What Do We Elect Committees For? A Voting Committee Model for Multi-Winner Rules

Transcription

1 What Do We Elect Commttees For? A Votng Commttee Model for Mult-Wnner Rules Potr Skowron Unversty of Warsaw Warsaw, Poland p.skowron@mmuw.edu.pl Abstract We present a new model that descrbes the process of electng a group of representatves (e.g., a parlament) for a group of voters. In ths model, called the votng commttee model, the elected group of representatves runs a number of ballots to make fnal decsons regardng varous ssues. The satsfacton of voters comes from the fnal decsons made by the elected commttee. Our results suggest that dependng on a sngle-wnner electon system used by the commttee to make these fnal decsons, dfferent mult-wnner electon rules are most sutable for electng the commttee. Furthermore, we show that f we allow not only a commttee, but also an electon rule used to make fnal decsons, to depend on the voters preferences, we can obtan an even better representaton of the voters. 1 Introducton There are many real-lfe stuatons n whch we want to select a subset of a gven set of canddates. Examples of such stuatons nclude selectng a set of moves for an arplane [Skowron et al., 015], choosng locatons for common facltes [Shmoys et al., 1997], short-lstng canddates for hrng n a new team, or electng a parlament. These examples dffer n ther nature and n each of them we mght desre the selected group of canddates to exhbt dfferent propertes. In ths paper we consder the case of selectng a commttee, also referred to as a group of representatves, for the agents. However, even f we restrct our analyss to the concrete case of choosng a commttee, we mght requre dfferent representatves to be elected, dependng on ther rghts and responsbltes n the elected body. Indeed, dfferent vews on what the desred propertes of commttees are have led to nventng a number of multwnner electon rules [Chamberln and Courant, 1983; Monroe, 1995; Fshburn, 1981; Gehrlen, 1985; Brams et al., 007; lgour, 010; Debord, 199]. In ths paper we argue that the desred propertes of mult-wnner rules strongly depend on what we elect the commttee for. One of our man conclusons says that for dfferent sngle-wnner rules used by the commttee to make fnal decsons, dfferent mult-wnner rules should be used to elect the decson-makng commttee. Specfcally, we obtan theoretcal justfcaton for the followng clams. We argue that mult-wnner rules that acheve fully proportonal representaton, such as Chamberln Courant rule [1983] and Monroe rule [1995] are partcularly well-suted for electng commttees that need to make unanmous decsons. Unanmty s often requred n stuatons where makng a wrong decson mples severe consequences, e.g., n case of jures votng on convctons. Indeed, n such cases we are partcularly wllng to ensure that mnortes are represented n the commttees to avod based decsons. Further, our analyss suggests that -top rules,.e., scorng rules [Young, 1975] that select canddates wth the hghest total utltes, are sutable for electng commttees that use the random dctatorshp rule to make decsons. It mght seem that the sgnfcance of ths observaton s compromsed by the low applcablty of randomzed electon rules. Such randomzed decson-makng processes, however, model stuatons where decsons are made by ndvdual members of the commttee but we are uncertan whch ssues wll be consdered by whch ndvduals. As an example of such a group of representatves consder a senor program commttee for a conference, where ndvdual experts makes fnal decsons regardng submtted papers. We observe that the medan OWA rule [Skowron et al., 015] s sutable for electng commttees that make fnal decsons by majorty votng. Informally speakng, n the medan OWA rule a voter s satsfed wth a commttee C f she s satsfed wth at least half of the members of C. We recall the formal defnton of the class of the OWA rules n Secton. We derve the above results by ntroducng and studyng a new formal model that allows us to valdate the applcablty of mult-wnner electon rules. In our model, herenafter referred to as the votng commttee model, we assume that an elected commttee runs a number of ballots, n whch t makes collectve decsons regardng varous ssues. Ths settng corresponds to, e.g., a parlament votng on law acts, a jury votng on convctons, a supervsory board votng on strategc busness decsons, etc. We assume that the ultmate satsfacton of the voters depends solely on the fnal decsons made by the commttee. Consequently, each voter ranks canddates for the commttee accordng to how lkely t s that they vote accordng to her preferences. Ths new model allows to numercally assess qualtes of commttees, dependng on what electon rule these commttees use to make the

