Colitionl Mnipultion for Shulz s Rul Srg Gsprs UNSW n NICTA Syny, Austrli srgg@s.unsw.u.u Thoms Klinowski Univrsity of Rostok Rostok, Grmny thoms.klinowski@unirostok. Toy Wlsh NICTA n UNSW Syny, Austrli toy.wlsh@nit.om.u Nin Nroytsk NICTA n UNSW Syny, Austrli nin.nroytsk@nit.om.u ABSTRACT Shulz s rul is us in th ltions of lrg numr of orgniztions inluing Wikimi n Din. Prt of th rson for its populrity is th lrg numr of xiomti proprtis, lik monotoniity n Conort onsistny, whih it stisfis. W intify potntil shortoming of Shulz s rul: it is omputtionlly vulnrl to mnipultion. In prtiulr, w prov tht omputing n unwight olitionl mnipultion (UCM) is polynomil for ny numr of mnipultors. This rsult hols for oth th uniqu winnr n th o-winnr vrsions of UCM. This rsolvs n opn qustion in [14]. W lso prov tht omputing wight olitionl mnipultion (WCM) is polynomil for oun numr of nits. Finlly, w isuss th rltion twn th uniqu winnr UCM prolm n th o-winnr UCM prolm n rgu tht thy hv sustntilly iffrnt nssry n suffiint onitions for th xistn of sussful mnipultion. Ctgoris n Sujt Dsriptors I.2. [Distriut Artifiil Intllign]: Multignt Systms; F.2 [Thory of Computtion]: Anlysis of Algorithms n Prolm Complxity Gnrl Trms Eonomis, Thory Kywors soil hoi, voting, mnipultion 1. INTRODUCTION On importnt issu with voting is tht gnts my st strtgi vots inst of rvling thir tru prfrns. Gir [] n Sttrtwhit [15] prov tht most voting ruls r mnipull in this wy. Brtholi, Tovy n Trik [3] suggst omputtionl omplxity my nvrthlss t s rrir to mnipultion. Intrstingly, it is NP-hr to omput mnipultion for mny ommonly us voting ruls, inluing mximin, rnk pirs [17], Apprs in: Proings of th 12th Intrntionl Confrn on Autonomous Agnts n Multignt Systms (AAMAS 2013), Ito, Jonkr, Gini, n Shhory (s.), My, 6, 2013, Sint Pul, Minnsot, USA. Copyright 2013, Intrntionl Fountion for Autonomous Agnts n Multignt Systms (www.ifms.org). All rights rsrv. Bor [6, 4], 2n orr Copln, STV [2], Nnson n Blwin [13]. A rnt survy on omputtionl omplxity s rrir ginst mnipultion in ltions n foun in []. W stuy hr th omputtionl omplxity of mnipulting Shulz s voting rul, whih is rguly th most wispr Conort voting mtho in us toy. Shulz s rul ws propos y Mrkus Shulz in 17, n ws quikly opt y mny orgniztions. It is, for xmpl, us y th Annox Assoition, Blitz, Cssnr, Din, th Europn Dmorti Eution Confrn, th Fr Softwr Fountion, GNU Privy Gur, th Hskll Logo Comptition, Knight Fountion, MTV, No, Opn Stk, th Pirt Prty, RLL- MUK, Sugr Ls, TopCor, Uuntu n th Wikimi Fountion. In ition to ing Conort onsistnt, Shulz s rul stisfis mny othr sirl xiomti proprtis, inluing Prto optimlity, monotoniity, Conort losr ritrion, inpnn to lons, rvrsl symmtry n th mjority ritrion. Shulz s rul is known y svrl othr nms inluing th Btpth Mtho n Pth Voting. Th mtho n sn s th invrs prour to nothr Conort onsistnt voting mtho, rnk pirs. Th rnk pirs mtho strts with th lrgst fts n uss s muh informtion out ths fts s it n without rting miguity. By omprison, Shulz s rul rptly rmovs th wkst ft until miguity is rmov. Shulz s rul hs numr of intrsting omputtionl proprtis. Whilst it is polynomil to omput th winnr of Shulz s rul, it rquirs fining pths in irt grph ll with th strngth of fts. Suh pths n foun using vrint of th ui tim Floy-Wrshll lgorithm [1]. Mor rntly, Prks n Xi initit th stuy of th omputtionl omplxity of mnipulting this voting rul [14]. Thy prov tht in th uniqu winnr UCM prolm it is polynomil for singl gnt to omput mnipulting vot if on xists. Thy lso invstigt th vulnrility of Shulz s rul to vrious typs of ontrol. Howvr, thy lft th omputtionl omplxity of UCM with mor thn on mnipultor s n opn qustion. In this ppr, w ontinu this stuy n show tht UCM rmins polynomil for n ritrry numr of mnipultors. For usrs of Shulz s rul, this rsult hs oth positiv n ngtiv onsquns. On th ngtiv si, this mns tht th rul is omputtionlly vulnrl to ing mnipult. On th positiv si, this mns tht whn liiting vots, w n omput in polynomil tim whn w hv ollt nough vots to lr th winnr. Our rsults lso highlight th importn of istinguishing rfully twn mnipultion prolms whr w r looking for singl winnr ompr to o-winnrs.
