Mostapha Diss
email: diss@gate.cnrs.fr.

Abdelmonaim Tlidi

Another perspective on Borda's paradox

Keywords: Voting, Geometry, Borda's Paradox, Condorcet Pairwise Procedure, Borda, Plurality, Negative Plurality, Weighted Scoring Rules. JEL codes: D71, D72 Voting, Geometry, Borda's Paradox, Condorcet Pairwise Procedure, Borda, Plurality, Negative Plurality, Weighted Scoring Rules. JEL classification: D71, D72

This paper presents the conditions required for a profile in order to never exhibit either the strong or the strict Borda paradoxes under all weighted scoring rules in three-candidate elections. The main particularity of our paper is that all the conclusions are extracted from the differences of votes between candidates in pairwise majority elections. This way allows us to answer new questions and provide an organized knowledge of the conditions under which a given profile never shows one of the two paradoxes.

Introduction

Collective rationality has been a central property of social choice in economic theory. The eighteenth century saw the first thinking about various counterintuitive or paradoxical events that could occur when a group of voters aim to select a best candidate from a set of available candidates. Marquis de Condorcet (1785) and Borda (1781) have exhibited a difficulty concerning collective rationality through a given configuration of individual preferences that could create non coherent social outcomes using the well-known Plurality rule. Under this rule, each voter ranks the candidates in her order of preference and then we count the number of times each candidate is ranked first. Finally, the candidate who receives a relative majority (i.e., more votes than any of her competitors) is elected. Condorcet (1785) proposed a system for electing candidates who truly command pairwise majority support. In this system, a voter has the opportunity to rank all candidates from best to worst and once the individual ballots are submitted, the system considers all of the different pairwise majorities of two candidates against one another. The winner, also known as the Condorcet winner, is the candidate that wins all of the pairwise elections she is involved in. Borda (1781) also had serious doubts about the Plurality system, and therefore, defended a method which is today named after him. If a group must rank m candidates such that each individual has already ranked them, the Borda voting rule determines the winner as follows. In each individual ranking, Borda rule assigns m -1 to the top-ranked candidate, m -2 to the second-ranked, until the candidate ranked last who receives zero points. For each of the candidates, the points are summed across all the ballots, and the candidate with the most points is the winner.

One of the particularly troublesome observation made by Borda ( 1781) is related to the possibility for Plurality rule to elect the Condorcet loser, i.e., the candidate who is defeated by every other candidate in pairwise majority elections. Indeed, when the alternative ranked first by more voters than any other alternative is defeated by every other alternative in pairwise majority elections, then we refer to such an outcome as the strong Borda paradox. This defines the Borda's major concern in his discussion initiated with Condorcet about such issues. However, Plurality rule can exhibit a more problematic collective outcome since it can completely reverse the ranking of the pairwise majority elections, i.e., the loser by the pairwise majority rule is elected and, at the same time, the winner by the pairwise majority rule is ranked last by Plurality rule. This phenomenon is called the strict Borda paradox in social choice literature.

The current study focuses on the conditions under which the two forms of Borda's paradox might be observed under all weighted scoring rules. In general, under a weighted scoring rule, each candidate gets some points from each voter according to her position in the voter's preference, and the candidate with the highest aggregated score is elected. As a direct consequence of this definition, both Borda and Plurality rules are considered as specific examples of this category of voting rules. We begin with a description of the related literature and then we present the originality of the contributions presented in our paper.

Related literature

Many research papers have already analyzed the occurrence of the two forms of Borda's paradox. Indeed, we know from the literature that Borda's paradox is not limited to elections that use Plurality rule. More precisely, all weighted scoring rules other than Borda rule can exhibit the strict and the strong Borda paradoxes. [START_REF] Daunou | A paper on elections by ballot[END_REF] showed that Borda rule cannot rank the Condorcet winner in last place in an election, which means that the strict Borda paradox cannot occur. In addition, both [START_REF] Gärdenfors | Positionalist voting functions[END_REF] and [START_REF] Smith | Aggregation of preferences with variable electorate[END_REF] proved that Borda rule is the only weighted scoring rule that has this property for sufficiently large number of voters. In other words, the strict Borda paradox can occur for all weighted scoring rules other than Borda rule. [START_REF] Fishburn | Borda's rule, positional voting, and Condorcet's simple majority principle[END_REF] extend these results and show that Borda rule is the only weighted scoring rule that cannot select the Condorcet loser for sufficiently large number of voters. That is to say that Borda rule is the only weighted scoring rule that cannot exhibit the strong Borda paradox. [START_REF] Gehrlein | On the probability of observing Borda's paradox[END_REF] examined the probability that the two paradoxes will be observed with Plurality rule and Negative Plurality rule (another specific weighted scoring rule, which is defined later) under a well-known assumption about the voters' preferences. [START_REF] Diss | Borda's paradox with weighted scoring rules[END_REF] extend that analysis by considering all the class of weighted scoring rules and taking into account different assumptions about the individuals' preferences. Notice that Gehrlein and Lepelley (2010b) also analyzed the impact that degrees of mutual coherence among voters' preferences will have on the likelihood of observing the two forms of Borda's paradox under Plurality rule and Negative Plurality rule. [START_REF] Kamwa | Scoring rules and preference restrictions: The strong Borda paradox revisited[END_REF] use an analytical approach to identify the range of all weighted scoring rules that never exhibit the strong Borda paradox under some restrictions on the individuals' preferences. In addition, they provide the likelihood of the strong Borda paradox given these restrictions of preferences. Finally, it is worthwhile to mention that some empirical studies have been conducted in order to discuss if the two forms of Borda's paradox have occurred in real-life elections. The reader is refereed to [START_REF] Diss | Borda's paradox with weighted scoring rules[END_REF] and Gehrlein and Lepelley (2010a).

Our contributions

We extend the results of [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] in order to answer questions about the required conditions for a profile1 to never exhibit either the strong or the strict Borda paradoxes under all weighted scoring rules and any given number of voters in threecandidate elections. The new technique developed in [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] identifies all profiles that satisfy any specified pairwise and weighted criteria, such as where Condorcet winner is ranked first by any weighted scoring rule. More exactly, all the conclusions are extracted from the differences of votes between candidates in pairwise majority elections.

By doing so, our results allow us to precise, for instance, if any weighted scoring rule can never exhibit the two forms of Borda's paradox when at least one candidate beats another by receiving over 65% of the vote, or where nobody receives more than 75% of the vote. The approach that we consider in this paper affords an opportunity to advance our understanding of which profiles create the strict and the strong Borda paradoxes. In other words, this paper provides an organized knowledge of the conditions for a profile to (never) show Borda's paradox. In sum, we have investigated a simple framework allowing us to answer a large variety of new questions. For instance, the results of this paper allow us to:

(i) Determine the minimum number of voters needed for a profile to show either the strong or the strict Borda paradoxes when the differences between candidates and the weighted scoring rule are already determined.

(ii) Give the differences between candidates in the pairwise election outcomes required for a profile to never exhibit one of the two paradoxes for a given weighted scoring rule and a fixed number of voters.

(iii)

Describe what range of weighted scoring rules could possibly accompany a given number of voters and specified differences between candidates in the pairwise election outcomes in order for a profile to never exhibit one of the two paradoxes.

The rest of this paper is structured as follows. Section 2 presents the basic terminology, including the geometric tools needed to understand the next developments of this paper. Section 3 describes our results by giving the conditions that allow to never exhibit Borda's paradox. These results are described in Theorems (1-4) and Corollaries (1-4). Section 4 presents some comments and conclusions, and finally proofs are given in Section 5.

Preliminaries

Basic Framework

Throughout the paper, we will consider the framework of three-candidate elections that can be defined by a set of candidates, or alternatives, {A, B, C} and a set of n voters {1, . . . , n}. A linear order is a transitive, antisymmetric, and total relation. Each voter is assumed to have a linear order on the set of candidates from the most desirable candidate to the least desirable one. In addition, each voter is assumed to act according to her true preferences (i.e., to vote sincerely). The six possible strict rankings that voters might have are presented as follows:

No. Ranking No. Ranking No. Ranking

1 A B C 2 A C B 3 C A B 4 C B A 5 B C A 6 B A C (1)
The notation A B C means that voters have candidate A as most preferred, candidate C as least preferred, and candidate B as middle-ranked. In this context, a profile is a sequence of n linear orders over {A, B, C}. Each profile can be defined by the vector ñ = (n 1 , n 2 , . . . , n 6 ), where in parentheses, we refer the number of voters n i endowed with each of the six orders such that 6

i=1 n i = n. For instance, n 1 is the number of voters endowed with the linear order A B C.

Given two candidates A and B, we shall write AMB to say that candidate A beats candidate B in a pairwise majority election. This occurs when a strict majority of voters prefer A to B. A given candidate is said to be a Condorcet winner if she wins in all of her pairwise elections. For instance, using our notations in (1), A is a Condorcet winner if AMB and AMC, which is equivalent to n 1 + n 2 + n 3 > n 4 + n 5 + n 6 and n 1 + n 2 + n 6 > n 3 + n 4 + n 5 , respectively. Conversely, a given candidate is said to be a Condorcet loser if she loses to everyone. For instance, C is a Condorcet loser if AMC and BMC, which is equivalent to

n 1 + n 2 + n 6 > n 3 + n 4 + n 5 and n 1 + n 5 + n 6 > n 2 + n 3 + n 4 , respectively.
Without loss of generality, we assume throughout the paper that AMB and BMC. With this assumption, if there is a strict transitive outcome of the paired comparisons, it has the form AMBMC. In other words, if the election has a Condorcet winner, it always is candidate A, and candidate C always is the Condorcet loser. In contrast, if there is a cyclical outcome, it always has the form AMB, BMC and CMA. This assumption does not affect the generality of our results since a name change converts any other situation into our framework.

Let XY denote the difference between X's and Y 's majority vote in the pairwise comparison between X and Y .2 Illustrating with a simple example with n = 85 voters distributed as follows: n 1 = 25, n 2 = 15, n 3 = 20, n 4 = 5, n 5 = 10 and n 6 = 10. It results that AB = 60 -25 = 35, BC = 45 -40 = 5 and AC = 50 -35 = 15. Naturally, three immediate consequences follow from the XY values. First, the larger the XY value is, the better X does against Y . Second, our assumption of AMB and BMC is equivalent to AB > 0 and BC > 0, respectively. Third, the equation ( 2) is always satisfied. XY = -Y X.

(2)

Similarly to [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF], we will use a technical assumption, which is needed to separate the different cases of our results. This condition is defined as follows:

Definition 1 A profile satisfies the strongly non-cyclic condition if AC ≥ min(AB, BC).

(3)

With our assumption of AB > 0 and BC > 0, it is obvious that if the condition (3) is satisfied, this leads to AC > 0. On the contrary, it is important to point out that if the condition (3) fails to hold because of weaker AC, our assumption of AB > 0 and BC > 0 does not necessarily define a cycle. A simple example with AB = 6, BC = 4 and AC = 2 illustrates this fact.

In this paper, we are interested in the well-known weighted scoring rules, also called positional rules. Any weighted scoring rule in three-candidate elections can be defined by the vector S = (1, s, 0) such that 0 ≤ s ≤ 1. In other words, each of the n voters ranks the 3 candidates and we assign 1 point to the one ranked first, s points to the one ranked second, and 0 to the candidate ranked last. Predictably, the candidate that accumulates the most points, summed over all voters, wins the election. In this setting, the most commonly studied weighted scoring rules are Plurality rule with s = 0, Negative Plurality rule with s = 1 and Borda rule with s = 1 2 . Given two candidates A and B, we shall write ASB to say that candidate A beats candidate B when the weighted scoring rule S is used. As a consequence, A is the overall winner of the election if ASB and ASC.

Geometry

We follow the Saari's methodology and some of his geometric arguments using the wellknown triangle. In the current section, we only outline a summary of the very basic steps needed to understand the next developments of this paper. For a detailed description of Saari's voting geometry, the reader is referred to [START_REF] Saari | A dictionary for voting paradoxes[END_REF][START_REF] Saari | Basic Geometry of Voting[END_REF][START_REF] Saari | Geometry of voting[END_REF]. 3 We particularly use the new way to analyze three-candidate relationships among pairwise and positional election outcomes that is developed in [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF]. In the following, we present an overview of this methodology, and full details about this procedure are given in the original paper.

The geometric approach starts by representing three-candidate profiles in an equilateral triangle, where the name of each candidate is identified with a distinct vertex. Then, we draw a line between each vertex and the midpoint of the opposite segment (perpendicular bisector). What results is a partitioning of the triangle into six ranking regions. A point in the triangle defines a ranking according to its distances to the three vertices, where closer is better. We place n i , the number of voters with the ith preference ranking, in each region as illustrated in Figure 1-a. For instance, any point in the small triangle with a n 1 value is closest to A, next closest to B, and farthest from C, so it has the A B C strict linear order. It follows from this figure that, in a pairwise majority election between A and B, for instance, we just need to sum the numbers on each side of perpendicular bisector of the edge between A and B. In other words, the vertical line separates preferences where A is preferred to B (on the left) from the others where B is preferred to A (on the right). The analysis for the remaining two pairs is similar. This summing process is illustrated by the example in the left-hand side of Figure 1-b. This profile, which corresponds to our example in Section 2.1 with n = 85 voters, leads to AMB (60 votes for A and 25 for B), BMC (45 votes for B and 40 for C) and AMC (50 votes for B and 35 for C). Again, this is equivalent to AB = 35, BC = 5 and AC = 15.

The next step is to use this triangle to specify the outcome under a weighted voting rule S = (1, s, 0). Using the triangle in Figure 1-a, the score of each candidate under Plurality rule is given by the sum of entries in the two regions sharing the corresponding vertex. For instance, the score of Plurality of A is n 1 + n 2 . To put it in more general terms, the score of each candidate under the weighted scoring rule S = (1, s, 0) is equal to her {Plurality score} plus {s× the number of voters who have her second ranked}. This leads to, respectively, n 1 + n 2 + s(n 3 + n 6 ), n 5 + n 6 + s(n 1 + n 4 ) and n 3 + n 4 + s(n 2 + n 5 ), the scores of A, B and C. In our example described in the left-hand side of Figure 1-b, the scores of A, B and C are, respectively, 40 + 30s, 20 + 30s and 25 + 25s. With these scores, the Condorcet winner A is the winner under any weighted scoring rule S and the Condorcet loser C always is ranked second, while B always is ranked in the last position, with the exception of s = 1, where B and C are tied in the second position.

Using the XY values, [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] present the following notions.

Definition 2 For specified AB, BC and AC values, the essential profile is the profile with the smallest number of voters that have the specified XY values.

Definition 3 For a given number of voters n, a supporting profile is any profile that has the specified XY values of the essential profile. [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] present all essential profiles (four of them) and the corresponding number of voters in each one. The results can be summarized as follows:4

1. With the strongly non-cyclic condition, the essential profile has one of the Figure 2 a-c forms. The three possible cases are:

• If AB is the largest value, then the essential profile has n 4 = n 5 = n 6 = 0 (Figure 2-a). The number of voters in this essential profile is AB.

• If AC is the largest value, then the essential profile has

n 3 = n 4 = n 5 = 0 (Figure 2-b).
The number of voters in this essential profile is AC.

• If BC is the largest value, then the essential profile has n 2 = n 3 = n 4 = 0 (Figure 2-c). The number of voters in this essential profile is BC.

2. If the strongly non-cyclic condition is not satisfied, the "cyclic" essential profile5 has n 2 = n 4 = n 6 = 0 (Figure 2-d). The number of voters in this essential profile is AB + BC -AC.

3. In each essential profile, the number of voters endowed with the ith linear ranking X Y Z is given by e i such that:

e i = 1 2 [XY + Y Z]. (4)
The essential profile of the original n = 85 voters example is described in the middle of Figure 1 The decomposition approach is based on the fact that, to remove terms from the initial profile without affecting XY values, the method called reversal pairs 6 is the only way 7 as shown in [START_REF] Saari | Explaining all three-alternative voting outcomes[END_REF][START_REF] Saari | Disposing Dictators, Demystifying Voting Paradoxes[END_REF]. More precisely, to find the essential profile just remove as many reversal pairs as possible. Illustrating with the original example (25,15,20,5,10,10) with n = 85 voters. Removing pairs (5, 0, 0, 5, 0, 0) 8 , (0, 10, 0, 0, 10, 0), and (0, 0, 10, 0, 0, 10) creates its essential profile (20, 5, 10, 0, 0, 0) with 35 voters. Using an essential profile, it becomes possible to identify all possible profiles having the specified XY values (i.e., the supporting profiles) by only adding reversal pairs to the essential profile. With the original n = 85 voter example, 50 (85-35) voters must be added in appropriate ways in order to never affect the XY values. Naturally, the initial example presented in the lefthand side of Figure 1-b is a possibility by adding pairs (5, 0, 0, 5, 0, 0), (0, 10, 0, 0, 10, 0), and (0, 0, 10, 0, 0, 10). Another possibility would be to add pairs (1, 0, 0, 1, 0, 0), (0, 2, 0, 0, 2, 0) and (0, 0, 22, 0, 0, 22) leading to the supporting profile (21,7,32,1,2,22) or also adding pairs (2, 0, 0, 2, 0, 0), (0, 5, 0, 0, 5, 0) and (0, 0, 18, 0, 0, 18) leading to the supporting profile (22,10,28,2,5,18). More generally, this can be done by adding pairs (α, 0, 0, α, 0, 0), (0, β, 0, 0, β, 0), and (0, 0, γ, 0, 0, γ) to the essential profile such that α + β + γ = 50 2 . In our example in the right-hand side of Figure 1-b, the decomposition method leads to α = 5 and β = γ = 10.

To summarize, it follows from [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] that, with the strongly noncyclic condition and if XY is the maximum pairwise victory, the number of pairs added to the essential profile is given by 2(α + β + γ) such that:

9 α + β + γ = 1 2 [n -XY ] (5)
This is true because, as mentioned above, the number of voters in an essential profile, when XY is the maximum pairwise victory, is XY . If the strongly non-cyclic condition is not satisfied, the number of pairs added to the essential profile is given by 2(α + β + γ) and the value of α + β + γ becomes:

α + β + γ = 1 2 [n -(AB + BC -AC)] (6)
6 For instance, the preference A B C type has a reverse given by C B A. It is clear that, removing the same number of voters with reversal pairs does not change XY values.

7 This is not true with more than three candidates.

8 It means to remove 5 voters of each of both A B C and C B A preference types. 9 Notice that, n and all XY values have the same parity. This is true because, for each pair of candidates X and Y , we have n = {X's vote + Y 's vote}, which leads to {X's vote} = 1 2 [n + XY ] since XY = {X's vote -Y 's vote}. In order for {X's vote} to be an integer value, it follows that n and all XY values either are odd integers, or all are even integers. In other words, α + β + γ always is an integer value and, even more, it has the same parity as n and all XY values. It is important to point out that this approach allows [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] to collect all profiles in equivalence classes by decomposing each profile into the part giving the essential profile -this part determines the XY values -and the part that affects only positional outcomes. Two profiles belong to the same equivalence class if they have in common the same specified XY values. It is clear that each equivalence class includes the class's essential profile as a component plus the other profiles, i.e., supporting profiles. As noted by [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF], this decomposition simplifies discovering and proving new conclusions. This approach will be used throughout this paper to tackle the strong and the strict Borda paradoxes.

Results

This section describes the restrictions on XY values that allow to never exhibit Borda's paradox under all weighted scoring rules S = (1, s, 0) and any number of voters n. Proofs are in Section 5. We begin by providing our results for the strong Borda paradox.

Strong Borda Paradox

In our setting, the strong Borda paradox occurs if the Condorcet loser C is the winner of the weighted scoring rule S. In other words, CSA and CSB. Theorems 1 and 2 specify what happens with the strong Borda paradox for all S. The particular cases of Plurality (s = 0) and Negative Plurality (s = 1) rules are described in Corollaries 1 and 2.

Theorem 1 If the strongly non-cyclic condition is satisfied, the supporting profiles that never exhibit the strong Borda paradox are described as follows:

1. When AB is the largest pairwise victory, then three cases arise:

a. If s > 1 2 and (3s -1)AB + 2(1 -s)AC + sBC ≥ (2s -1)n b. If s < 1 2 , (1 -s)(AC -AB) + 2sBC < 0 and (1 -s)AB + 2(1 -s)AC + sBC ≥ (1 -2s)n c. If s < 1 2 , (1 -s)(AC -AB) + 2sBC ≥ 0 and (1 -s)AC + sBC ≥ 1 -2s 3 (n -2 + 2F (U ))
2. When AC is the largest pairwise victory, then two cases arise:

a. If s > 1 2 and (1 -s)AB + 2sAC + sBC ≥ (2s -1)n b. If s < 1 2 and (1 -s)AC + sBC ≥ 1 -2s 3 (n -2 + 2F (U ))
3. When BC is the largest pairwise victory, then three cases arise:

a. If s > 1 2 , 2(1 -s)AB + s(AC -BC) ≥ 0 and (1 -s)AB + 2sAC + sBC ≥ (2s -1)n b. If s > 1 2 , 2(1 -s)AB + s(AC -BC) < 0 and (s -1)AB + sAC + 2sBC ≥ (2s -1)n c. If s < 1 2 and sAC + (1 -s)BC ≥ 1 -2s 3 (n -2 + 2F (U )) Such that U = (1 -s)(AC -AB) + 2sBC 2(1 -2s) and F (U ) is its fractional part. 10
If the strongly non-cyclic condition is not satisfied, the requirements for the non existence of Borda's paradox are similar in form.

Theorem 2 If the strongly non-cyclic condition is not satisfied, the supporting profiles that never exhibit the strong Borda paradox are described as follows:

a. If s > 1 2 , AB + (1 -s)AC -sBC ≥ 0 and (3s -1)AB + 2(1 -s)AC + sBC ≥ (2s -1)n b. If s > 1 2 , AB + (1 -s)AC -sBC < 0 and (3s -2)AB + (1 -s)AC + 2sBC ≥ (2s -1)n c. If s < 1 2 , (1 -s)AB -sAC -BC > 0 and (1 -s)AB + 2sAC + (2 -3s)BC ≥ (1 -2s)n d. If s < 1 2 , (1 -s)AB -sAC -BC ≤ 0 and (1 -s)BC + sAC ≥ 1 -2s 3 (n -2 + 2F (U ))
Such that U = (s -1)AB + sAC + BC 2(1 -2s) and F (U ) is its fractional part.

The immediate consequence is that these theorems allow to deduce the conditions under which the strong Borda paradox does not occur with Plurality (s = 0) and Negative Plurality (s = 1) rules. A listing of what can happen follows:

Corollary 1 For Plurality rule, the supporting profiles under which the strong Borda paradox never occurs are described as follows: 10 Recall that for all U in Z, we have U = U -F (U ) such that U stands for the greatest integer less than or equal to U . Notice that F (U ) is defined in the same way for positive and negative numbers. However, it is important to precise that for instance -1.3 = -2 leading to F (-1.3) = 0.7 and 1.3 = 1 leading to F (1.3) = 0.3.

11 Notice that, in this case, U = AC -AB 2 and F (U ) = 0 since AB and AC have the same parity (see discussion in footnote 9). The same remark can be used for some other parts of our corollaries.

Corollary 2 For Negative Plurality rule, the supporting profiles under which the strong Borda paradox never occurs are described as follows:

1. If the strongly non-cyclic condition is satisfied: a) When AB dominates and 2AB + BC ≥ n. b) When AC dominates and 2AC + BC ≥ n. c) When BC dominates and AC + 2BC ≥ n.

2. If the strongly non-cyclic condition is not satisfied: a) When AB ≥ BC and 2AB+BC ≥ n. b) When AB < BC and AB + 2BC ≥ n.

We can deduce the following facts from Theorems 1 and 2 and Corollaries 1 and 2.

• The first lesson to be drawn is that, for fixed values of XY and a given weighted scoring rule S, our results provide the minimum number of voters to have the strong Borda paradox. For instance, consider the XY values in Figure 1 supporting profiles can be constructed such that the strong Borda paradox occurs. In other words, for a profile to show the strong Borda paradox, the smallest value is α + β + γ = 16 leading to n = 67. In this case, suppose that γ = 16 and α = β = 0, which is equivalent to the profile ñ = (20, 5, 10, 0, 0, 0) + (0, 0, 16, 0, 0, 16) = (20, 5, 26, 0, 0, 16). As a consequence, it is easy to verify that this profile exhibits the strong Borda paradox under Plurality rule. For Negative Plurality rule and the same XY values, the strong Borda paradox never occurs if n ≤ 75, while some supporting profiles showing the strong Borda paradox can be found if n > 75, i.e., α + β + γ > (75 -35)/2 = 20. Suppose for example that α + β + γ = 21 leading to n = 77, which is, in this case, the smallest number of voters for a profile to show the paradox. Let assume that β = 21 and α = γ = 0, which is equivalent to ñ = (20, 5, 10, 0, 0, 0) + (0, 21, 0, 0, 21, 0) = (20, 26, 10, 0, 21, 0). Clearly, this profile gives rise to the strong Borda paradox under Negative Plurality rule.

• The second lesson to be learned is that, for a given weighted scoring rule S and a fixed number of voters n, our results provide the conditions over the XY values required for a supporting profile to never exhibit the strong Borda paradox. For instance, with Plurality rule and n = 85 (the same number of voters as Figure 1-b), we definitely are assured to only have supporting profiles that never exhibit the strong Borda paradox when these profiles verify the strongly non-cyclic condition with AB dominating and AB + 2AC ≥ 85. In other words, if AC = 15 and BC = 5 for instance, then all supporting profiles with AB ≥ 55 never show the strong Borda paradox. If AB < 55, some supporting profiles exhibiting the paradox can be found. For instance, the profiles ñ = (20, 5, 35, 0, 0, 25) and ñ = (23, 7, 30, 3, 2, 20) with AB = 35.

• Another lesson learned from this paper is that our results provide all tools needed to describe what weighted scoring rule S could possibly accompany specified XY values and n in order for a supporting profile to never exhibit the strong Borda paradox. Let us consider a simple example with AB = 4, AC = BC = 2 and n = 10 (i.e., the strongly non-cyclic condition is satisfied and AB dominating). Then it is easy to show that all weighted scoring rules S with 1 2 < s ≤ 1 never exhibit the strong Borda paradox. This is true because the part 1-a in Theorem 1 leads to s > 1 2 and s ≤ 1. In addition, with the same example, all weighted scoring rules in the range 1 7 ≤ s < 1 2 never show the strong Borda paradox. More precisely, the part 1-b in Theorem 1 leads to

1 7 ≤ s < 1 3 while the part 1-c gives rise to 1 3 ≤ s < 1 2
. Let us explain more precisely the latter case.

First, the condition (1 -s)(AC -AB) + 2sBC ≥ 0 in 1-c of Theorem 1 leads to s ≥ 1 3 .

In addition, in our example, the value of F (U ) in the condition (1 -s)AC + sBC ≥ 1 -2s 3 (n-2+2F (U )) does not matter. Indeed, we know that 0 ≤ F (U ) 1. In addition,

if

F (U ) = 0, the condition (1 -s)AC + sBC ≥ 1 -2s 3 (n -2 + 2F (U )) becomes s ≥ 1 8 . Moreover, if F (U ) 1, the condition (1 -s)AC + sBC ≥ 1 -2s 3 (n -2 + 2F (U )) becomes s 1 5 . Both s ≥ 1 8
and s 1 5 are relaxed in comparison with the first requirement

s ≥ 1 3 .
Notice that, if we consider our example in Figure 1-b where n = 85, AB = 35, AC = 15

and BC = 5, we find that all weighted scoring rules in the ranges 1 2 < s ≤ 8 9 and 2 11 ≤ s < 1 2 never show the strong Borda paradox.

• Our last remark deals with the comparison of all weighted voting rules S and how the conditions change when s varies. In Theorems 1 and 2, it is easy to see that when

s → 1 2
, then the right-hand sides (2s-1)n for s > 1 2 and either (1-2s)n or 1 -2s 3 (n-2+ 2F (U )) for s < 1 2 tend to zero. By avoiding a lower bound with n, the conditions become significantly more relaxed than those required for the other weighted scoring rules.

For instance, the right-hand side of each condition of Corollary 1 for Plurality rule and Corollary 2 for Negative Plurality rule involve the n value. In other words, the region for which the strong Borda paradox does not occur is much more important when s → 1 2 . This result is not new and is consistent with the existing literature dealing with the probability of the strong Borda paradox under various scenarios (e.g., [START_REF] Diss | Borda's paradox with weighted scoring rules[END_REF][START_REF] Gehrlein | On the probability of observing Borda's paradox[END_REF]. However, one of the particularity of our paper is that this result is general and it does not depend on the restrictions on the individuals' preferences.

Strict Borda Paradox

In our setting, the strict Borda paradox occurs if the Condorcet loser C is ranked first and the Condorcet winner A is ranked last by the weighted scoring rule S = (1, s, 0). In other words, the used weighted scoring rule would completely reverse the pairwise ranking on the candidates. Theorems 3 and 4 below generalize the restrictions on XY under which the strict Borda paradox occurs for all rules S and any number of voters n. The particular cases of Plurality (s = 0) and Negative Plurality (s = 1) rules are described in Corollaries 3 and 4.

Theorem 3 If the strongly non-cyclic condition is satisfied, the supporting profiles that never exhibit the strict Borda paradox are described as follows:

1. When AB is the largest pairwise victory, then three cases arise:

a. If s > 1 2 , sAB + (1 -s)AC ≥ 2s -1 3 (n -4 + 4F (U )) b. If s < 1 2 , (1-s)(AC -AB) +2sBC ≥ 0 and sBC + (1-s)AC ≥ 1 -2s 3 (n-4 +4F (V )) c. If s < 1 2 , (1 -s)(AC -AB) + 2sBC < 0 and 2(1 -s)AB -sBC + (1 -s)AC ≥ (1 -2s)n
2. When AC is the largest pairwise victory, then two cases arise:

a. If s > 1 2 and (1 -s)AB + sAC ≥ 2s -1 3 (n -4 + 4F (U )) b. If s < 1 2 and sBC + (1 -s)AC ≥ 1 -2s 3 (n -4 + 4F (V ))
3. When BC is the largest pairwise victory, then three cases arise:

a. If s > 1 2 , 2(1 -s)AB + s(AC -BC) ≥ 0 and (1 -s)AB + sAC ≥ 2s -1 3 (n -4 + 4F (U )) b. If s > 1 2 , 2(1 -s)AB + s(AC -BC) < 0 and (s -1)AB + 2sBC + sAC ≥ (2s -1)n c. If s < 1 2 and sAC + (1 -s)BC ≥ 1 -2s 3 (n -4 + 4F (V ))
Where U = AB+(1-s)AC-sBC 2(2s-1)

, V = (1-s)(AC-AB)+2sBC 2(1-2s)
and F (U ) and F (V ) are the fractional parts of U and V , respectively.

If the strongly non-cyclic condition is not satisfied, the requirements for the non existence of the strict Borda paradox are described below.

Theorem 4 If the strongly non-cyclic condition is not satisfied, the supporting profiles that never exhibit the strict Borda paradox are described as follows:

a. If s > 1 2 , AB + AC -s(BC + AC) ≥ 0 and sAB + (1 -s)AC ≥ 2s -1 3 (n -4 + 4F (U )) b. If s > 1 2 , AB + AC -s(BC + AC) < 0 and (3s -2)AB + (1 -s)AC + 2sBC ≥ (2s -1)n c. If s < 1 2 , BC -AB + s(AB + AC) < 0 and 2(1 -s)AB + (1 -3s)BC + sAC ≥ (1 -2s)n d. If s < 1 2 , BC -AB + s(AB + AC) ≥ 0 and (1 -s)BC + sAC ≥ 1 -2s 3 (n -4 + 4F (U ))
Where U = AB+(1-s)AC-sBC

2(2s-1) , V = (1-s)(AC-AB)+2sBC 2(1-2s)
and F (U ) and F (V ) are the fractional parts of U and V , respectively.

The conditions under which the strict Borda paradox never occurs with the well-known Plurality (s = 0) and Negative Plurality (s = 1) rules are consequently described in the next corollaries.

Corollary 3 For Plurality rule, the supporting profiles under which the strict Borda paradox never occurs are described as follows:

1. If the strongly non-cyclic condition is satisfied: a) When AB dominates and 2AB + AC ≥ n. b) When AC dominates and 3AC ≥ n -4. c) When BC dominates and 3BC ≥ n -4.

2. If the strongly non-cyclic condition is not satisfied: a) When AB > BC and 2AB+BC ≥ n. b) When AB ≤ BC and 3BC ≥ n -4.

Corollary 4 For Negative Plurality rule, the supporting profiles under which the strict Borda paradox never occurs are described as follows:

1. If the strongly non-cyclic condition is satisfied: a) When AB dominates and 3AB ≥ n -4. b) When AC dominates and 3AC ≥ n-4. c) When BC dominates and 2BC+AC ≥ n.

2. If the strongly non-cyclic condition is not satisfied: a)When AB ≥ BC and 3AB ≥ n-4. b) When AB < BC and AB + 2BC ≥ n.

Similarly to the previous section, we can deduce some interesting facts from Theorems 3 and 4 and Corollaries 3 and 4.

• For fixed values of XY and a given weighted scoring rule S, it is possible to deduce the minimum number of voters to ensure that a supporting profile exhibiting a strict Borda paradox can be constructed. It is clear that there exist supporting profiles for which the strict Borda paradox occurs if the inequalities are reversed in each case. For instance, let us again consider the XY values in Figure 1-b. We see from our results that the strict Borda paradox never occurs under Plurality rule if n ≤ 85, i.e., α + β + γ ≤ (85 -35)/2 = 25. Hence, if n > 85, i.e., α + β + γ > 25, supporting profiles can be constructed such that the strict Borda paradox occurs. In other words, the minimum number of reversal pairs α+β +γ for which problems occur is 26. This leads to n = 87, the minimum number of voters for a profile to exhibit the strict Borda paradox for our XY values. An instance of this paradox is where γ = 26 and α = β = 0, which is equivalent to the profile ñ = (20, 5, 10, 0, 0, 0) + (0, 0, 26, 0, 0, 26) = (20, 5, 36, 0, 0, 26). It is easy to verify that the pairwise ranking on the candidates (AMBMC and AMC) is completely reversed by the Plurality rule. For Negative Plurality rule and the same XY values, the strict Borda paradox never occurs if n ≤ 109, while some supporting profiles can be found if n > 109, i.e., α + β + γ > (109 -35)/2 = 37. Suppose for example that α + β + γ = 38, i.e., n = 111. Let assume that α = 16, β = 22 and γ = 0, which is equivalent to ñ = (20, 5, 10, 0, 0, 0) + (16, 22, 0, 16, 22, 0) = (36, 27, 10, 16, 22, 0). This profile clearly gives rise to the strict Borda paradox under Negative Plurality rule and n = 111 is the minimum number of voters for a supporting profile to exhibit this paradox.

• For a given weighted scoring rule S and a fixed number of voters n, our results give the constraints over the XY values required for a profile to never show the strict Borda paradox. An instance can be given with Plurality rule and n = 85. If AB dominates with the strongly non-cyclic condition verified and 2AB + AC ≥ 85, then no supporting profiles can be found such that the strict Borda paradox occurs. Suppose for instance that AC = 15 and BC = 5. In this case, all supporting profiles with AB ≥ 35 never show the strict Borda paradox. If AB < 35, some supporting profiles exhibiting the paradox can be constructed. For instance, the supporting profiles ñ = (19, 5, 35, 0, 0, 26) and ñ = (19, 6, 34, 0, 1, 25) with AB = 33.

• For specified XY values and a given number of voters n, we can deduce from our results all weighted scoring rules that never exhibit the strict Borda paradox. Let consider the same values as the example considered in Section 3.1 where AB = 4, AC = BC = 2 and n = 10. This means that the strongly non-cyclic condition is satisfied and AB dominates. Hence all weighted scoring rules S with 1 2 < s ≤ 1 never exhibit the strong Borda paradox. More precisely, the part 1-a in Theorem 3 leads to s > 1 2 and either s ≤ 2 with F (U ) = 0 or s 8 7 with F (U ) 1. In addition, with the same example, all weighted scoring rules in the range 0 ≤ s < 1 2 never show the strict Borda paradox.

With the example described in Figure 1-b where n = 85, AB = 35, AC = 15 and BC = 5, we find that all weighted scoring rules s = 1 2 never show the strict Borda paradox.

• The previous remark allows us to point out that the ranges of weighted scoring rules that never show the strict Borda paradox are much larger that those for the strong Borda paradox. This is equivalent to say that the conditions of observing a strict Borda paradox that are listed in Theorems 3 and 4 are significantly more relaxed than the corresponding conditions for observing a strong Borda paradox in Theorems 1 and 2. For instance, as shown in the previous remark, with the example with n = 85, AB = 35, AC = 15 and BC = 5, we find that all weighted scoring rules s = 1 2 never show the strict Borda paradox. However, at the same time, we have found in Section 3.1 that all weighted scoring rules in the ranges

1 2 < s ≤ 8 9 and 2 11 ≤ s < 1 2 never
show the strong Borda paradox. The same remark is valid for the example with n = 10 where AB = 4 and AC = BC = 2. This result is also consistent with the literature on the probability of Borda's paradox (e.g., [START_REF] Diss | Borda's paradox with weighted scoring rules[END_REF][START_REF] Gehrlein | On the probability of observing Borda's paradox[END_REF].

• Similarly to our remark for the strong Borda paradox, it is easy to see from Theorems Negative Plurality rules. This result is consistent with the existing literature dealing with the probability of the strict Borda paradox under various scenarios. Again, our result is general since it does not depend on the condition made on the voters' preferences.

Concluding remarks

We have used a new technique developed in [START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] to answer questions about the required conditions for the non-existence of either the strong or the strict Borda paradoxes under all weighted scoring rules S = (1, s, 0) in three-candidate elections. This paper is reporting completely new results allowing us, for example, to (i) give the conditions over the XY values required for a supporting profile to never exhibit the strong and the strict Borda paradoxes for a given weighted scoring rule S and a fixed number of voters n. (ii) determine the minimum number of voters n needed for a supporting profile to show one of the two paradoxes when the XY values and the weighted scoring rule S are already determined. (iii) describe what range of weighted scoring rule S could possibly accompany specified XY and n values in order for a supporting profile to never exhibit one of the two paradoxes.

In other reports, we expect to adapt this technique in order to deal with other interesting questions. Our next objective will be to investigate the impact of an increasing degree of mutual coherence among the preferences of voters on some voting paradoxes for which many open questions still remain unanswered. For more details about metrics of group mutual coherence, the reader is referred to the book by Gehrlein and Lepelley (2010a) or their recent article in [START_REF] Gehrlein | Refining measures of group mutual coherence[END_REF].

Proofs

Theorems 1 and 3 are proved for the strongly non-cyclic case where AB has the largest victory. The other cases of these theorems as well as the case where the strongly noncyclic condition is not satisfied (Theorems 2 and 4) are proved in the same manner. Proofs are described to let the reader understand what steps were taken in order to provide our results. Complete proofs are available upon request from the authors.

Proof of Theorem 1

Recall that, in our setting, the strong Borda paradox occurs if the Condorcet loser C is the winner of the weighted scoring rule defined by the vector S = (1, s, 0), i.e., CSA and CSB. Suppose AB is the largest pairwise victory, which means that our framework corresponds to the Figure 2.a. In other words, any profile ñ = (n 1 , n 2 , . . . , n 6 ) can be written as ñ = (e 1 + α, e 2 + β, e 3 + γ, α, β, γ). By using the weighted scoring rule S, the scores of A, B and C are, respectively, e 1 + e 2 + α + β + s(e 3 + 2γ), β + γ + s(e 1 + 2α), e 3 + α + γ + s(e 2 + 2β).

It follows from these scores that CSA and CSB are equivalent to, respectively, e 3 -e 1 -e 2 + s(e 2 -e 3 ) > (1 -2s)β + (2s -1)γ, (7)

e 3 + s(e 2 -e 1 ) > (1 -2s)β + (2s -1)α. (8)
The (1 -2s) and (2s -1) terms on the right-hand side of each inequality differ in sign depending on whether s > 1 2 or s < 1 2 . Then, we must consider two possible cases:

• Case 1: s > 1 2 . In this case, (2s -1) > 0 and the inequalities (7-8) become As a consequence, the minimal conditions in (11-14) become:

α min = 0, (16)
and

γ min = 0, (17)
and

β min = -e 3 + e 1 + e 2 -s(e 2 -e 3 ) 2s -1 + 1. (18)
U stands for the integer part of the number U . In all cases, the values of α, β and γ must satisfy α ≥ α min , β ≥ β min and γ ≥ γ min . In addition, we know from (5) that

α + β + γ = q = 1 2 (n -AB) = 1 2 (n -e 1 -e 2 -e 3 ).
Then, the inequalities (16-18) lead to

-e 3 + e 1 + e 2 -s(e 2 -e 3 ) 2s -1 + 1 ≤ 1 2 (n -e 1 -e 2 -e 3 ). (19)
Notice that U 1 ≤ U 2 ⇐⇒ U 1 < U 2 + 1 if U 2 is an integer. Taking into account the fact that 1 2 (n -e 1 -e 2 -e 3 ) = 1 2 (n -AB) is an integer value since n and AB have the same parity (see discussion in footnote 9), the inequality ( 19) is equivalent to

-e 3 + e 1 + e 2 -s(e 2 -e 3 ) 2s -1 < 1 2 (n -e 1 -e 2 -e 3 ). (20)
Collecting terms using (4) leads to the condition (21), which defines the minimal requirement to guarantee the occurrence of the strong Borda paradox.

(3s -1)AB + 2(1 -s)AC + sBC < (2s -1)n. (21)
In other words, if the inequality ( 21) is reversed, then there is no profile for which the strong Borda paradox occurs.

• Case 2: s < 1 2 . In this case, (2s -1) < 0 and the inequalities (7-8) become -If (1 -s)(AC -AB) + 2sBC < 0. In this case,

α min = 0. (28)
Using ( 24), and (27-28) and taking into account that the values of α, β and γ must satisfy α ≥ α min , β ≥ β min , γ ≥ γ min and the equality α

+ β + γ = q = 1 2 (n -AB) = 1 2 (n -e 1 -e 2 -e 3 ), it follows that e 1 + e 2 -e 3 -s(e 2 -e 3 ) 1 -2s + 1 ≤ 1 2 (n -e 1 -e 2 -e 3 ). (29)
Using again U 1 ≤ U 2 ⇐⇒ U 1 < U 2 + 1 and collecting terms lead to

(1 -s)AB + 2(1 -s)AC + sBC < (1 -2s)n. (30)
Finally, if this inequality is reversed, then there is no profile for which the strong Borda paradox occurs.

-If (1 -s)(AC -AB) + 2sBC ≥ 0. In this case,

α min = -e 3 -s(e 2 -e 1 ) 1 -2s + 1. (31)
Using ( 24), ( 27) and ( 31) and the fact that the values of α, β and γ must satisfy α ≥ α min , β ≥ β min , γ ≥ γ min and the equality α Notice that U = U -F (U ), such that F (U ) is the fractional part of U . Collecting terms of (34) leads to

+ β + γ = q = 1 2 (n -AB) = 1 2 (n -e 1 -
(1 -s)AC + sBC < 1 -2s 3 (n -2 + 2F (U )). (35)
Such that U = -e 3 -s(e 2 -e 1 ) 1 -2s = (1 -s)(AC -AB) + 2sBC 2(1 -2s) . In the same way, if the inequality ( 35) is reversed, then there is no profile for which the strong Borda paradox occurs.

Q.E.D.

Proof of Theorem 3

Recall that, in our setting, the strict Borda paradox occurs if the Condorcet loser C is ranked first and the Condorcet winner A is ranked last by the weighted scoring rule S = (1, s, 0). This is equivalent to CSB and BSA. Suppose AB is the largest pairwise victory, which again refers to the framework of Figure 2.a. In other words, any profile ñ = (n 1 , n 2 , . . . , n 6 ) can be written as ñ = (e 1 + α, e 2 + β, e 3 + γ, α, β, γ). Using the weighted scoring rule S, the scores of A, B and C are, respectively, e 1 + e 2 + α + β + s(e 3 + 2γ), β + γ + s(e 1 + 2α), e 3 + α + γ + s(e 2 + 2β).

It follows that CSB and BSA are equivalent to, respectively, e 3 + s(e 2 -e 1 ) > (1 -2s)β + (2s -1)α,

-(e 1 + e 2 ) + s(e 1 -e 3 ) > (1 -2s)α + (2s -1)γ.

(37)

The (1 -2s) and (2s -1) terms on the right-hand side of each inequality differ in sign depending on whether s > 1 2 or s < 1 2 . Then, we must consider two possible cases:

• Case 1: s > 1 2 . In this case, (2s -1) > 0 and the inequalities (36-37) become

Knowing that α, β and γ all are positive integers, to ensure that a profile can be constructed, it follows that the minimal conditions for such a profile are where

γ min = 0, (40)
This is true because e 1 + e 2 + s(e 3 -e 1 ) > 0 and α min + -e 3 + s(e 1 -e 2 ) 2s -1 > 0. U stands for the integer part of the number U . In all cases, the values of α, β and γ must satisfy α ≥ α min , β ≥ β min and γ ≥ γ min . In addition, we know from (5) that α + β + γ = q = 1 2 (n -AB) = 1 2 (n -e 1 -e 2 -e 3 ). Then, the equations (40-42) lead to

Finally, if the inequality (48) is reversed, then there is no profile for which the strict Borda paradox occurs.

• Case 2: s < 1 2 . In this case, (2s -1) < 0 and the inequalities (36-37) become

Notice that e 1 + e 2 + s(e 3 -e 1 ) 1 -2s always is positive. However, -e 3 + s(e 1 -e 2 ) 1 -2s , which is equivalent to (1 -s)(AC -AB) + 2sBC 2(1 -2s) , can be either positive or negative and depends on the sign of (1 -s)(AC -AB) + 2sBC. In other words, two cases arise.

Figure 1 :

 1 Figure 1: Example when AB dominates

Figure 2 :

 2 Figure 2: The four essential profiles

1.

  If the strongly non-cyclic condition is satisfied: a) When AB dominates and AB + 2AC ≥ n. b) When AC dominates and 3AC ≥ n -2. 11 c) When BC dominates and 3BC ≥ n -2 2. If the strongly non-cyclic condition is not satisfied: a) When AB > BC and AB+2BC ≥ n. b) When AB ≤ BC and 3BC ≥ n -2.

  . In other words, the conditions to never exhibit the strict Borda paradox are significantly more relaxed with s → 1 2 in comparison with the other weighted scoring rules such as Plurality and

e 3

 3 + s(e 2 -e 1 ) 2s -1 > α -β, and -(e 1 + e 2 ) + s(e 1 -e 3 ) 2s -1 > γ -α. e 1 + e 2 + s(e 3 -e 1 ) 2s -1 .

and α min = e 1

 1 + e 2 + s(e 3 -e 1 ) 2s -1 + 1,(41)and β min = α min + -e 3 + s(e 1 -e 2 ) 2s -1 + 1.

2 e 1

 1 + e 2 + s(e 3e 1 -e 2 -e 3 ) -3. (44)Taking into account the fact that 1 2 (n -e 1 -e 2 -e 3 ) is an integer value and using again U 1 ≤ U 2 ⇐⇒ U 1 < U 2 + 1, the inequality (44) leads to2 e 1 + e 2 + s(e 3 -e 1 ) 2s -1 + -e 3 + s(e 1 -e 2 ) 2s -1 < 1 2 (n -e 1 -e 2 -e 3 ) -2. (45)Or equivalently,e 1 + e 2 + s(e 3 -e 1 ) 2s -1 + -e 3 + s(e 1 -e 2 ) 2(2s -1) < 1 4 (n -e 1 -e 2 -e 3 ) -1. (46)Notice that U = U -F (U ), such that F (U ) is the fractional part of U . The inequality (46) can be written ase 1 + e 2 + s(e 3e 1 -e 2 -e 3 ) + F (U ) -1.(47)Where U = e 1 + e 2 + s(e 3 -e 1 ) 2s -1 = AB + (1 -s)AC -sBC 2(2s -1) . Collecting terms leads to (48), which defines the minimal requirement to guarantee the occurrence of the strict Borda paradox.sAB + (1 -s)AC < 2s -1 3 (n -4 + 4F (U )).

e 3

 3 + s(e 2 -e 1 ) 1 -2s > β -α, and -(e 1 + e 2 ) + s(e 1e 2 + s(e 3 -e 1 ) 1 -2s .

  -b where AB = 35, AC = 15 and BC = 5. It follows that the strong Borda paradox never occurs under Plurality rule if n ≤ 65, i.e., α + β + γ ≤ (65 -35)/2 = 15. However, if n > 65, i.e., α + β + γ > 15,

  Knowing that α, β and γ all are positive integers, to ensure that a profile can be constructed, it follows that minimal conditions for such a profile are where

								β min = 0,	(24)
				and		γ >	-e 3 + e 1 + e 2 -s(e 2 -e 3 ) 1 -2s	,	(25)
				and			α >	-e 3 -s(e 2 -e 1 ) 1 -2s	.	(26)
	Notice that	-e 3 + e 1 + e 2 -s(e 2 -e 3 ) 1 -2s	always is positive. In other words,
				γ min =	-e 3 + e 1 + e 2 -s(e 2 -e 3 ) 1 -2s	+ 1.	(27)
	However,	-e 3 -s(e 2 -e 1 ) 1 -2s	, which is equivalent to	(1 -s)(AC -AB) + 2sBC 2(1 -2s)	, can be
	e 3 -e 1 -e 2 + s(e 2 -e 3 ) 1 -2s	> β -γ,	and	e 3 + s(e 2 -e 1 ) 1 -2s	> β -α. (22)
	Or equivalently,
	β < γ +	e 3 -e 1 -e 2 + s(e 2 -e 3 ) 1 -2s	,	and	β < α +	e 3 + s(e 2 -e 1 ) 1 -2s	. (23)

either positive or negative and depends on the sign of (1 -s)(AC -AB) + 2sBC. In other words, two cases arise for α min .

  e 2 -e 3 ), it follows Using again U 1 ≤ U 2 ⇐⇒ U 1 < U 2 + 1, the inequality (33) is equivalent to e 1 + e 2 -e 3 -s(e 2 -e 3 ) 1 -2s < 1 2 (n -e 1 -e 2 -e 3 ) --e 3 -s(e 2 -e 1 ) 1 -2s -1. (34)

	e 1 + e 2 -e 3 -s(e 2 -e 3 ) 1 -2s	+	-e 3 -s(e 2 -e 1 ) 1 -2s	+ 2 ≤	1 2	(n -e 1 -e 2 -e 3 ). (32)
	Or equivalently,

e 1 + e 2 -e 3 -s(e 2 -e 3 ) 1 -2s ≤ 1 2 (n -e 1 -e 2 -e 3 ) --e 3 -s(e 2 -e 1 ) 1 -2s -2. (33)

A rigorous definition of this notion is given in Section

2.1.

[START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF] use the notation P (X, Y ). Instead, we use the notation XY since it allows us to reduce the length of our theorems and corollaries.

See also[START_REF] Nurmi | Voting Procedures under Uncertainty[END_REF].

Notice that these results are found under the standard assumption that AB > 0 and BC > 0. This is established without loss of generality.

Recall that, if the strongly non-cyclic condition is not satisfied, this does not necessarily define a cycle. Although we use the name "cyclic" for this essential profile (to be consistent with[START_REF] Saari | Connecting pairwise and positional election outcomes[END_REF]), we need to consider AC > 0, in our framework of Borda's paradox, in order for C to be a Condorcet loser.

* The first author gratefully acknowledges financial support by the National Agency for Research (ANR)research program "Dynamic Matching and Interactions: Theory and Experiments" (DynaMITE) ANR-BLANC.

-If (1 -s)(AC -AB) + 2sBC < 0. In this case, to ensure that a profile can be constructed, it follows that the minimal conditions are where

and

and

Using (51-53) and taking into account that the values of α, β and γ must satisfy α ≥ α min , β ≥ β min , γ ≥ γ min and α + β + γ = q = 1 2 (n -AB) = 1 2 (n -e 1 -e 2 -e 3 ), it follows that e 1 + e 2 + s(e 3 -e 1 )

1

Using

If this inequality is reversed, then there is no profile for which the strict Borda paradox occurs. -If (1 -s)(AC -AB) + 2sBC ≥ 0. In this case,

and

and

Using (56-58) and the fact that α ≥ α min , β ≥ β min , γ ≥ γ min and α + β + γ = q = 1 2 (n -AB) = 1 2 (n -e 1 -e 2 -e 3 ), it follows 2 -e 3 + s(e 1 -e 2 ) 1 -2s + e 1 + e 2 + s(e 3 -e 1 ) 1 -2s + 3 ≤ 1 2 (n -e 1 -e 2 -e 3 ). (59)

Or equivalently,

Or equivalently,

Using U = U -F (U ) and collecting terms lead to

Such that U = -e 3 + s(e 1 -e 2 ) 1 -2s = (1 -s)(AC -AB) + 2sBC 2(1 -2s) . Finally, if the inequality ( 63) is reversed, then there is no profile for which the strict Borda paradox occurs.

Q.E.D.


C Bocage

H Delebecque

O Friedel

C Timsit

Ese

ESE 1000 A hardware toolkit for teaching computer architecture

come

We have developed a modular toolkit for hands-on experimentation which enables students to build and test a wide variety of computer architectures. This toolkit is based on a decomposition into three hierarchical layers: a layer of the CPU's internal modules (e.g. an ALU), a layer of internal processor modules (e.g. a cache module), and one of processor modules (e.g. a 68000 motherboard). This toolkit allows experimentation on a complex multiprocessor array as easily as experimentation on a simple pedagogical computer, with the same user interface and minimal changes in the environment. The toolkit has been designed to be driven either locally, through a pedagogical and powerful front panel, or through a remote host (such as any workstation supporting X Window and Motif, or a Macintosh).

C o n t e x t

Pedagogical Requirements

The ESE 1000 has been designed with the pedagogical objective of supplying our students with convenient building blocks and debugging tools for practical experiments. The hardware toolkit has been preferred to its software counterpart for two reasons: firstly, the students work with real hardware, with all its inherent physical constraints (timing, fan-out, characteristics dispersion), and secondly because up to now, software simulators that take such factors into account have been professional products, which make large demands on computer time and require extensive training for their use. It's feared that the difficulty of running such a product would divert the student's attention from the goal of the experiment by putting too much emphasis on the software aspects.

User Interface

For the OPIP experiment, students use the ESE 1000 in its least expensive stand-alone configuration. It offers a pedagogical front panel, which allows the display of all the necessary modules (two 16-bit registers, an ALU, a 4K word memory module, a multi-phase clock, an instruction register and discrete NAND gates), and their control via user-accessible inputs.

The advanced architectures that students experiment with in their third year, involve complex modules which require remote control and analysis capabilities. This defines the extended configuration of the ESE 1000, which includes a workstation or any personal computer able to offer a good human/machine interface. This interface is the main link between the students and the computers they are trying to design, and it must therefore provide the representation of every information exchanged with the ESE 1000 in the most pedagogical and transparent way. This ability to display the internal state of every module without distortion is a key point of the ESE 1000 since this allows the student's attention not to be diverted by technological considerations.

The pedagogical aims

The OPIP lab session is organized to emphasize the fundamentals of computer architecture, through the design of a very simple but complete processor, starting from gates and other very simple modules. The student has to design the sequencer, using the timing signals provided by the multi-phase clock, and the information he picks from the data path. The student can monitor the execution of particular instructions either step-by-step, analyzing each information transfer, or instructionby-instruction when the processor begins to work properly.

