A continuous rating method for preferential voting. The incomplete case

A CONTINUOUS RA TING METHOD F OR PREFERE NTIAL V OTING. THE INCOMPLETE CASE R osa Camps, Xa vier Mora and Laia Saumell Departamen t de Matem` atiques, Univ ersitat Aut` ono ma de Barcelona, Catalonia, Spain xmora @ mat.uab.cat July 13, 2009; revised Octo b er 10, 2011 Abstract A metho d is giv en for quan titativ ely rating the so cial acceptance of differen t options whic h are the matter of a pr eferential v ote. In con trast to a previous article, here the individu al v otes are al- lo w ed to b e incomplete, that is, they need not express a comparison b et wee n every pair of options. Th is includ es the case where eac h vote r giv es an ordered list restricted to a subset of most preferred options. In this connection, the prop osed method (except for one of the giv en v aria nts) carefully distinguishes a lac k of information ab out a giv en pair of options f rom a prop er tie b etw een them. As in the sp ecial case of complete individ ual v otes, the p rop osed generalization is pro v ed to ha v e certain desirable prop erties, whic h includ e: the conti nuit y of the rates with resp ect to the data, a decomp osition p rop erty that c har- acterizes certain s itu ations opp osite to a tie, the Condorcet-Smith principle, and clone consistency . Keyw ords: pr efer ential voting, quantitative r ating, c ontinuous r a t- ing, majority principles, Condor c et-Smith principle, clone c onsistency, one-dimensional sc aling. appr o val voting, AMS sub ject classifications: 05 C20, 91B12, 91B14, 91C1 5, 91C20. In a previous article [ 3 ] w e intro duced a metho d for quan titativ ely rating the so cial a cceptance of differen t candidate optio ns whic h are the matter of a preferen tial vote. The quan titativ e c haracter of this metho d lies in a com bination of t w o properties: First, a prop ert y of contin uit y that allo ws to sens e the closeness b etw ee n t w o candidates, for instance the winner and the runner-up. Second, a prop ert y of decomp osition that allo ws to recognise certain situations that are o pp osite to a tie; in pa r t icular, a candidate gets the b est p ossible r a te if and only if it is placed first b y all v oters. These t w o prop erties are com bined with other o nes of a qualitativ e nature. Esp ecially outstanding among the latter is the Condorcet-Smith principle: Assume that 2 R. Camps, X. Mora, L. Sa umell the set of candidates is partitioned in to t w o classes X and Y suc h that f or eac h mem b er of X and ev ery member o f Y there are more than half o f the individual v otes where the former is preferred to the latter; in that case, the so cial ra nking prefers also eac h member of X to an y mem b er of Y . T o our kno wledge, the existing literature do es not offer an y other ra ting metho d that com bines the three mentioned prop erties, namely contin uit y , decomp osition and the Condorcet-Smith principle. Ho w ev er, in [ 3 ] we restricted ourselv es to the complete case, t ha t is, w e assumed that ev ery individual expres ses a comparison (a preference or a tie) ab out each pair of options. Suc h a restriction leav es out man y cases of in terest, lik e for instance truncated rankings. In this article we will extend that metho d to the incomplete case, where the indiv idual v otes need not express a comparison abo ut ev ery pair of options. This extens ion will b e done in suc h a w ay that a lack of information ab out the preference of a v oter betw ee n a giv en pair of options will b e carefully distinguishe d fro m a prop er tie b et w een them. In this connection, we will ha v e to cop e with the fact that a quan titativ e sp ecification of the collectiv e o pinion a b out a pair of options has then tw o degrees of freedom; in f act, kno wing ho w man y v oters preferred x to y do es not determine how man y of them preferred y to x . This intro duces a sp ecial difficult y that w as no t presen t in the complete case. An extreme case of incompleten ess is that where eac h v ote reduce s to c ho osing a single option. In that case our rates will be linearly related to the v ote fra ctions. Another imp ortant case is that of appro v al v oting. In that case, our rates will b e ess en tially differen t from the num ber o f receiv ed ap- pro v als; ho w ev er, one of the v arian ts of our metho d, namely the marg in- based v arian t, will b e sho wn to rank the options exactly in the same w a y as t he n um b er of receiv ed approv als. W e call our quan titative metho d the CLC rating metho d , where the capital letters stand for “ Con tin uous Llull Condorcet”. The reader in terested to try it can use the CLC calculator whic h has b een made av ailable a t [ 9 ]. Of course, an y rating automatically implies a ranking. In this connection, it should b e noticed that the CLC rating metho d is built up on certain v ariants of t he qualitativ e ranking method that w as in tro duced in 1997 b y Markus Sc h ulze [ 13 , 1 4 ; 15 : p. 228–232 ] . This article is organized as follo ws: Section 1 giv es a more precise state- men t of the problem together with some general remarks. Section 2 presen ts an outline of the prop o sed metho d, follow ed by a discussion of certain sp ecial cases, a summary of the pro cedure, and a discu ssion of certain v arian ts. Sec- tion 3 gives three represen tativ e examples. Finally , sections 4–9 give detailed mathematical pro ofs of the claimed prop erties. Continuous ra ting f or incomplete prefere ntial v oting 3 1 Statemen t of the problem and general remarks 1.1 Let us consider a set of N options which are the matter of a v ote. Let us assume that each v oter expresses his preferen ces ab out certain pairs of options. Our aim is to com bine suc h individual preferences so as to rate the so cial acceptance of eac h option o n a con tinuous scale. More sp ecifically , w e w ould like to do it in accordance with t he following conditions: A Sc ale invarianc e (homo geneity) . The rates dep end only on the rela- tiv e frequency of eac h p o ssible con ten t of an individual v ote. In other w ords, if ev ery individual v ote is replaced by a fixed num ber of copies of it, the rates remain exactly the same. B Pe rmutation e quivarianc e (neutr ality) . Applying a certain p erm uta- tion of the options to all of the individual v otes has no other effect than getting the same p erm uta tion in the so cial rating. C Co n tinuity . The rates dep end contin uously on the relative frequency of eac h p ossible conten t of an individual vote. The next t w o conditions consider a specific form of rating. F r om now on, w e will refer to it as ra nk-lik e r a ting. D R ank-like form . Eac h rank-lik e rate is a n umber, inte ger or frac- tional, b etw een 1 and N . The b est p ossible v alue is 1 and the w orst p ossible o ne is N . The a v erage ra nk-lik e rate is larger than or equal to ( N + 1) / 2 , with equalit y in the complete case. E R ank-like de c om p osition in the c omplete c ase . Assume that the in- dividual preferences are complete, i. e. a comparison (a preferenc e or a t ie) is expressed abo ut ev ery pa ir of options. Consider a split- ting of the options in tw o classes X and Y suc h that eac h mem b er of X is unanimously preferred to ev ery mem b er o f Y . Suc h a sit- uation translates into t he three following equiv alent facts, where | X | denotes the num b er of elemen ts of X : ( a ) The rank-lik e rates of X coincide with those that one obtains when the individual v otes are restricted to X . (b) After diminishing them b y the num b er | X | , the r a nk-lik e rates of Y coincide with those tha t one obtains when the individual v otes are restricted to Y . (c) The av erage rank-lik e rate of X is ( | X | + 1) / 2 . In particular, in the complete case an option will get a rank-lik e rate exactly equal to 1 [resp. N ] if and only if it is unanimously preferred to [resp. consid- ered w orse than] a ny other. As w e will se e later on, some of the implications con tained in the preceding condition will hold also in certain situations that allo w f o r incompleteness. 4 R. Camps, X. Mora, L. Sa umell The next condition considers the extre me case of incompleteness where eac h v ote reduces to c ho osing a single option. F R ank-like r ates for single-choic e voting . Assume that each v ote re- duces to c ho osing a single option. In tha t case, the rank-like rate of eac h o ption is the w eighted a verage of the num b ers 1 and N with w eigh ts giv en respectiv ely b y the fraction of the v ot e in fa vour of that option and the complemen ta ry f raction. Finally , w e require a condition that concerns only the concomitan t so cial ranking, that is, the o r dina l informatio n con tained in the so cial r ating: M Condor c et-Smith principle . Consider a splitting of the options in to t w o classes X and Y . Assume that for eac h member of X and ev ery mem b er of Y t here are more than half of the individual v otes where the fo rmer is preferred to t he latter. In t ha t case, t he social ranking also prefers eac h mem b er of X to eve ry mem b er of Y . 1.2 W e will adopt the p oin t of view of paired comparisons. In other w ords, w e will b e based up o n t he num b er of v oters who prefer x to y , where x and y v ar y o ver all ordered pairs of options. These n umbers will b e denoted b y V xy . Most of the time, how ev er, w e will b e dealing with the fractions v xy = V xy /V , where V denotes the total n umber of v otes. W e will refer to V xy and v xy resp ectiv ely as the absolute a nd relativ e scores of the pair xy , and the whole collection of these scores will be called the (absolute or relativ e) Llull matrix o f the vote. In the complete case one has v xy + v y x = 1 , so v xy automatically de- termines v y x . In con tr a st, in the incomplete case w e a re ensured only that v xy + v y x ≤ 1 , so v xy alone do es no t determine v y x . In particular, the con- ditions v xy > 1 2 and v xy > v y x are not equiv alent to eac h other, whic h giv es rise to tw o p ossible notions of ma jo rit y . Anyw a y , in the incomplete case a quantitativ e sp ecification of preference betw een t wo optio ns x and y requires the v alues of b oth v xy and v y x , or equiv alently , their sum and dif - ference, t xy = v xy + v y x and m xy = v xy − v y x , whic h w e will call resp ective ly the (relative ) turnout a nd margin asso ciated with the pair xy . 