Daniela Bubboloni
email: daniela.bubboloni@unifi.it

Mostapha Diss
email: diss@gate.cnrs.fr

Michele Gori
email: michele.gori@unifi.it

Extensions of the Simpson voting rule to the committee selection setting

Keywords: Multiwinner Elections, Committee Selection Rule, Simpson Voting Rule, Paradoxes, Probability JEL codes: D71, D72 Multiwinner Elections, Committee Selection Rule, Simpson Voting Rule, Paradoxes, Probability. JEL Classification Numbers: D71, D72

Committee selection rules are procedures selecting sets of candidates of a given size on the basis of the preferences of the voters. There are in the literature two natural extensions of the well-known singlewinner Simpson voting rule to the multiwinner setting. The first method gives a ranking of candidates according to their minimum number of wins against the other candidates. Then, if a fixed number k of candidates are to be elected, the k best ranked candidates are chosen as the overall winners. The second method gives a ranking of committees according to the minimum number of wins of committee members against committee nonmembers. Accordingly, the committee of size k with the highest score is chosen as the winner. We propose an in-depth analysis of those committee selection rules, assessing and comparing them with respect to several desirable properties among which unanimity, fixed majority, nonimposition, stability, local stability, Condorcet consistency, some kinds of monotonicity, resolvability and consensus committee. We also investigate the probability that the two methods are resolute and suffer the reversal bias, the Condorcet loser paradox and the leaving member paradox. We compare the results obtained with the ones related to further well-known committee selection rules. The probability assumption on which our results are based is the widely used Impartial Anonymous Culture.

Introduction

A voting rule is a procedure for associating with any preference profile, that is a collection of individual preferences expressed as linear orders on a set of candidates, one of the candidates to be interpreted as the winner of the election. There are many situations where instead of choosing a single winner a society is required to select a given number of candidates from the set of available options. This may begin with the choice of representatives in an elementary school class to the choice of national legislators or international representatives of a country (e.g., in the European Parliament). That leads to introduce the concept of committee selection rule (csr), namely a procedure which associates with any preference profile and any positive integer k a subset of candidates of size k. There are many csrs studied in the literature based on different ideas and principles. Most of them are extensions of well-known voting rules to the multiwinner setting, that is, they reduce to those voting rules when k = 1.

According to this approach, [START_REF] Barberà | How to choose a non-controversial list with k names[END_REF] take into consideration an extension of the Simpson voting rule,1 called here the Simpson csr and denoted by S. Recall that, given a preference profile, the Simpson voting rule first associates with any candidate x the number of voters preferring x to y for all other candidate y. The minimum of these quantities is called the Simpson score of x. Next, a candidate is selected if she has the highest Simpson score. If now k is the target size of the committee to be elected, a set W of candidates of size k is selected by S if the Simpson score of each candidate in W is at least as great as the one of each nonmember of W . [START_REF] Barberà | How to choose a non-controversial list with k names[END_REF] show that, even though the Simpson voting rule is Condorcet consistent, S is not stable, that is, it may fail to select weak k-Condorcet sets when k ≥ 2. Recall that, following [START_REF] Gehrlein | The Condorcet criterion and committee selection[END_REF], a subset of candidates of size k is called a weak k-Condorcet set if no candidate in this subset can be defeated by any candidate from outside the subset on the basis of pairwise majority comparisons. Obviously, this concept is a generalization to the committee selection setting of the well-known concept of the weak Condorcet winner. We further notice that S presents much more severe distortions. Assume, for instance, that only three candidates x, y and z are considered and that each voter has preferences given by x ≻ y ≻ z. If we focus on committees having size k = 2, S selects the two sets {x, y} and {x, z}, violating a very natural unanimity principle which should admit {x, y} as the unique selected set. [START_REF] Coelho | Understanding, evaluating and selecting voting rules through games and axioms[END_REF] proposes another possible extension of the Simpson voting rule to the committee selection setting,2 called the Minimal Size of External Opposition csr and here denoted by M. Given a preference profile, the method associates with any nonempty set W a score, called the Maximin score of W , computed as follows: for every x in W and y not in W the number of voters preferring x to y is considered and then the minimum of these numbers is taken. Once the size k of the committee is fixed, M selects the sets of candidates which obtain the greatest Maximin score among the ones of size k. The main difference between the two extensions lies in the fact that S determines its outcome on the basis of a ranking of candidates while the outcome of M is directly obtained by means of a ranking of committees. Notice that M selects the subset {x, y} as the unique winning committee in the above considered example. This depends on the fact that the Maximin score of {x, y} equals the total number of voters, while the Maximin score of the other two possible subsets of size two is zero.

The first contribution of this paper is an in-depth analysis of the properties of the two csr S and M. We mainly rely on the papers of Kamwa (2017a,b) as well as the paper of [START_REF] Elkind | Properties of multiwinner voting rules[END_REF]. In Kamwa (2017a) some of the properties considered in our paper are already studied for M and other stable csrs which can be considered as possible extensions of well-known voting rules to the committee selection setting. Such a paper is extended by Kamwa (2017b) who compares the considered csrs when the candidates are three and the size of the committee to be elected is two on the basis of what the author calls the 'divergence on outcomes' defined as the situation where a committee selected by a csr differs from the committee made by the best candidates of the corresponding voting rule from which the csr is adapted. [START_REF] Elkind | Properties of multiwinner voting rules[END_REF] represents a preliminary attempt to develop a formal framework for the study of the characteristics of csrs when the voters' preferences are linear orders, where several properties are introduced and discussed. We take into account all the properties considered in Kamwa (2017a,b) and [START_REF] Elkind | Properties of multiwinner voting rules[END_REF] and, in addition, we formulate further desirable properties against which the two csrs S and M are judged. Table 1 in Section 3 provides a summary of our results. It is worth noting that, in [START_REF] Elkind | Properties of multiwinner voting rules[END_REF], two families of csrs were introduced to describe many classic csrs in a unified framework: the best-k rules and the committee scoring rules (see Section 3.2 in [START_REF] Elkind | Properties of multiwinner voting rules[END_REF]. It can be easily seen that S is a best-k rule, while it is not a committee scoring rule (Proposition 17). Even though those families surely cover a lot of interesting csrs, there is room for other kinds of rules based on a quite different approach and which do not fall into one of those categories. Indeed, M is neither a best-k rule nor a committee scoring rule (Propositions 11 and 17).

In the course of our investigations, we also observe that both S and M fail to be resolute and immune to the reversal bias and suffer the Condorcet loser and the leaving member paradoxes. We compute then the probability of these particular situations and we compare the results with those obtained for the Borda, the Plurality, the Negative Plurality and the Bloc csrs. The probability assumption on which our results are based is the widely used Impartial Anonymous Culture (IAC). We are also concerned about the probability that S and M agree on the same committee as well as the probability of the agreement between each of them and the Borda csr. This identifies the second contribution of this paper. Our results show distinct performances of the csrs according to the number of candidates, the numbers of voters and the target size of the committee. In many cases, M does not have the highest level of performance, but in general it behaves better than S.

The paper is structured as follows. Section 2 is devoted to basic notations and definitions. Section 3 presents our results about the properties of S and M. Section 4 presents the computational probabilistic analysis related to S and M and the other considered csrs.

Definitions and notations

We assume 0 ∈ N and we set N 0 = N ∪ {0}. Let k ∈ N and X be a finite set. We denote by [k] the set {1, . . . , k} and by |X| the size of X. Moreover, we call a subset of X of size k a k-subset of X and denote by 2 X k the set of the k-subsets of X. If |X| = m ≥ 1, we denote by L(X) the set of linear orders on X. Every q ∈ L(X) is represented by a writing of the type x 1 ≻ q x 2 ≻ q . . . ≻ q x m (the subscript q is generally omitted) or by a column vector [x 1 , . . . , x m ] T , where x 1 , . . . , x m are the distinct elements of X.

Consider a countably infinite set C whose elements are to be interpreted as potential candidates and a countably infinite set V whose elements are to be interpreted as potential voters. For simplicity, we identify C and V with N. An election is a vector (C, V, p), where C is a finite subset of C having at least two elements, V is a nonempty and finite subset of V, and p is a function from V to L(C), the set of linear orders on C. Given an election (C, V, p), p(v) is interpreted as the preference relation of voter v on the set of candidates C. For x, y ∈ C with x = y, x ≻ p(v) y means that x is strictly preferred to y in p(v). The function p is called the preference profile of the election. When V = [n], p can be represented by a matrix whose v-th column is p(v) for all v ∈ V .

The set of elections is denoted by E. A committee selection rule (csr) R is a function which associates with every vector (C, V, p) ∈ E and k ∈

[|C| -1], a nonempty set of k-subsets of C. Let (C, V, p) ∈ E and x ∈ C. The Simpson score of x is defined by S(C, V, p, x) = min y =x c p (x, y),
where, for every x, y ∈ C with x = y,

c p (x, y) = |{v ∈ V : x ≻ p(v) y}|.
Following [START_REF] Barberà | How to choose a non-controversial list with k names[END_REF], the Simpson csr S is defined, for every (C,

V, p) ∈ E and k ∈ [|C| -1], by S(C, V, p, k) = W ∈ 2 C k : ∀x ∈ W, ∀y ∈ W, S(C, V, p, x) ≥ S(C, V, p, y) .
The Simpson csr restricted to the case k = 1 agrees, by definition, with the classical Simpson voting rule. Note that, according to Elkind et al. (2017, Definition 1), the Simpson csr is the best-k rule associated with the social preference function naturally induced by the Simpson score. That observation allows to obtain some properties for S as application of general results about best-k rules proved in [START_REF] Elkind | Properties of multiwinner voting rules[END_REF].

Note also that the relation on C defined as

R S (C, V, p) = {(x, y) ∈ C 2 : S(C, V, p, x) ≥ S(C, V, p, y)}
is an order and that S(C, V, p, k) = C k (R S (C, V, p)) within the notation in Bubboloni and Gori (2018, Section 3). That observation also allows to obtain some properties for S as an application of results proved in [START_REF] Bubboloni | The flow network method[END_REF]. [START_REF] Coelho | Understanding, evaluating and selecting voting rules through games and axioms[END_REF] proposes another csr which agrees with the Simpson voting rule when k = 1. Here we introduce it via an equivalent definition. Given (C, V, p) ∈ E and

W ⊆ C with |W | ∈ [|C| -1], the Maximin score of W is defined by M (C, V, p, W ) = min x∈W,y ∈W c p (x, y). The Minimal Size of External Opposition csr M is defined, for every (C, V, p) ∈ E and k ∈ [|C| -1], by 3 M(C, V, p, k) = arg max W ∈2 C k M (C, V, p, W ).
We stress that, even though S and M agree when k = 1, that is not true in general for other values of k.

Indeed, considering (C, V, p) ∈ E with C = [4], V = [9] and p =     1 1 1 1 3 3 4 4 4 2 2 2 4 1 2 1 1 2 4 4 4 2 2 4 2 2 3 3 3 3 3 4 1 3 3 1     ,
we have that M(C, V, p, 2) = {{1, 2}} and S(C, V, p, 2) = {{1, 4}}. As we are going to show in the next section, S and M present important differences also in terms of satisfied properties.

In what follows, when the reference to the set of candidates C and voters V is clear, we usually omit those symbols. For instance, the writing S(C, V, p, x) simply becomes S(p, x).

An analysis of the Simpson and the Maximin csrs

Anonymity, neutrality and homogeneity are well-known properties a csr may meet. The proof of the following result is straightforward and thus omitted.

Proposition 1. S and M are anonymous, neutral and homogeneous.

Let (C, V, p) ∈ E, k ∈ [|C| -1] and W ∈ 2 C k . Following Gehrlein (1985), we say that W is a weak k-Condorcet set if, for every x ∈ W and y ∈ C \ W , c p (x, y) ≥ |V | 2 , and that W is a k-Condorcet set if, for every x ∈ W and y ∈ C \ W , c p (x, y) > |V | 2 .
3 In [START_REF] Coelho | Understanding, evaluating and selecting voting rules through games and axioms[END_REF] M is denoted by SEO and it is defined as

M(C, V, p, k) = arg min W ∈2 C k max y ∈W,x∈W cp(y, x). Clearly W is a weak k-Condorcet set if and only if M (C, V, p, W ) ≥ |V | 2 ; W is a k-Condorcet set if and only if M (C, V, p, W ) > |V | 2 ; if W is a k-Condorcet set, then it is a weak k-Condorcet set.
Moreover, it is easily checked that if a k-Condorcet set exists, then it is unique. On the other hand, it is possible to have more than one weak k-Condorcet set.

