¸is Franco

Durand

Antonin Macé

AND Matías Núñez

Laurent Bouton

ANALYSIS OF APPROVAL VOTING IN POISSON GAMES *

Keywords: Approval voting, Poisson games, Stable equilibria, Monte-Carlo simulations. JEL CLASSIFICATION. D72, C72, C63

We analyze Approval Voting in Poisson games endowing voters with private values over three candidates. We first show that any stable equilibrium is discriminatory: one candidate is commonly regarded as out of contention. We fully characterize stable equilibria and divide them into two classes. In direct equilibria, best responses depend only on ordinal preferences.

In indirect equilibria, preference intensities matter. Counter-intuitively, any stable equilibrium violates the ordering conditions, a set of belief restrictions used to derive early results in the literature. We finally use Monte-Carlo simulations to estimate the prevalence of the different sorts of equilibria and their likelihood to elect a Condorcet winner.

a collective perspective. [START_REF] Laslier | The Leader Rule: Strategic Approval Voting in a Large Electorate[END_REF] and [START_REF] Myerson | A theory of voting equilibria[END_REF] deliver a very sharp prediction: namely, in equilibrium, this rule tends to uniquely select the Condorcet winner if it exists. This prediction holds in the model of [START_REF] Myerson | A theory of voting equilibria[END_REF], where voters' beliefs meet the following two properties: (i) the higher the perceived score of a candidate, the infinitely more likely that a vote for him becomes pivotal and (ii) pivot events involving two candidates are infinitely more likely than pivot events in which three or more candidates are involved. We refer in the sequel to these two properties as the ordering conditions. Building on it, voters' best responses are immediate since candidates' perceived scores are enough to determine the optimal ballots (see [START_REF] Laslier | The Leader Rule: Strategic Approval Voting in a Large Electorate[END_REF] for a precise statement). Figure 1 illustrates these best responses with three candidates 1 . Since a and b are perceived as the most serious contenders, voters who prefer a to b approve a and those who prefer b to a vote for b. Approving of c is a best response if and only if c is preferred to the perceived winner, candidate a. With this sharp description of best responses, the logic of the unique equilibrium is rather clear: only a Condorcet Winner can be elected. Yet, the status of AV in Myerson's Poisson games is rather uncertain. Poisson games are used for several applications in mechanism design and, in particular, they have become the classical model to understand large elections from a game-theoretical perspective. In these games, the size of the electorate follows a Poisson distribution, and the strategic decision of each voter is based on endogenous pivot probabilities, rather than on ad-hoc belief restrictions. [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF] presents an elegant example that shows a nice feature of AV in this model: when all voters unanimously prefer some candidate, AV uniquely selects the Condorcet winner in equilibrium whereas the Plurality rule fails to do so. However, as proved by [START_REF] Núñez | Condorcet Consistency of Approval Voting: A Counter Example on Large Poisson Games[END_REF],

one can find examples in which AV fails to induce the victory of a candidate ranked first by a majority of the electorate. Beyond these specific examples, little is known on AV equilibria in general 2 , and the comparison to first-past-the-post voting methods seems difficult to establish.

Our work is the first one to give a general description of the equilibria in this game. As a starting point, we present an in-depth description of the voter's decision problem in order to identify the precise events that matter to determine best responses. Focusing on stable equilibria, our first main contribution is to show that these equilibria must be discriminatory. In such an equilibrium, a pair of candidates (say a and b) acts as a coordinating device: each voter 1

We focus exclusively on a setting with three candidates as the vast majority of works in Poisson games.

2 [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF] only provides general results for bipolar elections, in which the electorate can be divided in two antagonist sides.

anticipates that, conditional on being pivotal, his vote will almost surely determine whether a or b wins the election. Thus, each voter approves either a or b but no voter approves both or disapproves both. 3 In contrast with first-past-the-post voting, the previous observation is not sufficient to determine best responses, as each voter may choose to approve c or not. This choice is driven by the relative probabilities of the other pivots (being decisive between a and c, or between b and c).

In a first class of direct equilibria, the ratio of these probabilities diverge, and best responses are straightforward. However, these equilibria often fail to exist. We then characterize the remaining class of indirect equilibria, exhibiting finite pivot probability ratios, and for which best responses may depend on cardinal utilities. We show that, for every ordinal preference profile, there is an open set of cardinal representations of these preferences for which such an indirect equilibrium exists. Moreover, if a Condorcet winner exists, the constructed indirect equilibrium elects him.

As a by-product of our analysis, we obtain an important and striking result: the ordering conditions are never satisfied in a stable equilibrium (both conditions (i) and (ii) are violated).

This finding underscores the discrepancy between the Myerson-Weber model and the model of Poisson games. Finally, we exploit our characterization of stable equilibria to analyze numerically the prevalence of direct and indirect equilibria, and to assess the likelihood of electing a

Condorcet winner when it exists.