1.3 In preferen tial v oting the individual preferences are usually assumed to b e expressed in the form of a ranking, that is, a list of options in order of preference. In this connection, it is quite natural to admit the p ossibility of ties as w ell as incomplete lists. When w e are dealing with incomplete lists, their translation in to paired comparisons admits of sev eral inte rpretations. In most cases, it is reasonable to use the follow ing o ne: Continuous ra ting f or incomplete prefere ntial v oting 5 (a) When x and y are b oth in the list and x is rank ed a b ov e y (without a tie), we certainly inte rpret that x is preferred to y . (b) When x and y are b oth in the list and x is rank ed as go o d a s y , w e interpret it as b eing equiv alent to half a v ote preferring x to y plus another half a vote preferring y to x . (c) When x is in t he list and y is not in it, w e inte rpret that x is preferred to y . (d) When neither x nor y a r e in the list, w e in terpret nothing ab out the preference o f the voter b et w een x and y . Instead o f rule (d), o ne can consider the p ossibilit y of using the following alternativ e: (d ′ ) When neither x nor y are in the list, w e in terpret that they are considered equally go o d (or equally bad), so we pro ceed a s in (b). This amo unts to complete eac h truncated ranking by app ending to it all the missing options tied to eac h other, which bring s the problem to the complete case considered in [ 3 ]. Generally sp eaking, how ev er, this in terpretatio n can b e criticized in that the added infor ma t io n might not b e really mean t b y the v oter. On the other hand, in the spirit of not adding an y information not really mean t b y the v o ter, in certain cases it may b e appropriate to r eplace rule (c) b y the follo wing one: (c ′ ) When x is in the list and y is not in it, we inte rpret nothing ab out the preference o f the voter b et w een x and y . Generally sp eaking, the individ ual v otes could b e arbitra r y binary rela- tions, in terpreted as it is men tioned in [ 3 : § 3.1 ] ; ev en more g enerally , t hey could b e v alued binary relations b elonging to Ω = { v ∈ [0 , 1] Π | v xy + v y x ≤ 1 } , where Π denotes the set of pairs xy ∈ A × A with x 6 = y [ 3 : § 3.3 ] . Suc h a p ossibilit y ma kes sens e in that the individual opinions may already be the result of agg regating a v a riet y of criteria. An yw ay , the collectiv e Llull matrix is simply the center of gravit y of a distribution of individual v otes: v xy = X k α k v k xy , (1) where α k are the relative frequencies or weigh ts o f the individual v otes v k ∈ Ω . 1.4 In the part icular case where the set X consists of a single option, the Condorcet-Smith principle M takes the following form: 6 R. Camps, X. Mora, L. Sa umell M1 Condor c et princi p le (maj ority form) . If an o ption x ha s the prop erty that for eve ry y 6 = x there are more t han half of the individual v otes where x is preferred to y , then x is the so cial winner. In the complete case (where the Condorcet principle w as originally prop osed) the preceding condition is equiv alent to the following one: M1 ′ Condor c et principle (mar gin fo rm ) . If an option x has the prop ert y that for ev ery y 6 = x there are more individual v otes where x is preferred to y than vice v ersa, then x is the so cial winner. Ho w ev er, generally sp eaking condition M1 is w eak er than M1 ′ , and the CLC metho d will satisfy only the w eak er v ersion. This lack of compliance with the stronger condition M1 ′ ma y b e consid- ered undesirable. Ho w ev er, other authors ha v e a lready remark ed the need to require only M1 in order to b e able to k eep ot her prop erties (see for instance [ 17 ]). In o ur case, M1 ′ seems to conflict with the con tinuit y prop- ert y C. 1.5 Conditions E and M refer to cases where all the voters or a t least half of them pro ceed in a certain w ay . Of course, it should b e clear whether all the v o t ers means all of the actual ones or ma yb e all the p oten tial ones (i. e. actual v oters plus abstainers). W e assume that one has made a choice in that connection, th us defining t he total num b er of v oters V . Mathematically sp eaking, w e only need V to b e larger than an y absolute turnout V xy + V y x . Increasing the v alue o f V ha s no other effect than contracting the final rating to w ards the p oint where a ll rates tak e the w orst p ossible v alue ( N for rank- lik e r a tes). 2 Outline of the metho d This section presen ts the pro p osed metho d at the same time that it introduces the asso ciated terminology . As in [ 3 ], the pro cedure in v olv es a pro j ection of the Llull matrix on to a sp ecial set of suc h matr ices. Steps 0–3, as w ell as the last o ne, will b e exactly the same as in the complete case. Ho we v er, steps 4 and 5 con tain new elemen ts. More sp ecifically , step 4 requires a quadratic optimization in connection with the turnouts, and step 5 tak es the union o f certain in terv als where the complete case tak es t he maxim um of certain margins. The reasons b ehind steps 1 –3 and 6 w ere explained in [ 3 : § 2 ]. Those b ehind steps 4 and 5 will b e briefly explained in § 2.2.3, after ha ving lo ok ed a t the particularities of the complete case as w ell as those of single-c hoice voting. Continuous ra ting f or incomplete prefere ntial v oting 7 2.1 Step 0 . T o b egin with, we m ust form the Llull matrix ( v xy ) . Its en tries are the relativ e scor es v xy = V xy /V , where V is the n umber of v oters, and V xy coun ts ho w many o f them prefer x to y . In t he case of ranking v otes, this coun t will usu ally make use of rules (a–d) of § 1.3, though in certain cases it may b e r easonable to replace rule (c) by (c ′ ), or rule (d) b y (d ′ ). Besides the scores themselv es, which will b e used in the next step, later on w e will also mak e use of the a sso ciated turnouts t xy = v xy + v y x . (2) Step 1. Concerning the marg in comp onent, w e will rely on the indirect scores v ∗ xy . They derive from the original scores through an op eratio n that generalizes the notion of transitiv e closure to v alued relations. More sp ecifi- caly , they are defined in the following w a y: v ∗ xy = max { v α | α is a path x 0 x 1 . . . x n from x 0 = x to x n = y } , (3) where the score v α of a path α = x 0 x 1 . . . x n is defined as v α = min { v x i x i +1 | 0 ≤ i < n } . (4) Ob viously , we ha v e 0 ≤ v ∗ xy ≤ 1 , (5) v ∗ xy ≥ v xy . (6) The indirect turnouts v ∗ xy + v ∗ y x can b e larger than 1 . Ho w ev er, the follow - ing steps will use the indirect scores only through the asso ciated indirect margins m κ xy = v ∗ xy − v ∗ y x . (7) Step 2. This step is the discrete core o f the pro cedure. It b egins b y consid- ering the indirect comparison relation κ = { xy | m κ xy > 0 } , (8) as w ell as its co dual ˆ κ = { xy | m κ xy ≥ 0 } . (9) The relation κ has the virtue of b eing transitiv e. This crucial f act w as remark ed in 1998 by Markus Sc h ulze [ 13 b ]. As a consequence, κ is a partia l order, and therefore one can alw ay s extend it t o a total o r der ξ . F o r instance, 8 R. Camps, X. Mora, L. Sa umell according to Prop o sition 5.2 of [ 3 ], it suffices to arrange the optio ns b y non- decreasing v alues of the “ t ie- splitting” Cop eland ranks r x = 1 + |{ y | y 6 = x, m κ y x > 0 } | + 1 2 |{ y | y 6 = x, m κ y x = 0 } | . (10) Suc h a total order extens ion ξ automatically satisfies not only κ ⊆ ξ but also ξ ⊆ ˆ κ . In t he followin g w e call it an admissible order . The following steps assume that one has fixed an admissible order ξ . The inte rmediate quantities computed in these steps may dep end on whic h admissible order is used, but the final results will b e independen t of it. F rom no w on, the situation xy ∈ ξ will b e express ed also by writing x ≻ ξ y . F urthermore, x ′ will mean the immediate succes sor of x in ξ . Step 3. Starting from the indirect mar g ins m κ xy , one computes the super - diagonal in termediate pro jected margin s m σ xx ′ = min { m κ pq | p ≻ − ξ x, x ′ ≻ − ξ q } . (11) Step 4. Starting from the o riginal turnouts t xy , one computes the inter- mediate pro jected turnouts t σ xy , These num b ers dep end not only on the original turnouts t xy , but also on the superdiago nal in termediate pro jected margins m σ xx ′ . More specifically , they are tak en as the v alues of τ xy that minimize the quan tit y Φ = X x X y 6 = x ( τ xy − t xy ) 2 (12) under the fo llo wing constraints: τ xy = τ y x , (13) m σ xx ′ ≤ τ xx ′ ≤ 1 , (14) 0 ≤ τ xz − τ x ′ z ≤ m σ xx ′ , whenev er z 6∈ { x, x ′ } . (15) The actual computation of the minimizer can b e carried out in a finite n um b er of steps by me ans of a quadratic programming algorithm [ 10 : c h. 16 ]. F or future reference, the set of matrices ( τ xy ) that satisfy (13–15) will be denoted as T , and the preceding minimizing op eration that defines the in termediate pro jected t urno uts ( t σ xy ) as a f unction of the original turnouts ( t pq ) and the sup erdiagonal in termediate pro jected margins ( m σ pp ′ ) will b e denoted as Ψ : t σ xy = Ψ[( t pq ) , ( m σ pp ′ )] xy . (16) Continuous ra ting f or incomplete prefere ntial v oting 9 Step 5. F orm t he in terv als γ xx ′ = [ ( t σ xx ′ − m σ xx ′ ) / 2 , ( t σ xx ′ + m σ xx ′ ) / 2 ] , (17) as w ell as their unions γ xy = [ γ pp ′ , with p v arying in the in terv al x ≻ − ξ p ≻ ξ y , (18) where xy is restricted t o satisfy x ≻ ξ y . The sets γ xy are still in terv a ls. The pro jected scores ar e the upp er and lo w er b ounds of these interv als: v π xy = max γ xy , v π y x = min γ xy . (19) Equiv alently to (17– 19), the pro jected scores can be computed also in the follo wing w a y , whic h ha s a more practical character: T ak e v π xx ′ = ( t σ xx ′ + m σ xx ′ ) / 2 , v π x ′ x = ( t σ xx ′ − m σ xx ′ ) / 2 , (20) and then, for x ≻ ξ y , v π xy = max { v π pp ′ | x ≻ − ξ p ≻ ξ y } , v π y x = min { v π p ′ p | x ≻ − ξ p ≻ ξ y } (21) Step 6. Finally , the rank-like rates are determined by the form ula R x = N − X y 6 = x v π xy (22) 2.2 Sp ecial cases and heuristic considerations. 