Rank Maximal Equal Contribution: a Probabilistic Social Choice Function

Transcription

1 Rank Maxmal Equal Contrbuton: a Probablstc Socal Choce Functon Hars Azz Data61, CSIRO and UNSW Sydney, Australa Pang Luo Data61, CSIRO and UNSW Sydney, Australa Chrstne Rzkallah Unversty of Pennsylvana Phladelpha, Unted States Abstract When aggregatng preferences of agents va votng, two desrable goals are to ncentvze agents to partcpate n the votng process and then dentfy outcomes that are Pareto effcent. We consder partcpaton as formalzed by Brandl, Brandt, and Hofbauer (2015) based on the stochastc domnance (SD) relaton. We formulate a new rule called RMEC (Rank Maxmal Equal Contrbuton) that s polynomal-tme computable, ex post effcent and satsfes the strongest noton of partcpaton. It also satsfes many other desrable farness propertes. The rule suggests a general approach to achevng very strong partcpaton, ex post effcency and farness. Introducton Makng collectve decsons s a fundamental ssue n multagent systems. Two fundamental goals n collectve decson makng are (1) agents should be ncentvzed to partcpate and (2) the outcome should be such that there exsts no other outcome that each agent prefers. We consder these goals of partcpaton (Fshburn and Brams, 1983; Mouln, 1988) and effcency (Mouln, 2003) n the context of probablstc socal choce. In probablstc socal choce, we study probablstc socal choce functons (PSCFs) whch take as nput agents preferences over alternatves and return a lottery (probablty dstrbuton) over the alternatves. 1 The lottery can also represent tme-sharng arrangements or relatve mportance of alternatves (Azz, 2013; Bogomolnaa, Mouln, and Stong, 2005). For example, agents may vote on the proporton of tme dfferent genres of songs are played on a rado channel. Ths type of preference aggregaton s not captured by tradtonal determnstc votng n whch the output s a sngle dscrete alternatve whch may not be sutable to cater for dfferent tastes. When defnng notons such as partcpaton, effcency, and strategyproofness, one needs to reason about preferences over probablty dstrbutons (lotteres). In order to defne these propertes, we consder stochastc domnance (SD). A lottery s preferred over another lottery wth respect Copyrght c 2018, Assocaton for the Advancement of Artfcal Intellgence ( All rghts reserved. 1 PSCFs are also referred to as socal decson schemes n the lterature. to SD, f for all utlty functons consstent wth the ordnal preferences, the former yelds as much utlty as the latter. Although effcency and strategyproofness wth respect to SD have been consdered n a seres of papers (Azz, 2013; Azz and Stursberg, 2014; Azz, Brandt, and Brll, 2013b; Azz, Brandl, and Brandt, 2014; Bogomolnaa, Mouln, and Stong, 2005; Cho, 2012; Gbbard, 1977; Procacca, 2010), three notons of partcpaton wth respect to SD were formalzed only recently by Brandl, Brandt, and Hofbauer (2015a). The three notons nclude very strong (partcpatng s strctly benefcal), strong (partcpatng s at least as helpful as not partcpatng) and standard (not partcpatng s not more benefcal). In contrast to determnstc socal choce n whch the number of possble outcomes are at most the number of alternatves, probablstc socal choce admts nfntely many outcomes whch makes partcpaton even more meanngful: agents may be able to perturb the outcome of the lottery slghtly n ther favour by partcpatng n the votng process. In sprt of the rado channel example, voters should deally be able to ncrease the fractonal tme of ther favorte musc genres by partcpatng n the vote to decde the duratons. One of the central results presented by Brandl, Brandt, and Hofbauer (2015a) was that there exsts a PSCF (RSD Random Seral Dctatorshp) that satsfes very strong SDpartcpaton and ex post effcency (Theorem 4, (Brandl, Brandt, and Hofbauer, 2015a)). In ths paper, we propose a polynomal-tme rule that satsfes the strongest noton of partcpaton and s also ex post effcent. We show that t also satsfes several other desrable propertes. Contrbutons Our central contrbuton s a new probablstc votng rule called Rank Maxmal Equal Contrbuton Rule (RMEC). RMEC satsfes very strong SD-partcpaton and ex post effcency. Moreover RMEC s polynomal-tme computable and also satsfes other mportant axoms such as anonymty, neutralty, far share, and proportonal share. Far share property requres that each agent gets at least 1/n of the maxmum possble utlty. Proportonal share s a stronger verson of far share. Whereas RMEC s ex post effcent, t s not SD-effcent. RMEC has two key advantages over RSD the known rule that satsfes very strong SD partcpaton. Frstly,

