Deepest voting on rankings

Deep est v oting on rankings Jean-Baptiste Aubin 1 , An toine Rolland 2 , Ioana Ga vra 3 , Ir ` ene Gannaz 4 , and Jacques Anderson Kouassi 4 1 Univ Ly on, INSA Ly on, UJM, UCBL, ECL, ICJ, UMR5208, 69621 Villeurbanne, F rance 2 Univ ersit ´ e Lumi ` ere Lyon 2, Univ ersite Claude Bernard Lyon 1, ERIC, 69007 , Ly on, F rance 3 Univ Rennes 2 , IRMAR - UMR CNRS 662 5, F-35000 Rennes, F rance, 4 Univ. Grenoble Alp es, CNRS, Grenoble INP , G -SCOP , 380 00 Grenoble, F rance F ebruary 2026 Abstract This article aims to present a unified framework f or r anking-based v oting rules ba sed on the use of depth funct ions on permutations, as a counterpart of deep est voting rules on ev aluation introduced in Aubin et al. [ 2022 ]. It intro duces the notion of depth functions, in contin uous sets a nd in per m utation sets, the later using the no tion o f F r´ echet means. Deep est voting pro cedures ar e then formally defined, and so me classical voting rules are express e d as de e pest voting pro cedures, using a larg e v ariet y o f distances on the set of permutations. Links are done b etw een the depth functions mathematical prop erties and some b eha viour s of the v oting rule, such a s Neutralit y , Anonymit y , Univ ers alit y , Condorc e t winner /loser pr oper t y and so on. 1 In tro duction A voting pro cedure, or so cia l choice function, is a pro cess that a ggregate individual judgmen ts int o co llectiv e outcomes. In case of uninominal voting, the r esult sho uld b e the election on a n unique winner selected in a finite set of candidates. T r aditionally , voting theory ha s bee n domi- nated b y preference-ba sed (ordina l) metho ds, wher e ea c h voter is suppo s ed to ranks candidates by preference, and the s ocia l c hoice function selects a winner or so cial ranking accordingly . Class ic examples include pluralit y voting or Borda count, which interpret voter r ankings to reflect colle c - tive preferences and satisfy nor mativ e cr iteria such as ma jority supp ort or mono tonicit y . So cial choice theory rigo r ously formalizes these mo dels, exploring their prop erties and limitations — mos t famously exemplified b y A rr o w’s imp ossibility theorem a nd related strategic co nc e rns such as the Gibbard–Satterthw aite result on manipulability of o rdinal rules. Another framework is a lthough p ossible, which concerns ev aluation-based (car dina l) voting pro ce- dures. V oter s ar e supp osed to as s ign sco res or gr ades to ea c h option, pr oviding r ic her informatio n 1 ab out int ensity of supp ort t han in the case o f r a nking. Car dina l metho ds, s uc h as sco res o r r ange voting, are more conv enient to capture n uances in voter sent iment, which should impr o ve co llectiv e welf ar e outcomes .. In the context of ev aluation-ba sed voting rules , Aubin et al. [ 2022 ] pr opo se an unified mo del o f so cial c hoice function based on the use of the depth function concept. A depth function is a measure of the ”cen trality” o f a point into a scatter-plot. The deep est point is therefor e the most central p oint of the scatter - plot. Of course there are man y differen t depth functions that lead to po ssibly different deep est p oints. F or example in a one- dimens ion space, b oth the sum of absolute differences or the sum os square differences to the other p oint s c an b e conside r ed as depth functions (strictly sp eaking, in v erse o f depth functions) lea ding to t he median a nd mean as deepest points. So ro ughly sp eaking, Aubin et al. [ 202 2 ] introduce deepest voting rules, that consider voters a s po in ts in the candidates’ space, and tries to recover the ”mos t central” voter in the voters set. This sp ecific voter is therefore cons idered as mo st representative of the co llectiv e ev aluations, and the choice o f this vote is considered to b e the c o llectiv e choice. The choice o f a specific depth function le ads to the definition o f sp ecific voting rule, including w ell-known ev aluation-based voting rules such as r a nge voting or ma jor it y judgment. The ob jective of this pap er is to genera lize the deepe s t voting appro a c h to ranking -based votes. Aubin et al. [ 20 22 ] o nly consider e v aluations based settings. W e prop ose in this pap er to forma lize ranking-ba sed voting rules thro ugh the use of dept h functions defined o n rankings seen as p ermu- tations. W e use the concept o f 𝑝 − F r´ echet mean to ge ne r alize the co ncept of depth functions to per m utations. W e show that some usual procedur es (n amely Bucklin, Borda, Kemeny , Plur alit y , An tiplurality) can b e written a s deep e st v oting pro cesses. W e also a na lyse some classica l v oting rules pro perties that are satisfied or no t by ra nking-based deep est v oting rules . Distance rationaliza tion o f voting rules is a lso an a xiomatic appro ach based on the use o f distance betw een r ankings to determine the result of a v oting pro cess , see for example Mesk anen and Nur mi [ 2008 ], E lkind et a l. [ 20 0 9 ] and E lkind et al. [ 201 0 ]. How ever the framework is differe n t. Ratio- nalization of voting rules is bas ed on the idea to compar e a globa l voting pro file to the clo sest unambiguous v oting profile (for example unanimous profiles in the early work of Nitzan [ 19 81 ]), that leads to an una m biguous winner of the electio n. F or example Do dgson’s voting r ule is the rationaliza tion of the Condor cet winner with re s pect to the Kendall distance; ho w muc h should one derive of the obser v ed election profile to reach a Condo r cet winner? Deepest voting, o n the contrary , only measure the inner distances b etw een voters to find the innermo st voter, which is the most represen tative of t he election profile. More r e cen tly the Level 𝑟 Consensus method pr opos ed in Maha jne et al. [ 2 015 ] deals with dis tance betw een preferenc e relations w.r.t. an election pro files, and not betw een voters as in the dee p est voting rules. F o rmally , we co ns ider a framework with 𝑛 v oters, V = { 1 , . . . 𝑛 } , and 𝑚 candidates C = { 1 , . . . 𝑚 } . A voting situatio n suppo se each voter 𝑣 gives an opinion on c andidate 𝑐 , denoted 𝑒 𝑐 𝑣 . The o pinio ns ( 𝑒 𝑐 𝑣 ) 𝑐 ∈ C take v alues in E . In pr e fer ences ra nking votes, the set E is the set of preferences. Tha t is, E = 𝔖 𝑚 , with 𝔖 𝑚 the set o f p ermutations of { 1 , . . . , 𝑚 } . The idea o f ev aluation-ba sed framework 2 is to c hang e the set of opinions to ev aluations ra ther than r ankings. In s uc h a case, E = Λ 𝑚 , with Λ the set of ev aluations of one voter for each candidate. Class ic a l choices are Λ = { 1 , . . . , 𝐾 } (discrete ev alua tions) a nd Λ = [ 0 , 𝑀 ] (contin uous ev aluations ). In the following, w e will consider that a n ev aluation decreases with the preference to a candida te. That is, if the voter 𝑣 prefers the candidate 𝑐 1 to the candidate 𝑐 2 , then 𝑒 𝑐 1 𝑣 < 𝑒 𝑐 2 𝑣 . This o rdering be ing the same a s the o ne on rankings allows to make parallelis ms. Hence, a voting framew ork consists in v otes Φ = ( 𝑒 𝑐 𝑣 ) 𝑣 ∈ V , 𝑐 ∈ C . F o r eac h v oter 𝑣 , the opinion ( 𝑒 𝑐 𝑣 ) 𝑐 ∈ C can b e seen a s the realisatio n of a 𝑚 -multiv ar iate random v aria ble 𝐸 , with 𝐸 at v alues in E . This sta tistical p oin t of view offers an in teresting p erspe ctiv e. It allows using the s tatistical to ols to de s cribe the set of votes. In particular, it offers the p ossibility of finding c enters of the distribution, thanks to the applica tion of sta tistical depth functions. Quoting Zuo and Serfling [ 2000 ]: “Asso ciated with a given distribution 𝐹 o n R 𝑚 , a depth function is designed to pr ovide a 𝐹 -based center-outw ard or dering [...] of points 𝑥 in R 𝑚 . High depth corres p onds to c entr ality , low depth to outlyingness ”. In o ther words, a depth funct ion t akes high (po sitiv e) v alues at the c enter of a scatter plot and v anishes out of it. A f or ma l de finitio n will b e given in Section 2 . A depth function 𝐷 on E is a function defined on E × F , where F is the set o f probability distributions on E . It has v alues in [ 0 , ∞) , and is built suc h that for any distribution 𝐹 on E , the function 𝐷 ( . , 𝐹 ) is maximal at a p oint whic h can be cons idered as the c enter o f the distribution 𝐹 . Applied to an observed vote situa tio n, Φ ∈ E 𝑛 , 𝐷 ( . , Φ ) is ma x imal at a p oint which can b e c onsidered a s the c enter of the empirical distribution r elated to Φ . Using this approa c h enables to recov er classical c enters o f distributions, such as the mean or the median, but also to define v arious c enters , having interesting prop erties. The application of a depth function 𝐷 ( ) on a v oting situation Φ ∈ E 𝑛 provides a (p ossibly fictive) central voter, 𝑣 ∗ with opinions 𝐸 ∗ ∈ E 𝑚 , 𝐸 ∗ = ( 𝑒 ∗ 1 , . . . 