A csr R is called stable if, for every (C, V, p) ∈ E and k ∈ [|C| -1] such that there exists a weak k-Condorcet set for (C, V, p), we have that every

W ∈ R(C, V, p, k) is a weak k-Condorcet set; 4 R is Condorcet consistent if, for every (C, V, p) ∈ E and k ∈ [|C| -1] such that there exists a k-Condorcet set W for (C, V, p), we have that R(C, V, p, k) = {W }; 5 R satisfies fixed majority (resp. strong unanimity) if, for every (C, V, p) ∈ E, k ∈ [|C| -1] and W ∈ 2 C
k such that a strict majority of the voters (resp. all the voters) rank all the members of W above all the non-members of W , we have that R(C, V, p, k) = {W };6 R satisfies weak unanimity if, for every (C,

V, p) ∈ E, k ∈ [|C| -1] and W ∈ 2 C
k such that all the voters rank all the members of W above all the non-members of W , we have that

W ∈ R(C, V, p, k); 6 R satisfies strong non-imposition if, for every C ⊆ C and V ⊆ V finite sets with |C| ≥ 2, |V | ≥ 1, k ∈ [|C| -1] and W ∈ 2 C k , there exists (C, V, p) ∈ E such that R(C, V, p, k) = {W }; R satisfies non-imposition if, for every finite set C ⊆ C with |C| ≥ 2, k ∈ [|C| -1] and W ∈ 2 C k , there exists (C, V, p) ∈ E such that R(C, V, p, k) = {W }.
It is easily proved that if R is stable, then it is Condorcet consistent. Indeed, the existence of a k-Condorcet set W for (C, V, p) ∈ E implies that the set of the weak k-Condorcet sets for (C, V, p) is {W }. Then the following chain of implications holds true:

Stability ⇒ Condorcet Consistency ⇒ Fixed Majority ⇒ Strong Unanimity ⇒ Weak Unanimity
(1) Note also that strong unanimity implies strong non-imposition. The next proposition shows not only that S fails stability, as shown by Barberà and Coelho (2008, Proposition 3), but that it even fails strong unanimity. On the contrary, M is stable. This is a crucial difference between the two csrs.

Proposition 2. S is not strongly unanimous. In particular, S does not satisfy fixed majority, is not Condorcet consistent and is not stable.

Proof. Consider (C, V, p) with C = {1, 2, 3} and 1 ≻ p(v) 2 ≻ p(v) 3 for all v ∈ V and k = 2. The strong unanimity principle implies that {1, 2} is the only selected committee. However, S(p, 1) = |V |, S(p, 2) = 0, S(p, 3) = 0 so that S(p, 2) = {{1, 2}, {1, 3}}.

Proposition 3. S is weakly unanimous, satisfies non-imposition and does not satisfy strong nonimposition.

Proof. Let (C, V, p) ∈ E and k ∈ [|C| -1]. Assume that W ∈ 2 C
k is such that all the voters rank all the members of W above all the non-members of W . We want to show that W ∈ S(p, k), that is, for every x ∈ W and every y ∈ C \ W , we have S(p, x) ≥ S(p, y). Fix then x ∈ W and y ∈ C \ W . Then S(p, y) = min z =y c p (y, z) ≤ c p (y, x) = 0. Thus, trivially, S(p, x) ≥ S(p, y) = 0. This shows that S is weakly unanimous.

In order to show non-imposition, let C be a finite set with

|C| = m ≥ 2, k ∈ [|C| -1] and W * ∈ 2 C k . Let W * = {x 1 , . . . , x k } and C \ W * = {y k+1 , . . . , y m }. If k = 1
, consider just a single voter having x 1 ranked first. Assume next that k ≥ 2. In this case, consider the election (C, [2], p), where

x 1 ≻ p1 x 2 ≻ p1 . . . ≻ p1 x k ≻ p1 y k+1 ≻ p1 . . . ≻ p1 y m and x k ≻ p2 x k-1 ≻ p2 . . . ≻ p2 x 1 ≻ p2 y k+1 ≻ p2 . . . ≻ p2 y m .
Then we have S(p, x i ) = 1 for all i ∈ [k], and S(p, y i ) = 1 for all i ∈ {k + 1, . . . , m}. Thus S(p, k) = {W * }.

Finally note that S does not satisfy strong non-imposition because the selection of two alternatives when a single voter is considered never leads to a unique outcome.

The next proposition shows, among other things, that M is stable. We note that [START_REF] Coelho | Understanding, evaluating and selecting voting rules through games and axioms[END_REF] just introduces M as an example of stable csr.

Proposition 4. M is stable. In particular, M is Condorcet consistent, satisfies fixed majority, strong unanimity, weak unanimity and strong non-imposition.

Proof. Let (C, V, p) ∈ E and k ∈ [|C| -1] be such that there exists a weak k-Condorcet set, say W * . We know that,

M (p, W * ) ≥ |V | 2 . Let now W ∈ M(p, k) and prove that W is a weak k-Condorcet set. Indeed, assume by contradiction that W is not a weak k-Condorcet set. Then M (p, W ) < |V | 2 , so that M (p, W ) < M (p, W * ) and W ∈ M(p, k), a contradiction.
Since M is stable, applying the chain of implications (1), all the remaining facts are immediately established. Kamwa (2017a) proposes the next property which refers to the behaviour of a csr when a Condorcet winner exists. A csr R satisfies the Condorcet winner criterion if, for every (C,

V, p) ∈ E, k ∈ [|C| -1], W ∈ R(C, V, p, k) and x ∈ C a Condorcet winner, we have that x ∈ W .
Proposition 5. S satisfies the Condorcet winner criterion.

Proof. Consider (C, V, p) ∈ E, k ∈ [|C|-1] and W ∈ S(C, V, p, k). Assume that x ∈ C is a Condorcet winner. Then S(p, x) > |V |
2 . Moreover, for every y ∈ C \ {x}, we have S(p, y) ≤ c p (y, x) < |V | 2 so that S(p, y) < S(p, x). Assume now that x ∈ W and pick y ∈ W . Then S(p, y) ≥ S(p, x), a contradiction Proposition 6. M does not satisfy the Condorcet winner criterion.

Proof. See Kamwa (2017a, Example 3).

Following [START_REF] Aziz | The Condorcet principle for multiwinner election: from short listing to proportionality[END_REF], given (C, V, p) ∈ E, k ∈ [|C| -1] and W ∈ 2 C k , we say that W is a k-locally stable set with respect to the Droop quota if, for every y ∈ C \ W , we have that

|{v ∈ V : ∀x ∈ W, y ≻ p(v) x}| ≤ |V | k + 1 .
We say then that a csr R is locally stable if, for every (C, V, p) ∈ E and k ∈ [|C| -1] such that there exists a k-locally stable set, we have that every W ∈ R(C, V, p, k) is a k-locally stable set.

Proposition 7. There exists no csr R which is stable and locally stable.

Proof. Assume by contradiction that R is a stable and locally stable csr and consider

C = [4], V = [7] and p =     1 2 3 2 1 3 1 3 3 4 3 3 4 3 4 4 1 4 4 2 4 2 1 2 1 2 1 2     Then {3,
p =     1 2 2 2 3 4 4 4 3 1 1 3 1 1 2 3 4 3 3 4 4 2 3 1 2 4 4 1 2 3 1 2     (2)
It is immediately checked that {2, 4} is the unique 2-locally stable set while S(C, V, p, 2) = {{1, 2}, {1, 3}, {1, 4}}.

Following [START_REF] Elkind | Properties of multiwinner voting rules[END_REF], we say that a csr R satisfies committee monotonicity if, for every (C, V, p) ∈ E with |C| ≥ 3 and k ∈ [|C| -2], we have that:

(1) if W ∈ R(C, V, p, k), then there exists W ′ ∈ R(C, V, p, k + 1) such that W ⊆ W ′ , (2) if W ∈ R(C, V, p, k + 1), then there exists W ′ ∈ R(C, V, p, k) such that W ′ ⊆ W .
By Theorem 2 in [START_REF] Elkind | Properties of multiwinner voting rules[END_REF], a best-k rule is committee monotone. As a consequence, we immediately get the following result.

Proposition 9. S satisfies committee monotonicity.

As shown by [START_REF] Barberà | How to choose a non-controversial list with k names[END_REF], stability and committee monotonicity cannot co-exist. In particular, from the example proposed in the proof of Proposition 8 in [START_REF] Barberà | How to choose a non-controversial list with k names[END_REF], we get the next result.

Proposition 10. M fails both condition (1) and condition (2) of committee monotonicity. In particular, M fails committee monotonicity.

As pointed out by [START_REF] Elkind | Properties of multiwinner voting rules[END_REF], the fact that committee monotonicity fails is not a matter of concern in itself, because the desirability of that property strongly depends on the application. The above negative result allows to shed light onto the nature of M.

Proposition 11. M is not a best-k rule.

Proof. By Theorem 2 in [START_REF] Elkind | Properties of multiwinner voting rules[END_REF], every best-k rule is committee-monotone. Since, by Proposition 10, M fails committee monotonicity, we deduce that M is not a best-k rule.

Let us consider now three further kinds of monotonicity. A csr R is said to satisfy candidate monotonicity (resp. non-crossing monotonicity; membership monotonicity) if, for every (C,

V, p) ∈ E, k ∈ [|C| -1], W * ∈ R(C, V, p, k), x * ∈ W * ,
and p ′ obtained by p by shifting x * one position forward in the preference relation of some voter by swapping its position with some x ∈ C, we have that there exists

W ′ ∈ R(C, V, p ′ , k) such that x * ∈ W ′ (resp. W * ∈ R(C, V, p ′ , k) provided that x ∈ W * ; W * ∈ R(C, V, p ′ , k)).
Notice that candidate monotonicity and non-crossing monotonicity are proposed in [START_REF] Elkind | Properties of multiwinner voting rules[END_REF], while membership monotonicity in Kamwa (2017a). Note also that membership monotonicity implies non-crossing monotonicity.

In order to study these three properties for S and M, we need to observe that, referring to the notation used in the above definition, for every W ∈ 2 C k , the following facts hold true:

c p ′ (x * , x) = c p (x * , x) + 1, (3)
c p ′ (x, x * ) = c p (x, x * ) -1, (4) if x, y ∈ C with x = y and {x, y} = {x * , x}, then c p ′ (x, y) = c p (x, y), (5)
S(p ′ , x) ∈ {S(p, x), S(p, x) -1}, (6)
S(p ′ , x * ) ∈ {S(p, x * ), S(p, x * ) + 1}, (7) if x ∈ C \ {x * , x}, then S(p ′ , x) = S(p, x), ( 8
) if {x * , x} ⊆ W or {x * , x} ⊆ C \ W, then M (p ′ , W ) = M (p, W ), (9) if x * ∈ W, then M (p ′ , W ) ≥ M (p, W ), (10) if x * / ∈ W, then M (p ′ , W ) ≤ M (p, W ). ( 11
)
The proofs of ( 3)-( 11) are elementary and thus omitted. Our next propositions show that S satisfies more monotonicity properties than M. This result is quite interesting. As noticed by [START_REF] Elkind | Properties of multiwinner voting rules[END_REF] and [START_REF] Barberà | How to choose a non-controversial list with k names[END_REF], monotonicity should be a desirable requirement in many situations like, for instance, when the finalists of a competition are to be chosen or when some applicants competing for a position at a university are to be short-listed for an interview. We believe that this expresses one of the main values of S with regards to M.

Proposition 12. S satisfies non-crossing monotonicity and candidate monotonicity.

Proof. Let (C, V, p) ∈ E, k ∈ [|C| -1], W * ∈ S(p, k), x * ∈ W * ,
and p ′ obtained by p by shifting x * one position forward in the preference relation of some voter by swapping its position with some x ∈ C.

We first show non-crossing monotonicity. Assume then that x / ∈ W * . We want to show that W * ∈ S(p ′ , k). Let x ∈ W * and y ∈ W * . Then, since x = x and y = x * , by ( 6)-( 8), we get

S(p ′ , x) ≥ S(p, x) ≥ S(p, y) ≥ S(p ′ , y).
We next show candidate monotonicity. This time we may have x / ∈ W * or x ∈ W * and we need to exhibit

W ′ ∈ S(p ′ , k) with x * ∈ W ′ . If x / ∈ W * , by non-crossing monotonicity, we know that W * ∈ S(p ′ , k) and thus it is enough to choose W ′ = W * . Let now x ∈ W * . Then, since W * ∈ S(p, k), we have S(p, x) ≥ S(p, y) for all y ∈ C \ W * .
Assume first that, for every y ∈ C \ W * , we have

S(p, x) > S(p, y). (12)
We show that, in this case, W * ∈ S(p ′ , k). Let x ∈ W * and y ∈ C \ W * . Then y / ∈ {x * , x} and thus, by (8), we have S(p, y) = S(p ′ , y). If x = x, using (7), ( 8) and the fact that W * ∈ S(p, k), we get

S(p ′ , x) ≥ S(p, x) ≥ S(p, y) = S(p ′ , y).
Moreover, by (6) and by ( 12), we also obtain

S(p ′ , x) ≥ S(p, x) -1 ≥ S(p, y) = S(p ′ , y).

Suppose next that there exists y

′ ∈ C \ W * such that S(p, x) = S(p, y ′ ). ( 13
) Note that y ′ / ∈ {x * , x}. Define W ′ = (W * \ {x}) ∪ {y ′ } ∈ 2 C k . Note that x * ∈ W ′ and that C \ W ′ = [C \ (W * ∪ {y ′ })] ∪ {x}. We show that W ′ ∈ S(p ′ , k). Let x ∈ W ′ and y ∈ C \ W ′ . Then y = x * . Consider first the case x = y ′ . Then x ∈ W * \ {x}. If y = x, we have that y / ∈ {x * , x} and y ∈ C \ W * .
Thus, using ( 7), ( 8) and the fact that 6)-( 8), ( 13) and the fact that W * ∈ S(p, k), we have

W * ∈ S(p, k), we get S(p ′ , x) ≥ S(p, x) ≥ S(p, y) = S(p ′ , y). If instead y = x, using (
S(p ′ , x) ≥ S(p, x) ≥ S(p, y ′ ) = S(p, x) ≥ S(p ′ , x). It remains to treat the case x = y ′ , showing that S(p ′ , y ′ ) ≥ S(p ′ , y). If y = x, then y ∈ C \ W * .
Thus, recalling that y ′ / ∈ {x * , x} and using ( 8), ( 13) and the fact that W * ∈ S(p, k), we have

S(p ′ , y ′ ) = S(p, y ′ ) = S(p, x) ≥ S(p, y) = S(p ′ , y).
Finally note that, by ( 6), ( 8) and ( 13), we also have

S(p ′ , y ′ ) = S(p, y ′ ) = S(p, x) ≥ S(p ′ , x).
Proposition 13. S does not satisfy membership monotonicity.