1. APPROVAL VOTING GAMES

Candidates and voters

We follow the model stated by [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF] and thus consider an election in which voters elect one candidate into office from a set of candidates K = {a, b, c}. The number of voters is drawn from a Poisson distribution with mean n. 4 It follows that the common knowledge probability that exactly p voters take part in the election equals e -n n p p! . A voter's type t determines its cardinal preferences. Throughout, we assume that preferences are strict and that two voters with the same ordinal preferences share the same cardinal preferences. For any preference order over the candidates i j k, the corresponding type is denoted by t := i j. Up to renormalization, we assume that voters of this type have utility 1 (resp. 0) for the victory of candidates i (resp. k), and we denote by u t ∈ (0, 1) their utility for the victory of candidate j. The type space is denoted by T = {ab, ac, ba, bc, ca, cb}. Each voter's type is independently drawn from T according to the preference profile denoted by r = (r t ) t∈T with t∈T r t = 1. We assume that for any subsets of types T, T ⊆ T , with T T , t∈T r t t∈T r t . Note that this implies that r t > 0 for each t ∈ T . This assumption is imposed to simplify the statements of the different results, yet it can be relaxed at the cost of more cumbersome notation and is indeed relaxed in most examples for the purpose of simplicity. To 3 The existence of discriminatory equilibria is not unattractive per se: under first-past-the-post, the problem lies in their multiplicity [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF]. 4 In these games [START_REF] Myerson | Population Uncertainty and Poisson Games[END_REF], the voters' actions uniquely depend on their private information t (common public information). Moreover, the number of voters who choose a given ballot is independent of the number of voters who choose another ballot (independence of actions). These two properties are used throughout.

assess the relevance of some results, we sometimes refer to a measure on the set of preference profiles r, which is computed with the uniform distribution on the simplex over T .

We focus on elections held under AV: voters cast a ballot for as many candidates as they want, and the candidate receiving the highest number of votes is elected (ties are broken with equal probability). Each voter simultaneously selects one message chosen in the set of approval voting ballots M := {a, b, c, ab, ac, bc}. For instance, a stands for the ballot that approves candidate a whereas bc is the one that approves both b and c. For any ballot m and candidate k,

we note k ∈ m if k is approved in m.

Strategies and Equilibria.

As usual in these models, strategies are defined type by type so that all agents of the same type play the same strategy. Namely, a pure strategy profile is a vector σ = (σ t ) t∈T , where σ t ∈ M for each type t. For example, if σ ab = c, then voters of type ab choose the ballot c. Then, for any ballot m ∈ M and any candidate k ∈ K , we write Share of ballot m :

Z m ∼ P (nτ m ) with expected share τ m = t|σ t =m r t , Score of candidate k : S k ∼ P (nγ k ) with expected score γ k = m|k∈m τ m .
Note that the ballot profile τ = (τ m ) m∈M meets m∈M τ m = 1. The ballot shares are independent random variables whereas the candidate scores may be correlated since each voter can approve of several candidates. We say that a candidate i is elected at a strategy profile σ if γ i > γ j , γ k , as this implies that the score of i is above those of both j and k with a probability approaching to one when n gets large.

The set Z ⊆ N M describes the possible ballot outcomes of the election, a typical element is a vector z = (z m ) m∈M where each component z m denotes the number of voters choosing ballot m. An event is a subset of ballot outcomes E ⊆ Z . Slightly abusing notation, we write z + m for the vector z + 1 m (when we add 1 to the m-component only) and Em for the event

{z ∈ Z | z + 1 m ∈ E}.
The probability of a ballot outcome z ∈ Z is then given by:

P[z | nτ] = m ∈ M e -nτ m (nτ m ) z m z m ! .
For each ballot outcome z ∈ Z and each candidate k, the score of k at z is noted s k (z) and the set K(z) stands for the set of candidates with the maximal score at z:

s k (z) = m|k∈m z m and K(z) = arg max k∈K s k (z).
Since ties are broken by a fair coin toss, the probability of a candidate winning the election at z equals 1 #K(z) for each candidate with a maximal score. Building on the previous notations, the expected utility for a voter of type t when voting m is denoted by

U t [m | nτ] = z∈Z U t [m | z] P[z | nτ],
where U t [m | z] denotes the average utility of candidates in K(z + m) for type t.

Therefore, the best response to a strategy profile σ depends only on nτ. It can be written

as BR(• | nτ) ⊆ M T , where, ∀t ∈ T , BR(t | nτ) = arg max m∈M U t [m | nτ].
We focus on a large electorate, that is, we consider the limit behaviour when its expected size n tends to infinity.

Voting equilibria are sometimes criticized in the literature for hinging on too precise beliefs held by voters [START_REF] Fey | Stability and coordination in duverger's law: A formal model of preelection polls and strategic voting[END_REF]. In particular, Poisson voting models with private values exhibit an absence of aggregate uncertainty, so that equilibria may rest on the knowledge of the precise ballot profile τ. To address this concern, we focus on stable equilibria, which are immune to the introduction of a small dose of aggregate uncertainty. Formally, this refinement coincides with the concept of strictly perfect equilibrium for normal-form games [START_REF] Okada | On Stability of Perfect Equilibrium Points[END_REF], already employed in Poisson games by [START_REF] Bouton | Majority runoff elections: strategic voting and duverger's hypothesis[END_REF]. Building on this observation, we introduce a graphical representation of the voting game we analyze throughout. We represent on Figure 2 the Approval game for an arbitrary preference profile r. Each type of voter t is located between the half-lines pointing to the ballots t 1 and t 1 t 2 . For instance, a voter of type ab chooses between ballot a (to her left) and ballot ab (to her right).