𝑒 ∗ 𝑚 ) ⊤ . The preferred ca ndidate of 𝑣 ∗ is then the winner o f the votes, a sso ciated to the depth function. This pr oce dur e is defined as deep est voting pro cedure by Aubin e t a l. [ 2022 ]. W e in vestigate in this pa per the ca se where E = 𝔖 𝑚 . The pap er is org anized as follows. Section 2 provides the definitions o f depth functions, in contin u- ous sets R 𝑚 and in p ermutation sets 𝔖 𝑚 . The la ter ar e in particular based on the notion of F r´ ech et means a nd on distances on 𝔖 𝑚 , that will b e r ecalled in this section. Deep est voting pro cedures are then defined in Section 3 . Section 4 then gives some results with contin uous depths, while Section 5 c o nsider depth b ase d on p ermutations. In particular, some cla ssical voting r ules are ex - pressed as deep est voting pro cedures, and links ar e done betw een the depth functions mathematica l prop erties and some b ehaviours of the voting rule, such as Neutrality , Ano nymit y or Universalit y . Pro ofs are given in the Appendix. 2 Definition of depth functions In this section, we provide for mal definitions of depths functions. W e first consider depths functions defined on co ntin uous m ultiv a riate sets, on a connected subset of R 𝑚 . N ext we pr e sen t depth functions o n the set o f p ermut ations 𝔖 𝑛 . 3 2.1 Depth functions in con tin uous case W e reca ll here the definition of a depth function on R 𝑚 . Suc h a depth function can b e applied in an y connected s et E 𝐷 ⊆ R 𝑚 . W e refer to Mosler [ 2013 ], Zuo and Serfling [ 2 000 ] and refere nc e s therein for a more detailed o verview. Definition 1. L et the ma pping 𝐷 : R 𝑚 × F → R + b e b ounde d, and satisfyi ng: (C1) Stability by Permutation. L et 𝐸 = ( 𝑒 1 , . . . , 𝑒 𝑚 ) b e a r andom ve ctor in R 𝑚 , 𝑥 ∈ R 𝑚 , and 𝜎 a p ermutation on { 1 , . . . , 𝑚 } . Le t 𝐸 𝜎 = ( 𝑒 𝜎 ( 1 ) , . . . , 𝑒 𝜎 ( 𝑚 ) ) and 𝑥 𝜎 = ( 𝑥 𝜎 ( 1 ) , . . . , 𝑥 𝜎 ( 𝑚 ) ) . Then 𝐷 ( 𝑥 𝜎 , 𝐹 𝐸 𝜎 ) = 𝐷 ( 𝑥 , 𝐹 𝐸 ) . (C2) Affine Invarianc e. F or al l 𝑎 ∈ R , 𝑏 ∈ R 𝑚 , for any r andom ve ctor 𝐸 ∈ R 𝑚 , argma x 𝑥 ∈ R 𝑚 𝐷 ( 𝑎 𝑥 + 𝑏 , 𝐹 𝑎 𝐸 + 𝑏 ) = argma x 𝑥 ∈ R 𝑚 𝐷 ( 𝑥 , 𝐹 𝐸 ) , wher e 𝐹 𝑋 denotes the distribution of the r andom variable 𝑋 . (C3) Maximality at c enter. F or a distribution 𝐹 ∈ F having a uniquely defin e d center 𝜃 ( e.g. t he p oint of symmetry ) , 𝐷 ( 𝜃 , 𝐹 ) = sup 𝑥 ∈ R 𝑚 𝐷 ( 𝑥 , 𝐹 ) . (C4) Quasi-c onc avity. F or any 𝐹 ∈ F , 𝐷 ( · , 𝐹 ) is quasi-c onc ave. Th at is, if 𝜃 ∈ argma x 𝑥 ∈ R 𝑚 𝐷 ( 𝑥 , 𝐹 ) , then 𝐷 ( 𝑥 , 𝐹 ) ≤ 𝐷 ( 𝜃 + 𝜆 ( 𝑥 − 𝜃 ) , 𝐹 ) for any 0 ≤ 𝜆 ≤ 1 . (C5) V anishing at Infinity. L et k . k denote the euclid e an norm on R 𝑚 . The n, 𝐷 ( 𝑥 , 𝐹 ) → 0 as k 𝑥 k → ∞ for e ach 𝐹 ∈ F . (C6) L et 𝐹 ∈ F b e a distribution on R 𝑚 with mar ginal distributions 𝐹 1 , . . . , 𝐹 𝑚 . Supp ose that for 𝑐 ∈ { 1 , . . . , 𝑚 } , 𝐹 𝑐 has su pp ort c ontaining a u n ique p oint { 𝛼 } . Then for al l 𝑥 ∗ ∈ argsup 𝑥 ∈ R 𝑚 𝐷 ( 𝑥 , 𝐹 ) , the 𝑐 th c o or dinate of 𝑥 ∗ is 𝑥 ∗ 𝑐 = 𝛼 . Then 𝐷 ( · , ·) is c al le d a s tatistical depth function . W e refer to Aubin et al. [ 2022 ] for a discussion on the assumptions. Note that the notio n of symmetry in (C3) is (volunt ar ily) not precisely fixed, and v arious notions of s ymmetry ar e po ssible such as, from the mo s t constraining to the w eakest, central symmetry , angular symmetry and halfspace sy mmetr y . In o ur c on text, w e will a pply the depth functions on empirical dis tribution. Consider a voting situation with 𝑛 voters a nd 𝑚 candidates, C = { 1 , . . . 𝑚 } , with an opinion matrix Φ ∈ E 𝑛 . Let 𝐷 ( ) be a depth function defined on E 𝐷 × F . I n the fo llowing, we will deno te 𝐷 ( . , Φ ) the depth function 𝐷 ( 𝑥 , 𝐹 Φ ) where 𝐹 Φ is the empirica l distr ibution of the voter’s opinions in E , { Φ ( ., 𝑣 ) = ( 𝑒 𝑐 𝑣 ) 𝑐 ∈ { 1 , . .. , 𝑚 } , 𝑣 = 1 , . . . , 𝑛 } . Let 𝑥 ∈ R 𝑚 and Φ ( ., 1 ) , . . . , Φ ( . , 𝑛 ) in R 𝑚 . E xamples of depth functions satisfying Definition 1 are: The weigh ted 𝐿 𝑞 depths. [ Zuo , 20 04 , Mo sler , 201 3 ] The w eighted 𝐿 𝑞 depth is defined by 𝑤 𝐿 𝑞 𝐷 ( 𝑥 , Φ ) = 1 1 + 1 𝑛 Í 𝑛 𝑣 = 1 𝜔 ( k Φ ( . , 𝑣 ) − 𝑥 k 𝑞 ) , 4 where 𝑞 > 0, 𝜔 is a non-decre a sing and contin uous function on [ 0 , ∞ ) with 𝜔 ( ∞) = ∞ and k 𝑥 − 𝑥 ′ k 𝑞 =  Í 𝑚 𝑐 = 1 | 𝑥 𝑐 − 𝑥 ′ 𝑐 | 𝑞  1 / 𝑞 . If 𝜔 : 𝑥 ↦→ 𝑥 𝑞 , then 𝐿 𝑞 𝐷 ( 𝑥 , Φ ) : = 1 1 + 1 𝑛 Í 𝑛 𝑣 = 1 Í 𝑚 𝑐 = 1 | Φ ( 𝑐, 𝑣 ) − 𝑥 𝑐 | 𝑞 will b e called a 𝐿 𝑞 depth. If 𝑞 = ∞ , the definition can b e ex tended to 𝐿 ∞ 𝐷 ( 𝑥 , Φ ) : = 1 /  1 + 1 𝑛 Í 𝑛 𝑣 = 1 max 𝑐 = 1 , .. . 𝑚 | Φ ( 𝑐, 𝑣 ) − 𝑥 𝑐 |  . The halfspace depth. [ T ukey , 19 7 5 ] The halfspace depth is defined by 𝐻 𝐷 ( 𝑥 , Φ ) : = minimum pr opo rtion o f v oters in a halfspace 𝐻 including 𝑥 . The pro jection depth [ Zuo , 20 0 3 ] The pro jectio n depth is defined b y 𝑃 𝐷 ( 𝑥 , Φ ) : = inf 𝑢 ∈ R 𝑚 , k 𝑢 k = 1 1 1 + | 𝑢 ⊤ 𝑥 − 𝜇 ( 𝐹 𝑢 ) | / 𝜎 ( 𝐹 𝑢 ) , where k · k denotes the euclidean no rm,, 𝜇 ( 𝐹 ) denotes a central statistic o f a distribution 𝐹 and 𝜎 ( 𝐹 ) a dis persio n statistic. 𝐹 𝑢 is the empiric al distribution o f 𝑢 ⊤ Φ . Typically , 𝜇 ( ·) is the median and 𝜎 ( ·) is t he median absolute deviation. Note that following 𝐿 𝑞 -depths, a depth function can be built from a distance 𝑑 ( ) on R 𝑚 , considering 𝐷 ( 𝑥 , Φ 𝑛 ) : = 1 /  1 + 𝑑 ( 𝑥 𝑐 , Φ ( 𝑐 , 𝑣 ) )  . 2.2 Depth functions on p er m utations Our ob jective is to extend the de e p est voting framew ork, defined in ev aluation-based framework by Aubin et al. [ 2022 ], to voting metho ds based on preference rankings. F or preference rankings, opinions belong to the set E = 𝔖 𝑚 . W e could co ns ider dept h functions defined on E 𝐷 = [ 0 , 𝑚 ] 𝑚 , since E ⊂ E 𝐷 , but it seems appropr iate to explor e depth functions defined on E 𝐷 = 𝔖 𝑚 . T o achiev e this go al, we wan t to define depth functions on p ermutation sets. Before defining depth function, let us recall some definitions of dis tances on 𝔖 𝑚 . 2.2.1 Distances o n p ermutations V a rious distances on the s et 𝔖 𝑚 hav e b een defined in the literature (see Deza and Huang [ 19 9 7 ] and Deza and Deza [ 2016 ] for complete rev iews). Es pecially , for 𝜎 , 𝜏 ∈ 𝔖 𝑚 , exa mples of distances are Kendall’s dis tance : 𝑑 𝐾 ( 𝜎 , 𝜏 ) = 𝑚 − 1 Õ 𝑐 = 1 𝑚 Õ 𝑐 ′ = 𝑐 + 1 1 { ( 𝜎 ( 𝑐 ) − 𝜎 ( 𝑐 ′ ) ) ( 𝜏 ( 𝑐 ′ ) − 𝜏 ( 𝑐 ) ) < 0 } . 5 Hamming distance : 𝑑 𝐻 ( 𝜎 , 𝜏 ) = 𝑚 Õ 𝑐 = 1 1 { 𝜎 ( 𝑐 ) ≠ 𝜏 ( 𝑐 ) } . Ca yley m e tric (or tr anspo sition distance): 𝑑 𝐶 ( 𝜎 , 𝜏 ) is the minimum num b er of transp ositions needed to obtain 𝜎 fro m 𝜏 (cf Diaconis [ 1988 ]), 𝑑 𝐶 ( 𝜎 , 𝜏 ) = 𝑚 − #cycles in ( 𝜎 ◦ 𝜏 − 1 ) , where 𝜎 ◦ 𝑠 denotes the successive applica tions of 𝑠 and 𝜎 : for a ll 𝑐 ∈ { 1 , . . . , 𝑚 } , 𝜎 ◦ 𝑠 ( 𝑐 ) = 𝜎 ( 𝑠 ( 𝑐 ) ) . Mink owski-H¨ older distances : ∀ 𝑞 ∈ [ 1 , +∞ ) , 𝑑 𝑞 ( 𝜎 , 𝜏 ) =  𝑚 Õ 𝑐 = 1 | 𝜎 ( 𝑐 ) − 𝜏 ( 𝑐 ) | 𝑞  1 𝑞 , 𝑑 ∞ ( 𝜎 , 𝜏 ) = max 𝑐 ∈ { 1 , . .. , 𝑚 }   𝜎 ( 𝑐 ) − 𝜏 ( 𝑐 )   . Among Minko wski-H¨ older distances , tw o special ca ses are when 𝑞 = 1 and 𝑞 = 2: when 𝑞 = 1 , Mink owski-H¨ older dista nce is r eferred to a s Spea rman fo otrule, when 𝑞 = 2 , Mink owski-H¨ older dista nce is r eferred to a s Spea rman 𝜌 distance. 2.2.2 W ei g h ted distances on p ermuta tions When measuring the discr epancy b e t ween tw o r anking lists in the co n text o f voting, one mig ht wan t to take so me additio na l infor mation in to a ccoun t: for example the p o sitions on which the per m utations differ o r, if av ailable, a similarity measure betw een the candidates. Some of the classical distances present ed in Section 2 .2.1 can b e slightly mo dified by incorpor ating weigh ts that balance the information in the distance. F or instance, o ne could wan t to g iv e more weigh t to the firs t ranked candidate with res pect to the others, estimating the first p osition is more emb lematic. Or one could wan t to build a distance wher e the weight of the differenc e b et ween t wo given ra nks increases with the difference. The idea is to in tro duce weigh ts in the distance which will dep ends o n the tw o ranks which are co mpared Kumar and V assilvitskii [ 2010 ]. The weights write as { 𝑤 𝑟 , 𝑟 ′ , 𝑟 , 𝑟 ′ ∈ { 1 , . . . , 𝑚 } } . Applying this to Hamming distance and Minko wski-Holder t yp e distance, w e obtain, for 𝜎 , 𝜏 ∈ 𝔖 𝑚 , • W eighted Hamming distance: 𝑑 𝑤 𝐻 ( 𝜎 , 𝜏 ) = 𝑚 Õ 𝑐 = 1 𝑤 ( 𝜎 ( 𝑐 ) , 𝜏 ( 𝑐 ) ) 1 { 𝜎 ( 𝑐 ) ≠ 𝜏 ( 𝑐 ) } , • W eighted Minkowski-Holder type d istance , for 𝑞 ∈ [ 1 , +∞) : 𝑑 𝑤 𝑞 ( 𝜎 , 𝜏 ) =  𝑚 Õ 𝑐 = 1 𝑤 ( 𝜎 ( 𝑐 ) , 𝜏 ( 𝑐 ) ) ( | 𝜎 ( 𝑐 ) − 𝜏 ( 𝑐 ) | ) 𝑞  1 𝑞 . 