0 \ 0 \ 40 0-0 40-40 40-40 40 0-40 40-20 - 0 - - - - Figur 1: Th wight mjority grph G P n th tl of S P(x, y), x, y {,,,, } from Exmpl 1. Figur 2: Th WMG n th tl of S P NM PM(x, y), x, y {,,,, } from Exmpl 2. 2. DEFINITIONS Voting systms. Consir n ltion with st of m nits C = { 1,..., m}. Avot is spifi y totl strit orr on C: i1 i2 im.ann-gnt profil P on C onsists of n vots, P =( 1, 2,..., n). Shulz s voting rul. Givn n n-gnt profil P on C, Shulz s rul trmins st of winnrs W P Cs follows. 1. For nits x y, ltn P (x, y) not th numr of gnts who prfr x ovr y, i.. th numr of inis i with x i y. 2. Th wight mjority grph (WMG) is irt grph G P whos vrtx st is C, n with n r of wight w P (x, y) = N P (x, y) N P (y,x) for vry pir (x, y) of istint nits. W not WMG ssoit with profil P y (G P,w P ). 3. Th strngth of irt pth π =(x 1,x 2,...,x k ) in G P is fin to th minimum wight ovr ll its rs, i.. w P (π) = wp (xi,xi+1). min 1 i k 1 4. For nits x n y, lts P (x, y) not th mximum strngth of pth from x to y,i.. S P (x, y) =mx{w P (π) : π is pth from x to y in G P }. A pth from x to y is ritil pth if its strngth is S P (x, y). 5. Th winning st is fin s W P = {x C : y C\{x} S P (x, y) S P (y,x)}. If S P (x, y) >S P (y,x) for two nits x, y, thn w sy tht x omints y. Thus, W P is th st of non-omint vrtis. Th winning st is lwys non-mpty [16]. Not tht ll wights w P (x, y), (x, y) G P r ithr o or vn, pning on th siz of th profil P. Convrsly, for ny wight igrph whr ll wights hv th sm prity, orrsponing profil n onstrut [7]. In th litrtur, for xmpl, [17] n [14] rfr to this s MGrvy s trik [12]. W us this rsult hr s w fin th non-mnipultors profil y thir wight mjority grph inst of y thir vots. EXAMPLE 1. Consir n ltion with 5 ltrntivs {,,,, }. Th wight mjority grph G P is shown in Figur 1. W omit rs with zro or ngtiv wight for lrity. Th tl shows vlus S P (x, y), x, y {,,,, }. As n sn from th tl, S P (, x) > S P (x, ), for ll x {,,, }. Hn, th winning st ontins singl ltrntiv W P = {}. Strtgi hvior. W istinguish twn gnts tht vot truthfully n gnts tht vot strtgilly. W ll th lttr mnipultors. W us th suprsript NM to not th nonmnipultors profil n th suprsript M to not th mnipultors profil. Th o-winnr unwight olitionl mnipultion (UCM) prolm is fin s follows. An instn is tupl (P NM,,M), whrp NM is th non-mnipultors profil, is th nit prfrr y th mnipultors n M is th st of mnipultors. W r sk whthr thr xists profil P M for th mnipultors suh tht W P NM P M.Thuniqu winnr UCM prolm is vrint of th o-winnr UCM whr w r looking for mnipultion suh tht {} = W P NM P M. Th wight olitionl mnipultion (WCM) is fin similrly, whr th wights of th gnts (oth non-mnipultors n mnipultors) r intgrs n r lso givn s inputs. 3. WEIGHTED COALITIONAL MANIPU- LATION W onsir th o-winnr WCM prolm for Shulz s voting rul. W show tht if thr xists sussful mnipultion P M thn thr xists sussful mnipultion P M whr ll mnipultors vot intilly. W prov this rsult in two stps. First, w onstrut kin of irt spnning tr of th WMG G P NM P M root t, whih givs us ritil pth from to ll othr ltrntivs. Thn, y trvrsing this tr, w uil nw linr orr of nits tht spifis vot for ll mnipultors. EXAMPLE 2. Consir th WMG G P from Exmpl 1. Suppos tht P orrspons to th non-mnipultors profil, so tht P NM = P. Suppos w hv 4 mnipultors with wights, 3, 2 n 5 tht vot in th following wy: th first thr mnipultors vot n th lst mnipultor vots. Hn, th totl wight of th vot in P M is 15 n th totl wight of th vot in P M is 5. Th upt WMG G P NM P M n th orrsponing tl tht shows th vlus of pirwis mximum strngths r shown in Figur 2. Not tht th ltrntiv is non-omint s wll s ltrntivs {,, }. Hn, th winning st W P NM P M = {,,, }. W show tht givn ny profil P, winning nit W P n sust P 0 of th st of vots,.g. P 0 = P M, w n moify th vots in P 0 to ll th sm, n is still in th winning st of th rsulting profil P. To o this, w onstrut vot Λ=( 1 m 1) suh tht is still winnr if w rpl vry vot in P 0 y Λ. Hn, in th ontxt of th mnipultion prolm w n think of P s P NM P M n P 0 s P M. An out-rnhing T of irt grph G root t vrtx r is onnt spnning suigrph of G in whih r hs in-gr 0 n ll othr vrtis hv in-gr 1. LEMMA 1. Lt G = G P th igrph ssoit with th givn profil P. Thr xists n out-rnhing T root t in G