2 fnal decsons. Intutvely, the numercal qualty of a commttee C estmates how lkely t s that C makes decsons consstent wth voters preferences. Our work extends the lterature on propertes of multwnner electon rules [Smeone and Pukelshem, 006; Barberá and Coelho, 008; Elknd et al., 014; Azz et al., 015; Gehrlen, 1985; Debord, 199]. Ths lterature ncludes, e.g., the works of Barberá and Coelho [008] and of Azz et al. [015], where the authors defne propertes that good mult-wnner rules should satsfy. Elknd et al. [014] argue that the desrablty of many natural propertes of multwnner rules must be evaluated n the context of ther specfc applcatons. Our paper extends ths dscusson by showng an ntutve model and concrete examples concernng ths model, for whch dfferent mult-wnner rules are partcularly well applcable. Many mult-wnner electon systems are defned as functons selectng such commttees that optmze certan metrcs, usually related to voters satsfacton [Bock et al., 1998; Chamberln and Courant, 1983; Monroe, 1995; Brams et al., 007; lgour, 010; Debord, 199]. These dfferent optmzaton metrcs capture certan desred propertes of electon systems. Our paper complements these works by presentng a specfc metrc that s motvated by the analyss of decsonmakng processes of commttees. Smlarly to Fshburn [1981], we explore the dea of comparng mult-wnner electon systems. Further, our perspectve s conceptually close to the ones gven by Chrstan et al. [007], who study computatonal problems related to lobbyng n drect democracy, where the decsons are made drectly by the voters who express ther preferences n open referenda (there are no representatves). Our model s also close to the one of oryama et al. [013], who alke assume that the elected commttee makes a number of decsons n a sequence of ballots. In such model, they compute the frequency, wth whch the wll of each ndvdual s mplemented. They further make some assumptons on how the socety evaluates the frequency values, to justfy degressve proportonalty, the prncple of proportonal apportonment. Our methodology s also close n sprt to the one gven by Young [1995], who ntroduced probablstc models to valdate optmalty of electon rules. In our of such models the decsons are taken by a group of experts these decsons mght be factually ether correct or wrong. Young shows the rule that s most adequate n such a settng. Fnally, we note that there exsts a broad lterature on votng on mult-attrbute domans, where agents need to make collectve decsons regardng a number of ssues [Brams et al., 1998; Lacy and Nou, 000; Xa et al., 008; 010]. We dffer from these works by consderng an ndrect process of decson-makng, through an elected commttee. Our contrbuton s the followng: () We ntroduce a new formal model that relates voter satsfacton from a commttee to ther satsfacton from the commttee s decsons. In our model we defne the noton of optmalty of a mult-wnner electon rule, gven that a certan sngle-wnner rule s used by the elected commttee to make the fnal decsons. () By studyng the determnstc varant of our model, we fnd optmal mult-wnner rules for three natural sngle-wnner rules: majorty, unanmty, and random dctatorshp. () After relaxng the assumptons regardng the determnsm of the model, we show that the class of OWA rules contans all optmal rules for most natural sngle-wnner electon systems. (v) We ntroduce a defnton of the ndrect electon rule that allows voters to elect not only commttees, but also the snglewnner rules used by these commttees to make decsons. We dscuss the exstence of optmal ndrect rules. Prelmnares In ths secton we frst defne the notaton and basc notons used n the paper. Next, we recall defntons of several known sngle-wnner and mult-wnner electon rules that we use n our analyss..1 Notaton and Basc Defntons For each setx, by1 X we denote the ndcator functon ofx, defned as: 1 X (x) = 1 x X, and1 X (x) = 0 x / X. For smplcty, we wrte1 X nstead of1 X (x) f x s clear from the context. For each fnte set X and for each voter, a utlty functon of over X, s a functon ut : X R that quantfes the level of satsfacton of the voterfrom the elements ofx. We denote the utlty that an agent assgns to a canddate a as u,a = ut (a). We are specfcally nterested n the approval model, where utltes come from the bnary set {0, 1}. In the approval model, we say that a voter approves of a canddate a f u,a = 1. Otherwse