On the reversal bias of the Minimax social choice correspondence
aa r X i v : . [ m a t h . C O ] D ec On the reversal bias of theMinimax social choice correspondence ∗ Daniela Bubboloni
Dipartimento di Scienze per l’Economia e l’ImpresaUniversit`a degli Studi di Firenzevia delle Pandette 9, 50127, Firenze, Italye-mail: daniela.bubboloni@unifi.ittel: +39 055 2759667
Michele Gori
Dipartimento di Scienze per l’Economia e l’ImpresaUniversit`a degli Studi di Firenzevia delle Pandette 9, 50127, Firenze, Italye-mail: michele.gori@unifi.ittel: +39 055 2759707
October 24, 2018
Abstract
We introduce three different qualifications of the reversal bias in the framework of socialchoice correspondences. For each of them, we prove that the Minimax social choice correspon-dence is immune to it if and only if the number of voters and the number of alternatives satisfysuitable arithmetical conditions. We prove those facts thanks to a new characterization of theMinimax social choice correspondence and using a graph theory approach. We discuss the sameissue for the Borda and Copeland social choice correspondences.
Keywords: reversal bias; Minimax social choice correspondence; directed graphs.
JEL classification:
D71.
MSC classification:
Consider a committee having h ≥ n ≥ h preferences, each of them associated with one of the individuals in the committee.Thus, a procedure to choose, whatever individual preferences are, one or more alternatives as socialoutcome can be represented by a social choice correspondence ( scc ), that is, a function from theset of preference profiles to the set of nonempty subsets of the set of alternatives. ∗ Daniela Bubboloni was partially supported by GNSAGA of INdAM. scc s and their comparison is usually based on which properties,among the ones considered desirable or undesirable under a social choice viewpoint, those scc sfulfil. Moving from the ideas originally proposed by Saari (1994) and then deepened by Saari andBarney (2003), we focus here on a quite unpleasant property that a scc may meet and that, in ouropinion, hasn’t deserved the right attention yet.In order to describe such a property, recall that the reversal of a preference profile is the prefer-ence profile obtained by it assuming a complete change in each committee member’s mind about herown ranking of alternatives (that is, the best alternative gets the worst, the second best alternativegets the second worst, and so on). Assume now that a given scc associates with a certain pref-erence profile a singleton, that is, it selects a unique alternative. If we next consider the outcomedetermined by the reversal of the considered preference profile, we would expect to have somethingdifferent from the previous singleton as it seems natural to demand a certain degree of differencebetween the outcomes associated with a preference profile and its reversal. As suggested by Saariand Barney (2003, p.17), suppose after the winner of an important departmental election was announced, it was discov-ered that everyone misunderstood the chair’s instructions. When ranking the three candidates,everyone listed his top, middle, and bottom-ranked candidate in the natural order first, second,and third. For reasons only the chair understood, he expected the voters to vote in the oppositeway. As such, when tallying the ballots, he treated a first and last listed candidate, respectively,as the voter’s last and first choice. Imagine the outcry if after retallying the ballots the chairreported that [...] the same person won.
In other words, common sense suggests that we should express doubts about the quality of a scc which associates the same singleton both with a preference profile and with its reversal, that is,which suffers what we are going to call the reversal bias.Among the classical scc s, such a bias is experienced by the Minimax scc , that is the scc which selects those alternatives whose greatest pairwise defeat is minimum. Indeed, assume thata committee having six members ( h = 6) has to select some alternatives within a set of fouralternatives denoted by 1, 2, 3 and 4 ( n = 4). Consider then a preference profile represented by thematrix where, for every i ∈ { , , , , , } , the i -th column represents the i -th member’s preferencesaccording to the rule that the higher the alternative is, the better it is. A simple check shows thatthe Minimax scc associates both with that preference profile and with its reversal the same set { } . On the other hand, if we consider two alternatives only, then the Minimax scc agrees withthe simple majority and it is immediate to verify that it is immune to the reversal bias whateverthe number of committee members is.For such a reason, we address the problem of finding conditions on the number of individualsand on the number of alternatives that make the Minimax scc immune to the reversal bias. Ourmain result is the following theorem. Theorem A.
The Minimax scc is immune to the reversal bias if and only if h ≤ or n ≤ or ( h, n ) ∈ { (4 , , (5 , , (7 , , (5 , } . Theorem A shows, in particular, that the Minimax scc does no exhibit the reversal bias notonly when there are two alternatives but also in other cases. Remarkably, that property holds true Also known as Simpson-Kramer or Condorcet scc . Theorem A is a rephrase of Theorem 2 for j = 1. . Indeed, standard social choice theoretical arguments naturally allow toprove that, for lots of pairs ( h, n ), the Minimax scc suffers the reversal bias . On the other hand,except for the trivial case n = 2, they turn out to be difficult to apply to prove that, for theremaining pairs, the Minimax scc is immune to the reversal bias. In particular, no simple intuitionindicates how to treat the cases ( h, n ) ∈ { (4 , , (5 , , (7 , , (5 , } . For such a reason, we firstpropose a new characterization of the Minimax scc showing that, for every preference profile, analternative x is selected by the Minimax scc if and only if, for every majority threshold µ notexceeding the number of individuals but exceeding half of it, if there is an alternative which ispreferred by at least µ individuals to x , then, for every alternative, there is another one which ispreferred by at least µ individuals to it (Proposition 1). We then associate with each preferenceprofile p and each majority threshold µ a directed graph Γ µ ( p ), called majority graph, whose verticesare the alternatives and whose arcs are the µ -majority relations among alternatives (Section 5.2).By the analysis of connection and acyclicity properties of those graphs, we find out a generaland unified method to approach the proof of Theorem A. That allows, in particular, to avoid therepetition of similar arguments and the discussion of very long lists of cases and subcases. Thegeometric representation of the graph Γ µ ( p ) is also a useful mental guidance in the tricky stepsneeded to carry on such an analysis as well as the proof of Theorem A. We emphasise that theresults related to graph theory deal with quite general majority issues so that they are not limited,in their meaning, to the specific problem considered in the paper. We are confident that thoseresults could be a smart tool to manage, in the future, many other problems related not only tothe Minimax scc .We also introduce two weaker versions of reversal bias. Namely, we say that a scc suffers thereversal bias of type 2 if there exists a preference profile such that the outcomes associated with itand its reversal are not disjoint and one of the two is a singleton; we say instead that a scc suffersthe reversal bias of type 3 if there exists a preference profile such that the outcomes associated withit and its reversal are not disjoint and none of the two is the whole set of the alternatives. It isimmediate to observe that the reversal bias (also called reversal bias of type 1) implies the reversalbias of type 2 which in turn implies the reversal bias of type 3. Using the same tools and techniquesused to prove Theorem A, we get the following results . Theorem B.
The Minimax scc is immune to the reversal bias of type 2 if and only if h = 2 or n ≤ or ( h, n ) = (4 , . Theorem C.
The Minimax scc is immune to the reversal bias of type 3 if and only if n = 2 or ( h, n ) = (3 , . We emphasize that there is an interesting link between the different qualifications of reversalbias above described and the concept of Condorcet loser. Indeed, let C be a scc satisfying theCondorcet principle, that is, always selecting the Condorcet winner as unique outcome when itexists. If C is immune to the reversal bias of type 1, then it never selects the Condorcet loseras unique outcome, that is, C fulfils the weak Condorcet loser property; if C is immune to thereversal bias of type 2, then it never selects the Condorcet loser, that is, C fulfils the Condorcetloser property. Thus, since the Minimax scc satisfies the Condorcet principle, Theorems A andB provide, in particular, conditions on ( h, n ) that are sufficient to make the Minimax scc satisfythe weak Condorcet loser property and the Condorcet loser property, respectively. Certainly, as it Note that the use of graphs in social choice theory is well established (see, for instance, Laslier (1997)). Propositions 23 and 24, which determine a large set of pairs ( h, n ) for which the Minimax scc suffers the reversalbias, are based on an intuitive argument which could be carried on without the machinery of graph theory. Theorems B and C are rephrases of Theorem 2 for j = 2 and j = 3, respectively.