For each size of the electorate n and each ballot profile τ, the decision of a voter of type t reduces to determining the sign of the following expression:

∆(t | nτ) = U t [t 1 | nτ] -U t [t 1 t 2 | nτ] = z∈Z P[z | nτ] U t [t 1 | z] -U t [t 1 t 2 | z] . (1)
As is well-known, for most election outcomes, altering a single ballot does not affect the winner of the election. It follows that relatively few outcomes determine voters' best responses.

Expression (1) can be greatly simplified by focusing only on pivot and tie events, for which the winner of the election can indeed be affected by a single ballot, which we define below.

5 More precisely, in a framework with a fixed number of voters, it is weakly dominant for a voter to approve of his best candidate and it is weakly dominated to vote for his worst candidate (see De [START_REF] Sinopoli | Approval Voting: Three Examples[END_REF], [START_REF] Laslier | The Basic Approval Voting Game[END_REF] and [START_REF] Bouton | One Person, Many Votes: Divided Majority and Information Aggregation[END_REF]). For any t ∈ T and any

z ∈ Z , U t [t 1 | z], U t [t 1 t 2 | z] ≥ U t [m | z]
for any m ∈ M \ {t 1 , t 1 t 2 }. Moreover, for any size n of the electorate and any ballot profile τ, the event in which there is exactly one voter occurs with positive probability. In this event, the previously mentioned dominance relations become strict since each additional point matters to determine the winner. For each pair i j of candidates (and with k denoting the remaining candidate), we write

piv i j = z ∈ Z | s i (z) = s j (z) > s k (z) ,
the pivot event where candidates i and j are tied for victory. For each K ⊆ K , we let

tie K = z ∈ Z | s i (z) = s j (z), ∀i, j ∈ K
stand for the event where all candidates in K are tied.

For each pair i j of candidates and each type t ∈ T , we denote by piv t i j the event where casting a ballot t 1 or t 1 t 2 can create a pivot between candidates i and j so that piv t i j = (piv i j -t 1 ) ∪ (piv i j -t 1 t 2 ).

For each type t ∈ T , we denote: tie t,1 abc = tie abc -t 1 and tie t,2 abc = tie abc -t 1 t 2 , the events where casting a ballot t 1 or t 1 t 2 (respectively) can create a tie between the three candidates.

Equipped with the previous notations, the following proposition shows how the utility difference ∆(t | nτ) can be expressed as a function of pivot and tie probabilities.

Proposition 1. For each ballot profile τ and each size n of the electorate, the utility difference ∆(t | nτ) for a type t can be expressed as:

P[piv t t 1 t 2 | nτ] 1 -u t 2 + P[piv t t 2 t 3 | nτ] -u t 2 + 2P[tie t,1 abc | nτ] + P[tie t,2 abc | nτ] 1 -2u t 6 .
(2)

The main lesson of the proposition is that the best response of a voter in this game simply depends on the sign of ∆, which is a decreasing function of u t . Therefore, when n is large enough, for each ballot profile τ and each type t, there is some value u * t ∈ [0, 1] such that the best responses are given by:

BR(t | nτ) = t 1 if ∆(t | nτ) > 0 ⇐⇒ u t < u * t t 1 t 2 if ∆(t | nτ) < 0 ⇐⇒ u t > u * t .
The precise value of u * t depends on the ballot profile τ and depends on the relative likelihood of pivot and tie events. For instance, consider a voter with type t who believes that piv abc exhibit finite pairwise probability ratios. In that case (and only in that case), these ratios deliver a cutoff u * t ∈ (0, 1), such that the voter's best response is to vote t 1 if u t < u * t , and to vote t 1 t 2 if u t > u * t . We provide an example of such a situation in Section 5.4.

PIVOT PROBABILITIES AND THEIR MAGNITUDES

It is well known that pivot probabilities are typically hard to compute explicitly. Early papers in the literature rely on combinatorial arguments (Palfrey andRosenthal [1983, 1985]) or avoid any computation by referring to the ordering conditions [START_REF] Myerson | A theory of voting equilibria[END_REF]). A great insight from [START_REF] Myerson | Large poisson games[END_REF] is to introduce the idea of a magnitude, which measures the rate of convergence of the probability of an event to zero: when two events have different magnitudes, their relative probabilities diverge.

Definition 2. The magnitude of an event E ⊆ Z is defined by

µ(E) = lim n→∞ log(P[E | nτ]) n . With this definition, if µ(E) > µ(F), then lim n→∞ P[F | nτ]/P[E | nτ] = 0 (but the converse is not true in general).
The important advantage of working with magnitudes of pivot and tie events is that they can be computed explicitly as the solution of a constrained minimization program, thanks to the Dual Magnitude Theorem in [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF]. Building on this result, we characterize the values that relevant magnitudes can take.