we say that dsapproves of a. We do not make any assumptons on where the utlty functons come from. For nstance, the utltes can be provded drectly by the voters, can be extracted from rankngs provded by the voters (cf. the work of Young [1975]), etc. For each two fnte setsx andy,wth Y = y, a utlty profle ofy overx s a vector of y utlty functons. We denote the set of all possble utlty profles ofy overx asπ y (X). For each set X and each N, by P(X) we denote the set of all subsets of X and by P (X) the set of all subsets of X of sze. We refer to the elements of P (X) as to commttees of sze, elected from X. A -mult-wnner electon rule for a set of voters Y, Y = y, and a set of canddatesx s a functonr 1 : Π y (X) P (X) that for a gven utlty profle of Y over X returns a commttee of sze. We denote the set of all -mult-wnner electon rules for sets Y and X as MW(Y, X, ). In general, a sngle-wnner electon rule for a set of voters Y, Y = y, and a set of canddates X s a functon R : Π y (X) X that for a gven utlty profle returns exactly one canddate from X. Thus, sngle-wnner rules can be seen as1-mult-wnner rules. In our further analyss we consder sngle-wnner electon rules for bnary sets of canddates only. We also study randomzed sngle-wnner rules that for each utlty profle, nstead of a sngle canddate from X, return a lottery over X. Thus, herenafter we use the followng defnton. For a set of voters Y, Y = y, and a bnary set of canddates X = {d 0,d 1 }, a (randomzed) sngle-wnner electon rule R s a functonr : {d 0,d 1 } y R, that for each vector of y bnary values, returns the probablty that the value d 0 wll

3 be selected. We denote a set of all randomzed sngle-wnner electon rules for sets Y and X as RSW(Y, X).. Overvew of Sngle-Wnner Rules A unformly random dctatorshp rule selects a canddate d wth probablty proportonal to the number of voters who prefer d over the other canddate. A majorty rule determnstcally selectsd 0 f at least half of the voters preferd 0 overd 1 (ths assumes that tes are resolved n favor ofd 0 ). A unanmty rule returns determnstcally d 0 f and only f all voters prefer d 0 over d 1 (n case of the unanmty rule t s convenent to thnk of votng for d 0 as of votng for pass and of votng ford 1 as of votng forveto)..3 Overvew of Mult-Wnner Rules A sgnfcant part of results that we provde n the paper concerns OWA rules, the broad class of mult-wnner electon systems defned by Skowron et al. [015]. Below, we recall the defnton ths class of rules and descrbe several of them. For each voter, each commttee C, and each number j, 1 j, let u,c (j) denote the utlty of the j-th most preferred canddate from C, accordng to. In other words: {u,c (1),...,u,C ()} = {u,a : a C}, and u,c (1) u,c (). For each voter, each commtteec, and each -element vector α, let α(, C) denote the α-satsfacton of fromc: α(,c) = j=1 α ju,c (j). Theα-rule s a rule that selects a commttee C that maxmzes total α-satsfacton of the voters α(,c). For a gven set of canddatesx, a set of votersy, and a sze of the commttee we want to elect,, let OWA(X,Y,) denote the set of all OWA rules that gven votes ofy overx return commttees of sze : OWA = {α-rule : α = α 1,α,...,α, α 0}. There are several specfcally nterestng OWA rules. For nstance, f α = 1,0,...,0 we get Chamberln Courant rule [1983], where each voter receves satsfacton scores from her most preferred member of the commttee only. If α = 1,1,...,1 we get a -top rule that selects canddates wth the hghest total utlty. If α has 1 on the k-th postons and 0 on the others, then we get a k-medan rule, n whch the satsfacton of a voter from a commttee C s her satsfacton from thek-th most preferred member ofc. 3 The Votng Commttee Model In ths secton we descrbe a votng commttee model that allows one to assess the qualtes of varous mult-wnner electon rules. Let N = {1,,...,n} be a set of voters, and let A = {a 1,a,...,a m } be a set of canddates. We assume there s a set D = {D 1,D,...,D r } of r ssues; each ssue D j s a bnary set consstng of two alternatves D j = {d 0,d 1 }. Voters have strct preferences over the alternatves wthn each ssue; by d j we denote the preferred alternatve from D j from the perspectve of the voter. The ssues mght dffer n ther mportance to dfferent voters. We consder two types of atttudes: a votercan consder an ssue D j ether as mportant or as nsgnfcant. We denote the set of all the ssues mportant for voterasd m () D. In the frst stage n our model, a commtteec of sze s selected through a mult-wnner electon ruler 1 ; the selected commttee conssts of members. In