Proof.

Consider C = [3], V = [3] and p =   1 2 3 2 3 2 3 1 1   Then {1, 2} ∈ S(p, 2) = {{1, 2}, {2, 3}}. If now 2 ∈ {1, 2} is swapped with 1 in the preferences of voter 1, then {1, 2} ∈ S(p ′ , 2) = {{2, 3}}.
Kamwa (2017a) proved that M does not satisfy membership monotonicity and that it satisfies candidate monotonicity. In Proposition 14 we show that also non-crossing monotonicity, a weaker property than membership monotonicity, fails. For the sake of completeness, we prove in Proposition 15 that M is candidate monotonic.

Proposition 14. M does not satisfy non-crossing monotonicity. In particular, it does not satisfy membership monotonicity. 4 1 1 1 2 2 2 1 4 3 3 3 3 4 3 3 4 4 4 4 3 2 2 2 2 1 1

Proof. Consider C = [4], V = [7] and p =    
1     Then {1, 2} ∈ M(p, 2) = 2 C 2 . If now 1 ∈ {1, 2} is swapped with 4 ∈ {1, 2} in the preferences of voter 1, then {1, 2} ∈ M(p ′ , 2) = {{1, 3}}. Proposition 15. M satisfies candidate monotonicity. Proof. Let (C, V, p) ∈ E, k ∈ [|C| -1], W * ∈ M(p, k), x * ∈ W * ,
and p ′ obtained by p by shifting x * one position forward in the preference relation of some voter by swapping its position with some x ∈ C. By ( 9)-( 11), we have that for every

W ∈ 2 C k such that x * ∈ W , we have M (p ′ , W ) ≥ M (p, W ); for every W ∈ 2 C k such that x * ∈ W , we have M (p ′ , W ) ≤ M (p, W
). Thus, using the fact that W * ∈ M(p, k), we obtain that, for every

W ∈ 2 C k with x * ∈ W, M (p ′ , W * ) ≥ M (p, W * ) ≥ M (p, W ) ≥ M (p ′ , W ).
Assume now, by contradiction, that k). By what shown above we have then that

x * / ∈ W ′ for all W ′ ∈ M(p ′ , k). Pick W ′ ∈ M(p ′ ,
M (p ′ , W * ) ≥ M (p ′ , W ′ ) ≥ M (p ′ , W * ). Thus M (p ′ , W * ) = M (p ′ , W ′ ), which says that W * ∈ M(p ′ , k) against the fact that x * ∈ W * .
Following [START_REF] Elkind | Properties of multiwinner voting rules[END_REF] and Kamwa (2017a), we say that R satisfies consistency (resp. weak consistency) if, for every pair of elections

E 1 = (C, V 1 , p 1 ) and E 2 = (C, V 2 , p 2 ) over a common set C of candidates and with V 1 ∩ V 2 = ∅ and every k ∈ [|C| -1], if R(E 1 , k) ∩ R(E 2 , k) = ∅, then R(E 1 , k) ∩ R(E 2 , k) = R(E 1 + E 2 , k) (resp. R(E 1 , k) ∩ R(E 2 , k) ⊆ R(E 1 + E 2 , k)), where E 1 + E 2 denotes the election (C, V 1 ∪ V 2 , p 1 + p 2 ) with p 1 + p 2 defined as the preference profile given by (p 1 + p 2 )(v) = p 1 (v) for all v ∈ V 1 and (p 1 + p 2 )(v) = p 2 (v) for all v ∈ V 2 .
Of course, consistency implies weak consistency. Kamwa (2017a, Theorem 6) proves that M fails consistency, even though from the proof we deduce that the weak consistency property fails too. Below we show, using a same example, that both S and M do not satisfy weak consistency.

Proposition 16. S and M fail weak consistency. In particular, they fail consistency. 3 1 2 3 1 2 4 4 4 4 4 2 3 1 2 3 1 3 1 2 2 1 1 2 3 1 2 3 2 3 1 3 3 4 4 4 4 4 4 1 2 3 1 1 2 3 1 2 3 4 4 4 4 4 2 3 1 2 3 1 1 2 3 1 2 3 1 2 3 1 2 2 3 1 3 3 4 4 4 4 4 4 3 1 2 2

Proof. Consider C = [4], V 1 = {1, . . . , 11} and V 2 = {12, . . . , 22}. Consider then p 1 =    
2     (14) p 2 =    
1     (15)
where, for every v ∈ {1, . . . , 11}, p 1 (v) is the v-th column of p 1 and, for every v ∈ {12, . . . , 22},

p 2 (v) is the (v -11)-th column of p 2 . Then S(C, V 1 , p 1 , 1) = M(C, V 1 , p 1 , 1) = S(C, V 2 , p 2 , 1) = M(C, V 2 , p 2 , 1) = {{4}} while S(C, V 1 ∪V 2 , p 1 +p 2 , 1) = M(C, V 1 ∪V 2 , p 1 +p 2 , 1) = {{1}, {2}, {3}}.
The nature of S and M can be now better enlightened.

Proposition 17. S and M are not committee scoring rules.

Proof. By Theorem 7 in [START_REF] Elkind | Properties of multiwinner voting rules[END_REF], every committee scoring rule satisfies consistency. Since, by Proposition 16, S and M fail consistency, we deduce that S and M are not committee scoring rules.

We say that R satisfies resolvability (resp. weak resolvability) if, for every (C,

V, p) ∈ E, k ∈ [|C| -1], W * ∈ R(C, V, p, k) and q * ∈ L(C) such that, for every x ∈ W * and y ∈ C \ W * , x ≻ q * y, we have that if v * ∈ V \ V and (C, V * , p * ) ∈ E is defined by V * = V ∪ {v * }, p * (v) = p(v) for all v ∈ V and p * (v * ) = q * , then R(C, V * , p * , k) = {W * } (resp. W * ∈ R(C, V * , p * , k)).
Of course, resolvability implies weak resolvability. Note that we are adapting here the standard notion of resolvability for single-winner voting rules (see, for instance, [START_REF] Schulze | A new monotonic, clone-independent, reversal symmetric, and Condorcetconsistent single-winner election method[END_REF][START_REF] Tideman | Independence of clones as a criterion for voting rules[END_REF][START_REF] Tideman | Collective decisions and voting: The potential for public choice[END_REF] to the committee selection setting.

Proposition 18. S does not satisfy weak resolvability. In particular, it does not satisfy resolvability.

Proof. Let p be the preference profile considered in Proposition 14. It is easily checked that S(C, V, p, x) = 3 for all

x ∈ C = [4]. Thus {2, 3} ∈ S(C, V, p, 2) = 2 C
2 . Let q * be the linear order 3 ≻ 2 ≻ 1 ≻ 4. Since S(C, V * , p * , 2) = {{1, 3}}, we see that {2, 3} ∈ S(C, V * , p * , 2). This shows that S does not satisfy weak resolvability.

Proposition 19. M satisfies weak resolvability and does not satisfy resolvability.

Proof. We show that, for every W ∈ 2 C k , we have M (C, V * , p * , W ) ≤ M (C, V * , p * , W * ). Note that, for every x, y ∈ C with x = y, we have c p * (x, y) ≤ c p (x, y)+1. Thus, we also have M (C, V * , p * , W ) ≤ M (C, V, p, W )+1. On the other hand, for every x ∈ W * and y ∈ C \W * we have c p * (x, y) = c p (x, y)+ 1 and therefore

M (C, V, p, W * ) + 1 = M (C, V * , p * , W * ). Since M (C, V, p, W ) ≤ M (C, V, p, W * ), we then get M (C, V * , p * , W ) ≤ M (C, V, p, W ) + 1 ≤ M (C, V, p, W * ) + 1 = M (C, V * , p * , W * ).
We now investigate resolvability. Let p be the preference profile considered in Proposition 14. Let {1, 2} ∈ M(C, V, p, 2) = 2 C 2 and let q * be the linear order 2 ≻ 1 ≻ 3 ≻ 4. Then {1, 2} is not the only element in M(C, V * , p * , 2) because, for instance, we also have {1, 3} ∈ M(C, V * , p * , 2). Thus M does not satisfy resolvability.

Note that, since M satisfies the property of fixed majority, if one keeps adding voters whose preferences are described by a linear order q * ∈ L(C) such that, for every x ∈ W * and y ∈ W * , x * ≻ q * y * , then in a finite number of steps we reach the goal of having W * as the only outcome of M. As it can be immediately understood by the proof of Proposition 2, this does not hold true, in general, for S.

According to Kamwa (2017a), we say that a csr R satisfies the Pareto criterion (resp. weak Pareto criterion) if, for every (C,

V, p) ∈ E, k ∈ [|C| -1], W * ∈ R(C, V, p, k) and x * , y * ∈ C with x * = y * and c p (x * , y * ) = |V |, having y * ∈ W * implies x * ∈ W * (resp. x * ∈ W ′ for some W ′ ∈ R(C, V, p, k)).
Note that if k = 1 the definition means that if y * is unanimously beaten by x * then it cannot be selected (resp. it cannot be the unique winner). Clearly if R satisfies the Pareto criterion, then it satisfies the weak Pareto criterion. Here we study those properties for S and M. Notice that the Pareto criterion has already been proved for M in Kamwa (2017a). Anyway, for the sake of completeness, we also provide the proof of that fact below.

Proposition 20. S satisfies the weak Pareto criterion but does not satisfy the Pareto criterion.

Proof. We first show that S satisfies the weak Pareto criterion.

Let (C, V, p) ∈ E, k ∈ [|C| -1], W * ∈ S(C, V, p, k), y * ∈ W * and x * ∈ C, with x * = y * , such that c p (x * , y * ) = |V |. We need to find W ′ ∈ S(C, V, p, k) such that x * ∈ W ′ . If x * ∈ W * we simply take W ′ = W * . Assume instead that x * / ∈ W * . Note that c p (x * , y * ) = |V | implies c p (y * , x * ) =
∈ C \ W * . Define then W ′ = (W * \ {y * }) ∪ {x * }. We show that W ′ ∈ S(C, V, p, k). Let z ∈ W ′ and t ∈ C \ W ′ . Since C \ W ′ = [C \ (W * ∪ {x * })] ∪ {y * }, we have that t ∈ C \ W * or that t = y * .
In both cases, we know that S(p, t) = 0 and thus, trivially, S(p, z) ≥ S(p, t).

In order to see that S does not satisfy the Pareto criterion, consider (C, V, p) with C = [3], k = 2 and assume that 1 ≻ p(v) 2 ≻ p(v) 3 for all v ∈ V . Then {1, 3} ∈ S(p, 2). Now we have that c p (2, 3) = |V |, and 3 ∈ {1, 3}. However 2 ∈ {1, 3}.

Proposition 21. M satisfies the Pareto criterion. In particular, M satisfies the weak Pareto criterion.

Proof. Assume, by contradiction, that there exists (C,

V, p) ∈ E, k ∈ [|C| -1], W * ∈ M(p, k) and x * , y * ∈ C with x * = y * such that c p (x * , y * ) = |V |, y * ∈ W * and x * ∈ W * . Since c p (y * , x * ) = 0, we have that M (p, W * ) = 0. Consider now v * ∈ V and W ∈ 2 C
k such that, for every x ∈ W and y ∈ C \ W , x ≻ p(v * ) y. Then, for every x ∈ W and y ∈ C \ W , we have c p (x, y) > 0 and hence M (p, W ) > 0 so that W * ∈ M(p, k) and the contradiction is found.

Let E = (C, V, p) ∈ E and R be a csr. Given k ∈ [|C| -1], we say that R is k-flat on E if R(C, V, p, k) = 2 C k . We say that R is flat on E if it is k-flat on E for all k ∈ [|C| -1]. Moreover, R is called symmetric if it is flat on every (C, V, p
) ∈ E such that c p (x, y) = c p (y, x) for all x, y ∈ C, with x = y. Note that a preference profile p satisfying the above symmetry request on c p is given, for instance, by any p such that p and p r agrees up to a reordering of the names of the voters. The definitions of flatness and symmetry here considered are inspired by the properties by the same name introduced by [START_REF] González-Díaz | Paired comparisons analysis: an axiomatic approach of ranking methods[END_REF] in the framework of ranking methods.

We stress that being often flat is considered a defect because if R is flat on an election, then it cannot be used to make an effective decision. On the other hand, being flat on the elections with profiles satisfying c p (x, y) = c p (y, x) is considered a value. Indeed if the voters collectively do not discriminate between any couple of candidates, it is natural to require that every k-set of candidates must occur as outcome of the voting procedure.

