs such as RSD and SML. We prove each of these results by reasonng about a set of preference profles and dervng a contradcton. In partcular, the proofs assume a specfc number of agents and alternatves, but can be generalzed to any (larger) number of agents and alternatves as follows. For more alternatves, we add the addtonal alternatves as ted for last rank n each agent s preference relaton and every preference profle. Notce that we do not leave the doman n case of dchotomous preferences. To show a statement for more agents, we add agents that are ndfferent between all alternatves. Both constructons do not affect the set of effcent lotteres and ncentves of agents. Hence, the proofs wth the same arguments carry through. Our frst result states that effcency and strategyproofness are ncompatble when preferences over lotteres are gven by the natural PC extenson. Theorem 2. There s no anonymous, neutral, PC -effcent, and PC -strategyproof SDS for n 3 and m 3. Proof. Assume for a contradcton that f s an SDS wth propertes as stated above and consder the followng preference profle. R : a, {b, c} R 2 : b, a, c R 3 : c, a, b Anonymty and neutralty mply that f(r )(b) = f(r )(c). The only PC -effcent lottery whch puts equal weght on b and c s the degenerate lottery a, snce every other lottery of ths form s domnated by a (agent 2 and 3 are ndfferent whle agent s strctly better off). Hence, f(r ) = a. Now consder the followng profle. R 2 : a, {b, c} R 2 2 : b, a, c R 2 3 : {a, c}, b In ths profle a Pareto domnates c, hence f(r 2 )(c) = 0. If agent 3 reports R 3 nstead of R 2 3, he receves one of hs most preferred alternatves, namely a, wth probablty. Therefore by PC -strategyproofness, f(r 2 ) = a. Next, we consder the profle R 3. R 3 : a, {b, c} R 3 2 : b, {a, c} R 3 3 : {a, c}, b PC -effcency mples that f puts probablty 0 on c when appled to R 3, snce a Pareto domnates c. If f(r 3 ) f(r 2 ), agent 2 would have an ncentve to devate n one drecton or the other. Thus, f(r 3 ) = a.

5 DL-effcency PC -effcency UL-effcency strong SD-strategyproofness SD-effcency DL-strategyproofness PC -strategyproofness UL-strategyproofness Pareto optmalty SD-strategyproofness BD-effcency BD-strategyproofness Fgure 2: Relatonshps between effcency and strategyproofness concepts. Snce we wll need t later, we state an observaton at ths pont. By anonymty and neutralty, f has to choose the unform lottery /3a + /3b + /3c n the followng profle. R 4 : c, a, b R 4 2 : a, b, c R 4 3 : b, c, a Also notce that n ths profle agent prefers any lottery wth hgher probablty on c than on b to the unform lottery accordng to (R 4 ) PC. Now we consder another preference profle. R 5 : {a, c}, b R 5 2 : a, b, c R 5 3 : b, c, a Here we dstngush two cases. Frst, we assume f(r 5 ) = a and consder a devaton by agent 3. R 6 : {a, c}, b R 6 2 : a, b, c R 6 3 : c, b, a Anonymty and neutralty mply that f(r 6 )(a) = f(r 6 )(c). Any lottery of ths form other than /2a + /2c s PC - domnated by the latter. Thus, f(r 6 ) = /2a + /2c by PC -effcency. But agent 3 prefers /2a + /2c to a f hs preferences are R3. 