Theorem 1. For any ballot profile τ, any type t and any pair i j of candidates (denoting by k the remaining candidate),

µ[piv t i j ] = µ[tie i j ] if δ i j (τ) > 0, µ[tie abc ] if δ i j (τ) ≤ 0, with δ i j (τ) = τ i τ j + τ jk τ i + τ ik + τ i j -τ jk τ i + τ ik τ j + τ jk -τ k ,
and

µ[tie t,1 abc ] = µ[tie t,2 abc ] = µ[tie abc ].
The theorem establishes two important properties: (i) the magnitudes of pivot events do not depend on types; and (ii) the magnitude of a pivot i j equals either the magnitude of tie i j or that of tie abc . The intuition for the second point stems from the interpretation of the threshold δ i j (τ) as the difference between the expected score of i and that of k, conditionally on tie i j . 6

6 To see why, recall that [START_REF] Myerson | Large poisson games[END_REF][START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF] apply large deviations techniques to show that the probability of a cone event E (such as tie i j ) is concentrated (for large electorates) in outcomes for which the proportion of either ballot is (approximatively) fixed. For each ballot m, this proportion can be expressed as the product of the ballot share τ m to its offset ratio φ m . The offset ratios can be obtained as the solutions of a constrained minimization program, whose value corresponds to the magnitude of the event E (this is the Dual Magnitude Theorem in [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF]). For E = tie i j , it is easy to see that φ k = φ i j = 1 (ballots i j and k do not influence the relative scores of i and j), while φ i = 1/φ jk = (τ j + τ jk )/(τ i + τ ik ). Therefore, the most likely difference in score between i and k in the event tie i j can be written as

δ i j (τ) = τ i φ i + τ i j φ i j -τ jk φ jk + τ k φ k .
If conditionally on tie i j , the expected score of k is below that of i, the probability of piv i j (i.e.

s i = s j > s k
) is equivalent to that of the tie i j . On the other hand, if conditionally on tie i j , the expected score of k is above that of i, then piv i j becomes much less likely than tie i j , and in fact equivalent to tie abc .

In the sequel, we write µ abc = µ[tie abc ] and, for each pair i j of candidates, µ i j = µ[piv i j ].

STABLE EQUILIBRIA

Stable equilibria are discriminatory

We establish a first result on the form of stable equilibria.

Theorem 2. Let σ be a stable equilibrium. Then, σ is discriminatory: µ ab > µ ac , µ bc , up to a relabelling of the candidates.

In a discriminatory equilibrium,7 voters perceive that one pivot event is infinitely more likely than any other. If µ ab > µ ac , µ bc , each voter approves of either a or b, depending on whether he prefers a to b or the opposite (and can of course approve of c as well). As preferences are strict, no voter approves both a and b (i.e. votes ab) or disapprove them both (i.e. votes c), so that τ ab = τ c = 0. Building on this observation, the following proposition provides formulas for the magnitudes of each pivot event in any stable equilibrium.

Proposition 2. Take any ballot profile τ with τ ab = τ c = 0. Letting

µ abc = -τ a -τ bc 2 -τ b -τ ac 2 ,
the magnitudes of the different pivot events are given by:

µ ab = -τ a + τ ac -τ b + τ bc 2 if τ ac τ bc < τ a τ b , µ abc if τ ac τ bc ≥ τ a τ b , µ ac = -τ a -τ bc 2 if τ b < τ ac , µ abc if τ b ≥ τ ac , µ bc = -τ b -τ ac 2 if τ a < τ bc , µ abc if τ a ≥ τ bc .
Proposition 2 is a central tool for our analysis of stable equilibria. As a first observation, the conditions for all two-candidate pivot magnitudes being jointly above µ abc are together incompatible, so that we always have

µ abc = min µ ab , µ ac , µ bc .
From this, we conclude that there are two sorts of stable equilibria: direct equilibria for which µ ab > µ ac > µ bc = µ abc , and indirect equilibria for which µ ab > µ ac = µ bc = µ abc , up to a relabelling of the candidates. While best responses can be straightforwardly inferred from the magnitude ranking in a direct equilibrium, they may depend on precise pivot probability ratios and on cardinal utilities in an indirect equilibrium.

Stable equilibria violate the ordering conditions

Before delving into the description of stable equilibria, we comment on the relation between candidates' expected scores and the relative likelihood of pivot events, which differs markedly in this game with respect to alternative models of strategic voting.

If candidates' scores were independent, one might expect that (i) pivot events involving candidates with higher expected scores should be infinitely more likely and (ii) a tie between the three candidates should be infinitely less likely than a pivot between any two candidates.

These intuitive properties, known as the ordering conditions, are the main ingredients of the early strategic voting model of [START_REF] Myerson | A theory of voting equilibria[END_REF], and can be expressed as follows.

(ordering conditions) (i)

γ i > γ j =⇒ µ ik > µ jk (ii) ∀i, j ∈ K , µ i j > µ abc
We know from a counter-example in [START_REF] Núñez | Condorcet Consistency of Approval Voting: A Counter Example on Large Poisson Games[END_REF] that condition (i) may be violated in Poisson games under AV, as candidates' scores can be correlated when some voters approve more than one candidate. We generalize this observation and obtain the surprising result that the condition is violated at any stable equilibrium.

Theorem 3. Every stable equilibrium violates the ordering conditions. More precisely, 1. Any direct equilibrium with µ ab > µ ac > µ bc (up to a relabelling of the candidates), satisfies

γ a > γ c > γ b . The ordering condition (i) is violated as γ c > γ b and yet µ ab > µ ac .
2. Any indirect equilibrium with µ ab > µ ac = µ bc (up to a relabelling of the candidates), satisfies

γ a , γ b ≥ γ c .
The ordering condition (i) is violated whenever γ a γ b , as µ ac = µ bc .