Stp 2 Stp 1 Stp 3 Stp 4 0 Stp 1 ( ) () Stp 2 ( ) > > > > () Figur 3: () Th out-rnhing root t tht is prou y Algorithm 1 n th orrsponing ritil -x-pths, x {,,, }; () Th Λ orring onstrut y Algorithm 2. suh tht for vry nit th uniqu pth from to in T is ritil - -pth in G. PROOF. W onstrut n out-rnhing T of G y Algorithm 1. At th initil stp th lgorithm mks th root of T. At h stp,wnwvrtxx, x V (G) \ V (T ), tothtrt iff th r (x, y),y V (T ) hs mximum vlu w(x, y) mong ll rs (x,y ), x V (G) \ V (T ),y V (T ). Algorithm 1 Out-rnhing onstrution Input: wight igrph (G, w) =(G P,w P ) n istinguish nit. Initiliz F 1 = {}, X 1 = C\{} n T 1 = {}. for i =1,...,m 1 o D =mx{w(x, y) : x F i,y X i} Choos F i n X i with w(, ) =D F i+1 = F i {} X i+1 = X i \{} T i+1 = T i {(, )} rturn T = T m Clrly, Algorithm 1 rturns n out-rnhing us th input igrph is omplt. So w just hv to show tht it stisfis th rquir proprty. W o this y inution on th siz of T. For i =1th lim is ovious, so ssum 1 <i<m, n lt thvrtxinstpi, i.. {} = F i+1 \ F i. Lt π = ( = 0, 1,..., k 1, k = ) th --pth in T,nltj th inx of th first r on tht pth rlizing its strngth, i.. j = min{t : w( t, t+1) = w(π)}. Lt q th stp in whih th r ( t, t+1) is to T. Now suppos tht thr is --pth π =(, f 1,...,f r,...,) in G with w(π ) >w(π). Bus π n π T, thr xists som r (f r,f r+1) π with f r F q n f r+1 X q. Thn, w(π) =w( t, t+1) w(f r,f r+1) w(π ), ontriting th ssumption n thus onluing th proof. EXAMPLE 3. Figur 3() shows th out-rnhing for G P NM P M n ritil -x-pths, x {,,, }, of th WMG from Exmpl 2. Consir, for xmpl, th pth (,, ) in th out-rnhing. This pth hs strngth n it orrspons to th mximum strngth --pth in G P NM P M. LEMMA 2. Lt G = G P th grph ssoit with th givn profil P n lt T n out-rnhing root t s in Lmm 1. Thn thr xists n orring Λ=( 1 m 1) on th st of nits with th following proprtis. 0 Proprty 1: For h i th uniqu - i-pth in T rspts th orring Λ, i.. itisofthform(, j1,..., jk = i) with 1 j 1 <j 2 < <j k. Proprty 2: Th strngth of ritil pth from i, i [1,m) to is noninrsing long th orring Λ: S P ( i,) S P ( j,) for 1 i<j m 1. Th intuition for Proprty 1 is tht th strngth of h ritil pth from to i, i [1,m) os not rs if w hng ll vots in P 0 to Λ. PROOF. Algorithm 2 rturns totl orr on th st of nits. Th lgorithm trvrss th out-rnhing T otin y Algorithm 1. At h stp, w intify vrtx x with th lrgst vlu of th strngth S P (x, ). Thn w fin th pth π from to x in T whih is ritil pth y Lmm 1. A prfix of th pth π might to Λ t this point. Hn, w only fous on th suffix of π tht os not ontin vrtis to Λ. Thn w th vrtis in this suffix of π to Λ in th orr in whih thy ppr in π. W trmint whn Λ is totl orr ovr ll ltrntivs. Algorithm 2 Constrution of th orring Λ Input: wight igrph (G, w) =(G P,w P ), istinguish nit n th out-rnhing T with root from Algorithm 1. Initiliz Λ=(), X = C\{} whil X o D =mx{s P (x, ) : x X} Lt X ny vrtx with S P (, ) =D. Lt π th uniqu --pth in T. A th vrtis in π X to Λ in th orr in whih thy ppr on π. Upt X := X \ π. rturn Λ W show tht it stisfis th two proprtis y inution on th lngth of Λ. Forth initilλ =() it is oviously tru. So suppos w r in th whil loop ing π X for --pth π =( = g 0,g 1,...,g k,). Not tht π X is suffix of π, i..π X = {g j+1,...,g k,} for som j. To s this, lt g j th lst vrtx on π tht is lry in Λ. Thn y onstrution, ll th vrtis g 1,g 2,...,g j hv n to Λ in th stp in whih g j ws or rlir. By th inution hypothsis th -g j-pth in T rspts Λ, n us th suffix g j+1,...,g k, is to Λ n g j+1,...,g k, is su-pth of π, th onition of Proprty 1 is stisfi for ll ths vrtis. Nxt w osrv tht S P (g t,) S P (, ) for t = j +1,...,k. To s this, lt π n --pth of strngth S P (, ). W hv w(g t,g t+1) S P (, ) S P (, ) for ll t, whr th first inqulity is tru us (g t,g t+1) is n r on th --pth in T whih is ritil pth, n th son inqulity us is winnr. Thus th ontntion of g t,g t+1,...,g k, n π provis g t--pth of strngth S P (, ). Now Proprty 2 follows from th osrvtion tht S P (x, ) S P (, ) for ll x X \ π whih follows from th mximlity onition in th stp whr is hosn. EXAMPLE 4. W onstrut n orring Λ s on th outrnhing otin in Exmpl 3. Th ltrntivs {, } r suh tht S P NM P M (, ) = S P NM P M (, ) = mx{s P NM P M (x, ) : x {,, }}. W rk th ti