5 Hence, a contradcton to PC - strategyproofness. The second case s f(r 5 ) a. If f(r 5 )(c) > f(r 5 )(b), then by the above observaton, agent prefers f(r 5 ) to f(r 4 ) f hs preferences are R. 4 A contradcton to PC -strategyproofness. Hence, f(r 5 )(c) f(r 5 )(b) and thus, by the assumpton n the second case, f(r 5 )(b) > 0. R 7 : {a, c}, b R 7 2 : a, b, c R 7 3 : b, {a, c} It follows from f(r 5 )(b) > 0 that f(r 7 )(b) > 0, otherwse agent 3 would devate from R 7 3 to R 5 3. In partcular f(r 7 ) a. Fnally, consder the followng profle. R 8 : {a, c}, b R 8 2 : a, {b, c} R 8 3 : b, {a, c} By anonymty, f(r 8 ) = f(r 3 ) = a. But ths mples that agent 2 can successfully devate from R 7 2 to R 8 2 and receve a nstead. Hence, the desred contradcton. It can be shown that random dctatorshp volates PC - effcency, even when preferences are lnear. Ths emphaszes the effcency problems of random dctatorshp. Prevously, t was only known that RSD volates SD-effcency for weak preferences. Stll, PC -effcency s not unduly restrctve as SML s known to satsfy t. Next, we prove a smlarly negatve result for the ULextenson, whch only requres two agents. Theorem 3. There s no anonymous, UL-effcent, and ULstrategyproof SDS for n 2 and m 3. The proof of ths theorem s omtted to meet space constrants. Interestngly, DL-effcency the dual noton of ULeffcency s compatble wth SD-strategyproofness (and hence DL-strategyproofness) because random dctatorshp satsfes both condtons when preferences are lnear. The next result s a strengthenng of Theorem by Azz, Brandt, and Brll (203b) n two respects: t uses a weaker noton of strategyproofness and t holds for the set of all parwse SDSs rather than only majortaran SDSs. 7 Theorem 4. There s no parwse, Pareto optmal, and BDstrategyproof SDS for n 4 and m 4. Proof. Let f be a parwse, Pareto optmal, and BDstrategyproof SDS. We frst consder the preference profle R and ts (weghted) majorty graph depcted n Fgure 3 (). R : a, c, {b, d} R 2 : b, d, {a, c} Snce f s Pareto optmal and parwse, f(r ) = p = /2a + /2b. Now we consder the profle R 2 wth majorty graph as n Fgure 3 (). R 2 : a, c, {b, d} R 2 2 : {b, d}, {a, c} Both agents are ndfferent between b and d and agan c s Pareto domnated. Hence, f(r 2 ) = q = ( 2λ)a+λb+λd for some λ [0, ]. Frst assume for a contradcton λ > /3. We consder profle R 3. R 3 : a, {b, c, d} R 3 2 : {b, d}, {a, c} The majorty graph of R 3 s as n Fgure 3 (). Hence, by anonymty and neutralty, f(r 3 ) = r = /3a + /3b + /3d. But r (P 2 ) BD q f λ > /3, whch contradcts BDstrategyproofness of f snce voter n R 2 can manpulate by reportng R 3 nstead of R 2. 7 Note, however, that the proof of Theorem by Azz, Brandt, and Brll (203b) only requres lnear preferences.

6 a c b () d a c () b d a c () Fgure 3: Graphs depctng parwse comparsons. An edge from x to y s labeled wth the number of agents preferrng x to y mnus the number of agents preferrng y to x. All mssng edges denote majorty tes. Now assume for a contradcton that λ = /3. R 4 : a, c, b, d R 4 2 : {b, d}, {a, c} R 4 3 : a, c, {b, d} R 4 4 : {b, d}, c, a The profle R 4 has a majorty graph as n Fgure 3 (), thus f(r 4 ) = p. R 5 : a, c, b, d R 5 2 : {b, d}, {a, c} R 5 3 : a, c, {b, d} R 5 4 : d, b, c, a The majorty graph of R 5 s as n Fgure 3 () and hence f(r 5 ) = q. But then voter 4 n R 4 can manpulate by reportng R4 5 nstead of R4 4 snce q (P4 4 ) BD p. Ths agan contradcts BD-strategyproofness of f. Fnally, we assume λ < /3 and consder profle R 6. R 6 : a, c, {b, d} R 6 2 : {b, d}, {a, c} R 6 3 : {b, c, d}, a R 6 4 : a, {b, c, d} Its majorty graph s as n Fgure 3 () and therefore f(r 6 ) = q. We consder one last profle. R 7 : a, c, {b, d} R 7 2 : {b, d}, {a, c} R 7 3 : {b, d}, c, a R 7 4 : a, {b, c, d} The majorty graph of ths profle s as n Fgure 3 () whch mples f(r 7 ) = r. But r (P3 6 ) BD q f λ < /3. Thus, agent 3 n R 6 benefts from reportng R3 7 nstead of R3. 