We illustrate now the quite different behaviour of S and M with respect to flatness and show that they are both symmetric.

Proposition 22. Let (C, V, p) ∈ E. The following facts are equivalent:

(i) S is flat on (C, V, p); (ii) S is k-flat on (C, V, p) for some k ∈ [|C| -1]; (iii) the function S(C, V, p, •) : C → N 0 is constant. Proof. (i) ⇒ (ii) is obvious.
(ii) ⇒ (iii) By assumption, we have that S(p, k) = 2 C k . Since this can be interpreted as

C k (R S (C, V, p)) = 2 C
k , we can apply Proposition 26 in [START_REF] Bubboloni | The flow network method[END_REF] and deduce that

R S (C, V, p) = C 2 , that is, S(C, V, p, •) is constant. (iii) ⇒ (i) is trivial. Proposition 23. Let E = (C, V, p) ∈ E.
If M is flat on E, then S is flat on E. The converse does not hold.

Proof. Let M be flat on E. Then M, in particular, is 1-flat on E. Since S and M for k = 1 coincide, we have that S is 1-flat on E. Thus, by Proposition 22, we deduce that S is flat.

In order to show that S flat on E does not imply M flat on E, we consider the election E = (C, V, p) given by C = [4], V = [5] and

p =     2 1 3 4 4 1 2 1 3 2 3 3 2 1 3 4 4 4 2 1     .
It is immediately checked that S(p, x) = 2 for all x ∈ C and so, by Proposition 22, S is flat on E while M(p, 3) = {{1, 2, 3}}. Thus M is not 3-flat on E. In particular, M is not flat on E.

By the above result, the set of elections in which S is flat is larger than the set of elections in which M is flat. This represents a critical issue for S with respect to M.

Corollary 24. The property M is flat on E = (C, V, p) is not equivalent to the property M is k-flat on E for some k ∈ [|C| -1].
Proof. The example used in Proposition 23 exhibits a case in which M is 1-flat but not 3-flat. We finally show that S and M both fail to be resolute and suffer the reversal bias, the Condorcet loser paradox and the leaving member paradox.7 Recall that a csr R is resolute if, for every (C, V, p) ∈ E and k ∈

[|C|-1], |R(C, V, p, k)| = 1; R suffers the reversal bias if there exist (C, V, p) ∈ E and k ∈ [|C| -1] such that |R(C, V, p, k)| = 1 and R(C, V, p, k) = R(C, V, p r , k); R suffers the Condorcet loser paradox if there exist (C, V, p) ∈ E, k ∈ [|C| -1] and W * ∈ 2 C k such that W * is a k-Condorcet loser set for (C, V, p), that is, for every x ∈ W * and y ∈ C \ W * , c p (x, y) < |V | 2 , and R(C, V, p, k) = {W * }; R suffers the leaving member paradox if there exist (C, V, p) ∈ E, k ∈ [|C| -1] with k = 1, and x * ∈ C such that R(C, V, p, k) = {W * }, x * ∈ W * , R(C ′ , V, p ′ , k) = {W ′ } and W ′ ∩ W * = ∅,
p =     1 1 1 2 2 2 3 3 3 3 4 4 4 4 1 1 4 2 2 4 4 1 2 3 3 3 3 1 1 1 2 2 2 3 2 2 4 4 3 4 4 1 1 3    
Then M(p, 2) = {{1, 2}}. Assume now that 1 drops out of the competition for some reason and consider (C ′ , V, p ′ ) ∈ E, where C ′ = {2, 3, 4} and p ′ is obtained by p by erasing candidate 1 in each column. Then, we get M(p ′ , 2) = {{3, 4}}.

A summary of our results is provided in Table 1. The properties marked with * are new in the committee selection literature. The other properties have already been introduced and studied for other csrs.

Some further propositions

Let us present here other results that will turn out to be useful in the next section. Then, for every x ∈ W and y ∈ C \ {x}, c p (x * , y m ) < c p (x, y) < c p (x * , y M ). In particular, we have that

c p (x * , y m ) < c p (y m , x * ) < c p (x * , y M ) and c p (x * , y m ) < c p (y M , x * ) < c p (x * , y M ), so that c p (x * , y m ) < |V | -c p (x * , y m ) < c p (x * , y M ) and c p (x * , y m ) < |V | -c p (x * , y M ) < c p (x * , y M ). Thus, we have that |V | -c p (x * , y m ) < c p (x * , y M ) and c p (x * , y m ) < |V | -c p (x * , y M ) from which we deduce the contradiction |V | < c p (x * , y m ) + c p (x * , y M ) < |V |. Lemma 34. Let (C, V, p) ∈ E and W ⊆ C with |W | ∈ [|C| -1]. Then M (p, W ) = M (p r , C \ W ). Proof. Simply observe that C \ W ∈ C, |C \ W | ∈ [|C| -1] and that M (p, W ) = min x∈W,y∈C\W c p (x, y) = min x∈W,y∈C\W c p r (y, x) = M (p r , C \ W ). Proposition 35. Let (C, V, p) ∈ E, k ∈ [|C| -1] and W * ∈ 2 C k . Then W * ∈ M(p, k) if and only if C \ W * ∈ M(p r , |C| -k). In particular, M(p, k) = {W * } if and only if M(p r , |C| -k) = {C \ W * }.
Proof. Consider the following statements:

1. W * ∈ M(p, k); 2. for every W ∈ 2 C k , M (p, W * ) ≥ M (p, W ); 3. for every W ∈ 2 C k , M (p r , C \ W * ) ≥ M (p r , C \ W ); 4. for every W ∈ 2 C |C|-k , M (p r , C \ W * ) ≥ M (p r , W ); 5. C \ W * ∈ M(p r , |C| -k).
It is immediate to show that, for every i ∈ {1, 2, 3, 4}, statement i is equivalent to statement i + 1.

Then statement 1 is equivalent to statement 5 and the proof is completed.

Corollary 36. Let (C, V, p) ∈ E be such that |C| is even and

|M(p, |C|/2)| = |M(p r , |C|/2)| = 1. Then M(p, |C|/2) = M(p r , |C|/2).
Proof. Assume that M(p, |C|/2) = {W * }. Then by Proposition 35 we have that

M(p r , |C|-|C|/2) = M(p r , |C|/2) = {C \ W * }. Since W * = C \ W * , the proof is complete. Corollary 37. Let (C, V, p) ∈ E and k ∈ [|C| -1].
The following conditions are equivalent:

(i) M(p, k) = M(p r , k) and |M(p, k)| = 1; (ii) M(p, |C| -k) = M(p r , |C| -k) and |M(p, |C| -k)| = 1.
Proof. Assume that (i) holds true. Then there exists

W * ∈ 2 C k such that M(p, k) = M(p r , k) = {W * }. Then, by Proposition 35, we have that M(p, |C| -k) = M(p r , |C| -k) = {C \ W * } which implies (ii). The proof that (ii) implies (i) is analogous.

Computational results

In this section we better investigate the resoluteness as well as the properties of suffering the reversal bias, the Condorcet loser paradox and the leaving member paradox for S and M. We fix the number m ∈ {3, 4} of candidates, the number n ≥ 2 of the voters and the size k ∈ [m -1] of the committee to be selected and we focus only on elections (C, V, p) with |C| = m and |V | = n.9 We then study each of the considered properties through a computational approach estimating the probability of its occurrence for S and M and providing a comparison of the results obtained with those obtained for the Borda, the Plurality, the Negative Plurality and the Bloc csrs. 10 We emphasize that the Bloc csr agrees with the Plurality csr when k = 1 and with the Negative Plurality csr when k = m -1. Recall also that S agrees with M when k = 1. In the last part of the section we also study the probability that S and M agree on the same unique committee as well as the probability of the agreement of each of them with the Borda csr. Before giving the results of our analysis, we first describe in Section 4.1 the methodology applied in order to calculate our probabilities.

Evaluating the probability of voting situations

Let us first present our main computational assumption, that is, the Impartial Anonymous Culture (IAC). With m candidates, there are m! possible individual preferences (linear orders) which we order in some fashion. A voting situation is a vector (n 1 , . . . , n i , . . . , n m! ) such that m! i=1 n i = n, where the integer n i is the number of individuals voting the i-th linear order. The IAC condition stipulates that all voting situations are equiprobable. This assumption, introduced by [START_REF] Gehrlein | The probability of the paradox of voting: A computable solution[END_REF], is one of the most used assumption in social choice theory when computing the theoretical probability of electoral events. For more details on the IAC condition and other well-known assumptions, we refer the reader to the recent books by [START_REF] Gehrlein | The Condorcet criterion and committee selection[END_REF]Lepelley (2011, 2017). Under IAC, obtaining the probability of an electoral event is accomplished by the computation of two elements. The first one is the total number of voting situations, that is, n+m!-1 m!-1 . The second one is the number of voting situations associated with the property under examination. This can be in general reduced to a finite system of linear constraints with rational coefficients. For instance, if C = {x, y, z}, there exist 3!=6 linear orders on

C: x ≻ y ≻ z (n 1 ), x ≻ z ≻ y (n 2 ), y ≻ x ≻ z (n 3 ), y ≻ z ≻ x (n 4 ), z ≻ x ≻ y (n 5
), and z ≻ y ≻ x (n 6 ). In this setting, evaluating the situations for which the Plurality csr selects {x, y} as unique outcome for a given number of voters n is equivalent to finding the number of voting situations (n 1 , . . . , n 6 ) subject to the following conditions: n 1 + n 2n 5n 6 > 0 (the Plurality score of candidate x is greater than the one of z), n 3 + n 4n 5n 6 > 0 (the Plurality score of y is greater than the one of z), n i ≥ 0 for each i ∈ [6], and

6 i=1 n i = n.
As recently pointed out in the literature of social choice theory, Ehrhart polynomials are the appropriate mathematical tool to study such problems [START_REF] Gehrlein | The Condorcet criterion and committee selection[END_REF]Lepelley, 2011, 2017;[START_REF] Lepelley | On Ehrhart polynomials and probability calculations in voting theory[END_REF][START_REF] Wilson | Probability calculations under the IAC hypothesis[END_REF]. In fact, they have been widely used in numerous studies analyzing the probability of electoral events in the case of three-candidate elections under IAC assumption [START_REF] Courtin | The q-majority efficiency of positional rules[END_REF][START_REF] Diss | Strategic manipulability of self-selective social choice rules[END_REF]Diss et al., 2012;[START_REF] Gehrlein | The Condorcet criterion and committee selection[END_REF]Lepelley, 2011, 2017;[START_REF] Gehrlein | Voters' preference diversity, concepts of agreement and Condorcet's paradox[END_REF][START_REF] Gehrlein | A note on Approval Voting and electing the Condorcet loser[END_REF][START_REF] Gehrlein | An Evaluation of the Benefit of Using Two-Stage Election Procedures[END_REF][START_REF] Kamwa | Scoring rules and preference restrictions: The strong Borda paradox revisited[END_REF][START_REF] Gehrlein | Elections, voting rules and paradoxical outcomes[END_REF][START_REF] Smaoui | Borda elimination rule and monotonicity paradoxes in three-candidate elections[END_REF]. There exist strong algorithms that enable to specify the Ehrhart polynomials for many problems in the case of three-candidate elections. In this paper, all our results for the case of three candidates are obtained by using the parameterized Barvinok's algorithm.11 After obtaining the Ehrhart polynomials corresponding to each of the considered properties, we evaluate these polynomials for any needed number of voters and we get the desired probabilities. As noticed in [START_REF] Lepelley | On Ehrhart polynomials and probability calculations in voting theory[END_REF], the different algorithms that can be used in the three-candidate framework do not allow to deal with four-candidate elections, where the total number of variables, i.e., possible linear orderings, is 4! = 24. However, recent developments within the polytope theory may allow to obtain exact results for the case of m = 4 and a small number of voters. These results are obtained using the algorithms of Normaliz (Bruns et al., 2017a) which is, to the best of our knowledge, the only program able to compute the number of voting situations in polytopes corresponding to elections with up to four candidates. The reader interested in a deeper understanding of the algorithms of Normaliz is refereed to Bruns et al. (2017b) who describe several results obtained in four-candidate elections. This software can do all computations in dimension 24, but the limitation of the method is that all voting situations (n 1 , . . . , n m! ) that verify the required conditions of the considered voting event are enumerated and stored. Thus, this method needs relatively high memory when the number of voters increases. Moreover, the computation time rises accordingly. Consequently, exact results are obtained with n ∈ {2, . . . , 9} and m = 4 and we use computer simulations in order to evaluate the probabilities for n > 9. 12 We describe now the Monte-Carlo simulation methodology applied in order to estimate our probabilities in the same spirit as the Impartial Anonymous Culture condition. Let us consider as an example the probability of resoluteness.

1. At the beginning of the evaluation, we randomly generate a voting situation of length m!.

2. In the second step, we check whether the conditions of the resoluteness are fulfilled or not.

3. These two steps are iterated 1, 000, 000 times to obtain the number of voting situations for which resoluteness holds.