3. Any stable equilibrium violates the ordering condition (ii) as µ abc = min{µ ab , µ ac , µ bc }.

The result is important as it suggests that, for the analysis of strategic voting under AV, the model of Poisson games is fundamentally different from models where the ordering conditions prevail. It appears that the ordering conditions are useful but somewhat arbitrary belief restrictions, that do not capture the real relative likelihood of pivot events obtained in models where pivot probabilities are endogenous. In the sequel, we show how to characterize equilibria in such a model, without imposing any belief restriction.

DIRECT EQUILIBRIA

This section considers the simplest stable equilibria: direct equilibria, in which the magnitudes of pivot events fully determine voters' best responses. We first characterize preference profiles admitting a direct equilibrium, for a given ordering of pivot magnitudes. Then, we

show that many preference profiles do not admit any direct equilibrium.

Equilibrium Characterization.

In order to illustrate the logic of direct equilibria in any preference ordering, assume for instance that µ ab > µ ac > µ bc . In any such equilibrium, the strict ordering of the magnitudes implies that:

P[piv ab | nτ] n→∞ P[piv ac | nτ] n→∞ P[piv bc | nτ], P[piv abc | nτ].
In turn, this implies that best responses are uniquely determined when n is large, and that cardinal utilities do not matter. The best responses for each type are represented in Figure 3, where, for instance, the arrow from r ab to a means that BR(ab

| nτ) = a. c ab b ac bc a r ba r ab r bc r ac r cb r ca BR(ab | nτ) = BR(ac | nτ) = a BR(bc | nτ) = BR(cb | nτ) = bc BR(ba | nτ) = b BR(ca | nτ) = ac FIGURE 3. Best responses (for n large) in a direct equilibrium with µ ab > µ ac > µ bc .
In words, as pivot ab dominates, each voter approves of his favorite candidate between a and b. Next, as pivot ac comes second, a voter approves of c if and only if he prefers c to a. It follows that

τ a = r ab + r ac , τ b = r ba , τ ac = r ca , τ bc = r bc + r cb , τ c = τ ab = 0. ( * )
The constructed strategy profile σ is indeed a direct equilibrium if the associated ballot profile τ triggers the magnitude ordering µ ab > µ ac > µ bc .

As τ ab = τ c = 0, Proposition 2 implies that the magnitude ordering is indeed satisfied if and only if:

τ a τ bc > τ ac τ b > 1 and -τ a + τ ac -τ b + τ bc 2 > -τ a -τ bc 2 . ( * * )
To conclude, we obtained a characterization of direct equilibria:8 a preference profile r admits a direct equilibrium if and only if conditions ( * ) and ( * * ) can be jointly satisfied for some labelling of the candidates.

Existence

In this section we show that, for many preference profiles, a direct equilibrium does not exist.

To see why, observe that, by combining conditions ( * ) and ( * * ), we obtain that a necessary condition to have a direct equilibrium with µ ab > µ ac > µ bc is to have:

r a r -a > r ca r ba > 1,
where r a denotes the share of voters who rank a first and r -a denotes the share of voters who rank a last.

Consider the preference profile r, denoting a symmetric Condorcet cycle: r ab = r bc = r ca = 1/3 so that r k /r -k = 1 for k = a, b, c. It is then immediate to see that the preference profile r does not admit a direct equilibrium. By extension, we can prove that the non-existence of a direct equilibrium remains true in a neighborhood of this preference profile. We thus obtain the following result.

Proposition 3. The set of preference profiles admitting no direct equilibrium is of positive measure.

This result is important for the analysis of Poisson voting games: in many cases, there is no equilibrium with a strict pivot magnitude ordering. This implies that we often need to look beyond pivot magnitudes to derive voters' best responses, a step that is undertaken in Section 5.

Quantitative Assessment

To assess quantitatively the prevalence of direct equilibria, we performed Monte-Carlo simulations consisting of 100,000 draws of the preference profile r, the draws being independent and uniform on the simplex over T .

As a first observation, we note that direct equilibria exist in 27.7% of the preference profiles.

This reinforces the conclusion of Proposition 3: only looking at direct equilibria would not allow to understand strategic behavior under AV in a large number of situations.

To go further, we would like to identify preference profiles for which direct equilibria exist (or do not). As a first step, we might hint from earlier models operating with the ordering conditions that direct equilibria are unlikely to be found when a Condorcet cycle exists. 9 Numerically, we find that this intuition is almost exact: there appears to be around 0.1% of preference profiles admitting a Condorcet cycle for which a direct equilibrium exists.

When the preference profile admits a Condorcet winner, we observe that a direct equilibrium exists only for 29.8% of the preference profiles (as a reference, such an equilibrium always exists under the ordering conditions). Focusing on the candidate that can be elected at a direct equilibrium, we observe that it is almost always the Condorcet winner. More precisely, 29.7% of the profiles admit a direct equilibrium electing the Condorcet winner, but no direct equilibrium electing another candidate. Less than 0.1% of the profiles admit a direct equilibrium electing a non-Condorcet winner, but no direct equilibrium electing the Condorcet winner. 10 Finally, only 3 profiles over our 100,000 draws admit both a direct equilibrium electing the Condorcet winner, and another direct equilibrium electing another candidate.

An Example of Direct Equilibrium

We consider an electorate composed of three types of voters: ab, ba and ca with respective shares r ab = 3 10 , r ba = 3 10 and r ca = 4 10 . 9 The logic is quite straightforward under the ordering conditions. As argued in the introduction, any equilibrium with γ a > γ b > γ c satisfying the ordering conditions is such that

γ a = t : a b r t , γ b = t : b a r t and γ c = t : c a r t .
As γ a > γ b , we obtain that a majority prefers a to b, so that γ a > 1/2 > γ b . As we have a Condorcet cycle, a majority prefers c to a, so that γ c > 1/2, hence a contradiction with γ b > γ c . 10 Such a situation, with both one equilibrium electing the Condorcet Winner and one not electing him, was first presented in [START_REF] Núñez | Condorcet Consistency of Approval Voting: A Counter Example on Large Poisson Games[END_REF].

Since we look for a direct equilibrium, preference intensities do not play any role and we leave the values u ab , u ba and u ca unspecified. This preference profile admits a unique direct equilibrium σ with σ ab = a, σ ba = b and σ ca = ac.

The strategy profile σ is depicted in Figure 4, together with candidates' expected scores. the strategy profile σ is a direct equilibrium since, for each 0 ≤ η < 3,

µ ab = -( 7 + η 10 - 3 -η 10 ) 2 > µ ac = - 3 + η 10 > µ bc = µ abc = - 3 + η 10 -   3 -η 10 - 4 10   2 .

INDIRECT EQUILIBRIA

This section considers the second kind of stable equilibria: indirect equilibria in which the magnitudes of pivot events do not fully determine voters' best responses. We first show how to derive best responses by estimating the likelihood ratios of pivot events. Then, we provide a general existence result: for any ordinal preference profile, there exists a specification of cardinal preferences for which an indirect equilibrium exists.

Equilibrium Characterization

Let σ be a strategy profile such that µ ab > µ ac = µ bc = µ abc . Looking at Proposition 2, a ballot profile τ generates this ordering of magnitudes if and only if τ b ≥ τ ac , τ a ≥ τ bc and τ a τ b > τ ac τ bc .

As illustrated on Figure 5, the best response is fully determined for types ab, ba, cb and ca (solid arrows), because the pivot ab dominates. However, types ac and bc need to assess the relative likelihood of pivot events ac, bc and abc: their best response is indeterminate (dashed arrows). To determine the best responses of types ac and bc, we compute for each type the relative probabilities of pivots ac, bc and tie abc. A central tool for this analysis is the notion of an offset ratio. Intuitively, the offset ratio φ m of a ballot m at an event E corresponds to the most likely share of ballots m when E occurs, divided by its expected value nτ m . As shown in the next result, the offset ratios in the event tie abc constitute sufficient statistics to compute the relative likelihood of any relevant pivot event with respect to tie abc .

The values of these offset ratios are obtained as follows. Applying the Dual Magnitude Theorem [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF], the magnitude µ[tie abc ] equals the optimal value of min x,y∈R τ a e x+y + τ ac e x + τ b e -x + τ bc e -(x+y) -1.

Moreover, denoting by (x * , y * ) the values of x, y at the optimum, the offset ratios are given by:

φ a = e x * +y * = τ bc τ a , φ b = e -x * = τ ac τ b , φ ac = 1/φ b and φ bc = 1/φ a . ( 3
)
The following result derives the relative likelihood of pivot and tie events and characterizes indirect equilibria.

Theorem 4. Let σ be a stable equilibrium such that µ ab > µ ac = µ bc (indirect equilibrium).

Then, for t ∈ {ac, bc}:

1. The probabilities of pivot and tie events satisfy:

∀{i, j} = {a, b},

lim n→∞ P[piv t ic | nτ] P[tie abc | nτ] = φ j 1 -φ j φ t 1 + φ t 1 t 2 , lim n→∞ P[tie t,1 abc | nτ] P[tie abc | nτ] = φ t 1 , lim n→∞ P[tie t,2 abc | nτ] P[tie abc | nτ] = φ t 1 t 2 .
2. There exists a cutoff u * t ∈ (0, 1) such that u t < u * t ⇒ σ t = t 1 , u t > u * t ⇒ σ t = t 1 t 2 , and given by

u * t = f t (φ t 1 t 2 ) f t (φ t 1 t 2 ) + f t (1/φ t 1 ) with f t (y) = φ t 1 + (φ t 1 + φ t 1 t 2 ) y + 2 y -1 ,
where the offset ratios are the ones associated to tie abc .

Existence and Condorcet consistency

Contrary to direct equilibria, we establish that indirect equilibria exist for any ordinal preference profile r (at least for some values of the preference intensities). Moreover, we obtain that the equilibrium we construct elects the Condorcet winner when he exists, a property referred to as Condorcet consistency.

Let r be an ordinal preference profile and let a be a candidate such that r a > r -a (the share of voters who rank a first is larger than the share of voters who rank a last). Let b be a candidate such that r ba > r ca . 11 We will show that there exists some u ∈ (0, 1) T for which an equilibrium exists with µ ab > µ ac = µ bc . Let σ be the strategy profile defined by:

σ ab = σ ac = a, σ bc = σ cb = bc, σ ba = b and σ ca = ac.
Under this strategy profile, we know that τ a = r a , τ b = r ba , τ ac = r ca , τ bc = r -a and τ c = τ ab = 0. By assumption, we thus have τ a > τ bc , τ b > τ ac and τ a τ b > τ ac τ bc . Therefore, we obtain from Proposition 2 that the ordering µ ab > µ ac = µ bc is indeed satisfied under the strategy profile σ.