20 20 120 20 \ - 120 - - - - Figur 4: An upt WMG n th tl of S P NM PM(x, y), x, y {,,,, } from Exmpl 4 using th Λ orring onstrut y Algorithm 2. twn n ritrrily n slt. Hn, w uil prtil orr. Th nxt ltrntivs tht w onsir r {, } s S P NM P M (, ) = S P NM P M (, ) = mx{s P NM P M (x, ) : x {, }}. W slt n th suffix to th prtil orr, so tht w gt Λ=( ). Hn, 4 mnipultors n vot with rspt to Λ. Figur 3() shows th xution of Algorithm 2. Figur 4 shows th nw WMG n th orrsponing tl of mximum strngths. It is sy to s tht is still winnr ftr th mnipultors hng thir vots. For our givn profil P n istinguish nit, w onstrut n orring Λ s sri in th proof of Lmm 2. THEOREM 1. Lt P ny profil with nit in th winning st, lt P 0 P ny suprofil, n st P 1 = P \ P 0.Lt P th profil givn y P = P 1 P 0 i=1 {Λ}, whrλ is th orring onstrut in Lmm 2. Thn isstill in th winning st W P. PROOF. Dnot th WMGs ssoit with th two profils y (G, w) =(G P,w P ) n (G,w )=(G P,w P ). W rll tht w us th out-rnhing T with root otin y Algorithm 1. Th thorm is s on th following two lims. CLAIM 1. For h pth π in T strting from th strngth of π os not rs in th grph G,i..w (π) w(π). By onstrution of Λ, whvw (x, y) w(x, y) for vry r (x, y) T, n this implis Clim 1. CLAIM 2. For vry --pth π, th strngth of π in G os not x th strngth of ritil --pth in G, i.. w (π) S P (, ). To prov Clim 2, ssum, for th sk of ontrition, tht is vrtx suh tht thr is n --pth π =( = 1,..., k = ) with w (π) >S P (, ), n w.l.o.g. w ssum tht for ll i-pths σ, 1 i k 1, whvw (σ) S P ( i,). Bus is winnr with rspt to P, π must ontin n r (x, y) of wight w(x, y) suh tht w(x, y) S P (, ). Lt(, ) =( i, i+1) th first r with this proprty, i.. i =min{j : w( j, j+1) S P (, )}. Nxt w show th hin of inqulitis w (π) (1) >S P (, ) (2) S P (, ) (3) S P (, ) (4) S P (, ) (5) w (π), whih is ontrition n thus provs th lim. Th following rgumnts for th singl inqulitis ov r illustrt in Figur 5. (1) By ssumption. 0 i= 30 40 30 i+1= Figur 5: A igrm illustrting th rgumnts for stps (2),(3) n (4) of th inqulity hin in th proof of Clim 2. (2) As is winnr for P,vry--pth must ontin n r (x, y) with w(x, y) S P (, ). By th hoi of, w know tht (, ) is th first r suh tht w(, ) S P (, ). Hn, th strngth of th --pth is grtr thn th strngth of th --pth, S P (, ) > S P (, ). Now from S P (, ) min{s P (, ),S P (, )} it follows tht S P (, ) S P (, ). (3) From th ssumption w (π) > S P (, ) it follows tht w (, ) > w(, ) whih implis tht oms for in th orring Λ, n thn th inqulity (3) follows from Lmm 2. (4) By ssumption, w (σ) S P (, ) for ll --pths σ, hn S P (, ) S P (, ). (5) Lt π 1 th --supth of π. ThnS P (, ) w (π 1) w (π). Togthr, Clims 1 n 2 prov th thorm. COROLLARY 1. Th o-winnr WCM prolm for Shulz s rul is polynomil if th numr of nits is oun. PROOF. As th numr of nits is oun w n numrt ll possil istint vots in polynomil tim. From Thorm 1 it follows tht it is suffiint to onsir mnipultions whr ll mnipultors vot intilly. 4. UNWEIGHTED COALITIONAL MA- NIPULATION In this stion w prsnt our min rsult: o-winnr UCM is polynomil for ny numr of mnipultors. This loss n opn qustion ris in [14]. By Thorm 1, (P NM,,M) is Ysinstn for o-winnr UCM if n only if thr is vot suh tht W P NM P M whr vots in P M orrspons to. It rmins to i if suh vot xists. As in th wight s, w not (G, w) =(G P,w P ) n (G,w ) = (G P,w P ) th WMGs of th voting profils P = P NM n P = P NM P M with r wight funtions w n w, rsptivly, n S P (x, y) nots th mximum strngth of pth from x to y in G. First, w giv high-lvl sription of th two-stg lgorithm. In th first stg, w run prprossing prour on G tht ims to intify st of nssry onstrints on th strngths S P (x, y), suh s S P (x, y) must qul to S P (x, y)+ M. Th prour is s on st of ruls tht nfor nssry onitions for to win, nmly, S P (, x) S P (x, ) must hol. If th prprossing prour tts filur thn thr is no st of vots for M suh tht oms winnr. Th psuoo for th first stg of th lgorithm is givn in Algorithm 3. Stion 4.1 provs th orrtnss of Algorithm 3. If no filur is tt y pplying ths ruls uring th prprossing stg, w show tht mnipultion xists n provi onstrutiv prour tht fins mnipultion. Th psuoo for th son stg of th lgorithm is givn