2 Propertes Seral dctator RSD SML BO ES R RMEC SD-effcent ex post effcent Very strong SD-partcpaton Strong SD-partcpaton SD-partcpaton Anonymous Proportonal share + + Strategyproof for dchotomous and strct preferences Polynomal-tme computable Table 1: A comparson of axomatc propertes of dfferent PSCFs: RSD (random seral dctatorshp), SML (strct maxmal lotteres), BO (unform randomzaton over Borda wnners), ES R (egaltaran smultaneous reservaton) and RMEC (Rank Maxmal Equal Contrbuton). RMEC s polynomal-tme computable 2 whereas computng the RSD probablty shares s #P-complete. The computatonal tractablty of RMEC s a sgnfcant advantage over RSD especally when PSCFs are used for tme-sharng purposes where computng the tme shares s mportant. For RSD, t s even open whether there exsts an FPRAS (Fully Polynomal-tme Approxmaton Scheme) for computng the outcome shares/probabltes. Secondly, RMEC s much more effcent n a welfare sense than RSD. In partcular, RMEC domnates RSD n the followng sense: for any profle on whch RMEC s not SD-effcent, RSD s not SD-effcent as well. 3 In fact we show that for most preference profles wth small number agents and alternatves (for whch arbtrary lotteres can be SD-neffcent), RMEC almost always returns an SD-effcent outcome. For 4 or less agents and 4 or less alternatves, all RMEC outcomes are SD-effcent whereas ths s not the case for RSD. Our formulaton of RMEC suggests a general computatonally-effcent approach to achevng ex post effcency and very strong SD-partcpaton. We dentfy MEC (Maxmal Equal Contrbuton) a general class of rules that all satsfy the propertes satsfed by RMEC: sngle-valued, anonymty, neutralty, far share, proportonal share, ex post effcency, very strong SD-partcpaton, and a natural monotoncty property. They are also strategyproof under strct and dchotomous preferences. A relatve comparson of dfferent probablstc votng rules s summarzed n Table 1. 2 Unlke other desrable rules such as maxmal lotteres (Azz, Brandt, and Brll, 2013b; Brandl, Brandt, and Seedg, 2016) and ESR (Azz and Stursberg, 2014), RMEC s relatvely smple and does not requre any lnear programs to fnd the outcome lottery. 3 Ths dea of comparng two mechansms wth respect to a property may be of ndependent nterest. When two mechansms f and g do not satsfy a property φ n general, one can stll say that that f domnates g wth respect to φ f for any nstance on whch f does not satsfy φ, g does not satsfy t ether. We prove that RMEC domnates RSD wrt SD-effcency. Related Work One of the frst formal works on probablstc socal choce s by Gbbard (1977). The lterature n probablstc socal choce has grown over the years although t s much less developed n comparson to determnstc socal choce (Brandt, 2017). The man result of Gbbard (1977) was that random dctatorshp n whch each agent has unform probablty of choosng hs most preferred alternatve s the unque anonymous, strategyproof and ex post effcent PSCF. Random seral dctatorshp (RSD) s the natural generalzaton of random dctatorshp for weak preferences but the RSD lottery s #P-complete to compute (Azz, Brandt, and Brll, 2013a). RSD s defned by takng a permutaton of the agents unformly at random and then nvokng seral dctatorshp: each agent refnes the workng set of alternatves by pckng hs most preferred of the alternatves selected by the prevous agents). Bogomolnaa and Mouln (2001) ntated the use of stochastc domnance to consder varous notons of strategyproofness, effcency, and farness condtons n the doman of random assgnments whch s a specal type of socal choce settng. They proposed the probablstc seral mechansm a desrable random assgnment mechansm. Cho (2012) extended the approach of Bogomolnaa and Mouln (2001) by consderng other lottery extensons such as ones based on lexcographc preferences. Partcpaton has been studed n the context of determnstc votng