A continuous rating method for preferential voting. The complete case
AA CONTINUOUS RATING METHODFOR PREFERENTIAL VOTING. THE COMPLETE CASE
Rosa Camps, Xavier Mora and
Laia Saumell
Departament de Matem`atiques, Universitat Aut`onoma de Barcelona,Catalonia, Spain xmora @ mat.uab.cat
July 13, 2009; revised February 24, 2011
Abstract
A method is given for quantitatively rating the social acceptanceof different options which are the matter of a complete preferentialvote. Completeness means that every voter expresses a comparison(a preference or a tie) about each pair of options. The proposedmethod is proved to have certain desirable properties, which include:the continuity of the rates with respect to the data, a decomposi-tion property that characterizes certain situations opposite to a tie,the Condorcet-Smith principle, and clone consistency. One can viewthis rating method as a complement for the ranking method intro-duced in 1997 by Markus Schulze. It is also related to certain methodsof one-dimensional scaling or cluster analysis.
Keywords: preferential voting, quantitative rating, continuous rat-ing, majority principles, Condorcet-Smith principle, clone consistency,one-dimensional scaling, ultrametrics.
AMS subject classifications:
The outcome of a vote is commonly expected to specify not only a winnerand an ordering of the candidates, but also a quantitative estimate of thesocial acceptance of each of them. Such a quantification is expected evenwhen the individual votes give only qualitative information.The simplest voting methods are clearly based upon such a quantifi-cation. This is indeed the case of the plurality count as well as that ofthe Borda count. However, it is well known that these methods do notcomply with basic majority principles nor with other desirable conditions.In order to satisfy certain combinations of such principles and conditions,one must resort to other more elaborate methods, such as the celebrated ruleof Condorcet, Kem´eny and Young [ ; : p. 182–190 ], the method of ranked a r X i v : . [ m a t h . O C ] M a r R. Camps, X. Mora, L. Saumell pairs [ , ; : p. 219–223 ], or the method introduced in 1997 by MarkusSchulze [ , ; : p. 228–232 ], which we will refer to as the method of paths.Now, as they stand, these methods rank the candidates in a purely ordinalway, without properly quantifying the social acceptance of each of them.So, it is natural to ask for a method that combines the above-mentionedprinciples and conditions with a quantitative rating of the candidates.A quantitative rating should allow to sense the closeness between twocandidates, such as the winner and the runner-up. For that purpose, it isessential that the rates vary in a continuous way, especially through situationswhere ties or multiple orders occur.On the other hand, it should also allow to recognise certain situationsthat are opposite to a tie. For instance, a candidate should get the bestpossible rate if and only if it has been placed first by all voters. This is aparticular case of a more general condition that we will call decomposition .This condition, that will be made precise later on, places sharp constraintson the rates that should be obtained when the candidates are partitioned intwo classes X and Y such that each member of X is unanimously preferredto every member of Y .In this article we will produce a rating method that combines such aquantitative character with other desirable properties of a qualitative nature.Among them we will be especially interested in the following extension of theCondorcet principle introduced in 1973 by John H. Smith [ ]: Assume thatthe set of candidates is partitioned in two classes X and Y such that foreach member of X and every member of Y there are more than half of theindividual votes where the former is preferred to the latter; in that case, thesocial ranking should also prefer each member of X to any member of Y .This principle is quite pertinent when one is interested not only in choosinga winner but also in ranking all the alternatives (or in rating them).To our knowledge, the existing literature does not offer any other ratingmethod that combines this principle with the above-mentioned quantitativeproperties of continuity and decomposition. We will refer to our method asthe CLC rating method , where the capital letters stand for “ContinuousLlull Condorcet”. The reader interested to try it can use the
CLC calculator which has been made available at [ ].Of course, any rating automatically implies a ranking. In this connection,it should be noticed that the CLC rating method is built upon Schulze’smethod of paths as underlying ranking method. As we will remark in theconcluding section, we doubt that any of the other ranking methods men-tioned above could be extended to a rating method with the properties of ontinuous rating for complete preferential voting ]. In anotherseparate article we deal with another class of rates that have a fraction-likecharacter [ ].The present article is organized as follows: Section 1 gives a more pre-cise statement of the problem and finishes with a general remark. Section 2presents an heuristic outline of our proposal, ending with a summary of theprocedure and an illustrative example. Section 3 introduces some mathemat-ical language. Sections 4–10 give detailed mathematical proofs of the claimedproperties. Sections 11–12 are devoted to other interesting properties of theconcomitant social ranking, namely clone consistency and two weak forms ofmonotonicity. Finally, section 13 makes some concluding remarks and posesa few open questions. Let us consider a set of N options which are the matter of a vote. Letus assume that each voter expresses his preferences in a qualitative way, forinstance by listing those options in order of preference. Our aim is to combinesuch individual preferences so as to rate the social acceptance of each optionon a continuous scale. More specifically, we would like to do it in accordancewith the following conditions:A Scale invariance (homogeneity) . The rates depend only on the rela-tive frequency of each possible content of an individual vote. In otherwords, if every individual vote is replaced by a fixed number of copiesof it, the rates remain exactly the same.B
Permutation equivariance (neutrality) . Applying a certain permuta-tion of the options to all of the individual votes has no other effectthan getting the same permutation in the social rating.C
Continuity . The rates depend continuously on the relative frequencyof each possible content of an individual vote.
R. Camps, X. Mora, L. Saumell
The next two conditions choose a specific form of rating. From now onwe will refer to it as rank-like rating. In the complete case considered in thepresent paper, these two conditions take the following form:D
Rank-like form (complete case) . Each rank-like rate is a number,integer or fractional, between 1 and N . The best possible valueis 1 and the worst possible one is N . The average rank-like rateis ( N + 1) / Rank-like decomposition (complete case) . Consider a splitting of theoptions in two classes X and Y . Consider the case where each mem-ber of X is unanimously preferred to every member of Y . This factis equivalent to each of the following ones, where | X | denotes thenumber of elements of X : (a) The rank-like rates of X coincide withthose that one obtains when the individual votes are restricted to X .(b) After diminishing them by the number | X | , the rank-like ratesof Y coincide with those that one obtains when the individualvotes are restricted to Y . (c) The average rank-like rate of X is( | X | + 1) / N if and only if it is unanimously consideredworse than any other.Finally, we require a condition that concerns only the concomitant socialranking, that is, the ordinal information contained in the social rating:M Condorcet-Smith principle . Consider a splitting of the options intwo classes X and Y . Assume that for each member of X andevery member of Y there are more than half of the individual voteswhere the former is preferred to the latter. In that case, the socialranking also prefers each member of X to every member of Y . Let us emphasize that the individual votes that we are dealing with do not have a quantitative character (at least for the moment): each voter isallowed to express a preference for x rather than y , or vice versa, or maybea tie between them, but he is not allowed to quantify such a preference.This contrasts with “range voting” methods, where each individual vote isalready a quantitative rating [ : p. 174–176 ; ]. Such methods are free frommany of the difficulties that lurk behind the present setting. However, theymake sense only as long as all voters mean the same by each possible valueof the rating variable. This hypothesis may be reasonable in some cases, butin many others it is hardly valid. In fact, voting is often used in connection ontinuous rating for complete preferential voting This section presents our proposal as the result of a quest for the desiredproperties. Hopefully, this will communicate the main ideas that lie behindthe formulas.