the second stage, the commttee C runs r ndependent ballots, for each ballot usng the same sngle-wnner electon ruler. In the-th ballot, 1 r, the commttee makes a collectve decson regardng ssued ford the commttee can make ether decson d 0 or decsond 1. The fnal outcome of ths two-stage process s descrbed by a vector ofr decsons. Intutvely, the frst stage of electons mght correspond to e.g., parlamentary electons, electons for supervsory or faculty board, etc. The second stage mght be vewed as e.g., a sequence of parlamentary ballots regardng varous matters, such as fnancal and monetary economcs, educaton poltcs, changes n natonal health-care system, etc. The ultmate satsfacton of voter depends solely on the fnal outcome of the r ballots. Nevertheless, the voters need to select the commttee members frst and, thus, they need a way of judgng the qualtes of the commttees. Intutvely, voter consders commttee C as good, f for s mportant ssues,.e., ssues fromd m (),C s lkely to make decsons consstent wth s preferences. More formally, we ntroduce the followng probablstc model. For each voter N and each canddate a A we set p,a, the probablty of representaton ofbya, to be the probablty that a for each s mportant ssue D j D m () wll vote for d j,.e., accordng to s preferences. For the sake of smplcty of notaton, we also use q,a = 1 p,a. We assume that the voters know these probabltes, and that n the consdered utlty functons the values that voters assgn to the canddates are functon of the probabltes of representaton only. More formally, we assume there exsts an njectve functon P : R R, such that a voter assgns utlty u to the canddatea, f and only fu = P(p,a ). We are specfcally nterested n the approval model, where there are only two allowed values of utltes, and thus two values of probabltes of representaton: p = P 1 (1) and q = P 1 (0). As a dstngushed specal case of the approval model, we consder a varant n whch the values of probabltes of representaton come from the bnary set{0,1} (p = 1 and q = 0). In such case, herenafter referred to as the determnstc model, for each voter there are two types of canddates: the canddates that perfectly represent,.e., determnstcally vote accordng to s preferences n all ssues mportant for (and these canddates are approved of by ) and the others that perfectly msrepresent,.e., determnstcally vote contrarly to s preferences (and so, are dsapproved of). For each voter, each mportant ssue D j D m () for, each commttee C, and each commttee vote v Dj, let P C (v) denote the probablty that members ofc cast votev: P C (v) = ( ) 1 v[j]=d jp,a +1 v[j] d jq,a. a C Let R : {d 0,d 1 } R be a randomzed sngle-wnner electon rule used by the commttee to make decsons over ssues. By P R ( v) we denote the probablty that the rule R, gven votev, makes decsond j : P R ( v) =1 d0=d j R (v)+1 d1=d j (1 R (v)).

4 For each commtteec, each voter, and each mportant ssue D j D m (), by P C,R () we denote the probablty that C makes decson consstent wth s preferences: P C,R () = P C (v)p R ( v). v D j Now, we can defne the expected ultmate satsfacton of a voter from the commttee C, as P C,R () ths s the expected fracton of ssues mportant for, for whch the commttee C would make decsons consstent wth s preferences. Fnally, we can defne the central noton of ths paper. Defnton 1. LetR be a sngle-wnner electon rule used by the commttee to make fnal decsons, and let denote the sze of the commttee to be elected. A commttee C s optmal n the utltaran sense f: C = argmax C A: C = P C,R (). (1) Analogously, we can defne commttees optmal n the egaltaran sense, by replacng sum wth max n Defnton 1. For the sake of concreteness, n ths paper we focus on the utltaran case and refer to the optmalty n the utltaran case as to the optmalty. 4 Optmalty for nown Sngle-Wnner Rules In ths secton we show that several known mult-wnner electon rules can be vewed as optmal n our votng commttee model. Each such a rule s optmal for dfferent sngle-wnner electon system R, used by the selected commttee to make fnal decsons. Thus, our results gve ntuton regardng the applcablty of dfferent mult-wnner electon rules, dependng on for what knd of decson makng the commttee s elected for. Defnton. Let R be a sngle-wnner electon rule used n the second stage of the electon model. A mult-wnner electon rule R 1 s optmal for R f for each preferences of voters t elects an optmal commttee. For mult-wnner approval rules and determnstc model, we wll add the addtonal word determnstcally to the noton of optmalty. We start wth presentng