3s not known whether such conditions are also necessary, determining all the pairs ( h, n ) makingthe Minimax scc satisfy those properties is an interesting problem which, in our opinion, can befruitfully attacked using the methods described in this paper. Finally note that, given a scc C always selecting the Condorcet winner (not necessarily as unique outcome) when it exists, we havethat if C is immune to the reversal bias of type 2, then it fulfils the weak Condorcet loser property;if C is immune to the reversal bias of type 3, then it never selects the Condorcet loser when the setof outcomes is different from the whole set of alternatives.Observe now that, even though the main concepts of our paper are mainly inspired to theideas of Saari and Barney (2003), the framework we consider, as well as the terminology we use,is different from the one used by those authors. Indeed, they deal with election methods, namely,functions from the set of finite sequences of individual preferences (still called preference profiles) tothe set of complete and transitive relations on the set of alternatives. In that framework, they saythat an election method suffers the reversal bias if it associates the same relation with a preferenceprofile and its reversal, provided that such a relation is not a complete tie, so that in their paper theexpression reversal bias is used with a different meaning. For every k ≤ n −
1, they also introduce theconcept of k -winner reversal bias (called top-winner bias when k = 1), that is that phenomenon thatoccurs when an election method associates with a preference profile and its reversal two relationshaving the property to have the same k top ranked alternatives . Anyway, despite the differences,it is obvious that any result of theirs about the top-winner reversal bias of a certain election methodimplies some information about the reversal bias of type 1 for the scc generated by that methodrestricting its domain to those sequences of individual preferences having h terms and looking onlyat those alternatives that are top ranked. On the other hand, it is clear that none of their theoremsimplies a result about the reversal biases of type 2 and 3 as an immediate by-product. In particular,from Theorem 8 in Saari and Barney (2003), we deduce that the Borda and Copeland scc s areimmune to the reversal bias of type 1, but nothing can be deduced about the other types of reversalbias. That makes interesting the following result . Theorem D.
The Borda and Copeland scc s are immune to the reversal bias of type 3.
We conclude with an observation. Recall that a positional method is an election method whereeach time an alternative is ranked k -th by one individual it obtains w k points and alternatives arethen ranked according to the final score they get; the vector w = ( w k ) nk =1 ∈ R n associated withthe method is called its voting vector and is assumed to satisfy the relations w ≥ w ≥ . . . ≥ w n and w > w n . Assume now that n ≥ w such that there exist k , k ∈ { , . . . , n } with w k + w n − k = w k + w n − k . Then, from Theorem 1 in Saari and Barney(2003), we deduce that the scc generated by the positional method associated with w suffers thereversal bias of type 1, provided that the number h of individuals is large enough. That fact isremarkable because it implies the existence of many scc s different from the Minimax scc , likeplurality and anti-plurality scc s, which suffer that bias. Certainly, as the considered theorem givesno information about the exact values of h for which the reversal bias of type 1 really occurs, findingthose values of h is an interesting issue that deserves to be carefully investigated. More generally,given any classical scc C and any social choice property, one can consider the problem to determineconditions on the number of individuals and alternatives which are necessary and sufficient to make C fulfil the property. We believe that investigating those problems is an interesting and promisingresearch project since, as particularly shown by our results on the reversal bias, comparing different Saari (1994) introduces for election methods another interesting concept, called reversal symmetry, which isrelated to the ones now discussed. Namely, an election method is said to be reversal symmetric if the outcomesassociated with any preference profile and its reversal are one the reversal of the other. Of course, if an electionmethod is reversal symmetric it cannot suffer either the reversal bias or the k -winner reversal bias. Reversal symmetryhas been recently studied by Llamazares and Pe˜na (2015) for positional methods and by Bubboloni and Gori (2015)for social welfare functions with values in the set of linear orders. Theorem D is a rephrase of Proposition 3. cc s on the basis of their properties cannot ignore how many individuals and alternatives areinvolved in the decision process. Let N ⋄ = { a ∈ N : a ≥ } . From now on, let n, h ∈ N ⋄ be fixed, and let N = { , . . . , n } be the setof alternatives and H = { , . . . , h } be the set of individuals.A preference relation on N is a linear order on N , that is, a complete, transitive and antisym-metric binary relation on N . The set of linear orders on N is denoted by L ( N ). Let q ∈ L ( N )be fixed. Given x, y ∈ N , we usually write x ≥ q y instead of ( x, y ) ∈ q , and x > q y insteadof ( x, y ) ∈ q and x = y . The function rank q : N → { , . . . , n } defined, for every x ∈ N , byrank q ( x ) = |{ y ∈ N : y > q x }| + 1 is bijective. We identify q with the function rank − q and denoteit still by q . We also identify q with the column vector [ q (1) , . . . , q ( n )] T . Moreover, we define q r asthe element in L ( N ) such that, for every x, y ∈ N , ( x, y ) ∈ q r if and only if ( y, x ) ∈ q . Of course,( q r ) r = q . For instance, let n = 3 and q ∈ L ( N ) be such that 2 > q > q
3. Then q (1) = 2, q (2) = 1, q (3) = 3 and we identify q with [2 , , T and q r with [3 , , T . A preference profile is an element of L ( N ) h . The set L ( N ) h is denoted by P . Let p ∈ P be fixed. Given i ∈ H , the i -th component of p is denoted by p i and represents the preferencesof individual i . The preference profile p can be naturally identified with the matrix whose i -thcolumn is [ p i (1) , . . . , p i ( n )] T . Define p r ∈ P as the preference profile such that, for every i ∈ H ,( p r ) i = ( p i ) r . Of course, ( p r ) r = p . We will write the i -th component of p r simply as p ri , insteadof ( p r ) i . Given µ ∈ N ∩ ( h/ , h ] and x, y ∈ N , we write x > pµ y if |{ i ∈ H : x > p i y }| ≥ µ . Notethat x > pµ y if and only if y > p r µ x. Elements in N ∩ ( h/ , h ] are called majority thresholds. We call minimal majority threshold the integer µ = ⌈ h +12 ⌉ . Further details about preference relations andpreference profiles can be found in Bubboloni and Gori (2015).A social choice correspondence ( scc ) is a function from P to the set of the nonempty subsets of N . The set of scc s is denoted by C . Let C ∈ C . We say that C suffers the reversal bias (of type 1) if, there exists p ∈ P and x ∈ N such that C ( p ) = C ( p r ) = { x } ;the reversal bias of type 2 if, there exists p ∈ P such that | C ( p ) | = 1 and C ( p ) ∩ C ( p r ) = ∅ ;the reversal bias of type 3 if, there exists p ∈ P such that | C ( p ) | < n and C ( p ) ∩ C ( p r ) = ∅ . Clearly if C ∈ C suffers the reversal bias of type 1, then C suffers also the reversal bias of type 2,and if C suffers the reversal bias of type 2 then C suffers also the reversal bias of type 3. We define,for every j ∈ { , , } , the sets C j = { C ∈ C : C is immune to the reversal bias of type j } . Note that C ⊆ C ⊆ C . 5 The Minimax scc
In this section we focus on the Minimax scc , denoted by M and defined, for every p ∈ P , by M ( p ) = argmin x ∈ N max y ∈ N \{ x } |{ i ∈ H : y > p i x }| . According to the above definition, M ( p ) is then the set of those alternatives which minimize thegreatest pairwise defeat, with respect to the individual preferences described by p . However, theoutcomes of the Minimax scc admit an alternative interpretation in terms of majority thresholds.Given µ ∈ N ∩ ( h/ , h ], define, for every p ∈ P , the set D µ ( p ) = { x ∈ N : ∀ y ∈ N, |{ i ∈ H : y > p i x }| < µ } . Thus, an alternative x belongs to D µ ( p ) if and only if it cannot be found another alternative whichis preferred to x by at least µ individuals, according to the preference profile p . Note that the set D µ ( p ) corresponds to the set of µ -majority equilibria associated with p as defined by Greenberg(1979) in the more general setting where individual preferences are represented via complete andtransitive relations. Observe that if µ ≤ µ ′ , then D µ ( p ) ⊆ D µ ′ ( p ) for all p ∈ P . Moreover, as animmediate consequence of Corollary 3 in Greenberg (1979) and its proof, for every µ ∈ N ∩ ( h/ , h ],we have that D µ ( p ) = ∅ for all p ∈ P if and only if µ > n − n h. (1)Since h ∈ N ∩ ( h/ , h ] and h > n − n h , it is well defined the Greenberg majority threshold given by µ G = min (cid:26) m ∈ N ∩ ( h/ , h ] : m > n − n h (cid:27) . For every p ∈ P , we consider the integer µ ( p ) = min { µ ∈ N ∩ ( h/ , h ] : D µ ( p ) = ∅ } . Note that, since (1) implies D µ G ( p ) = ∅ , we have that µ ( p ) is well defined and µ ≤ µ ( p ) ≤ µ G .We can now prove the following proposition. Proposition 1.
For every p ∈ P , M ( p ) = D µ ( p ) ( p ) .Proof. We show first that M ( p ) ⊆ D µ ( p ) ( p ) proving that N \ D µ ( p ) ( p ) ⊆ N \ M ( p ). Let x ∈ N \ D µ ( p ) ( p ) . Then there exists y ∈ N \ { x } such that |{ i ∈ H : y > p i x }| ≥ µ ( p ). Picking now x ∈ D µ ( p ) ( p ), we have that, for every y ∈ N \ { x } , |{ i ∈ H : y > p i x }| ≤ µ ( p ) −
1. Thus,max y ∈ N \{ x } |{ i ∈ H : y > p i x }| ≤ µ ( p ) − < |{ i ∈ H : y > p i x }| ≤ max y ∈ N \{ x } |{ i ∈ H : y > p i x }| , which says x / ∈ M ( p ).We next show that D µ ( p ) ( p ) ⊆ M ( p ) . Let x ∈ D µ ( p ) ( p ). Then, we have thatmax y ∈ N \{ x } |{ i ∈ H : y > p i x }| ≤ µ ( p ) − . Assume, by contradiction, that there exists x ∈ N such thatmax y ∈ N \{ x } |{ i ∈ H : y > p i x }| < max y ∈ N \{ x } |{ i ∈ H : y > p i x }| . Fishburn (1977) presents the equivalent definition M ( p ) = argmax x ∈ N min y ∈ N \{ x } |{ i ∈ H : x > p i y }| . x = x and, for every y ∈ N \ { x } , we have |{ i ∈ H : y > p i x }| < µ ( p ) −
1. If µ ( p ) − > h/
2, that says x ∈ D µ ( p ) − ( p ) = ∅ and the contradiction is found. Assume instead that µ ( p ) − ≤ h/
2. Then µ ( p ) = µ = (cid:6) h +12 (cid:7) ≤ h +22 and since |{ i ∈ H : x > p i x }| ≤ µ − , we get |{ i ∈ H : x > p i x }| ≥ h − µ + 2 . Now we observe that, due to µ ≤ h +22 , we have h − µ + 2 ≥ µ ,against x ∈ D µ ( p ).Define the sets T = { ( h, n ) ∈ N ⋄ : h ≤ } ∪ { ( h, n ) ∈ N ⋄ : n ≤ } ∪ { (4 , , (5 , , (7 , , (5 , } ,T = { ( h, n ) ∈ N ⋄ : h = 2 } ∪ { ( h, n ) ∈ N ⋄ : n ≤ } ∪ { (4 , } ,T = { ( h, n ) ∈ N ⋄ : n = 2 } ∪ { (3 , } . and note that T ( T ( T . We can now state the main result of the paper. Its proof is technicaland will be presented in Section 5. We stress that it relies on Proposition 1 and the use of languageand methods of graph theory (Sections 5.1 and 5.2). Theorem 2.
Let j ∈ { , , } . Then, M ∈ C j if and only if ( h, n ) ∈ T j . scc s In this section we show that, differently from the case of the Minimax scc , the analysis of thereversal bias is easy for the Borda and Copeland scc s. Those scc s are respectively denoted by
Bor and
Cop , and defined , for every p ∈ P , as Bor ( p ) = argmax x ∈ N P hi =1 (cid:0) n − rank p i ( x ) (cid:1) ,Cop ( p ) = argmax x ∈ N (cid:0) |{ y ∈ N : x > pµ y }| − |{ y ∈ N : y > pµ x }| (cid:1) . The following results show that they are immune to the reversal bias of type 3.
Proposition 3.
Bor, Cop ∈ C .Proof. We start considering the Borda scc . We need to show that, for every p ∈ P , Bor ( p ) ∩ Bor ( p r ) = ∅ implies Bor ( p ) = N . Fix then p ∈ P and x ∈ N such that x ∈ Bor ( p ) ∩ Bor ( p r ).Let f , g and u be the functions from N to R defined, for every x ∈ N , by f ( x ) = h X i =1 (cid:0) n − rank p i ( x ) (cid:1) , g ( x ) = h X i =1 (cid:0) n − rank p ri ( x ) (cid:1) , u ( x ) = h X i =1 rank p i ( x ) . Note that
Bor ( p ) = argmax x ∈ N f ( x ) , Bor ( p r ) = argmax x ∈ N g ( x ) , and that, for every x ∈ N , f ( x ) = hn − u ( x ) and g ( x ) = u ( x ) − h , due to the fact that rank p r ( x ) = n + 1 − rank p ( x ) . Then x realises both the minimum and the maximum of u , so that u is constant.It follows that f is constant too and therefore Bor ( p ) = N. With Borda scc we mean the well-known Borda count. The definition of the Copeland scc can be found, forinstance, in Fishburn (1977).
7e next consider the Copeland scc . We need to show that, for every p ∈ P , Cop ( p ) ∩ Cop ( p r ) = ∅ implies Cop ( p ) = N . Fix then p ∈ P and x ∈ N such that x ∈ Cop ( p ) ∩ Cop ( p r ). Let f and g be functions from N to R defined, for every x ∈ N , by f ( x ) = |{ y ∈ N : x > pµ y }|−|{ y ∈ N : y > pµ x }| , g ( x ) = |{ y ∈ N : x > p r µ y }|−|{ y ∈ N : y > p r µ x }| . Note that
Cop ( p ) = argmax x ∈ N f ( x ) , Cop ( p r ) = argmax x ∈ N g ( x ) . Moreover, since x > p ri y is equivalent to y > p i x for all x, y ∈ N and i ∈ H , we have that, for every x ∈ N , g ( x ) = − f ( x ). Then x realises both the minimum and the maximum of f . It follows that f is constant and therefore Cop ( p ) = N. Next corollaries show how the results proved about the reversal bias of M , Bor and
Cop can beused to establish conditions on the number of individuals and alternatives that are necessary andsufficient to have the equalities M = Bor and M = Cop . Corollary 4. M = Bor if and only if n = 2 .Proof. If n = 2, then we surely have M = Bor . Assume now that n ≥
3. If ( h, n ) T , then,by Theorem 2 and Proposition 3, M C and Bor ∈ C so that M = Bor . If ( h, n ) ∈ T , then( h, n ) = (3 ,
3) and we still have M = Bor since the two scc s differ, for instance, on the preferenceprofile p = Corollary 5. M = Cop if and only if ( h, n ) ∈ T .Proof. If n = 2, then we surely have M = Cop . Assume now that n ≥
3. If ( h, n ) T , then,by Theorem 2 and Proposition 3, M C and Cop ∈ C so that M = Cop . If ( h, n ) ∈ T , then( h, n ) = (3 , p ∈ P . Since both M and Cop are neutral, without loss of generality, wecan assume that p = [1 , , T . Recalling that both M and Cop satisfy the Condorcet principle,they surely coincide when a Condorcet winner exists. Thus, we can assume that the alternativesranked first are different among themselves. Since both M and Cop are anonymous, we can assumethat p (1) = 2 and p (1) = 3. That leaves just four cases to treat. By a case by case computationon those, it can be finally proved that M = Cop . From Proposition 1 we immediately have that Theorem 2 is implied by the following three propo-sitions. Their tricky proofs, based on graph theory, are presented in the Sections 5.3, 5.4 and5.5.
Proposition 6.
There exist p ∈ P and x ∈ N such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } if and onlyif ( h, n ) ∈ N ⋄ \ T . Proposition 7.
There exist p ∈ P and x ∈ N such that D µ ( p ) ( p ) = { x } ⊆ D µ ( p r ) ( p r ) if and onlyif ( h, n ) ∈ N ⋄ \ T . Proposition 8.
There exists p ∈ P such that D µ ( p ) ( p ) = N and D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ if andonly if ( h, n ) ∈ N ⋄ \ T . .1 Graphs In this section, we recall some basic facts and notation from graph theory, which we are going touse in the sequel . All the considered graphs are directed. A graph is a pair ( V, A ), where V is anonempty set called vertex set and A is a subset of { ( x, y ) ∈ V : x = y } called arc set. Note that ifΓ = ( V, A ) is a graph and | V | = 1, then A = ∅ . Given two graphs Γ = ( V , A ) and Γ = ( V , A ),we say that Γ is a subgraph of Γ if V ⊆ V and A ⊆ A . If Γ is a subgraph of Γ , we writeΓ ≤ Γ .Let now Γ = ( V, A ) be a graph. Γ is called complete if for every x, y ∈ V with x = y , we have( x, y ) ∈ A or ( y, x ) ∈ A . We say that x ∈ V is maximal [ minimal ] for Γ if there exists no y ∈ V suchthat ( y, x ) ∈ A [( x, y ) ∈ A ]. We denote by max(Γ) [min(Γ)] the set of maximal [minimal] verticesfor Γ. Note that those sets may be empty. We say that x ∈ V is a maximum [ minimum ] of Γ if, forevery y ∈ V \ { x } , we have that ( x, y ) ∈ A [( y, x ) ∈ A ] . We denote by Max(Γ) [Min(Γ)] the set ofmaxima [minima] of Γ. We say that x ∈ V is isolated in Γ if, for every y ∈ V \ { x } , ( x, y ) , ( y, x ) A .We denote by I(Γ) the set of the isolated vertices of Γ. It is useful to note thatmax(Γ) ∩ min(Γ) = I(Γ) . (2)Note also that if x ∈ Max(Γ) ∪ Min(Γ) and | V | ≥
2, then x I(Γ) . Γ is said to be connected if, for every x, y ∈ V with x = y , there exist k ≥ x , . . . , x k of distinct elements of V such that x = x , x k = y, and, for every j ∈ { , . . . , k − } , ( x j , x j +1 ) ∈ A or ( x j +1 , x j ) ∈ A . Note that if Γ has a maximum [minimum], then Γ is connected.It is well known that there exist a uniquely determined c ∈ N and connected subgraphs Γ =( V , A ) , . . . , Γ c = ( V c , A c ) of Γ such that ∪ ci =1 V i = V , ∪ ci =1 A i = A , and for every i, j ∈ { , . . . , c } with i = j , V i ∩ V j = A i ∩ A j = ∅ . Those subgraphs Γ , . . . , Γ c are called the connected components of Γ. They are maximal among the connected subgraphs of Γ, that is, if Γ ′ ≤ Γ is connected andΓ ′ ≥ Γ i for some i ∈ { , . . . , c } , then Γ ′ = Γ i . In particular, for every i ∈ { , . . . , c } , x ∈ V i and y ∈ V \ V i imply ( x, y ) , ( y, x ) / ∈ A ; x, y ∈ V i and ( x, y ) ∈ A imply ( x, y ) ∈ A i . Note that x ∈ N isisolated in Γ if and only if the connected component of Γ containing x is ( { x } , ∅ ) . Given l ≥
2, Γis said to be a l -cycle if | V | = l and there exists an ordered sequence x , . . . , x l of the elements of V such that, once defined x l +1 = x , we have that A = { ( x j , x j +1 ) : 1 ≤ j ≤ l } . Γ is said to bea cycle if it is a l -cycle for some l ≥
2. Fixed l ≥
2, Γ is said to be l -cyclic if there exists a l -cycleΓ ≤ Γ, and l -acyclic otherwise. Γ is said to be acyclic if it is l -acyclic for all l ≥
2. Note that if | V | = 1, then Γ is acyclic. Let p ∈ P and µ ∈ N ∩ ( h/ , h ] . In a natural way, we associate with the relation on N given byΣ µ ( p ) = { ( x, y ) ∈ N × N : x > pµ y } , the graph Γ µ ( p ) = ( N, Σ µ ( p )), called the µ -majority graph of p . Note that, if µ, µ ′ ∈ N ∩ ( h/ , h ] with µ ′ ≤ µ , then Γ µ ( p ) ≤ Γ µ ′ ( p ) . In particular, Γ µ ( p ) ≤ Γ µ ( p )holds for all µ ∈ N ∩ ( h/ , h ] . The concept of majority graph has been considered by many authorsessentially in relation to the case when h is odd and µ = µ = h +12 (see, for instance, Miller (1977)).For the purpose of our paper that case is interesting because Γ h +12 ( p ) is complete (see Lemma 12),but we are not focussed only on that particular majority graph.The properties of the relation Σ µ ( p ) translates easily into graph theoretical properties for Γ µ ( p ).Moreover, considering Γ µ ( p ) we gain the advantage of using concepts like l -acyclicity and connect-edness which typically belongs to graph theory. That gives very soon a better comprehension ofthe sets D µ ( p ) and D µ ( p ) ( p ) = M ( p ). All unexplained notation is standard. See, for instance, Diestel (2010). Note that if x is a maximum [minimum] of Γ it is not necessarily maximal [minimal] for Γ. In fact, givenΓ = ( { , } , { (1 , , (2 , } ), we have that 1 and 2 are both a maximum [minimum] but none of them is maximal[minimal]. emma 9. Let µ ∈ N ∩ ( h/ , h ] and p ∈ P . Then D µ ( p ) = max(Γ µ ( p )) = min(Γ µ ( p r )) . Moreover D µ ( p ) ∩ D µ ( p r ) = I(Γ µ ( p )) = I(Γ µ ( p r )) . Proof.
The equalities D µ ( p ) = max(Γ µ ( p )) = min(Γ µ ( p r )) follow from the definitions of D µ ( p ) and p r . As a consequence, since ( p r ) r = p , we also have D µ ( p r ) = max(Γ µ ( p r )) = min(Γ µ ( p )), so that(2) completes the proof. Lemma 10.
Let µ ∈ N ∩ ( h/ , h ] and p ∈ P . Then Γ µ ( p ) is -acyclic and Γ µ ( p ) has at most onemaximum. Moreover, if Γ µ ( p ) has a maximum x ∈ N , then D µ ( p ) = Max(Γ µ ( p )) = { x } .Proof. The 2-acyclicity follows immediately from µ > h/ x, y ∈ Max(Γ µ ( p )). Then ( x, y ) ∈ Σ µ ( p ) and( y, x ) ∈ Σ µ ( p ), so that Γ = ( { x, y } , { ( x, y ) , ( y, x ) } ) ≤ Γ µ ( p ). Since Γ is a 2-cycle, that contradictsthe fact that Γ µ ( p ) is 2-acyclic.Assume now that there exists x ∈ Max(Γ µ ( p )) and show that x ∈ max(Γ µ ( p )) = D µ ( p ). Bycontradiction, let x max(Γ). Then there is y ∈ N such that ( y, x ) ∈ Σ µ ( p ). Since also ( x, y ) ∈ Σ µ ( p ), the 2-cycle Γ = ( { x, y } , { ( x, y ) , ( y, x ) } ) is a subgraph of Γ µ ( p ) and the contradiction is found.We complete the proof simply noticing that, being x a maximum of Γ µ ( p ), for every y ∈ N \ { x } ,we have that ( x, y ) ∈ Σ µ ( p ) so that y max(Γ µ ( p )). Lemma 11.
Let µ ∈ N ∩ ( h/ , h ] . Then Γ µ ( p ) is acyclic for all p ∈ P if and only if µ ≥ µ G . Inparticular Γ h ( p ) is acyclic for all p ∈ P . Proof.
It is an immediate consequence of Proposition 6 and 7 in Bubboloni and Gori (2014).
Lemma 12. If h is odd, then, for every p ∈ P , Γ µ ( p ) is complete and I(Γ µ ( p )) = ∅ . Moreover,if D µ ( p ) = ∅ then µ ( p ) = µ , Γ µ ( p ) admits maximum x ∈ N and D µ ( p ) = { x } .Proof. Let us fix p ∈ P and note that, being h odd, we have µ = h +12 . Assume now, by con-tradiction, that there exist x, y ∈ N with x > pµ y and y > pµ x . Then, we get the impossiblerelation h = |{ i ∈ H : x > p i y }| + |{ i ∈ H : y > p i x }| ≤ µ − µ − (cid:18) h + 12 (cid:19) − h − . Thus Γ µ ( p ) is complete and, as an immediate consequence, I(Γ µ ( p )) = ∅ . In order to prove the second part, assume that D µ ( p ) = ∅ . Then µ ( p ) ≤ µ and so µ ( p ) = µ .Next, pick x ∈ D µ ( p ). Since Γ µ ( p ) is complete, then we have x > pµ y for all y ∈ N \ { x } , that is, x is a maximum in Γ µ ( p ). Then, by Lemma 10, D µ ( p ) = { x } .Let us denote by C (Γ µ ( p )) the set of the connected components of Γ µ ( p ) and define A (Γ µ ( p )) = { Γ ∈ C (Γ µ ( p )) : Γ is acyclic } . We are ready for a key proposition giving a lower bound for | D µ ( p ) | and leading to some interesting consequences. Proposition 13.
Let µ ∈ N ∩ ( h/ , h ] and p ∈ P . Then D µ ( p ) = [ Γ ∈C (Γ µ ( p )) max(Γ) ⊇ [ Γ ∈A (Γ µ ( p )) max(Γ) ⊇ I(Γ µ ( p )) , and | D µ ( p ) | = X Γ ∈C (Γ µ ( p )) | max(Γ) | ≥ |A (Γ µ ( p )) | ≥ | I(Γ µ ( p )) | . roof. Let Γ = (
V, A ) ∈ C (Γ µ ( p )) . Then Γ ≤ Γ µ ( p ), so that V ⊆ N and A ⊆ Σ µ ( p ). Since Γ isa connected component of Γ µ ( p ), we have that, for every x ∈ V and y ∈ N \ V , y > pµ x . Thisimmediately gives that each x ∈ max(Γ) belongs to D µ ( p ) , so that D µ ( p ) ⊇ S Γ ∈C (Γ µ ( p )) max(Γ) . Theother inclusion is trivial and thus D µ ( p ) = S Γ ∈C (Γ µ ( p )) max(Γ) . Since A (Γ µ ( p )) ⊆ C (Γ µ ( p )) and, forevery x ∈ I(Γ µ ( p )), ( { x } , ∅ ) ∈ A (Γ µ ( p )), we also get [ Γ ∈C (Γ µ ( p )) max(Γ) ⊇ [ Γ ∈A (Γ µ ( p )) max(Γ) ⊇ I(Γ µ ( p )) . In particular, since there is no overlap between vertices of different connected components, wededuce | D µ ( p ) | = P Γ ∈C (Γ µ ( p )) | max(Γ) | . We complete the proof showing that for every Γ ∈ A (Γ µ ( p )),we have max(Γ) = ∅ . Pick x ∈ V . If y > pµ x for all y ∈ V , then we have x ∈ max(Γ) and we havefinished. Assume instead there exists x ∈ V with x > pµ x . Obviously, we have x = x . Then,repeat the argument for x . Since the set N is finite and Γ contains no cycle, in a finite number k ≤ n of steps, we obtain an element x k ∈ max(Γ) . Corollary 14.
Let µ ∈ N ∩ ( h/ , h ] and p ∈ P . If Γ µ ( p ) admits at least an acyclic connectedcomponent, then µ ( p ) ≤ µ. Proof.
By Proposition 13, we have | D µ ( p ) | ≥ , so that D µ ( p ) = ∅ . Corollary 15.
Let µ ∈ N ∩ ( h/ , h ] and p ∈ P . If Γ µ ( p ) is acyclic and D µ ( p ) is a singleton, then Γ µ ( p ) is connected.Proof. Since Γ µ ( p ) is acyclic, we have that C (Γ µ ( p )) = A (Γ µ ( p )). Then, using Proposition 13, weget 1 = | D µ ( p ) | ≥ |C (Γ µ ( p )) | ≥
1. That implies |C (Γ µ ( p )) | = 1, that is, Γ µ ( p ) is connected. Lemma 16.
Let p ∈ P such that µ ( p r ) ≤ µ ( p ) . Then:(i) D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) ⊆ I(Γ µ ( p ) ( p )) . In particular, if Γ µ ( p ) ( p ) is connected, then D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ . (ii) If | D µ ( p ) ( p ) | = 1 and Γ µ ( p ) ( p ) is acyclic, then D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ . Proof. ( i ) From µ ( p r ) ≤ µ ( p ) we get D µ ( p r ) ( p r ) ⊆ D µ ( p ) ( p r ) and thus, by Lemma 9, we deduce D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) ⊆ D µ ( p ) ( p ) ∩ D µ ( p ) ( p r ) = I(Γ µ ( p ) ( p )) . If Γ µ ( p ) ( p ) is connected, then I(Γ µ ( p ) ( p ))is empty and thus also D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ . ( ii ) By Corollary 15, (i) applies giving D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ . Corollary 17.
Let µ ∈ N ∩ ( h/ , h ] and p ∈ P . If Γ µ ( p ) is acyclic, then, for every x ∈ N , we donot have D µ ( p ) = D µ ( p r ) = { x } . Proof.
Assume by contradiction that D µ ( p ) = D µ ( p r ) = { x } , for some x ∈ N . Then, by Lemma9, we have that x is isolated in Γ µ ( p ). On the other hand, by Corollary 15, Γ µ ( p ) is connected sothat its only vertex is x , against n ≥ . Lemma 18.
Let p ∈ P and assume that both Γ µ ( p ) ( p ) and Γ µ ( p r ) ( p r ) admit an acyclic connectedcomponent. Then:(i) µ ( p ) = µ ( p r ) .(ii) If Γ µ ( p ) ( p ) is connected, then D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ . (iii) If Γ µ ( p ) ( p ) is acyclic, then there exists no x ∈ N such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } . roof. ( i ) The fact that Γ µ ( p ) ( p ) admits an acyclic connected component implies that also Γ µ ( p ) ( p r )admits an acyclic connected component and therefore Corollary 14 gives µ ( p r ) ≤ µ ( p ). The sameargument applied to Γ µ ( p r ) ( p r ) gives µ ( p ) ≤ µ ( p r ) . ( ii ) Assume that Γ µ ( p ) ( p ) is connected. Since, by ( i ), we have that µ ( p ) = µ ( p r ), then Lemma16 applies, giving D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ . ( iii ) Let Γ µ ( p ) ( p ) be acyclic and assume, by contradiction, that there exists x ∈ N such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } . By Corollary 15, Γ µ ( p ) ( p ) is connected and thus, by ( ii ), we havethat D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ , a contradiction. Lemma 19. If h is odd, then, for every p ∈ P , µ ( p ) = µ and D µ ( p ) ( p ) = D µ ( p r ) ( p r ) imply µ ( p r ) > µ .Proof. Let p ∈ P and assume that µ ( p ) = µ and D µ ( p ) ( p ) = D µ ( p r ) ( p r ). Then D µ ( p ) = ∅ and,using Lemma 12, Γ µ ( p ) ( p ) has a maximum x ∈ N and D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } . Assume bycontradiction that µ ( p r ) = µ . By Lemma 9, we get that x is isolated in Γ µ ( p ) ( p ), against the factthat x is the maximum of Γ µ ( p ) ( p ). Corollary 20.
Let p ∈ P such that µ ( p ) = µ . If Γ µ ( p ) ( p ) is acyclic, then there exists no x ∈ N such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } .Proof. The acyclicity of Γ µ ( p ) ( p ) implies that of Γ µ ( p ) ( p r ), so that, by Corollary 14, we have µ ≤ µ ( p r ) ≤ µ ( p ) = µ . It follows that µ ( p r ) = µ ( p ) = µ and Corollary 17 applies.Due to the previous results, it is important to understand which conditions guarantee the acyclic-ity of Γ µ ( p ) ( p ). By Lemma 10, we know that, for every µ ∈ N ∩ ( h/ , h ], Γ µ ( p ) is 2-acyclic. Anyway,it can admit l -cycles for some l ≥
3. We explore this possibility through Propositions 6 and 7 inBubboloni and Gori (2014).
Proposition 21.
Let µ ∈ N ∩ ( h/ , h ] and l ∈ N ∩ [2 , n ] . Then there exists p ∈ P such that Γ µ ( p ) is l -cyclic if and only if µ ≤ l − l h. Proof.
Consider µ > l − l h and assume by contradiction that there exists p ∈ P and an l -cycleΓ ≤ Γ µ ( p ) with vertex set V . Then V ⊆ N and | V | = l ≤ n . Consider the preference profile p ′ onthe set of l alternatives V obtained from p eliminating (if any) those entries in N \ V. By Proposition6 in Bubboloni and Gori (2014), we have that Γ µ ( p ′ ) is acyclic, against the fact that Γ ≤ Γ µ ( p ′ ) . Let now µ ≤ l − l h and let V ⊆ N with | V | = l . By Proposition 7 in Bubboloni and Gori (2014),there exists a preference profile p ′ on the set of alternatives V such that Γ µ ( p ′ ) contains an l -cycleΓ whose set of vertices is V . Consider a preference profile p on the set of alternatives N, in whichevery individual i ∈ H ranks in the first l positions the alternatives in V as p ′ i and those in N \ V as she likes. Then Γ ≤ Γ µ ( p ).Let us consider now µ a = min (cid:26) m ∈ N ∩ ( h/ , h ] : m > n − n − h (cid:27) , and note that µ a is well defined because h ∈ N ∩ ( h/ , h ] and h > n − n − h . Moreover, we have that µ ≤ µ a ≤ µ G and, when n ∈ { , } , µ a = µ . Corollary 22.
Let p ∈ P . If µ ( p ) ≥ µ a , then Γ µ ( p ) ( p ) is acyclic. In particular, for every n ∈ { , } , Γ µ ( p ) ( p ) is acyclic.Proof. Consider Γ µ ( p ) ( p ). It admits no n -cycle, because having such a cycle obviously implies thecontradiction D µ ( p ) ( p ) = ∅ . On the other hand, by Proposition 21, it does not have l -cyclesfor all l ∈ { , . . . , n − } because µ ( p ) > n − n − h ≥ l − l h. Finally note that, if n ∈ { , } , then µ ( p ) ≥ µ = µ a . 12ue to the previous result, we call µ a the acyclicity threshold . Proposition 23. If n ≥ and h is odd and such that h ≥ n − n − , then there exist p ∈ P such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { n } . Proof.
First of all, note that µ = h +12 . Define then µ = h +32 = µ + 1 and V = N \ { n } . Theassumption h ≥ n − n − is equivalent to µ ≤ ( n − − n − h and thus, by Proposition 21, there exists p ′ , a preference profile on the set of alternatives V, such that Γ µ ( p ′ ) has an ( n − p ∈ P defining, for every i ∈ H , the preference p i as follows. If i ≤ µ , then let p i (1) = n and p i ( j ) = p ′ i ( j −
1) for all j ∈ { , . . . , n } ; if µ < i ≤ h, then let p i ( j ) = p ′ i ( j ) for all j ∈ { , . . . , n − } and p i ( n ) = n. Note that in p , the alternative n is rankedfirst µ times and last h − µ times. Thus, n is a maximum in Γ µ ( p ) . By Lemma 12, we then get µ ( p ) = µ and D µ ( p ) ( p ) = { n } . Moreover Γ ≤ Γ µ ( p ) so that also Γ µ ( p r ) contains an ( n − r , with inverted orientation, whose vertex set is V . That implies that D µ ( p r ) = ∅ . Indeed, n isnot maximal in Γ µ ( p r ), being beaten µ times by any other alternative, and each alternative in V is not maximal in Γ µ ( p r ) because, due to the presence of the cycle Γ r , it is beaten µ > µ timesby a suitable alternative in V . Anyway D µ ( p r ) = { n } , because n is isolated and thus maximal inΓ µ ( p r ) , by (2); no other alternative is maximal because involved in Γ r . It follows that µ ( p r ) = µ and D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { n } . Proposition 24. If n ≥ and h is even and such that h ≥ n − n − , then there exist p ∈ P suchthat D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { n } . Proof.
First of all, note that µ = h +22 and define V = N \ { n } . The assumption h ≥ n − n − isequivalent to µ ≤ ( n − − n − h and thus, by Proposition 21, there exist a preference profile p ′ on theset of alternatives V such that Γ µ ( p ′ ) has an ( n − p ∈ P , defining, for every i ∈ H , the preference p i as follows. If i ≤ h , then let p i (1) = n and p i ( j ) = p ′ i ( j −
1) for all j ∈ { , . . . , n } ; if h < i ≤ h, then let p i ( j ) = p ′ i ( j ) for all j ∈ { , . . . , n − } and p i ( n ) = n. Note that in p , the alternative n is ranked first h times and last h times. Thus, by(2), n is isolated and maximal both in Γ µ ( p ) and in Γ µ ( p r ). Moreover, no further alternative ismaximal in Γ µ ( p ) because each element in V is involved in the cycle Γ ≤ Γ µ ( p ). Since each cyclein Γ µ ( p ) determines a cycle with inverted orientation in Γ µ ( p r ), the same consideration holds forΓ µ ( p r ), as well. Then, we conclude that µ ( p ) = µ ( p r ) = µ and D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { n } . We conclude the section with a further lemma which is useful to manage the case with threeindividuals and three alternatives.
Lemma 25.
Let ( h, n ) = (3 , and p ∈ P . Then:(i) the two following conditions are equivalent:(a) the alternatives ranked first as well as those ranked third in p are distinct;(b) Γ ( p ) is a -cycle.Moreover, if one of the above conditions holds true, then the arc set of Γ ( p ) is empty.(ii) µ ( p ) = µ ( p r ) . Proof. ( i ) We start showing that ( a ) implies ( b ). Assume that p i (1) = p j (1) and p i (3) = p j (3) forall i, j ∈ H = { , , } with i = j . Without loss of generality we can assume that p (1) = 1 , p (1) =13 , p (1) = 3. Thus p (3) ∈ { , } . If p (3) = 2, then, since the alternatives ranked third are distinct,we necessarily have p (3) = 3 and p (3) = 1. That implies that p = Similarly, if p (3) = 3 , we get p = In both cases we have that Γ ( p ) is a 3-cycle and the arc set of Γ ( p ) is empty.We next show that ( b ) implies ( a ) . Let p ∈ P . If there exists x ∈ N which is ranked first in p by at least two individuals, then x is a maximum for Γ ( p ) and so it cannot be involved in a cycleof Γ ( p ). If there exists x ∈ N which is ranked third in p by at least two individuals, consider p r .By what shown above, Γ ( p r ) is acyclic, so that Γ ( p ) is acyclic too.( ii ) By contradiction, assume that µ ( p r ) = µ ( p ) , say µ ( p r ) > µ ( p ) . Then µ ( p ) = 2 and µ ( p r ) = 3.Thus Γ ( p r ) admits a cycle. By Lemma 10, we then get that Γ ( p r ) is a 3-cycle. Using ( i ) we deducethat the alternatives ranked first as well as those ranked third in p r are distinct. But then, thesame property holds for p , so that also Γ ( p ) is a 3-cycle. Thus D ( p ) = ∅ , against µ ( p ) = 2. First of all, let us prove that if ( h, n ) ∈ N ⋄ \ T , then there exist p ∈ P and x ∈ N such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } . We obtain the proof showing that the assumptions of Propositions23 or 24 hold true. First of all, note that ( h, n ) ∈ N ⋄ \ T implies h ≥ n ≥ . If n = 4, theneither h is even with h ≥ h ≥ n − n − , or h is odd with h ≥ h ≥ n − n − . If n = 5, then the same argument applies. If n ≥ , then we have n − n − ≤ ≤ h forall h even, as well as n − n − ≤ ≤ h for all h odd.Assume now that ( h, n ) ∈ T and prove that it cannot be found p ∈ P and x ∈ N such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } . Consider then ( h, n ) ∈ T and assume, by contradiction, that thereexist p ∈ P and x ∈ N such that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { x } . Since the Minimax scc is neutral,we can assume that x = n so that D µ ( p ) ( p ) = D µ ( p r ) ( p r ) = { n } . (3)There are several cases to study.If n ∈ { , } , then, by Corollary 22, we have that Γ µ ( p ) ( p ) and Γ µ ( p r ) ( p r ) are both acyclic sothat Lemma 18 (iii) applies contradicting (3).If h = 2, then µ ( p ) = µ ( p r ) = µ a = 2 and, by Corollary 22, we have that Γ µ ( p ) ( p ) and Γ µ ( p r ) ( p r )are both acyclic so that Lemma 18 (iii) applies contradicting (3).If h = 3, then µ = 2 and µ ( p ) , µ ( p r ) ∈ { , } . If µ ( p ) = 2, then, by Lemma 19, we have that µ ( p r ) = 3 so that D µ ( p r ) ( p r ) = { n } and D µ ( p ) ( p r ) = ∅ . Let V = N \ { n } . Since n is the onlymaximal element in Γ µ ( p r ) ( p r ), for every x ∈ V , there exists y ∈ N with y > p r µ ( p r ) x . Note thatif y were equal to n , then from n > p r µ ( p r ) x we would get x > pµ ( p r ) n against the maximality of n in Γ µ ( p ) ( p ). Thus, there exists a cycle in Γ µ ( p r ) ( p r ) involving some vertices of V. That leads to acontradiction since, by Lemma 11, Γ µ ( p r ) ( p r ) is acyclic. If µ ( p r ) = 2, then the previous argumentapplies to p r . If µ ( p ) = µ ( p r ) = 3 , then we reach a contradiction applying Lemma 11 and Corollary17. 14f ( h, n ) = (4 , µ = µ a = 3 and µ ( p ) , µ ( p r ) ∈ { , } . Thus, by Corollary 22, Γ µ ( p ) ( p )and Γ µ ( p r ) ( p r ) are both acyclic, so that Lemma 18 (iii) applies contradicting (3).If ( h, n ) = (5 , µ = 3, µ a = µ G = 4, and µ ( p ) , µ ( p r ) ∈ { , } . If µ ( p ) = µ ( p r ) = 4,then by Corollary 22, Γ µ ( p ) ( p ) and Γ µ ( p r ) ( p r ) are both acyclic and we contradict (3), using Lemma18 (iii). If µ ( p ) = 3 = µ , then, by Lemma 19, µ ( p r ) = 4. By Corollary 22 we have that Γ µ ( p r ) ( p r )is acyclic and then, by Corollary 15, connected. Assume there exists x ∈ V = { , , } such that4 > p r µ ( p r ) x . Then x > pµ ( p r )
4, against 4 ∈ D µ ( p ) ( p ). So, we have 4 > p r µ ( p r ) x , for all x ∈ V . Onthe other hand, from 4 ∈ D µ ( p r ) ( p r ), we deduce that x > p r µ ( p r )