4. Finally, the probability of resoluteness is calculated as the quotient of the number obtained in step 3 over the total number of simulated voting situations, i.e., 1, 000, 000. 13

Another technique is used in this paper in order to obtain exact results when the number of candidates is m = 4 and the number of voters tends to infinity. In this case, the calculations of the limiting probability under IAC condition are reduced to computation of volumes of convex polytopes. For this, our volumes are found with the use of the algorithm Convex which is a Maple package for convex geometry by [START_REF] Franz | Convex -a Maple package for convex geometry[END_REF]. This package works with the same general procedure that was implemented in [START_REF] Cervone | Which scoring rule maximizes Condorcet efficiency under IAC?[END_REF] and recently used in other studies, e.g., [START_REF] Diss | Multi-winner scoring election methods: Condorcet consistency and paradoxes[END_REF], [START_REF] Diss | Strategic manipulability of self-selective social choice rules[END_REF]Gehrlein (2012, 2015), [START_REF] Gehrlein | Voters' preference diversity, concepts of agreement and Condorcet's paradox[END_REF], and [START_REF] Moyouwou | A note on Approval Voting and electing the Condorcet loser[END_REF]. 14 Finally note that in what follows the probability values 0 and 0.0000 have different meanings. In fact, the first one corresponds to an exact value while the second one is obtained using our simulation method.

The probability of resoluteness

Our first concern is the probability of resoluteness. The results of our computations are provided in Tables 2 to 6. Recall that a csr is resolute if it always selects a single committee. We stress that the analysis of the probability of resoluteness of the classical scoring csrs considered here is new in the literature.

It is intuitive that ties are highly unlikely if the number of voters is large so that the probability of resoluteness is expected to approach 1 as n increases. Moreover, it is natural to understand that the probability of ties highly depends on whether the number of voters is odd or even, with ties more probable for an even number of voters. Those facts are confirmed by our results.

We observe very different behaviours of the considered csrs for odd and even numbers of voters. For instance, with m = 3 and k = 1, both S and M (which agree in this case) perform better than the other csrs when the number of voters is odd, but the scoring csrs have a superior performance when the number of voters is even. In this case, the Borda csr very well in comparison with the other scoring csrs. When m = 3, k = 2 and n is odd, our results show that M is the best csr, while S performs the worst. With n even, S still is the worst scenario, but it is found that some scoring csrs can perform better than M. Finally, when the number of candidates increases to m = 4, it appears that usually the Borda csr has a greater probability of resoluteness than M which in turn leads to a better performance than some scoring csrs. Note that S performs the worst in most cases. This remains in general true independently of the target size of the committee.

It is worth mentioning that some theoretical facts about the resoluteness of the considered csrs help in building the tables. In particular, via very simple arguments involving the concept of reversal of a preference profile, it can be proved that the probability of resoluteness of the Borda csr is the same for k = 1 and k = m -1; the probability of resoluteness of the Plurality csr with k = 1 is the same as the one of the Negative Plurality csr with k = m -1; the probability of resoluteness of Negative Plurality csr with k = 1 is the same as the one of Plurality csr with k = m -1; when m is even and k = m 2 , the probability of resoluteness of the Plurality csr and the one for Negative Plurality csr are the same. Moreover, from Proposition 35, we also get that the probability of resoluteness of M is the same for k = 1 and k = m -1. 13 The MATLAB code of our simulations is available upon request. 14 Once again this technique can not be used for S and M for which obtaining the probabilities in the limit case is only possible with computer simulations. We consider for them a number of voters n = 100, 000.

The probability of suffering the reversal bias

In Tables 9 to 11 we collect our results about the probability of suffering the reversal bias. Recall that a csr suffers the reversal bias when for a given preference profile a unique committee is selected and reversing the preferences of all the voters the same committee is still selected as unique outcome. [START_REF] Saari | Consequences of reversing preferences[END_REF] and [START_REF] Bubboloni | On the reversal bias of the Minimax social choice correspondence[END_REF] prove several results about the reversal bias when k = 1. On the basis of our results and those in [START_REF] Bubboloni | On the reversal bias of the Minimax social choice correspondence[END_REF] we can deduce interesting facts concerning the computations we are going to perform. In particular, from Theorem A in [START_REF] Bubboloni | On the reversal bias of the Minimax social choice correspondence[END_REF], we get that S and M are immune to the reversal bias for m = 3 and k = 1; from Proposition 33, S is immune to the reversal bias for m = 3 and k = 2 as well as for m = 4 and k = 3; when m = 4, from Theorem A in [START_REF] Bubboloni | On the reversal bias of the Minimax social choice correspondence[END_REF], we have that S and M are immune to the reversal bias if and only if n ∈ {2, 3, 4, 5, 7}; from Theorem A in [START_REF] Bubboloni | On the reversal bias of the Minimax social choice correspondence[END_REF] and Corollary 37, M is immune to the reversal bias for m = 3 and k = 2; from Corollary 36, M is immune to the reversal bias for m = 4 and k = 2; from Corollary 37, the probability for M to be immune to the reversal bias for m = 4 and k = 1 is the same as for m = 4 and k = 3. It is also immediate to show that the Borda csr is always immune to the reversal bias; the probability of suffering the reversal bias of the Plurality csr (Bloc csr) with k = 1 is the same as the one of the Negative Plurality csr (Bloc csr) with k = m -1; the probability of suffering the reversal bias of the Negative Plurality csr with k = 1 is the same as the one of the Plurality csr with k = m -1; when m is even and k = m 2 , the probability of suffering the reversal bias of the Plurality csr and the one of the Negative Plurality csr are the same; the Bloc csr is immune to the reversal bias.

It turns out that the most interesting cases to study for comparing S and M to the other csrs correspond to m = 4 with k = 1 and k = 3. Our computations reveal that, yet again, S and M perform better than the other csrs (except Borda). However, M appears to be more vulnerable to the reversal bias than S for m = 4 and k = 3.

Finally, the computations made suggest to conjecture that S is immune to the reversal bias if m = 4 and k = 2. This is, at the moment, an open problem.

The probability of suffering the Condorcet loser paradox

Tables 12 to 16 describe our results about the probability of suffering the Condorcet loser paradox provided a Condorcet loser set exists. Recall that such a paradox occurs when, for a given preference profile, the unique winning committee is a Condorcet loser set, that is, a set having the property that each candidate which does not belong to it is preferred by the majority of voters to every of its elements. It is important to notice that this paradox is also known in the literature as the strong Borda paradox. This type of problem for k = 1 is studied by [START_REF] Diss | Borda's paradox with weighted scoring rules[END_REF]; [START_REF] Diss | Another perspective on Borda's paradox[END_REF]; [START_REF] Fishburn | Borda's rule, positional voting, and Condorcet's simple majority principle[END_REF]; Gehrlein and Lepelley (2010b), among others.

It is important to notice that, since a k-Condorcet loser set for (C, V, p) is a k-Condorcet winner for (C, V, p r ), if a csr is Condorcet consistent and immune to the reversal bias then it is immune to the Condorcet loser paradox. Since M is Condorcet consistent, the analysis made in the previous section can be used to get some information about the Condorcet loser paradox for M. For instance, with m = 4 and k = 1, Table 9 shows that M is immune to the reversal bias when n ∈ {2, 3, 4, 5, 7}. This remains true for the immunity to the Condorcet loser paradox as shown in Table 14. It is well known that the Borda csr is immune to the Condorcet loser paradox when k = 1 and it is immediate to show that the probability of suffering the Condorcet loser paradox of the Plurality csr (Bloc csr) with k = 1 is the same as the one of the Negative Plurality csr (Bloc csr) with k = m -1; the probability of suffering the Condorcet loser paradox of the Negative Plurality csr with k = 1 is the same as the one of the Plurality csr with k = m -1; when m is even and k = m 2 , the probability of suffering the Condorcet loser paradox of the Plurality csr and the one of the Negative Plurality csr are the same. Notice that the results in Table 12 for Plurality and Negative Plurality when the number of voters is large are in agreement with results in Gehrlein and Lepelley (2010b).

From the tables we understand that the probability of observing the Condorcet loser paradox can never exceed 0.0315 with m = 3 and 0.0240 with m = 4 under all the csrs that we consider in this paper. In other words, the probability of observing this paradox should be rare, but not impossible to observe particularly when the number of voters increases. Our results also show that the use of csrs like Borda, M and S will clearly tend to minimize the probability of observing the Condorcet loser paradox, with a certain advantage for S and Borda when m = 4 and k = 3.

Our probability calculations suggest to conjecture that the Borda csr is immune to the Condorcet loser paradox whatever the values of m, n, and k. In addition, except for the situation with m = 4 and k = 1, S seems to be immune to the Condorcet loser paradox for all the other values of m and k that we consider in this paper. Again these conjectures are, at the moment, open problems.

The probability of suffering the leaving member paradox

The probability of suffering the leaving member paradox is illustrated in Table 17. Recall that the leaving member paradox occurs when for a certain election a unique committee of size k out of m candidates is elected and if an elected candidate leaves the office for some reason and a new election on the m -1 remaining candidates is hold, then a unique committee is elected and such a committee and the original one are disjoint. This paradox has been introduced by [START_REF] Staring | Two paradoxes of committee elections[END_REF]. [START_REF] Kamwa | Scoring rules over subsets of alternatives: Consistency and paradoxes[END_REF] considered the probability of this paradox under the four scoring csrs presented in this paper with the Impartial Culture (IC)15 assumption for large electorates. Considering the IAC condition, [START_REF] Diss | Multi-winner scoring election methods: Condorcet consistency and paradoxes[END_REF] extended the results of [START_REF] Kamwa | Scoring rules over subsets of alternatives: Consistency and paradoxes[END_REF] in many directions. In particular, the results we propose for the limit case of the scoring csrs are taken from [START_REF] Diss | Multi-winner scoring election methods: Condorcet consistency and paradoxes[END_REF]. Since we are focusing only on m ∈ {3, 4}, the leaving member paradox only occurs in the case m = 4 and k = 2. We deduce then from our results that M has a better performance than the other csrs except for some small number of voters; in most cases S performs better than the Plurality csr and the Bloc csr. The Bloc csr appears to be the worst csr according to this criterion.

The probability of agreement

We know that for certain voting situations the set of committees selected by S differs from the set of committees selected by M. However, in some cases, they both select the same unique committee.

Here we study the probability of such a situation, which we shortly refer to as an agreement between S and M. Recall that S equals M when k = 1. Moreover, when m = 3, k = 2 and under the IAC hypothesis, Kamwa (2017b) computes the probability of having voting situations where S and M coincide.

The results of our calculations are described in Table 18. We can make some observations based upon this table. First of all, the probability of agreement is smaller when there is an even number of voters than when there is an odd number of voters. One of the possible reasons for that is that having ties for S or M, which are more likely in the even case, implies no agreement between the two csrs. We also observe that, our results indicate that increasing an odd (even) number of voters implies an increase in the probability of having the agreement. Moreover, when the number of voters is large the probability of the agreement reaches 70% for m = 3 and k = 2 and 60% with m = 4 and k = 2 or k = 3. Finally, it seems from the tables that, for fixed m and n, with n large, the probability of the agreement decreases as the size of the committee to be elected increases.

We study also the probability of agreement between S and the Borda csr (Table 19), and between M and the Borda csr (Table 20). Our results show that there exists, at least in three-candidate and four-candidate elections, an important set of voting situations where the Borda csr and one of the two extensions of the Simpson voting rule agree. Surprisingly, this probability reaches more than 80% for many cases with only 101 voters. Moreover, our results indicate that the Borda csr is more likely to agree with M than with S except for a small even number of voters (n ≤ 6). Since the probability of resoluteness tends to rapidly increase as the number of voters increases, we know that each of the studied csrs usually selects a unique committee as the number of voters is large. Hence, for large electorates, the probability of the agreement here considered should be close to the probability of equality. Note that, using a simple argument based on the concept of reversal of a preference profile, it can be proved that the probability of agreement between M and the Borda when k = 1 is the same as the one when k = m -1.

n m = 3 m = 4 k = 1 k = 2 k = 1 k = 2 k = 3 2 0.

  0 and thus S(p, y * ) = 0. By y * ∈ W * we also have S(p, y * ) ≥ S(p, y) for all y ∈ C \ W * . It follows that S(p, y) = 0 for all y

Proposition 25 .