Algorithm 3 PREPROCESSINGBOUNDS. Input: wight igrph (G =(V,E),w)=(G P,w P ), th strngths S P n istinguish nit. for (x, y) V V o w(x, y) =w(x, y)+ M w(x, y) =w(x, y) M U(x, y) =S P (x, y)+ M whil no onvrgn o /* Rul 1 */ for x V \{} o U(x, ) =min{u(x, ), U(, x)} /* Rul 2 */ for x V \{} o V r = {y V : U(y,) <U(x, ),y } E r = {(f,g) ( E : w(f,g) ) <U(x, )} V r V V V r G x = (V \ V r), (E \ E r) if G x ontins no -x-pth thn U(x, ) =U(x, ) 2 /* Rul 3 */ for x V \{} o for y V \{x, } o if U(x, ) <w(x, y) U(y, ) =min(u(y, ),U(x, )) for x V \{} o if U(x, ) <S P (x, ) M thn rturn FAIL rturn U in Algorithm 4. Hr, th lgorithm trvrss vrtis in G in spifi orr, whih fins th mnipultors vots. Stion 4.2 provs th orrtnss of Algorithm 4. 4.1 Stg 1. Prprossing Algorithm 3 uss funtion U(x, y), whih for ny two nits x n y, givs n uppr oun for S P (x, y). Initilly, U(x, y) := S P (x, y) + M for h pir (x, y). W lso us th following nottion for n uppr n lowr oun of w (x, y): w(x, y) :=w(x, y)+ M n w(x, y) :=w(x, y) M. In th first stg, Algorithm 3 rss U(x, y) whn it tts nssry onitions implying S P (x, y) < U(x, y). Th lgorithm is s on th following thr rution ruls. W show tht ths ruls r soun in th sns tht n pplition of rul os not hng th st of solutions of th prolm. Rul 1. If thr is nit x suh tht U(, x) < U(x,),thn st U(x, ) :=U(, x). PROPOSITION 1. Rul 1 is soun. PROOF. To s tht Rul 1 is soun, suppos S P (x, ) > S P (, x). Butthn,/ W P. To stt th nxt rution rul, fin for ny nit x th irt grph G x otin from G y rmoving ll vrtis y with U(y, ) <U(x, ) n ll rs (y, z) suh tht w(y,z) <U(x, ). Rul 2. If thr is nit x suh tht G x hs no irt pth from to x,thnstu(x, ) :=U(x, ) 2. PROPOSITION 2. Rul 2 is soun. PROOF. Suppos th prmiss of th rul hol, n, for th sk of ontrition, suppos thr xists pth in G from x to with 1 f1 1 1 2 6 1 Figur 6: Th WMG G P from Exmpl 5 strngth s, whrs quls U(x, ) for th pplition of th rul. Sin G x hs no irt pth from to x, ll irt pths in G from to x pss ithr through vrtx y with U(y, ) <s or through n r (y, z) suh tht w(y, z) <s. Sin ny suh pth hs strngth lss thn s, whvthts P (, x) <s. But, sin longs to th winning st in G,whvthtS P (, x) S P (x, ) s, ontrition. Thus, S P (x, ) <s. Th sounnss of Rul 2 now follows from th ft tht ll S P (y, z) hv th sm prity s NM + M, y, z V, n w mintin th invrint tht ll U(, ) hv this prity. Rul 3. If thr r nits x, y suh tht U(x, ) < w(x, y) n U(y, ) > U(x, ), thn st U(y, ) := U(x, ). PROPOSITION 3. Rul 3 is soun. PROOF. Suppos S P (y, ) >U(x, ) n π is ritil pth from y to in G. But thn, th pth x π, otin y ontnting x n π, hs strngth min{w (x, y),s P (y,)}. Sinw (x, y) w(x, y) >U(x, ), th strngth of this irt pth from x to is stritly grtr thn U(x, ), ontriting our ssumption tht U(x, ) is nssry uppr oun for S P (x, ). W rmrk tht Ruls 1-3 rmnt U(,) whn nssry onitions r foun tht rquir smllr uppr oun for S P (,). Shoul t ny tim suh vlu U(x, ) om smllr thn S P (x, ) M, thn thr r no vots for M tht mk winnr. In this s, th prprossing lgorithm rturns FAIL. THEOREM 2. Algorithm 3 is soun. PROOF. Algorithm 3 implmnts Ruls 1 3. As ths ruls r soun, th lgorithm is soun. Consir how Algorithm 3 works on n xmpl. EXAMPLE 5. Consir n ltion with lvn ltrntivs { 1, 2, 1, 2,, 1, 2, 1, 2,f 1,f 2} with th WMG in Figur 6, whr M =1n is th prfrr nit. W not tht thr r two nits 1 n 2 suh tht S P (, x) =S P (x, ) 2, x { 1, 2}. For nit 1 thr r two wys to inrs S P (, 1). Th first wy is to inrs th strngth of th - 1-1- 1-pth y rnking 1 1. Th son wy is to inrs th strngth of th - 2-2- 1-pth y rnking 2 2. If w slt th first wy thn n xtnsion of 1 1 to ny totl orr ls to / W P. If w slt th son wy thn w n uil sussful mnipultion. W show tht Algorithms 3 4 sussfully onstrut vli mnipultion. W strt with Algorithm 3. Tl 1 shows xution of Algorithm 3 on this prolm instn. 2 2 f2 2