6 The w eight 𝑤 𝑟 , 𝑟 ′ > 0 for each p ositions 𝑟 , 𝑟 ′ ∈ { 1 , . . . , 𝑚 } balance the impor tance of s wap, that is of moving from p osition 𝑟 to positio n 𝑟 ′ in the ranking. While 𝑑 𝑤 𝑟 𝐻 ( ) is indeed a distance o n 𝔖 𝑚 , in all generality , this is no long er the case for the functions 𝑑 𝑤 𝐻 ( ) and 𝑑 𝑤 𝑞 ( ) in tro duced ab ov e. In particular the triangle inequality do e s not necessa rily ho ld. If the weigh ts a re symmetric, in the se ns e that 𝑤 𝑟 , 𝑟 ′ = 𝑤 𝑟 ′ ,𝑟 , for all 𝑟 , 𝑟 ′ ∈ { 1 , . . . , 𝑚 } , then the t wo dissimilarity functions ar e semi-metrics (they satisfy a ll the a xioms of a distance ex cept the triangle inequality). How ever, a so called 𝜌 -r e laxed triangle inequa lit y do es still hold and we have 𝑑 𝑤 ( 𝜎 1 , 𝜎 2 ) ≤ 𝜌 ( 𝑑 𝑤 ( 𝜎 1 , 𝜎 3 ) + 𝑑 𝑤 ( 𝜎 3 , 𝜎 2 ) ) , with 𝜌 = max 𝑟 𝑎 ,𝑟 𝑏 ,𝑟 𝑐 𝑤 𝑟 𝑎 , 𝑟 𝑏 𝑤 𝑟 𝑎 , 𝑟 𝑐 , and 𝑑 𝑤 ( ) e quals 𝑑 𝑤 𝐻 ( ) o r 𝑑 𝑤 𝑝 ( ) . 2.2.3 Depth functions asso ciated to a distance Let P 𝑚 denote the set of probabilit y distributions on 𝔖 𝑚 . Given a dista nce on 𝔖 𝑚 , Goib ert et a l. [ 2022 ] pro pos e d the follo wing definitio n of a depth function on p ermutations. Definition 2 ( Goib ert et al. [ 202 2 ]) . L et 𝑑 ( ) b e a distanc e on t he set 𝔖 𝑚 of p ermutations. 𝐷 : 𝔖 𝑚 × P 𝑚 ↦→ R + is a d epth function if it satisfies the fol lowing pr op erties: (P1) Invarianc e. F or al l 𝑠 ∈ 𝔖 𝑚 and Π ∈ P 𝑚 , define Π 𝑠 the pr ob ability distribution s uch t hat Π 𝑠 ( 𝜎 ) : = Π ( 𝜎 ◦ 𝑠 − 1 ) for al l 𝜎 ∈ 𝔖 𝑚 . 𝐷 is said to b e invaria nt if and only if, for al l 𝜎 , 𝑠 ∈ 𝔖 𝑚 , for all Π ∈ P 𝑚 , 𝐷 ( 𝜎 , Π ) = 𝐷 ( 𝜎 ◦ 𝑠 , Π 𝑠 ) . (P2) Maximality at Center. F or a distribution Π with a c enter of symmetry (defin e d later), the function 𝐷 ( ., Π ) r e aches i ts maximum at this c enter. (P3) L o c al Monotonicity. L et Π ∈ P 𝑚 and define 𝜎 ∗ = argmax 𝜎 ∈ 𝔖 𝑚 𝐷 ( 𝜎 , Π ) . Supp ose that the de ep est p ermutation 𝜎 ∗ is unique. Then for any 𝜏 , 𝜎 ∈ 𝔖 𝑚 such that 𝑑 ( 𝜎 ∗ , 𝜎 ◦ 𝜏 ) = 𝑑 ( 𝜎 ∗ , 𝜎 ) + 1 , we have 𝐷 ( 𝜎 ◦ 𝜏, Π ) ≤ 𝐷 ( 𝜎 , Π ) . (P4) Glob al Monotonicity. Le t Π ∈ P 𝑚 and define 𝜎 ∗ = argmax 𝜎 ∈ 𝔖 𝑚 𝐷 ( 𝜎 , Π ) . Supp ose that the de ep est p ermutation 𝜎 ∗ is unique. The n we have 𝑑 ( 𝜎 ∗ , 𝜎 ) ≤ 𝑑 ( 𝜎 ∗ , 𝜏 ) ⇒ 𝐷 ( 𝜏 , Π ) ≤ 𝐷 ( 𝜎 , Π ) . The pr o perties are quite similar to Definition 1 . First, (C2) do es not apply in th e case of per m u- tations. Indeed, a ffine transforms are undefined on the set 𝔖 𝑚 . Also, since the set is countable, (C5) do es no t a pply either. Next (P1) is similar to (C1) , and (P 3) and (P 4) corres pond to (C4) . Finally (P2) , as (C3) in the contin uous ca s e, states that the depth function is maximal a t a natur al center if s uch a center ex ists. Goib ert et a l. [ 2022 ] define a H-center (in Pro pos itio n 11), which corres p onds to a halfspace symmetr y . The a uthors also define a M -center (in D efinition 6), bas ed on the distance 𝑑 ( ) on 𝔖 𝑚 . W e refer to the la ter for pre cise definitions and discussions. These prop erties allo w the depth function to b e indeed maximal at the c ent er of a sample of per m utations, a nd to behave similarly t o a depth function defined on R 𝑚 . 7 Depth functions in Definition 2 are highly related to F r´ ec het means. F r´ echet [ 1948 ] intro duces the notion of typical p osition o f order 𝑝 , 𝑝 ≥ 1, for distributions on g e neral metric spa c es as an extension of the cla ssical moments of order 𝑝 as s ocia ted with distributions on Euclidean spa ces. This is kno wn a s 𝑝 - F r´ ec het mean. In the case o f p ermutations, the function a sso ciated to the 𝑝 -F r´ ec het mean can b e seen a s a depth function on the metric space ( 𝔖 𝑚 , 𝑑 ) as sho wn b elow. Definition 3. L et ( 𝔖 𝑚 , 𝑑 ) b e a metric sp ac e endowe d with a pr ob ability me asur e Π . F or 𝑝 ≥ 1 , the 𝑝 -F r ´ echet me an on ( 𝔖 𝑚 , 𝑑 ) with r esp e ct to Π is define d as argmin 𝜎 ∈ 𝔖 𝑚 𝑈 𝑑 , Π , 𝑝 ( 𝜎 ) , with 𝑈 𝑑 , Π , 𝑝 ( 𝜎 ) = E 𝑆 ∼ Π [ 𝑑 𝑝 ( 𝜎 , 𝑆 ) ] . Based on this notion, Goib ert et a l. [ 2022 ] defined some r anking depth functions, asso ciated to metrics on 𝔖 𝑚 . Definition 4 ( Goiber t et a l. [ 20 2 2 ]) . L et 𝔖 𝑚 b e the set of p ermutations on { 1 , . . . , 𝑚 } . L et 𝑑 ( ) b e a distanc e on 𝔖 𝑚 and 𝑝 ∈ R , 𝑝 ≥ 1 a given p ar ameter. The depth function asso ciate d to ( 𝑑 , 𝑝 ) is define d a s fol lows: ∀ 𝜎 ∈ 𝔖 𝑚 , for all Π ∈ P 𝑚 , 𝐷 ( 𝜎 , Π ) = k 𝑑 𝑝 k ∞ − E 𝑆 ∼ Π [ 𝑑 𝑝 ( 𝜎 , 𝑆 ) ] = k 𝑑 𝑝 k ∞ − 𝑈 𝑑 , Π , 𝑝 ( 𝜎 ) , with k 𝑑 𝑝 k ∞ = max 𝜎 , 𝜏 ∈ 𝔖 𝑚 𝑑 𝑝 ( 𝜎 , 𝜏 ) , 𝑆 a r andom variable of law Π , and 𝑈 ( ) is define d in D efinition 3 . The authors show ed that, under sufficient conditions on the distribution Π , the depth function given in the definition a bov e a sso ciated to the Kendall distance (see ( 2.2.1 )) sa tisfies the prop erties of Definition 2 for any 𝑝 ≥ 1. They also prov ed that (P1) and (P2) hold when using Sp earman fo otrule or S p earman 𝜌 - dis tance (( 2.2.1 ) with respectively 𝑞 = 1 and 𝑞 = 2 ). The deepest p oint can then b e defined as the p ermutation ma ximizing the depth function. Goib ert et al. [ 2022 ] intro duce the c onsensus r anking and the b aryc en t er r anking , which corre s ponds to the deep- est p oint s obtained resp ectively with 𝑝 = 1 and with 𝑝 = 2 in Definition 4 . W e will fo cus o n thes e t wo ca s es in the follo wing. They a re related to the 𝑝 -F r´ echet mea ns as follows. Definition 5 ( Go ibert et al. [ 202 2 ]) . L et ( 𝔖 𝑚 , 𝑑 ) b e a metric sp ac e endow e d with a pr ob abili ty me asur e Π . Denote 𝐷 1 ( ) and 𝐷 2 ( ) t he dep th functions asso ciate d r esp e ctively to ( 𝑑 , 1 ) and to ( 𝑑 , 2 ) as define d i n Definition 4 . Then, (i) a co nsensus r anking 𝜎 ∗ ∈ 𝔖 𝑚 is a p ermutation such that 𝐷 1 ( 𝜎 ∗ , Π ) = max 𝜎 ∈ 𝔖 𝑚 𝐷 1 ( 𝜎 , Π ) = k 𝑑 k ∞ − min 𝜎 ∈ 𝔖 𝑚 𝑈 𝑑 , Π , 1 ( 𝜎 ) ; (ii) a bar ycen ter rank ing 𝜎 ∗ ∈ 𝔖 𝑚 is a p ermut ation such that 𝐷 2 ( 𝜎 ∗ , Π ) = max 𝜎 ∈ 𝔖 𝑚 𝐷 2 ( 𝜎 , Π ) = k 𝑑 2 k ∞ − min 𝜎 ∈ 𝔖 𝑚 𝑈 𝑑 , Π , 2 ( 𝜎 ) . R emark 1 . No te that E Π [ 𝑑 ( 𝜎 , S ) ] = 𝑈 𝑑 , Π , 1 ( 𝜎 ) mea ns that, for any distance 𝑑 ( ) a nd any pr obability distribution Π on 𝔖 𝑚 , the ra nking given by maximizing the depth function 𝐷 1 ( 𝜎 , Π ) can b e retrieved by minimizing the functional a sso ciated to the 1-F r´ echet mean on the same spac e . In 8 particular, the co nsensus ranking g iv en by the depth function 𝐷 1 ( 𝜎 , Π ) is a 1 -F r´ ec het mean on ( 𝔖 𝑚 , 𝑑 , Π ) a nd ca n b e seen as a median r anking on 𝔖 𝑚 . R emark 2 . Using the ter minology introduced by Zuo and Serfling [ 2000 ], 𝑈 𝑑 , Π , 𝑝 corres p onds to the inv erse o f a T yp e B depth function . Similarly to the contin uous case, we will apply the depth functions on the empirical distribution of v oting opinio ns. Co ns ider a v oting situation wit h 𝑛 v oter s and 𝑚 candidates , with an opinions matrix Φ ∈ E 𝑛 . If the opinions a r e rankings, Φ writes as Φ = ( 𝜎 𝑣 ) 𝑣 = 1 , .. . , 𝑛 , a nd E = 𝔖 𝑚 . L e t 𝐷 ( ) b e a depth function defined on E 𝐷 × P 𝑚 . In the follo wing, we will denote 𝐷 ( ., Φ ) the depth function 𝐷 ( 𝑥 , 𝐹 Φ ) where 𝐹 Φ is the empirical distribution of the v oter ’s opinio ns in E = 𝔖 𝑚 , { Φ ( . , 𝑣 ) = ( 𝑒 𝑐 𝑣 ) 𝑐 ∈ { 1 , . .. , 𝑚 } , 𝑣 = 1 , . . . , 𝑛 } . 3 Deep est v oting The depth functions b eing defined, deepe st voting pro cedures can b e built as follows. Definition 6 (Deep est V o ting ) . Consider a voting situation with 𝑛 voters and 𝑚 c andida tes, C = { 1 , . . . 