four theorems characterzng four mult-wnner electon rules n whch voters express ther preferences by approvng subsets of canddates. These rules are optmal for majorty, random dctatorshp, and unanmty sngle-wnner electon systems. Theorem 1. Assume s odd. The +1 -medan electon system s determnstcally optmal for the majorty rule. Proof. We calculate P C,R (), the ultmate satsfacton of a voter from a commttee C, assumng C uses majorty rule to make fnal decsons. Snce the model s fully determnstc, and a commttee member a C votes accordng to s preferences f and only f a s approved by, a commttee C makes decsons consstent wth s preferences, f t contans at least +1 members approved by. Thus, the satsfacton P C,R () offromc s equal to1f C contans at least +1 members approved by, otherwse t s equal to 0. The same formula defnes satsfacton of from C n the +1-medan electon system. Fnally, we note that n the +1 -medan electon system, the commttee that maxmzes voters total satsfacton s selected, whch completes the proof. Theorem. The -approval electon system s optmal for the unformly random dctatorshp rule. Proof. For a commtteec, let apprv (C) denote the number of canddates from C that are approved of by. Let p and q denote the probablty of representaton of by canddates approved of and dsapproved of by, respectvely. Naturally, p > q. For each mportant for ssue D j D m () and each commttee C let P(C, a) denote the probablty that, durng the unformly random dctatorshp ballot regardng D j, the fnal decson wll be made by the commttee membera. Naturally, P(C,a) = 1. The probablty that C wll vote accordng to s preferences s equal to: P C,R () = a CP(C,a) (p1 approves of a +q1 dsapproves of a ) = apprv (C) = apprv (C) p q +q. p +( apprv (C)) q Consequently, a commtteec s optmal f t has the maxmal value apprv (C). Exactly such commttee s elected by the-approval electon system. Theorems 1, and are qute powerful n a sense that they clam the optmalty of the +1 -medan and -approval mult-wnner electon rules, for the correspondng snglewnner rules, rrespectvely of the preferences of the voters over the ssues. Unfortunately, ths s not always the case, whch s expressed by the followng two theorems. Before we proceed further, we ntroduce two new defntons that descrbe two extreme classes of voters preferences over the ssues. The ntutve meanng of these two classes can be better expressed f we rename the alternatves d 0 and d 1 to pass and veto, respectvely. We say that voters are rejecton-orented f for each voter, and eachd j D m (), d j = d 1 (after renamng, d j = veto), meanng that each voter gets utlty only from rejectng mportant ssues. Analogously, we say that the voters are acceptance-orented f for each voter, and eachd j D m (),d j = pass. Theorem 3. For rejecton-orented voters, Chamberln Courant system wth approval votes s determnstcally optmal for the unanmty system. Proof. Here, the reasonng s very smlar to the proof of Theorem 1. The ultmate satsfacton of a voter from a commttee C, P C,R (), s equal to 1 f C contans at least one canddate approved by, and s equal to 0 otherwse. Ths s equvalent to the defnton of voters satsfacton n the Chamberln Courant electon rule, whch completes the proof.

5 Theorem 4. For acceptance-orented voters, the -medan system wth approval votes s determnstcally optmal for the unanmty system. Theorems 3 and 4 suggest that Chamberln Courant s rule and -medan rule are sutable for electng commttees that have veto rghts. One should choose one of them dependng on the voters satsfacton model. For nstance, f passng a wrong decson has much more severe consequences than rejectng a good one (whch s captured by modelng the voters as rejecton-orented), a commttee should be selected wth Chamberln Courant s rule, and t should use unanmty rule to make fnal decsons. If passng a wrong decson has relatvely low cost compared to rejectng a good one, the commttee should be selected wth a -medan rule Below, we show a more general result that characterzes a class of optmal electon systems for most natural snglewnner electon rules R : {0,1} R. These natural rules are those whch satsfy neutralty, anonymty and monotoncty (for the defntons of these propertes we refer the reader to the work of Endrss [015]). Intutvely, the outcomes of such rules depend on the number of votes gven to certan alternatves only. Defnton 3. A nondetermnstc rule R : {0,1} R s normal f there exsts a non-decreasng functonr n : N R such that R (v) = R n ( { : v[] = 0} ), and such that for eachj, R n (j) = 1 R n ( j). In Defnton 