Altrntivs C\{} f 1 f 2 1 2 1 2 1 2 1 2 Initil vlus U(, ) 7 U(,) Rul 1 upts U( 2,), U( 1,) n U( 2,) U(, ) 7 U(,) 7 Rul 3 upts U(f 1,) s U( 2,)=7n w( 2,f 1)= U(, ) 7 U(,) 7 7 Rul 2 upts U( 1,) (Figur 7() for G 1 ) U(, ) 7 U(,) 7 7 7 Rul 3 upts U( 1,) n U( 1,) U(, ) 7 U(,) 7 7 7 7 7 Tl 1: Exution of Algorithm 3 on Exmpl 5. U(, )/U(, ) stns for th uppr oun vlu U(, )/U(, ), whr is th ltrntiv in th orrsponing olumn, C\{}. 1 f1 1 1 2 6 1 () 2 2 f2 2 1 7 f1 1 1 2 7 7 Figur 7: () G 1. Dlt rs n vrtis r in gry. Thr is no pth from to 1;()WMGG P {Λ} from Exmpl 5 whr Λ is vli mnipultion. 4.2 Stg 2. Construt mnipultors vots Algorithm 4 onstruts linr orr Λ s on th following gry prour. Initilly, Λ={}, is th top nit, lstv =, th frontir F = {} n th st of unrh vrtis X = C\{}. During th xution of th lgorithm, Λ is linr orr on F n ontins n lmnt x y for ny two onsutiv vrtis x, y in this orr. Th vrtx lstv is th lst vrtx in this orr lstv. Whil Λ is not totl orr, th lgorithm s on of th unrh vrtis y to th n of prtil orr Λ stisfying th following onitions: x F, y X, U(y, ) = D n w(x, y) D, whrd is th mximum vlu U(y, ) mong ll unrh vrtis y X. THEOREM 3. Algorithm 4 onstruts totl orr Λ with top lmnt. Morovr, for ny vrtx x V \{}, thris-xpth π =( = x 1,...,x p = x) suh tht w(x i,x i+1) U(x, ) n x i x i+1 Λ, i =1,...,p 1. PROOF. First, w n to prov tht th lgorithm n lwys vrtx y to th orr Λ stisfying th onitions ov. Lt z ny nit from X suh tht U(z, ) =D. Sin Rul 2 os not pply, th sugrph G z hs irt pth from to z. Lt (x, y) th r on this pth with x F n y X (w oul 1 () 2 2 f2 2 Algorithm 4 Constrution of orring Λ Input: wight igrph (G =(V,E),w)=(G P,w P ), th strngths S P, istinguish nit n th funtion U rturn y Algorithm 3. for (x, y) V V o w(x, y) =w(x, y)+ M Initiliz F = {}, X = C\{}, lstv = n Λ={}. for i =1,..., V 1 o D =mx{u(y, ) : y X} Choos x F n y X with w(x, y) U(y, ) =D F = F {y} X = X \{y} Λ=Λ {lstv y} lstv = y rturn Λ possily hv tht y = z). Also, y Rul 2 w hv tht U(y, ) U(z, ) n tht w(x, y) U(z, ). Thus, U(y,) = D n w(x, y) D, whih mns tht (x, y) stisfis th onitions of th ltrntiv y to to Λ. W prov th son sttmnt y inution. In th s s, F = {} n w y suh tht w(, y) U(y,). Hn, π =( = x 1,x 2 = y), w(, y) U(y,) n y Λ. Suppos, th sttmnt hols for i 1 stps. Lt (x, y) th r suh tht x F n y X, w(x, y) U(y, ) =D tht w t th ith stp. By th inution hypothsis, w know tht thr is -x-pth π =( = x 1,...,x p = x) suh tht w(x j,x j+1) U(x, ) n x j x j+1 Λ, j =1,...,p 1, p i 1. Byth sltion of y, w gt tht w(x, y) U(y, ). By Algorithm 4, w know tht U(x, ) U(y, ). Hn, w(x j,x j+1) U(x, ) U(y, ), j =1,...,p 1. As w x y to Λ w gt tht thr is -y-pth π =( = x 1,...,x p = x, x p+1 = y) suh tht w(x, y) U(y, ) n x j x j+1 Λ, j =1,...,p. This orr Λ fins th vot of th mnipultors. THEOREM 4. Consir th orr Λ rturn y Algorithm 4. Thn W P,whrP = P NM M i=1 {Λ}. PROOF. Du to th onstrution of Λ, w know tht S P (, x) U(x, ), x V \{} s for h vrtx x thr is -x-pth σ =( = x 1,x 2,...,x p = x) suh tht w(x i,x i+1) U(x, ) n x i x i+1 Λ, i =1,...,p 1. Lt us mk sur tht S P (x, ) U(x, ) for h nit x V \{}. On th ontrry, suppos thr is nit x suh tht S P (x, ) >U(x, ) n suppos mong ll suh vrtis, x hs th shortst ritil pth to. Dnot y π =(x, x 1,x 2,...,) shortst ritil pth from x to. Consir two ss pning on whthr x 1 = or x 1. Suppos tht x 1. W hv tht S P (x 1,) S P (x, ) sin th pth π is ritil. Thrfor, U(x 1,) >U(x, ) y th sltion of x n π. Sin nits r y non-inrsing vlus of U(,) to Λ, x 1 ws for x, sothtx 1 x. Thus, w (x, x 1)=w(x, x 1). By Rution Rul 3, w hv tht w(x, x 1) U(x, ). Thus, w (x, x 1) U(x, ), ontriting tht π hs strngth >U(x, ) in G. Suppos tht x 1 =. In this s, π =(x, ) n S P (x, ) = w (x, ). As is th top lmnt of Λ w hv tht w (x, ) = w(x, ) =w(x, ) M. As Algorithm 3 i not tt filur, w know tht U(x, ) S P (x, ) M. Morovr, S P (x, ) w(x, ), y finition of th ritil pth. Thrfor, U(x, ) S P (x, ) M w(x, ) M = w (x, ) =S P (x, ). Hn,
S P (x, ) U(x, ), ontriting tht π hs strngth >U(x, ) in G. Not tht Corollry 1 os not follow from Thorm 4, us Algorithm 3 tks O(w mx V 3 ) tim, whr w mx = mx (x,y) V V w(x, y). As w mx n O(2 V ), Algorithm 3 tks xponntil numr of stps in WCM. EXAMPLE 6. Consir how Algorithm 4 works on Exmpl 5. Th lgorithm trvrss G y vrtis orr y th vlu U(x, ), x C \ {}. Initilly, w strt t, n F = {} n X = C\{}. W omput D =mx{u(y, ) : y X}, D =.W onsir ll vrtis y X suh tht U(y,) =, whih is th st Q = {f 2, 2, 2, 2, 1}. W slt on of thos vrtis, f 2, tht stisfis th onition on th vlu w(x, y), x F, y X: w(, f 2)= U(f 2,)=. In th nxt four stps w ll lmnts of Q n otin prtil orr Λ= f 2 2 2 1. Th nxt mximum vlu D =mx{u(y, ) : y C\ {, f 2, 2, 2, 2, 1} is 7. Th st of vrtis suh tht U(y, ) =7 is Q = {f 1, 1, 1, 1, 2}. Hn, w ths vrtis to Λ on y on n otin totl orr Λ= f 2 2 2 1 f 1 1 2 1 1. Figur 7() shows th WMG G P {Λ}. W omitt ll rs of wight 1 for lrity. 