 25 The property M is k-flat on E = (C, V, p), for some k ∈ [|C| -1], does not imply the property S is flat on E.Proof. Consider the election E = (C, V, p) given by C = [4], V = [8] and p defined in (2). It is immediately checked that M(p, 2) = 2 C 2 so that M is 2-flat on E. On the other hand, we have S(p, 1) = 4, S(p, 2) = S(p, 3) = S(p, 4) = 3. Thus, S is not flat on E. Proposition 26. S and M are symmetric. Proof. Let (C, V, p) ∈ E such that c p (x, y) = c p (y, x) for all x, y ∈ C, with x = y, and let k ∈ [|C| -1]. Let x, y ∈ C with x = y. Since the preferences in the profile p are linear orders, we have c p (x, y) + c p (y, x) = |V |. Thus |V | is even and c p (x, y) = |V | 2 . In particular, c p is constant. As a consequence, for every W ∈ 2 C k and for every x ∈ C, we have that M (p, W ) = |V | 2 and S(p, x) = |V | 2 so that M(p, k) = S(p, k) = 2 C k . Following Elkind et al. (2017), we say that R satisfies consensus committee if, for every (C, V, p) ∈ E, k ∈ [|C| -1] and W ∈ 2 C k such that each voter ranks some member of W first and each member of W is ranked first by either |V | k or |V | k voters, we have that R(C, V, p, k) = {W }. Proposition 27. S and M do not satisfy consensus committee. Proof. Consider C = [3], V = [2], {1, 2}. Any csr satisfying consensus committee should select W as unique winning committee for the election E = (C, V, p). However, both S and M are 2-flat on E.

  where C ′ = C \ {x * } and p ′ is the preference profile obtained by p erasing the candidate x * . 8 Proposition 28. S and M do not satisfy resoluteness.Proof. Consider C = [2], V = [2p, 1) = M(p, 1) = {{1}, {2}}.Proposition 29. S and M suffer the reversal bias. Proof. Consider C = [4], V = [11], and p as in (14). Then S(p, 1) = M(p, 1) = {{4}} and S(p r , 1) = M(p r , 1) = {{4}}. Proposition 30. S and M suffer the Condorcet loser paradox. Proof. Consider C = [4], V = [11], and p as in (14). Then {4} is a 1-Condorcet loser set for (C, V, p) and S(p, 1) = M(p, 1) = {{4}}. Note that, according to Kamwa (2017a), a csr R satisfies the Condorcet loser criterion if, for every (C, V, p) ∈ E, k ∈ [|C| -1], W ∈ R(C, V, p, k) and x ∈ C a Condorcet loser, we have that x ∈ W . The above proposition shows that S and M fail the Condorcet loser criterion. This fact for M was proved in Kamwa (2017a, Theorem 2). Proposition 31. S and M suffer the leaving member paradox. Proof. Consider first S with C = [4], V = [4p, 2) = {{1, 2}}. Assume now that 1 drops out of the electoral competition for some reason and consider (C ′ , V, p ′ ) ∈ E, where C ′ = {2, 3, 4} and p ′ is obtained by p by erasing candidate 1 in each column. Then, we get S(p ′ , 2) = {{3, 4}}. Consider now M with C = [4], V = [11], and

Proposition 32 .

 32 Let (C, V, p) ∈ E, k ∈ [|C| -1] and W * ∈ 2 C k . Then S(p, k) = {W * } ifand only if, for every x ∈ W * and y ∈ C \ W * , S(p, x) > S(p, y).

  Proof. Let S(p, k) = {W * }. Assume, by contradiction, that there exists x * ∈ W * and y * ∈ C \ W * such that S(p, x * ) ≤ S(p, y * ). Since for every x ∈ W * and y ∈ C \ W * , we surely have S(p, x) ≥ S(p, y), we then have S(p, x * ) = S(p, y * ). (16)Define W ′ = (W * \ {x * })∪{y * } ∈ 2 C k . We reach the desired contradiction showing that W ′ ∈ S(p, k). Let x ∈ W ′ and y ∈ C \W ′ = C \(W * ∪{x * })∪{x * }.We see that s(p, x) ≥ S(p, y). Let first x = y * and y = x * . Then we have x ∈ W * and y ∈ C \W * and thus, since W * ∈ S(p, k), we get S(p, x) ≥ S(p, y). Let next x = y * and y = x * . Then by (16), S(p, x) = S(p, y * ) = S(p, x * ) = S(p, y). Let now x = y * and y = x * . Then y ∈ C \ W * and, by (16) and W * ∈ S(p, k), we obtain S(p, x) = S(p, y * ) = S(p, x * ) ≥ S(p, y). Let finally be x = y * and y = x * . Then x * ∈ W * and y ∈ C \ W * . Hence, by W * ∈ S(p, k) and (16), we get S(p, x) ≥ S(p, y * ) = S(p, x * ) = S(p, y). Assume next that W * ∈ 2 C k is such that for every x ∈ W * and y ∈ C \ W * , S(p, x) > S(p, y). (17) Then W * ∈ S(p, k). We want to show that W * is the only element in S(p, k). Pick W ∈ 2 C k \ {W * }. Since W * and W have the same size k, there exist y ∈ W \ W * and x ∈ W * \ W. Thus, by (17), we have S(p, x) > S(p, y), which excludes W ∈ S(p, k). Proposition 33. Let (C, V, p) ∈ E be such that |S(p, |C| -1)| = |S(p r , |C| -1)| = 1. Then S(p, |C| -1) = S(p r , |C| -1). Proof. Assume by contradiction that there exists W ⊆ C with |W | = |C| -1 such that S(p, |C| -1) = S(p r , |C| -1) = {W }. Let x * be the unique element of the set C \ W . Then, by Proposition 32, we have that, for every x ∈ W , S(p, x * ) < S(p, x) and S(p r , x * ) < S(p r , x). Since, for every x, y ∈ C with x = y, c p (y, x) = |V |c p (x, y) and c p r (x, y) = c p (y, x), we obtain that, for every x ∈ W , min y =x * c p (x * , y) < min y =x c p (x, y) and max y =x * c p (x * , y) > max y =x c p (x, y). Thus, there exist y m , y M ∈ W such that, for every x ∈ W , c p (x * , y m ) < min y =x c p (x, y) and c p (x * , y M ) > max y =x c p (x, y).

Table 1

 1

	: Summary
	Properties	M	S
	Anonymity	✓	✓
	Neutrality	✓	✓
	Homogeneity	✓	✓
	Non-Imposition	✓	✓
	Strong Non-Imposition*	✓	✗
	Weak Unanimity	✓	✓
	Strong Unanimity	✓	✗
	Fixed Majority	✓	✗
	Consensus Committee	✗	✗
	Weak Pareto criterion*	✓	✓
	Pareto criterion	✓	✗
	Condorcet Consistency	✓	✗
	Stability	✓	✗
	Local Stability	✗	✗
	Condorcet Winner Criterion	✗	✓
	Immunity to the Condorcet Loser Paradox	✗	✗
	Condorcet Loser Criterion	✗	✗
	Committee Monotonicity	✗	✓
	Membership Monotonicity	✗	✗
	Non-crossing Monotonicity	✗	✓
	Candidate Monotonicity	✓	✓
	Weak Consistency	✗	✗
	Consistency	✗	✗
	Weak Resolvability*	✓	✗
	Resolvability*	✗	✗
	Symmetry*	✓	✓
	Resoluteness	✗	✗
	Immunity to the Reversal Bias	✗	✗
	Immunity to the Leaving Member Paradox	✗	✗

Table 2 :

 2 The probability of resoluteness: m = 3 and k = 1

	n	S = M	Plurality Borda = Bloc	Negative Plurality
	2	0.4286	0.4286	0.7143	0.5714
	3	0.9643	0.8571	0.8571	0.6429
	4	0.6190	0.7857	0.8571	0.5952
	5	0.9524	0.7857	0.8810	0.7381
	6	0.7143	0.8377	0.8831	0.7662
	7	0.9545	0.8788	0.9091	0.7424
	8	0.7692	0.8298	0.9044	0.8089
	9	0.9590	0.8931	0.9201	0.8242
	50	0.9530	0.9703	0.9806	0.9588
	51	0.9888	0.9725	0.9810	0.9595
	100	0.9758	0.9854	0.9900	0.9782
	101	0.9941	0.9851	0.9901	0.9788
	1000 0.9975	0.9985	0.9990	0.9978
	1001 0.9994	0.9985	0.9990	0.9978
	∞	1	1	1	1

Table 3 :

 3 The probability of resoluteness: m = 3 and k = 2

	n	S	M	Plurality Borda	Negative Plurality = Bloc
	2	0.4286 0.4286	0.5714	0.7143	0.4286
	3	0.4286 0.9643	0.6429	0.8571	0.8571
	4	0.5714 0.6190	0.5952	0.8571	0.7857
	5	0.5952 0.9524	0.7381	0.8810	0.7857
	6	0.6623 0.7143	0.7662	0.8831	0.8377
	7	0.6818 0.9545	0.7424	0.9091	0.8788
	8	0.7226 0.7692	0.8089	0.9044	0.8298
	9	0.7373 0.9590	0.8242	0.9201	0.8931
	50	0.9413 0.9530	0.9588	0.9806	0.9703
	51	0.9423 0.9888	0.9595	0.9810	0.9725
	100 0.9697 0.9758	0.9782	0.9900	0.9854
	101 0.9700 0.9941	0.9788	0.9901	0.9851
	1000 0.9969 0.9975	0.9978	0.9990	0.9985
	1001 0.9969 0.9994	0.9978	0.9990	0.9985
	∞	1	1	1	1	1

Table 4 :

 4 The probability of resoluteness: m = 4 and k = 1

	n	S = M	Plurality Borda = Bloc	Negative Plurality
	2	0.2800	0.2800	0.7200	0
	3	0.9015	0.6677	0.8185	0.3323
	4	0.4807	0.7754	0.8274	0.5169
	5	0.8733	0.6769	0.8481	0.5692
	6	0.5155	0.6819	0.8618	0.5491
	7	0.8259	0.7799	0.8730	0.6063
	8	0.5787	0.7772	0.8829	0.6608
	9	0.8346	0.7652	0.8901	0.6863
	50	0.8841	0.8899	0.9615	0.9245
	51	0.9468	0.8886	0.9619	0.9244
	100	0.9411	0.9441	0.9804	0.9591
	101	0.9716	0.9443	0.9805	0.9601
	1000 0.9942	0.9930	0.9993	0.9949
	1001 0.9955	0.9933	0.9996	0.9966
	∞	1	1	1	1

Table 5 :

 5 The probability of resoluteness: m = 4 and k = 2

	n	S	M	Plurality Borda	Negative Plurality	Bloc
	2	0.4400 0.2000	0.7200	0.6400	0.7200	0.2000
	3	0.2769 0.8492	0.5815	0.8031	0.5815	0.8031
	4	0.5115 0.4048	0.3805	0.7774	0.3805	0.4769
	5	0.4508 0.8083	0.5590	0.8227	0.5590	0.7460
	6	0.5197 0.4219	0.7017	0.8245	0.7017	0.6103
	7	0.4825 0.7292	0.6367	0.8477	0.6367	0.7454
	8	0.5495 0.4857	0.6119	0.8517	0.6119	0.6808
	9	0.5422 0.7416	0.6841	0.8665	0.6841	0.7603
	50	0.8563 0.8448	0.8812	0.9500	0.8825	0.8939
	51	0.8593 0.9065	0.8817	0.9477	0.8819	0.8983
	100 0.9219 0.9106	0.9340	0.9719	0.9341	0.9459
	101 0.9262 0.9455	0.9380	0.9779	0.9383	0.9483
	1000 0.9908 0.9739	0.9916	0.9966	0.9989	0.9946
	1001 0.9964 0.9825	0.9920	0.9973	0.9949	0.9962
	∞	1	1	1	1	1	1
				20

Table 6 :

 6 The probability of resoluteness: m = 4 and k = 3

	n	S	M	Plurality Borda	Negative Plurality = Bloc
	2	0.2400 0.2800	0	0.7200	0.2800
	3	0.4000 0.9015	0.3323	0.8185	0.6677
	4	0.4356 0.4807	0.5169	0.8274	0.7754
	5	0.5245 0.8733	0.5692	0.8481	0.6769
	6	0.5414 0.5155	0.5491	0.8618	0.6819
	7	0.5871 0.8259	0.6063	0.8730	0.7799
	8	0.6051 0.5787	0.6608	0.8829	0.7772
	9	0.6297 0.8346	0.6863	0.8901	0.7652
	50	0.8767 0.8841	0.9245	0.9615	0.8899
	51	0.8800 0.9468	0.9244	0.9619	0.8886
	100 0.9341 0.9411	0.9591	0.9804	0.9441
	101 0.9374 0.9716	0.9601	0.9805	0.9443
	1000 0.9920 0.9942	0.9949	0.9993	0.9930
	1001 0.9945 0.9955	0.9966	0.9996	0.9933
	∞	1	1	1	1	1

Table 7 :

 7 The probability of suffering the reversal bias: m = 3 and k = 1

	n	S = M Plurality Negative & Borda = Bloc Plurality
	2	0	0	0.1429
	3	0	0	0.1071
	4	0	0.0238	0.0714
	5	0	0	0.1190
	6	0	0.0260	0.1234
	7	0	0.0379	0.1061
	8	0	0.0210	0.1352
	9	0	0.0420	0.1379
	50	0	0.0776	0.1856
	51	0	0.0789	0.1859
	100	0	0.0852	0.1938
	101	0	0.0850	0.1942
	1000	0	0.0918	0.2027
	1001	0	0.0918	0.2027
	∞	0	0.0926	0.2037

Table 8 :

 8 The probability of suffering the reversal bias: m = 3 and k = 2

	n	S & M Plurality & Borda	Negative Plurality = Bloc
	2	0	0.1429	0
	3	0	0.1071	0
	4	0	0.0714	0.0238
	5	0	0.1190	0
	6	0	0.1234	0.0260
	7	0	0.1061	0.0379
	8	0	0.1352	0.0210
	9	0	0.1379	0.0420
	50	0	0.1856	0.0776
	51	0	0.1859	0.0789
	100	0	0.1938	0.0852
	101	0	0.1942	0.0850
	1000	0	0.2027	0.0918
	1001	0	0.2027	0.0918
	∞	0	0.2037	0.0926

Table 9 :