rules n great detal. Fshburn and Brams (1983) formalzed the paradox of a voter havng an ncentve to not partcpate for certan votng rules. Mouln (1988) proved that Condorcet consstent votng rules are susceptble to a no show. We pont out that no determnstc votng rule can satsfy very strong partcpaton. Consder a votng settng wth two agents and two alternatves a and b. Agent 1 prefers a over b and agent 2 prefers b over a. Then whatever the outcome of votng rule, one agent wll get a least preferred outcome despte partcpatng. The example further motvates the study of PSCFs wth good partcpaton ncentves. The tradeoff of effcency and strategyproofness for PSCFs was formally consdered n a seres of papers (Azz, 2013; Azz and Stursberg, 2014; Azz, Brandt, and Brll, 2013b; Azz, Brandl, and Brandt, 2014; Bogomolnaa, Mouln, and Stong, 2005). Azz and Stursberg (2014) presented a generalzaton Egaltaran Smultaneous Reservaton (ES R) of the probablstc seral mechansm to the doman of socal choce. Azz (2013) proposed the maxmal recursve (MR) PSCF whch s smlar to the random seral dctatorshp but for whch the lottery can be computed n polynomal tme. Brandl, Brandt, and Hofbauer (2015b) study the connecton between welfare maxmzaton and partcpaton and show how welfare maxmzaton acheves SD-partcpaton. However the approach does not necessarly acheve very strong SD-partcpaton or even strong SD-partcpaton. In very recent work, Gross, Anshelevch, and Xa (2017) presented an elegant rule called 2-Agree that satsfes very strong SD-partcpaton, ex post effcency, and varous other

3 propertes. However, the rule s defned for strct preferences. 4 Prelmnares Consder the socal choce settng n whch there s a set of agents N = {1,..., n}, a set of alternatves A = {a 1,..., a m } and a preference profle = ( 1,..., n ) such that each s a complete and transtve relaton over A. Let R denote the set of all possble weak orders over A and let R N denote all the possble preference profles for agents n N. Let F (N) denote the set of all fnte and non-empty subsets of N. We wrte a b to denote that agent values alternatve a at least as much as alternatve b and use for the strct part of,.e., a b ff a b but not b a. Fnally, denotes s ndfference relaton,.e., a b f and only f both a b and b a. The relaton results n equvalence classes E 1, E2,..., Ek for some k such that a a f and only f a E l and a E l for some l < l. Often, we wll use these equvalence classes to represent the preference relaton of an agent as a preference lst : E 1, E2,..., Ek. For example, we wll denote the preferences a b c by the lst : {a, b}, {c}. For any set of alternatves A, we wll refer by max (A ) to the set of most preferred alternatves accordng to preference. An agent s preferences are dchotomous f and only f he parttons the alternatves nto at most two equvalence classes,.e., k 2. An agent s preferences are strct f and only f s antsymmetrc,.e. all equvalence classes have sze 1. Let (A) denote the set of all lotteres (or probablty dstrbutons) over A. The support of a lottery p (A), denoted by supp(p), s the set of all alternatves to whch p assgns a postve probablty,.e., supp(p) = {x A p(x) > 0}. We wll wrte p(a) for the probablty of alternatve a and we wll represent a lottery as p 1 a p m a m where p j = p(a j ) for j {1,..., m}. For A A, we wll (slghtly abusng notaton) denote a A p(a) by p(a ). A PSCF s a functon f : R n (A). If f yelds a set rather than a sngle lottery, we call f a correspondence. Two mnmal farness condtons for PSCFs are anonymty and neutralty. Informally, they requre that the PSCF should not depend on the names of the agents or alternatves respectvely. In order to reason about the outcomes of PSCFs, we need to determne how agents compare lotteres. A lottery extenson extends preferences over alternatves to (possbly ncomplete) preferences over lotteres. Gven over A, a lottery extenson extends to preferences over the set of lotteres (A). We now defne stochastc domnance (SD) whch s the most establshed lottery extenson. Under stochastc domnance (SD), an agent prefers a lottery that, for each alternatve x A, has a hgher probablty of selectng an alternatve that s at least as good as x. Formally, p SD q f and only f y A: x A:x y