3, the requrement that for each j, R n (j) = 1 R n ( j) enforces that the probablty of selectng the value 0 gven j votes for 0 s equal to the probablty of selectng1, gvenj votes for1, thus, t follows from neutralty. Theorem 5. For each normal electon system R, n the approval model there exsts a -element vector α, such that α-rule wth approval votes s optmal forr. Proof. LetR n be as n Defnton 3. LetC l, denote the commttee that has exactly l members approved of by, and let P(,s,l) be the probablty that exactly s members of C l, wll vote accordngly to s preferences. We have: P(,s,l) = 1 x l 1 x s 1 s x l x=1 ( ) ( ) l l p x q l x q s x p l s+x. x s x We can see that P(,s,l) does not depend on. Further, let P R ( s) be the probablty that the rule R makes decson consstent wth s preferences, assumng s members of C l, votes accordngly to s preferences. P R ( s) = R n (s)1 d j +(1 R n( s))1 =dj,0 d j =dj,1. Snce ether d j = d j,0 or d j = d j,1 and snce R n (s) = (1 R n ( s)), we get that P R ( s) = R n (s), and thus P R ( s) does not depend on. Consequently, for each mportant forssued j D m () we can calculate the expected ultmate satsfactonp Cl,,R () of a voterfrom a commttee C l,, so that ths value does not depend on: P Cl,,R () = P(,s,l)P R ( s). s=1 BecauseP Cl,,R () does not depend on, we wll denote t as P Cl,R. Naturally, sncep q, we havep Cl+1,R P Cl,R, for eachl. Now, we can see that the followng vector: α = P C1,R, (P C, R P C1,R ), (P C3,R P C,R ),...,(P C,R P C 1,R ) satsfes the requrement from the thess. Indeed, n the α- rule the satsfacton of a voter from a commttee wth l approved members s the sum of frst l coeffcents of α, whch sp Cl,R. Ths completes the proof. The above theorem can be vewed as an evdence of the expressveness and power of OWA electon rules. Unfortunately, OWA electon rules are not suffcently expressve to descrbe the non-approval model. To get characterzaton smlar to the one gven n Theorem 5, but for arbtrary utltes, we would need to consder a more general class of electon rules, that s rules n whch the satsfacton of a sngle voter from a commttee C s expressed as a lnear combnaton of products of utltes from ndvduals: C C a C u (a) (n contrast to a lnear combnaton of utltes only). Such rules that consder nseparable commttees were consdered e.g., by Ratlff [006; 003]. Nevertheless, for the case of arbtrary utltes, we can get a result smlar to the one gven n Theorem. Theorem 6. There exsts a utlty functon P : R R transformng voters utltes to representaton probabltes, such that -top electon system s optmal for the unformly random dctatorshp rule. Proof. Let us defne P(C,a) as n the proof of Theorem. The probablty that C wll vote accordng to s preferences s equal to: P C,R () = P(C,a) p,a = 1 p,a. a C a C Let P be the dentty functon. The rule that maxmzes the total score of selected canddates, also maxmzes ther total expected ultmate satsfacton. Ths completes the proof. 5 Optmalty of Indrect Electon Rules In the prevous secton we studed the optmalty of multwnner electon rules, gven nformaton on what snglewnner electon system the commttee wll use to make fnal decsons. In ths secton we show that n our probablstc model we can compare qualtes of the pars of mult-wnner and sngle-wnner systems. Thus, our results not only suggest whch mult-wnner electon rule s sutable for electng a commttee, but also ndcates whch sngle-wnner rule should be used by the commttee to make fnal decsons. We start wth ntroducng the defnton of the ndrect electon rule. Ths defnton uses an nterestng concept: t allows the voters to elect not only commttees, but also sngle-wnner rules that these commttees use to make fnal decsons. Defnton 4. An ndrect electon rule s a functon F : Π n (N) P(A) SW that for each utlty profle π n Π n (N) returns a par F(π n ) = (C,R ), where C P(A)

6 s the elected commttee, andr SW s the sngle-wnner electon rule that C wll use when makng fnal decsons. We use the notatonf(π n )[1] = C andf(π n )[] = R. Defnton 5. LetP : R R be a functon transformng voters utltes to representaton probabltes. An ndrect electon rule F weakly domnates an ndrect electon rule G for P, f for each utlty profle π n Π n (N), F returns a commttee and a sngle-wnner rule that gves the total expected ultmate satsfacton of the voters at least as hgh as the one gven by a commttee and a sngle-wnner rule returned byg: P F(πn )[1],F(π n )[]() P G(πn )[1],G(π n )[](). () F strongly domnatesg forp f t weakly domnatesg and f there exsts profleπ n Π n (N) for whch: P F(πn )[1],F(π n )[]() > P G(πn )[1],G(π