The aim of complying with condition M calls for the point of view of paired comparisons . So our starting point will be the numbers V xy thatcount how many voters prefer x to y . In this connection we will take theview that each vote that ties x with y is equivalent to half a vote preferring x to y plus another half a vote preferring y to x . In order to achievescale invariance, we will immediately switch to the corresponding fractions v xy = V xy /V , where V denotes the total number of votes. The numbers v xy will be called the binary scores of the vote, and their collection will becalled the Llull matrix of the vote. Since we are considering the case ofcomplete votes, these numbers are assumed to satisfy v xy + v yx = 1 . (1)Besides the scores v xy , in the sequel we will often deal with the margins m xy , which are defined by m xy = v xy − v yx . (2)Obviously, their dependence on the pair xy is antisymmetric, that is m yx = − m xy . (3)It is clear also that the equality (1) allows to recover the scores from themargins by means of the following formula: v xy = (1 + m xy ) / . (4) R. Camps, X. Mora, L. Saumell
A natural candidate for defining the social preference is the following: x is socially preferred to y whenever v xy > v yx . Of course, it can happen that v xy = v yx , in which case one would consider that x is socially tied with y .The binary relation that includes all pairs xy for which v xy > v yx will bedenoted by µ ( v ) and will be called the comparison relation ; together withit, we will consider also the relation ˆ µ ( v ) defined by the non-strict inequality v xy ≥ v yx .As it is well known, the main problem with paired comparisons is thatthe comparison relations µ ( v ) and ˆ µ ( v ) may lack transitivity even if theindividual preferences are all of them transitive [
11, 20 ]. The next developments rely upon an operation ( v xy ) → ( v ∗ xy ) thattransforms the original system of binary scores into a new one. This operationis defined in the following way: for every pair xy , one considers all possiblepaths x x . . . x n going from x = x to x n = y ; every such path is associatedwith the score of its weakest link, i. e. the smallest value of v x i x i +1 ; finally, v ∗ xy is defined as the maximum value of this associated score over all pathsfrom x to y . In other words, v ∗ xy = max x = xx n = y min i ≥ i < n v x i x i +1 , (5)where the max operator considers all possible paths from x to y , and themin operator considers all the links of a particular path. The scores v ∗ xy willbe called the indirect scores associated with the (direct) scores v xy .If ( v xy ) is the table of 0’s and 1’s associated with a binary relation ρ (by putting v xy = 1 if and only if xy ∈ ρ ), then ( v ∗ xy ) is exactly the tableassociated with ρ ∗ , the transitive closure of ρ . So, the operation ( v xy ) (cid:55)→ ( v ∗ xy ) can be viewed as a quantitative analogue of the notion of transitiveclosure (see [ : Ch. 25 ]).The main point, remarked in 1998 by Markus Schulze [
15 b ], is that thecomparison relation associated with a table of indirect scores is always tran-sitive (Theorem 4.3). So, µ ( v ∗ ) is always transitive, no matter what thecase is for µ ( v ). This is true in spite of the fact that µ ( v ∗ ) can easily differfrom µ ∗ ( v ) (the transitive closure of µ ( v )). In the following we put κ = µ ( v ∗ ) , ˆ κ = ˆ µ ( v ∗ ) , m κxy = v ∗ xy − v ∗ yx . (6) ontinuous rating for complete preferential voting xy ∈ κ if and only if v ∗ xy > v ∗ yx , i. e. m κxy >
0, and xy ∈ ˆ κ if and only if v ∗ xy ≥ v ∗ yx , i. e. m κxy ≥
0. From now on we will refer to κ as the indirectcomparison relation , and to m κxy as the indirect margin associated withthe pair xy .As it has been stated above, the relation κ is transitive. Besides that,it is clearly asymmetric (one cannot have both v ∗ xy > v ∗ yx and vice versa).On the other hand, it may be incomplete (one can have v ∗ xy = v ∗ yx ). When itdiffers from κ , the complete relation ˆ κ is not asymmetric and —somewhatsurprisingly— it may be not transitive either. However, one can always finda total order ξ that satisfies κ ⊆ ξ ⊆ ˆ κ (Theorem 5.1). From now on,any total order ξ that satisfies this condition will be called an admissibleorder . Let us remark that such a definition is redundant: in fact, one easilysees that each of the required inclusions implies the other one.The rating that we are looking for will be based on such an order ξ .More specifically, it will be compatible with ξ in the sense that the rank-like rates R x will satisfy the inequality R x ≤ R y whenever xy ∈ ξ . If κ isalready a total order, so that ξ = κ , the preceding inequality will be satisfiedin the strict form R x < R y , and this will happen if and only if xy ∈ κ .The following steps assume that one has fixed an admissible order ξ .From now on the situation xy ∈ ξ will be expressed also by x (cid:31) ξ y . Accordingto the definitions, the inclusions κ ⊆ ξ ⊆ ˆ κ are equivalent to saying that v ∗ xy > v ∗ yx implies x (cid:31) ξ y and that the latter implies v ∗ xy ≥ v ∗ yx . In otherwords, if the different options are ordered according to (cid:31) ξ , the matrix v ∗ xy has then the property that each element above the diagonal is larger than orequal to its symmetric over the diagonal. Rating the different options means positioning them on a line. Besidescomplying with the qualitative restriction of being compatible with ξ in thesense above, we want that the distances between items reflect the quantitativeinformation provided by the binary scores. However, a rating is expressedby N numbers, whereas the binary scores are N ( N −
1) numbers. So we arebound to do some sort of projection. Problems of this kind have a certaintradition in combinatorial data analysis and cluster analysis [
10, 8 ]. In fact,some of the operations that will be used below can be viewed from that pointof view.Let us assume for a while that the votes are total orders, i. e. each vote listsall the options by order of preference, without any ties. This is the standardcase for the application of Borda’s method, which is linearly equivalent torating each option by the mean value of its ranks, i. e. the ordinal numbers
R. Camps, X. Mora, L. Saumell that give its position in these different orders. As it was noticed by Bordahimself (in his setting linearly related to ours), these mean ranks, which wewill denote by ¯ r x , can be obtained from the Llull matrix by means of thefollowing formula: ¯ r x = N − (cid:88) y (cid:54) = x v xy , (7)or equivalently, ¯ r x = ( N + 1 − (cid:88) y (cid:54) = x m xy ) / . (8)Let us look at the meaning of the margins m xy in connection with theidea of projecting the Llull matrix into a rating: If there are no other itemsthan x and y , we can certainly view the sign and magnitude of m xy as givingrespectively the qualitative and quantitative aspects of the relative positionsof x and y on the rating line, that is, the order and the distance betweenthem. When there are more than two items, however, we have several piecesof information of this kind, one for every pair, and these different piecesmay be incompatible with each other, quantitatively or even qualitatively,which motivates indeed the problem that we are dealing with. In particular,the mean ranks ¯ r x often violate the desired compatibility with the relation ξ .In order to construct a rating compatible with ξ , we will use a formulaanalogous to (7) where the scores v xy are replaced by certain projectedscores v πxy to be defined in the following paragraphs. Together with them,we will make use of the corresponding projected margins m πxy = v πxy − v πyx .Like the original scores (but not necessarily the indirect ones) the projectedscores will be required to satisfy the equality v πxy + v πyx = 1, from which itfollows that v πxy = (1 + m πxy ) /
2. So, the rates that we are looking for will beobtained in the following way: R x = N − (cid:88) y (cid:54) = x v πxy , (9)or equivalently, R x = ( N + 1 − (cid:88) y (cid:54) = x m πxy ) / . (10)Such formulas will be used not only in the case where the votes are totalorders, but also in more general situations. Our goals will be achieved by defining the projected margins in thefollowing way, where we assume x (cid:31) ξ y and x (cid:48) denotes the item that imme- ontinuous rating for complete preferential voting x in the total order ξ : m κxy = v ∗ xy − v ∗ yx , (11) m σxy = min { m κpq | p (cid:31)− ξ x, y (cid:31)− ξ q } , (12) m πxy = max { m σpp (cid:48) | x (cid:31)− ξ p (cid:31) ξ y } , (13) m πyx = − m πxy . (14)As one can easily check, this construction ensures that m πxz = max ( m πxy , m πyz ) , whenever x (cid:31) ξ y (cid:31) ξ z . (15)From this equality it follows that the absolute values d xy = | m πxy | satisfy thefollowing inequality, which makes no reference to the relation ξ : d xz ≤ max ( d xy , d yz ) , for any x, y, z . (16)This condition, called the ultrametric inequality, is well known in clusteranalysis, where it appears as a necessary and sufficient condition for thedissimilarities d xy to define a hierarchical classification of the set under con-sideration [ : § ; : § ]. Remark . The operation ( m κxy ) → ( m πxy ) defined by (12–13) is akin to thesingle-link method of cluster analysis, which can be viewed as a continuousmethod for projecting a matrix of dissimilarities onto the set of ultrametricdistances; such a continuous projection is achieved by taking the maximalultrametric distance which is bounded by the given matrix of dissimilarities[ : § ]. The operation ( m κxy ) → ( m πxy ) does the same kind ofjob under the constraint that the clusters be intervals of the total order ξ .
0. Form the Llull matrix ( v xy ) ( § v ∗ xy defined by (5). An efficient way todo it is the Floyd-Warshall algorithm [ : § ]. Work out the indirectmargins m κxy = v ∗ xy − v ∗ yx .2. Consider the indirect comparison relation κ = { xy | m κxy > } .Fix an admissible order ξ , i. e. a total order that extends κ . For in-stance, it suffices to arrange the options by non-decreasing values ofthe “tie-splitting” Copeland scores r x = N − |{ y | y (cid:54) = x, m κxy > }|− |{ y | y (cid:54) = x, m κxy = 0 }| (Proposition 5.2).0 R. Camps, X. Mora, L. Saumell