p(x) x A:x y q(x). SD (Bogomolnaa and Mouln, 2001) s par- 4 Under strct preferences, random dctatorshp satsfes all the propertes examned n ths paper. tcularly mportant because p SD q f and only f p yelds at least as much expected utlty as q for any von-neumann- Morgenstern utlty functon consstent wth the ordnal preferences (Cho, 2012). Note that n such utlty functons, agents are nterested n maxmzng expected utlty. We defne the RSD PSCF because we wll especally compare our PSCF wth RSD. Let Π N be the set of permutatons over N and π() be the -th agent n permutaton π Π N. Then, RSD(N, A, ) = π Π 1 N n! U(Pro(N, A,, π)) where Pro(N, A,, π) = max π(n) (max π(n 1) ( (max π(1) (A)) )) and U(B) s the unform lottery over the gven set B. Effcency A lottery p s SD-effcent f and only f there exsts no lottery q such that q SD p for all N and q SD p for some N. A PSCF s SD-effcent f and only f t always returns an SD-effcent lottery. A standard effcency noton that cannot be phrased n terms of lottery extensons s ex post effcency. A lottery s ex post effcent f and only f t s a lottery over Pareto effcent alternatves. Partcpaton Brandl, Brandt, and Hofbauer (2015a) formalzed three notons of partcpaton. Formally, a PSCF f satsfes SD-partcpaton f there exsts no R N for some N F (N), and some N such that f ( ) SD f ( ). A PSCF f satsfes strong SD-partcpaton f f ( ) SD f ( ) for all N F (N), R N, and for all N. A PSCF f satsfes very strong SD-partcpaton f for all N F (N), R N, and for all N, f ( ) SD f ( ) and f ( ) SD f ( ) whenever p (A): p SD f ( ). Informally speakng, SD-partcpaton avods the ncentve to abstan; strong SD-partcpaton gves voters at least as much beneft n partcpatng as abstanng; and very strong SD-partcpaton gves voters a strct beneft n partcpatng. The frst two concepts are dfferent because the SD relaton may not be complete. Very strong SD-partcpaton s a desrable property because t gves an agent strctly more expected utlty for each utlty functon consstent wth hs ordnal preferences. We already ponted out that no determnstc votng rule can satsfy very strong SD-partcpaton. Strategyproofness A PSCF f s SD-manpulable f and only f there exsts an agent N and preference profles and wth j = j for all j such that f ( ) SD f ( ). A PSCF s weakly SD-strategyproof f and only f t s not SD-manpulable. It s SD-strategyproof f and only f f ( ) SD f ( ) for all and wth j = j for all j. Note that SD-strategyproofness s equvalent to strategyproofness n the Gbbard sense. Rank Maxmal Equal Contrbuton We present Rank Maxmal Equal Contrbuton (RMEC). The rule s based on the noton of rank maxmalty that s well-establshed n other contexts such as assgnment (Mchal, 2007; Featherstone, 2011).

4 For any alternatve a, ts rank n agent s preference lst s j f a E j.e., t s n s j-th equvalence class. For any alternatve a, ts correspondng rank vector s r(a) = (r 1 (a),..., r m (a)) where r j (a) s the number of agents who have a n ther j-th equvalence class. For a lottery p, ts correspondng rank vector s r(p) = (r 1 (p),..., r m (p)) where r j (p) s N a E j p(a). We compare rank vectors lexcographcally. One rank vector r = (r 1,..., r m ) s better than r = (r 1,..., r m) f for the smallest such that r r, t must hold that r > r. The noton of rank vectors leads to a natural PSCF: randomze over alternatves that have the best rank vectors. However such an approach does not even satsfy strong SDpartcpaton. It can also lead to perverse outcomes n whch mnorty s not represented at all: Consder the followng preference profle. 1 : a, b 2 : a, b 3 : b, a For the profle, the rank maxmal rule smply selects a wth probablty 1. Ths s unfar to agent 3 who s n a mnorty. Agent 3 does not get any beneft of partcpatng. Let F(, A, ) be the set of most preferred alternatves of agent that have best rank vector among all hs most preferred alternatves. In the RMEC rule, each agent N contrbutes 1/n probablty weght to a subset of hs most preferred alternatves. Precsely, he gves probablty weght 1 /n F(,A, ) to each alternatve n F(, A, ). The resultant lottery p s the RMEC outcome. We formalze the RMEC rule as Algorthm 1. We vew RMEC outcome lottery p as consstng of n components p 1,..., p n where p = a F(,A, ) 1 n F(,A, ) a. Input: (N, A, ) Output: lottery p over A. 1 Intalze probablty p(a) of each alternatve a A to zero. 2 for = 1 to N do 3 Identfy F(, A, ) the subset of alternatves n max (A) wth the best rank vector. 