From the magnitude ordering, we know directly that σ t = BR(t | nτ) for any type t ∈ {ab, ba, ca, cb}.

Following the proof of Theorem 4, there exists for each t ∈ {ac, bc} a number u * t ∈ (0, 1) such that, for n large enough:

u t < u * t ⇒ BR(t | nτ) = t 1 and u t > u * t ⇒ BR(t | nτ) = t 1 t 2 .
We conclude that, for u ac small enough and u bc large enough, the strategy profile σ is an indirect equilibrium. 12Finally, let us consider the case where a profile r admits a Condorcet winner, say a, and let us show that r a > r -a . As a majority prefers a to b, we have that r a + r ca > r -a + r ba ; as a majority prefers a to c, we have that r a + r ba > r -a + r ca . Thus, r a -r -a > |r ca -r ba | > 0. Since γ a = r a + r ca , γ b = r -a + r ba and γ c = r -a + r ca , the candidate a is elected, and we thus obtain the following result. Proposition 4. For any ordinal preference profile r, there exists an open set of cardinal representations u ∈ (0, 1) T such that an indirect equilibrium exists. If r admits a Condorcet winner, it is elected at the constructed equilibrium.

Quantitative Assessment

As for direct equilibria, we conducted Monte-Carlo simulations to assess indirect equilibria.

Building on the equilibrium characterization in Theorem 4, we can describe the set of cardinal preferences allowing the existence of any potential indirect equilibrium, for each ordinal preference profile r. Thus, we first draw 200,000 ordinal preference profiles r: 100,00 draws uniformly on the subset of the simplex on T for which a Condorcet winner exists; and 100,000 draws uniformly on the subset of the simplex on T for which a Condorcet cycle exists. Then, for each draw, we can compute explicitly the probability of existence (resp. of co-existence) of any indirect equilibrium (resp. of any set of indirect equilibria), assuming that utilities (u t ) t∈T are drawn uniformly on [0, 1], and independently across types.

We first present the distribution of the probabilities of existence of a stable equilibrium over the universe of ordinal preference profiles, when a Condorcet winner exists. This distribution is plotted in Figure 6. We observe on the right the 29.8% of ordinal preference profiles for which an equilibrium exists with probability one, which corresponds to direct equilibria (note that indirect equilibria may also exist at these profiles). For all other ordinal preference profiles, we observe that an (indirect) equilibrium exists with strictly positive probability, as shown in Proposition 4. Furthermore, we see that this probability of existence is significant, above 30% for most ordinal preference profiles. FIGURE 6. Distribution of the probabilities of existence of a stable equilibrium when a Condorcet winner exists As we know that direct and indirect equilibria may coexist at a given preference profile, one may wonder whether different candidates may be elected for the same profile. We draw on Figure 7 the distribution of the number of candidates that can be elected at a stable equilibrium for a given (cardinal) preference profile, when a Condorcet winner exists. We observe that whenever a stable equilibrium exists (at least one possible winner), it is very rare that 2 candidates be potential winners of the election (1.9% of all cardinal preference profiles).

Then, we investigate whether the likely winner of the election is the Condorcet winner or not. To that aim, we plot on Figure 8 two distributions: the probabilities of existence of a stable equilibrium electing the Condorcet winner (resp. electing a non-Condorcet winner) over the universe of ordinal preference profiles, when a Condorcet winner exists. We observe that for most ordinal preference profiles, only a Condorcet winner can be elected. For around 15% of all ordinal profiles, a non-Condorcet winner may be elected, but with a small probability, Finally, we consider profiles for which no Condorcet winner exists: there is a Condorcet cycle. We report on Figure 9 the distribution of the probabilities of existence of an equilibrium for these preference profiles. We observe that there always is a significant probability that an (indirect) equilibrium exists. This can be seen as an improvement over earlier models which didn't provide any prediction for these preference profiles [START_REF] Laslier | The Leader Rule: Strategic Approval Voting in a Large Electorate[END_REF]. FIGURE 9. Distribution of the probabilities of existence of an equilibrium when a Condorcet cycle exists Finally, we report on Figure 10 the distribution of the number of candidates that can be elected at a stable equilibrium for a given (cardinal) preference profile, when a Condorcet cycle exists. We observe that when a stable equilibrium exists, the possible winner is most likely unique. Moreover, there are rare cases for which any of the three candidates may be elected at a stable equilibrium (for around 1.9% of all cardinal preference profiles). This preference profile admits an indirect equilibrium σ defined by

σ ab = a, σ bc = b, σ ca = ac and σ cb = bc.
The strategy profile σ is depicted in Figure 11, together with candidates' corresponding expected scores.