2.2.1 In the complete case the original turnouts t xy are all of them equal to 1 . One easily sees that in these circumstances step 4 results in t σ xy also equal to 1 for all pairs xy . In fact, this choice clearly satisfies conditions (1 3– 15) at the same t ime that it certainly minimizes (12 ) to 0 . As a consequenc e, the in terv als γ xx ′ defined by (17) are all of t hem cen tred at 1 / 2 . Ob viously , this pro p ert y will b e inherited by their unions γ xy . On the other hand, one easily se es t ha t the width of γ xy , in other w ords the pro jected marg in m π xy = v π xy − v π y x , will b e the follo wing: m π xy = max { m σ pp ′ | x ≻ − ξ p ≻ ξ y } . So, in the c omplete c as e the ab ove-de scrib e d pr o c e dur e r e duc es to the o ne that was pr esen te d in [ 3 ]. 10 R. Camps, X. Mora, L. Sa umell 2.2.2 Let us see what we get in the case of single-c hoice v oting. T o b e- gin with, rules (c–d) of § 1.3 result in v xy = f x for ev ery y 6 = x , where f x is the fraction of v oters who c ho ose x . This implies that v ∗ xy = v xy = f x . In fact, an y path γ fro m x to y starts with a link of the for m xp , whose asso ciated score is v xp = f x . So v γ ≤ f x and therefore v ∗ xy ≤ f x . But on the other ha nd f x = v xy ≤ v ∗ xy . Conseque ntly , w e get m κ xy = v ∗ xy − v ∗ y x = v xy − v y x = f x − f y , and the admissible o rders ar e those for whic h the f x are non-increasing. Owing to this non- increasing character, the in termediate pro jected margins are m σ xx ′ = m xx ′ = f x − f x ′ . On the other hand, the in termediate pro jected turnouts are t σ xy = t xy = f x + f y . In fact these n um b ers ar e easily seen to satisfy conditions (14–15) and they ob viously minimize (12). As a con- sequence , γ xx ′ = [ f x ′ , f x ] . In particular, the in terv als γ xx ′ and γ x ′ x ′′ are adjacen t to eac h other (the righ t end of t he latter coincides with the left end of the former). This fact en t a ils that γ xy = [ f y , f x ] whenev er x ≻ ξ y . So, the pro jected scores are the end p oints of these in terv als, namely v π xy = f x and v π y x = f y . In part icular, they coincide with t he o r ig inal scores. F inally , the r ank-like r ates ar e R x = 1 + ( N − 1) f x = f x + (1 − f x ) N , as state d by c ondition F. 2.2.3 In § 2.2.1 we ha v e seen that in the complete case, the pro jected margins are obtained from the sup erdiagonal intermediate ones b y means of a maxim um op eration: m π xy = max { m σ pp ′ | x ≻ − ξ p ≻ ξ y } (whenev er x ≻ ξ y ). In con trast, in the case of single-choice voting w e ha v e a sum: m π xy = P { m σ pp ′ | x ≻ − ξ p ≻ ξ y } (whenev er x ≻ ξ y , since w e hav e both m π xy = m xy = f x − f y and m σ pp ′ = m pp ′ = f p − f p ′ ); as w e hav e just seen in § 2.2.2, it m ust necess arily b e so if we are to satisfy condition F. So a general metho d requires a n op eration that r educes to maxim um in one case and to a dditio n in the other. This leads to t he idea that this general op eration should b e t hat of tak- ing the union o f suitable “score interv als”. A score in terv al can b e view ed as giving a pair of scores ab out t w o options, these scores b eing resp ectiv ely in fa v our and against a sp ecified preference ab out the tw o options. Alterna- tiv ely , it can b e view ed as giving a certain margin together with a certain turnout. F ollowing on this line, one can b e tempted to also replace the min- im um o p eration (11) of step 3 by an inte rsection of the score interv als that com bine the original turnouts t xy with the indirec t margins m κ xy . Suc h a pro cedure w orks as des ired b ot h in the case o f complete votes a nd that o f single-c hoice v oting. Ho w ev er, it breaks dow n in other case s of incomplete v otes t ha t pro duce empty in tersections and disjoint unions. In order to a v oid these problems, w e w ere led t o replace the original turnouts t xy b y the in termediate pro jected ones t σ xy . As we will see, the Continuous ra ting f or incomplete prefere ntial v oting 11 constrain ts that are imp osed on the latter hav e the virtue of ensuring a non- empt y in tersection for an y tw o consecutiv e interv als γ xx ′ and γ x ′ x ′′ . On the other hand, they ensure also the inequalit y t π xx ′ ≥ t π x ′ x ′′ . These f acts will b e crucial for ac hieving the following prop erties for the final ra nk-lik e rates R x : (a) b eing the same for any admissible order ξ ; and (b) being consisten t with an y suc h order ξ , i. e. ha ving R x ≤ R y whenev er x ≻ ξ y . 2.3 Summary 0. F o rm the Llull matrix ( v xy ) . W o r k out the turnouts t xy = v xy + v y x . 1. Compute the indirect scores v ∗ xy defined b y (3–4) . An efficien t w a y to do it is the Flo yd-W arshall algorithm [ 4 : § 25.2 ]. F or small v alues of N , one can do it b y ha nd in successiv e steps that progressiv ely increase the length of the paths under consideration. Ha ving computed the indirect scores, one w orks out the indirect margins m κ xy = v ∗ xy − v ∗ y x . 2. Consider the indirect comparison relation κ = { xy | m κ xy > 0 } . Fix an admissible order ξ , i. e. a total order that extends κ . F o r instance, it suffices to arr ange the options b y non-decreasing v alues of the “tie- splitting” Cop eland ranks (10 ). 3. Sta rting from the indirect margins m κ xy , work out the superdiag onal in termediate pr o jected margins m σ xx ′ as defined in (11) . 4. Sta rting fro m the original turnouts t xy , and taking in to account the su- p erdiagonal in termediate pro jected margins m σ xx ′ , determine the inter- mediate pro jected turnouts t σ xy b y minimizing (12) under t he constrain ts (13 –15). This can b e carried out in a finite n umber of steps by means of a quadratic programming algorithm [ 10 : c h. 16 ]. 5. F o rm the in terv als γ xx ′ defined b y (17 ) , deriv e their unions γ xy as defi- ned b y ( 1 8), and read off the pro jected scores v π xy (19). Or equiv alen tly: Compute the sub- and sup er-diago nal pro jected scores as defined b y ( 2 0), and then deriv e all the others a ccording to (21). 6. Compute the rank-like rates R x according to (22). 2.4 V arian ts. The preceding pro cedure admits of ce rtain v ariants whic h migh t b e appropriate to some special situat io ns. Next w e will distinguish four of them, namely 1. Main 2. Co dual 12 R. Camps, X. Mora, L. Sa umell 3. Ba la nced 4. Marg in-based The ab ov e-describ ed pro cedure is included in this list as the main v aria n t. The four v arian ts are exactly equiv alen t to eac h other in the complete case, but in the incomplete case they can pro duce differen t results. In spite of this, they all share the main properties. Ha ving said that, the pro o fs giv en in this pap er assume either the main v a r ian t or the marg in- based one. The codual v a rian t is analo g ous to the main one except that the max-min indirect scores v ∗ xy are replaced by the fo llo wing min-ma x ones: ∗ v xy = min x 0 = x x n = y max i ≥ 0 i < n v x i x i +1 . (23) Equiv alently , ∗ v xy = 1 − ˆ v ∗ y x where ˆ v xy = 1 − v y x . In the complete case one has ∗ v xy = 1 − v ∗ y x , so that ∗ v xy − ∗ v y x = v ∗ xy − v ∗ y x ; as a consequence, the co dual v ariant is then equiv alent to the main one. The balanced v aria n t take s κ = { xy | v ∗ xy > v ∗ y x , ∗ v xy > ∗ v y x } together with m κ xy =      min ( v ∗ xy − v ∗ y x , ∗ v xy − ∗ v y x ) , if xy ∈ κ , − m κ y x , if y x ∈ κ , 0 , otherwise. (24) The remarks made in connection with the co dual v ar ian t show that in the complete case the balanced v ariant is also equiv alent to the preceding ones. The margin- ba sed v aria nt follows the pro cedure of [ 3 ] eve n if o ne is not originally in the complete case. Equiv alently , it corr esp o nds to replacing the original scores v xy b y the follo wing ones: v ′ xy = (1 + m xy ) / 2 , where m xy = v xy − v y x . This amoun ts to replacing an y lac k of informat io n ab out a pair of o pt io ns b y a prop er tie betw een them, whic h brings the problem in to the complete case. In the case of ranking v ot es, it corresp onds to in terpreting the v otes using rule (d ′ ) instead of rule (d) as men tioned in § 1.3. So, the sp ecific c haracter of this v ar ian t lies only in its interpretation of incomplete v otes. R emark . O ther v a rian ts —in the incomplete case— arise when equation (22) is replaced b y the follo wing one: R x = 1 + X y 6 = x v π y x . (25) Continuous ra ting f or incomplete prefere ntial v oting 13 2.5 The main ideas that underlie the indirect scores of step 1 and the asso ciated indirect compar ison relation κ are the same as in the ranking metho d of Markus Sc h ulze [ 14 ]. Lik e us, he allo ws for the Llull matrix to b e incomple te, and he distinguis hes sev eral v aria n t methods. One o f them is a margin v ar ia n t that coincides essen tially with ours; more sp ecifically , b oth of them give exactly the same indirect comparison relation κ —Sc h ulze’s O — although they ma y differ in the subsequen t treatmen t of t ies. Ho w ev er, aside from the margin approa c h, which do es not prop erly face incompleteness , none of o ur other v aria n ts coincides with an y of Sc hulze’s ones. As a matter of fact, they can result in differen t indirect comparison relations κ and therefore, b y Theorem 5.2 b elow, in differen t final rankings. In this connection, w e ma y add that none of Sch ulze’s v ariants except the margin one is appropriate for b eing exte nded to a con tinuous rating metho d. In fact, to this effec t, the strength comparison relation ≻ D that Sc h ulze uses to compare the pairs ( v xy , v y x ) with eac h other should remain unc hanged under small p erturbat ions of the scores. Ho w ev er, this is not the case for an y of those v aria nts, namely D = ratio , D = win , and D = lo s . More sp ecifically , b y loo king at the corresp onding definitions of ≻ D , one easily c hec ks that one can ha ve , for instance, ( v , v ′ ) ≻ D ( w , w ) but, in contrast, ( w + ǫ, w ) ≻ D ( v , v ′ ) for arbitrarily small ǫ > 0 ; to this effect it suffices to assume 1 > v > v ′ > 0 and to c ho o se w as follows dep ending on D : w > v for D = win ; w < v ′ for D = los ; w = 0 for D = ratio . 