4. Thus, 4 is isolated in Γ µ ( p r ) ( p r ),against the connection of Γ µ ( p r ) ( p r ). If µ ( p r ) = 3 = µ , then the previous argument applies to p r .If ( h, n ) = (7 , µ = 4 , µ a = 5, µ G = 6 and µ ( p ) , µ ( p r ) ∈ { , , } . If µ ( p ) , µ ( p r ) ∈ { , } ,then by Corollary 22, Γ µ ( p ) ( p ) and Γ µ ( p r ) ( p r ) are both acyclic and we contradict (3), using Lemma18 (iii). If µ ( p ) = 4, then, by Lemma 19, µ ( p r ) ∈ { , } . By Corollary 22, Γ µ ( p r ) ( p r ) is acyclic andthen, by Corollary 15, connected. Assume there exists x ∈ V = { , , } such that 4 > p r µ ( p r ) x . Then x > pµ ( p r )
4, against 4 ∈ D µ ( p ) ( p ). So, we have 4 > p r µ ( p r ) x for all x ∈ V . On the other hand, from4 ∈ D µ ( p r ) ( p r ) we deduce that x > p r µ ( p r ) x ∈ V. Thus, 4 is isolated in Γ µ ( p r ) ( p r ), against theconnection of Γ µ ( p r ) ( p r ). If µ ( p r ) = 4, then the previous argument applies to p r .If ( h, n ) = (5 , µ = 3 , µ a = 4, µ G = 5 and µ ( p ) , µ ( p r ) ∈ { , , } . If µ ( p ) , µ ( p r ) ∈ { , } ,then by Corollary 22, Γ µ ( p ) ( p ) and Γ µ ( p r ) ( p r ) are both acyclic and we contradict (3), using Lemma18 (iii). If µ ( p ) = 3, then, by Lemma 19, µ ( p r ) ∈ { , } . By Corollary 22, Γ µ ( p r ) ( p r ) is acyclic andthen, by Corollary 15, connected. Assume there exists x ∈ V = { , , , } such that 5 > p r µ ( p r ) x .Then x > pµ ( p r )
5, against 5 ∈ D µ ( p ) ( p ). On the other hand, from 5 ∈ D µ ( p r ) ( p r ) we deduce that x > p r µ ( p r ) x ∈ V. Thus, 5 is isolated in Γ µ ( p r ) ( p r ), against the connection of Γ µ ( p r ) ( p r ). If µ ( p r ) = 3, then the previous argument applies to p r . First of all, let us prove that if ( h, n ) ∈ N ⋄ \ T , then there exists p ∈ P such that D µ ( p ) ( p ) = { } ⊆ D µ ( p r ) ( p r ). If ( h, n ) ∈ N ⋄ \ T , then we can apply Proposition 6. Assume then that ( h, n ) ∈ T \ T and note that T \ T = { ( h, n ) ∈ N ⋄ : h = 3 , n ≥ } ∪ { (5 , , (5 , , (7 , } . If ( h, n ) ∈ N ⋄ is such that h = 3 and n ≥
4, then consider p ∈ P defined by p = [1 , (5) , . . . , ( n ) , , , T , p = [1 , (5) , . . . , ( n ) , , , T , p = [4 , , , ( n ) , . . . , (5) , T Thus, µ ( p ) = 2 and D µ ( p ) ( p ) = { } , while µ ( p r ) = 3 and D µ ( p r ) ( p r ) = N. If ( h, n ) = (5 , p ∈ P defined by Thus, µ ( p ) = 3 and D µ ( p ) ( p ) = { } , while µ ( p r ) = 4 and D µ ( p r ) ( p r ) = { , , } . If ( h, n ) = (5 , p ∈ P defined by µ ( p ) = 3 and D µ ( p ) ( p ) = { } , while µ ( p r ) = 4 and D µ ( p r ) ( p r ) = { , } . If ( h, n ) = (7 , p ∈ P defined by Thus, µ ( p ) = 4 and D µ ( p ) ( p ) = { } , while µ ( p r ) = 5 and D µ ( p r ) ( p r ) = { , , } . Assume now that ( h, n ) ∈ T and prove that it cannot be found p ∈ P and x ∈ N such that D µ ( p ) ( p ) = { x } ⊆ D µ ( p r ) ( p r ). By Lemmata 18(i) and 16(ii), it is enough to show that Γ µ ( p ) ( p )and Γ µ ( p r ) ( p r ) are both acyclic. This comes applying Corollary 22 in all the possible cases. Theapplication is obvious when n ∈ { , } ; if h = 2 note that µ ( p ) = µ ( p r ) = µ a = 2; if ( h, n ) = (4 , µ = µ a = 3 and thus µ ( p ) , µ ( p r ) ≥ . First of all, let us prove that if ( h, n ) ∈ N ⋄ \ T , then there exists p ∈ P such that D µ ( p ) ( p ) = N and D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ . If ( h, n ) ∈ N ⋄ \ T then we can apply Proposition 7. Assume thenthat ( h, n ) ∈ T \ T and note that T \ T = { ( h, n ) ∈ N ⋄ : h = 2 , n ≥ } ∪ { ( h, n ) ∈ N ⋄ : h = 3 , n = 3 } ∪ { (4 , } . If ( h, n ) ∈ N ⋄ is such that h = 2 and n ≥
3, then consider p ∈ P defined by p = [1 , , , . . . , n − , n ] T , p = [ n, , , . . . , n − T . Thus, µ ( p ) = µ ( p r ) = 2 and, since n ≥
3, we have D µ ( p ) ( p ) = { , n } 6 = N . Moreover D µ ( p r ) ( p r ) = { n − , n } , so that D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = { n } . If ( h, n ) ∈ N ⋄ is such that h = 3 and n = 3, then consider the partition of N ⋄ \ { } given by H = { h = 2 + 3 k : k ≥ } , H = { h = 1 + 3 k : k ≥ } , H = { h = 3 + 3 k : k ≥ } . If h ∈ H , thenconsider any p ∈ P such that |{ i ∈ H : p i = [1 , , T }| = 1 + k, |{ i ∈ H : p i = [3 , , T }| = 1 + k, |{ i ∈ H : p i = [2 , , T }| = k. If h ∈ H , consider any p ∈ P such that |{ i ∈ H : p i = [1 , , T }| = k, |{ i ∈ H : p i = [2 , , T }| = k, |{ i ∈ H : p i = [3 , , T }| = k, |{ i ∈ H : p i = [1 , , T }| = 1 . If h ∈ H , consider any p ∈ P such that |{ i ∈ H : p i = [1 , , T }| = k, |{ i ∈ H : p i = [3 , , T }| = k, |{ i ∈ H : p i = [2 , , T }| = k + 1 , |{ i ∈ H : p i = [1 , , T }| = 2 . In all the above situations, it is easily checked that D µ ( p ) ( p ) = { , } 6 = N and D µ ( p r ) ( p r ) = { , } so that D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = { } 6 = ∅ .If ( h, n ) = (4 , p ∈ P defined by µ ( p ) = µ ( p r ) = 3, D µ ( p ) ( p ) = { , , } 6 = N and D µ ( p r ) ( p r ) = { , , } . Assume now that ( h, n ) ∈ T and prove that it cannot be found p ∈ P such that D µ ( p ) ( p ) = N and D µ ( p ) ( p ) ∩ D µ ( p r ) ( p r ) = ∅ .If n = 2, then the condition D µ ( p ) ( p ) = N is equivalent to | D µ ( p ) ( p ) | = 1. Since ( h, n ) ∈ T Proposition 7 applies.Finally let ( h, n ) = (3 , p ∈ P , we have M ( p ) ∩ M ( p r ) = ∅ or M ( p ) = N . Fix p ∈ P and note that µ = 2. Assume first that there exists x ∈ N such that { i ∈ H : p i (1) = x } has at least two elements. By Lemma 25 (i) and Lemma 10, Γ ( p ) is acyclic.Thus µ ( p ) = 2 and M ( p ) = D ( p ) = { } . By Lemma 25 (ii) we also have µ ( p r ) = 2. Since in p r thealternative 1 is beaten by the alternative 2 at least two times, we have that 1 / ∈ D ( p r ) = M ( p r )and so M ( p ) ∩ M ( p r ) = ∅ . If there exists x ∈ N such that { i ∈ H : p i (3) = x } has at least twoelements we apply the argument above to p r , obtaining again M ( p ) ∩ M ( p r ) = ∅ . We are then leftwith assuming that the alternatives ranked first as well as those ranked third are distinct in p . Inthis case, by Lemma 25 (i), Γ ( p ) is a 3-cycle and µ ( p ) = 3. Moreover, the arc set of Γ ( p ) is emptyso that M ( p ) = D ( p ) = N . References
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