6 In any case, we found a successful manpulaton, contradctng BD-strategyproofness of f. As a consequence of the prevous theorem, the parwse SDS SML does not satsfy BD-strategyproofness and, as a matter of fact, no reasonable parwse SDS can. For our fnal result we agan consder the rather weak noton of BD-strategyproofness, but also consder devatons by groups of agents. It turns out that BD-groupstrategyproofness s already ncompatble wth Pareto optmalty when also requrng anonymty and neutralty. 8 8 Bogomolnaa, Mouln, and Stong (2005) (Proposton 3) proved that for n 4 and m 6, there exsts no anonymous, neutral, Pareto optmal, and SD-group-strategyproof SDS for dchotomous preferences. We strengthen ther result by weakenng SDgroup-strategyproofness to the sgnfcantly weaker noton of BDgroup-strategyproofness and by usng less alternatves and agents. b d Theorem 5. There s no anonymous, neutral, Paretooptmal, and BD-group-strategyproof SDS for n 3 and m 3, even when preferences are dchotomous. Proof. Assume for contradcton that f s an SDS wth the propertes as stated. Consder a settng wth three agents and three alternatves and the followng preference profle. R : {a, b}, c R 2 : {a, c}, b R 3 : {b, c}, a By neutralty and anonymty, f(r ) = /3a + /3b + /3c. Now let agents and 2 change ther preferences and consder the profle R 2. R 2 : a, {b, c} R 2 2 : a, {b, c} R 2 3 : {b, c}, a Agan by neutralty and anonymty, f(r 2 ) = ( 2λ)a + λb + λc. If λ > /3, then agents and 2 would rather report R and R 2 respectvely f ther true preferences were R 2 and R 2 2. On the other hand, f λ < /3 and ther true preferences were R and R 2, they would rather report R 2 and R 2 2. Hence, λ = /3 and f(r 2 ) = /3a + /3b + /3c. R 3 : a, {b, c} R 3 2 : {a, b}, c R 3 3 : b, {a, c} In R 3, c s Pareto-domnated, thus by neutralty and anonymty, f(r 3 ) = /2a + /2b. To ths end, we consder the followng profle. R 4 : a, {b, c} R 4 2 : {a, b}, c R 4 3 : {b, c}, a If agent 3 changes hs preferences from R 3 3 to R 4 3, c s stll Pareto-domnated and hs preferences over a and b reman unchanged. Hence, by BD-strategyproofness, f(r 4 ) = f(r 3 ). But then agent 2 n R 2 would have an ncentve to report R 4 2 nstead of R 2 2, a contradcton. For the stronger (but less reasonable) noton of groupstrategyproofness n whch only one of the devatng agents has to be strctly better off, we were able to show the prevous mpossblty even wthout requrng anonymty and neutralty. The proof s omtted due to lmted space. Theorem 5 mples that RSD volates BD-groupstrategyproofness. As a matter of fact, both RSD and SML only satsfy the rather weak noton of ST -groupstrategyproofness where ST s a weakenng of BD ntroduced by Azz, Brandt, and Brll (203b). Put n a nutshell, RSD does better n terms of ndvdual strategyproofness (strong SD-strategyproofness vs. ST -strategyproofness) whle SML s more effcent (PC -effcency vs. Pareto optmalty). Acknowledgments Ths materal s based upon work supported by Deutsche Forschungsgemenschaft under grants BR 232/7-2 and BR 232/0- and by NICTA, whch s funded by the Australan Government as represented by the Department of Broadband, Communcatons and the Dgtal Economy and the Australan Research Councl through the ICT Centre of Excellence program. The authors thank Markus Brll, Hervé Mouln, Shas Nandebam, and Arunava Sen for helpful dscussons.