3. Starting from the indirect margins m κxy , work out the superdiagonalintermediate projected margins m σxx (cid:48) as defined in (12).4. Compute the projected margins m πxy according to (13–14). The pro-jected scores are then determined by the formula v πxy = (1 + m πxy ) / R x according to (9) (here equivalent to (10)).The computing time is of order N , where N is the number of options.The CLC calculator made available at [ ] allows to follow the details of theprocedure by choosing the option “Detailed mode”. As an illustrative example we will consider the final roundof a dancesport competition. Specifically, we have chosen the ProfessionalLatin Rising Star section of the 2007 Blackpool Dance Festival (Blackpool,England, 25th May 2007). The data were taken from . As usual, the final was con-tested by six couples, whose competition numbers were , , , , and . Eleven adjudicators ranked their simultaneous performances infour equivalent dances.The all-round official result was (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) . This re-sult comes from the so-called “Skating System”, whose name reflects a prioruse in figure-skating. The Skating System has a first part which producesa separate result for each dance. This is done mainly on the basis of themedian rank obtained by each couple (by the way, this criterion underliesthe “practical” method that Condorcet was proposing in 1792/93 [ : ch. 8 ]).However, the fine properties of this criterion are lost in the second part ofthe Skating System, where the all-round result is obtained by adding up thefinal ranks obtained in the different dances.From the point of view of paired comparisons, it makes sense to base theall-round result on the Llull matrix which collects the 44 rankings producedby the 11 adjudicators over the 4 dances. As one can see below, in thepresent case this matrix exhibits several Condorcet cycles, like for instance (cid:31) (cid:31) (cid:31) and (cid:31) (cid:31) (cid:31) , which means that the competitionwas closely contested. In consonance with it, the CLC rates obtained beloware quite close to each other, particularly for the couples , , and .By the way, the CLC result orders the contestants differently than the Bordarule, whose associated ordering is (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) .Instead of relative scores and margins, the following tables show theirabsolute counterparts, i. e. without dividing by the total number of votes.This has the virtue of staying with small integer numbers. ontinuous rating for complete preferential voting x Original scores V xy ∗
23 28 23 28 ∗
30 24
16 21 ∗
24 29 ∗
28 23
16 14 19 16 ∗ ∗ Indirect scores V ∗ xy ∗ ∗
30 24
21 21 ∗
24 24 29 ∗
28 24
19 19 19 19 ∗ ∗ Ranks r x x Indirect margins M κxy
122 4 264 3 31 238 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Projected margins M πxy
122 4 264 3 31 238 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Rates R x We consider a finite set A . Its elements represent the options which are thematter of a vote. The number of elements of A is N .In order to deal with preferences we must consider (ordered) pairs of ele-ments of A . The pair formed by a and b , in this order, will be denoted simplyas ab . The pairs that consist of two copies of the same element, i. e. thoseof the form aa , are not relevant for our purposes. So, we will systematicallyexclude them from our considerations. This will help towards a more efficientlanguage. The set of all proper pairs, i. e. the pairs ab with a (cid:54) = b , will bedenoted as Π . Unless we say otherwise, from now on any statement will beunderstood to imply the assumption that all the pairs that appear in it areproper pairs, i. e. they belong to Π .Besides pairs, we will be concerned also with longer sequences a a . . . a n ,which will be referred to as paths . As one could see in the heuristic outline, our developments make use ofthe mathematical concept of (binary) relation . Stating that two elements a and b are in a certain relation ρ is equivalent to saying that the pair ab is a member of a certain set ρ . Because of what has been said, unless we2 R. Camps, X. Mora, L. Saumell say otherwise we will restrict our attention to strict relations, i. e. relationscontained in Π . Under the convention made above, we can keep the followingstandard definitions. A relation ρ ⊆ Π will be called : asymmetric when ab ∈ ρ implies ba (cid:54)∈ ρ ; total , or complete , when ab (cid:54)∈ ρ implies ba ∈ ρ ; transitive when the simultaneous occurrence of ab ∈ ρ and bc ∈ ρ implies ac ∈ ρ ; a partial order when it is transitive and asymmetric; a totalorder when it is transitive, asymmetric and total; a total preorder whenit is transitive and total.Given a relation ρ , we will often consider the relation ˆ ρ that consists ofall pairs ab such that ba (cid:54)∈ ρ ; ˆ ρ is called the codual of ρ . The followinglemma collects several properties which are immediate consequences of thedefinitions: Lemma 3.1. (a) ˆˆ ρ = ρ . (b) ρ ⊂ σ ⇐⇒ ˆ σ ⊂ ˆ ρ . (c) ρ is asymmetric ⇐⇒ ρ ⊆ ˆ ρ ⇐⇒ ˆ ρ is total. (d) ρ is total ⇐⇒ ˆ ρ ⊆ ρ ⇐⇒ ˆ ρ is asymmetric. The transitive closure of ρ , which we will denote as ρ ∗ , is defined asfollows: ab ∈ ρ ∗ if and only if there exists a path a a . . . a n from a = a to a n = b such that a i a i +1 ∈ ρ for every i . ρ ∗ is the minimum transitive rela-tion that contains ρ . The transitive-closure operator is easily seen to havethe following properties: ρ ∗ ⊆ σ ∗ whenever ρ ⊆ σ ; ( ρ ∩ σ ) ∗ ⊆ ( ρ ∗ ) ∩ ( σ ∗ );( ρ ∗ ) ∪ ( σ ∗ ) ⊆ ( ρ ∪ σ ) ∗ ; ( ρ ∗ ) ∗ = ρ ∗ .A subset C ⊆ A is said to be autonomous for a relation ρ when,for any x (cid:54)∈ C , having ax ∈ ρ for some a ∈ C implies bx ∈ ρ for any b ∈ C , and similarly, having xa ∈ ρ for some a ∈ C implies xb ∈ ρ for any b ∈ C (see for instance [ ]). On the other hand, C ⊆ A will be said to bean interval for a relation ρ when the simultaneous occurrence of ax ∈ ρ and xb ∈ ρ with a, b ∈ C implies x ∈ C . The following facts are easyconsequences of the definitions: If ρ is asymmetric and C is autonomousfor ρ then C is an interval for ρ . If ρ is total and C is an interval for ρ then C is autonomous for ρ . As a corollary, if ρ is total and asymmetric, then C is autonomous for ρ if and only if it is an interval for that relation. Lateron we will make use of the following fact, which is also an easy consequenceof the definitions: An equivalent way to put it is that we accept a relation ρ ⊆ Π as total / transitive,whenever ρ ∪ { xx | x ∈ A } has the corresponding property in the standard sense. ontinuous rating for complete preferential voting Lemma 3.2. C is autonomous for ρ ⇐⇒ C is autonomous for ˆ ρ . When C is an autonomous set for ρ , it is natural to consider a new set (cid:101) A and a new relation (cid:101) ρ by proceeding in the following way: (cid:101) A is obtained from A by replacing the set C by a single element (cid:101) c , i. e. (cid:101) A = ( A \ C ) ∪ { (cid:101) c } ;for every x ∈ A , let us denote by (cid:101) x the element of (cid:101) A defined by (cid:101) x = (cid:101) c if x ∈ C and by (cid:101) x = x if x (cid:54)∈ C ; with this notation, (cid:101) ρ is defined byputting (cid:101) x (cid:101) y ∈ (cid:101) ρ if and only if xy ∈ ρ whenever (cid:101) x (cid:54) = (cid:101) y (this definition is notambiguous since C is autonomous for ρ ). We will refer to this operation asthe contraction of ρ by the autonomous set C .Any relation can be interpreted as expressing a system of qualitativepreferences: having xy ∈ ρ and yx (cid:54)∈ ρ means that x is preferred to y ;having both xy ∈ ρ and yx ∈ ρ means that x is tied with y ; having neither xy ∈ ρ nor yx ∈ ρ means that no information is given about the preferencebetween x and y .From this point of view, it is quite natural to rank the different x ∈ A by taking into account the number of y such that x is preferred to y as wellas the number of y such that x is tied with y . More precisely, it makes senseto define the rank of x in a relation ρ by the formula r x = N − (cid:12)(cid:12) { y | xy ∈ ρ, yx (cid:54)∈ ρ } (cid:12)(cid:12) − (cid:12)(cid:12) { y | xy ∈ ρ, yx ∈ ρ } (cid:12)(cid:12) . (17)Ranking by r x is often considered in connection with tournaments. Such amethod is known as the Copeland rule (see for instance [ : p. 206–209 ], wherethe tie-splitting term is not present —since ties are not occurring— and anequivalent formulation is used). The next lemma is an easy consequence ofthe definitions. Its second part justifies using the term ‘rank’. Lemma 3.3. If ρ is a partial order, then having xy ∈ ρ implies r x < r y .If ρ is a total order, then having xy ∈ ρ is equivalent to r x < r y ; in fact, r x coincides then with the ordinal number that gives the position of x in ρ . The Llull matrix of a vote, as well as the analogous matrices formedrespectively by the indirect scores and by the projected scores, are all of themparticular instances of the abstract notion of valued relation (also called‘fuzzy relation’). In fact, a valued relation on A means simply a mapping v whereby every pair xy ∈ Π is assigned a score v xy in the interval [0 , v xy measures “how much” is x related to y . Here and in thefollowing we keep the notation and terminology introduced in the precedingsections.4 R. Camps, X. Mora, L. Saumell
Most of the notions that are associated with relations can be generalizedto valued relations (sometimes in several different ways). Some of thesegeneralized notions have already appeared in the heuristic outline of § v ∗ xy generalizethe notion of transitive closure. Another generalized notion, namely that ofautonomous set, will appear in § v xy + v yx ≤ v xy + v yx ≥ In general terms, the problem of preference aggregation deals withvalued relations whose scores satisfy the condition v xy + v yx ≤
1. The set ofall such objects will be denoted by Ω. So, Ω = { v ∈ [0 , Π | v xy + v yx ≤ } .The complete case corresponds to the subset Γ determined by the equality v xy + v yx = 1. So, Γ = { v ∈ [0 , Π | v xy + v yx = 1 } .The sets Ω and Γ are respectively the fields of variation of the collectiveLlull matrix of a vote in the general case and in the complete one. On theother hand, the individual votes can also be viewed as belonging to these sets.In fact, any qualitative expression of preferences (not necesarily transitive)between the elements of A can be represented as a Llull matrix, i. e. anelement of Ω, by putting v xy = , if x is preferred to y, / , if x is tied with y, , if either y is preferred to x or no informationis given about the preference between x and y . (18)Such a mapping satisfies v xy + v yx = 1 whenever we are in the complete case,i. e. when either a preference or a tie is expressed about each pair of options.According to its definition, the collective Llull matrix is simply the center ofgravity of the distribution of individual votes: v xy = (cid:88) k α k v kxy , (19)where v k are the Llull matrices associated with the individual votes, and α k are the corresponding relative frequencies or weights.Since the individual votes play no other role than contributing to thecollective Llull matrix as described by the preceding equation, generallyspeaking there is no need to restrict them to express qualitative prefer-ences only, but one can allow them to express valued preferences, i. e. to be ontinuous rating for complete preferential voting ]. Let us recall that the indirect scores v ∗ xy are defined in the following way: v ∗ xy = max { v α | α is a path x x . . . x n from x = x to x n = y } , where the score v α of a path α = x x . . . x n is defined as v α = min { v x i x i +1 | ≤ i < n } . Remark . The matrix of indirect scores v ∗ can be viewed as a power of v (supplemented with v xx = 1 ) for a matrix product defined in the followingway: ( vw ) xz = max y min( v xy , w yz ). More precisely, v ∗ coincides with such apower for any exponent greater than or equal to N − Lemma 4.1.
The indirect scores satisfy the following inequalities: v ∗ xz ≥ min ( v ∗ xy , v ∗ yz ) for any x, y, z . (20) Proof.
Let α be a path from x to y such that v ∗ xy = v α ; let β be a pathfrom y to z such that v ∗ yz = v β . Consider now their concatenation αβ . Since αβ goes from x to z , one has v ∗ xz ≥ v αβ . On the other hand, the definitionof the score of a path ensures that v αβ = min ( v α , v β ). Putting these thingstogether gives the desired result. Lemma 4.2.