3 Examples 3.1 As a n example of a v ote whic h in v olv ed tr uncated rankings, w e lo ok at a n election whic h to ok place the 16t h of F ebruary of 1652 in the Span- ish roy al househ old. This elec tion is quoted in [ 12 ], but w e use the more reliable and slightly different data whic h ar e giv en in [ 11 : vol. 2, p. 263 –264 ] . The office under election was that of “ap o sen ta dor may or de palacio”, and the king was asse ssed b y six noblemen, who expressed resp ectiv ely the fol- lo wing preference s abo ut six candidates a – f : b ≻ e ≻ d ≻ a ; b ≻ a ≻ f ; a ≻ f ≻ b ≻ d ; e ≻ b ≻ f ≻ c ; e ≻ a ≻ b ≻ f ; b ≻ d ≻ a ≻ f . The CLC computations are sho wn below. Instead of relative scores , mar- gins and turnouts, we will show their absolute coun terparts, i. e. without dividing them by the total n um b er of v otes. This has the virtue of sta ying with small in teger n umbers. Pairwis e info rmation will b e giv en by means of square matrices; a s usual, the cell in row x and column y corresponds to the pair xy . Since w e consider only pairs xy with x 6 = y , w e use the diagonal cells for sp ecifying the sim ultaneous lab elling of ro ws and columns by the 14 R. Camps, X. Mora, L. Sa umell mem b ers o f A . W e m ust specify the lab elling sinc e at a certain stag e w e will rearrange the o ptio ns in a ccordance with an admissible order. W e b egin b y forming the Llull matrix ( V xy ) (step 0) and deriving the corresp onding indirect scores ( V ∗ xy ) (step 1): ( V xy ) = a 2 5 3 3 5 4 b 6 6 4 5 1 0 c 1 0 0 2 0 3 d 2 2 3 2 3 3 e 3 1 1 5 4 3 f ; ( V ∗ xy ) = a 2 5 4 3 5 4 b 6 6 4 5 1 1 c 1 1 1 2 2 3 d 2 2 3 2 3 3 e 3 3 2 5 4 3 f . (26) In t he matrix of the indirect scores w e hav e visualized the indirect comparison relation κ b y marking in black b old face those pairs xy that b elong to it, i. e. t ha t satisfy V ∗ xy > V ∗ y x . Those tha t satisfy V ∗ xy = V ∗ y x are sho wn in grey b old fa ce. In accordance with (1 0 ), the Cop eland rank of each option is then easily w ork ed out as the num b er of blac k b old faces in the corresp onding column plus ha lf the nu mber of grey b old faces in it plus one. The resulting v alues are sho wn next: ( r x ) = a b c d e f 2 1 2 1 6 5 3 3 1 2 . (27) Arranging the options b y non-decreasing v alues of r x giv es an admissible order. In this case there is only one p ossibilit y , namely b ≻ a ≻ e ≻ f ≻ d ≻ c . This completes step 2. F rom no w on, the options will b e arranged in this order. W e now tak e the indirect margins M κ xy = V ∗ xy − V ∗ y x to w ork out the sup erdiagonal in termediate pro jected mar g ins M σ xx ′ (step 3) . Ha ving arranged the options in the adopted a dmissible order, M σ xx ′ is simply the minim um M κ v alue in the rectangle that lies to the righ t of x and ab ov e x ′ : ( M κ xy ) = b 2 2 3 4 5 ∗ a 0 2 2 4 ∗ ∗ e 0 1 2 ∗ ∗ ∗ f 2 4 ∗ ∗ ∗ ∗ d 2 ∗ ∗ ∗ ∗ ∗ c ; ( M σ xx ′ ) = b 2 ∗ ∗ ∗ ∗ ∗ a 0 ∗ ∗ ∗ ∗ ∗ e 0 ∗ ∗ ∗ ∗ ∗ f 1 ∗ ∗ ∗ ∗ ∗ d 2 ∗ ∗ ∗ ∗ ∗ c . (2 8) W e no w tak e the origina l turnouts T xy to determine the in termediate pro- jected ones T σ xy (step 4). This in v olv es also the sup erdiago nal in termediate pro jected margins M σ xx ′ and requires solving (the absolute coun terpart of ) Continuous ra ting f or incomplete prefere ntial v oting 15 the problem of minimizing (12 ) under the constraints (1 3–15). The results are sho wn b elo w: ( T xy ) = b 6 6 6 6 6 ∗ a 6 6 5 6 ∗ ∗ e 6 5 3 ∗ ∗ ∗ f 6 5 ∗ ∗ ∗ ∗ d 4 ∗ ∗ ∗ ∗ ∗ c ; ( T σ xy ) = b 6 6 6 6 6 ∗ a 6 6 5 1 3 4 2 3 ∗ ∗ e 6 5 1 3 4 2 3 ∗ ∗ ∗ f 5 1 3 4 2 3 ∗ ∗ ∗ ∗ d 4 ∗ ∗ ∗ ∗ ∗ c . (29) Although the last step in volv ed all of the in termediate pro jected turnouts, the next one uses only the superdiag onal ones T σ xx ′ . These n umbers together with the sup erdiag onal in termediate margins M σ xx ′ determine the in terv als Γ xx ′ = (( T σ x ′ x − M σ xx ′ ) / 2 , ( T σ x ′ x + M σ xx ′ ) / 2) = ( V π x ′ x , V π xx ′ ) , whose unions as in (18) giv e all the ot her in terv als Γ xy = ( V π y x , V π xy ) (step 5): ( V π xy ) = b 4 4 4 4 4 2 a 3 3 3 1 6 3 1 6 2 3 e 3 3 1 6 3 1 6 2 3 3 f 3 1 6 3 1 6 2 2 1 6 2 1 6 2 1 6 d 3 1 1 1 1 1 c . (30) Finally , the rank-lik e rates are obta ined fr om the pro jected scores b y means of formula (22) with v π xy = V π xy /V (step 6): ( R x ) = b a e f d c 2.6667 3.6111 3.611 1 3.6111 4.08 33 5.1667 . (31) According to these results, the office should ha v e been giv en to candi- date b , who is also the winner by most other metho ds. In the CLC metho d, this candidate is f ollo w ed b y three runners-up tied to eac h other, namely can- didates a , e and f . In spite of the clear adv an tage of candidate b , the king app ointe d candidate f , whic h w as the celebrated painte r Diego V el´ azquez. 3.2 As a second example of an election in v olving truncated rankings w e tak e the Debian Pro ject leader election, whic h is using the qualitative metho d of Markus Sc h ulze since 20 03. So far, the winners of these elections hav e b een clear enough. How ev er, a quan titativ e measure of this clearness w as lac king. In the following we consider the 2006 election, which had a participatio n of 16 R. Camps, X. Mora, L. Sa umell V = 4 2 1 actual v oters out of a total p opulatio n of 972 mem b ers. The indi- vidual v otes w ere tak en fro m http ://www.d ebian.or g/vote/2006/vote - 002 . The next tables show the Llull matrix of that election a nd the resulting rank-lik e rates: ( V xy ) = 1 321 144 159 1 2 193 1 2 347 1 2 246 320 51 2 42 5 3 50 262 65 163 251 340 3 198 1 2 253 362 300 345 245 1 2 341 204 1 2 4 256 371 1 2 291 1 2 339 1 2 193 1 2 325 144 149 5 357 254 321 1 2 26 1 2 77 24 22 1 2 21 6 30 74 1 2 137 292 90 109 1 2 131 330 7 296 76 207 54 71 1 2 75 1 2 302 1 2 89 8 , (3 2) ( R x ) = 1 2 3 4 5 6 7 8 4.1105 5.9145 3.692 6 3.6784 4.11 05 6.7197 4.5 720 5.810 0 . (33) In this case, the CLC results are in full agreemen t with the Cop eland ra nks of the original Llull matrix. In particular, b o th of them give an exact tie b et w een candidates 1 and 5 . Ev en so, the CLC rates yield a quan titative information whic h is not presen t in the Cop eland r anks. F or the computation of the rat es w e ha v e tak en V = 421 (the actual n um b er of votes) instead of V = 972 (the num b er of p eople with the righ t to v ote). This is esp ecially justified in D ebian elections since they systematically include “none of the ab ov e” as one of the alternativ es, so it is reasonable to in terpret that a bsten tio n do es not hav e a critical c haracter. In this case, “none of the ab ov e” was alternat ive 8 , whic h obtained a better result than t w o of the real candidates. 3.3 Finally , we lo ok at an example of approv al voting. Specifically , w e consider the 2006 Public Choice So ciet y ele ction [ 1 ]. Besides an approv al v ote, here the voters w ere also a sk ed for a preferen tial v ote “in t he spirit of researc h on public c hoice”. Ho w ev er, here w e will limit ourselv es to the appro v al v ote, whic h w as the official one. The v ote had a participation of V = 3 7 v oters, most of whic h appro v ed more than one candidate. Continuous ra ting f or incomplete prefere ntial v oting 17 The actual ba llots are listed in the follo wing table, 1 where w e giv e not only the appro v al v o t ing data but also the the express ed preferences. The appro v ed candidates are the ones whic h lie at t he left o f the slash. A ≻ B / A ≻ C ≻ B / D / A ≻ B ≻ E ≻ C B ≻ A / D ≻ C ≻ E D ≻ A ≻ B ≻ C / E C ≻ B ≻ A / E / D C ≻ A ≻ B ≻ E / D ≻ E / C ≻ A ≻ B E / B ≻ C / D ≻ C / B ≻ E ≻ A B / A / A / D / A ∼ B ∼ C ∼ E A ∼ C / / B ≻ E ≻ A ≻ D ≻ C A ∼ B ∼ E / A ∼ B ∼ C ∼ D ∼ E / D ≻ A ≻ B / B ≻ D ≻ A / C ≻ E A / B ≻ E ≻ C ≻ D D / A ∼ C ≻ B / D ≻ E A / D ≻ B ≻ C ≻ E C / B ≻ D ≻ A ≻ E C / D ∼ E / A ≻ B ∼ C B / C ≻ A ≻ D ≻ E D ≻ C ≻ E / C / A ≻ B ∼ D ∼ E C / B ≻ D / E ≻ C ≻ A B ≻ C / A ≻ E ≻ D D ≻ A ≻ C ≻ B / D ≻ E / A ≻ B The next table giv es the n um b er of receiv ed approv als A x together with the r a nk-lik e rates resulting from the four v ariants of the CLC metho d ( the sup erindices 1 , 2 , 3 , 4 indicate resp ectiv ely the main, co dual, bala nced and margin-based v ar ian ts). x A B C D E A x 17 16 17 14 9 R 1 x 3.6014 3.6486 3.614 9 3.7720 4.1689 R 2 x 3.6081 3.6486 3.608 1 3.7568 4.2162 R 3 x 3.6081 3.6486 3.608 1 3.7703 4.1622 R 4 x 2.8919 2.9324 2.891 9 3.0135 3.2703 As one can see, the appro v al scores result in a tie fo r the first place b et w een candidates A and C , whic h are follow ed at a minimum distance by can- didate B and then by candidates D and E . Exactly the same ranking is found in the results of the co dual, balanced and margin-based v ariants of the CLC metho d, but not in those of the main v ariant, whic h discriminates b e- t w een candidates A and C , giving the victory to A . In § 9 we will see that the ranking giv en by the margin-based v ariant is alwa ys in full agreemen t with that given b y the approv al score. 