𝑚 } , with an opinion matrix Φ ∈ E 𝑛 . L et 𝐷 ( . , . ) b e a depth function define d on E 𝐷 × F , wher e E ⊆ E 𝐷 ⊆ R 𝑚 and F is the set o f pr ob abi lity distributions on E 𝐷 . Denote E ∗ 𝐷 : = { 𝐸 ∈ R 𝑚 : 𝐷 ( 𝐸 , Φ ) = sup ( 𝐷 ( . , Φ ) ) } the set o f de ep est p oints (opinions) with r esp e ct to Φ . L et 𝑐 ∗ 𝐷 : = ar gmin 𝑐 ∈ C { 𝑒 ∗ 𝐷 , 𝑐 , 𝐸 ∗ 𝐷 = ( 𝑒 ∗ 𝐷 , 1 , . . . , 𝑒 ∗ 𝐷 , 𝑚 ) ⊤ ∈ E ∗ 𝐷 } . The de ep est voting pr o c ess wi th r esp e ct to the depth 𝐷 ( ) is de fine d as the function which map s { Φ ( 𝑐 , 𝑣 ) , 𝑐 = 1 , . . . , 𝑚 , 𝑣 = 1 , . . . , 𝑛 } t o 𝑐 ∗ 𝐷 ⊆ C . If 𝑐 ∗ 𝐷 is unique, t hen the winner of the ele ction is t he c andidate 𝑐 ∗ 𝐷 . If 𝑐 ∗ 𝐷 is not unique, ther e is no unique wi nner of the ele ction. Aubin et al. [ 2 022 ] consider the framework of ev aluations. In suc h a case, the set of opinions E satisfies E = Λ 𝑚 , with Λ = { 0 , . . . , 𝐾 } or Λ = [ 0 , 𝑀 ] . In that cases, the authors considere d depth f unctions defined on E 𝐷 = R 𝑚 . Statistical depths functions are w ell d efined in suc h spaces, which c orresp ond to 𝑚 -multiv aria te con tinuous distributions, as r ecalled in Section 2 .1 . Some results in this fra mew ork are g iven in Section 4 . The rela tion b et ween usual voting pr ocedur es and given depth functions will b e displa yed. Note that in the discrete case depth functions defined on E 𝐷 = N 𝑚 could hav e b een co nsidered, but, t o our knowledge, such depth functions are not defined in litera ture. The most common voting metho ds a r e ba sed o n preference rankings . This includes ma jority voting systems as w ell as those ba sed on the Condor cet o r Borda pr inc iples , among others. A review of these voting metho ds c a n b e found in F els en thal and Nurmi [ 2019 ]. In that ca ses, the set of opinions E satisfies E = 𝔖 𝑚 , with 𝔖 𝑚 the set of p ermutations of { 1 , . . . , 𝑚 } . The first p ossibility is 9 to consider depth functions defined on E 𝐷 = [ 0 , 𝑚 ] 𝑚 , which indeed includes E = 𝔖 𝑚 . Usual depth functions used in the ev alua tion-based framework can then b e used. An alterna tiv e is to consider depth functions defined o n E 𝐷 = E = 𝔖 𝑚 , as defined in Section 2 .2 . This choice resp ects more the nature of the data. Section 5 show that s ome usual pr o cedur e s can b e written as p ermut ation deep e st voting. Relations b etw een voting pro cedures and per m utation depth functions are studied in Section 5.2 . 4 Con tin u ous deep est v oting Let us consider fir st e v aluatio n-based voting. Recall that in such a case, the v oters give opinions on candidates on t he form of ev alua tion, tha t is, for eac h v oter 𝑣 and candidate 𝑐 , the opinion is 𝑒 𝑐 𝑣 ∈ Λ , with Λ = { 1 , . . . , 𝐾 } or Λ = [ 0 , 𝑀 ] . Hence, e ac h voter has a vector o f opinions in E = Λ 𝑚 . With 𝑛 v oters and 𝑚 candidates, the opinion matrix , denoted Φ , belong s to E 𝑛 . In that case, we consider depth functions defined on R 𝑚 , with prop erties giv en in Definition 1 . F or a given depth function 𝐷 ( ) , the a sso ciated deepest voting rule is obtained by maximizing 𝐷 ( . , Φ ) , as des c r ibed in Definition 6 . In this context, Aubin et al. [ 20 22 ] established parallelism betw een conditions o n the depth func- tions a nd usua l a xioms on v oting pro cedures. The authors pr o ve that a voting procedure defined in Definition 6 with a depth function defined in Definition 1 s atisfies the proper ties of Neutralit y , Univ ers alit y and Unanimity . They also study Monotonic ity for given depths functions, and they provide a c ondition on t he depth function to satisfy Indep endence to Irrelev an t alternative. Next, Aubin et al. [ 2022 ] sho wed that cla ssical ev aluatio n-based v oting rules can be expressed as deep e st voting procedure s with 𝐿 𝑞 depths, 𝑞 ≥ 1. If ev a luation grades are in Λ = { 0 , 1 } , as seen in Dort and friends [ 20 25 ], for all 𝑞 ≥ 1, 𝐿 𝑞 depth voting is equiv alent to approv al voting [ Brams and Fish burn , 2007 ]. If ev aluation grades are in Λ = [ 0 , 1 ] , 𝐿 𝑞 deep e st v oting with 𝑞 = 1 and 𝑞 = 2 re c o vers resp ectiv ely ma jor it y judgment [ Balinski a nd La raki , 2 007 ] and r ange voting [ Smith , 200 0 ]. T hes e results are summar ized in T able 1 . Depth V oting pro cess o n [0,1] 𝐿 1 V o te to the highest median (ma jor ity judgment) 𝐿 2 V o te to the highest mean (ra nge voting) Depth V oting pro cess o n { 0,1 } 𝐿 𝑞 , 𝑞 ≥ 1 Approv al voting Depth V oting pro cess o n 𝔖 𝑚 𝐿 1 Bucklin’s voting 𝐿 2 Borda’s voting T a ble 1: (Con tinuous) 𝐿 𝑞 deep e st voting rules . Considering r anking-based pro cedure, we can also wr ite some pr oce dures as c o n tinuous deep est voting process . Indeed, let us consider a framework where, for each v oter 𝑣 , the opinions on the 10 candidates are a vector ( 𝑒 𝑐 𝑣 ) 𝑐 = 1 , .. . , 𝑚 which is a p ermutation of { 1 , . . . , 𝑚 } . The opinions b elong to E = 𝔖 𝑚 , the s et o f p ermutations of { 1 , . . . , 𝑚 } , which is included in { 1 , . . . , 𝑚 } 𝑚 . Consequently , we ca n see the ra nkings as ev aluatio ns and apply depth functions defined on R 𝑚 . W e can rec over existing voting rules. In particular, let conside r Borda ’s pro cedure and Bucklin’s pro cedure. McCab e-Dansted and Slink o [ 2006 ] defines the Bucklin’s voting r ule (also called Ma joritaria n Compro mise) winner as the can- didate with the lowest median rank. W e c an prov e that these so cial choice functions a r e related resp ectively to 𝐿 2 and 𝐿 1 deep e st voting. Prop osition 1. Consider 𝑛 voters and 𝑚 c andidates. L et 𝑒 𝑐 𝑣 b e the r ank in { 1 , . . . , 𝑚 } given by voter 𝑣 to c andidate 𝑐 , and denote Φ = ( 𝑒 𝑐 𝑣 ) 𝑣 , 𝑐 ∈ 𝔖 𝑚 the obtaine d r anks. Then, (i) Bor da’ s pr o c e dur e is the (c ontinuous) 𝐿 2 -de ep est voting applie d on Φ ; (ii) Bucklin ’s winner(s) i s include d i n t he (c ontinuous) 𝐿 1 -de ep est vo ting set on Φ , 𝑐 ∗ 𝐷 define d in Definition 6 . When the winner is unique Bucklin ’s winner c oincides with t he winner of 𝐿 1 de ep est voting. The proof is straig h tforward and th us omitted. Noted that with 𝐿 1 pro cedure, w e distinguish whether the set 𝑐 ∗ 𝐷 is a singleton o r not. Indee d, consider the c ase of 3 candidates and 4 voters with Φ equal to 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑐 1 2 2 1 1 𝑐 2 1 1 3 3 𝑐 3 3 3 2 2 In this configuration, Bucklin’s winner is candidate 𝑐 1 . But all vector ( 𝑎 1 , 𝑎 2 , 𝑎 3 ) ⊤ with 𝑎 1 ∈ [ 1 , 2 ] , 𝑎 2 ∈ [ 1 , 3 ] , 𝑎 3 ∈ [ 2 , 3 ] is in the deep est set. Hence 𝑐 1 , 𝑐 2 and 𝑐 3 belo ng to 𝑐 ∗ 𝐷 . 5 Deep est v oting on p erm utations Let us now deal with ranking- based voting. The voters give opinions on candidates on the form o f rankings, that is, for each voter, the vector of o pinio ns b elongs to E = 𝔖 𝑚 , the set of p ermutations of { 1 , . . . , 𝑚 } . With 𝑛 v oters a nd 𝑚 candidates, the opinion matrix, denoted Φ , b elongs t o E 𝑛 . In that ca se, we consider depth functions defined on 𝔖 𝑚 , with pr oper ties given in Definition 2 . F or a given depth function 𝐷 ( ) , the ass o cia ted voting rule is obtained ma ximizing 𝐷 ( . , Φ ) , a s described in Definition 6 . In particular, based on Definition 4 , we prop ose to define the 𝑝 -F r´ ech et mean voting rule on rankings. 11 Definition 7. Consider a voting situ ation Φ ∈ 𝔖 𝑛 𝑚 , and a distanc e 𝑑 ( ) on 𝔖 𝑚 . F or al l 𝑝 ≥ 1 , denote 𝔖 ∗ ( 𝑑 , Φ , 𝑝 ) : = argmin 𝜎 ∈ 𝔖 𝑚 𝑈 𝑑 , Φ , 𝑝 ( 𝜎 ) , with 𝑈 𝑑 , Φ , 𝑝 ( 𝜎 ) = 1 𝑛 𝑛 Õ 𝑣 = 1 𝑑 𝑝 ( 𝜎 , Φ ( ., 𝑣 ) ) the set o f 𝑝 -F r´ echet me ans of ( 𝔖 𝑚 , Φ ) with r esp e ct to 𝑑 . L et 𝑐 ∗ 𝑝 , 𝑑 : = { 𝑐 ∈ { 1 , . . . , 𝑚 } , 𝜎 ∗ ( 𝑐 ) = 1 , 𝜎 ∗ ∈ 𝔖 ∗ ( 𝑑 , Φ , 𝑝 ) } . If 𝑐 ∗ 𝑝 , 𝑑 is unique, t hen the winner of the ele ction asso ciate d to ( 𝑑 , 𝑝 ) is the c andidate 𝑐 ∗ 𝑝 , 𝑑 . If 𝑐 ∗ 𝑝 , 𝑑 is not u nique, ther e is no unique winner of the ele ction asso ciate d to ( 𝑑 , 𝑝 ) . As for co n tinuous deep est v oting, we will show that we rec o ver usual voting rules . 5.1 Classical rules as F r´ ec het v oting Combining distances b etw een p ermutations, depth functions on p ermutations lea d to a huge n um- ber of voting rules, as defined in definition 6 . Some of these rules co rresp ond to well-kno wn voting rules. 5.1.1 Un w eighted dis tances Let us first consider dista nc e s without w eights (or with w eights all e qual to 1 / 𝑚 ). W e can es tablish links b et ween Kemeny’s procedure and Borda’s pro cedure a nd some deep est voting pro cedure defined in Definition 7 . Recall that Kendall’s distance was defined in ( 2.2.1 ) and that Sp earman 𝜌 distance is 𝑞 -Minko wski-H¨ o lder distance (( 2.2.1 )) with 𝑞 = 2. Prop osition 2. The de ep est voting rule asso ciate d to c onsensus r anking and Kendal l distanc e on 𝔖 𝑚 ele cts t he Kemeny’ s winner of the ele ction. Prop osition 3. The de ep est voting rule asso ciate d t o b aryc enter r anking and Sp e arman 𝜌 distanc e is e quivalent to Bor da’ s voting metho d. Pro ofs o f these prop ositions can be found in Appendix A . No te that Prop osition 1 shows that Borda’s pro cedure ca n a lso b e seen as a con tinuous 𝐿 2 -deep est voting. Borda’s voting method is known to be the ranking -mean v oting rule. It is not surpris ing that it corres p onds to the Sp earman 𝜌 distance deep est voting. Similar ly , Buc klin voting metho d is known to b e the r a nking-median voting rule, and Sp earman fo otrule ( 𝑞 -Minko wski-H¨ older distance ( 2.2.1 ) with 𝑞 = 1) se e ms a natural dista nce to asso ciate with. Ho wev er, it is not the case, as sta ted in the following prop osition. Prop osition 4. Bucklin ’s voting rule do es not c orr esp ond to the de ep est voting rule a sso ciate d to c onsensus r anking and Sp e arman fo otrule di stanc e, nor to the de ep est voting rules asso ciate d t o b aryc enter r anking and Sp e arman fo otrule distanc e as p ermut ation distanc e. 12 A proo f is provided in App endix A . Remark that Propo sition 1 expr esses Buc klin’s pro cedure as a result of a con tinuous deep est voting, where ranks are seen as ev aluations. 