7 References Azz, H.; Brandt, F.; and Brll, M. 203a. The computatonal complexty of random seral dctatorshp. Economcs Letters 2(3): Azz, H.; Brandt, F.; and Brll, M. 203b. On the tradeoff between economc effcency and strategyproofness n randomzed socal choce. In Proceedngs of the 2th Internatonal Conference on Autonomous Agents and Mult-Agent Systems (AAMAS), IFAAMAS. Azz, H.; Brandt, F.; and Seedg, H. G Computng desrable parttons n addtvely separable hedonc games. Artfcal Intellgence 95: Azz, H Maxmal Recursve Rule: A New Socal Decson Scheme. In Proceedngs of the 22nd Internatonal Jont Conference on Artfcal Intellgence (IJCAI), Barberà, S A note on group strategy-proof decson schemes. Econometrca 47(3): Barberà, S Strategy-proof socal choce. In Arrow, K. J.; Sen, A. K.; and Suzumura, K., eds., Handbook of Socal Choce and Welfare, volume 2. Elsever. chapter 25, Brrell, E., and Pass, R. 20. Approxmately strategy-proof votng. In Proceedngs of the 22nd Internatonal Jont Conference on Artfcal Intellgence (IJCAI), Blyth, C. R Some probablty paradoxes n choce from among random alternatves. Journal of the Amercan Statstcal Assocaton 67(338): Bogomolnaa, A., and Mouln, H A new soluton to the random assgnment problem. Journal of Economc Theory 00(2): Bogomolnaa, A.; Mouln, H.; and Stong, R Collectve choce under dchotomous preferences. Journal of Economc Theory 22(2): Bouveret, S., and Lang, J Effcency and envyfreeness n far dvson of ndvsble goods: logcal representaton and complexty. Journal of Artfcal Intellgence Research 32(): Chatterj, S.; Sen, A.; and Zeng, H Random dctatorshp domans. Games and Economc Behavor. Forthcomng. Cho, W. J Probablstc assgnment: A two-fold axomatc approach. Unpublshed manuscrpt. Contzer, V., and Sandholm, T Nonexstence of votng rules that are usually hard to manpulate. In Proceedngs of the 2st Natonal Conference on Artfcal Intellgence (AAAI), AAAI Press. Elknd, E., and Wooldrdge, M Hedonc coalton nets. In Proceedngs of the 8th Internatonal Conference on Autonomous Agents and Mult-Agent Systems (AAMAS), Fshburn, P. C Lotteres and socal choce. Journal of Economc Theory 5: Fshburn, P. C Nontranstve measurable utlty. Journal of Mathematcal Psychology 26():3 67. Fshburn, P. C Domnance n SSB utlty theory. Journal of Economc Theory 34(): Fshburn, P. C Nonlnear preference and utlty theory. The Johns Hopkns Unversty Press. Gbbard, A Manpulaton of votng schemes. Econometrca 4: Gbbard, A Manpulaton of schemes that mx votng wth chance. Econometrca 45(3): Postlewate, A., and Schmedler, D Strategc behavour and a noton of ex ante effcency n a votng model. Socal Choce and Welfare 3: Procacca, A Can approxmaton crcumvent Gbbard-Satterthwate? In Proceedngs of the 24th AAAI Conference on Artfcal Intellgence (AAAI), AAAI Press. Satterthwate, M. A Strategy-proofness and Arrow s condtons: Exstence and correspondence theorems for votng procedures and socal welfare functons. Journal of Economc Theory 0: Servce, T. C., and Adams, J. A Strategyproof approxmatons of dstance ratonalzable votng rules. In Proceedngs of the th Internatonal Conference on Autonomous Agents and Mult-Agent Systems (AAMAS), Sönmez, T., and Ünver, M. U. 20. Matchng, allocaton, and exchange of dscrete resources. In Benhabb, J.; Jackson, M. O.; and Bsn, A., eds., Handbook of Socal Economcs, volume. Elsever. chapter 7, Stone, P. 20. The Luck of the Draw: The Role of Lotteres n Decson Makng. Oxford Unversty Press. Walsh, T., and Xa, L Lot-based votng rules. In Proceedngs of the th Internatonal Conference on Autonomous Agents and Mult-Agent Systems (AAMAS), Young, H. P An axomatzaton of Borda s rule. Journal of Economc Theory 9: Zeckhauser, R Majorty rule wth lotteres on alternatves. Quarterly Journal of Economcs 83(4): Zwcker, W. S. 99. The voter s paradox, spn, and the Borda count. Mathematcal Socal Scences 22: