Incentives for Participation and Abstention in Probabilistic Social Choice

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1 Inentives for Prtiiption n Astention in Proilisti Soil Choie Florin Brnl Institut für Informtik TU Münhen, Germny rnlfl@in.tum.e Felix Brnt Institut für Informtik TU Münhen, Germny rntf@in.tum.e Johnnes Hofuer Institut für Informtik TU Münhen, Germny hofuej@in.tum.e ABSTRACT Voting rules re powerful tools tht llow multiple gents to ggregte their preferenes in orer to reh joint eisions. A ommon flw of some voting rules, known s the no-show prox, is tht gents my otin more preferre outome y stining n eletion. Whenever rule oes not suffer from this prox, it is si to stisfy prtiiption. In this pper, we initite the stuy of prtiiption in proilisti soil hoie, i.e., for voting rules tht yiel proility istriutions over lterntives. We onsier three egrees of prtiiption se on expete utility, the strongest of whih even requires tht n gent is stritly etter off y prtiipting t n eletion. While the ltter onition is prohiitive in non-proilisti soil hoie, we show tht it n e met y resonle proilisti funtions. More generlly, we stuy to whih extent prtiiption n Preto effiieny re omptile. To the est of our knowlege, this is the first work in this iretion. Ctegories n Sujet Desriptors [Theory of Computtion]: Algorithmi mehnism esign; [Theory of Computtion]: Algorithmi gme theory; [Theory of Computtion]: Rnomness, geometry, n isrete strutures Generl Terms Eonomis, Theory Keywors Voting theory; Preto-optimlity; no-show prox; strtegi mnipultion; stohsti ominne; rnomiztion 1. INTRODUCTION Whenever group of multiple gents ims t rehing joint eision in fir n stisftory wy, they nee to ggregte their (possily onfliting) preferenes. Preferene Moifie in August The previous version ontine n inorret exmple of pirwise SDS tht stisfies very strong prtiiption (Theorem 2). Appers in: Proeeings of the 14th Interntionl Conferene on Autonomous Agents n Multigent Systems (AAMAS 2015), Borini, Elkin, Weiss, Yolum (es.), My 4 8, 2015, Istnul, Turkey. Copyright 2015, Interntionl Fountion for Autonomous Agents n Multigent Systems ( All rights reserve. ggregtion rules re stuie in etil in soil hoie theory n re oming uner inresing srutiny from omputer sientists who re intereste in their omputtionl properties or wnt to utilize them in omputtionl multigent systems. A ommon flw of mny suh rules, first oserve y Fishurn n Brms [22], who lle it the no-show prox, is tht gents my otin more preferre outome y stining n eletion. In seminl pper, Moulin [26] hs shown tht ll resolute, i.e., single-vlue, Conoret extensions re suseptile to the no-show prox. Conoret extensions omprise lrge lss of voting rules tht stisfy otherwise rther esirle properties. In this pper, we initite the stuy of prtiiption in proilisti soil hoie, i.e., we onsier funtions tht mp the orinl preferenes of the gents to proility istriution (or lottery) over the lterntives. Gir [23] lle these funtions eision shemes n they re now usully referre to s soil eision shemes (SDSs). Previous work on prtiiption fouse on resolute voting rules [see, e.g., 26, 32, 27, 25] n set-vlue voting rules [see, e.g., 28, 24, 10]. To the est of our knowlege, this is the first time prtiiption is explore in the ontext of proilisti soil hoie. Rnomize voting rules hve surprisingly long trition going k to nient Greee n hve reently gine inrese ttention in politil siene [see, e.g., 17, 35] n soil hoie theory [see, e.g., 9, 13]. Within omputer siene, rnomiztion is very suessful tehnique in lgorithm esign n is eing onsiere more n more often in the ontext of voting [see, e.g., 15, 31, 37, 34, 7, 3, 1, 4, 5]. In orer to reson out the outomes of SDSs, we nee to efine how gents ompre lotteries when only their preferenes over lterntives re known. A ommon ssumption is tht gents re equippe with von Neumnn-Morgenstern (vnm) utility funtions, i.e., funtions tht ssign rinl utility vlue to eh lterntive. These funtions re usully unknown to the soil plnner n my even e unknown to the gent himself. Uner these ssumptions, preferenes over lterntives re extene to preferenes over lotteries using stohsti ominne (SD) [see, e.g., 23, 30, 8]. One lottery stohstilly omintes nother if, for every lterntive x, the former is t lest s likely to yiel n lterntive t lest s goo s x s the ltter. It is well-known ft tht lottery stohstilly omintes nother if the former yiels t lest s muh expete utility s the ltter for every vnm funtion tht is omptile with the orinl preferenes over lterntives.

2 Bse on stohsti ominne, we introue three novel notions of prtiiption tht form hierrhy. The wekest notion is SD-prtiiption. It presries tht no gent n otin n SD-preferre outome y leving the eletorte. In gme-theoreti terms, n SDS stisfies SD-prtiiption if voting is stritly unominte (y not voting). Sine the preferenes over lotteries otine from stohsti ominne re inomplete, SD-prtiiption oes not rule out tht n gent otins n inomprle lottery y leving the eletorte. Hene, oring to his (unknown) vnm funtion, he my reeive higher expete utility y stining. Strong SD-prtiiption requires tht, for every omptile vnm funtion, n gent hs to reeive t lest s muh expete utility y voting thn y not voting, i.e., voting yiels wekly SD-preferre lottery ompre to not voting. In gme-theoreti terms, strong SD-prtiiption emns tht voting is wekly ominnt strtegy. Still, in mny ses, single gent my not e le to ffet the eletion outome y prtiipting. Hene, if there is some mrginl ost involve in sting one s llot, gents re etter off y not voting. This onern is nnihilte y very strong SD-prtiiption whih requires tht n gent lwys otins stritly SD-preferre outome y voting (unless his most preferre outome is lrey hosen). In gme-theoreti terms, voting is stritly ominnt strtegy (whenever this is possile). Very strong prtiiption is remrkly strong property tht nnot e met y resonle non-proilisti voting rules. In ft, the ommon phenomenon tht iniviul gents usully fe no inentive to vote euse they nnot ffet the outome t ll is sometimes ue the Downs prox or the prox of voting. Interestingly, we fin tht very strong prtiiption n e stisfie ex nte y resonle SDSs in proilisti soil hoie, i.e., gents o fe (potentilly very smll) inentive to inrese their expete utility y voting. Also for the weker notions of prtiiption, we otin more positive results thn in the non-proilisti setting. In ontrst to Moulin s negtive result for resolute Conoret extensions mentione t the eginning of this setion, there re proilisti Conoret extensions tht stisfy prtiiption. More generlly, this pper stuies to whih extent prtiiption is omptile with vrious notions of (Preto) effiieny. We onsier SD-effiieny improving the stisftion of one gent with respet to stohsti ominne will hurt nother gent, ex post effiieny Preto-ominte lterntives reeive proility zero, n unnimity ny lterntive tht is uniquely top-rnke y ll gents will e selete with proility one. Some of our results pertin to limite lsses of SDSs. In prtiulr, we onsier pirwise SDSs n mjoritrin SDSs tht only tke into ount the weighte n unweighte mjority omprisons etween pirs of lterntives, respetively. Our impossiility results re s follows. Very strong SD-prtiiption nnot e stisfie y mjoritrin SDSs (Theorem 1). Very strong SD-prtiiption is inomptile with unnimity for pirwise SDSs (Theorem 3). Strong SD-prtiiption is inomptile with unnimity for mjoritrin SDSs (Theorem 6). SD-prtiiption is inomptile with ex post effiieny for mjoritrin SDSs (Theorem 8). We lso otin two positive results. There re SDSs tht stisfy very strong SDprtiiption n ex post effiieny (Theorem 4). There re pirwise SDSs tht stisfy strong SDprtiiption n SD-effiieny (Theorem 7). 2. RELATED WORK Fishurn n Brms [22] first oserve tht gents who shre the sme lest-preferre lterntive n mke extly this lterntive win y joining the eletorte uner the single trnsferle vote (STV) rule n referre to this phenomenon s the no-show prox. Ry [32] showe tht the no-show prox for STV is likely to our in prtie. Moulin [26] efine prtiiption s the property tht requires tht no gent is ever worse off y joining n eletorte n prove tht ll resolute Conoret extensions fil to stisfy prtiiption. Prtiiption ws stuie in more etil y Pérez [27] n Lepelley n Merlin [25]. However, ll these ppers onsier resolute, i.e., single-vlue, voting rules. Pérez [28], Jimeno et l. [24], n Brnt [10] hve otine vrious results on prtiiption for set-vlue voting rules using iffering ssumptions on how gents ompre sets of lterntives. Astention in slightly ifferent ontexts thn the one stuie in this pper hs reently lso ught the ttention of omputer sientists working on voting equiliri n mpigning [see, e.g., 16, 6]. To our est knowlege, prtiiption hs never een onsiere in the ontext of proilisti soil hoie. Prtiiption is similr to, ut logilly inepenent from, strtegyproofness. An SDS is strtegyproof if no gent n otin more preferre outome y misrepresenting his preferenes. Reently, numer of theorems tht illustrte the treoff etween strtegyproofness n effiieny in proilisti soil hoie hve een shown [8, 9, 3, 4]. Bogomolni n Moulin [8] hve prove tht strong SDstrtegyproofness n SD-effiieny re inomptile. Aziz et l. [3] onjeture tht this inomptiility even hols for (wek) SD-strtegyproofness n prove this lim for mjoritrin SDSs. This sttement ws lter strengthene to pirwise SDSs [4]. We show in this pper tht no orresponing sttement for prtiiption hols y proposing n SDS tht stisfies strong SD-prtiiption, SD-effiieny, n even pirwiseness (Theorem 7). Mnipultion y stention is rguly more severe prolem thn mnipultion y misrepresenttion for two resons. First, gents might not e le to fin suessful strtegi misrepresenttion of their preferenes. It ws shown in vrious ppers tht the orresponing omputtionl prolem n e intrtle [see, e.g., 20]. Fining suessful mnipultion y strtegi stention, on the other hn, is never hrer thn omputing the outome of the voting rule. Seonly, one oul rgue tht gents will not lie out their preferenes euse this is onsiere immorl (Bor fmously exlime my sheme is intene only for honest men ), while strtegi stention is eeme eptle. 1 1 Alterntively, one oul lso rgue tht mnipultion y misrepresenttion is more ritil euse gents re

3 3. PRELIMINARIES Let A e finite set of lterntives n N = {1, 2,... } set of gents. F(N) enotes the set of ll finite n nonempty susets of N. A (wek) preferene reltion is omplete, reflexive, n trnsitive inry reltion on A. The preferene reltion of gent i is enote y. The set of ll preferene reltions is enote y R. In orne with onventionl nottion, we write P i for the strit prt of, i.e., x P i y if x y ut not y x n I i for the inifferene prt of, i.e., x I i y if x y n y x. A preferene reltion is liner if x P i y or y P i x for ll istint lterntives x, y A. For X F(N), we enote y mx Ri (X) the set of gent i s most preferre lterntives in X, i.e., mx Ri (X) = {x X : x y for ll y X}. We will omptly represent preferene reltion s omm-seprte list with ll lterntives mong whih n gent is inifferent ple in set. For exmple x P i y I i z is represente y : x, {y, z}. A preferene profile s funtion from set of gents N to the set of preferene reltions R. The set of ll preferene profiles is enote y R F(N). For preferene profile R R N n S N, T N, i N, j N we efine R i = R \ {(i, )}, R S = R \ k S {(k, R k )}, n R +j = R {(j, R j)}, R +T = R {(k, R k )}. k T By n R(x, y) we enote the numer of gents who wekly prefer x to y, i.e., n R(x, y) = {i N : x y}. Whenever s ler from the ontext we only write n(x, y). In ition, we efine the mjority mrgin g R(x, y) s g R(x, y) = n R(x, y) n R(y, x). The mjority reltion R M of preferene profile s given y the mjority omprisons etween eh pir of lterntives, i.e., x R M y iff n(x, y) n(y, x). An lterntive x is Conoret winner if x P M y for ll y A \ {x}. Let furthermore (A) enote the set of ll lotteries (or proility istriutions) over A, i.e., { } (A) = p(x) x: p(x) = 1, x A: p(x) 0. x A x A We usully write lotteries s onvex omintions of lterntives, i.e., 1 /2 + 1 /2 enotes the uniform istriution over {, }. For lottery p (A) n n lterntive x A, p(x) enotes the proility tht p ssigns to x. Similrly, for set of lterntives S A, p(s) enotes the sum of proilities tht p ssigns to lterntives in S. The support of p, enote supp(p), is the set of ll lterntives to whih p ssigns positive proility, i.e., supp(p) = {x A: p(x) > 0}. Our entrl ojets of stuy re soil eision shemes (SDSs), i.e., funtions tht mp preferene profile to lottery. Formlly, n SDS is funtion f : R F(N) (A). A soil hoie funtion (SCF), on the other hn, is funtion f : R F(N) 2 A \ tht mps preferene profile to suset of lterntives. A miniml firness onition for SDSs (n SCFs) is nonymity, whih requires tht f(r) = f(r ) for ll N, M F(N), R R N, R R M, n ijetions π : N M suh tht = R π(i) for ll i N. Another firness requirement is neutrlity. For permuttion π of A n preferene reltion, efine π(x) Ri π π(y) if n only tempte to t immorlly, whih is vli, ut ifferent, onern. if x y. Then, n SDS (or SCF) f is neutrl if for ll permuttions π n ll R R F(N), f(r)(x) = f(r π )(π(x)) for ll x A. An SDS (or SCF) f is pirwise if it is neutrl n for ll preferene profiles R, R R F(N), f(r) = f(r ) whenever g R(x, y) = g R (x, y) for ll lterntives x, y. 2 In other wors, the outome of pirwise SDS only epens on the nonymize omprisons etween pirs of lterntives [see, e.g., 38, 39]. Mny ommon SCFs re pirwise [see, e.g., 21]. Typil exmples re Bor s rule, Kemeny s rule, or the Simpson-Krmer rule (k mximin). An SDS (or SCF) f is mjoritrin (or neutrl C1 funtion) if it is neutrl n for ll preferene profiles R, R R F(N), f(r) = f(r ) whenever R M = R M. Even the seemingly nrrow lss of mjoritrin SCFs ontins vriety of funtions (sometimes lle tournment solutions). Exmples inlue Copeln s rule, the top yle, or the unovere set [see, e.g., 11]. It is esy to see tht the three lsses form hierrhy: every mjoritrin SDS is pirwise n every pirwise SDS is nonymous n neutrl. An SDS is Conoret extension if it puts proility one on Conoret winner whenever one exists. Exept Bor s rule, ll of the pirwise n mjoritrin SDSs mentione ove re Conoret extensions. In orer to reson out the outomes of SDSs, we nee to mke ssumptions on how gents ompre lotteries. A ommon wy to exten preferenes over lterntives to preferenes over lotteries is stohsti ominne (SD). A lottery SD-omintes nother if, for every lterntive x, the former is t lest s likely to yiel n lterntive t lest s goo s x s the ltter. Formlly, p R SD i q iff for ll x A, y : y x p(y) y : y x q(y). It is well-known tht p Ri SD q iff the expete utility for p is t lest s lrge s tht for q for every vnm funtion omptile with. Thus, for the preferene reltion :,,, we for exmple hve tht ( 2 /3 + 1 /3 ) P SD i ( 1 /3 + 1 /3 + 1 /3 ) while 2 /3 + 1 /3 n re inomprle. 4. EFFICIENCY AND PARTICIPATION In this setion we efine the notions of effiieny n prtiiption onsiere in this pper. The three notions of effiieny efine elow re generliztions of estlishe effiieny notions for SCFs. A rther si requirement is unnimity [see e.g., 33, 29, 13]. For SCFs, unnimity presries tht if ll gents report the sme lterntive s their (unique) top hoie, this lterntive is hosen uniquely. The rguly most nturl generliztion of unnimity to proilisti soil hoie is tht if ll gents report the sme lterntive s their top hoie, this lterntive is hosen with proility one. An lterntive is Preto-ominte if there exists nother lterntive suh tht ll gents wekly prefer the ltter to the former with strit preferene for t lest one gent. 2 Aprt from some tehnil sutleties, this is wht Fishurn lls C2 funtions [21].

4 An SDS is ex post effiient if it ssigns proility zero to ll Preto-ominte lterntives [see e.g., 23, 9, 19]. Finlly, we efine effiieny with respet to stohsti ominne. A lottery p is SD-effiient if there is no other lottery q tht is wekly preferre y every gent with strit preferene for t lest one gent, i.e., q Ri SD p for ll i N n q Pi SD p for some i N. An SDS is SD-effiient if it returns n SD-effiient lottery for every preferene profile [see e.g., 8, 4, 5]. These notions of effiieny form hierrhy (see Figure 1). It is well-known tht SD-effiieny implies ex post effiieny n from the efinitions it follows strightforwrly tht ex post effiieny implies unnimity. Moreover, it is esily seen tht every Conoret extension stisfies unnimity. For etter illustrtion onsier A = {,,, } n the preferene profile R = (R 1,..., R 4), R 1 :,,,, R 2 :,,,, R 3 :,,,, R 4 :,,,. Oserve tht no lterntive is Preto-ominte, i.e., for instne the uniform lottery 1 /4 + 1 /4 + 1 /4 + 1 /4 is ex post effiient. On the other hn, the uniform lottery is not SD-effiient s ll gents stritly SD-prefer 1 /2 + 1 /2. Prtiiption presries tht no gent n otin more preferre outome y stining the eletion. We otin vrying egrees of this property ssoite with stohsti ominne se on the interprettion of inomprilities n ties (see Figure 1). The wekest notion of prtiiption we onsier is SD-prtiiption, i.e., no gent n otin n SD-preferre outome y not voting. Formlly, n SDS f is SD-mnipulle (y strtegi stention) if there exist R R N for some N F(N) n i N suh tht f(r i) Pi SD f(r). If n SDS is not SD-mnipulle it stisfies SD-prtiiption. However, it my e interprete s suessful mnipultion y stention if n gent n otin lottery tht is inomprle (oring to stohsti ominne) to the lottery he otins y voting, sine the former yiels more expete utility thn the ltter for some (rther thn ll) omptile vnm funtions. Strong SD-prtiiption requires tht voting is wekly ominnt strtegy. SDS f stisfies strong SD-prtiiption if f(r) Ri SD Formlly, n f(r i) for ll N F(N), R R N, n i N. We otin the strongest notion of prtiiption onsiere in this pper y requesting tht voting is stritly ominnt strtegy (whenever this is possile). An SDS f stisfies very strong SD-prtiiption if for ll N F(N), R R N, n i N, f(r) R SD i f(r) P SD i f(r i) n f(r i) whenever p (A): p P SD i f(r i) Hene, voting hs to e stritly ominnt strtegy for every gent unless he lrey reeives one of his most preferre outomes when stining. By the sme token, we otin three notions of groupprtiiption. Note tht SD-group-mnipulility requires ll gents to e stritly etter off fter hving left the eletorte. 5. RESULTS AND DISCUSSION In the following, we look t the ifferent notions of prtiiption introue ove together with vrying types of effiieny n firness. 5.1 Very Strong SD-prtiiption Very strong SD-prtiiption is the strongest prtiiption-property onsiere in this pper. As every single gent who joins n eletorte hs to e le to influene the outome in his fvor, it n e esily seen tht no mjoritrin SDS n stisfy this property. Theorem 1. There is no mjoritrin SDS stisfying very strong SD-prtiiption even when preferenes re require to e liner. Proof. Let R = (R 1, R 2, R 3) suh tht R 1, R 2 :,, R 3 :,. Note tht R M = (R 3) M. Therefore, for every mjoritrin SDS f it hols tht f(r) = f(r 3) giving tht it never n e the se tht f(r) P3 SD f(r 3). In omprison to mjoritrin SDSs, pirwise SDSs llow for more sensitivity with respet to vritions in the set of gents. Still, when itionlly requiring unnimity, we otin nother impossiility. Theorem 3. There is no pirwise n unnimous SDS stisfying very strong SD-prtiiption even when preferenes re require to e liner. Proof. Let R = (R 1, R 2, R 3) e preferene profile suh tht R 1, R 2 :,, R 3 :,. Note tht ny pirwise n unnimous SDS f hs to hoose f((r 1, R 2)) = f((r 1)) = n y the ft tht g (R1 )(, ) = g R(, ) we get f((r 1)) = f(r). Put together, it n never e the se tht f(r) P3 SD f(r 3). Within the unrestrite omin of SDSs, there o exist funtions tht stisfy very strong SD-prtiiption n ertin notions of effiieny, in prtiulr rnom seril ittorship (RSD) the nonil generliztion of rnom ittorship [see e.g., 23] to wek preferenes. RSD is efine y piking sequene of the gents uniformly t rnom n then letting eh gent nrrow own the set of lterntives y piking his most preferre of the lterntives selete y the previous gents. Formlly, we otin the following reursive efinition. RSD(R, X) = R x X i=1 1 x if R =, X 1 R RSD(R i, mx (X)) otherwise, n RSD(R) = RSD(R, A). For efinition of RSD employing permuttions we refer to Aziz et l. [4]. Theorem 4. RSD stisfies nonymity, neutrlity, ex post effiieny, n very strong SD-prtiiption. Proof. We only prove very strong SD-prtiiption here n refer to Aziz et l. [3] for ex post effiieny. Let N F(N), R R N, n i N. A first step for showing very strong SD-prtiiption is to prove tht RSD(R) Ri SD RSD(R i). It is lrey known tht RSD stisfies strong SD-strtegyproofness, i.e., RSD(R) Ri SD RSD(R ) for every preferene profile R R N where R j = R j for ll j i [see, e.g., 3]. If gent i is ompletely inifferent etween ll lterntives in A, it trivilly hols tht RSD(R) = RSD(R i). We otin s iret onsequene tht RSD stisfies strong SD-prtiiption. In orer to see tht the even stronger notion pplies, ssume tht R i llows for strit improvement for i, i.e.,

5 SD-effiieny open prolem very strong SD-prtiiption ex post effiieny unnimity BOR (Theorem 7) RSD (Theorem 4) no pirwise SDS (Theorem 3) no mjoritrin SDS (Theorem 6) no mjoritrin SDS (Theorem 8) strong SD-prtiiption SD-prtiiption Figure 1: Reltionships etween effiieny n prtiiption onepts. An rrow from one notion of effiieny or prtiiption to nother enotes tht the former implies the ltter. A soli line inites tht there exist SDSs with the given properties. A she line inites tht no SDSs with the given properties exists. The otte line mrks n open prolem. there is p (A) suh tht p Pi SD RSD(R i). Thus, RSD(R i)(mx Ri (A)) < 1. We hve RSD(R)(mx(A)) = RSD(R, A)(mx(A)) = j N 1 n = 1 n + 1 n 1 n + n 1 n RSD(R j, mx R j j N\{i} (A))(mx(A)) RSD(R j, mx(a))(mx(a)) R j }{{} RSD(R {i,j},mx Rj (A))(mx Ri (A)) RSD(R i, A)(mx(A)) }{{} <1 y ssumption > RSD(R i, A)(mx(A)) = RSD(R i)(mx(a)). We onlue tht RSD(R) Pi SD RSD(R i) for ll i N whih mens RSD stisfies very strong SD-prtiiption. It is noteworthy tht RSD oes not even stisfy SD-groupprtiiption. This n for instne e oserve when looking t A = {,,, }, R = (R 1, R 2, R 3, R 4), with R 1 : {, },,, R 2 : {, },,, R 3 : {, },,, R 4 : {, },,. Here, we hve RSD(R) = 1 /3 + 1 /3 + 1 /6 + 1 /6 n RSD(R {1,2} ) = 1 /2 + 1 /2. Yet, for oth gents i {1, 2}, it hols tht RSD(R {1,2} ) Pi SD RSD(R). While RSD stisfies ex post effiieny n the strongest notion of prtiiption onsiere in this pper, it ws reently shown tht omputing RSD is #P-omplete [2]. Note tht RSD is y fr not the only SDS stisfying very strong SD-prtiiption n ex post effiieny. Further SDSs tht stisfy these properties n e otine y tking the onvex omintion of RSD n other SDSs. Theorem 5. Let f 1, f 2 e two ex post effiient SDSs suh tht f 1 stisfies very strong SD-prtiiption n f 2 stisfies strong SD-prtiiption. Moreover, let λ (0, 1). Then, f = λf 1 + (1 λ)f 2 stisfies ex post effiieny n very strong SD-prtiiption. Proof. Let f 1, f 2, f, n λ e s ove. First note tht if oth f 1, f 2 put proility zero on ll Preto-ominte lterntives x A, so oes f. Aitionlly, we hve for ll y A, i N, l {1, 2} f l (R)(x) f l (R i)(x) x A : x y x A : x y n for some ȳ A it hols for ll i N f 1(R)(x) > f 1(R i)(x). x A : x ȳ x A : x ȳ We iretly eue tht for ll y A, i N f(r)(x) f(r i)(x) x A : x y n for ȳ lso x A : x ȳ f(r)(x) > x A : x y x A : x ȳ Therefore, f(r) P SD i f(r i) for ll i N. f(r i)(x). As onsequene, every proper onvex omintion of RSD n BOR (whih will e efine in the next setion) stisfies very strong SD-prtiiption n ex post effiieny. By ontrst, very strong SD-prtiiption is prohiitive in generl if we onsier stention y groups of gents. More preisely, there is no SDS tht stisfies very strong SD-group-prtiiption. The proof is omitte ue to spe onstrints. 5.2 Strong SD-prtiiption Sine very strong SD-prtiiption implies strong SDprtiiption, positive results from the previous setion rry over. However, in ontrst to Theorem 1, there o exist mjoritrin SDSs tht stisfy strong SD-prtiiption. The rguly simplest exmple is onstnt funtion tht lwys hooses the uniform istriution over ll lterntives. In the se tht ex post effiieny is require s well, the more generl impossiility shown in Theorem 8 (whih only requires SD-prtiiption) pplies. With respet to unnimity, we otin the following result.

6 Theorem 6. When A 4, there is no mjoritrin n unnimous SDS stisfying strong SD-prtiiption, even when preferenes re require to e liner. Proof. Let A = {,,, }. For ontrition, suppose f is n SDS stisfying mjoritrinness, unnimity n strong SD-prtiiption. For the ske of reility, we slightly use nottion n write f(t ) inste of f(r) whenever n implition hols for ll R R F(N) inuing T, i.e., T = R M. First look t the tournment T α s epite elow n note tht T α is trnsitive n oul thus e inue y only one gent with preferenes :,,,. By unnimity, we otin f(t α) =. T α T β Now, let two gents α, α with ientil preferenes R α, R α :,,, join n eletorte with preferene profile R R F(N) inuing T α. Note tht s suppose to e of the form tht g R(x, y) 3 for ll (x, y) A A\{(, ), (, )}, x y, n g R(, ) = 1. The itionl gents lter the mjority grph in wy suh tht it equls T for R +{α,α }. Using strong SD-prtiiption, we eue tht f(t )() = f(t )() = 0. As tournment T β is the mjority grph inue y (R j1, R j2, R j3 ), R j1 :,,,, R j2 :,,,, R j3 :,,,, unnimity yiels f(t β ) =. Anlogously to efore, let two gents β, β with preferenes R β, R β :,,, join n eletorte with preferene profile R R F(N) inuing T β. We suppose R to e of the form tht g R (x, y) 3 for ll (x, y) A A \ {(, ), (, )}, x y n g R (, ) = 1. This hnges the mjority grph of R suh tht it equls T for R +{β,β }. Here, strong SD-prtiiption gives f(t )() = f(t )() = 0. Consequently, we iretly get f(t ) =. Due to neutrlity, we know tht T γ f(t γ) = 1 /4 + 1 /4 + 1 /4 + 1 /4. A single gent γ, R γ :,,, to n eletorte with preferene profile R R F(N) inuing T γ where g R (, ) = g R (, ) = g R (, ) = g R (, ) 2. The mjority grph is thus ltere suh tht it equls T for R +γ. Note tht for γ, f(t ) n f(t γ) re inomprle oring to the SD-extension ontriting tht f stisfies strong SDprtiiption n onluing the proof. As orollry of this theorem, there is no mjoritrin Conoret extension stisfying strong SD-prtiiption. While RSD stisfies very strong SD-prtiiption (Theorem 4), it fils to stisfy SD-effiieny for wek preferenes [see, e.g., 8, 3]. We now efine n SDS tht stisfies SDeffiieny, strong SD-prtiiption, n pirwiseness. The SDS, lle BOR, yiels the uniform istriution over ll Bor winners. For preferene profile R R N efine the Bor sore of lterntive x s s R(x) = i N {y A: x P i y} + 1 /2 {y A \ {x}: x I i y} T A Bor winner is n lterntive with the highest Bor sore. BOs efine s the SDS, tht returns the uniform istriution over ll Bor winners, i.e., BOR(R) = 1 rg mx y s R(y) x rg mx y s R (y) If preferenes re liner, this efinition of s R(x) oinies with the stnr efinition of Bor sores. Whenever n gent is inifferent etween some lterntives X A, the sores tht woul hve een wre to those lterntives h they een rnke linerly re summe up n eqully ivie mong them. Theorem 7. BOR stisfies pirwiseness, SD-effiieny, n strong SD-prtiiption. Proof. First we show tht BOs pirwise. Oserve tht {i N : x I i y} = n(x, y) + n(y, x) n. Rewriting the efinition of s R(x) n using the ft stte efore yiels s R(x) = {i N : x P i y} + 1 /2 {i N : x I i y} y A\{x} = y A\{x} = 1 /2 n + 1 /2 x. n(x, y) 1 /2 (n(x, y) + n(y, x) n) y A\{x} (n(x, y) n(y, x)). Hene, the orer of the s R(x) n thus lso the outome of BOR only epens on n(x, y) n(y, x). Consequentilly, BOs pirwise. In orer to see tht BOR stisfies strong SD-prtiiption onsier the following: if y joining n eletorte N i, some gent i n fore n lterntive into the set of Bor winners without ropping ny other, then hs to e rnke ove the other Bor winners in, i.e., BOR(R) Ri SD BOR(R i). On the other hn, if i n fore n lterntive out of the set of Bor winners y prtiipting, hs to e rnke elow the other Bor winners in giving one more BOR(R) Ri SD BOR(R i). A omintion of oth rguments yiels tht even for some lterntives joining the set of Bor winners s well s others leving when i prtiiptes t the eletion, we lwys get BOR(R) Ri SD BOR(R i). Finlly, suppose BOR oes not stisfy SD-effiieny, i.e., there exists some eletorte N F(N), preferene profile R R(A) N, n lottery p (A) suh tht p Ri SD BOR(R) for ll i N. First, note tht therefore p(x)s R(x) BOR(R)(x)s R(x). x supp(p) x supp(bor(r)) Stte ifferently n tking into ount tht one gent hs to stritly SD-prefer p, the (weighte) verge Bor sore of the lterntives in supp(p) woul hve to e greter thn the one of the lterntives in supp(bor(r)) ontriting the ft tht BOR hooses the lterntives with mximl Bor sore. This onlues the proof of the theorem. With the sme proof s for Theorem 7 it n e shown tht rnomizing uniformly over the winners of ny soring rule with stritly monotoni eresing sore vetor stisfies SD-effiieny n strong SD-prtiiption. Also note tht BOR stisfies SD-group-prtiiption. The proof uses verge Bor sores within groups n works nlogously. The

7 stronger notion of strong SD-group-prtiiption nnot e stisfie y ny nonymous, neutrl, n unnimous SDS. 5.3 SD-prtiiption In ontrst to strong SD-prtiiption, SD-prtiiption llows for mjoritrin n effiient SDSs. Interestingly, this oes not only hol for single gents ut for groups of gents s well. The SDS tht returns Conoret winner whenever one exists n the uniform lottery over ll lterntives otherwise is oth mjoritrin n unnimous n itionlly stisfies SD-group-prtiiption. We fin tht it is not possile to further strengthen the egree of effiieny to ex post effiieny without losing either SD-prtiiption or mjoritrinness. In orer to simplify the proof of Theorem 8, we introue some itionl nottion n stte n uxiliry lemm linking ex post effiieny n prtiiption to the (MKelvey) unovere set [see 18]. We sy tht n lterntive x (MKelvey) overs n lterntive y if x is t lest s goo s y ompre to every other lterntive. Formlly, x overs y if x P M y n, for ll z A, oth y R M z implies x R M z, n z R M x implies z R M y. The unovere set of R M, enote UC (R M), is the set of ll lterntives tht re not overe y ny other lterntive. By efinition, UC is mjoritrin SCF. Brnt et l. [12] hve shown tht for ll preferene profiles R R F(N) n lterntives x, y A, if x overs y in R, then there is preferene profile R R F(N) suh tht R M = R M n x Preto-omintes y in R. Hene, every mjoritrin n Preto-optiml SCF hs to selet suset of the unovere set. Or, phrse in terms of proilisti soil hoie, every mjoritrin n ex post effiient SDS hs to put ll proility on lterntives in the unovere set. Lemm 1. Let f e mjoritrin n ex post effiient SDS. Then, supp(f(r)) UC (R M) for ll R R F(N). With Lemm 1 t hn, we n now ontinue with our min theorem regring the omptiility of effiieny n prtiiption for mjoritrin SDSs. Theorem 8. When A 4, there is no mjoritrin SDS stisfying ex post effiieny n SD-prtiiption. Proof. Let A = {,,, }. Anlogously to the proof of Theorem 6, we slightly use nottion for the ske of reility n write f(t ) inste of f(r) whenever n implition hols for ll R R F(N) inuing T, i.e., T = R M. In orer to show Theorem 8 ssume for ontrition there exists mjoritrin SDS f stisfying oth ex post effiieny n SD-prtiiption. First note tht y Lemm 1, lterntives not in UC (R M) reeive proility zero in f(r). Strt the proof y exmining tournments of the struture T s epite elow. Therefore, we egin with preferene profile R R F(N) inuing mjority grph of type T α. As is overe (e.g., y ), ue to neutrlity, f hs to yiel the lottery f(t α) = 1 /3 + 1 /3 + 1 /3. Suppose this preferene profile s of the form tht g R(, ) = 1 n g R(x, y) 3 for ll (x, y) A A \ {(, ), (, )}, x y. T α T β Thus, two susequently e gents α, α enowe with preferenes P α n P α le to the preferene profile R +{α,α }. The mjority grph of R +{α,α } is of type T n lterntive remins overe y. This is the se inepenent of ll other preferenes, n, s f stisfies SD-prtiiption, it nnot e tht f(r) Pα SD f(r +α) or f(r +α) P SD α f(r +{α,α }). Due to the flexiility left in R α n R α, we otin f(t ) = f(t α). Continue with some preferene profile R R F(N) tht inues mjority grph of type T β. Without loss of generlity, let g R (x, y) 2 for ll (x, y) A A \ {(, ), (, )}, x y. Sine the MKelvey unovere set onsists of ll lterntives n no symmetries exist in T β, it is not possile to sy immeitely whih lterntives f hs to hoose. Yet, using two uxiliry tournments, we will show tht f(t β ) n e etermine extly. If n gent β with preferenes P β leves the eletorte, the mjority grph of R β equls T, regrless of how n re rnke. We know from efore tht T γ f(t ) = 1 /3 + 1 /3 + 1 /3. Sine f stisfies SD-prtiiption, it my not e the se tht f(r β) Pβ SD f(r ), hene, y the flexiility of n, we onlue tht f(t β )() 1 /3. On the ontrry, if nother gent β equippe with preferenes P β leves the eletorte, the mjority grph of R hnges to tournment tht is isomorphi to T n in whih is overe. Consequently, f(r β ) = 1 /3 + 1 /3 + 1 /3. As for β, it must lso hol for β tht f(r SD β ) Pβ f(r ) is not the se. Employing this, we n eue two (in)equlities: f(t β )() = f(t β )() f(t β )() + f(t β )() 1 /3 The equlity follows sine we n freely rrnge n in β s preferene reltion n the ft tht f(t β )() 1 /3 y the proeeing greement. Together with f(t β )() 1 /3 we therefore iretly get f(t β ) = 1 /3 + 1 /3 + 1 /3. On the other hn, ny preferene profile R R F(N) inuing mjority grph T γ neessrily results in the lottery f(t γ) = 1 /4 + 1 /4 + 1 /4 + 1 /4 euse of neutrlity. Finlly note tht n gent γ with preferenes R γ : {, },, joining suitle eletorte with preferene profile R hnges the mjority grph of R in T

8 SD-effiient ex post effiient unnimous unrestrite mjoritrin very strong SD-prt. pirwise? nonymous n neutrl? mjoritrin ++ strong SD-prt. pirwise nonymous n neutrl mjoritrin SD-prt. pirwise nonymous n neutrl Tle 1: Existene of SDSs omining ertin notions of effiieny n prtiiption. + n ++ inite the existene of SDSs stisfying single-gent prtiiption n group-prtiiption, respetively. extly the wy, tht it equls T β fterwrs. From ove, we know tht, f(t β ) = 1 /3 + 1 /3 + 1 /3. Sine f(t γ) Pγ SD f(t β ), gent γ hs the possiility of SDmnipultion y strtegi stention ontriting the initil ssumption tht f stisfies SD-prtiiption. This onlues the proof. 6. CONCLUSIONS We nlyze to whih extent effiieny n prtiiption re omptile in proilisti soil hoie. Our results re summrize in Tle 1 for stention y single gents n groups of gents. Positive results rry over from groupprtiiption to prtiiption, from stronger to weker notions of effiieny n prtiiption, n from mjoritrinness to pirwiseness to nonymity/neutrlity. As for impossiilities, the implitions re extly the other wy roun. Briefly summrize, we hve seen tht the SDS BOR (whih yiels the uniform lottery over ll Bor winners) stisfies importnt esirle properties. Aprt from the strongest notion of effiieny exmine in this pper, SDeffiieny, BOR lso fres well in terms of resistne ginst mnipultion y strtegi stention oth for single gents s well s for groups of gents. This result is of speil interest sine it seprtes strong SD-prtiiption from the relte notion of strong SD-strtegyproofness, whih is inomptile with nonymity, neutrlity, n SD-effiieny [8]. 3 For RSD, we were le to show very strong SDprtiiption, i.e., ny gent (who is not lrey perfetly hppy with the outome) n improve his expete utility y prtiipting. It seems s if this property is stisfie y no SDS other thn vritions of RSD n onvex omintions of severl SDSs inluing RSD. It remins open whether this oservtion n le to hrteriztion of RSD n thus eeper unerstning of very strong SDprtiiption. Interestingly n in ontrst to BOR, RSD oes not stisfy SD-group-prtiiption. 3 Interestingly, BOs known to e prtiulrly vulnerle to strtegi mnipultion s it is single-winner mnipulle [36], whih mens tht it not only violtes strong SDstrtegyproofness ut strtegyproofness with respet to ny lottery extension. Theorem 8 hs estlishe tht, when restriting ttention to mjoritrin SDSs, ex post effiieny n SDprtiiption re inomptile. This oviously lso hols for the weker ut omplete ownwr lexiogrphi n upwr lexiogrphi lottery extension [14] s well s the reently efine pirwise omprison lottery extension [4, 5]. It is unknown whether this result still hols when further strengthening the lottery extension pplie to, e.g., iliner ominne. Two other importnt open prolems tht remin re whether there is n SDS tht stisfies SD-effiieny n very strong SD-prtiiption, n whether there is pirwise SDS tht stisfies very strong SD-prtiiption. Aknowlegments This mteril is se upon work supporte y Deutshe Forshungsgemeinshft uner grnts BR 2312/7-2 n BR 2312/10-1. REFERENCES [1] H. Aziz. Mximl Reursive Rule: A New Soil Deision Sheme. In Pro. of 22n IJCAI, pges AAAI Press, [2] H. Aziz, F. Brnt, n M. Brill. The omputtionl omplexity of rnom seril ittorship. Eonomis Letters, 121(3): , [3] H. Aziz, F. Brnt, n M. Brill. On the treoff etween eonomi effiieny n strtegyproofness in rnomize soil hoie. In Pro. of 12th AAMAS Conferene, pges IFAAMAS, [4] H. Aziz, F. Brnl, n F. Brnt. On the inomptiility of effiieny n strtegyproofness in rnomize soil hoie. In Pro. of 28th AAAI Conferene, pges AAAI Press, [5] H. Aziz, F. Brnl, n F. Brnt. Universl Preto ominne n welfre for plusile utility funtions. In Pro. of 15th ACM-EC Conferene, pges ACM Press, [6] D. Bumeister, P. Fliszewski, J. Lng, n J. Rothe. Cmpigns for lzy voters: trunte llots. In Pro. of 11th AAMAS Conferene, pges IFAAMAS, 2012.

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