4 for each a F(, A, ) do 5 p(a) p(a) + 1 /(n F(,A, ) ) {we wll denote by p the probablty weght of 1/n allocated by agent unformly to alternatves n F(, A, )} 6 return lottery p. Algorthm 1: The Rank Maxmal Equal Contrbuton rule Example 1 Consder the followng preference profle. 1 : {a, b, c, f }, d, e 2 : {b, d}, e, {a, c, f } 3 : {a, e, f }, d, b, c 4 : c, d, e, {a, f }, b 5 : {c, d}, {e, a, b, f } The rank vectors of the alternatves are as follows: a : (2, 1, 1, 1, 0); b : (2, 1, 1, 0, 1); c : (3, 0, 1, 1, 0); d : (2, 3, 0, 0, 0); e : (1, 2, 2, 0, 0); and f : (2, 1, 1, 1, 0). Each agent selects the most preferred alternatves wth the best rank vector to gve hs 1/5 probablty unformly to the followng alternatves: 1 : c; 2 : d; 3 : a, f ; 4 : c; and 5 : c. So the outcome s 1 10 a c d f. Propertes of RMEC We observe that RMEC s both anonymous and neutral. The RMEC outcome can be computed n tme polynomal n the nput sze. Snce the contrbuton to an alternatve by an agent s 1/yn for some y {1,..., m}, the probabltes are ratonal. Proposton 1 RMEC s anonymous and neutral. The RMEC outcome can be computed n polynomal tme O(m 2 n) and conssts of ratonal probabltes. Next we note that f preferences are strct, then RMEC s equvalent to random dctatorshp. As a corollary, RMEC satsfes both SD-effcency and very strong SD-partcpaton under strct preferences. More nterestngly, RMEC satsfes very strong SD-partcpaton even for weak orders. Proposton 2 RMEC satsfes very strong SDpartcpaton. Proof: Let us consder the RMEC outcome p when abstans and compare t wth the RMEC outcome q when votes. When abstans, agent j N \ {} contrbutes probablty weght 1/(n 1) unformly to alternatves n F( j, A, ). Now consder the stuaton when also votes. We want to dentfy the alternatves j wll contrbute to. Our central clam s that for each a F( j, A, ) and b max (F( j, A, )), t s the case that a b. To prove the clam, assume for contradcton that when votes, j contrbutes to some alternatve b less preferred by to a max (F( j, A, ). But ths s not possble because b had at most the same rank as a when dd not vote but snce a b, a wll have strctly more rank than b when votes. Hence when votes, agent j sends all hs probablty weght to ether alternatves n max (F( j, A, )) or alternatves even more preferred by. Thus we have proved the clam. By provng the clam, we have shown that when partcpates, any change n the relatve contrbuton of some agent j s n favour of agent. Take any b A and consder {a : a b}. Assume j s any agent n N \ {}. If j contrbutes anythng (at most 1/(n 1)) to {a : a b} when agent abstans, then when votes, j wll contrbute 1/n to {a : a b} because of the central clam proved above. Now, for the two scenaros where votes or abstans, the contrbuton dfference from j to {a:a b} s at most 1/n(n 1), and the total contrbuton dfference from N\ {} to {a:a b} s at most 1/n, whch would be compensated by the contrbuton of to {a:a b} when votes. Therefore for each b A, q({a : a b}) p({a : a b}). Thus q SD p so RMEC satsfes strong SD-partcpaton. We now show that RMEC satsfes very strong SDpartcpaton. Suppose that p = RMEC(N, A, ) s such that p(max (A)) < 1. It s suffcent to show that for q =

5 RMEC(N, A, ), q(max (A)) > p(max (A)). If some other agent j s relatve contrbuton changes n favour of agent, we are already done. So let us assume that each j, F( j, A, ) = F( j, A, ). When votes, the total contrbuton to max (A) by agents other than s p(max (A)) n 1 n. The contrbuton of agent to max (A) s 1 n. Hence q(max(a)) = n 1 n p(max (A)) + 1 n (1) = n 1 n p(max (A)) + 1 n (p(max (A)) + 1 p(max(a))) = p(max(a)) + 1 (1 p(max(a))) > p(max(a)) n The last nequalty holds because we supposed that p(max (A)) < 1 so that 1 p(max (A)) > 0. Thus RMEC satsfes very strong SD-partcpaton. The fact that RMEC satsfes very strong SD-partcpaton s one the central results of the paper. We note here that very strong SD-partcpaton can be a trcky property to satsfy. For example the followng smple varants of RMEC volate even strong SD-partcpaton: (1) each agent contrbutes to a most preferred Pareto optmal alternatve or (2) each agent contrbutes unformly to Pareto optmal alternatves most preferred by her. Next, we prove that RMEC s also ex post effcent.e., randomzes over Pareto optmal alternatves. Proposton 3 RMEC s ex post effcent. Proof: Each alternatve a n the support s an alternatve that s the most preferred alternatve of an agent wth the best rank vector. Suppose the alternatve a s not Pareto optmal. Then there exsts another alternatve b such that b j a for all j N and b j a for some j N. Note that snce a s the most preferred alternatve of, t follows that b a. Snce b Pareto domnates a, b s a most preferred alternatve of wth a better rank vector than a. But ths contradcts the fact that a s a most preferred alternatve of wth the best rank vector. Although RMEC s ex post effcent, t unfortunately does not satsfy the stronger effcency property of SD-effcency. Example 2 Consder the followng preference profle wth dchotomous preferences. 1, 2, 3, 4 : d 5, 6 : {d, c} 7, 8 : {d, b} 9 : {a, b} 10 : {a, c} The RMEC outcome s 8 10 d c+ 1 10b but s SD-domnated by 9 10 d a. In the example above, although each agent chooses those most preferred alternatves that are most benefcal to other agents, the agents do not coordnate to make these mutually benefcal decsons. Ths results n a lack of SD-effcency. Although RMEC s not SD-effcent just lke RSD, t has a dstnct advantage over RSD n terms of SD-effcency. Proposton 4 For any profle, f the RSD outcome s SDeffcent, then the RMEC outcome s also SD-effcent. Furthermore, there exst nstances for whch the RSD outcome s not SD-effcent but the RMEC s not only SD-effcent but SD-domnates the RSD outcome. Proof: Due to the result of Azz, Brandl, and Brandt (2015) that SD-effcency depends on the support, t s suffcent to show that supp(rsd(n, A, )) supp(rmec(n, A, )). Now suppose that a supp(rmec(n, A, )). We also know that a F(, A, ) for some N. We prove that a supp(rsd(n, A, )) by showng that there exsts one permutaton π under whch seral dctatorshp gves postve probablty to a. The frst agent n the permutaton π s. We buld the permutaton π so that a s an outcome of seral dctatorshp wth respect to π. The workng set W s ntalzed to A. Agent refnes W to max (A). Now suppose for contradcton that each remanng agent strctly prefers some other alternatve n W to a. In that case, a s not the rank maxmal alternatve from max (A) whch s a contradcton to a F(, A, ). Thus for some agent j not consdered yet, a s a most preferred alternatve n W. We can add such an agent to the permutaton and let hm refne and update W. In W, a stll remans rank maxmal (wth respect to agents who have not been added to the permutaton) among alternatves n W. We can contnue dentfyng a new agent who maxmally prefers a n the latest verson of W and appendng the agent to the permutaton π untl π s fully specfed. Note that a stll remans n the workng set whch mples that a supp(rsd(n, A, )). Ths completes the proof that f the RSD outcome s SD-effcent, then the RMEC outcome s also SD-effcent. Next we prove the second statement. Consder the followng preference profle. 1 : {a, c}, b, d 2 : {a, d}, b, c 3 : {b, c}, a, d 4 : {b, d}, a, c The unque RSD lottery s p = 1 /3 a+ 1 /3 b+ 1 /6 c+ 1 /6d, whch s SD-domnated by 1 /2 a + 1 /2 b. Ths was observed by Azz, Brandt, and Brll (2013b). We now compute the RMEC outcome. The rank vectors are as follows: a : (2, 2, 0, 0); b : (2, 2, 0, 0); c : (2, 0, 2, 0); and d : (2, 0, 2, 0). The agents choose alternatves as follows: 1 : a, 2 : a, 3 : b, 4 : b RMEC returns the followng lottery whch s SD-effcent and SD-domnates the RSD lottery: 1 /2 a + 1 /2 b. Ths completes the proof. Although RMEC s not SD-effcent n general, we gve expermental evdence that t returns SD-effcent outcomes for most profles. An exhaustve experment shows that RMEC s SD-effcent for every profle wth 4 agents and 4 alternatves. Further experments show that RMEC s SDeffcent for almost all the profles wth n, m 8. In the experment, we generated profles unformly at random for specfed numbers of agents and alternatves so that each preference s equprobable, and examned whether the correspondng RMEC lottery s SD-effcent. The results are shown n Table 2.