 9 The probability of suffering the reversal bias: m = 4 and k = 1

	n	S = M	Plurality Negative Borda = Bloc Plurality
	2	0	0	0	0
	3	0	0	0.0246	0
	4	0	0.0328	0.0492	0
	5	0	0.0215	0.0513	0
	6	0.0009	0.0163	0.0426	0
	7	0	0.0380	0.0548	0
	8	0.0011	0.0417	0.0670	0
	9	0.0003	0.0381	0.0714	0
	50	0.0025	0.0790	0.1326	0
	51	0.0022	0.0792	0.1333	0
	100	0.0030	0.0924	0.1458	0
	101	0.0031	0.0924	0.1471	0
	1000 0.0042	0.1086	0.1599	0
	1001 0.0044	0.1078	0.1605	0
	∞	0.0044	0.1105	0.1608	0

Table 10 :

 10 The probability of suffering the reversal bias: m = 4 and k = 2

	n	S	M & Borda & Bloc	Plurality & Negative Plurality
	2	0	0	0.0800
	3	0	0	0.0277
	4	0	0	0.0085
	5	0	0	0.0254
	6	0.0000	0	0.0466
	7	0.0000	0	0.0325
	8	0.0000	0	0.0271
	9	0.0000	0	0.0374
	50	0.0000	0	0.0656
	51	0.0000	0	0.0653
	100 0.0000	0	0.0759
	101 0.0000	0	0.0750
	1000 0.0000	0	0.0879
	1001 0.0000	0	0.0872
	∞	0.0000	0	0.0889

Table 11 :

 11 The probability of suffering the reversal bias: m = 4 and k = 3

	n	S & Borda	M	Plurality	Negative Plurality = Bloc
	2	0	0	0	0
	3	0	0	0.0246	0
	4	0	0	0.0492	0.0328
	5	0	0	0.0513	0.0215
	6	0	0.0009	0.0426	0.0163
	7	0	0	0.0548	0.0380
	8	0	0.0011	0.0670	0.0417
	9	0	0.0003	0.0714	0.0381
	50	0	0.0025	0.1326	0.0790
	51	0	0.0022	0.1333	0.0792
	100	0	0.0030	0.1458	0.0924
	101	0	0.0031	0.1471	0.0924
	1000	0	0.0042	0.1599	0.1086
	1001	0	0.0044	0.1605	0.1078
	∞	0	0.0044	0.1608	0.1105

Table 12 :

 12 The probability of suffering the Condorcet loser paradox: m = 3 and k = 1

	n	S = M Plurality Negative & Borda = Bloc Plurality
	2	0	0	0
	3	0	0	0
	4	0	0	0
	5	0	0	0.0250
	6	0	0	0
	7	0	0.0160	0.0080
	8	0	0	0.0100
	9	0	0.0159	0.0222
	50	0	0.0209	0.0245
	51	0	0.0258	0.0287
	100	0	0.0250	0.0276
	101	0	0.0275	0.0300
	1000	0	0.0291	0.0311
	1001	0	0.0294	0.0313
	∞	0	0.0296	0.0315

Table 13 :

 13 The probability of suffering the Condorcet loser paradox: m = 3 and k = 2

	n	S & M Negative Plurality Plurality & Borda = Bloc
	2	0	0	0
	3	0	0	0
	4	0	0	0
	5	0	0	0.0250
	6	0	0	0
	7	0	0.0160	0.0080
	8	0	0	0.0100
	9	0	0.0159	0.0222
	50	0	0.0209	0.0245
	51	0	0.0258	0.0287
	100	0	0.0250	0.0276
	101	0	0.0275	0.0300
	1000	0	0.0291	0.0311
	1001	0	0.0294	0.0313
	∞	0	0.0296	0.0315

Table 14 :

 14 The probability of suffering the Condorcet loser paradox: m = 4 and k = 1

	n	S = M	Plurality Negative Borda = Bloc Plurality
	2	0	0	0	0
	3	0	0	0.0137	0
	4	0	0	0	0
	5	0	0.0078	0.0059	0
	6	0.0000	0	0.0005	0
	7	0	0.0081	0.0105	0
	8	0.0000	0.0031	0.0052	0
	9	0.0000	0.0090	0.0116	0
	50	0.0002	0.0115	0.0154	0
	51	0.0004	0.0165	0.0208	0
	100	0.0004	0.0166	0.0194	0
	101	0.0005	0.0194	0.0222	0
	1000 0.0006	0.0221	0.0237	0
	1001 0.0007	0.0224	0.0236	0
	∞	0.0007	0.0228	0.0240	0

Table 15 :

 15 The probability of suffering the Condorcet loser paradox: m = 4 and k = 2

	n	S	M	Plurality	Negative Plurality	Bloc	Borda
	2	0	0	0	0	0	0
	3	0	0	0	0	0	0
	4	0	0	0	0	0	0
	5	0	0	0.0027	0.0027	0.0003	0
	6	0.0000	0	0.0013	0.0013	0	0
	7	0.0000	0	0.0026	0.0026	0.0005	0
	8	0.0000	0	0.0003	0.0003	0	0
	9	0.0000	0	0.0038	0.0038	0.0006	0
	50	0.0000	0	0.0055	0.0051	0.0005 0.0000
	51	0.0000	0	0.0068	0.0069	0.0012 0.0000
	100 0.0000	0	0.0071	0.0070	0.0009 0.0000
	101 0.0000	0	0.0078	0.0076	0.0013 0.0000
	1000 0.0000	0	0.0089	0.0089	0.0014 0.0000
	1001 0.0000	0	0.0090	0.0089	0.0015 0.0000
	∞	0.0000	0	0.0090	0.0090	0.0016 0.0000

Table 16 :

 16 The probability of suffering the Condorcet loser paradox: m = 4 and k = 3

	n	S	M	Plurality	Negative Plurality = Bloc	Borda
	2	0	0	0	0	0
	3	0	0	0.0137	0	0
	4	0	0	0	0	0
	5	0	0	0.0059	0.0078	0
	6	0.0000 0.0000	0.0005	0	0
	7	0.0000	0	0.0105	0.0081	0
	8	0.0000 0.0000	0.0052	0.0031	0
	9	0.0000 0.0000	0.0116	0.0090	0
	50	0.0000 0.0002	0.0160	0.0116	0.0000
	51	0.0000 0.0004	0.0202	0.0162	0.0000
	100 0.0000 0.0003	0.0196	0.0160	0.0000
	101 0.0000 0.0005	0.0221	0.0189	0.0000
	1000 0.0000 0.0007	0.0233	0.0222	0.0000
	1001 0.0000 0.0007	0.0235	0.0225	0.0000
	∞	0.0000 0.0007	0.0240	0.0228	0.0000

Table 17 :

 17 The probability of suffering the leaving member paradox: m = 4 and k = 2

	n	S	M	Plurality Borda	Negative Plurality	Bloc
	2	0	0	0	0	0	0
	3	0	0	0	0	0	0
	4	0.0021	0	0	0	0	0.0055
	5	0.0011	0	0.0033	0	0	0
	6	0.0038 0.0000	0.0018	0.0002	0	0.0085
	7	0.0020 0.0002	0.0008	0.0006	0	0.0252
	8	0.0049 0.0000	0.0082	0.0008	0	0.0081
	9	0.0035 0.0006	0.0060	0.0014	0	0.0253
	50	0.0201 0.0032	0.0321	0.0058	0.0137	0.0485
	51	0.0205 0.0041	0.0324	0.0059	0.0154	0.0512
	100 0.0262 0.0051	0.0386	0.0073	0.0199	0.0612
	101 0.0267 0.0060	0.0399	0.0076	0.0193	0.0605
	1000 0.0338 0.0076	0.0480	0.0090	0.0278	0.0738
	1001 0.0344 0.0082	0.0485	0.0090	0.0286	0.0756
	∞	0.0350 0.0084	0.0485	0.0094	0.0286	0.0757

Table 18 :

 18 The probability of agreement between S and M

	n	m = 3 k = 1 k = 2 k = 1 k = 2 m = 4	k = 3
	2	1	0.1429	1	0.0800 0.0400
	3	1	0.3214	1	0.1846 0.2615
	4	1	0.2857	1	0.1593 0.1730
	5	1	0.4286	1	0.2794 0.3206
	6	1	0.3766	1	0.1806 0.2305
	7	1	0.4848	1	0.2840 0.3502
	8	1	0.4336	1	0.2197 0.2766
	9	1	0.5215	1	0.3146 0.3749
	50	1	0.6346	1	0.4908 0.4900
	51	1	0.6524	1	0.5168 0.5193
	100	1	0.6603	1	0.5442 0.5355
	101	1	0.6694	1	0.5603 0.5517
	1000	1	0.6847	1	0.5983 0.5807
	1001	1	0.6856	1	0.6045 0.5832
	∞	1	0.6875	1	0.6074 0.5866

Table 19 :

 19 The probability of agreement between S and the Borda csr

Table 20 :

 20 The probability of agreement between M and the Borda csr

		4286 0.4286 0.2800 0.3600 0.2400
	3	0.8571 0.3214 0.7631 0.2123 0.3077
	4	0.6190 0.5238 0.4561 0.4075 0.3651
	5	0.8333 0.4762 0.7287 0.3509 0.4120
	6	0.6883 0.5762 0.4793 0.4147 0.4362
	7	0.8409 0.5530 0.6944 0.3749 0.4607
	8	0.7273 0.6107 0.5286 0.4323 0.4789
	9	0.8452 0.5964 0.6999 0.4198 0.4937
	50	0.8624 0.7528 0.7523 0.6344 0.6526
	51	0.8812 0.7522 0.7861 0.6348 0.6554
	100 0.8787 0.7716 0.7908 0.6706 0.6866
	101 0.8880 0.7714 0.8055 0.6753 0.6887
	1000 0.8941 0.7896 0.8253 0.7118 0.7199
	1001 0.8950 0.7896 0.8245 0.7146 0.7228
	∞	0.8958 0.7917 0.8272 0.7168 0.7262

The Simpson voting rule is also known as the Simpson-Kramer, the Condorcet and the Minimax voting rule. See[START_REF] Kramer | A dynamical model of political equilibrium[END_REF] and[START_REF] Simpson | On defining areas of voter choice: Professor Tullock on stable voting[END_REF].

Another extension of the Simpson voting rule, called Minimax Approval Voting, has been introduced by[START_REF] Brams | A minimax procedure for electing committees[END_REF] in the context of approval-based preferences.

Barberà and Coelho (2008, Definition 3).

[START_REF] Gehrlein | The Condorcet criterion and committee selection[END_REF].

[START_REF] Elkind | Properties of multiwinner voting rules[END_REF].

Those properties are largely studied in the literature. See for instance,[START_REF] Bubboloni | On the reversal bias of the Minimax social choice correspondence[END_REF],[START_REF] Diss | Multi-winner scoring election methods: Condorcet consistency and paradoxes[END_REF],[START_REF] Diss | Borda's paradox with weighted scoring rules[END_REF],[START_REF] Diss | Another perspective on Borda's paradox[END_REF],[START_REF] Duggan | Strategic manipulability without resoluteness or shared beliefs: Gibbard-Satterthwaite generalized[END_REF],[START_REF] Fishburn | Borda's rule, positional voting, and Condorcet's simple majority principle[END_REF],Gehrlein and Lepelley (2010b),[START_REF] Jeong | Resolute majority rules[END_REF],[START_REF] Kamwa | Scoring rules over subsets of alternatives: Consistency and paradoxes[END_REF],[START_REF] Saari | Consequences of reversing preferences[END_REF][START_REF] Saari | Consequences of reversing preferences[END_REF][START_REF] Staring | Two paradoxes of committee elections[END_REF].

Note that if k were allowed to be 1, then every csr would suffer the leaving member paradox.

Note that when, m ≤ 2 or n = 1, the analysis of the considered properties turns out to be straightforward.

See Diss and Doghmi (2016) and[START_REF] Elkind | Properties of multiwinner voting rules[END_REF] for the definitions of these csrs.

For a detailed description of algorithms computing Ehrhart polynomials, we recommend the report by[START_REF] Verdoolaege | Computation and manipulation of enumerators of integer projections of parametric polytopes[END_REF].

This is really possible for all the scoring csrs we are considering here but, unfortunately, this is not true when we consider S and M. For those csrs exact probabilities can be obtained only for n ∈ {2, 3, 4, 5}.

This is another assumption widely used for analyzing the probability of electoral events. Under this assumption, the preference relation of each voter is drawn uniformly at random from the set of all possible linear orders.


Jean-François Santucci

Ghjasippina Thury-Bouvet

Antoine Ottavi

Hadi El

Khoumeri

El Hadi

ORIENTATIONS OF NEOLITHIC TOMBS IN Khoumeri Astro- Nomical

Monte A Revincu

J F Aiello

San

Santucci Jean-François

Thury-Bouvet Ghjasippina

Ottavi Antoine

El Hadi Khoumeri

ASTRONOMICAL ORIENTATIONS OF NEOLITHIC TOMBS IN MONTE REVINCU

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction.