5.1.2 W ei g h ted distances Let us supp ose now tha t the dista nc e s are weigh ted, as prop osed in Section 2 .2.2 , with no n uni- form weigh ts. Specia l set of w eights can b e used to represen t plurality and anti-plurality rules with weigh ted Hamming distance ( 2.2.2 ) or w eighted Spearman fo otrule distance (w eighted 𝑞 - Minko wski-H¨ older distance ( 2 .2.2 ) with 𝑞 = 1). In the follo wing we denote 𝐷 𝑉 ( 𝑝 , 𝑑 , 𝑊 ) the v oting rule on per mutations asso ciated to the 𝑝 - F r´ echet mean as depth function, the p ermutation distance 𝑑 ( ) and the weigh ts 𝑊 . Reca ll that, as discussed previously , the ass ocia ted w eighted di stanc e functions ar e not true distance s on the set o f p erm utations. F or th e sake o f simplicity , w e denote the optimum p ermut ation given by the voting rule as the de ep est p ermut ation . Denote 𝑊 ( 1 , ( 1 ) ) the weight s suc h that 𝑤 𝑟 , 𝑟 ′ = 1 if 𝑟 = 1 or 𝑟 ′ = 1 and 𝑤 𝑟 , 𝑟 ′ = 0 elsewhere. Denote also 𝑊 ( − 1 , ( 𝑚 ) ) the weigh ts such that 𝑤 𝑟 , 𝑟 ′ = − 1 if 𝑟 = 𝑚 or 𝑟 ′ = 𝑚 and 𝑤 𝑟 , 𝑟 ′ = 0 elsewhere. These weigh ts allow us to link some well known voting metho ds to the depth framework on per m utations. Prop osition 5. 𝐷 𝑉 ( 1 , Hamming , 𝑊 ( 1 , ( 1 ) ) ) ar e e quivalent to Plur ality voting met ho d. Prop osition 6. 𝐷 𝑉 ( 1 , Hamming , 𝑊 ( − 1 , ( 𝑚 ) ) ) ar e e quivalent to Antiplur ality voting metho d. Pro ofs are given in Appendix B . Distance W eig h ts 𝑝 in 𝑝 -F r´ ec het mean V oting rule Kendall none 1 Kemeny Spea rman 𝜌 none 2 Borda Hamming 𝑊 ( 1 , ( 1 ) ) 1 P luralit y Hamming 𝑊 ( − 1 , ( 𝑚 ) ) 1 Antiplurality T a ble 2: Connections b et ween deep est voting r ules on permutations a nd c la ssical voting r ules. Obtained links b etw een clas sical voting r ule s a nd dee p est voting rules are summarized in T a ble 2 . The ob jective of this pap er is no t to pr o vide an exhaustive list of links b etw een usual so cial choice functions and deep est voting pro cedures, but to highligh t the connections. This works shows that several s ocia l choice functions can b e asso ciated with a depth function, and hence write as an optimization pro blem. 5.2 V oting rules prop erties Studying forma l pro perties o f voting rules is a matter of interest for an a xiomatic approa c h o f elections. [ F els en thal and Nurmi , 2018 , Chapter 2] contains a go o d r eview of classical prop erties of a voting rule. W e prop ose hereafter to discuss so me of them. Aubin et al. [ 2 022 ] has related 13 some prop erties of deepest voting rules with t he ma thematical prop erties o f t he co n tinuous depth functions. W e extend their result to p ermutation-based deep est v oting. Definition 8 (V o ting rules prop erties) . • Neutrality . The so cial cho ic e f un ction gives the same r esult by p ermuting the r ows of Φ ( i.e. by p ermut ing the c andidates). • Anon ymity . The vo ting pr o c e dure gives the same r esu lt b y p ermuting the c olumns of Φ ( i.e. by p ermut ing the voters). • Univ ersa lit y . F or al l Φ ∈ 𝔖 𝑛 𝑚 , the vo ting pr o c e dur e pr ovides a subset of C . • Unanimit y . If a c andidate is in first p osition for al l voters, it should b e in first p osition in the voting p r o c e dur e’s r esults. • Monotonicity . S upp ose that a voting situation Φ gives a winner 𝑐 ∗ , and that ther e exists a voter 𝑣 0 ∈ { 1 , . . . , 𝑛 } such t hat Φ ( 𝑐 ∗ , 𝑣 0 ) = 𝛼 ≠ 1 . L et 𝑐 0 b e the c andidate such that Φ ( 𝑐 0 , 𝑣 0 ) = 𝛼 − 1 . Consid er another voting situation e Φ such that e Φ = Φ ex c ept tha t e Φ ( 𝑐 ∗ , 𝑣 0 ) = 𝛼 − 1 and e Φ ( 𝑐 0 , 𝑣 0 ) = 𝛼 . Then the winner of the v oting pr o c e dur e is stil l 𝑐 ∗ . • Independence to Losers . Consider a voting situation Φ with 𝑚 c andidates C = { 𝑐 1 , . . . , 𝑐 𝑚 } and 𝑛 voters, with a winner 𝑐 ∗ . Consider another voting situation e Φ with 𝑚 − 1 c andida tes and 𝑛 voters, wher e a c andid ate 𝑐 0 ≠ 𝑐 ∗ as b e en r emove d. Su pp ose that e Φ is i dentic al on the r e duc e d set of c andidates. That is, f or any voter 𝑣 ∈ { 1 , . . . , 𝑛 } , the or der of { Φ ( 𝑐 , 𝑣 ) , 𝑐 ∈ C \ { 𝑐 0 } } is the same as the or der of { e Φ ( 𝑐 , 𝑣 ) , 𝑐 ∈ C \ { 𝑐 0 } } . Then the winner for e Φ is stil l 𝑐 ∗ . • Condorcet Winner . A c andida te 𝑐 0 ∈ C is a Condor c et winner if for al l c andidates 𝑐 ∈ C \ { 𝑐 0 } , the numb er of voters 𝑣 such that Φ ( 𝑐 0 , 𝑣 ) < Φ ( 𝑐 , 𝑣 ) is strictly higher t han the numb er of voters 𝑣 such that Φ ( 𝑐 0 , 𝑣 ) > Φ ( 𝑐, 𝑣 ) . A voting rule satisfies t he Condor c et Winner pr op ert y if it ele cts the Condor c et winner whe n it exists. • Condorcet Lo ose r . A c andidate 𝑐 0 ∈ C is a Condor c et lo oser if for al l c andidates 𝑐 ∈ C \ { 𝑐 0 } , the nu mb er of voters 𝑣 such that Φ ( 𝑐 0 , 𝑣 ) < Φ ( 𝑐 , 𝑣 ) is strictly lower than the nu mb er of voters 𝑣 su ch that Φ ( 𝑐 0 , 𝑣 ) > Φ ( 𝑐 , 𝑣 ) . A voting rule satisfies the Condor c et lo oser pr op ert y if it n ever ele cts the Condor c et lo oser when it exists. Our ob jective in this section is to study if the pro perties stated ab ov e ar e satisfied with deep est vot- ing rules. As sa id pr eviously , some results were o btained in Aubin et al. [ 20 22 ] a nd we concentrate here on permutation-based deep est v oting. Pro ofs of this section can b e found in Appendix C . First, Neutra lit y , Anonymit y and Universality ar e satisfied wha tev er the depth function used. As for Una nimity , we can prove it is satisfie d by deepest voting rules for a larg e category of per m utation-bas ed depth f unctions. Prop osition 7. Any de ep est voting p r o c ess satisfies Neutr ality, Anonymity and Universality. 14 Prop osition 8 . Consider a distanc e on p ermutations which is either H amming, Kendal l or a 𝑞 - Minkowski-H¨ older distanc e on p ermutations, with 𝑞 ∈ [ 1 , +∞ [ . Then, for al l 𝑝 ≥ 1 , the asso ciate d 𝑝 -F r´ echet me an voting satisfies Unanimity. When 𝑞 = ∞ , 𝑝 -F r´ echet me an voting satisfies Unanimity when the winner is unique, but it may not satisfy Unanimity otherwise ( i.e. ther e ar e at le ast two p ermutations with a differ ent t op-r anke d c andid ate in the de ep est set). The Co ndorcet-winner pr oper t y and the Monotonicity a r e sa tisfied by some deep est voting pro ce- dures, as stated in the tw o following pro pos itio ns. Prop osition 9. 1. F or 𝑝 = 1 , Kendall-b ase d c onsensus r anking satisfies the Condor c et-winner pr op ert y. 2. F or 𝑝 > 1 , Kenda l l-b ase d 𝑝 -F r´ echet me an do es not sa tisfy the Condor c et-winner pr op erty. 3. F or al l 𝑝 ≥ 1 , Cayley and Hamming b ase d voting ru les do not satisfy the Condor c et-winner pr op erty. 4. F or 𝑝 = 1 and al l 𝑞 ≥ 1 , 𝑞 -Minkowski-H¨ older b ase d voting rules d o not sa tisfy the Condor c et- winner pr op erty. Prop osition 10. 1. The c onsensus r anking ( 𝑝 = 1 ) asso ciate d to the Sp e arman fo otrule ( 1 -Minkowski -H¨ older) satisfies the Mono tonicity pr op erty. 2. The c onsensus r anking ( 𝑝 = 1 ) asso ciate d t o the Hamming distanc e do es not satisfy t he Monotonicity pr op erty. Finally , I ndep endence to Lo s ers prop erty is no t satisfied by a large categor y o f p ermutation-based voting rules. Prop osition 11. Hamming-b ase d, C ayley-b ase d, Kendal l-b ase d and M inkowski-H¨ older b ase d c on- sensus ( 𝑝 = 1 ) de ep est voting rules do not satisf y Indep endenc e to L osers pr op erty. T a ble 3 b elow summar izes the r esults of this section. Distance Neutrality Anonymit y Unanimit y Monotonicit y In d.Losers Condorcet winner Hamming    N for 𝑝 = 1 N for 𝑝 = 1 N Kendall    . N  only for 𝑝 = 1 Ca yley   . . N N Minko wski-H¨ older q=1     for 𝑝 = 1 N N for 𝑝 = 1 𝑞 > 1    . N N for 𝑝 = 1 T a ble 3 : The table s yn thesizes if the deep est voting pro cedure as socia ted to several depth functions the prop erties given in Definition 8 . Deep est voting rules are 𝑝 -F r´ ec het means, with differen t distances on p erm utations.  