The ESE 1000 can work in two different modes to allow the student to start and/or stop the processor at will. In the first and normal mode, the student's hand-wired logic circuitry controls all the modules, and the multi-phase clock delivers its signals continuously. In the second mode, called the debugging mode, the student can interact with every module via the ESE 1000 pilot processor, which is able to sample the content of each module and to display it on the front panel. Moreover, in order to allow a workstation to sample the state of the ESE 1000 , the pilot processor can interpret its commands as well.

During the advanced experiments, students apply the concept of microprogramming, and discover its advantages and drawbacks by building various processors on the same hardware base. Moreover, they can investigate the design of a well-know industrial computer, namely a PDP-11, as well as other, more specialized processors, like a Lisp processor. The extended configuration also includes addressable registers and a microprogramming module, composed of an AMD 2910 and 4K of 112-bit word memory.

A three-layer hierarchical design

The ESE 1000 has been designed around three hierarchical levels, each of which corresponds to a bus. The first and most internal bus is divided into five 16-bit wide buses, and a 192-bit wide state/control bus. Three of the 16-bit wide buses are called argument buses and the two others result buses, based on their use by the ALU module. The state/control bus conveys every signal coming from, or going to, a module plugged into the internal bus. The meaning of each state/control signal is not predefined and is determined by the module.that uses it; therefore two modules requesting the same signal for different uses can't be plugged in at the same time. The amorphous nature of this bus is the key to the E S E 1000 's versatility. The internal bus requires a double-europe sized backplane, and plays the role of the on-chip bus of commercial microprocessors.

The second bus, called the external bus, is the main link between the internal bus and the outside world. In some advanced experiments, it can be used as a processor bus, connecting the processor to its central memory module and peripheral devices. The internal and external buses exchange information through a gateway. By default, the processor built around the internal bus is the master of the external bus, but it may, in some cases, have to support DMA and allow the pilot processor to access the memory modules.. The third and more widely accessible bus is a well-known industrial standard, the NuBus, designed by Texas Instruments. Its main purpose is to allow the design of multiprocessor arrays, by connecting several external buses together through special interface cards.

Related work

C o n c l u s i o n

It is still too early to draw general conclusions on its effectiveness. However, the students who have used it have greatly appreciated the clearness and precision of the tools developed for on-line analysis and debugging.

R e f e r e n c e s

  A previous hardware toolkit, called the ESE 10 , developed at the Computer Science Department in 1968, was in use until last June. The OPIP lab session described above was originally designed for this machine. The microprogramming module was added in 1973, allowing the development of the advanced projects, but also revealing the limitations of the ESE 10 . The ESE 10 was based on 16-bit wide modules connected together by students using 16 different wires, each one conveying a bit, whereas the ESE 1000 uses three-state 16-bit wide outputs, connected to general purpose buses. With the ESE 10 , the time spent by students in wiring the data path of their processors increased dramatically as experiments became more complex leading the students to focus their attention on multiple point-to-point connections, instead of considering whole computer design. Moreover, the DTL technology used in the ESE 10 , chosen for its good noise tolerance and its robustness, is no longer supported by integrated circuit manufacturers. Its excellent modularity and the versatility of its internal bus make ESE 1000 an open and expandable toolkit for experimenting with computer architectures. The differences between our approach, and those described elsewhere [Morgan.88], is the ability of the ESE 1000 t o support a broad variety of architectures, ranging from a simple 16-bit processor to complex multiprocessor configurations. There are other versatile toolkits which allow an equally wide range of experiments[Cutler & Ecker.87]. However these are strictly software simulators with their own specific properties and they serve a purpose which is different from our ESE 1000 as has already been pointed out above.


Laurent Dufossé

Mireille Fouillaud

Jean-Pascal Quod

Pr E B Gareth

Jones / Peter Pr

Proksch

Abdel - Raheem
email: a.abdel-raheem@mailcity.com

Ahmed Egypt

Bertrand Samuel
email: samuel.bertrand@univ-nantes.fr

Blanchet
email: eblanchet@obs-banyuls.fr

Burgaud Gaëtan
email: gaetan.burgaud@univ-brest.fr

Capon Rob

Chaurand Ludivine
email: ludivine.chaurand@gmail.com

Daoud Sahar
email: daoud.sahar@hotmail.com

Dias Camila
email: acamiladias@gmail.com

Dufosse Laurent
email: laurent.dufosse@univ-reunion.fr

Ebel Rainer

Grovel Olivier
email: olivier.grovel@univ-nantes.fr

Grunwald Alyssa
email: agrunwald@upei.ca

Hayashi Aiko
email: aiko.hayashi@utas.edu.au

Herrenknecht Christine
email: christine.herrenknecht@univ-nantes.fr

Hess Philip
email: philipp.hess@ifremer.fr

Hoang Thi

Mohammed Rizmanmalaysia

Imhoff Johannes

Kalyanaraman Rajagopal

Kerr Russell

König Gabriele

Lagarde Aurélie
email: aurelie.lagarde@etu.unilim.fr

Le Bellec

Le Lamer

Liu Lan

Luo Zhu-Hua
email: luozhuhua@hotmail.com

Ory

Overy David

Ka - Pang
email: klpang@ntou.edu.twpouchus

Lai

Yves François

Proksch Peter
email: peter.proksch@uni-duesseldorf.derämä

Heinrich Heine

Roullier Catherine
email: catherine.roullier@univ-nantes.fr

Ruiz Nicolas
email: nicolas.ruiz@univ-nantes.fr

Tasdemir Deniz
email: dtasdemir@geomar.de

Tomasi Sophie
email: sophie.tomasi@univ-rennes1.fr

Van Bohemen
email: ai.vanbohemen@gmail.com

Wang Bingui
email: wangbg@ms.qdio.ac.cn

REUNION ISLAND, WEST INDIAN OCEAN: AN ACTIVE VOLCANO AS AN UNUSUAL SAMPLING SITE FOR MARINE-ASSOCIATED FUNGI

Marine ecosystems in active volcanic areas are regularly subject to natural hazards such as being covered by incandescent lava flows, acoustic phenomena caused by microseisms, temporary changes in physicochemical conditions of water bodies and exceptional elevations of temperature. The Piton de la Fournaise (Reunion Island) is one of the most active effusive volcanoes in the world with 27 eruptions between 1998 and 2007 and a mean frequency, over a century, of an eruptive phase every 9 months. It provides a natural laboratory to study the colonization of a blank substrate. Therefore, a project called BIOLAVE was launched in 2010 in order to study species and habitat diversity on the coastal marine area of the volcano, including the first taxonomic inventory for the southeast side of the Island [1]. One hundred fungal isolates were obtained from 1977 lava flow (-25m depth), 2004 lava flow (-70m) and free water (-70m) as preliminary sampling. Density of propagules was estimated as follows: 0.085/g free water (-70m), 20/g 2004 lava (-70m) and 80/g 1977 lava (-25m) (MEA, PDA and Sabouraud media). These isolates should now be identified using phenotypic characteristics and molecular biology. One hundred fungal isolates were obtained from 1977 lava flow (-25m depth), 2004 lava flow (-70m) and free water (-70m) as preliminary sampling. Density of propagules was as follows: 0.085/g free water (-70m), 20/g 2004 lava (-70m) and 80/g 1977 lava (-25m) (MEA, PDA and Sabouraud media). These isolates should now be identified using phenotypic characteristics and molecular biology.

[1] Bollard et al., Cah Biol Mar, 2013, 54, 491.

First

  Bollard S. et al., Cah. Biol. Mar., 2013, 54, 491-497 [2] Natural hydroxyanthraquinoid pigments as potent food grade colorants: an overview. Caro Y., Anamale L., Fouillaud M., Laurent P., Petit T., Dufossé L. Natural products and Bioprospecting, 2012, 2(5), 174-193. [3] Filamentous fungi are large-scale producers of pigments and colorants for the food industry. Dufossé L., Fouillaud M., Caro Y., Mapari S.A.S., Sutthiwong N. Current Opinion in Biotechnology, 2014, 26, 56-61. [4] Anthraquinones, the Dr Jekyll and Mr Hyde of the food pigment family. Dufossé L. Food Research International, 2014, 65, 132-136.

1

  Nectria haematococca (A) Nectria haematococca (B) Aspergillus sydowiiMyrothecium atroviride (Sabouraud)MaFNaP -Nantes 2015 -811:45-12:05 -Flash communications (5 min/poster) Dr. Amaria Matallah-Boutiba -"Fungal isolates from western algerian marine environment" Ms. Marion Navarri -"Antimicrobial activities of deep-sea marine fungi" Dr. Elodie Blanchet -"Intra-and interspecific communication of bacterial and fungal endophytic communities of Posidonia oceanica" Pr. Lan Liu -"Isolation, identification and cytotoxic activity analysis of symbiotic filamentous fungus from Himerometra magnipinna" 12:05-13:45 -LunchThird sessionAdaptation / chemical response of marine fungiChair: Pr. Soizic Prado 13:45-14:35 -Invited lecture -Pr. Peter Proksch -"Co-cultivation -A powerful tool for enhancing the chemical diversity of fungi" 14:35-14:55 -Ms. Marine Vallet -"Diversity, metabolome and ecological role of endophytic fungi associated with brown algae" 14:55-15:15 -Dr. David Overy -"Affects of osmotic stress upon secondary metabolism in Aspergillus aculeatus" 15:15-16:45 -Coffee break / Poster session / exhibition Fourth session Richness of marine fungi from unique environments Chair: Dr. Gaëtan Burgaud 16:45-17:35 -Invited lecture -Pr. Bin-Gui Wang -"Enhancing chemical diversity of secondary metabolites produced by marine-derived endophytic fungi from chinese sea waters" 17:35-17:55 -Dr. Vanessa Rédou -"Marine fungi from deep subseafloor sediments: species richness and adaptation" 17:55-18:15 -Dr. Zhu-Hua Luo -"Fungal diversity and abundance in a hydrothermal vent site at the Mid-Atlantic Ridge of the South Atlantic Ocean" 09:50 -Invited lecture -Pr. Johannes Imhoff -"Marine Fungi -still an underrepresented source of marine natural products" 09:50-10:10 -Pr. Laurent Dufossé -"Anthraquinones from marine-derived fungi, challengers for natural colors in industrial fields?" 10:10-10:30 -Ms. Camila Dias -"Lipids from marine-derived fungi as source of bioactive compounds" 10:30-10:45 -Coffee break Second sessionMetabolomics applied to marine fungal extractsChair: Pr. Bin-Gui Wang / Dr. Rainer Ebel 10:45-11:05 -Dr. Samuel Bertrand -"Aggressive LC-MS/MS data annotation: application to extract priorisation" 11:05-11:25 -Ms. Alyssa Grunwald -"Characterization of tetramic acid containing compounds from Tolypocladium species isolated from Canada's Arctic" 11:25-11:45 -Shimadzu sponsor communication 11:45-12:05 -Flash communications (5 min/poster) Ms. Sahar Daoud -"Identification of potential bioactive compounds against breast cancer cells from the metabolic profiles of marine-derived Penicillium strains" Dr. Henrik Harms -"Epipolythiodiketopiperazines from the marine-derived fungus Dichotomomyces cejpii with NF-κb inhibitory potential" Mr. Peter Hufendiek -"Epipyrones from the marine-derived fungus Epicoccum nigrum Link inhibit the proteases Cathepsin K and S" Mr. Jan Schrör -"Secondary metabolites of the fungus Stemphylium globuliferum and their biological activity" 14:35 -Invited lecture -Dr. Rainer Ebel -"Induction of secondary metabolites by/in marine-derived fungi" 14:35-14:55 -Dr. Olivier Grovel -"Combining dereplication and semi-synthesis to reveal marinesourced fungal chemodiversity" 14:55-15:15 -Ms. Anne-Isaline van Bohemen -"Marine-derived fungi, a new source for biocontrol agents: antifungal activity of peptaibols and of a peptaibol-producing strain, Trichoderma atroviride" 15:15-16:45 -Coffee break / Poster session / exhibition 16:45-18:15 -Round table / Marine Fungal Natural Products Consortium discussion Chair: Pr. Russell G. Kerr / Pr. E.B. Gabriele König / Pr. Johannes Imhoff 09:20-09:40 -Ms. Aïko Hayashi -"Australian dust storm associated with extensive Aspergillus sydowii fungal "bloom" in coastal waters" 09:40-10:00 -Dr. Giovanna Cristina Varese -"Phylogenetic diversity and chemical profiling of algicolous marine fungi active against mulidrug resistant bacteria" 10:00-10:20 -Pr. Rob J. Capon -"Polyketides from marine mollusc-associated fungi" Université de La Réunion, ESIROI & LCSNSA, Réunion Island, Indian Ocean, France. 2 ARVAM, Réunion Island, Sainte-Clotilde, France. *E-mail: laurent.dufosse@univ-reunion.fr Marine ecosystems in active volcanic areas are regularly subject to natural hazards such as being covered by incandescent lava flows, acoustic phenomena caused by microseisms, temporary changes in physicochemical conditions of water bodies and exceptional elevations of temperature. The Piton de la Fournaise (Reunion Island) is one of the most active effusive volcanoes in the world with 27 eruptions between 1998 and 2007 and a mean frequency, over a century, of an eruptive phase every 9 months. It provides a natural laboratory to study the colonization of a blank substrate. Therefore, a project called BIOLAVE was launched in 2010 in order to study species and habitat diversity on the coastal marine area of the volcano, including the first taxonomic inventory for the southeast side of the Island [1].

Second session

Metabolites of marine-derived fungi


Fanny Rome
email: frome@dedale.net

Guillaume Adam

Johan Condette

Frédéric Dehais

Mickaël Causse

Go-around manoeuver: a simulation study

Keywords: Go-around, eye tracking, situation awareness, aviation safety

Events during which a safe go-around was not achieved have been brought to the attention of the BEA (Bureau Enquête Analyse pour la Sécurité de l'Aviation civile -French Safety Board) due to their noticeable increase since the 2000s. Considering the greater risk associated with the go-around manoeuvre, currently defined as a normal phase of flight, the BEA has decided to undertake a study named ASAGA (Aircraft State Awareness during Go Around). The study aimed at identifying the weaknesses of the system during goaround manoeuvres regardless of the type of aircraft and the airline operational procedures. In this context, the analysis relied mainly on three components: (1) Safety events (incidents / accidents); (2) Large scale survey; and (3) Flight simulations. Experiments were conducted using two long-range full flight simulators with operational pilots. This paper presents the method used to conduct the experiment and some of the resulting data, as well as some initial conclusions. The data included: video camera recordings for task analysis; recordings of the ocular activity of the two pilots; and debriefings. The eye trackers were used to analyze the pilot's scanning patterns and the distribution of their attention during this particular flight phase. The debriefings consisted of interviews, based on the self-confrontation technique, which enable the pilots to selfassess their actions and situational awareness. Eleven crews from three French airlines participated to the simulations. The scope was not explained to them to prevent any anticipation. Each crew performed the same scenario from take off to landing that included three go-around manoeuvres. The first results indicate some concerns related to the cooperation between controllers and aircrews in such demanding occurrences, some lack of understanding of cockpit automatisms and failures in crew resource management. Results are discussed in terms of potential improvements of the Human Machine Interface, pilots-controllers interactions and task sharing within the aircrew.

Introduction,

Events during which a safe go-around was not achieved have been brought to the attention of the BEA (Bureau Enquête Analyse pour la Sécurité de l'Aviation civile -French Safety Board) due to their noticeable increase since the 2000s. Considering the greater risk associated with the go-around manoeuvre, currently defined as a normal phase of flight, the BEA has decided to undertake a study named ASAGA (Aircraft State Awareness during Go Around). The study aims at identifying the weaknesses of the system (composed by the crew, the aircraft and the controller) during go-around manoeuvres regardless of the type of aircraft and the airline operational procedures. In this context, the analysis relies mainly on three components: (1) Safety events (incidents / accidents) analysis; (2) a large scale survey and (3) a flight simulator experiment.

The objective of the simulations lied on the confirmation of the results uncovered by the event analysis and the large scale survey. The observation of the crew during the performance of the real activity, as well as data related to their visual scan, are necessary to understand the mechanisms that lead to unstabilized go-around. Eye tracking offers a fruitful perspective to assess the pilot's performance since visual perception is a key for pilot to control the flight [START_REF] Kasarskis | Comparison of expert and novice scan behaviors during VFR flight[END_REF]Mumaw&al., 2001) In addition, go-around is a rare and risky manoeuvre, the observation in the real context was obviously unrealistic. Therefore, the use of high-fidelity simulation was essential to recreate the dynamics of the situation. This paper presents the method used to conduct the simulations, the analysis of the resulting data, and some initial conclusions. A global report of the ASAGA study will be later published by the BEA, it will detail all these elements as well as the other components of the study.

Method

Simulation settings

The experiments were conducted using two longrange full flight simulators (B777 and A330).

Scenario

A realistic flight scenario was built with the participating airlines from take off to landing. The flight lasted 2 hours and contained three go around manoeuvres: -

A first one at low level with a low altitude gain, initiated at the request of the ATC. The trajectory was modified by the ATC; -A second one initiated by the crew (due to downwind); -A third one due to a lack of visibility.

Detailed scenario

The flight was planned to depart from Bordeaux inbound to Lyon. The aircraft has several hours of autonomy and no particular meteorological conditions are expected. ATC instructs the crew to perform a go around at 200ft due to an occupied runway. Contrary to the procedure, ATC requests that the crew turn left 340 degree and climb to 2,500ft. The crew diverts to Marseille. They perform a standard ILS approach during which the simulated wind progressively turns downwind its strength increases up to 15 to 20knots. ATC only informs the crew about this change in wind direction during short final. Controllers also limit the altitude to 2,000 ft instead of 3,500ft during the go around. The crew continue their flight (LOC DME).

Visibility is then nil, leading the crew to perform a third and last go around.

Data collected

Direct observations (simultaneous and video recordings) Video (Hd) was used to record all the displays and all the communications to allow an accurate post-analysis.

Observations were carried on during the entire scenario by two human factors specialists inside the cockpit and by two investigators with the help of a remote video streaming.

Subjective assessment

The debriefings consisted of interviews, based on the self-confrontation technique, which enabled the pilots to self-assess their actions and their situational awareness.

Visual scan

The objective was to analyze the pilots' scanning patterns and the distribution of their attention during the go-around. Two synchronised 50Hz Pertech head-mounted eye trackers were used (0.25 degree of accuracy). 12 areas of interest (AOI) on the cockpits were defined as follows: 1) primary flight display, 2) speed, 3) artificial horizon, 4) altitude, 5) heading, 6) flight mode annunciator (FMA), 7) navigation display (ND), 8) electronic centralized aircraft monitor (ECAM), 9) left multipurpose control display unit (MCDU), 10) right MCDU, 11) visual scene, 12) flight control unit (FCU). A 13th AOI was considered in order to collect the ocular fixations out of the other twelve previous (out of zone, OZ).

Sample

Eleven crews (one captain and one first officer) were recruited from three French airlines. Each crew performed the same scenario, i.e the three go-around manoeuvres. This represents a sample of 33 go around (GA) according to the following experimental plan:

A/C GA1 GA2 GA3 Crew 1-6 (1 airline)

Results

The results presented in this paper are limited to the first 11* go around (GA1) which contain the most interesting results. Temporal aspects, visual patterns and task analysis are described and then discussed. *Some results are for only 10 GA out of 11 because of a small modification of the scenario for one crew.

Temporal aspects

The timing of actions during GA1 was measured monitored from the beginning of the controller's clearance to the stabilisation at heading 340° and altitude 2500 ft (Figure 1). Regarding the 10 GA1, the total duration ranges from 1min00s to 1min40s. Figure 1 indicates the elapsed times to select the altitude and the heading modes and values corresponding to the controller's request. They respectively range from 20 to 51s and from 16 to 37s.The duration ranges emphasise the variability of crews' strategies and the resulting variety of situation dynamics. The PF were more focused on the horizon (50% average) compared to the PNF (12.5% average).

Visual Scan

Conversely, the PNF spent more time gazing Out of Zone (23.9% vs 4.1%), on the FCU (17.1% vs 4.6%) and on the ND (11% vs 4.4%).

The distribution of the time spent on the several cockpit equipments were also analysed for both PF and PNF. Distinct homogeneous groups, based on one or several areas of interest (some AOI cannot be dissociated), have been identified. The figures below (Figure 4 and Figure 5) present the hierarchical classification of these groups based on the percentage of time spent on them.

Regarding the PF (Figure 4), the artificial horizon elicits the higher fixation times (Group 1). The second mostly fixated group (Group 2) is composed by SPEED and ALT/VARIO. Regarding the PNF, the OZ elicits the higher fixation times (Group 1), followed by the FCU (Group 2) and the AOI group composed of speed/ND / artificial horizon. The statistical analysis indicates that the PF's and PNF's eye movements were significantly different. Although they did not pay the same attention in all AOIs, all PF constitute a homogeneous group in terms of eye movements as well as all PNF. This means that PF tended to spend the same percentage of time on the various AOI during GA1. This applies to PNF as well. However, the trajectories/paths of the eye movement for PNF are not homogeneous. PNF did not watch the AOIs in the same way, nor according to the same sequence.

PNF % time spent

Task analysis

The go-around manoeuvre consists in rapidly bringing the aircraft in an ascending profile according to an assigned trajectory. To manage this trajectory, once the go-around is initiated, crews had to take into account the vertical and heading constraints given by the controller while managing automatisms. The results of the task analysis corresponding to these different actions are presented in the following paragraphs. Their complete description falls beyond the scope of this paper; hence, the results are limited to the noticeable deviations or adaptations that are safety-relevant.

Initialisation

The Go-around initialisation consists in applying TOGA mode, in setting the adequate thrust and in retracting the flaps one step. For the eleven sessions, this initialisation was performed without major difficulties or failures by the crews: all of them initiated the go-around at the request of the ATC, all applied the adequate thrust, the TOGA mode, and retracted flaps one step. A normal variability can be observed: absence of the go-around announcement, error of step in the flaps retracting request etc... All these approximations were nevertheless compensated within the crew.

Positive climb callout and Gear up

The expected sequence of actions is as follows: the PNF tells the PF when the variometer becomes positive, the PF orders "gear up", and the action is executed by the PNF. The eleven crews moved the gear up; seven of them followed exactly the expected sequence. For the others, small adaptations can be noticed : in two cases the PF (instead of PNF) announced the positive climb, one PF ordered the gear up not by an announcements but just by pointing it with the finger and one PF moved the gear up without any announcement.

Management of ATC clearances (heading and altitude)

Ten GA1 out of the eleven were requested by ATC with the following communication: "XXX goaround, runway occupied , turn left heading 340 and climb 2500ft". The expected response consisted of: (1) `read back by the PNF; (2) selection of 340 and of the heading mode, selection of 2500 and altitude mode. The ten crews reacted in different ways; most of them broke their reaction in two steps. Indeed more than half of the crews re-contacted the ATC to obtain clarifications either at the PNF's (two cases) or at the PF's (four cases) initiative. For them, the heading and altitude selection was performed within 30-40s. One crew read back the go-around adding "standby" and contacted the ATC 30s after. For the three others who took into account the clearances in one step, some deviations and approximations can be noticed: An error in the altitude read back and an error in the altitude mode selection (directly detected by the PF); An an error of the lateral mode selection; An immediate selection of the clearances provided by the PF but the read back by the PNF only one minute later.

Modes management Initial modes announcement

In case of a mode change, the expected behaviour is the announcement by the PF and the check by the PNF. This expected behaviour, depending on the airline, is either described in the Go-around procedure or in the general operation procedures.

The figure 6 presents for each rank of change (1 st mode change, 2 nd mode change...), the number of sessions concerned by this rank. It also presents, for these sessions, the number of announcements and checks performed by the crew. For example, 9 sessions had at least 3 mode changes, the third change was announced by 5 PF out of the 9 and checked by 5 PNF. The initial modes were announced by seven PF (out of ten) and were checked (with an announcement) by five PNF. One PNF announced the initial modes. Thus, two crews did not announce at all the initial modes.

Figure 6 : Mode changes and announcements

Announcement of the following modes

The expected sequences of modes for the go around were disrupted by the ATC request and the adaptations performed by pilots to adhere to this request. These disturbances induced in certain cases a consequent number of mode changes in a rapid sequence. For example, in one GA situation, up to nine successive mode changes can be observed (considering the active modes only). These changes (especially the fifth and the followings) were only succinctly announced by crews (figure 6).

Mode selection

In order to modify the go around trajectory as required by the ATC, the crews had to select the "heading" and "altitude" modes. Some crews had difficulties in the selection of these modes. Twice, the HDG mode was not selected by the PNF, the two PF detected the error of selection: "heading sel", "please activate the hdg", For one crew, the heading mode was selected during a short time, but the PF (unaware of this activation) ordered the PNF to "pull" the button (to activate the mode), the PNF pushed the button and the navigation mode was thus engaged. The error mode selection remained undetected.

Trajectory management

The following of the requested trajectory is subjected to variability inherent to the high constraints imposed on the crew. Small altitude deviations occurred due to the aircraft inertia or due to the more or less accurate following of the Flight Director. Approximations of the heading also occurred. For example, some crews immediately acknowledged the ATC request to turn left and anticipated the selection of a precise heading. Besides this small variability, more considerable horizontal and vertical trajectory deviations occurred. In the horizontal plan, there were 3 major deviations from the requested 340°:

Until 300° for a first crew The PNF read back 340 and input 240 in the FCU. The PF checks the heading mode selection and follows the FD. The error in the value is not detected. The deviation is detected by ATC who informs the crew. Until 320° a the second crew The 340 heading is correctly selected by the PNF. The PF repeats the modification requests "left turn, 2500ft". He initially executes a left turn, without following the Flight director He said during the debriefing "I had adequately understood the altitude but I was not sure about the heading. I could see the FD on the right while we had to go on the left, so I overshot". The PNF says during the debriefing that he monitored the speed and the altitude: "I didn't see the .FD.". From 325° to 030°fo a third crew Heading 340 was adequately displayed but the heading mode was not selected. The PF initiates a left turn until 325 without any FD following, detects the first lateral deviation ("heading 340, I overshot"), follows the FD, which indicates the steps to follow the FMS GA procedure (NAV mode). The aircraft heading reaches 030. The PF realises that the A/P is re-engaged and that the aircraft starts a right turn as expected in the initial trajectory.

Large deviations from the requested 2500ft also occurred in the vertical dimension: until 3000ft for two crews The crews acknowledge the 2500ft limit while the aircraft is at 3500. They chose two different strategies: i) altitude 3000 selected in the FCU and ask 3000 to the ATC ii) 2500 selected and execution of an open descent. until 3900ft for a third crew An overshooting of 1400ft that can be decomposed in three phases: +400ft before the confirmation of the altitude with ATC +600ft with Vertical Speed +3800 +400ft due to the "aircraft inertia" Vzmax >3800ft/min, C-Chord alarm (40 secondes)

Subjective assessment

During the debriefings, crews were systematically interviewed about their priorities and difficulties encountered during the first go around. Their testimonies are not transcribed in this paper but parts of them are used in the following discussion.

Discussion

Results show that all crews performed the goaround manoeuvre at the request of the controller. The initialisation and the move of the gear up were executed in a homogeneous manner within the crews. On the other side, the execution of the manoeuvre differed depending on the strategy chosen by the crew and on their execution of the expected tasks.

Interesting measures of the pilot's activity were extracted from eye tracker data. They reveal that the PF was mainly focused on the trajectory execution since he assimilated mostly information related to artificial horizon, speed, altitude/variometer. The assimilation of such information was achieved within a reduced space (i.e. on the PFD) and required up to 70% of the visual attention. This indicates that the task of flying the trajectory was quite exclusive for the PF.

On the contrary, the PNF's visual attention was more dispersed. The PNF mostly captured the necessary information within the out of zone, which roughly corresponds to the aircraft configuration management (gears, flaps, radio panel or overhead panel). This is coherent with the different tasks affected to the PNF: « positive climb » announcement, gear up; read back of ATC request and consideration of the constraints (selection of the heading, of the altitude values, and of the corresponding modes); monitoring the initialization of the GA; monitoring the PF actions and the FMA modes.

During debriefing, almost all PNF reported difficulties related to the management of all these actions:

related to the ATC request : « I didn't retain all information hopefully you did» ; « a heading left but I don't remember» ; « I couldn't answer him» related to the (auditory) overload and the actions' prioritization : "the info of colleague, at the same time, ATC request heavy, painful". "capt requests added to the ATC demands, so much." "the management of priorities between thrust, trajectory, FMA and "ear".

related to the impacts of overload on actions or announces to be performed : « I took time, I delay, I did not announce the vario for the gear up » ; « I mix everything » To cope with the peak of workload, crews reduced the demand, and/or adapted their response. Some of them chose for example to differ their consideration of the ATC modification. This led, in certain cases, to large vertical deviations. The adaptations consisted mainly in simplifying some tasks and prioritising actions. For example, the announcement of the FMA modes was frequently omitted by pilots, whereas they previously announced them. This is a typical situation where an increase of workload can divert resources from maintaining the situation awareness. One crew, for example, erroneously believed that the heading mode was engaged whereas they were in navigation mode. A subsequent deviation occurred in their management of the horizontal trajectory (from 325° to 030° compared to the 340° requested). This kind of issues have already been largely discussed in the literature, especially by [START_REF] Sarter | Pilot's Monitoring Strategies and Performance on Automated Flight Decks: An Empirical Study Combining Behavioral and Eye-Tracking Data[END_REF]: "A loss of mode awareness can lead to mode errors, which occurs when a pilot performs an action appropriate for the assumed, but not the actual system state". Besides, the observed horizontal deviations highlight the multiplicity of information sources related to the management of the lateral trajectory: -"to the left" requested by ATC -"340" requested by ATC -The Flight Director -The heading information under the horizon -The trajectory monitoring indicated on the ND Under nominal situations, all these information are coherent and some are even redundant. Under critical situations, the precision and validity of these data may differ. The PF might not have the necessary resources to apprehend the various sources and the PNF's (or even ATC's) monitoring role is therefore crucial.

The diversity in PNF visual scan patterns and the fact that some deviations were undetected, as well as the difficulties expressed by PNF during the debriefing confirmed that the monitoring task was not easily performed during go-around. Considerable mental effort is required to keep attention moving between the tasks. As expressed by Dismukes & al. (2001), "the most commonly [reported] category of neglected tasks is monitoring (of aircraft position or status or of actions of the pilot flying/taxiing)".

PNF had difficulties to prioritise their attention focuses. When PF were asked about their priorities in debriefing, they are quite unanimous: « pitch, thrust, trajectory management ». PNF answers were more complex and differed as they reported: « aircraft climbing, trajectory, pitch, monitoring, thrust, announcements… ».

The CRM (Cockpit Resource Management) plays a crucial role during go around as a barrier against accidents. Firstly, the procedure relies on an effective teamwork, and particularly on the monitoring of deviations by the PNF. Secondly, previous events have shown the existence of potential attention tunneling during Go around maneuver; CRM is one of the current countermeasures against this phenomenon. However as explained in previous paragraphs, PF and PNF might have different priorities especially during the initialization. If the go around becomes not nominal, it might be difficult for them to exchange within the crew and share the same situation awareness. The CRM can fail in such demanding situations.

Conclusion

The survey emphasized potential issues when the ATC changes the published missed-approach track during the go-around manoeuvre. With the experiment, we confirm that in this case crews can be destabilised by the difference with the briefing during the approach. The consequences of these changes can be significant, particularly when they prevent the use of certain automated systems and increase the temporal pressure. In a flight phase where the workload is already high, additional actions further disrupt teamwork, and monitoring by the PNF in particular.

The results of the simulations highlight the importance of monitoring by the PNF during the go around. They also show that PNF have considerable difficulties to monitor all the parameters requested for the procedure.

The simulations show that, because of the great diversity of go-around situations, the number of changes in FMA modes during the go-around can be unexpectedly high. This makes it difficult for crew members to detect and follow all the mode changes. Thus, the robustness of go-around

Figure

  Figure 1 : Temporal aspects 0 20 40 60 80 100

Figure 2 :Figure 3 :

 23 Figure 2 : Percentages of time spent per AOI during GA1.The error bars represent the standard deviation.

Figure 2

 2 Figure 2 presents the percentages of time spent per area of interest during the first Go-around.The PF were more focused on the horizon (50% average) compared to the PNF (12.5% average). Conversely, the PNF spent more time gazing Out of Zone (23.9% vs 4.1%), on the FCU (17.1% vs 4.6%) and on the ND (11% vs 4.4%).

Figure 4 :

 4 Figure 4 : percentage of fixation time on the various instruments of the cockpit for the PF

Acknowledgement.

The authors would like to thank the participating airlines, the pilots an d the manufacturers. The authors would like to thank the BEA for its authorization of publishing this paper. All and opinions in this paper are the responsibility of the authors alone.

procedures cannot be assessed based solely on the assumption that FMA mode changes have been comprehensively read and understood.

The analysis of accidents or serious incidents performed in the ASAGA study showed that crews are often no longer aware of the basic parameters -pitch, thrustand their correlation with changes in calibrated airspeed and vertical speed. Results of the simulations confirm, if need was, that performing the go-around requires a high number of actions. Crews may have difficulty in prioritizing their actions and may omit to continuously monitor these parameters.

In summary, the experiment in simulator confirms and expands on earlier findings providing from the events analysis and the safety survey.

Besides, it emphasized the potential gain obtained by using combined approaches (eye tracking data, task analysis and debriefing) and raises the question of the use of such a method for training purposes.