1 W e a re g rateful to P rof. Steven J. B r ams, who was the president of the Public Cho ice So ciety when that election too k place, for his kind p ermissio n to repr o duce these data. 18 R. Camps, X. Mora, L. Sa umell 4 The pro jection. W ell-definedn ess and structural prop erties As it was seen in § 2, the CLC metho d in v olv es in a crucial wa y a pro jection ( v xy ) 7→ ( v π xy ) of the original Llull mat rix on to a special set of suc h matrices. In this section w e will see that t his pro jection is w ell defined and we will lo ok at certain structural prop erties of the resulting ma t r ix. The pro jection comprises steps 1–5 of the CLC pro cedure. In order to en- sure that it is well defined, one must c hec k t he follo wing p oin ts: (a) The existence (and effectiv e pro duction) of an admissible order ξ . Since it in v olv es only steps 1–2 , b efore an y div ergence of t he pro cedure fro m the com- plete case, this p oint requires no other considerations than those made in [ 3 ]. (b) The existence and uniquenes s of a minimizer of (12) under the condi- tions (13 – 15). This is a conseq uence of the fact that the se t T defined b y these conditio ns is a closed con v ex set [ 7 : c h . I, § 2 ] . In this connection, one can say that ( t σ xy ) is t he or t ho gonal pro jection of ( t xy ) on to the con ve x set T . And (c) that the final results are indep enden t of the admissible order ξ when there a r e sev eral p ossibilities for it. This will b e dealt with by Theorem 4.2 b elo w, whose pro o f is rather long, but not difficult. Before em bar king on that theorem, how ev er, w e will lo ok at certain pro p- erties of the in terv als γ xy . Besides b eing used in that theorem, these prop- erties will b e seen later on to b e a t the core o f t he structure o f the pro jected Llull matrix. W e will use the follo wing notation: | γ | means the length o f an in terv al, and • γ means its barycen tre, or cen troid, i. e. t he n um b er ( a + b ) / 2 if γ = [ a, b ] . Lemma 4.1. The sets γ xy have the fo l lowing p r op erties for x ≻ ξ y ≻ ξ z : (a) γ xy is a clos e d interval. (b) γ xy ⊆ [0 , 1] . (c) γ xz = γ xy ∪ γ y z . (d) γ xy ∩ γ y z 6 = ∅ . (e) | γ xz | ≥ max ( | γ xy | , | γ y z | ) . (f ) • γ xy ≥ • γ xz ≥ • γ y z . (g) • γ xy − | γ y z | / 2 ≤ • γ xz ≤ • γ y z + | γ xy | / 2 . Pr o of. Let us b egin b y noticing that the sup erdiagonal in termediate turnouts and margins are ensured to satisfy the following inequalities: 0 ≤ m σ xx ′ ≤ t σ xx ′ ≤ 1 (34) 0 ≤ t σ xx ′ − t σ x ′ x ′′ ≤ m σ xx ′ + m σ x ′ x ′′ . (35) Continuous ra ting f or incomplete prefere ntial v oting 19 The inequalities o f (34) are those of (14), with t σ xx ′ substituted for τ xx ′ , plus the f act that m σ xx ′ ≥ 0 . Those o f (35) a re the result of adding up (1 5) with x and z replaced resp ectiv ely b y x ′ and x plus the same inequalit y with z replaced by x ′′ , and using the symmetric c haracter of the t ur no uts. F rom (34) it follow s tha t 0 ≤ ( t σ xx ′ − m σ xx ′ ) / 2 ≤ ( t σ xx ′ + m σ xx ′ ) / 2 ≤ 1 . So, ev ery γ xx ′ is an in terv al (p ossibly reduced to one p oin t) and this in terv al is con tained in [0 , 1] . Also, the inequalities of (35) ensure o n the one hand that • γ xx ′ ≥ • γ x ′ x ′′ , and on the ot her hand tha t the in terv als γ xx ′ and γ x ′ x ′′ o v erlap eac h other. In t he follo wing w e will see t hat these facts ab out the elemen tary interv als γ xx ′ en tail the stated prop erties of the sets γ xy defined b y (18) . P art ( a ). This is an obv ious consequence of the fa ct that γ pp ′ and γ p ′ p ′′ o v erlap eac h o ther. P art ( b). This f o llo ws from the fa ct that γ pp ′ ⊆ [0 , 1] . P art ( c). This is a consequence of the asso ciativ e prop ert y enjoy ed b y the set-union op eration. P art ( d). This is again a n o b vious consequence of the fact that γ pp ′ and γ p ′ p ′′ o v erlap eac h other (take p ′ = y ) . P art ( e). This follo ws from (c) b ecause γ ⊆ η implie s | γ | ≤ | η | . P art ( f ). This follo ws from the inequalit y • γ pp ′ ≥ • γ p ′ p ′′ b ecause of the follo wing general fact: If γ and η are t w o ov erlapping interv als with • γ ≥ • η then • γ ≥ ( γ ∪ η ) • ≥ • η . This follow s immediately from the definition of the barycen tre. P art ( g ). This follows fr o m ( c) and ( d) b ecause of the follo wing general fact: If γ a nd η are tw o o verlapping interv als, then • γ − | η | / 2 ≤ ( γ ∪ η ) • ≤ • γ + | η | / 2 . Again, this fo llows easily from the definitions. Theorem 4.2. The pr oje cte d s c or es do not dep end on the admissible or der ξ use d for their c alculation, i. e. the value of v π xy is in dep enden t of ξ for every xy ∈ Π . On the other hand, the matrix of the pr oje cte d sc or es i n an admissible or d e r ξ is also indep endent of ξ ; i. e. i f x i denotes the element of r ank i in ξ , the v a lue of v π x i x j is indep ende n t o f ξ for every p air of indic es i, j . R emark . The tw o statemen ts say different things since the iden t it y o f x i and x j ma y dep end on the admissible order ξ . Pr o of. F or the purp oses of this pro of it b ecomes necessary to c hange our set-up in a certain w ay . In fact, un til no w the inte rmediate ob jects m σ xy , t σ xy 20 R. Camps, X. Mora, L. Sa umell and γ xy w ere considered only for x ≻ ξ y , i. e. xy ∈ ξ . How ev er, since w e ha v e to deal with c hanging the a dmissible order ξ , here w e will allo w their argumen t xy to b e any pair (of differen t elemen ts), no matter whethe r it b elongs to ξ or not. In this connection, w e will certainly put m σ y x = − m σ xy and t σ y x = t σ xy . On the o ther hand, concerning γ xy and γ y x , w e will pro ceed in the fo llo wing w a y: if γ xy = [ a, b ] then γ y x = [ b, a ] . So, generally sp eaking the γ xy are here “oriente d inte rv als”, i. e. ordered pairs of real num b ers. Ho w ev er, γ xy will alw ays b e “ p ositiv ely orien ted” when xy b elongs to an admiss ible order (but it will be reduced to a p oint whenev er there is another admissib le order which includes y x ). In particular, the γ pp ′ whic h are com bined in (18) are alw a ys p ositiv ely orien ted interv als; so, the union op eration p erformed in that equation can alw ay s be unders to o d in the usual sense. In the fo llo wing, γ ’ denotes t he orien ted in terv al “ rev erse” to γ , i. e. γ ’ = [ b, a ] if γ = [ a, b ] . So, let us consider the effect of replacing ξ b y anot her admissible order e ξ . In the following, the tilde is systematically used to distinguish b et w een hom- ologous ob jects whic h are asso ciated resp ectiv ely with ξ and e ξ ; in particular, suc h a no tation will b e used in connection with the lab els of the equations whic h ar e f orm ulated in terms of the a ssumed admissible order. With this t erminolo g y , we will pro v e the tw o follow ing equalities. First, γ xy = e γ xy , for any pair xy ( x 6 = y ) , (36) where γ xy are the interv als produced b y (16–18) together with the op eration γ y x = γ ’ xy , and e γ xy are those pro duced b y ( e 16– e 18) together with the op eration e γ y x = e γ ’ xy . Secondly , we will see a lso that γ x i x j = e γ ˜ x i ˜ x j , for an y pair of indices ij ( i 6 = j ) , (37) where x i denotes the elemen t of rank i in ξ , and a nalogously for ˜ x i in e ξ . These equalities con tain the statemen ts of the theorem since the pro jected scores are nothing else than the end p oints of the interv als γ xy . No w, b y a w ell-known result, prov ed for instance in [ 6 ], it suffices to deal with the case of t w o admissible orders ξ and e ξ whic h differ from eac h other b y one in vers ion only . So, w e will assume that there are t w o elemen ts a and b suc h that the only difference b et w een ξ and e ξ is that ξ contains ab whereas e ξ con tains ba . According to the definition of an admissible order, this implies that m κ ab = m κ ba = 0 . In order to control the effect o f the differences b et w een ξ and e ξ , we will mak e use of the follow ing notation: p will denote the immediate predecessor of a in ξ ; in this connection, an y stat ement ab out p will b e understo o d to imply the assumption t ha t the set of predecessors of a in ξ is not empty . Continuous ra ting f or incomplete prefere ntial v oting 21 Similarly , q will denote the immediate succ essor of b in ξ ; here to o, any statemen t a b out q will b e understo o d to imply the assumption that the set of successors of b in ξ is not empt y . So , ξ and e ξ contain resp ective ly the paths pabq and pbaq . Let us lo ok first at the sup erdiagona l interm ediate pro jected margins m σ hh ′ . Since their definition is the same as in the complete case, one can in vok e the same argumen ts as in the pro o f of Theorem 6.2 of [ 3 ] to o bta in the equality m σ x i x i +1 = e m σ ˜ x i ˜ x i +1 , for any i = 1 , 2 , . . . N − 1 . (38) In more sp ecific terms, m σ xx ′ = e m σ xx ′ , whenev er x 6 = p, a, b, (39) m σ pa = e m σ pb , (40) m σ ab = e m σ ba = 0 , (41) m σ bq = e m σ aq . (42) In connection with equation (39) it should be clear that for x 6 = p, a, b the imme diate suc c essor x ′ is the same in b oth o r ders ξ and e ξ . Next w e will see that the in termediate pro j ected turnouts t σ xy are in v ariant with resp ect to ξ : t σ xy = e t σ xy , for an y pair xy ( x 6 = y ) , (43) where t σ xy are the num b ers pro duced by (16) together with the symmetry t σ y x = t σ xy , and e t σ xy are those pro duced b y ( e 16) together with the symmetry e t σ y x = e t σ xy . W e will prov e (43) b y see ing that the set T determined b y conditions (13,14,15) coincides exactly with the set e T determined by (13, e 14, e 15). In other w ords, conditions (14–15 ) are exactly equiv alen t to ( e 14– e 15) under condition (13), whic h do es not dep end on ξ . In o rder to pro v e this equiv alence w e b egin b y noticing that condition (14 ) coincides exactly with ( e 14) when x 6 = p, a, b . This is true b ecause, on the one hand, x ′ is then the same in b oth orders ξ and e ξ , and, on the other hand, (39) ensures that the right-hand sides ha v e the same v alue. Similarly happ ens with conditions (15) and ( e 15) when z 6 = p, a, b . So, it remains to deal with conditions (14) and ( e 14) for x = p, a, b , and with conditions (1 5) and ( e 15) for z = p, a, b . Now, on a ccoun t of the sy mmetry (13), one easily sees that condition (1 4 ) with x = a is equiv alent to ( e 14) with x = b . In fact, b o th 22 R. Camps, X. Mora, L. Sa umell of them reduce to 0 ≤ τ ab ≤ 1 since m σ ab = e m σ ba = 0 , as it w as obtained in (41). This last equalit y ensures also the equiv alence b et w een condition (15 ) with z = a a nd condition ( e 15) with z = b . In this case b oth o f them reduce to τ xa = τ xb . (44) This common equalit y pla ys a central role in the equiv alence b et w een the re- maining conditions. Thus , its combin ation with (42) ensures the equiv alence b et w een (14) with x = b and ( e 14) with x = a , as w ell as the equiv alence b et w een (15) with z = b and ( e 15) with z = a when x 6 = a, b . On the other hand, its combination with (40) ensures the equiv alence betw een (14) and ( e 14) when x = p , as w ell as the equiv alence b etw een (15) a nd ( e 15) when z = p and x 6 = a, b . Finally , w e hav e the tw o following equiv a lences: (15) with z = p and x = b is equiv alen t to ( e 15) with z = p and x = a b ecause of the same equalit y (44) together with (40) and the symmetry (1 3); and similarly , (15) with z = b and x = a is equiv alen t to ( e 15) with z = a and x = b b ecause of (44) t ogether with (42) and (13). This completes the pro of of (43). Ha ving seen that condition (44) is included in b o th (15) a nd ( e 15), it follo ws that the inte rmediate pro jected turnouts satisfy t σ xa = t σ xb , e t σ xa = e t σ xb . (45) By taking x = p, q and using a lso (43), it follow s that t σ xx ′ = e t σ xx ′ , whenev er x 6 = p, a, b, (46) t σ pa = e t σ pb , (47) t σ ab = e t σ ba , (48) t σ bq = e t σ aq . (49) In other words , the sup erdiagonal in termediate turnouts satisfy t σ x i x i +1 = e t σ ˜ x i ˜ x i +1 , for any i = 1 , 2 , . . . N − 1 . (50) On accoun t of the definition of γ x i x i +1 and e γ ˜ x i ˜ x i +1 , the combin ation of (38) and (50) results in γ x i x i +1 = e γ ˜ x i ˜ x i +1 , for any i = 1 , 2 , . . . N − 1 , (51) from whic h the union op eration ( 1 8) pro duces (37). Continuous ra ting f or incomplete prefere ntial v oting 23 Finally , let us see that (36) holds to o. T o this effect, w e b egin by noticing that (41) together with (48) are sa ying not only that γ ab = e γ ba but also that this in terv al reduces to a p oin t. As a consequence, w e ha v e γ ba = γ ab = e γ ba = e γ ab . (52) Let us consider no w the equation γ pa = e γ pb , whic h is con t a ined in (51) . Since γ ab reduces to a p oint, parts (c) and (d) of Lemma 4.1 give γ pb = γ pa ∪ γ ab = γ pa . Ana lo gously , e γ pa = e γ pb ∪ e γ ba = e γ pb . Altogether, this gives γ pb = γ pa = e γ pb = e γ pa . (53) By means of an analo g ous argumen t, one o btains also that γ aq = γ bq = e γ aq = e γ bq . (54) On the other hand, (5 1) ensures that γ xx ′ = e γ xx ′ , whenev er x 6 = p, a, b . (55) Finally , part (c) of Lemma 4.1 allows to go from (52–55) to the desired general equalit y (3 6). Theorem 4.3. T h e pr oje cte d sc or es and their assso ciate d mar gins a nd turn- outs satisfy the fol lowing pr op erties with r esp e ct to an y admissible or der ξ : (a) The fol l o w ing in e qualities hold fo r x ≻ ξ y ≻ ξ z : v π xy ≥ v π y x , i. e. m π xy ≥ 0 , (56) v π xz = max ( v π xy , v π y z ) , (57) v π z x = min ( v π z y , v π y x ) , (58) m π xz ≤ m π xy + m π y z , (59) t π xz − t π y z ≤ m π xy , t π xy − t π xz ≤ m π y z . (60) (b) The fo l lowing ine qualities hold for x ≻ ξ y an d z 6∈ { x, y } : v π xz ≥ v π y z , v π z x ≤ v π z y , (61) m π xz ≥ m π y z , m π z x ≤ m π z y , (62) t π xz ≥ t π y z , t π z x ≥ t π z y , (63) (c) If v π xy = v π y x , or e quivalently m π xy = 0 , then (6 1 –63) ar e satisfie d a l l of them with an e quality sign. (d) The abs o lute pr oje cte d mar gins d π xy = | m π xy | satisfy the triangular ine qual- ity d π xz ≤ d π xy + d π y z for any x, y , z . 24 R. Camps, X. Mora, L. Sa umell Pr o of. W e will see that these prop erties deriv e from those satisfied b y the in terv als γ , whic h are collected in Lemma 4.1. F or the deriv atio n one has to b ear in mind that v π xy and v π y x are resp ectiv ely the righ t and left end p oin ts of the in terv al γ xy , and that m π xy = − m π y x and t π xy = t π y x are resp ective ly t he width and t wice the barycen tre of γ xy . P art (a ) . First of all, (56) holds as so on as γ xy is an interv al, as it is ensured by part (a) of L emma 4.1 . On the o ther hand, (57– 58) are nothing else than a pa raphrase of Lemma 4.1.(c), in the same w ay as (57–59) is a paraphrase of Lemma 4.1.(e). Finally , the inequalities of (60) are those of Lemma 4.1.(g) . P art (b). Let us b egin b y noticing that (62) will be an immediate conse- quence of (61), since m π xz = v π xz − v π z x and m π y z = v π y z − v π z y . On the other hand, (63.2) is equiv alent to (63.1). This equiv alence holds b ecause the turnouts are symmetric. So, it remains to prov e the ineq ualities (61) and either (63.1) or (63.2). In or der to prov e them w e will distinguish three cases, namely: (i) x ≻ ξ y ≻ ξ z ; (ii) z ≻ ξ x ≻ ξ y ; (iii) x ≻ ξ z ≻ ξ y . Case (i) : By part (c) of Lemma 4 .1, in this case w e ha v e γ xz ⊇ γ y z . This immediately implies (61) b ecause [ a, b ] ⊇ [ c, d ] is equiv alent to saying that b ≥ d and a ≤ c . O n t he other hand, the inequality (63.1) is contained in part (f ) of L emma 4.1 . Case (ii) is analogous t o case (i). Case (iii) : In this case, (61) follo ws from part (d) of Lemma 4.1 since [ a, b ] ∩ [ c, d ] 6 = ∅ is equiv alent to sa ying that b ≥ c a nd a ≤ d . On the other hand, (63.1 ) is still con tained in part (f ) of Lemma 4.1 (b ecause of the symmetric character of the turnouts). P art ( c). The h yp othesis that v π xy = v π y x is equiv alen t to sa ying that γ xy reduces to a p oint, i. e. γ xy = [ v , v ] for some v . W e will distinguish the same three cases a s in par t (a). Case (i) : On account of the o v erlapping prop ert y γ xy ∩ γ y z 6 = ∅ (part (d) of Lemma 4.1), the one-p o int interv al γ xy = [ v , v ] m ust b e con tained in γ y z . So, γ xz = γ xy ∪ γ y z = γ y z (where w e used part (c) of L emma 4.1 ). Case (ii) is again a nalogous to case (i). Case (iii) : By part (c) of Lemma 4.1 (with y and z in terc hanged with eac h other), the fact that γ xy reduces to t he one-p o in t in terv al [ v , v ] implies that b oth γ xz and γ z y reduce also to this o ne-p oint in t erv al. P art (d). It suffices to consider the particular ordering x ≻ ξ y ≻ ξ z and c hec k that eac h of the three n umbers d π xy = d π y x = m π xy , d π y z = d π z y = m π y z , d π xz = d π z x = m π xz is less than the sum of the other t w o. This is so since w e know that m π xz ≤ m π xy + m π y z , b y (59), and also t hat m π y z ≤ m π xz and m π xy ≤ m π xz , by (62), or mo r e directly , b y Lemma 4 .1.(e). Continuous ra ting f or incomplete prefere ntial v oting 25 In § 2.2.2 w e ha ve seen that in the case of single-c ho ice voting the pro- jected scores coincide with the o r ig inal ones. In that connection, one has the follo wing general result: Prop osition 4.4. Assume that ther e exists a total or de r ξ such that the original s c or es and their asso ciate d mar gins an d turnouts satisfy the fol lowin g c onditions: v xy ≥ v y x , i.e. m xy ≥ 0 , whenever x ≻ ξ y , (64) v xz = max ( v xy , v y z ) , whe n ever x ≻ ξ y ≻ ξ z , (65) v z x = min ( v z y , v y x ) , whenever x ≻ ξ y ≻ ξ z , (66) 0 ≤ t xz − t x ′ z ≤ m xx ′ , whenever z 6∈ { x, x ′ } . (67) In that c ase, the pr oje cte d sc o r es c oincide with the original ones. Pr o of. Let us b egin by noticing that condition (67 ) implies, as in the pro of of Lemma 4.1, that the interv als [ v x ′ x , v xx ′ ] a nd [ v x ′′ x ′ , v x ′ x ′′ ] ov erlap eac h other. In other w ords, o ne has the following inequalities “accross the diagona l” : v x ′ x ≤ v x ′ x ′′ and v x ′′ x ′ ≤ v xx ′ . No w, (65–66) together with the preceding inequalities imply that v xz ≥ v y z , v z x ≤ v z y , whenev er x ≻ ξ y and z 6∈ { x, y } . (68) F rom these fa cts, one can deriv e that v xz ≥ min ( v xy , v y z ) , for any x, y , z . (69) In fact, for z ≻ ξ y ≻ ξ x this inequalit y is guara n teed b y (66) (with x and z in terc hanged with eac h o ther), whereas for any other ordering of x, y , z one easily arrives at (69) a s a consequence of (68). No w, according to Lemma 4.2 of [ 3 ], (69) implies that the indirect scores coincide with the original ones: v ∗ xy = v xy , whic h en ta ils that m κ xy = m xy . In particular, ξ is ensured to b e an admissible order. Pro ceeding with the CLC algorithm, one easily c hec ks that m σ xy = m xy (b ecause the patt ern of growth (68) gets transmitted from t he scores to the margins) and t σ xy = t xy (since t he turnouts are a ssumed to satisfy (67) and one certainly has m xy ≤ t xy ≤ 1 ). As a consequenc e o f these facts, (20) results in v π xx ′ = v xx ′ and v π x ′ x = v x ′ x , from whic h (21) and (6 5–66) lead to conclude that v π xy = v xy for any pair xy . 26 R. Camps, X. Mora, L. Sa umell Since conditions (64–67) of Prop osition 4.4 are included among t he prop- erties of the pro jected Llull matrix according to Theorem 4.3, one can con- clude that they fully c haracterize the pro jected Llull matrices, and that the op erator ( v xy ) 7→ ( v π xy ) really deserv es b eing called a pro jection: Theorem 4.5. T he op er ator P : Ω ∋ ( v xy ) 7→ ( v π xy ) ∈ Ω i s idemp otent, i. e. P 2 = P . Its ima g e P Ω c onsists exactly of the Llul l matric es ( v xy ) that satisfy ( 6 4 – 67) for some total or der ξ . 5 The rank-lik e rates Let us recall that the rank-like rates R x are determined f rom the pro jected scores by formu la (22). The prop erties of the pro jected scores obtained in Theorem 4.3 imply the follo wing facts, whic h admit the same pro of as in [ 3 ] (the only c hange is that Theorem 4 .3 of the presen t article m ust b e inv oke d instead of [ 3 : Thm . 6.3 ] ): Lemma 5.1 (Same pro of as in [ 3 : Lem. 7.1 ]) . (a) If x ≻ ξ y in an a d missible or der ξ , then R x ≤ R y . (b) R x = R y if and only if v π xy = v π y x . (c) R x ≤ R y implies the ine qualities (56) an d (61) . (d) R x < R y if and only if v π xy > v π y x . (e) v π xy > v π y x implies x ≻ ξ y in any admissib l e or d er ξ . Theorem 5.2 (Same pro of as in [ 3 : Thm. 7.2 ]) . The r an k - like r ating give n by (22) is r elate d to the indir e ct c omp aris on r elation κ in the fo l lowing way: R x < R y ⇐ ⇒ y x 6∈ ( ˆ κ ) ∗ , (70) R x ≤ R y ⇐ ⇒ xy ∈ ( ˆ κ ) ∗ , (71) wher e ˆ κ is the c o dual of κ , namely ˆ κ = { xy | v ∗ xy ≥ v ∗ y x } . Corollary 5.3 (Same pro of as in [ 3 : Cor. 7.3 ] ) . (a) R x < R y ⇒ xy ∈ κ . (b) If ˆ κ is tr ansitive (in p articular, if κ is total), then R x < R y ⇔ xy ∈ κ . (c) If κ c on tains a set of the f o rm X × Y with X ∪ Y = A , then R x < R y for any x ∈ X and y ∈ Y . In the complete case, the fact that t π xy = v π xy + v π y x = 1 implies that P x ∈ A R x = N ( N + 1) / 2 . Related to it, one has the following general fact: Continuous ra ting f or incomplete prefere ntial v oting 27 Lemma 5.4. F o r any X ⊆ A one has X x ∈ X R x ≥ | X | ( | X | + 1) / 2 . (72) This ine q uali ty b e c omes a n e quality w h en a nd o n ly when the two fol lowing c onditions ar e satisfie d: t π x ¯ x = 1 , for al l x, ¯ x ∈ X (73) v π xy = 1 , for al l x ∈ X and y 6∈ X . (74) Pr o of. Starting fro m form ula (22), w e obtain X x ∈ X R x = N | X | − X x, ¯ x ∈ X ¯ x 6 = x v π x ¯ x − X x ∈ X y 6∈ X v π xy ≥ N | X | − | X | ( | X | − 1) / 2 − | X | ( N − | X | ) = | X | ( | X | + 1) / 2 , where the inequalit y deriv es from the fo llo wing ones: t π x ¯ x = v π x ¯ x + v π ¯ xx ≤ 1 for x, ¯ x ∈ X , and v π xy ≤ 1 for x ∈ X and y 6∈ X . 6 Con tin uity W e claim that the rank-lik e rates R x are con tin uous functions of the binary scores v xy . The main difficulty in proving this statemen t lies in the admissible order ξ , whic h pla ys a central r o le in t he computations. Since ξ v aries in a discrete set, its dep endence on the data cannot b e contin uous a t all. Ev en so, w e claim that the final result is still a con tin uous function o f the data . In this connection, one can consider as da ta the normalized Llull ma- trix ( v xy ) , its domain of v ar ia tion being the set Ω in tro duced in § 1.3. Al- ternativ ely , one can consider as data the relat ive frequencies of t he po ssible v otes, i. e. t he co efficien ts α k men t io ned a lso in § 1.3. Theorem 6.1. The pr oj e cte d sc or es v π xy and the r ank-like r ates R x dep end c ontinuously on the Llul l matrix ( v xy ) . Pr o of. Let us b egin b y conside ring the dep endence of the rank-lik e ra tes on the pro jected scores. This dependence is giv en by formu la (22), whic h is not only contin uous but ev en linear (non-homogeneous). So w e are left with the problem of showing that the pro jection P : ( v xy ) 7→ ( v π xy ) is con tinuous. By arguing as in [ 3 : Theorem 8.1 ], the problem reduces 28 R. Camps, X. Mora, L. Sa umell to sho wing the con tinuit y of P ξ : Ω ξ → Ω fo r an arbitr a ry total or der ξ , where Ω ξ means the su bset o f Ω which consists of the Llull matrices for whic h ξ is an a dmissible or der, and P ξ means the restriction of P to Ω ξ . In or der to c hec k that P ξ is contin uous for ev ery tota l order ξ , one has to go ov er the differen t mappings whose comp osition defines P ξ (see § 2.1), namely: ( v xy ) 7→ ( v ∗ xy ) 7→ ( m κ xy ) , ( v xy ) 7→ ( t xy ) , ( m κ xy ) 7→ ( m σ xy ) , Ψ : (( m σ xx ′ ) , ( t xy )) 7→ ( t σ xy ) , and fina lly (( m σ xx ′ ) , ( t σ xx ′ )) 7→ ( v π xy ) . Except for Ψ , all of these mappings in v olv e only the operatio ns of addition, subtraction, m ultiplication by a constan t, maxim um and minim um, whic h are certainly con tin uous. Concerning the op erator Ψ , let us recall that its output is the o r- thogonal pro jection of ( t xy ) on to a certain con vex set determine d b y ( m σ xx ′ ) ; a general result of contin uity for suc h an op eration can b e found in [ 5 ]. Corollary 6.2. T he r ank- l i k e r ates dep end c ontinuously on the r elative fr e- quency of e ach p ossi b le c ontent of an individual vote. Pr o of. It suffices to notice tha t the Llull matrix ( v xy ) is simply the cen ter of gra vit y of the distribution sp ecified b y these relative fr equencies (for mula ( 1) of § 1.3). 7 Decomp osition The decomp osition prop erty E is concerned with havin g a part it ion of A in to t w o non-empty sets X and Y suc h that eac h mem b er o f X is unanimously preferred to an y member of Y , that is: v xy = 1 (and therefore v y x = 0 ) whenev er xy ∈ X × Y . (75) More specifically , prop ert y E, whic h w as prov ed in [ 3 : § 9 ], ensures that in the complete case suc h a situation is c haracterized b y an y of the following equalities: R x = e R x , for all x ∈ X , (76 ) R y = e R y + | X | , for all y ∈ Y , (77) X x ∈ X R x = | X | ( | X | + 1) / 2 , (78) where e R x and e R y denote the ra nk-lik e rates whic h are determined resp ec- tiv ely fr o m the submatrices asso ciated with X and Y . Continuous ra ting f or incomplete prefere ntial v oting 29 In this section w e will see that some of these implications are still v a lid under certain assumptions that allow f o r incompleteness . In par t icular, The- orem 7.2 b elo w en tails that under the assumption of transitiv e individual preferences an o pt io n gets a rank-like rate exactly equal to 1 if and o nly if it is unanimously preferred to an y ot her. Lemma 7.1. Given a p artition A = X ∪ Y into two disjoint nonem pty sets, one ha s the fol low ing im plic ations: v xy = 1 ∀ xy ∈ X × Y  = ⇒  m κ xy = 1 ∀ xy ∈ X × Y  ⇐ ⇒  v π xy = 1 ∀ xy ∈ X × Y (79) If the individual pr efer enc es ar e tr a nsitive, then the c onverse of the first implic ation holds to o. Pr o of. Here w e will only pro v e the conv erse of the first implication in the case of transitive individual preferences. All the other statemen ts are prov ed b y the argumen ts giv en in [ 3 : Lemma 9.1 ]. Assume that m κ xy = 1 fo r all xy ∈ X × Y . Since m κ xy = v ∗ xy − v ∗ y x and b oth terms of this difference b elong to [0 , 1 ] , the only p ossibilit y is v ∗ xy = 1 (and v ∗ y x = 0 ). This implies the existence of a path x 0 x 1 . . . x n from x to y suc h that v x i x i +1 = 1 for all i . But this means tha t all of the v otes include eac h of t he pairs x i x i +1 of this path. So, if they are tra nsitive , all of them include also the pair xy , i. e. v xy = 1 . Theorem 7.2. Condition (75) im plies ( 78) . If the indivi dual pr efer enc es ar e tr ansitive, then the c onverse impl i c ation holds to o. Pr o of. On account of Lemmas 7.1 and 5.4, it suffic es to see that (74) im- plies (73). So, let us assume that v π xy = 1 for any x ∈ X a nd y ∈ Y . By Lemma 5 .1 .(e), an y a dmissible order ξ includes all pairs xy with x ∈ X and y ∈ Y . Let ℓ b e the last elemen t of X in a fixed admissible order. The n v π ℓℓ ′ = 1 and therefore t π ℓℓ ′ = 1 . On a ccoun t of the patt ern of g r owth of the pro jected turnouts, that is inequalities (63), this implies that t π x ¯ x = 1 for an y x, ¯ x ∈ X . Corollary 7.3. If the individual pr efer enc es ar e tr ans i tive, then R x = 1 if and only if v xy = 1 for every y 6 = x . Theorem 7.4. If the ind i v idual votes ar e r ank i n gs (p ossibly trunc ate d or with ties), then (75) implies (76 ) . 30 R. Camps, X. Mora, L. Sa umell Pr o of. As in ( 7 6) w e will con tin ue using a tilde to distinguish b et ween ho m- ologous ob j ects asso ciated respectiv ely with the whole matrix a nd with the submatrix asso ciat ed with X . First of all w e will show that t xy = 1 , for all xy ∈ X × A . (80) In fact, the r ules that w e are using for translating v otes in to binary pref- erences —namely , rules (a–d) o f § 1.3— entail the follo wing implications: (i) v xy = 1 fo r some y ∈ A implies that x is explicitly men tioned in all of the ranking v o t es; and (ii) x b eing explicitly men tioned in all of the r anking v otes implies that t xy = 1 for any y ∈ A. In particular, (80) ensures that the Llull matrix restricted to X is com- plete, from whic h it follo ws that e t π x ¯ x = 1 , for a ll x, ¯ x ∈ X . Concerning the non-restricted matrix, w e know, b y Lemma 7.1, that con- dition (75 ) implies v π xy = 1 and therefore t π xy = 1 for all xy ∈ X × Y . As in the pro of of Theorem 7.2, this implies that t π x ¯ x = 1 , for a ll x, ¯ x ∈ X . On the other hand, Lemma 9.2 of [ 3 ], whose pro of is v alid in the general case, ensures that e m σ x ¯ x = m σ x ¯ x for all x, ¯ x ∈ X . Altogether, w e get e v π x ¯ x = v π x ¯ x for a ll x, ¯ x ∈ X . Fina lly , (76 ) is a direct consequence o f this equality together with the a b ov e-remark ed fact that v π xy = 1 for all xy ∈ X × Y . 8 Other prop erties In this section w e collect sev eral other pr o p erties whose pro o f giv en in [ 3 ] remains v alid in the general case. The only cav eat to b ear in mind is that here they ultimately rely on Theorem 4.3 of the presen t article instead of [ 3 : Thm. 6.3 ]. One of t hese prop erties is here complemen ted b y an additional result that w as not men tioned in [ 3 ]. The first of these prop erties is the Condorcet-Smith principle M: Theorem 8.1 (Same pro of as in [ 3 : Thm. 10.1 ] ) . Both the indir e ct majority r elation κ and the r anking de term i n e d by the r a nk-like r a tes c om ply with the Condor c et-Smith pri n ciple: If A is p artitione d into two sets X and Y with the pr op erty that v xy > 1 / 2 for any x ∈ X an d y ∈ Y , then one has als o xy ∈ κ and R x < R y for any such x and y . Continuous ra ting f or incomplete prefere ntial v oting 31 The next results ar e concerned with clone consistency . In this connection w e mak e use of the no tion of auto nomous sets. A subset C ⊆ A is said to b e autonomous for a relation ρ when each elemen t fr om outside C relates to all elemen ts of C in the same w a y; in other w ords, when, for any x 6∈ C , havin g ax ∈ ρ fo r some a ∈ C implies bx ∈ ρ for an y b ∈ C , and similarly , ha ving xa ∈ ρ for some a ∈ C implies xb ∈ ρ for an y b ∈ C . More generally , a subset C ⊆ A will b e said to b e autonomous for a v alued relation ( v xy ) when the equalities v ax = v bx and v xa = v xb hold whe nev er a, b ∈ C and x 6∈ C . F or more details ab out the notion of auto nomous set and the prop erty of clone consistency we refer the reader to [ 3 : § 11 ]. Theorem 8.2 (Same pro of as in [ 3 : Thm. 11.5 and 11.7 ] ) . Assume that C ⊂ A is autonomous for the Llul l matrix ( v xy ) . Then C is autonomous for the indir e c t c omp arison r elation κ as wel l as for the r ankin g determine d by the r ank-like r ates, i. e. for the r elation ˆ ω = { xy ∈ Π | R x ≤ R y } . Besides, c ontr acting C to a single option in the Llul l matrix has n o other effe ct in κ and ˆ ω than getting the same c ontr ac tion. R emark . In con trast to [ 3 : Thm. 11.6 ] , in the incomple te case C is not en- sured to b e autonomous for the pro jected sc ores ( v π xy ) . I n fact, although the in termediate pro jected margins ( m σ xy ) do ha v e this prop ert y , the in ter- mediate pro j ected turnouts ( t σ xy ) can do a wa y with it. It w ould ce rtainly b e interes ting to find a n alternative to step 4 (quadra t ic minimization with constrain ts) so that the pro jected scores ( v π xy ) k eep the autonomo us sets of the original Llull matrix ( v xy ) (in addition to the presen t prop erties). T og ether with the fa cts stated in the preceding theorem, one could expect that the restriction