co rresp onds to verified pro perties , N to non verified ones, . to cases where no re sult was obtained. Wh en results ar e only obtained for a given v a lue of 𝑝 , it is precised in the corresp onding entry . 15 6 Conclusion In this pape r, we ex tended the deepest voting framew ork to ranking- ba sed voting rules b y in tro- ducing depth functions defined on the space of p ermutations. This reinforces links b etw een s ocia l choice theor y a nd statistical depth. This approa c h pro vides a unified statistical in terpretation of several classica l so cial choice pro cedure s as so lutions to optimization problems base d o n dis ta nces betw een rankings. Note that we focused in this w ork on distance - based depth functions. Other depth functions ca n be used, such as half-space or pro jection depths, that r ely more on the geometrical prop erties of the data . It may b e considered in future work. W e show ed tha t w ell-known rules suc h as Kemen y , Borda, P lurality a nd Antipluralit y can be ex- pressed a s deep est voting pro cedures asso ciated with a ppropriate (weigh ted) distances and F r´ echet means. Moreover, we analyzed key axio matic pr oper ties (univ ersa lity , monotonicity , etc.) of dee p- est voting pro cedures. Beyond these co nnections, the prop osed framework op e ns the wa y to the design of new voting r ules based on weigh ted or pro blem-specific distances be t ween r ankings. F uture work will fo cus on a deep e r axiomatic analysis or computational a spects. 7 Ac kno wledgemen t This work has b een supp orted by ANR F r a nce 20 30 agency through the PEPR Maths-Vives Con- dorcet pro ject ANR-24-E XMA-0001. References J.-B. Aubin, I. Gannaz, S. Leoni, and A. Rolland. Deep est v oting: a new way of electing . Mathe- matic al S o cial Scienc es , 11 6:1–16, 2022 . M. Balinski and R. Lara ki. A theor y of measuring , electing, and ranking. Pr o c e e dings of the National A c ademy of Scienc es , 104(21):872 0 –8725, 2007. S. Br ams and P . C. Fishburn. Appr oval voting . Spr inger, 2007. M. Deza and E. Deza. Encyclop e dia of Distanc es . Spring er B erlin Heidelber g, 2016 . ISBN 97836 625284 40. URL https: //books .google.fr/books?id=KQHdDAAAQBAJ . M. Deza and T. Huang. Metrics on p ermutations, a sur v ey . J . Co mb. Inf. Sys. Sci. , 23, 02 1997. P . Dia conis. Group represe ntations in proba bilit y and statistics. 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Pro of of Prop osition 2 Quoting F elsenthal and Nurmi [ 2019 ], p. 26, Kemeny‘s so cial c hoice pro cedure can also b e viewed as finding the most likely (or the best predictor, or the best compromise) true so cia l preference ordering, ca lled the median pr e ference ordering , i.e., that so cial pr eference ordering 𝑆 that minimizes the sum, ov er all voters 𝑖 , of the num ber o f pa ir s of candida tes that ar e or dered opp ositely by 𝑆 and b y the 𝑖 𝑡 ℎ voter. It is exactly the definition of the co nsensus r anking deep e st voting rule with Kendall distance. Pro of of Prop osition 3 . W e consider here that the Borda voting rule elects a unique candidate. Suppo se that 𝑐 ∗ is chosen with the Borda voting r ule. It mea ns that ∀ 𝑐 ∈ C , 𝑐 ≠ 𝑐 ∗ , 𝑛 Õ 𝑣 = 1 𝜎 𝑣 ( 𝑐 ∗ ) < 𝑛 Õ 𝑣 = 1 𝜎 𝑣 ( 𝑐 ) , where 𝜎 𝑣 ( 𝑐 ) = 𝑒 𝑐 𝑣 denotes the r ank given by voter 𝑣 to candidate 𝑐 . Then { 𝜎 𝑣 ) 𝑐 } is a finite family of cardina l 𝑛 ≥ 2 . Denote Π 𝑛 its empirical distribution. Let 𝜎 ∗ ∈ 𝔖 𝑚 a p ermutation and 𝑐 ∗ ∈ { 1 , . . . , 𝑚 } such that 𝜎 ∗ ( 𝑐 ∗ ) = 1. Let 𝑐 0 ∈ C , 𝑐 0 ≠ 𝑐 ∗ . No te 𝜏 = ( 𝑐 ∗ , 𝑐 0 ) the transp osition ex c hanging 𝑐 ∗ and 𝑐 0 . Let us show that 𝜎 ∗ 18 has a lo wer 2-F rechet mean than 𝜎 ∗ ◦ 𝜏 . That is, we wan t to p rove that E [ 𝑑 2 ( 𝜎 ∗ , Π 𝑛 ) ] < E [ 𝑑 2 ( 𝜎 ∗ ◦ 𝜏 , Π 𝑛 ) ] , with 𝑑 ( ) the 𝑞 -Mink owski-Holder dis tance for 𝑞 = 2. Using success iv ely the d efinition of Π 𝑛 and the definition of the distance 𝑑 ( ) , we hav e E [ 𝑑 2 ( 𝜎 ∗ , Π 𝑛 ) ] − E [ 𝑑 2 ( 𝜎 ∗ ◦ 𝜏 , Π 𝑛 ) ] = 1 𝑛 𝑛 Õ 𝑣 = 1 𝑑 2 ( 𝜎 ∗ , 𝜎 𝑣 ) − 1 𝑛 𝑛 Õ 𝑣 = 1 𝑑 2 ( 𝜎 ∗ ◦ 𝜏 , 𝜎 𝑣 ) = 1 𝑛 𝑛 Õ 𝑣 = 1 𝑚 Õ 𝑐 = 1 ( 𝜎 ∗ ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) ) 2 − 1 𝑛 𝑛 Õ 𝑣 = 1 𝑚 Õ 𝑐 = 1 ( 𝜎 ∗ ◦ 𝜏 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) ) 2 . Definition of 𝜏 yields E [ 𝑑 2 ( 𝜎 ∗ , Π 𝑛 ) ] − E [ 𝑑 2 ( 𝜎 ∗ ◦ 𝜏 , Π 𝑛 ) ] = 1 𝑛 𝑛 Õ 𝑣 = 1  ( 𝜎 ∗ ( 𝑐 ∗ ) − 𝜎 𝑣 ( 𝑐 ∗ ) ) 2 + ( 𝜎 ∗ ( 𝑐 0 ) − 𝜎 𝑣 ( 𝑐 0 ) ) 2 − ( 𝜎 ∗ ( 𝑐 0 ) − 𝜎 𝑣 ( 𝑐 ∗ ) ) 2 − ( 𝜎 ∗ ( 𝑐 ∗ ) − 𝜎 𝑣 ( 𝑐 0 ) ) 2  = 2 𝑛 𝑛 Õ 𝑣 = 1 ( 𝜎 ∗ ( 𝑐 ∗ ) − 𝜎 ∗ ( 𝑐 0 ) ) ( 𝜎 𝑣 ( 𝑐 0 ) − 𝜎 𝑣 ( 𝑐 ∗ ) ) = 2 𝑛 ( 𝜎 ∗ ( 𝑐 ∗ ) − 𝜎 ∗ ( 𝑐 0 ) )  𝑛 Õ 𝑣 = 1 𝜎 𝑣 ( 𝑐 0 ) − 𝑛 Õ 𝑣 = 1 𝜎 𝑣 ( 𝑐 ∗ )  . Using Equa tio n ( A ) a nd the fact that 𝜎 ∗ was defined such that 𝜎 ∗ ( 𝑐 ∗ ) − 𝜎 ∗ ( 𝑐 0 ) < 0, we deduce tha t the rig h t hand side is strictly negative. Hence the deep est p ermutations are s uc h that 𝜎 ∗ ( 𝑐 ∗ ) = 1, and therefor e the deep e s t voting rule elects the winner of Borda r ule. Pro of of Prop osition 4 Consider a case with 5 voters and 4 candidates, with the voting table is prop osed in T able 4 b elow. In this s etting, the median rankings are (3,2 ,3,4) a nd there fo re 𝑐 2 is the w inner of Buc klin’s vote 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑐 1 1 1 4 4 3 𝑐 2 2 2 2 2 2 𝑐 3 3 3 3 3 1 𝑐 4 4 4 1 1 4 T a ble 4: Matrix Φ of t he v oting situa tio n for the pro of o f P rop osition 4 . as it has the smallest median ra nking. But the optimum of the 𝑝 -F r´ echet mean with the Spear man fo otrule distance is ( 1 , 2 , 3 , 4 ) ⊤ , fo r b oth ca ses 𝑝 = 1 and 𝑝 = 2. Therefore 𝑐 1 is the winner o f 𝐷𝑉 ( 1 , Sp earman f o otrule ) , and 𝐷𝑉 ( 2 , Sp earman f o otrule ) . This co ncludes the pr oo f. 19 B Pro ofs for w eigh ted distances The pro ofs dealing with re s ults on depths base d on weigh ted p ermutation distance, given in Section 5.1.2 , are displayed in this section. Pro of of Prop osition 5 The weigh ted Hamming distance with the w eights 𝑊 ( 1 , ( 1 ) ) writes a s for all 𝜎, 𝜏 ∈ 𝔖 𝑚 , 𝑑 ( 𝜎 , 𝜏 ) =        2 if 𝜎 − 1 ( 1 ) ≠ 𝜏 − 1 ( 1 ) , 0 if 𝜎 − 1 ( 1 ) = 𝜏 − 1 ( 1 ) . Therefore, 𝜎 ∗ ∈ 𝔖 𝑚 minimizes Í 𝑛 𝑣 = 1 𝑑 ( 𝜎 , 𝜎 𝑣 ) if 𝜎 ∗ − 1 ( 1 ) = argma x 𝑐 = 1 , .. . , 𝑚 𝑛 Õ 𝑣 = 1 1 ( 𝜎 − 1 𝑣 ( 1 ) = 𝑐 ) . It concludes the pro of since it is exactly the definition of Plurality voting rule. Pro of of Prop osition 6 The weigh ted Hamming distance with the w eights 𝑊 ( − 1 , ( 𝑚 ) ) writes as for all 𝜎, 𝜏 ∈ 𝔖 𝑚 , 𝑑 ( 𝜎 , 𝜏 ) =        − 1 if 𝜎 − 1 ( 𝑚 ) ≠ 𝜏 − 1 ( 𝑚 ) , 0 if 𝜎 − 1 ( 𝑚 ) = 𝜏 − 1 ( 𝑚 ) . Therefore, 𝜎 ∗ ∈ 𝔖 𝑚 minimizes Í 𝑛 𝑣 = 1 𝑑 ( 𝜎 ∗ , 𝜎 𝑣 ) , if 𝜎 ∗ − 1 ( 𝑚 ) = a rgmin 𝑐 = 1 , .. . , 𝑚 𝑛 Õ 𝑣 = 1 1 ( ( 𝜎 − 1 𝑣 ( 𝑐 ) = 𝑚 ) ) . It concludes the pro of since it is exactly the definition of An tiplurality voting r ule. C Pro ofs of S ection 5.2 This section provides the pr oo fs dealing with the properties of the deep est voting pr ocedur es, stated in Section 5.2 . The pro p erties are defined in Definition 8 . 20 Pro of of Prop osition 7 Anonymity . F o r contin uous depth functions, the prop ert y has b een pr o ven in [ Aubin et al. , 2022 , Prop osition 1]. Consider a depth on p erm utations, defined in Definition 4 . F o r any v oting situation Φ , with empirical dis tr ibution Π 𝑛 , the deepest voting r elates o nly o n the function which asso ciate a p ermutation 𝜎 ∈ 𝔖 𝑚 to 𝑈 𝑑 , Π 𝑛 , 𝑝 ( 𝜎 ) = 1 𝑛 Í 𝑛 𝑣 = 1 𝑑 𝑝 ( 𝜎 , Φ ( ., 𝑣 ) ) . It is straight- forward that, 𝑈 𝑑 , Π 𝑛 , 𝑝 ( ) is not modified by p ermuting the columns of Φ ( i.e. p ermuting the voters), a nd henc e the deep est voting pro cedure re mains identical. Neutr ality . F or contin uous depth functions, t he prop erty has b een prov en in [ Aubin et al. , 2022 , Prop osition 1]. Co nsider a depth on p ermutations, defined in Definition 4 . Similarly , fo r any voting situation Φ , with empirical distribution Π 𝑛 , the deep est voting rela tes only on the function whic h asso ciate a p ermu tation 𝜎 ∈ 𝔖 𝑚 to 𝑈 𝑑 , Π 𝑛 , 𝑝 ( 𝜎 ) = 1 𝑛 Í 𝑛 𝑣 = 1 𝑑 𝑝 ( 𝜎 , Φ ( ., 𝑣 ) ) . Per- m uting the rows of Φ ( i.e. pe r m uting the candidates ) b y a p ermu tation 𝜏 , 𝑑 ( 𝜏 ◦ 𝜎 , Φ ( 𝜏 ( . ) , 𝑣 ) ) = 𝑑 ( 𝜎 , Φ ( ., 𝑣 ) ) . Hence the deep est v oting pro cedure provides the same argmax set, up to the per m utation 𝜏 , and co nsequen tly the same candidate will be winning. Universality Universalit y is o b vious as the definition domain E 𝐷 of the depth functions is 𝑚 - dimensional. Pro of of Prop osition 8 Recall that Unanimity mea ns that if there exists 𝑐 ∗ ∈ { 1 , . . . , 𝑚 } such that ∀ 𝑣 ∈ { 1 , . . . 𝑛 } , Φ ( 𝑐 ∗ , 𝑣 ) = 1, then 𝜎 ∗ ( 𝑐 ∗ ) = 1, with 𝜎 ∗ the deep est permutation. Suppo se that a ll v oters a gree on t he same bes t candidate, i.e. supp ose t hat ∃ 𝑐 ∗ ∈ { 1 , . . . , 𝑚 } suc h that ∀ 𝑣 ∈ { 1 , . . . 𝑛 } , Φ ( 𝑐 ∗ , 𝑣 ) = 1. Let 𝜎 ∈ 𝔖 𝑚 such that 𝜎 ( 𝑐 ∗ ) ≠ 1. Without loss o f g enerality , consider 𝑐 ∗ = 1 and 𝜎 ( 2 ) = 1. Consider 𝜏 the p ermutation suc h that 𝜏 ( 1 ) = 1, 𝜏 ( 2 ) = 𝜎 ( 1 ) , and 𝜏 ( 𝑐 ) = 𝜎 ( 𝑐 ) if 𝑐 = 3 , . . . , 𝑚 . Let us prov e that then 𝜏 as a lo wer 𝑝 -F r´ ec het mea n tha n 𝜎 . Denote Δ : = 1 𝑛 𝑛 Õ 𝑣 = 1  𝑑 𝑝 ( 𝜎 , 𝜎 𝑣 ) − 𝑑 𝑝 ( 𝜏 , 𝜎 𝑣 )  the difference of the 𝑝 -F r´ ec het means of 𝜎 a nd 𝜏 . Let us prove that Δ ≥ 0 . More precisely , w e are going to pro ve that for all 𝑣 ∈ { 1 , . . . , 𝑛 } we have 𝑑 ( 𝜎 , 𝜎 𝑣 ) > 𝑑 ( 𝜏 , 𝜎 𝑣 ) . Let us distinguish with r e spect to the distance. F or the Hamming distance. Let 𝑣 ∈ { 1 , . . . , 𝑛 } . Since 𝜎 ( 1 ) ≠ 𝜎 𝑣 ( 1 ) , 𝜎 2 ≠ 𝜎 𝑣 ( 2 ) , and 𝜏 ( 1 ) ≠ 21 𝜎 𝑣 ( 1 ) , w e hav e 𝑑 ( 𝜎 , 𝜎 𝑣 ) = 2 + 𝑚 Õ 𝑐 = 3 1 { 𝜎 ( 𝑐 ) ≠ 𝜎 𝑣 ( 𝑐 ) } , 𝑑 ( 𝜏 , 𝜎 𝑣 ) = 1 { 𝜎 ( 1 ) ≠ 𝜎 𝑣 ( 2 ) } + 𝑚 Õ 𝑐 = 3 1 { 𝜎 ( 𝑐 ) ≠ 𝜎 𝑣 ( 𝑐 ) } . With thes e equations, it is straightforw ard that 𝑑 ( 𝜏 , 𝜎 𝑣 ) < 𝑑 ( 𝜎 , 𝜎 𝑣 ) . F or the Kendall distance. Let 𝑣 ∈ { 1 , . . . , 𝑛 } . By d efinition of the Kendall distance, 𝑑 ( 𝜎 , 𝜎 𝑣 ) = 𝑚 − 1 Õ 𝑐 = 1 𝑚 Õ 𝑐 ′ = 𝑐 + 1 1 { ( 𝜎 ( 𝑐 ) − 𝜎 ( 𝑐 ′ ) ) ( 𝜎 𝑣 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) < 0 } . Using the pro perties of 𝜎 and 𝜏 , and dec ompos ing the s um on ca ndidates with resp ect to candidate 𝑐 = 1, candidate 𝑐 = 2 and ca ndidates 𝑐 ≥ 3, 𝑑 ( 𝜎 , 𝜎 𝑣 ) = 𝑚 Õ 𝑐 ′ = 2 1 { ( 𝜎 ( 1 ) − 𝜎 ( 𝑐 ′ ) ) > 0 } + 𝑚 Õ 𝑐 ′ = 3 1 { ( 𝜎 𝑣 ( 2 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) > 0 } + 𝑚 − 1 Õ 𝑐 = 3 𝑚 Õ 𝑐 ′ = 𝑐 + 1 1 { ( 𝜎 ( 𝑐 ) − 𝜎 ( 𝑐 ′ ) ) ( 𝜎 𝑣 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) < 0 } , 𝑑 ( 𝜏 , 𝜎 𝑣 ) = 𝑚 Õ 𝑐 ′ = 3 1 { ( 𝜎 ( 1 ) − 𝜎 ( 𝑐 ′ ) ) ( 𝜎 𝑣 ( 2 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) < 0 } + 𝑚 − 1 Õ 𝑐 = 3 𝑚 Õ 𝑐 ′ = 𝑐 + 1 1 { ( 𝜎 ( 𝑐 ) − 𝜎 ( 𝑐 ′ ) ) ( 𝜎 𝑣 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) < 0 } . Hence 𝑑 ( 𝜎 , 𝜎 𝑣 ) − 𝑑 ( 𝜏 , 𝜎 𝑣 ) = Í 𝑚 𝑐 ′ = 2 1 { ( 𝜎 ( 1 ) − 𝜎 ( 𝑐 ′ ) ) > 0 } + 𝛿 𝑣 with 𝛿 𝑣 = 𝑚 Õ 𝑐 ′ = 3  1 { ( 𝜎 ( 1 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) > 0 } + 1 { ( 𝜎 𝑣 ( 2 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) > 0 } − 1 { ( 𝜎 ( 1 ) − 𝜎 ( 𝑐 ′ ) ) ( 𝜎 𝑣 ( 2 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) < 0 }  . It is straightforward that each term of the s um abov e cannot b e negative (since if 1 { ( 𝜎 ( 1 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) > 0 } + 1 { ( 𝜎 𝑣 ( 2 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) > 0 } = 0, t hen 1 { ( 𝜎 ( 1 ) − 𝜎 ( 𝑐 ′ ) ) ( 𝜎 𝑣 ( 2 ) − 𝜎 𝑣 ( 𝑐 ′ ) ) < 0 } = 0). It res ults tha t 𝑑 ( 𝜎 , 𝜎 𝑣 ) − 𝑑 ( 𝜏 , 𝜎 𝑣 ) ≥ 𝑚 Õ 𝑐 ′ = 2 1 { ( 𝜎 ( 1 ) − 𝜎 ( 𝑐 ′ ) ) > 0 } . Since 𝜎 ( 1 ) > 𝜎 ( 2 ) = 1, we deduce that 𝑑 ( 𝜎 , 𝜎 𝑣 ) > 𝑑 ( 𝜏 , 𝜎 𝑣 ) . F or 𝑞 -M i nk o wski-H¨ o lder distances, 1 ≤ 𝑞 < ∞ . Let 𝑣 ∈ { 1 , . . . , 𝑛 } . First o bserve that: 𝑑 𝑞 𝑞 ( 𝜎 , 𝜎 𝑣 ) = 𝑚 Õ 𝑐 = 1 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | 𝑞 = | 𝜎 ( 1 ) − 1 | 𝑞 + | 1 − 𝜎 𝑣 ( 2 ) | 𝑞 + 𝑚 Õ 𝑐 = 3 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | 𝑞 . 22 Similarly , 𝑑 𝑞 𝑞 ( 𝜏 , 𝜎 𝑣 ) = | 𝜎 ( 1 ) − 𝜎 𝑣 ( 2 ) | 𝑞 + 𝑚 Õ 𝑐 = 3 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | 𝑞 . Hence, 𝑑 𝑞 𝑞 ( 𝜎 , 𝜎 𝑣 ) − 𝑑 𝑞 𝑞 ( 𝜏 , 𝜎 𝑣 ) = | 𝜎 ( 1 ) − 1 | 𝑞 + | 1 − 𝜎 𝑣 ( 2 ) | 𝑞 − | 𝜎 ( 1 ) − 𝜎 𝑣 ( 2 ) | 𝑞 . Next, • if 1 < 𝜎 ( 1 ) ≤ 𝜎 𝑣 ( 2 ) , 𝑑 𝑞 𝑞 ( 𝜎 , 𝜎 𝑣 ) − 𝑑 𝑞 𝑞 ( 𝜏 , 𝜎 𝑣 ) ≥ ( 𝜎 ( 1 ) − 1 ) 𝑞 > 0; • if 1 < 𝜎 𝑣 ( 2 ) ≤ 𝜎 ( 1 ) , 𝑑 𝑞 𝑞 ( 𝜎 , 𝜎 𝑣 ) − 𝑑 𝑞 𝑞 ( 𝜏 , 𝜎 𝑣 ) ≥ ( 𝜎 𝑣 ( 2 ) − 1 ) 𝑞 > 0. Hence, 𝑑 𝑞 ( 𝜎 , 𝜎 𝑣 ) > 𝑑 𝑞 ( 𝜏 , 𝜎 𝑣 ) . This concludes the proo f. F or 𝑞 -M i nk o wski-H¨ o lder distances, 𝑞 = ∞ . Here w e sho w that w he n there is unique solution 𝜎 ∗ then 𝑝 -F r´ ec het mean of ∞ -Minko wski- H¨ older sa tisfies Unanimity a nd o therwise that’s not always the case. Let’s r ecall that : Δ : = 1 𝑛 𝑛 Õ 𝑣 = 1  ( max 𝑐 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) 𝑝 − ( max 𝑐 | 𝜏 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) 𝑝  . Then Δ = 1 𝑛 𝑛 Õ 𝑣 = 1  ( max ( 𝜎 ( 1 ) − 𝜎 𝑣 ( 1 ) , 𝜎 ( 2 ) − 𝜎 𝑣 ( 1 ) , max 𝑐> 2 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) ) 𝑝 − ( max ( 𝜏 ( 1 ) − 𝜎 𝑣 ( 1 ) , 𝜏 ( 2 ) − 𝜎 𝑣 ( 2 ) , max 𝑐> 2 | 𝜏 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) ) 𝑝  . Since 𝜎 ( 2 ) = 1, ∀ 𝑣 , 𝜎 𝑣 ( 1 ) = 1, 𝜏 ( 1 ) = 𝜎 ( 2 ) = 1 and 𝜏 ( 2 ) = 𝜎 ( 1 ) > 1 , Δ = 1 𝑛 𝑛 Õ 𝑣 = 1  ( max ( 𝜎 ( 1 ) − 1 , 𝜎 𝑣 ( 2 ) − 1 , max 𝑐> 2 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) ) 𝑝 − ( max ( 𝜏 ( 2 ) − 𝜎 𝑣 ( 2 ) , max 𝑐> 2 | 𝜏 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) ) 𝑝  . Now, considering that max 𝑐> 2 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | = max 𝑐> 2 | 𝜏 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | and 𝜎 ( 1 ) − 1 > 𝜏 ( 2 ) − 𝜎 𝑣 ( 2 ) , it le a ds that for all 𝑝 ≥ 1 ( max ( 𝜎 ( 1 ) − 1 , 𝜎 𝑣 ( 2 ) − 1 , max 𝑐> 2 | 𝜎 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) ) 𝑝 ≥ ( max ( 𝜏 ( 2 ) − 𝜎 𝑣 ( 2 ) , max 𝑐> 2 | 𝜏 ( 𝑐 ) − 𝜎 𝑣 ( 𝑐 ) | ) ) , and Δ ≥ 0. Note that it doesn’t mean that Δ ≠ 0 but, necessarily a r anking with 𝑐 ∗ in first place minimises the 𝑝 -F r´ echet mean for the 𝐿 ∞ distance. Thus, if the solution is unique then that is this ranking. Otherwise, counter-example below sho ws that there is no winner therefore una nimit y pr oper t y is not satisfied. Consider a setting with 5 voters and 6 candidates, with the voting pre fer ences g iv en are according to T a ble 5 . 23 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑐 1 1 1 1 1 1 𝑐 2 4 6 4 2 6 𝑐 3 6 4 6 4 5 𝑐 4 3 5 3 6 4 𝑐 5 5 2 5 5 3 𝑐 6 2 3 2 3 2 T a ble 5 : Counter-example for the pro of that 𝑝 -F r´ e chet mean o f ∞ -Minko wski-H¨ olde r dista nce may not satisfy unanimity if there are ma n y so lutio ns. The 1-F r´ ec het means are (1,4,6,3,5,2), (1,5,6,3,4,2 ) and (2,5,6,3 ,4,1). Both the 2-F r´ ec het means and the 3-F r´ echet means are (1,5,6,3,4,2) and (2,5,6,3,4 ,1). Hence, the winner is not unique, a nd candidate 𝑐 6 is a lso a preferred candidate in the deepest set. Pro of of Prop osition 9 1. Kendall -based 𝑝 -F r´ ec het for 𝑝 = 1 Let Φ = ( 𝜎 𝑣 ) 𝑣 = 1 , .. . , 𝑛 ∈ 𝔖 𝑛 𝑚 be the set of rankings given by 𝑛 voters on 𝑚 candidates. Let 𝜎 ∗ be a median p ermutation of Φ bas ed on the Kendall distance 𝑑 𝐾 . That is, we consider consensus r anking, with 𝜎 ∗ ∈ a rgmin 𝜎 ∈ 𝔖 𝑚 Í 𝑛 𝑣 = 1 𝑑 𝐾 ( 𝜎 , 𝜎 𝑣 ) . Suppo se t hat 𝑐 0 is a Condor c et winner of the election and that 𝑐 0 is not ra nk ed firs t in 𝜎 ∗ . Thu s 𝜎 ∗ ( 𝑐 0 ) = 𝑟 ≠ 1 . There exis ts 𝑐 1 ∈ C such that 𝜎 ∗ ( 𝑐 1 ) = 𝑟 − 1 . Let 𝜏 ∈ 𝔖 𝑚 be suc h that 𝜏 = 𝜎 ∗ except that 𝜏 ( 𝑐 0 ) = 𝑟 − 1 a nd 𝜏 ( 𝑐 1 ) = 𝑟 . Observe that for all 𝑠 ∈ 𝔖 𝑚 , 𝑑 𝐾 ( 𝜎 ∗ , 𝑠 ) = 𝑑 𝐾 ( 𝜏 , 𝑠 ) + 𝛿 𝑠 ( 𝑐 0 , 𝑐 1 ) , where 𝛿 𝑠 ( 𝑐 0 , 𝑐 1 ) = 1 if 𝑠 ( 𝑐 0 ) < 𝑠 ( 𝑐 1 ) and 𝛿 𝑠 ( 𝑐 0 , 𝑐 1 ) = − 1 if 𝑠 ( 𝑐 0 ) > 𝑠 ( 𝑐 1 ) . Therefore, 𝑛 Õ 𝑣 = 1 𝑑 𝐾 ( 𝜎 ∗ , 𝜎 𝑣 ) = 𝑛 Õ 𝑣 = 1 𝑑 𝐾 ( 𝜏 , 𝜎 𝑣 ) + 𝑛 Õ 𝑣 = 1 𝛿 𝜎 𝑣 ( 𝑐 0 , 𝑐 1 ) . As 𝑐 0 is a Condorcet winner, we have more elements in the set { 𝑣 ∈ { 1 , . . . 𝑛 } , 𝜎 𝑣 ( 𝑐 0 ) < 𝜎 𝑣 ( 𝑐 1 ) } than in the set { 𝑣 ∈ { 1 , . . . 𝑛 } , 𝜎 𝑣 ( 𝑐 0 ) < 𝜎 𝑣 ( 𝑐 1 ) } . This implies that Í 𝑛 𝑣 = 1 𝛿 𝜎 𝑣 ( 𝑐 0 , 𝑐 1 ) > 0 a nd thus that Í 𝑛 𝑣 = 1 𝑑 𝐾 ( 𝜎 ∗ , 𝜎 𝑣 ) > Í 𝑛 𝑣 = 1 𝑑 𝐾 ( 𝜏 , 𝜎 𝑣 ) . Hence, 𝜎 ∗ cannot b e a median. Ther efore, by a bsurd, it means that 𝜎 ∗ ( 𝑐 0 ) = 1. So the consensus K endall-based voting rule sa tisfies the Co ndorcet-winner prop erty . Note that it is well-kno wn that Kemen y voting sa tisfies Condorcet-w inner p ro perty . 2. Kendall-based 𝑝 -F r´ echet for 𝑝 > 1 , Ca yley , Hamming -based 𝑝 -F r´ echet for 𝑝 ≥ 1 and 𝑞 -Mi nk o wski-H¨ ol der -based 𝑝 -F r´ ec het for 𝑝 = 1 • Kendal l-b ase d 𝑝 -F r´ echet for 𝑝 > 1 Consider a setting with 5 voters and 3 candidates, with the voting preference s given are according to T a ble 6 . 24 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑐 1 1 1 1 3 2 𝑐 2 2 2 2 2 1 𝑐 3 3 3 3 1 3 T a ble 6: Coun ter-exa mple for the pro of o f p oint 2. (K endall) of Prop o sition 9 . 