n )[](). (3) An ndrect electon rule F s optmal forp f t weakly domnates every other ndrect electon rule. Defnton 6. For a mult-wnner ruler 1 and a sngle-wnner rule R, by R 1 followed by R we call the ndrect functon that selects commttee usngr 1 and, ndependently of voters preferences, always usesr to make fnal decsons. We show how to apply Defnton 5 n perhaps the smplest varant, that s n the determnstc approval model. Frst, we defne a new electon rule COMB as a combnaton of +1 - medan rule followed by majorty rule wth the -approval rule followed by unformly random dctatorshp rule. Defnton 7. The rule COMB s defned as follows. Let C and C be commttees elected by -approval rule and by +1 -medan rule, respectvely. Let apprv be the total approval score of C and let owa be the total OWA score of C. If apprv > owa, then COMB returns the par (C, unformly random dctatorshp). Otherwse, COMB returns the par (C,majorty). In our determnstc model, the new rule COMB strongly domnates both rules that t s derved from. Proposton 1. In the determnstc model, COMB strongly +1 domnates -medan followed by majorty, and - approval followed by unformly random dctatorshp. Proof. Let apprv andowa be defned as n Defnton 7. Repeatng the analyss from the proofs of Theorems 1 and, we get that the total ultmate satsfacton of voters under +1 -medan followed by majorty s equal toowa, underapproval followed by unformly random dctatorshp s equal to apprv, and under COMB s equal tomax(apprv,owa). Ths completes the proof. The natural queston s whether we can fnd an optmal rule n our general probablstc model. Unfortunately, ths seems unlkely, as the socetal qualty of a rule also depends on the underlyng voters utlty model. For nstance, f for each j, D j = {pass,veto}, and the voters are rejecton-orented, they prefer rules n whch commttees can more easly make a collectve decson veto. In fact, for the rejecton-orented voters we can show that any mult-wnner rule followed by the constant sngle-wnner rule that always returns veto s optmal. Such rules, on the other hand, are hghly dslked by the acceptance-orented voters. Interestngly, f we restrct ourselves to normal sngle-wnner rules to make fnal decsons, and to the determnstc model, then there exsts an optmal ndrect rule. Theorem 7. In the determnstc model, f we consder only normal sngle-wnner rules used for makng the fnal decsons, then there exsts an optmal ndrect electon rule. Proof. The proof s constructve. From Defnton 3 we recall that a normal sngle-wnner rule R can be descrbed by values R n (1),...,R n (). We use notaton from the proof of Theorem 5. In the determnstc model P(,s,l) = 1 s=l. Consequently, for a commtteec l wthlapproved members, we havep Cl,R = R n (l). Let1 l, (C) denote a functon that returns 1 f C contans l elements approved of by, and 0 otherwse. For a gven commttee C we can fnd an optmal sngle-wnner electon ruler by solvng the followng lnear program: maxmze P C,R () = subject to: (a) : R n (l+1) R n (l), (b) : 0 R n (l) 1, (c) : R n (l) = 1 R n ( l), R n (l)1 l, (C) 1 l 1 1 l 1 l The optmal ndrect rule tres all commttees and selects such that gves the best soluton to the nteger program. From the soluton of the nteger program we can extract values R n (1),...,R n () that descrbe the optmal sngle-wnner electon rule that should be used to make fnal decsons. 6 Concluson We defned a new model, called the votng commttee model, whch explores scenaros where a group of representatves s elected to make decsons on behalf of the voters. Ths model lnks utltes of the voters from the elected commttee to ther utltes from the commttee s decsons regardng varous matters. Intutvely, a satsfacton of voter from a commttee C s proportonal to the probablty that for the ssues mportant for, C s decsons are consstent wth s preferences. Most of our results are for a restrcted verson of the model, called the approval model, n whch each voter dstngush only two types of canddates: the canddates who represent her well and those who do not. Our man concluson says that dfferent mult-wnner electon systems are sutable for choosng commttees, dependng on the rule used by the elected commttee to make decsons. In partcular, we justfed whch known mult-wnner rules are sutable for whch sngle-wnner ones. Further, under some smplfyng assumptons, we showed that the class of OWA rules contans all optmal mult-wnner electon systems. We ntroduced the noton of an ndrect rule, n whch voters elect a decsve commttee along wth the sngle-wnner

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