of the final ranking to C should coincide with the “lo cal” result that one obtains when starting f rom the restriction of the original Llull matrix to C × C . A requiremen t of this sort (though concerning only the winner) is included for instance in the pr o p ert y of “comp osition consistency” considered in [ 8 ]. Our metho d do es not en tirely satisfy it. This is due to the fact that C b eing autonomous do es not prev en t the indirect score v ∗ ab for a, b ∈ C to come from o ut side C . Ho w ev er, the next r esult still ensures that the differences b etw een the lo cal ranking and the glo bal one are limited to some strict preferences of the former b eing replaced b y ties in the latter. Theorem 8.3. Assume that C ⊂ A is autonomous fo r the Llul l matrix ( v xy ) . L et e R x denote the r an k-like r ates that ar e obtaine d on C when starting fr om the r estriction o f ( v xy ) to xy ∈ C × C . The r an k i ng that these r ates d etermine on C is r el a te d to that determine d by the glob al r ates R x in the fol low i n g way: e R a ≤ e R b = ⇒ R a ≤ R b , whenever a, b ∈ C . (81) 32 R. Camps, X. Mora, L. Sa umell Pr o of. W e will systematically use a tilde to denote the ob jects that a re obtained when starting from the restriction of ( v xy ) to xy ∈ C × C . Let a, b b e tw o arbitrary elemen ts of C . The implication (81) will b e a consequence of the fo llo wing o ne: e v ∗ ab ≥ e v ∗ ba = ⇒ v ∗ ab ≥ v ∗ ba , whenev er a, b ∈ C . (82) In fact, this is sa ying that ˆ e κ ⊆ ˆ κ , whic h obv iously en tails ( ˆ e κ ) ∗ ⊆ ( ˆ κ ) ∗ , a nd therefore give s (81) by virtue of Theorem 5.2. In order to pro ve (82) w e b egin by noticing that the definition of the indirect scores ensu res t he inequalit y v ∗ ab ≥ e v ∗ ab , as well a s the analogo us one for ba instead of ab . This immediately settles (82) in the case v ∗ ba = e v ∗ ba . In t he case v ∗ ba > e v ∗ ba one can a rgue as follo ws. Hav ing this strict inequality means that the maxim um that defines v ∗ ba is achie v ed by a path β from b to a that contains some x ∈ A \ C . Let β 1 and β 2 denote the segmen ts of this path that go resp ectiv ely from b to x and from x to a . The desired result is then obtained in the follo wing w ay , where the second step makes use o f the fact that C is autonomous for the indirect scores [ 3 : Prop. 11.3 ] : v ∗ ab ≥ min( v ∗ ax , v ∗ xb ) = min( v ∗ bx , v ∗ xa ) ≥ min( v β 1 , v β 2 ) = v β = v ∗ ba . (83) Finally , w e hav e the following (rather w eak) prop erty of monotonicit y: Theorem 8.4 (Same pro o f as in [ 3 : T h m. 12 .1 and Cor. 12.2 ] ) . Assume that ( v xy ) and ( e v xy ) ar e r ela te d to e ach other in the fol lowin g way: e v ay ≥ v ay , e v xa ≤ v xa , e v xy = v xy , ∀ x, y 6 = a. ( 8 4) In this c ase, the fol lo wing p r op erties ar e satisfie d fo r any x, y 6 = a : e v ∗ ay ≥ v ∗ ay , e v ∗ xa ≤ v ∗ xa , (85) R a < R y = ⇒ e R a ≤ e R y , (86) ( R a < R y , ∀ y 6 = a ) = ⇒ ( e R a < e R y , ∀ y 6 = a ) . (8 7) 9 Appro v al v oting In appro v al v o ting, eac h v oter is ask ed for a list of appro v ed options, without an y expression o f preference b et wee n them, and eac h option x is then rated Continuous ra ting f or incomplete prefere ntial v oting 33 b y the num b er of appro v als for it [ 2 : § 1 and 2 ] . In the follo wing w e will refer to t his n um b er as the appro v al score of x , and its v alue relativ e to the total num b er of votes V will b e denoted b y α x . F rom the p oin t of view of paired comparisons, an individual v ote of ap- pro v al t yp e can b e view ed as a truncated ranking where all the options that app ear in it are tied. In this section, w e will see that t he margin- ba sed v arian t orders the options exactly in the same wa y as the a pprov al scores. In other words, the metho d of approv al v oting agrees with ours in a qualita- tiv e w ay under interpretation (d ′ ) o f § 1.3, i. e. under the in terpretation t ha t the non-appro v ed options of eac h individual vote are tied. This interpreta- tion is in congruence with the hypothesis of dichotomous preferences, under whic h approv al v oting has esp ecially go o d prop erties [ 16 ]. Ha ving said that, the preliminary r esult 9.1 will hold not only under inter- pretation (d ′ ) but also under interpretation ( d), i. e. that there is no informa- tion ab out t he preference of the v oter b et w een tw o non-approv ed options, and also under the analogous in terpretation that there is no informatio n ab out his preference b et w een t w o appro ved options. Inte rpretation (d ′ ) do es not pla y an essen tia l role un til Theorem 9.3, where we use the fact that it alwa ys brings the problem in to the complete case. In the f ollo wing w e use the following notation: µ ( v ) = { xy | v xy > v y x } , ˆ µ ( v ) = { xy | v xy ≥ v y x } , (88) µ ( α ) = { xy | α x > α y } , ˆ µ ( α ) = { xy | α x ≥ α y } . (89) Notice that µ ( v ∗ ) is the indirect comparison relation κ . Prop osition 9.1. In the ap p r oval voting situation, the f o l lowing e quality holds: v xy − v y x = α x − α y . (90) As a c ons e quenc e, µ ( v ) = µ ( α ) . Pr o of. Obv iously , the p ossible ballots ar e in one-to-o ne corr esp o ndence with the subsets X of A . In the following, v X denotes the relativ e n um b er of v otes that approv ed exactly the set X . With this notat ion it is ob vious that α x = X X ∋ x v X = X X ∋ x X 6∋ y v X + X X ∋ x X ∋ y v X , (91) for any y ∈ A . On the other hand, one has v xy = X X ∋ x X 6∋ y v X +  1 2 X X ∋ x X ∋ y v X + 1 2 X X 6∋ x X 6∋ y v X  , (92) 34 R. Camps, X. Mora, L. Sa umell where eac h of the terms in brack ets is presen t or not depending on whic h in terpretation is used . An ywa y , the preceding expressions, together with the analogous ones where x and y are in terc hanged with eac h other, result in the equalit y (90) indep endently of tho se alternative in terpretations. Lemma 9.2. Assume that ther e exists a total or der ξ such that the sc or es ( v xy ) satisfy µ ( v ) ⊆ ξ to gether with the c o ndition v xz ≥ v y z , v z x ≤ v z y , whenever x ≻ ξ y and z 6∈ { x, y } . (93) Then one has also µ ( v ∗ ) ⊆ ξ , i. e. ξ is an admis sible o r der. In the c om plete c ase one h a s the e quality µ ( v ∗ ) = µ ( v ) . Pr o of. Let us b egin by r ecalling that µ ( v ) ⊆ ξ is equiv alent to ξ ⊆ ˆ µ ( v ) , and similarly for µ ( v ∗ ) instead o f µ ( v ) . The first statemen t of the lemma will b e obtained by sho wing that under it s h yp otheses one has v ∗ xy = v xy ≥ v ∗ y x ≥ v y x , whenev er x ≻ ξ y . (94) On accoun t of t he definition of v ∗ xy and the fact that v xy ≤ v ∗ xy (and analo- gously for y x ), in order to prov e (94) it suffices to show that x ≻ ξ y implies v γ ≤ v xy , v η ≤ v xy , (95) for an y path γ = x 0 x 1 . . . x n from x 0 = x to x n = y , and for an y path η = y 0 y 1 . . . y n from y 0 = y to y n = x . Witho ut loss of generality , in the following w e will assume n > 1 and w e will let η b e the rev erse of γ , i. e. y i = x n − i . Let us assume tha t x ≻ ξ y . In order to pro ve (95 ) w e will distinguish three cases, namely: (i) x ≻ − ξ x i ≻ − ξ y for all i ; (ii) x i ≻ ξ x for some i ; (iii) y ≻ ξ x i for some i . Case (i) : It suffices to notice that the definition o f the score of a path and the assumptions of the lemma allo w to write v γ ≤ v xx 1 ≤ v xy and also v η ≤ v x 1 x ≤ v xx 1 ≤ v xy . F or future reference, let us notice also that in the complete case with v xy > 1 / 2 one can write v η ≤ v x 1 x ≤ 1 / 2 < v xy , so v η is then strictly less than v xy . Case (ii) : Let i b e the first time that one has x i ≻ ξ x , and let j b e the first time after i that one has x ≻ − ξ x j . Ob viously , 0 < i < j ≤ n . By construction, w e ha v e x ≻ − ξ x i − 1 and x i ≻ ξ x ≻ ξ y , whic h en tail that v γ ≤ v x i − 1 x i ≤ v xx i ≤ v xy . On the o ther hand, w e ha ve x j − 1 ≻ ξ x ≻ ξ y and x ≻ − ξ x j , whic h entail v η ≤ v x j x j − 1 ≤ v xx j − 1 ≤ v xy . Similarly as ab o v e, in the complete case with v xy > 1 / 2 w e get the strict inequality v η < v xy . Continuous ra ting f or incomplete prefere ntial v oting 35 Case (iii) is analog o us to case ( ii). The stateme nt about the complete case will be pro v ed if w e sho w the follo wing implications v xy > v y x = ⇒ v ∗ xy > v ∗ y x , (96) v xy = v y x = ⇒ v ∗ xy = v ∗ y x . (97) In order to obtain (96) it suffices to use the already remark ed fa ct that in the presen t circumstances the second inequ ality o f (95) is strict, whic h implies that one has also a strict inequality in the middle of (94 ). Finally , in order to obtain (97) it suffices to use (94) with v xy = v y x = 1 / 2 . Theorem 9.3. In the app r oval voting situation, the mar gin-b as e d varia nt r esults in a ful l qualitative c omp atibil i ty b etwe en the r ank-like r ates R x and the appr oval sc or es α x , in the sense that R x < R y ⇐ ⇒ α x > α y . Pr o of. Recall that the margin- based v ariant amounts to using in terpreta- tion (d ′ ), which alw ay s brings the problem in to the complete case (when the terms in brac k ets are included, equation (92) has indeed the prop erty that v xy + v y x = 1 ). Let ξ b e an y total or dering of the elemen ts of A b y non-increasing v alues of α x . In other w ords, ξ is an y total order contained in ˆ µ ( α ) , whic h is equiv alent to sa y , any total order con taining µ ( α ) . W e claim that we are under the h yp othesis of t he preceding lemma. This is a consequenc e of Prop osition 9 .1. In fa ct, on the one hand it immediately gives µ ( v ) ⊆ ξ . On the other hand, it allow s to deal with the margins m xy = v xy − v y x in the follo wing w a y: m xz = α x − α z = ( α x − α y ) + ( α y − α z ) = m xy + m y z ≥ m y z , m z x = α z − α x = ( α z − α y ) + ( α y − α x ) = m z y + m y x ≤ m z y , where w e hav e assumed x ≻ ξ y and w e hav e used that m xy = α x − α y ≥ 0 and m y x = − m xy ≤ 0 . The o btained inequalities a re certainly equiv a len t to those of (93 ). By virtue of Lemma 9.2, w e are therefore ensured that κ := µ ( v ∗ ) = µ ( v ) . Since w e know that µ ( v ) = µ ( α ) , w e can a lso write κ = µ ( α ) and ˆ κ = ˆ µ ( α ) , whic h guarantees that ˆ κ is transitiv e. 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