𝑐 1 is obviously the Condo rcet winner of the election. Then, fo r 𝑝 ≥ 1, 5 Õ 𝑣 = 1 𝑑 𝑝 𝐾 ( ( 𝑐 1 , 𝑐 2 , 𝑐 3 ) , 𝑣 𝑣 ) = 1 + 3 𝑝 5 Õ 𝑣 = 1 𝑑 𝑝 𝐾 ( ( 𝑐 1 , 𝑐 3 , 𝑐 2 ) , 𝑣 𝑣 ) = 3 + 2 × 2 𝑝 5 Õ 𝑣 = 1 𝑑 𝑝 𝐾 ( ( 𝑐 2 , 𝑐 1 , 𝑐 3 ) , 𝑣 𝑣 ) = 3 + 2 𝑝 5 Õ 𝑣 = 1 𝑑 𝑝 𝐾 ( ( 𝑐 2 , 𝑐 3 , 𝑐 1 ) , 𝑣 𝑣 ) = 1 + 4 × 2 𝑝 5 Õ 𝑣 = 1 𝑑 𝑝 𝐾 ( ( 𝑐 3 , 𝑐 1 , 𝑐 2 ) , 𝑣 𝑣 ) = 2 + 3 × 2 𝑝 5 Õ 𝑣 = 1 𝑑 𝑝 𝐾 ( ( 𝑐 3 , 𝑐 2 , 𝑐 1 ) , 𝑣 𝑣 ) = 2 𝑝 + 3 × 3 𝑝 If 𝑝 = 1, 𝑐 1 is the winner of Kendall consensus ranking. But ∀ 𝑝 ≥ 2, Í 5 𝑣 = 1 𝑑 𝐾 ( ( 𝑐 1 , 𝑐 2 , 𝑐 3 ) , 𝑣 𝑣 ) 𝑝 > Í 5 𝑣 = 1 𝑑 𝐾 ( ( 𝑐 2 , 𝑐 1 , 𝑐 3 ) , 𝑣 𝑣 ) 𝑝 and then 𝑐 2 is the winner o f the election with the voting rule (Kendall, 𝑝 ) fo r 𝑝 ≥ 2. Other calcula tions are left to the reader. 3. F or the following with Cayley and Hamming, consider a setting with 7 voters and 3 candi- dates, wit h the voting preferences giv en ar e acco rding to T a ble 7 . 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 6 𝑣 7 𝑐 1 1 1 2 2 2 2 3 𝑐 2 2 2 1 1 3 3 1 𝑐 3 3 3 3 3 1 1 2 T a ble 7: Coun ter-exa mple for the pro of o f p oint 2 of Pr opo sition 9 . In this setting, the Condorce t winner is candidate 𝑐 1 as 4 voters prefer 𝑐 1 to 𝑐 2 and 4 voters prefer 𝑐 1 to 𝑐 3 . • Hamming-b ase d 𝑝 -F r ´ ech et for 𝑝 ≥ 1 With the voting preference s g iv en a re accor ding to 25 T a ble 7 , 7 Õ 𝑗 = 1 𝑑 𝑝 𝐻 ( ( 1 , 2 , 3 ) 𝑇 , 𝑣 𝑗 ) = 2 𝑝 + 2 𝑝 + 3 𝑝 + 3 𝑝 + 3 𝑝 7 Õ 𝑗 = 1 𝑑 𝑝 𝐻 ( ( 2 , 1 , 3 ) 𝑇 , 𝑣 𝑗 ) = 2 𝑝 + 2 𝑝 + 2 𝑝 + 2 𝑝 + 2 𝑝 7 Õ 𝑗 = 1 𝑑 𝑝 𝐻 ( ( 2 , 3 , 1 ) 𝑇 , 𝑣 𝑗 ) = 3 𝑝 + 3 𝑝 + 2 𝑝 + 2 𝑝 + 3 𝑝 7 Õ 𝑗 = 1 𝑑 𝑝 𝐻 ( ( 3 , 1 , 2 ) 𝑇 , 𝑣 𝑗 ) = 3 𝑝 + 3 𝑝 + 2 𝑝 + 2 𝑝 + 3 𝑝 + 3 𝑝 Other calcula tions a re left to the r e ader and we note tha t ( 2 , 1 , 3 ) 𝑇 minimizes the Hamming-based 𝑝 -F r´ ec het mean f or 𝑝 ≥ 1 and then 𝑐 2 is the winner of the election. • Cayley-b ase d 𝑝 -F r´ echet fo r 𝑝 ≥ 1 With the v oting prefer e nces given are according to T a ble 7 , 7 Õ 𝑗 = 1 𝑑 𝑝 𝐶 ( ( 1 , 2 , 3 ) 𝑇 , 𝑣 𝑗 ) = 1 + 1 + 2 𝑝 + 2 𝑝 + 2 𝑝 7 Õ 𝑗 = 1 𝑑 𝑝 𝐶 ( ( 2 , 1 , 3 ) 𝑇 , 𝑣 𝑗 ) = 1 + 1 + 1 + 1 + 1 7 Õ 𝑗 = 1 𝑑 𝑝 𝐶 ( ( 2 , 3 , 1 ) 𝑇 , 𝑣 𝑗 ) = 2 𝑝 + 2 𝑝 + 1 + 1 + 2 𝑝 7 Õ 𝑗 = 1 𝑑 𝑝 𝐶 ( ( 3 , 1 , 2 ) 𝑇 , 𝑣 𝑗 ) = 2 𝑝 + 2 𝑝 + 1 + 1 + 2 𝑝 + 2 𝑝 Other calcula tions a re left to the reader a nd we no te that ( 2 , 1 , 3 ) 𝑇 minimizes the Cayley- based 𝑝 -F r´ ec het mean for 𝑝 ≥ 1 and then 𝑐 2 is the winner of the election. 4. F or 𝑞 -Minkowski-H¨ older b ase d 𝑝 -F r ´ echet fo r 𝑝 = 1, let’s consider the previo us co un ter ex am- ple. 26 One can sho w that, f or 𝑞 ≥ 1, and the pr eferences expres sed in table 7 : 7 Õ 𝑗 = 1 𝑑 𝑞 ( ( 1 , 2 , 3 ) 𝑇 , 𝑣 𝑗 ) = 2 × 2 1 / 𝑞 + 3 × ( 2 𝑞 + 2 ) 1 / 𝑞 + 0 × 2 ( 𝑞 + 1 ) / 𝑞 7 Õ 𝑗 = 1 𝑑 𝑞 ( ( 2 , 1 , 3 ) 𝑇 , 𝑣 𝑗 ) = 3 × 2 1 / 𝑞 + 0 × ( 2 𝑞 + 2 ) 1 / 𝑞 + 2 × 2 ( 𝑞 + 1 ) / 𝑞 7 Õ 𝑗 = 1 𝑑 𝑞 ( ( 1 , 3 , 2 ) 𝑇 , 𝑣 𝑗 ) = 4 × 2 1 / 𝑞 + 2 × ( 2 𝑞 + 2 ) 1 / 𝑞 + 1 × 2 ( 𝑞 + 1 ) / 𝑞 7 Õ 𝑗 = 1 𝑑 𝑞 ( ( 3 , 1 , 2 ) 𝑇 , 𝑣 𝑗 ) = 3 × 2 1 / 𝑞 + 4 × ( 2 𝑞 + 2 ) 1 / 𝑞 + 0 × 2 ( 𝑞 + 1 ) / 𝑞 7 Õ 𝑗 = 1 𝑑 𝑞 ( ( 2 , 3 , 1 ) 𝑇 , 𝑣 𝑗 ) = 0 × 2 1 / 𝑞 + 3 × ( 2 𝑞 + 2 ) 1 / 𝑞 + 2 × 2 ( 𝑞 + 1 ) / 𝑞 7 Õ 𝑗 = 1 𝑑 𝑞 ( ( 3 , 2 , 1 ) 𝑇 , 𝑣 𝑗 ) = 3 × 2 1 / 𝑞 + 2 × ( 2 𝑞 + 2 ) 1 / 𝑞 + 2 × 2 ( 𝑞 + 1 ) / 𝑞 This sum is minimum for ( 2 , 1 , 3 ) 𝑇 for all 𝑞 ≥ 1. Ba s ic consider ations as 2 1 / 𝑞 ≤ ( 2 𝑞 + 2 ) 1 / 𝑞 ≤ 2 ( 𝑞 + 1 ) / 𝑞 mean to the act that only ( 1 , 2 , 3 ) 𝑇 and ( 2 , 1 , 3 ) 𝑇 can minimises the sum. Finally , ( 2 , 1 , 3 ) 𝑇 is the permutation minimizing t he sum. Pro of of Prop osition 10 1. Consider Φ a voting situation with a winner 𝑐 ∗ with a given voting pro cedure. Supp ose that there exists a voter 𝑣 0 such that Φ ( 𝑐 ∗ , 𝑣 0 ) = 𝛼 ≠ 1. Let 𝑐 0 such that Φ ( 𝑐 0 , 𝑣 0 ) = 𝛼 − 1 . Consider ano ther voting situatio n e Φ = Φ ex c ept that e Φ ( 𝑐 ∗ , 𝑣 0 ) = 𝛼 − 1 and e Φ ( 𝑐 0 , 𝑣 0 ) = 𝛼 . The voting pro cedure sa tisfies Monotonicity if the winner for ˜ Φ is still 𝑐 ∗ . Denote Π 𝑛 and ˜ Π 𝑛 the empirica l distributions assoc ia ted to the columns of Φ and ˜ Φ resp ec- tively . Without loss o f gener alit y , we supp ose that the voter for whom w e swap the ranks of 𝑐 ∗ and 𝑐 0 is the voter 𝑣 0 = 𝑛 . W e deno te 𝜎 𝑛 and ˜ 𝜎 𝑛 the asso ciated ra nkings. Consider 𝑈 𝑑 , Π , 𝑝 ( · ) the 𝑝 -F r´ ec het functional a sso ciated to the initial v otes, and 𝑈 𝑑 , ˜ Π 𝑛 , 𝑝 ( · ) the func- tional obtained after the sw ap. Let 𝜎 ∗ be a F r´ echet mean ass ocia ted to the distance 𝑑 ( ) in the initial v oting setting. W e would like to show th at 𝜎 ∗ is still a F r´ ec het m ean in the new setting, na mely that it minimizes 𝑈 𝑑 , ˜ Π 𝑛 , 𝑝 . Let 𝜎 ∈ 𝔖 𝑚 . W e ha ve 𝑛𝑈 𝑑 , ˜ Π 𝑛 , 𝑝 ( 𝜎 ∗ ) − 𝑛𝑈 𝑑 , ˜ Π 𝑛 , 𝑝 ( 𝜎 ) = 𝑛𝑈 𝑑 , Π 𝑛 , 𝑝 ( 𝜎 ∗ ) − 𝑛𝑈 𝑑 , Π 𝑛 , 𝑝 ( 𝜎 ) +  𝑑 𝑝 ( 𝜎 ∗ , ˜ 𝜎 𝑛 ) − 𝑑 𝑝 ( 𝜎 ∗ , 𝜎 𝑛 )  −  𝑑 𝑝 ( 𝜎 , ˜ 𝜎 𝑛 ) − 𝑑 𝑝 ( 𝜎 , 𝜎 𝑛 )  . W e w ant to prove that 𝑛 𝑈 𝑑 , ˜ Π 𝑛 , 𝑝 ( 𝜎 ∗ ) − 𝑛𝑈 𝑑 , ˜ Π 𝑛 , 𝑝 ( 𝜎 ) < 0. W e no w disting uish with respec t to the distance used and t he v alue o f 𝑞 . Let 𝑝 = 1. Suppose that 𝜎 ∗ is a F r´ ec het mean asso ciated to 𝑈 𝑑 1 , Π 𝑛 , 1 ( · ) . Then 𝑈 𝑑 1 , Π , 1 ( 𝜎 ∗ ) < 𝑈 𝑑 1 , Π 𝑛 , 1 ( 𝜎 ) for all 𝜎 ∈ 𝔖 𝑚 \ 𝔖 ∗ 1 , 𝑑 1 , Π 𝑛 . 27 First observe that, since the 1 -Mink owski-H¨ older dista nce b etw een tw o permutations is alwa ys even, this implies in pa rticular that 𝑛𝑈 𝑑 1 , Π 𝑛 , 1 ( 𝜎 ∗ ) ≤ 𝑛𝑈 𝑑 1 , Π 𝑛 , 1 ( 𝜎 ) − 2. Secondly , as ther e is only one swap betw een 𝜎 𝑛 and ˜ 𝜎 𝑛 , with a difference o ne b et ween the v a lues, we hav e, for any p erm utation 𝜎 , | 𝑑 1 ( 𝜎 , ˜ 𝜎 𝑛 ) − 𝑑 1 ( 𝜎 , 𝜎 𝑛 ) | ≤ 2 . Replacing this in ( 1 ), we obtain 𝑛𝑈 𝑑 1 , ˜ Π 𝑛 , 1 ( 𝜎 ∗ ) − 𝑛𝑈 𝑑 1 , ˜ Π 𝑛 , 1 ( 𝜎 ) ≤ 𝑑 1 ( 𝜎 ∗ , ˜ 𝜎 𝑛 ) − 𝑑 1 ( 𝜎 ∗ , 𝜎 𝑛 ) Lastly , observe that 𝑑 ( 𝜎 ★ , ˜ 𝜎 𝑛 ) − 𝑑 ( 𝜎 ★ , 𝜎 𝑛 ) dep ends only on the ra nkings of 𝑐 ∗ and 𝑐 0 . Indeed, 𝑑 ( 𝜎 ★ , ˜ 𝜎 𝑛 ) − 𝑑 ( 𝜎 ★ , 𝜎 𝑛 ) = − 2 if 𝜎 ★ ( 𝑐 0 ) ≥ 𝛼 and 0 o therwise. Thus 𝑈 𝑑 1 , ˜ Π , 1 ( 𝜎 ★ ) ≤ 𝑈 𝑑 1 , ˜ Π , 1 ( 𝜎 ) for all 𝜎 ∈ 𝔖 𝑚 \ 𝔖 ∗ 1 , 𝑑 1 , Π and all p ermut ations in 𝔖 ∗ 1 , 𝑑 1 , Π are also a F r´ ec het mean after the v ote swap. If, in addition 𝜎 ★ ( 𝑐 0 ) ≥ 𝛼 then the ine q ualit y is s tr ict implying that 𝔖 ∗ 1 , 𝑑 1 , Π ⊆ 𝔖 ∗ 1 , 𝑑 1 , ˜ Π and thus that 𝑐 ∗ remains the un ique winner of the election. 2. Consider the following voting preferences: 𝑎 = ( 𝐴 , 𝐵 , 𝐶 , 𝐷 , 𝐸 ) , 𝑏 = ( 𝐷 , 𝐶 , 𝐴 , 𝐵, 𝐸 ) , 𝑐 = ( 𝐸 , 𝐷 , 𝐶 , 𝐴, 𝐵 ) . Supp ose that we have 𝑛 = 5 voters and they vote a s follows: 𝜎 1 = 𝜎 2 = 𝑎 , 𝜎 3 = 𝜎 4 = 𝑏 a nd 𝜎 5 = 𝑐 . Then one ca n chec k that the F r´ echet mea n as socia ted to the consensus ranking for the Hamming distance is 𝑎 and thus the winner of the election is 𝐴 . Now if the fifth v oter changes his rankings, swapping the place o f 𝐴 and 𝐶 , ˜ 𝜎 5 bec omes ( 𝐸 , 𝐷 , 𝐴, 𝐶 , 𝐵 ) and 𝑏 bec o mes a F r´ echet mea n for ˜ Π , lea ding to a new election winner , 𝐷 . Pro of of Prop osition 11 Independenc e to Losers pr oper t y: 1. Kendall-based consensus deepes t voting is K emen y voting rule, which is known to not s a tisfy Independenc e to Los ers a s all Condo r cet metho ds (see Y oung and Levenglic k [ 197 8 ]). 2. W e wan t to prov e that all deep est voting rules based on 𝑞 -Minko wski-H¨ older distance, for all 𝑞 ≥ 1 , do not satisfy Independence to Los ers. As a counterexample, let supp ose the situatio n with 2 candidates and 7 voters, with ra nkings given by T a ble 8 . 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 6 𝑣 7 𝑐 1 1 1 1 1 2 2 2 𝑐 2 2 2 2 2 1 1 1 T a ble 8 : Counter-example for the pro of that voting r ules based on a Minko wski-H¨ older distance do not s atisfy Indep endence to Losers. Cas e 1, with 2 candida tes. It is obvious that for all 𝑞 ≥ 1, the deep e st per m utation is ( 1 , 2 ) ⊤ , and therefor e c a ndidate 𝑐 1 is the winner of the deepest voting with 𝑞 -Minko wski-H¨ older distance. Let int ro duce a candidate 𝑐 3 such that the preferences are now the ones in T a ble 9 . Note that the pairwise compar isons b et ween 𝑐 1 and 𝑐 2 do not c hange. 28 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 6 𝑣 7 𝑐 1 1 1 2 2 3 3 3 𝑐 2 2 2 3 3 1 1 1 𝑐 3 3 3 1 1 2 2 2 T a ble 9 : Counter-example for the pro of that voting r ules based on a Minko wski-H¨ older distance do not satisfy Indep endence to Losers. Case 2, with 3 candidates . The voting situation f or the first 2 candidates is similar to Ca se 1 in T a ble 8 . It is easy to chec k that for a ll 𝑞 ≥ 1 (ca lculus ar e le ft to the re a der) the deep e st per m utation is ( 3 , 1 , 2 ) ⊤ and therefore 𝑐 2 is the deep est winner of the election. Introducing 𝑐 3 changes the winner f ro m 𝑐 1 to 𝑐 2 , which prov es that Independence to Losers is no t satisfied. 3. T o prov e that Hamming - based deepest voting do not satisfy Indep endence to Losers, the same counterexample of the pro of of Mink owski-Holder-based deep est voting ca n b e considered. 4. The same counterexample can be used for the proof with Cayley-bas ed deep est voting, even if the winner is not unique in this case. 29