This paper presents how legends, megaliths and astronomy [START_REF] Von Keysserlingk | Und sie erstaten in Stein[END_REF] are linked in the Nebbiu Region in the north part of Corsica. From an oral story from north Corsica we first describe the megaliths involve in this story before presenting how the orientations of the megaliths seems to have astronomical reasons. The megalithic culture of Corsica [START_REF] Cesari | Les dolmens de Corse[END_REF][START_REF] Grosjean | La Corse avant l'histoire[END_REF][START_REF] De Lanfranchi | Art et croyances, Pratiques funéraires et symboliques des populations préhistoriques corses[END_REF][START_REF] De Lanfranchi | Le secrets des Mégalithes[END_REF][START_REF] De Lanfranchi | L'âmes des pierres[END_REF][START_REF] Leandri | Les megalithes de Corse[END_REF][START_REF] Mérimée | Notes d'un voyage en Corse[END_REF][START_REF] Mortillet | Rapport sur les monuments mégalithiques de la Corse[END_REF] flourished in the first half of the fourth millennium B.C. As regards burials, there seem in early neolithic times to have different customs in the different regions of Corsica : Nebbiu, Alta Rocca, Purti Vecchju, Aiacciu, Sartè, Taravu. In this paper we are interested in the megaliths of the Nebbiu Region [START_REF] Leandri | Les mégalithes du Monte Revincu[END_REF][START_REF] Leandri | Les sites du Monte Revincu et de Mamucci à Santu Petru Di Tenda : premiers résultats, actes des Xième Rencontres culturelles Interdisciplinaires organisées par le[END_REF][START_REF] De Lanfranchi | Le secrets des Mégalithes[END_REF] and specially those linked with the legend of the Lurcu situated around the Monte Revincu. One of our task here is to report the orientations that the builders selected for the megaliths which can be found near the Monte Revincu. We desire : (i) to show that the orientations of these megalithic sepulchres are highly nonrandom, so that the range of orientations for each group of monuments represents a burial custom characteristic of the culture that built these particular monuments ; (ii) to show that we have to consider possible astronomical motives. Although we are dealing with only less than 10 monuments, it is most unlikely that their orientations would be so similar purely by chance, and the signature in azimuth must result from some intention on the part of the builders. The rest of the paper is organize as follows. In section 2 we briefly present the oral story issued from the Corsican culture : the legend of the Giant, u Lurcu. Section 3 describe the megaliths which are involved in the previous legend. In section 4 we present some aspects of methodology concerning megaliths orientations measurements. This section is also dedicated to the presentation and the analysis of the orientation measurements performed on the megaliths of the legend. Conclusion and perspectives are drawn in the last section.

The legend : a fola di u Lurcu

This legend is belonging to the North Corsica oral culture. It is a legend from the Nebbiu country and involves people from Santu Petru di Tenda village. The place of the legend is around a set of megaliths, which can be found near the Monte Revincu hill. The story as told by the people from Santu Petru di Tenda is described below. Near the place named Casta was living the Lurcu, a sheppard, giant, cyclop, with long hairs. He was living there alone with his mother. We have pointed out on figure 1 the places where they were living as it is said in the legend coming from centuries and centuries by oral transmission. This giant was living in a house called a Casa di u Lurcu. It is a dolmen whose picture is given in Figure 1. Another dolmen where was living his mother according to the legend can be found 500m around the previous one. Figure 1 gives also a picture of this smaller one dolmen. The two dolmens are separated by a plateau named "Cima di Suarella" where a set of megaliths rectangular or circular structures can be found. In Figure 2 we present the northwest view from the plateau looking in direction of the Casa di u Lurcu. On the back of the picture one can see a sacred mountain called Monte Ghjenuva. The giant was very clever. It is said that he knew everything that could be learned at the time. Furthermore he was very powerful so that the people from Santu Petru di Tenda village living near the place called Casta decided to kill him. But he was difficult to catch so that they had to draw the giant into a trap.

Figure 2 : northwest view from "Cima di Suarella"

There is a place near Bocca Pivanosa where the Lurcu was used to come to drink the water of a spring. At night they put a pair of shoes full of pitch in order to capture him. The next morning the Lurcu came to drink and tried the shoes. The people from the village took him easily because of the shoes he was not able to run away. They were going to kill him near the place where there was the water, when the Lurcu told them a secret in order not to be killed: how to do a special cheese (called Brocciu) with sheep milk. After telling them this secret, he was going to tell another secret: what to do with the rest of the milk when the Brocciu has been done. But the mother from the plateau shouted in Corsican language: "Ùn palisà, chì tantu sì mortu" that means "Don't tell them anything because in any case they are going to kill you". They killed both the Lurcu and his mother. They buried the Lurcu near Bocca Pivanosa and the mother near Bocca Murellu. In both places one can find two early non-dolmenic tombs. As you read above this legend coming through ages by oral transmission is closely linked with the megaliths. In the next section we describe the megaliths of the legend around Monte Revincu.

The Megaliths around Monte Revincu

Different investigators of Corsican prehistory classify megalithic monuments according to datation, shape and functions [START_REF] Leandri | Les mégalithes du Monte Revincu[END_REF][START_REF] Leandri | Les sites du Monte Revincu et de Mamucci à Santu Petru Di Tenda : premiers résultats, actes des Xième Rencontres culturelles Interdisciplinaires organisées par le[END_REF][START_REF] De Lanfranchi | Le secrets des Mégalithes[END_REF] :

We give in figure 3 the positions of the two dolmens of the Monte revincu area and the two non dolmenic tombs (also called cists) associated with these dolmens. We pointed out the two dolmens of the legend (called Lurcu dolmen and Orca dolmen) and a third one on the top of Monte Revincu not involved in the legend (called Monte Revincu dolmen). We have also indicated in the figure the non-dolmenic tomb on Bocca Pivanosa and the one which has been destroyed named Tomba di Lurca.. We have also to mention two non-dolmenic tombs near the Capu Castincu area western than the Monte Revincu one's which will be measured and analysed in some future work.

Figure 3 : The megaliths near Monte Revincu

The first writing about these megaliths can be found in [START_REF] Mortillet | Rapport sur les monuments mégalithiques de la Corse[END_REF]. Since 1995 the excavations are due to F. Leandri [START_REF] Leandri | Les mégalithes du Monte Revincu[END_REF][START_REF] Leandri | Les sites du Monte Revincu et de Mamucci à Santu Petru Di Tenda : premiers résultats, actes des Xième Rencontres culturelles Interdisciplinaires organisées par le[END_REF]. The following dating has been given by F. Leandri :

•

Orientations of the megaliths

In this section we present the orientations of the megaliths involved in the legend presented in section 2 and detailed in section 3. We first give in sub-section 4.1 some aspects of methodology about measuring orientation and astronomical concepts.

In sub-section 4.2 the results of the measurements are given while the analysis of the collected orientations are detailed in sub-section 4.3

Some aspects of methodology

Our task in this paper is to report the orientations that the builders selected for the megaliths of Monte Revincu. We define these orientations to be the azimuths of the principal axis of the rectangular chamber of the measured tombs, in the direction from the closed end to the entrance. These directions (azimuths) were measured in June 2004 with compasses whose errors had already been established. In the case of dolmens there is no doubt that the relevant direction is from the interior towards the entrance [START_REF] Hoskin | Tombs, Temples and their orientations[END_REF]. However we have to precise our choice about the cists because the decision about which of the two directions of the principal axis of the cist becomes important. In the course of our fieldwork we also measured a number of additional non-dolmenic early tombs and the relevant data are also listed in sub-section 4.2.

In this case we choose the direction which seems to be the most obvious for us.

If and this is no more than a possibility the intention of the builders was astronomical and the megalithic sepulchres were constructed to face the setting or rising points of celestial objects, then angular altitude of the skyline must of course be taken into account when we calculate the declinations corresponding to these azimuths. The angular altitude has been measured with a hand-held clinometer. A third datum is needed for an astronomical interpretation : the latitude of the studied site. The use of a GPS is very useful for obtaining this last datum.

Given azimuth, angular altitude of the skyline, and latitude, the corresponding astronomical declination is easily found from a simple trigonometrical formula. In our case we used a computer program developed by C. Ruggles at the University of Leicester,England and which can be obtained using the web [START_REF] Ruggles | declination computation[END_REF].

Our measurements of the dolmens and early non-dolmenic tombs (azimuth, angular altitude, latitude and corresponding declination) are set out in the next sub-section.

In our fieldwork we came across the measurements of the three dolmens and four early nondolmenic tombs (cists) listed in section 3.. Their orientations are listed in Table 1 and Table 2. We have to point out that the dolmens 'Casa di u Lurcu' (Lurcu dolmen) 'Casa di l'Orca' (Orca dolmen) and involved two orientations because it has a clear defined entrance but also a clear defined passage with a different orientation. Once the data have been collected, the first question we have to answer is whether the previous collected orientations fall within a range. After this first analysis we have consider whether or not the motivation originated in the sky or whether it was terrestrial or even meteorological.

Orientations of the dolmens involved in the legend.

Orientations Analysis

In this sub-section we detail that although we are dealing only with 7 monuments and nine orientations, it is most unlikely that their orientations would be so similar purely by chance, and the signature in azimuth must result from some astronomical intention on the part of the builders.

From the two tables we can see that the azimuth are highly non random (from 60° to 130°) measures about 1/6 th of a circle (see figure 5). Such a concentration of axes cannot have come about by chance.

Furthermore the declination of the tables 1 and 2 show that all the nine orientations are in the correct range to face the rising sun or moon. The declinations are showed in figure 6 through an histogram pointed out clearly that all the declinations correspond to the rising of the sun or the moon. We point out in this article that the orientations of the great majority of these dolmens confirm the conclusions already described by Michael Hoskin [START_REF] Hoskin | Orientations of Corsican dolmens[END_REF][START_REF] Hoskin | Further orientations of Corsican dolmens[END_REF] and concerning the south Corsican dolmens. The orientation customs observed by builders of communal tombs in Corsica have been presented in detail in [START_REF] Hoskin | Tombs, Temples and their orientations[END_REF]. Eight dolmens of southern Corsica have been measured by Michael Hoskin [START_REF] Hoskin | Tombs, Temples and their orientations[END_REF][START_REF] Hoskin | Orientations of Corsican dolmens[END_REF][START_REF] Hoskin | Further orientations of Corsican dolmens[END_REF]. He showed that these dolmens faced easterly or southerly except one facing west. Furthermore he gave the following classification when dealing with astronomical orientations of funerary tombs in the Mediterranean basin :

0 0,5 1 1,5 2 2,
• SR (sun rising) :characterization of tombs having orientations within the range 60-130° or thereabouts. • SR/SC (sun rising/ climbing) :characterization of tombs having a wider range , from 60° to due south or thereabouts, • SS (sun setting) : characterization of tmbs facing the western half of the horizon within a range 240°-300°, • SD/SS (sun descending-setting) : characterization of tombs having a wider range from due south to 300°. The seven megalithic sepulchres of the Nebbiu region we investigated face roughly between north east and south east ; more exactly between azimuth 60° and 130° (see figure 5). We already point out that the builders seems to orient these monuments for reasons of astronomy. Furthermore we can deduce from figure 6 that all the tombs are SR according to Michael Hoskin classification. We can also pointed out that the Lurcu dolmen is faced the rising sun around midwinter sunrise while the orca dolmen is facing the rising of the sun around the midsummer sunrise. around Nebbiu Megaliths involving archaeology, Anthropology, GIS, 3D modelling representation, astronomy, Geology) Furthermore we are performing a set of crossanalysis in order to study the links between the megaliths structures of the plateau, the megaliths evidence that archeologists found on the top of Monte Ghjenuva (see figure 2) , the menhirs of Mamucci and the legends around the two dolmens. We give figures 7 an example of a structure which can be found on the plateau. Figure 8 points out a rectangular structure which seems to be an habitation and found on Cima Suarella.

Figure 1 :

 1 Figure 1 : the places of the legend

Figure 4 :

 4 Figure 4 : dolmens involved in the legend

FréquenceFigure 5 :

 5 Figure 5 : Histogram of the orientations of tombs in Nebbiu Area

Figure 6 :

 6 Figure 6 : Histogram of declination of tombs in Monte Revincu

Figure 7 :

 7 Figure 7 : Measurement of azimuth and angular altitude on Cima suarella • Collaboration with University of Cambridge and University of Leicester on a comparison between orientations of Corsican dolmens and dolmens in Brittany, Minorca and Sardinia will be undertaken through a common fieldwork involving one week in each region in spring 2005. • Collaboration with the DRAC of Corsica (Monte Revincu excavation by F. Leandri) : (i) Participation to the Summer 2005 Exposition in St Florent about Monte Revincu site organized by F. Leandri ; (ii) Definition of a common interdisciplinary project

Figure 8 :

 8 Figure 8 : Megalithic structure on Cima Suarella, Monte Revincu

Table 1 : Orientations of dolmens in Nebbiu zone

 1

	Az.	Alt.	Lat.	Dec.	Tomb
	°	°	°	°
	128	2	42.6	-25 1/2	Dolmen Monte Revincu
	110	4	42.6	-12	Casa di u Lurcu -entrance (Lurcu dolmen)
	130	4	42.6	-25 1/2	Casa di u Lurcu -passage
	75	21/2	42.6	12 1/2	Casa di l'Orca entrance (Orca dolmen)
	60	6°	42.6	25 ½	Casa di L'Orca-Passage

Table 2 : Orientation of early non-dolmenic tombs

 2

	Az.	Alt.	Lat.	Dec.	Tomb
	°	°	°	°
	87	3	42.6	5	Cist near u Lurcu
	95	4	42.6	-1	Cist near l'Orca -Tozzola 1
	108	4	42.6	-10 1/2	Cist near l'Orca -Tozzola 2
	68	2	42.6	17	Tomba di u Lurcu -Pivanosa