Assume that the original scores satisfy the following inequal-ities: v xz ≥ min ( v xy , v yz ) for any x, y, z . (21) In that case, the indirect scores coincide with the original ones.Proof.
The inequality v ∗ xz ≥ v xz is an immediate consequence of the defini-tion of v ∗ xz . The converse inequality can be obtained in the following way:6 R. Camps, X. Mora, L. Saumell
Let γ = x x x . . . x n be a path from x to z such that v ∗ xz = v γ . By virtueof (21), we havemin (cid:0) v x x , v x x , v x x , . . . , v x n − x n (cid:1) ≤ min (cid:0) v x x , v x x , . . . , v x n − x n (cid:1) . So, v ∗ xz ≤ v γ (cid:48) where γ (cid:48) = x x . . . x n . By iteration, one eventually gets v ∗ xz ≤ v xz . Remark . On the basis of the preceding results it makes sense to take condi-tion (21) as the definition of transitivity for a valued relation ( v xy ).The matrix of indirect scores v ∗ can be characterized as the lowest one thatlies above v and satisfies such a notion of transitivity; a proof of this fact—in a more general setting— will be found in [ : Theorem 3.3 ]. Theorem 4.3 (Schulze, 1998 [
15 b ]; see also [ : p. 228–229 ]) . The indirectcomparison relation κ = µ ( v ∗ ) is a partial order.Proof. Since µ ( v ∗ ) is clearly asymmetric, it is only a matter of showing itstransitivity. We will argue by contradiction. Let us assume that xy ∈ µ ( v ∗ ) and yz ∈ µ ( v ∗ ), but xz / ∈ µ ( v ∗ ). This means respectivelythat (a) v ∗ xy > v ∗ yx and (b) v ∗ yz > v ∗ zy , but (c) v ∗ zx ≥ v ∗ xz . On the otherhand, Lemma 4.1 ensures also that (d) v ∗ xz ≥ min ( v ∗ xy , v ∗ yz ). We will dis-tinguish two cases depending on which of the last two quantities is smaller:(i) v ∗ yz ≥ v ∗ xy ; (ii) v ∗ xy ≥ v ∗ yz .Case (i) : v ∗ yz ≥ v ∗ xy . We will see that in this case (c) and (d) entail acontradiction with (a). In fact, we have the following chain of inequalities: v ∗ yx ≥ min ( v ∗ yz , v ∗ zx ) ≥ min ( v ∗ yz , v ∗ xz ) ≥ min ( v ∗ yz , v ∗ xy ) = v ∗ xy , where we areusing successively: Lemma 4.1, (c), (d) and (i).Case (ii) : v ∗ xy ≥ v ∗ yz . An entirely analogous argument shows that in thiscase (c) and (d) entail a contradiction with (b). In fact, we have v ∗ zy ≥ min ( v ∗ zx , v ∗ xy ) ≥ min ( v ∗ xz , v ∗ xy ) ≥ min ( v ∗ yz , v ∗ xy ) = v ∗ yz , where we are usingsuccessively: Lemma 4.1, (c), (d) and (ii). Let us recall that an admissible order is a total order ξ such that κ ⊆ ξ ⊆ ˆ κ .Let us recall also that this definition is redundant since each of the twoinclusions implies the other one. So an admissible order is simply a totalorder that extends the partial order κ . The results that follow deal with theexistence and efficient finding of such extensions. ontinuous rating for complete preferential voting Theorem 5.1 (Szpilrajn, 1930 [ ]) . Given a partial order ρ on a finiteset A , one can always find a total order ξ such that ρ ⊆ ξ ⊆ ˆ ρ . If ρ contains neither xy nor yx , one can constrain ξ to include the pair xy . Proposition 5.2.
Let ρ be a partial order. Let r x denote the rank of x in ρ as defined by equation (17) of § A by non-decreasing values of r x is an extension of ρ .Proof. Let ξ be a total order of A for which x (cid:55)→ r x does not decrease.This means that xy ∈ ξ implies r x ≤ r y . Now, the contrapositive of the firststatement in Lemma 3.3 ensures that r x ≤ r y implies yx (cid:54)∈ ρ , i. e. xy ∈ ˆ ρ .So ξ ⊆ ˆ ρ , from which it follows that also ρ ⊆ ξ . Remark . The preceding proposition replaces the problem of finding a totalorder that contains ρ by the similar problem of finding a total order containedin the total preorder ˆ ω = { xy ∈ Π | r x ≤ r y } . However, from a practicalpoint of view the latter is a much easier thing to do, since one is guided bythe function x (cid:55)→ r x . Let us recall that our rating method is based upon certain projected scores v πxy .These quantities are obtained through the corresponding margins m πxy bymeans of the procedure (11–14). That procedure makes use of an admissibleorder ξ , whose existence has been dealt with in the preceding section, andit assumes xy ∈ ξ . Lemma 6.1.
The projected margins m πxy have the following properties: ≤ m πxy ≤ , whenever x (cid:31) ξ y. (22) (15) m πxz = max ( m πxy , m πyz ) , whenever x (cid:31) ξ y (cid:31) ξ z. (23) Proof.
Both properties are immediate consequences of (11–14) and the factthat 0 ≤ m κxy ≤ Theorem 6.2.
The projected scores do not depend on the admissible order ξ used for their calculation, i. e. the value of v πxy is independent of ξ forevery xy ∈ Π . On the other hand, the matrix of the projected scores in anadmissible order ξ is also independent of ξ ; i. e. if x i denotes the elementof rank i in ξ , the value of v πx i x j is independent of ξ for every pair ofindices i, j . R. Camps, X. Mora, L. Saumell
Remark . The two statements say different things since the identity of x i and x j may depend on the admissible order ξ . Proof.
Let us consider the effect of replacing ξ by another admissible order (cid:101) ξ .In the following, the tilde is systematically used to distinguish between hom-ologous objects which are associated respectively with ξ and (cid:101) ξ ; in particular,such a notation will be used in connection with the labels of the equationswhich are formulated in terms of the assumed admissible order.With this terminology, we will prove the two following equalities, whichamount to the two statements of the theorem. First, m πxy = (cid:101) m πxy , for any pair xy ( x (cid:54) = y ). (24)Secondly, we will see also that m πx i x j = (cid:101) m π ˜ x i ˜ x j , for any pair of indices ij ( i (cid:54) = j ) , (25)where x i denotes the element of rank i in ξ , and analogously for ˜ x i in (cid:101) ξ .Now, it is well known that the set of total order extensions of a givenpartial order is always connected through transpositions of consecutive elem-ents (see for instance [ : p. 30 ]). Therefore, it suffices to deal with the caseof two admissible orders ξ and (cid:101) ξ which differ from each other only by thetransposition of two consecutive elements. So, we will assume that thereare two elements a and b such that the only difference between ξ and (cid:101) ξ isthat ξ contains ab whereas (cid:101) ξ contains ba . According to the definition of anadmissible order, this implies that m κab = m κba = 0.In order to control the effect of the differences between ξ and (cid:101) ξ , we willmake use of the following notation: p will denote the immediate predecessorof a in ξ ; in this connection, any statement about p will be understoodto imply the assumption that the set of predecessors of a is not empty.Similarly, q will denote the immediate successor of b in ξ ; here too, anystatement about q will be understood to imply the assumption that the setof successors of b is not empty. So, ξ and (cid:101) ξ contain respectively the paths pabq and pbaq . Finally, x (cid:48) means here the immediate successor of x in ξ ,which is the same as in (cid:101) ξ if x (cid:54) = p, a, b .Let us look first at the superdiagonal intermediate projected margins m σhh (cid:48) .According to their definition, namely equation (12), m σhh (cid:48) is the minimumof a certain set of values of m κxy . In a table where x and y are orderedaccording to ξ , this set is an upper-right rectangle with lower-left vertex at hh (cid:48) . Using (cid:101) ξ instead of ξ amounts to interchanging two consecutive columnsand the corresponding rows of that table, namely those labeled by a and b . ontinuous rating for complete preferential voting h = a inthe order ξ , or h = b in the order (cid:101) ξ ; but then the minimum is still the samebecause the underlying set includes m κab = m κba = 0. So, m σx i x i +1 = (cid:101) m σ ˜ x i ˜ x i +1 , for any i = 1 , , . . . N −
1, (26) m σab = (cid:101) m σba = 0 . (27)On account of the definition of m πx i x j and (cid:101) m π ˜ x i ˜ x j , (26) results in (25).Finally, let us see that (24) holds too. To this effect, we begin by noticingthat (27) is saying that m πab = (cid:101) m πba = 0 (28)Let us consider now the equation m πpa = (cid:101) m πpb , which is contained in (25).On account of (23), these equalities entail m πpb = m πpa = (cid:101) m πpb = (cid:101) m πpa . (29)By means of an analogous argument, one obtains also that m πaq = m πbq = (cid:101) m πaq = (cid:101) m πbq . (30)On the other hand, (25) ensures also that m πxx (cid:48) = (cid:101) m πxx (cid:48) , whenever x (cid:54) = p, a, b . (31)Finally, (23) allows to go from (28–31) to the desired general equality (24). Theorem 6.3.
The projected scores and their asssociated margins satisfythe following properties with respect to any admissible order ξ : (a) The following inequalities hold whenever x (cid:31) ξ y and z (cid:54)∈ { x, y } : v πxy ≥ v πyx , m πxy ≥ , (32) v πxz ≥ v πyz , v πzx ≤ v πzy , (33) m πxz ≥ m πyz , m πzx ≤ m πzy , (34)(b) If v πxy = v πyx , or equivalently m πxy = 0 , then (33) and (34) are satisfiedall of them with an equality sign. R. Camps, X. Mora, L. Saumell
Proof.