5. UNIQUE WINNER VS CO-WINNER UCM In this stion w onsir th unwight olitionl mnipultion prolm with singl mnipultor tht ws onsir in [14]. Prks n Xi show tht th uniqu winnr UCM for Shulz s rul with singl mnipultor n solv in polynomil tim. W mphsiz tht in this vrint th im is to mk th prfrr nit th uniqu winnr. Th im of this stion is to show tht th proof from [14] nnot xtn to th owinnr UCM prolm with on mnipultor. This monstrts tht th o-winnr UCM prolm with on mnipultor ws not rsolv in [14]. W lso xtn our lgorithm for o-winnr UCM to th uniqu winnr s. Anothr rson to invstigt th rltion twn proprtis of uniqu winnr n o-winnr mnipultion prolms is tht thy r losly rlt to th hoi of tirking ruls. If th ti-rking rul rks tis ginst th mnipultors thn th mnipultors hv to nsur tht th prfrr nit is th uniqu winnr of n ltion. If th ti-rking rul rks tis in fvor of th mnipultors thn it is suffiint for th mnipultors to gurnt tht th prfrr nit is on of th o-winnrs of th ltion to hiv th sir outom. Th proof tht th uniqu winnr UCM is polynomil is s on th rsolvility proprty [16, Stion 4.2.2]. Th rsolvility ritrion stts tht ny o-winnr n m uniqu winnr y ing singl vot. Rsolvility. If S P (, x) S P (x, ) for ll nits x C\{}, thn thr is vot v suh tht S P {v} (, x) > S P {v} (x, ) for ll nits x C\{}. Th proof of th proprty is onstrutiv. Clrly, n th uniqu winnr in P {v} only if is o-winnr in P.Thvot v is onstrut using two ruls tht w sri low. W not P = P NM n {v} = P M to simplify nottions. (1) For vry ltrntiv x C \{}, w rquir y x in th mnipultor s vot v whr y isth prssor ofx on som strongst pth from to x. (2) For ny x, y C\{} with S P (x, ) >S P (y,) w rquir x y in th mnipultor s vot v. \ 12 12 12 12 () () - - 12 - - \ - - 12-12 12-12 - Figur : () Th WMG G P n th tl of S P(x, y), x, y {,,,, } from Exmpl 7; ()/() Th WMG G P/G P {v} n th tl of S P(x, y)/s P {v} (x, y), x, y {,,, } from Exmpl. It ws shown in [16] tht th rsulting st of prfrn rltions os not ontin yls n thus n xtn to linr orr whih mks th uniqu winnr. Howvr, it ws lso shown in [16] tht th sm pproh nnot rsolv tis twn nits tht o not long to th winning st. It is nturl qustion if nit tht is not in th winning st n m winnr y ing singl vot. Clrly nssry onition is S P (, x) S P (x, ) 2 for ll x C\{}. So w n formult th following prolm. Singl vot UCM. Givn profil P n nit with S P (x, ) S P (, x) +2for ll x C\{}, os thr xist singl vot v suh tht W P {v}? Hr, w show tht th strightforwr ption of th ov ruls os not solv this prolm, vn if thr is singl vot mnipultion tht mks winnr. A mjor iffrn twn th uniqu winnr n th o-winnr UCM prolms is tht th mnipultion lwys xists in th formr prolm n it might not xist in th lttr s th following xmpl monstrts. EXAMPLE 7. Consir n ltion with fiv ltrntivs {,,,, }. Figur () shows th WMG n th orrsponing tl of mximum strngths. Th uniqu winnr is. Howvr, th iffrn S P (x, ) S P (, x) 2, x {,,, }. Hn, stisfis th trivil nssry onition for ing m winnr y ing singl vot. To s tht thr is no sussful mnipultion w noti tht S P (, ) =S P (, ) 2. Hn th mnipultion must inrs th wight of t lst on ritil --pth. As thr is only on ritil --pth this fors in th mnipultor s vot. But on th othr hn S P (, ) =S P (, ) 2 rquirs tht th wight of vry ritil --pth rss whih implis tht or, whih givs ontrition. Consir th prfrn rltions tht r output y th ruls. Following th first rul w n. Following th son rul, w {,, }. This rts yl n thus nnot omplt to linr orr. Nxt, w show tht th ruls o not fin th mnipultor vot vn if suh mnipultion xists for th o-winnr UCM prolm using Exmpls. EXAMPLE. Consir n ltion with four ltrntivs {,,, }. Figur () shows its WMG n th orrsponing ()