6 A N ,000 10,000 10,000 9,999 10, ,999 10,000 10,000 9,998 9, ,999 10,000 9,996 10,000 9, ,000 9,999 9,997 9,998 9, ,999 9,996 9,998 9,997 9,996 Table 2: The number of profles for whch the RMEC outcome s SD-effcent out of 10,000 profles generated unformly at random for specfed numbers of agents and alternatves. Note that n general for any gven preference profle wth some tes, a sgnfcant proporton of lotteres are not SDeffcent. On the other hand, RMEC almost always returns an SD-effcent lottery. A smlar experment on RSD shows that the proporton of profles for whch RSD generates an SD-effcent lottery s consstently lower than that of RMEC. Table 3 shows the outcome of RSD for 1000 profles generated unformly at random for specfed numbers of agents and alternatves. We only ran t on 1000 profles nstead of 10,000 as RSD s sgnfcantly slower to run than RMEC. For the experment for RSD, the program also checks f the RMEC outcome s SDeffcent when the RSD outcome s not for a profle. There s only one generated profle (7 agents, 4 alternatves) for whch the RMEC outcome s not SD-effcent. A N Table 3: The number of profles for whch the RSD outcome s SD-effcent out of 1000 profles generated unformly at random for specfed numbers of agents and alternatves. We say that a lottery satsfes far welfare share f each agent gets at least 1/n of the maxmum possble expected utlty he can get from any outcome. Far welfare share was orgnally defned by Bogomolnaa, Mouln, and Stong (2005) for dchotomous preferences. We observe that snce RMEC gves at least 1/n probablty to each agent s frst equvalence class, t follows that each RMEC outcome satsfes far welfare share. Under dchotomous preferences, a compellng property s that of proportonal share (Duddy, 2015). We defne t more generally for weak orders as follows. A lottery p satsfes proportonal share f for any set S N, a A: S s.t. a max (A) p(a) S /n. We note that proportonal share mples far share. 5 It s easy to establsh that RMEC satsfes proportonal share. Proposton 5 RMEC satsfes the proportonal share property and hence the far share property. 5 ESR does not satsfy proportonal share and the maxmal lottery rule does not satsfy far welfare share. A dfferent farness requrement s that each agent fnds the outcome at least as preferred wth respect to SD as the unform lottery. A PSCF f satsfes SD-unformty f for each profle, f ( ) SD 1 m a m a m for each N. RMEC does not satsfy SD-unformty. However, we show that SD-unformty s ncompatble wth very strong SDpartcpaton. Proposton 6 There exsts no PSCF that satsfes very strong SD-partcpaton and SD-unformty. Proof: Consder the followng preference profle. 1 : a, b, c 2 : c, b, a 3 : a, b, c When 1 and 2 vote, SD-unformty demands, that the outcome s unform. When 1, 2, 3 vote, SD-unformty stll demands that the outcome s unform. However very strong- SD-partcpaton demands that 3 should get strctly better outcome wth respect to SD. Whereas RMEC satsfes the strongest noton of partcpaton, t can be shown to be vulnerable to strategc msreports. On the other hand, f n 2, we can prove that RMEC satsfes SD-strategyproofness. Also f preferences are strct or f they are dchotomous, RMEC s SD-strategyproof. We also note that RMEC satsfes a natural monotoncty property: renforcng an alternatve n the agent s preferences can only ncrease ts probablty. Dscusson In ths paper, we contnued the lne of research concernng strategc aspects n probablstc socal choce (see e.g., (Azz, 2013; Azz, Brandl, and Brandt, 2014; Azz, Brandt, and Brll, 2013b; Brandl, Brandt, and Hofbauer, 2015a; Gbbard, 1977; Procacca, 2010)). We proposed the RMEC rule that satsfes very strong SD-partcpaton and ex post effcency as well as varous other desrable propertes. In vew of ts varous propertes, t s a useful PSCF wth tw key advantages over RSD. Unlke maxmal lotteres (Brandt, 2017) and ESR (Azz and Stursberg, 2014), RMEC s relatvely smple and does not requre lnear programmng to fnd the outcome lottery. The use of rank maxmalty also makes t easer to deal wth weak orders n a prncpled manner. A general approach. Consder a scorng vector s = (s 1,..., s m ) such that s 1 > > s m. An alternatve n the j-th most preferred equvalence class of an agent s gven score s j. An alternatve wth the hghest score s the one that receves the maxmum total score from the agents (see for e.g., (Fshburn and Gehrlen, 1976) for dscusson on postonal scorng vectors). Note that an alternatve s rank maxmal f t acheves the maxmum total score for a sutable scorng vector (n m, n m 1,..., 1). We also note that RMEC s defned n a way so that each agent gves 1/n probablty to hs most preferred alternatves that have the best rank vector. The same approach can also be used to select the most preferred alternatves that have the best Borda score or score wth respect to any decreasng postonal scorng vector. We refer to s-mec as the maxmal equal contrbuton rule wth