Part (a). Let us begin by noticing that (32) reduces to (22). Noticealso that (33) follows from (34), and that (34.1) and (34.2) are equivalent toeach other. So, it suffices to prove either (34.1) or (34.2). We will distinguishthree cases, namely: (i) x (cid:31) ξ y (cid:31) ξ z ; (ii) z (cid:31) ξ x (cid:31) ξ y ; (iii) x (cid:31) ξ z (cid:31) ξ y .In case (i), (34.1) follows from (23). In case (ii), (34.2) follows from (23).Finally, in case (iii) it suffices to use (22) to see that m πxz ≥ ≥ m πyz .Part (b). Similarly to part (a), it suffices to prove the statement cor-responding to (34.1), i. e. that m πxy = 0 implies m πxz = m πyz . This followsimmediately from (23) in cases (i) and (ii). In case (iii), (23) allows to derivethat m πxz = m πzy = 0, and therefore also the equality m πxz = m πyz . Proposition 6.4.
Assume that there exists a total order ξ such that theoriginal scores and the associated margins satisfy the following conditions: v xy ≥ v yx , i.e. m xy ≥ , whenever x (cid:31) ξ y, (35) m xz = max ( m xy , m yz ) , whenever x (cid:31) ξ y (cid:31) ξ z. (36) In that case, the projected scores coincide with the original ones.Proof.
We will begin by showing that m xz ≥ min( m xy , m yz ) , for any x, y, z . (37)In order to prove this inequality we will distinguish six cases depending onthe relative position of x, y, z according to ξ : (a) If x (cid:31) ξ y (cid:31) ξ z , then (37)is an immediate consequence of (36). (b) If z (cid:31) ξ y (cid:31) ξ x , then (36) (with x and z interchanged with each other) gives m zx = max ( m zy , m yx ), whichowing to the antisymmetric character of the margins is equivalent to (37)with an equality sign. (c) If x (cid:31) ξ z (cid:31) ξ y , then condition (35) guaranteesthat m xz ≥ ≥ m yz = min( m xy , m yz ). (d) If z (cid:31) ξ x (cid:31) ξ y , then we have m xz ≥ m yz = min( m xy , m yz ), where the inequality holds because (36) ensuresthat m zy ≥ m zx , and the equality derives from the hypothesis upon ξ .(e,f) The two remaining cases, namely y (cid:31) ξ x (cid:31) ξ z and y (cid:31) ξ z (cid:31) ξ x , areanalogous respectively to (c) and (d).Now, since we are in the complete case, the scores v xy and the margins m xy are related to each other by the monotone increasing transformation v xy = (1+ m xy ) / . Therefore, the inequality (37) on the margins is equivalentto the following one on the scores: v xz ≥ min( v xy , v yz ) , for any x, y, z . (38) ontinuous rating for complete preferential voting v ∗ xy = v xy and therefore m κxy = m xy . In particular, ξ is ensured to be an admissible order.Let us now consider any pair xy contained in ξ . By making use of (36)we see that m σxy = m κxy = m xy . As a consequence, the equality m πxy =max { m σpp (cid:48) | x (cid:31)− ξ p (cid:31) ξ y } becomes m πxy = max { m pp (cid:48) | x (cid:31)− ξ p (cid:31) ξ y } . Fromhere, a second application of (36) allows to derive that m πxy = m xy , andtherefore v πxy = v xy .Since conditions (35–36) of Proposition 6.4 are included among the prop-erties of the projected Llull matrix according to Lemma 6.1, one can concludethat they fully characterize the projected Llull matrices, and that the oper-ator ( v xy ) (cid:55)→ ( v πxy ) really deserves being called a projection: Theorem 6.5.
The operator P : Γ (cid:51) ( v xy ) (cid:55)→ ( v πxy ) ∈ Γ is idempotent,i. e. P = P . Its image P Γ consists of the complete Llull matrices ( v xy ) that satisfy (35 – for some total order ξ . Let us recall that the rank-like rates R x are given by the formula (9), orequivalently by (10). From these formulas one easily checks that they satisfycondition D. Lemma 7.1. (a) If x (cid:31) ξ y in an admissible order ξ , then R x ≤ R y . (b) R x = R y if and only if v πxy = v πyx . (c) R x ≤ R y implies the inequalities (32 – . (d) R x < R y if and only if v πxy > v πyx . (e) v πxy > v πyx implies x (cid:31) ξ y in any admissible order ξ .Proof. Part (a). It is an immediate consequence of formula (9) togetherwith the inequalities (32) and (33.1) ensured by Theorem 6.3.Part (b). From (9) it follows that R y − R x = ( v πxy − v πyx ) + (cid:88) z (cid:54) = xz (cid:54) = y ( v πxz − v πyz ) . (39)Let ξ be an admissible order. By symmetry we can assume xy ∈ ξ . As aconsequence, Theorem 6.3 ensures that the terms of (39) which appear in2 R. Camps, X. Mora, L. Saumell parentheses are all of them greater than or equal to zero. So the only pos-sibility for their sum to vanish is that each of them vanishes separately,i. e. v πxy = v πyx and v πxz = v πyz for any z (cid:54)∈ { x, y } . Finally, part (b) of The-orem 6.3 ensures that all of these equalities hold as soon as the first one issatisfied.Part (c). When the hypothesis is satisfied as a strict inequality, the resultfollows by combining the contrapositive of (a) with part (a) of Theorem 6.3.In the case of equality, it suffices to combine (b) with part (b) of that theorem.Part (d). It follows from (c) and its contrapositive on account of (b).Part (e). It follows from (d) and the contrapositive of (a).The next theorem characterizes the preference relation determined by therank-like rates in terms of the indirect comparison relation κ defined in § Theorem 7.2 ( ) . The rank-like rating given by (9) is related to the indirectcomparison relation κ = µ ( v ∗ ) in the following way: R x < R y ⇐⇒ yx (cid:54)∈ (ˆ κ ) ∗ , (40) R x ≤ R y ⇐⇒ xy ∈ (ˆ κ ) ∗ . (41) Proof.
The statements (40) and (41) are equivalent to each other (via thecontrapositive of each implication plus a swap between x and y ). So itsuffices to prove (40). On the other hand, to establish the latter it sufficesto prove the two following statements: xy ∈ (ˆ κ ) ∗ = ⇒ R x ≤ R y , (42) yx (cid:54)∈ (ˆ κ ) ∗ = ⇒ R x < R y . (43)Proof of (42). By transitivity, it suffices to consider the case xy ∈ ˆ κ .Now, from Theorem 5.1 one easily sees that xy ∈ ˆ κ implies that xy belongsto some admissible order ξ . The conclusion that R x ≤ R y is then ensuredby Lemma 7.1.(a).Proof of (43). Since (ˆ κ ) ∗ is complete, yx (cid:54)∈ (ˆ κ ) ∗ implies xy ∈ (ˆ κ ) ∗ andtherefore, according to (42), R x ≤ R y . So, (43) will follow if we show that R x = R y = ⇒ yx ∈ (ˆ κ ) ∗ whenever x (cid:54) = y . (44)Let ξ be an admissible order. Since ξ ⊆ ˆ κ , the right-hand side of (44) isautomatically true if yx ∈ ξ ; so, it remains to consider the case where xy ∈ ξ . We thank an anonymous reviewer for certain remarks that led to the present versionof this theorem, which is stronger than the original one. ontinuous rating for complete preferential voting y = x (cid:48) . According to Lemma 7.1.(b), theequality R x = R x (cid:48) implies that m πxx (cid:48) = 0, that is m σxx (cid:48) = 0, which means thatthere exist a, b such that a (cid:31)− ξ x (cid:31) ξ b and m κab = 0. Now, the latter impliesthat ba ∈ ˆ κ , which can be combined with the fact that x (cid:48) b, ax ∈ ξ ⊆ ˆ κ to derive that x (cid:48) x ∈ (ˆ κ ) ∗ (with the obvious modifications if x = a or x (cid:48) = b ).Finally, if we only know that xy ∈ ξ , we can use Lemma 7.1.(a) to see thatthe equality R x = R y implies R p = R p (cid:48) for any p such that x (cid:31)− ξ p (cid:31) ξ y ,which reduces the problem to the preceding case. Corollary 7.3. (a) R x < R y ⇒ xy ∈ κ . (b) If ˆ κ is transitive (which is ensured whenever κ is total),then R x < R y ⇔ xy ∈ κ . (c) If κ contains a set of the form X × Y with X ∪ Y = A ,then R x < R y for any x ∈ X and y ∈ Y .Proof. Part (a). This is an immediate consequence of (40) since yx (cid:54)∈ (ˆ κ ) ∗ ⇒ yx (cid:54)∈ ˆ κ ⇔ xy ∈ κ .Part (b). It is just a matter of noticing that under the hypothesis thatˆ κ is transitive the right-hand side of (40) reduces to xy ∈ κ .Part (c). Let x ∈ X and y ∈ Y . Since X × Y ⊂ κ ⊆ ˆ κ , part (a) ensuresthat R x ≤ R y . So, it suffices to exclude the possibility that R x = R y .By using (40) one easily sees that this equality would imply yx ∈ (ˆ κ ) ∗ .In other words, there would be a path from y ∈ Y to x ∈ X entirelycontained in ˆ κ . Such a path would have to include a pair ab ∈ ˆ κ with a ∈ Y and b ∈ X , which is not possible since ab ∈ ˆ κ means ba (cid:54)∈ κ . Proposition 7.4.
Assume that the votes are total orders. Assume also thatthe Llull matrix satisfies the hypothesis of Proposition 6.4. In that case, therank-like rates R x coincide exactly with the mean ranks ¯ r x .Proof. Recall that the rank-like rates are related to the projected scoresin the same way as the mean ranks are related to the original scores whenthe votes are total orders ( § R. Camps, X. Mora, L. Saumell
We claim that the rank-like rates R x are continuous functions of the binaryscores v xy . The main difficulty in proving this statement lies in the admissibleorder ξ , which plays a central role in the computations. Since ξ varies in adiscrete set, its dependence on the data cannot be continuous at all. Evenso, we claim that the final result is still a continuous function of the data.In this connection, one can consider as data the normalized Llull ma-trix ( v xy ), its domain of variation being the set Γ introduced in § α k mentioned also in § Theorem 8.1.
The projected scores v πxy and the rank-like rates R x dependcontinuously on the Llull matrix ( v xy ) .Proof. The dependence of the rank-like rates on the projected scores isgiven by formula (9), which is not only continuous but even linear (non-homogeneous). So we are left with the problem of showing that the projection P : ( v xy ) (cid:55)→ ( v πxy ) is continuous. As it has been mentioned above, this is notso clear, since the projected margins are the result of certain operationswhich are based upon an admissible order ξ which is determined separately.However, we will see, on the one hand, that P is continuous as long as ξ remains unchanged, and on the other hand, that the results of § P is continuous on the whole of Γ in spite of the factthat ξ can change. In the following we will use the following notation:for every total order ξ , we denote by Γ ξ the subset of Γ which consistsof the Llull matrices for which ξ is an admissible order, and we denote by P ξ the restriction of P to Γ ξ .We claim that the mapping P ξ is continuous for every total order ξ .In order to check the truth of this statement, one has to go over the differentmappings whose composition defines P ξ (see § v xy ) (cid:55)→ ( m κxy ),( m κxy ) (cid:55)→ ( m σxy ), and finally ( m σxx (cid:48) ) (cid:55)→ ( m πxy ) (cid:55)→ ( v πxy ). All of these mappingsare certainly continuous since they involve only additions and substractionsas well as the max and min operations.Finally, the continuity of P (and the fact that it is well-defined) is a con-sequence of the following facts (see for instance [ : § ]): (a) Γ = (cid:83) ξ Γ ξ ;this is true because of the existence of ξ (Theorem 5.1). (b) Γ ξ is a closedsubset of Γ; this is true because Γ ξ is described by a set of non-strict inequal-ities which concern quantities that are continuous functions of ( v xy ) (namelythe inequalities m κxy ≥ xy ∈ ξ ). (c) ξ varies over a finite set.(d) P ξ coincides with P η at Γ ξ ∩ Γ η , as it is proved in Theorem 6.2. ontinuous rating for complete preferential voting Corollary 8.2.
The rank-like rates depend continuously on the relative fre-quency of each possible content of an individual vote.Proof.
It suffices to recall that the Llull matrix ( v xy ) is simply the center ofgravity of the distribution specified by these relative frequencies (formula (19)of § Property E is concerned with having a partition of A in two sets X and Y such that each member of X is unanimously preferred to any member of Y ,that is: v xy = 1 (and therefore v yx = 0) whenever xy ∈ X × Y . (45)According to property E, to be proved in the present section, in the com-plete case considered in this article such a situation is characterized by thefollowing equalities: R x = (cid:101) R x , for all x ∈ X, (46) R y = (cid:101) R y + | X | , for all y ∈ Y , (47) (cid:88) x ∈ X R x = | X | ( | X | + 1) / , (48)where (cid:101) R x and (cid:101) R y denote the rank-like rates which are determined respec-tively from the submatrices associated with X and Y . More specifically,each of preceding equalities is separately equivalent to (45).In the following we will continue using a tilde to distinguish between hom-ologous objects associated respectively with the whole matrix and with itssubmatrices associated with X and Y . Lemma 9.1.
Given a partition A = X ∪ Y in two disjoint nonempty sets,one has the following equivalences: v xy = 1 ∀ xy ∈ X × Y (cid:27) ⇐⇒ (cid:26) m κxy = 1 ∀ xy ∈ X × Y (cid:27) ⇐⇒ (cid:26) v πxy = 1 ∀ xy ∈ X × Y (49) Proof.
Assume that v xy = 1 for all xy ∈ X × Y . Then v yx = 0, for allsuch pairs, which implies that v γ vanishes for any path γ which goes from Y to X . This fact, together with the inequality v ∗ xy ≥ v xy , entails the following6 R. Camps, X. Mora, L. Saumell equalities for all x ∈ X and y ∈ Y : v ∗ yx = 0, v ∗ xy = 1, and consequently m κxy = 1.Assume now that m κxy = 1 for all xy ∈ X × Y . Let ξ be an admissibleorder. As an immediate consequence of the definition, it includes the set X × Y . Let (cid:96) be the last element of X according to ξ . From the presenthypothesis it is clear that m σ(cid:96)(cid:96) (cid:48) = 1, which entails that v πxy = 1 for every xy ∈ X × Y .Assume now that v πxy = 1 for all xy ∈ X × Y . Let ξ be an admissibleorder. Here too, we are ensured that it includes the set X × Y ; this is soby virtue of Theorem 6.3.(a). Let (cid:96) be the last element of X accordingto ξ . From the fact that m σ(cid:96)(cid:96) (cid:48) = m π(cid:96)(cid:96) (cid:48) = 1, one infers that m κxy = 1 for all xy ∈ X × Y .Finally, let us assume again that m κxy = 1 for all xy ∈ X × Y . Since m κxy = v ∗ xy − v ∗ yx and both terms of this difference belong to [0 , v ∗ xy = 1 and v ∗ yx = 0, which implies that v yx = 0. This equalityis equivalent to v xy = 1. Lemma 9.2.
Condition (45) implies, for any admissible order, the follow-ing equalities: m σxx (cid:48) = (cid:101) m σxx (cid:48) , whenever x, x (cid:48) ∈ X , (50) m σyy (cid:48) = (cid:101) m σyy (cid:48) , whenever y, y (cid:48) ∈ Y , (51) Proof.
As we saw in the proof of Lemma 9.1, condition (45) implies thevanishing of v γ for any path γ which goes from Y to X . Besides theconclusions obtained in that lemma, this implies also the following equalities: v ∗ x ¯ x = (cid:101) v ∗ x ¯ x , m κx ¯ x = (cid:101) m κx ¯ x , for all x, ¯ x ∈ X , (52) v ∗ y ¯ y = (cid:101) v ∗ y ¯ y , m κy ¯ y = (cid:101) m κy ¯ y , for all y, ¯ y ∈ Y . (53)Let us fix an admissible order ξ . The second equality of (49) not only ensuresthat ξ includes the set X × Y , but it can also be combined with (52) and(53) to obtain respectively (50) and (51). Theorem 9.3.
Conditions (45) , (46) , (47) and (48) are equivalent to eachother.Proof. Part (a): (45) = ⇒ (46), (47) and (48). As a consequence of (50)and (51) we get the following equalities: v πx ¯ x = (cid:101) v πx ¯ x , for all x, ¯ x ∈ X , (54) v πy ¯ y = (cid:101) v πy ¯ y , for all y, ¯ y ∈ Y . (55) ontinuous rating for complete preferential voting v πxy = 1 , for all xy ∈ X × Y . (56)When the projected scores are introduced in (9) these equalities result in(46) and (47). Finally, (48) is an immediate consequence of (46).Part (b): (46) ⇒ (45); (47) ⇒ (45). On account of formula (9), con-ditions (46) and (47) are easily seen to be respectively equivalent to thefollowing equalities: (cid:88) y ∈ Ay (cid:54) = x v πxy = (cid:88) ¯ x ∈ X ¯ x (cid:54) = x (cid:101) v πx ¯ x + | Y | , for all x ∈ X, (57) (cid:88) x ∈ Ax (cid:54) = y v πyx = (cid:88) ¯ y ∈ Y ¯ y (cid:54) = y (cid:101) v πy ¯ y for all y ∈ Y . (58)Let us add up respectively the equalities (57) over x ∈ X and the equalities(58) over y ∈ Y . Since v πpq + v πqp = (cid:101) v πpq + (cid:101) v πqp = 1, we obtain (cid:88) x ∈ Xy ∈ Y v πxy = | X | | Y | , (59) (cid:88) y ∈ Yx ∈ X v πyx = 0 . (60)Since the projected scores belong to [0 , v πxy = 1 , for all xy ∈ X × Y , (61) v πyx = 0 , for all xy ∈ X × Y , (62)which are equivalent to each other since v πxy + v πyx = 1. Finally, Lemma 9.1allows to arrive at (45).Part (c): (48) ⇒ (45). From the definition of R x and the fact that v πxy ≤ v πxy + v πyx = 1 , one easily derives the inequality (cid:80) x ∈ X R x ≥ | X | ( | X | + 1) / Corollary 9.4. (a) R x = 1 if and only if v xy = 1 for all y (cid:54) = x . (b) R x = N if and only if v xy = 0 for all y (cid:54) = x .Proof. It suffices to apply Theorem 9.3 to the special cases X = { x } and X = A \ { x } . The result can also be obtained directly from Lemma 9.1.8 R. Camps, X. Mora, L. Saumell
10 The Condorcet-Smith principleTheorem 10.1.
Both the indirect majority relation κ = µ ( v ∗ ) and the pref-erence relation determined by the rank-like rates comply with the Condorcet-Smith principle: If A is partitioned in two sets X and Y with the propertythat v xy > / for any x ∈ X and y ∈ Y , then one has also xy ∈ κ and R x < R y for any such x and y .Proof. Assume that x ∈ X and y ∈ Y . Since v ∗ xy ≥ v xy , the hypothesisof the theorem entails that v ∗ xy > /
2. On the other hand, let γ be a pathfrom y to x such that v ∗ yx = v γ ; since it goes from Y to X , this path mustcontain at least one link y i y i +1 with y i ∈ Y and y i +1 ∈ X ; now, for thislink we have v y i y i +1 ≤ − v y i +1 y i < /
2, which entails that v ∗ yx = v γ < / v ∗ yx < / < v ∗ xy , i. e. xy ∈ κ . Finally, the fact that thisholds for any x ∈ X and y ∈ Y implies, by Corollary 7.3.(c), that one hasalso R x < R y for any such x and y .
11 Clone consistency
Clone consistency (also known as independence of clones) refers to the effectof adding or deleting similar options. For many voting methods, this maychange the outcome in a substantial way. For instance, replacing a singleoption c by a set C of several options similar to c may change the resultfrom c being the winner to giving the victory to some option outside C .This does not seem right: if c deserves being chosen when going alone, thenin the second situation the right choice should be some member of C .The notion of similarity that is relevant here can be formalized by theconcept of autonomous set that was introduced in § C beingautonomous for a given binary relation means that each element from out-side C relates to all elements of C in the same way. In the context of votingtheory, autonomous sets are often called sets of clones. So, it makes senseto ask for the following property, which we call clone consistency: If a setof options is autonomous for each of the individual votes, then: (a) this setis also autonomous for the social ranking; and (b) contracting it to a singleoption in all of the individual votes has no other effect in the social rankingthan getting the same contraction.This requirement was introduced in 1986–87 by Thomas M. Zavist andT. Nicolaus Tideman, who also devised a method that satisfies it, namelythe rule of ranked pairs [
19, 21 ]. ontinuous rating for complete preferential voting κ as well as the preference relation determined by the rank-like rates. The core results were obtained by Markus Schulze [
15 c , ]. In order to prepare the ground, we need to deal first with certaingeneralities. To begin with, the notion of an autonomous set will be extendedto apply not only to a relation, as defined in § v xy ): A subset C ⊆ A will be said to be autonomous for ( v xy )when v ax = v bx , v xa = v xb , whenever a, b ∈ C and x (cid:54)∈ C . (63)This definition can be viewed as an extension of that given in § Lemma 11.1.
Given a binary relation ρ , let u xy and v xy be the binaryscores defined respectively by u xy = (cid:40) , if xy ∈ ρ, , if xy / ∈ ρ ; v xy = , if xy ∈ ρ and yx / ∈ ρ, / , if xy ∈ ρ and yx ∈ ρ, , if xy / ∈ ρ. (64) One has the following equivalences: (a) C is autonomous for ρ if and only if C is autonomous for ( u xy ) . (b) C is autonomous for ρ if and only if C is autonomous for ( v xy ) . Lemma 11.2.
Assume that C ⊂ A is autonomous for ( v xy ) . Assume alsothat either x or y , or both, lie outside C . In this case v ∗ xy = max { v γ | γ contains no more than one element of C } Proof.
It suffices to see that any path γ = x . . . x n from x = x to x n = y which contains more than one element of C can be replaced by anotherone (cid:101) γ which contains only one such element and satisfies v (cid:101) γ ≥ v γ . Con-sider first the case where x, y (cid:54)∈ C . In this case it will suffice to take (cid:101) γ = x . . . x j − x k . . . x n , where j = min { i | x i ∈ C } and k = max { i | x i ∈ C } ,which obviously satisfy 0 < j < k < n . Since x j − (cid:54)∈ C and x j , x k ∈ C ,we have v x j − x j = v x j − x k , so that v γ = min (cid:0) v x x , . . . , v x n − x n (cid:1) ≤ min (cid:0) v x x , . . . , v x j − x j , v x k x k +1 , . . . , v x n − x n (cid:1) = min (cid:0) v x x , . . . , v x j − x k , v x k x k +1 , . . . , v x n − x n (cid:1) = v (cid:101) γ . R. Camps, X. Mora, L. Saumell
The case where x (cid:54)∈ C but y ∈ C can be dealt with in a similar wayby taking (cid:101) γ = x . . . x j − x n , and analogously, in the case where x ∈ C and y (cid:54)∈ C it suffices to take (cid:101) γ = x x k +1 . . . x n . Proposition 11.3. If C ⊂ A is autonomous for the scores ( v xy ) , then C isautonomous also for the indirect scores ( v ∗ xy ) .Proof. Consider a, b ∈ C and x (cid:54)∈ C . Let γ = x x x . . . x n be a path from a to x such that v ∗ ax = v γ . By Lemma 11.2, we can assume that a is the onlyelement of γ that belongs to C . In particular, x (cid:54)∈ C , so that v ax = v bx ,which allows to write v ∗ ax = v γ = min (cid:0) v ax , v x x , . . . , v x n − x (cid:1) = min (cid:0) v bx , v x x , . . . , v x n − x (cid:1) ≤ v ∗ bx . By interchanging a and b , one gets the reverse inequality v ∗ bx ≤ v ∗ ax and there-fore the equality v ∗ ax = v ∗ bx . An analogous argument shows that v ∗ xa = v ∗ xb . Corollary 11.4. If C ⊂ A is autonomous for a relation ρ , then C isautonomous also for the transitive closure ρ ∗ .Proof. Because of Proposition 11.3 and Lemma 11.1.(a).
The next results assume that C ⊂ A is autonomous for the Llullmatrix of a vote. Obviously, this assumption is satisfied whenever C is au-tonomous for all of the individual votes (which can be allowed to be arbitraryelements of Γ as mentioned in § Theorem 11.5.
Assume that C ⊂ A is autonomous for the Llull matrix ( v xy ) . Then C is autonomous for the indirect comparison relation κ = µ ( v ∗ ) as well as for the total preorder determined by the rank-like rates ( i. e. forthe relation ˆ ω = { xy ∈ Π | R x ≤ R y } ) .Proof. Proposition 11.3 ensures that C is autonomous for the indirect scores( v ∗ xy ), from which one easily derives that C is autonomous for the relation κ .Now, according to Theorem 7.2, ˆ ω = (ˆ κ ) ∗ . So the statement about ˆ ω followsby virtue of Lemma 3.2 and Corollary 11.4. Theorem 11.6.
Assume that C ⊂ A is autonomous for the Llull matrix ( v xy ) . Then C is autonomous also for the projected scores ( v πxy ) . ontinuous rating for complete preferential voting Proof.
Since v πxy = (1 + m πxy ) /
2, it suffices to show that C is autonomousfor the projected margins ( m πxy ). By Theorem 11.3, we know that C is au-tonomous for the indirect scores ( v ∗ xy ), which immediately implies its beingautonomous also for the indirect margins ( m κxy ). So the problem lies at show-ing that the autonomy of C is maintained when going from ( m κxy ) to ( m πxy )via the procedure (11–14). In the following we let ξ be an admissible orderand we distinguish two cases depending on whether C is or not an intervalfor ξ .Assume first that C is an interval of ξ . Since margins are antisymmetric,in order to prove that C is autonomous for ( m πxy ) it suffices to show that m πxa = m πxb , for any a, b ∈ C and x (cid:31) ξ C , (65) m πay = m πby , for any a, b ∈ C and C (cid:31) ξ y . (66)In the following we prove (65), the proof of (66) being entirely analogous.If there are no x (cid:31) ξ C there is nothing to prove. Otherwise, let k be theimmediate predecessor of the first element of C . By using (13), one easilysees that (65) will follow if we show that m σcc (cid:48) ≤ m σkk (cid:48) , for any c ∈ C . (67)Now, this inequality holds because m σcc (cid:48) = min { m κpq | p (cid:31)− ξ c, c (cid:48) (cid:31)− ξ q } ≤ min { m κpq | p (cid:31)− ξ k, c (cid:48) (cid:31)− ξ q } = min { m κpq | p (cid:31)− ξ k, k (cid:48) (cid:31)− ξ q } = m σkk (cid:48) , where the inequality is due to the fact that we pass to a smaller set, and theequality that starts the second line holds because k (cid:48) (cid:31)− ξ q (cid:31)− ξ c implies q ∈ C ,whereas p (cid:31)− ξ k implies p (cid:54)∈ C , so that m κpq = m κpk (cid:48) for such p and q .Assume now that C is not an interval of ξ . That is, there exist a, b ∈ C and x (cid:54)∈ C such that a (cid:31) ξ x (cid:31) ξ b . This implies that ax, xb ∈ ˆ κ . Since weknow that C is autonomous for κ , it follows that cx, xc ∈ ˆ κ for all c ∈ C ,that is, m κcx = m κxc = 0 for all c ∈ C . From this fact one easily derives,using (12), that m σpp (cid:48) = 0 for all p ∈ A such that p, p (cid:48) ∈ ¯ C , where ¯ C meansthe minimum interval of ξ that contains C . Finally, this entails, using (13),that m πxy = 0 for all x, y ∈ ¯ C , and that any subset of ¯ C , in particular theset C , is autonomous for ( m πxy ). Finally, we consider the effect of contracting C to a single element.In this connection we will make use of the notation and definitions of § R. Camps, X. Mora, L. Saumell if ( v xy ) admits C as an autonomous set, the contracted binary scores ( (cid:101) v (cid:101) x (cid:101) y )are characterized by the equality (cid:101) v (cid:101) x (cid:101) y = v xy whenever (cid:101) x (cid:54) = (cid:101) y . In the follow-ing, a tilde is systematically used to distinguish between homologous objectsassociated respectively with ( A, v ) and ( (cid:101) A, (cid:101) v ). Theorem 11.7.
Assume that C ⊂ A is autonomous for the Llull matrix ( v xy ) . Then the relation (cid:101) κ coincides with the contraction of κ by the au-tonomous set C . Similarly, the relation ˆ (cid:101) ω = { xy ∈ (cid:101) Π | (cid:101) R x ≤ (cid:101) R y } coincideswith the contraction of ˆ ω = { xy ∈ Π | R x ≤ R y } by C .Proof. We begin by noticing that the operation of taking indirect scorescommutes with that of contraction by the autonomous set C , i. e. (cid:101) v ∗ (cid:101) x (cid:101) y = v ∗ xy whenever (cid:101) x (cid:54) = (cid:101) y . This is a consequence of Lemma 11.2. As a consequence, (cid:101) κ = µ ( (cid:101) v ∗ ) coincides with the contraction of κ = µ ( v ∗ ) by C , and simi-larly for ˆ (cid:101) κ and ˆ κ . On the other hand, a second application of Lemma 11.2—to the binary scores associated with the relation ˆ κ by formula (64.1)—ensures that (ˆ (cid:101) κ ) ∗ is the contraction of (ˆ κ ) ∗ . Finally, in order to see that ˆ (cid:101) ω is the contraction of ˆ ω it is just a matter of applying Theorem 7.2: For any x, y ∈ A such that (cid:101) x (cid:54) = (cid:101) y , we have (cid:101) R (cid:101) x ≤ (cid:101) R (cid:101) y ⇔ (cid:101) x (cid:101) y ∈ (ˆ (cid:101) κ ) ∗ ⇔ xy ∈ (ˆ κ ) ∗ ⇔ R x ≤ R y .
12 About monotonicity
In this section we consider the effect of raising a particular option a to amore preferred status in the individual ballots without any change in thepreferences about the other options . More generally, we consider the casewhere the scores v xy are modified into new values (cid:101) v xy such that (cid:101) v ay ≥ v ay , (cid:101) v xa ≤ v xa , (cid:101) v xy = v xy , ∀ x, y (cid:54) = a. (68)In such a situation, one would expect the social rates to behave in the fol-lowing way, where y is an arbitrary element of A \ { a } : (cid:101) R a ≤ R a , (69) R a < R y = ⇒ (cid:101) R a < (cid:101) R y , R a ≤ R y = ⇒ (cid:101) R a ≤ (cid:101) R y , (70)where the tilde indicates the objects associated with the modified scores.Unfortunately, the rating method proposed in this paper does not satisfythese conditions, but generally speaking it satisfies only the following weakerones: R a < R y = ⇒ (cid:101) R a ≤ (cid:101) R y . (71)( R a < R y , ∀ y (cid:54) = a ) = ⇒ ( (cid:101) R a < (cid:101) R y , ∀ y (cid:54) = a ) . (72) ontinuous rating for complete preferential voting a was the only winner for the scores v xy , then it is still the only winner for the scores (cid:101) v xy .An example exhibiting the lack of property (70) can be found in [ : num-ber 10 of “Example inputs” ].In this connection, it is interesting to remark that the method of rankedpairs enjoys also property (72) [ : p. 221–222 ], but it fails at (71). A profilewhich exhibits this failure of (71) for the method of ranked pairs is given in[ : number 9 of “Example inputs” ].The remainder of this section is devoted to proving properties (71) and (72). Theorem 12.1.
Assume that ( v xy ) and ( (cid:101) v xy ) are related to each other inaccordance with (68) . In this case, the following properties are satisfied forany x, y (cid:54) = a : (cid:101) v ∗ ay ≥ v ∗ ay , (cid:101) v ∗ xa ≤ v ∗ xa , (73) (71) R a < R y = ⇒ (cid:101) R a ≤ (cid:101) R y . (74) Proof.
Let κ = µ ( v ∗ ) and (cid:101) κ = µ ( (cid:101) v ∗ ). Observe that (73) implies (cid:101) m κay ≥ m κay , (cid:101) m κya ≤ m κya , ∀ y (cid:54) = a. (75)We begin by seeing that (74) will be a consequence of (73). In fact, wehave the following chain of implications: R a < R y ⇒ ay ∈ κ ⇔ m κay > ⇒ (cid:101) m κay > ⇔ ay ∈ (cid:101) κ ⇒ (cid:101) R a ≤ (cid:101) R y , where the central implication isprovided by (75) and the other two strict implications are guaranteed byCorollary 7.3.(a). So, the problem has been reduced to proving (73).Now, in order to obtain the indirect score for a pair of the form ay itis useless to consider paths involving xa for some x (cid:54) = a , since such pathscontain cycles whose omission results in paths having a larger or equal score.So, the maximum which defines v ∗ ay is realized by a path which does notinvolve any pair xa . For such a path γ we have v ∗ ay = v γ ≤ (cid:101) v γ ≤ (cid:101) v ∗ ay , where the first inequality follows directly by (68). An analogous argumentgives (cid:101) v ∗ xa ≤ v ∗ xa . Corollary 12.2 ( ) . Under the hypothesis of Theorem 12.1 one has also theproperty (72) . We thank Markus Schulze for pointing out this fact. R. Camps, X. Mora, L. Saumell
Proof.
According to Corollary 7.3.(a), the left-hand side of (72) impliesthe strict inequality v ∗ ay > v ∗ ya for all y (cid:54) = a . Now, this inequality canbe combined with (73) to derive that (cid:101) v ∗ ay > (cid:101) v ∗ ya for all y (cid:54) = a . Finally,Corollary 7.3.(c) with X = { a } and Y = A \ { a } guarantees that the right-hand side of (72) is satisfied.
13 Concluding remarks and open questions
To our knowledge, the existing literature does not offer any other ratingmethod that combines the quantitative properties of continuity and decom-position with the Condorcet-Smith principle. As it has been pointed out inthe introduction, the latter is quite pertinent when one is interested not onlyin choosing a winner but in ranking all the alternatives (or in rating them).In particular, the Borda rule (linearly equivalent to rating the options bytheir mean ranks) satisfies those two quantitative properties, but it doesnot comply with the Condorcet principle. Quite interestingly, the maximinrates, namely σ x = min y (cid:54) = x v xy , combine continuity and a certain form ofdecomposition with the standard Condorcet principle, that is, condition Mrestricted to the case where the subset X reduces to a single option. How-ever, they easily fail at this condition when the subset X contains severaloptions; a counterexample is given for instance in [ : p. 212–213 ].One may ask whether the CLC rating method is the only one that sat-isfies properties A–E and M. The answer to this question is surely negative:Although we are imposing sharp constraints on the rates to be obtained atcertain special points of the Llull matrix space, in between these points thereis still some degree of freedom (in particular, because rates vary in a con-tinuum). This remaining freedom might allow for imposing some additionalproperty. In this connection, we find especially interesting the following Open question 1.
Can one give a rating method that satisfies A – E and M together with quantitative monotonicity in the sense that new Llull scoressatisfying (68) implies new rates satisfying (69–70) ? It is also natural to ask whether an alternative rating method with thesame properties could be given where the underlying ranking method werenot Schulze’s method of paths, but some other one. In this connection, weconsider especially interesting, because of their properties, the celebratedrule of Condorcet, Kem´eny and Young [ ; : p. 182–190 ], as well as theranked-pairs one [ , ; : p. 219–223 ]: We thank Salvador Barber`a for having called our attention to the maximin rating. ontinuous rating for complete preferential voting Open question 2.
Is the rule of Condorcet, Kem´eny and Young compatiblewith a continuous rating method satisfying A – E ? Open question 3.
Is the ranked-pairs rule compatible with a continuousrating method satisfying A – E ? On the basis of a previous exploratory work, we believe that the answer toquestions 2 and 3 is in both cases negative. More deeply into the question,one can ask:
Open question 4 ( ) . Which properties characterize those ranking methodsthat can be extended into rating methods satisfying properties A – E ? References [ ] Michel Balinski, Rida Laraki, 2007–2011 [ a ] A theory of measuring, electing, and ranking.
Proceedings of the NationalAcademy of Sciences of the United States of America , 104 : 8720–8725(2007). [ b ] Majority Judgment · Measuring, Electing and Ranking . MIT Press (2011).[ ] Hermann Buer, Rolf H. M¨ohring, 1983. A fast algorithm for the decomposi-tion of graphs and posets. Mathematics of Operations Research , 8 : 170–184.[ ] Rosa Camps, Xavier Mora, Laia Saumell, 2009. A continuous rating methodfor preferential voting. The incomplete case. (Submitted for publication).[ ] Rosa Camps, Xavier Mora, Laia Saumell, 2009. Fraction-like rates for pref-erential voting. (Submitted for publication).[ ] Rosa Camps, Xavier Mora, Laia Saumell, 2010. A general method for de-ciding about logically constrained issues. (Submitted for publication).[ ] Thomas H. Cormen, Charles L. Leiserson, Ronald L. Rivest, Clifford Stein,1990 , 2001 . Introduction to Algorithms . MIT Press.[ ] Jacqueline Feldman-H¨ogaasen, 1969. Ordres partiels et permuto`edre. Math´ematiques et Sciences Humaines , 28 : 27–38.[ ] Lawrence J. Hubert, Phipps Arabie, Jacqueline Meulman, 2001. Combina-torial Data Analysis: Optimization by Dynamic Programming . Society forIndustrial and Applied Mathematics.[ ] Eric Jacquet-Lagr`eze, 1971. Analyse d’opinions valu´ees et graphes depr´ef´erence. Math´ematiques et Sciences Humaines , 33 : 33–55.[ ] Nicholas Jardine, Robin Sibson, 1971. Mathematical Taxonomy . Wiley. We thank an anonymous reviewer for having raised this interesting question. R. Camps, X. Mora, L. Saumell [ ] Iain McLean, Arnold B. Urken (eds.), 1995. Classics of Social Choice . TheUniversity of Michigan Press, Ann Arbor.[ ] Bernard Monjardet, 1990. Sur diverses formes de la “r`egle de Condorcet”d’agr´egation des pr´ef´erences. Math´ematiques et Sciences Humaines , 111 :61–71.[ ] Xavier Mora, 2008–2010. CLC calculator . http://mat.uab.cat/~xmora/CLC calculator/ .[ ] James R. Munkres, 1975. Topology . Prentice-Hall.[ ] Markus Schulze, 1997–2003. [ a ] Posted in the
Election Methods Mailing List . http://lists.electorama.com/pipermail/election-methods-electorama.com/1997-October/001570.html and (see also ibidem /1998-January/001577.html ). [ b ] Ibidem /1998-August/002045.html . [ c ] A new monotonic and clone-independent single-winner election method.