tl of mximum strngths. Th st of winnrs is {,, } n S P (x, ) S P (, x) 2, x W P. Following th first rul w. Howvr, y th son rul, w whih rts yl. Not tht sussful mnipultion v xists v =( ) (Figur ()). EXAMPLE. Consir th ltion with ltrntivs from Exmpl 5. Following th first rul w 1 1 1 to th mnipultor vot s π =(, 1, 1, 1) is strongst pth from to 1. As w show in Exmpl 5, thr os not xist n xtnsion of this prtil orr to totl orr tht mks o-winnr. Howvr, sussful mnipultion v xists (Figur 7()). Thrfor, our stuy highlights iffrn twn uniqu winnr n o-winnr UCM unr Shulz s rul with singl mnipultor n monstrts tht o-winnr UCM with singl mnipultor ws not rsolv. Morovr, w liv tht Shulz s rul is n intrsting xmpl, whr th ti-rking in fvor of mnipultor, whih orrspons to o-winnr UCM, mks th prolm non-trivil ompr to ti-rking ginst mnipultors, whih orrspons to uniqu winnr UCM. Two ruls with similr proprtis hv n onsir in th litrtur. Conitzr, Snholm n Lng [5] show tht Copln s rul is polynomil with 3 nits in uniqu winnr WCM, whil it is NP-hr with 3 nits in o-winnr WCM []. Th most rnt rsult is u to Hmspnr, Hmspnr n Roth [] who show tht th onlin mnipultion WCM is polynomil for plurlity in th owinnr mol, whil it is onp-hr in th uniqu winnr mol. Our lgorithm from Stion 5 n still us s suroutin to solv th uniqu winnr UCM prolm. COROLLARY 2. Th uniqu winnr UCM prolm n solv in polynomil tim. PROOF. Run th lgorithm from Stion 5 with M 1 mnipultors n rturn th nswr. To show th orrtnss of this prour, w n to show tht is o-winnr with M 1 mnipultors iff is uniqu winnr with M mnipultors. ( ): Suppos n m o-winnr with M 1 mnipultors. Us th Rsolvility proprty to on mor vot to mk uniqu winnr. ( ): Suppos n m uniqu winnr with M mnipultors. Thrfor, S P (, x) S P (x, ) +2for vry nit x C\{} in th profil P = P NM P M. Now, rmov n ritrry vot of mnipultor n otin th profil P.Whvtht S P (, x) S P (, x) 1 n S P (x, ) S P (x, )+1for vry nit x C\{}. Thrfor, S P (, x) S P (, x) 1 S P (x, )+1 S P (x, ) for vry nit x C\{}, showing tht is o-winnr with M 1 mnipultors. 6. CONCLUSIONS W hv invstigt th omputtionl omplxity of th olitionl wight n unwight mnipultion prolms unr Shulz s rul. W prov tht it is polynomil to mnipult Shulz s rul with ny numr of mnipultors in th wight o-winnr mol n in th unwight s in oth uniqu n o-winnr mols. This rsolvs n opn qustion rgring th omputtionl omplxity of unwight olition mnipultion for Shulz rul [14]. This vulnrility to mnipultion my of onrn to th mny supportrs of Shulz s rul. is lso fun y AOARD grnt 124056. Srg Gsprs knowlgs support from th Austrlin Rsrh Counil (grnt DE1201761).. REFERENCES [1] R.K. Ahuj, T.L. Mgnnti, n J.B. Orlin. Ntwork Flows: Thory, Algorithms n Applitions. Prnti Hll, 13. [2] J.J. Brtholi n J.B. Orlin. Singl trnsfrl vot rsists strtgi voting. Soil Choi n Wlfr, (4):341 354,. [3] J.J. Brtholi, C.A. Tovy, n M.A. Trik. Th omputtionl iffiulty of mnipulting n ltion. Soil Choi n Wlfr, 6(3):227 241, 1. [4] N. Btzlr, R. Nirmir, n G.J. Wogingr. Unwight olitionl mnipultion unr th Bor rul is NP-hr. In Pro. of 22n Intrntionl Joint Confrn on AI. 20. [5] V. Conitzr, T. Snholm, n J. Lng. Whn r ltions with fw nits hr to mnipult? Journl of th ACM (JACM), 54(3):14, 2007. [6] J. Dvis, G. Ktsirlos, N. Nroytsk, n T. Wlsh. Complxity of n lgorithms for Bor mnipultion. In Pro. of 25th AAAI Confrn on AI. 20. [7] B. Dor. Axiomtistion proéurs grégtion préférns. PhD thsis, Univrsité Sintifiqu, Thnologiqu t Méil Grnol, 17. [] L. Hmspnr, E. Hmspnr, n J. Roth. Th Complxity of Onlin Mnipultion of Squntil Eltions In Pro. of th 5r Int. Workshop on Computtionl Soil Choi (COMSOC-12). [] P. Fliszwski, E. Hmspnr, n H. Shnoor. Copln voting: tis mttr. In Pro. of 7th Int. Joint Confrn on Autonomous Agnts n Multignt Systms, 200. [] P. Fliszwski, n A. Proi. AI s Wr on Mnipultion: Ar W Winning? AI Mgzin, 31(4):53 64, 20. [] A. Gir. Mnipultion of voting shms: gnrl rsult. Eonomtri, 41(4):57 1, 173. [12] D.C. MGrvy. A thorm on th onstrution of voting proxs. Eonomtri, 21: 6, 153. [13] N. Nroytsk, T. Wlsh, n L. Xi. Mnipultion of Nnson s n Blwin s ruls. In Pro. of 25th AAAI Confrn on AI. 20. [14] D.C. Prks n L. Xi. A omplxity-of-strtgi-hvior omprison twn Shulz s rul n rnk pirs. In Pro. of 26th AAAI Confrn on AI, 2012. [15] M.A. Sttrthwit. Strtgy-proofnss n Arrow s onitions: Existn n orrsponn thorms for voting prours n soil wlfr funtions. Journl of Eonomi Thory, :17 217, 175. [16] M. Shulz. A nw monotoni, lon-inpnnt, rvrsl symmtri, n Conort-onsistnt singl-winnr ltion mtho. Soil Choi n Wlfr, 36:267 303, 20. [17] L. Xi, M. Zukrmn, A.D. Proi, V. Conitzr, n J.S. Rosnshin. Complxity of unwight olitionl mnipultion unr som ommon voting ruls. In Pro. of 21st Int. Joint Confrn on AI. 200. 7. ACKNOWLEDGMENTS NICTA is fun y th Austrlin Govrnmnt s rprsnt y th Dprtmnt of Bron, Communitions n th Digitl Eonomy n th Austrlin Rsrh Counil. This rsrh