7 respect to scorng vector s. In the rule, each agent dentfes F(, A, ) the subset of alternatves n max (A) wth the best total score and unformly dstrbutes 1/n among alternatves n F(, A, ). The argument for very strong SD-partcpaton and ex post effcency stll works for any s-mec rule. Any s-mec rule s also anonymous, neutral, sngle-valued, and proportonal share far. It wll be nterestng to see how RMEC fares on more structured preferences (Anshelevch and Postl, 2016). Random assgnment rules (Bogomolnaa and Mouln, 2001; Katta and Sethuraman, 2006) can be seen as applyng a PSCF to a votng problem wth more structured preferences (see e.g., (Azz and Stursberg, 2014)). It wll be nterestng to see how RMEC wll fare as a random assgnment rule especally n terms of SD-effcency. Acknowledgments Hars Azz s supported by a Julus Career Award. References Anshelevch, E., and Postl, J Randomzed socal choce functons under metrc preferences. In Proceedngs of the 25th Internatonal Jont Conference on Artfcal Intellgence (IJCAI), AAAI Press. Azz, H., and Stursberg, P A generalzaton of probablstc seral to randomzed socal choce. In Proceedngs of the 28th AAAI Conference on Artfcal Intellgence (AAAI), AAAI Press. Azz, H.; Brandl, F.; and Brandt, F On the ncompatblty of effcency and strategyproofness n randomzed socal choce. In Proceedngs of the 28th AAAI Conference on Artfcal Intellgence (AAAI), AAAI Press. Azz, H.; Brandl, F.; and Brandt, F Unversal Pareto domnance and welfare for plausble utlty functons. Journal of Mathematcal Economcs 60: Azz, H.; Brandt, F.; and Brll, M. 2013a. The computatonal complexty of random seral dctatorshp. Economcs Letters 121(3): Azz, H.; Brandt, F.; and Brll, M. 2013b. On the tradeoff between economc effcency and strategyproofness n randomzed socal choce. In Proceedngs of the 12th Internatonal Conference on Autonomous Agents and Mult- Agent Systems (AAMAS), IFAAMAS. Azz, H Maxmal Recursve Rule: A New Socal Decson Scheme. In Proceedngs of the 23nd Internatonal Jont Conference on Artfcal Intellgence (IJCAI), AAAI Press. Bogomolnaa, A., and Mouln, H A new soluton to the random assgnment problem. Journal of Economc Theory 100(2): Bogomolnaa, A.; Mouln, H.; and Stong, R Collectve choce under dchotomous preferences. Journal of Economc Theory 122(2): Brandl, F.; Brandt, F.; and Hofbauer, J. 2015a. Incentves for partcpaton and abstenton n probablstc socal choce. In Proceedngs of the 14th Internatonal Conference on Autonomous Agents and Mult-Agent Systems (AAMAS), IFAAMAS. Brandl, F.; Brandt, F.; and Hofbauer, J. 2015b. Welfare maxmzaton entces partcpaton. Techncal report, Brandl, F.; Brandt, F.; and Seedg, H. G Consstent probablstc socal choce. Econometrca 84(5): Brandt, F Rollng the dce: Recent results n probablstc socal choce. In Endrss, U., ed., Trends n Computatonal Socal Choce. AI Access. chapter 1. Forthcomng. Cho, W. J Probablstc assgnment: A two-fold axomatc approach. Mmeo. Duddy, C Far sharng under dchotomous preferences. Mathematcal Socal Scences 73:1 5. Featherstone, C. R A rank-based refnement of ordnal effcency and a new (but famlar) class of ordnal assgnment mechansms. Fshburn, P. C., and Brams, S. J Paradoxes of preferental votng. Mathematcs Magazne 56(4): Fshburn, P. C., and Gehrlen, W. V Borda s rule, postonal votng, and Condorcet s smple majorty prncple. Publc Choce 28(1): Gbbard, A Manpulaton of schemes that mx votng wth chance. Econometrca 45(3): Gross, S.; Anshelevch, E.; and Xa, L Vote untl two of you agree: Mechansms wth small dstorton and sample complexty. In Proceedngs of the 31st AAAI Conference on Artfcal Intellgence (AAAI), Katta, A.-K., and Sethuraman, J A soluton to the random assgnment problem on the full preference doman. Journal of Economc Theory 131(1): Mchal, D Reducng rank-maxmal to maxmum weght matchng. Theoretcal Computer Scence 389(1-2): Mouln, H Condorcet s prncple mples the no show paradox. Journal of Economc Theory 45(1): Mouln, H Far Dvson and Collectve Welfare. The MIT Press. Procacca, A. D Can approxmaton crcumvent Gbbard-Satterthwate? In Proceedngs of the 24th AAAI Conference on Artfcal Intellgence (AAAI), AAAI Press.