The strategy profile σ leads to the victory of a since γ a > γ b > γ c for any η. As τ ab = τ c = 0, pivot magnitudes can be computed as in Proposition 2. We obtain:

µ ab = (5 + η)(5 -η) 5 -1 since τ ac τ bc < τ a τ b , µ ac = µ bc = µ abc = 3(2 + η) + 3(2 -η) 5 -1 since τ b ≥ τ ac and τ a ≥ τ bc .
One can check that µ ab > µ ac = µ bc = µ abc for any 0 ≤ η < 1 as required. Building on Proposition 1, the best responses of the different types of voters are as follows: (i) ab-voters approve only a since µ ab is maximal , (ii) ca-voters approve of c since c is their most preferred candidate and approve of a since µ ab is maximal and (iii) cb-voters approve of c since c is their most preferred candidate and approve of b since µ ab is maximal. The decision of bc-voters relies on the exact relative likelihoods of pivot events, which are obtained from the offset ratios. Recall that, for t = bc, ∆(t | nτ) equals:

P[piv t ac | nτ] 1 -u bc 2 + P[piv t bc | nτ] -u bc 2 + 2P[tie t,1 abc | nτ] + P[tie t,2 abc | nτ] 1 -2u bc 6 .
As shown by Theorem 4, the limit ratio of pivot probabilities equals, for t = bc,

lim n→∞ P[piv t ac | nτ] P[tie abc | nτ] = φ b 1 -φ b φ b + φ bc , lim n→∞ P[piv t bc | nτ] P[tie abc | nτ] = φ a 1 -φ a φ b + φ bc , lim n→∞ P[tie t,1 abc | nτ] P[tie abc | nτ] = φ b , lim n→∞ P[tie t,2 abc | nτ] P[tie abc | nτ] = φ bc .
where the offset ratios are given by:

φ a = τ bc τ a = 2 -η 3 , φ b = τ ac τ b = 2 + η 3 , φ ac = 3 2 + η and φ bc = 3 2 -η .
Therefore, we can conclude that BR(bc

| nτ) = b if and only if u bc < u * (η) with u * (η) = φ b + (φ b + φ bc ) φ bc + 2 φ bc -1 2φ b + (φ b + φ bc ) φ bc + 2 φ bc -1 + φ ac + 2 φ ac -1 .
It can be shown that u * (0) = 0.5, u * (1) = 0 and that u * (η) is decreasing in η for 0 ≤ η < 1. To conclude, σ is an equilibrium as long as u bc < u * (η).

As shown in Theorem 3, the indirect equilibrium σ violates the ordering condition (i), since we have µ ac = µ bc , and yet γ a > γ c . We even observe more precisely that the ratio P[piv t ac | nτ]/P[piv t bc | nτ] tends to a positive constant as n gets large. This observation extends to indirect equilibria in general, by direct application of Theorem 4.

CONCLUDING REMARKS

This paper gives an in-depth account of Approval Voting with private values in the main setting for understanding strategic voting: Poisson games. It first proves that stable equilibria are discriminatory, which means that a pivot is infinitely more likely than the other ones, so that a pair of candidates focuses the attention of the electorate. It then divides stable equilibria into two main classes: direct and indirect ones. In a direct equilibrium, pivot magnitudes differ and best responses are immediate, whereas in an indirect one, some pivot magnitudes coincide which renders best responses dependent on preference intensities. Above all, the main result of this work is that no stable equilibrium satisfies the ordering conditions, a set of belief restrictions used in early works to derive predictions on AV. This violation is partly a consequence of the correlation of candidates' scores, arising under AV in Poisson games, but which does not occur in the models of [START_REF] Myerson | A theory of voting equilibria[END_REF] and [START_REF] Laslier | The Leader Rule: Strategic Approval Voting in a Large Electorate[END_REF]. Building on the previous description of equilibria, we have run Monte-Carlo simulations that give us global figures on the structure of stable equilibria under AV. Among the different findings, we can underline two of them.

The first finding concerns the existence of indirect equilibria. These equilibria are, to our knowledge, novel in the literature. Quantitatively, indirect equilibria are important. First, there always exists an indirect equilibrium, at least for some preference intensities, while there are many ordinal preference profiles for which no direct equilibrium exists, even for profiles admitting a Condorcet winner. Second, we observe that the actual probabilities with which an indirect equilibrium exists are significant, both when a Condorcet winner exists and when there is a Condorcet cycle.

The second finding concerns the Condorcet consistency of AV. When a Condorcet winner exists, there always exists an indirect equilibrium that elects him, at least for some preference intensities. If in addition, there exists a direct equilibrium, it is very likely that this equilibrium elects the Condorcet winner. Overall, the probabilities of having a stable equilibrium electing the Condorcet winner are much higher than having a stable equilibrium electing another candidate (Figure 8).

As a final remark on this work, we investigate what would happen in Poisson games if the correlation in candidates' scores was assumed away. We observe that our methods can be profitably exploited to analyse that case. More precisely, we introduce a behavioral twist in the analysis: voters neglect correlation, that is, they compute pivot probabilities and magnitudes as if the scores of the different candidates were independent. 13 As we show in the appendix (Section B), this alternative model yields the following result.

Theorem 5. In the presence of correlation neglect, any equilibrium with γ a > γ b > γ c is either:

1. a direct equilibrium with µ ab > µ ac > µ bc , if γ 2 b < γ a γ c . 2. an indirect equilibrium with µ ab > µ ac = µ bc , if γ 2 b ≤ γ a γ c .
13 While this extension is here developed as a sanity check, the literature on correlation neglect is now firmly established in the economics literature (see [START_REF] Levy | Correlation Neglect, Voting Behavior, and Information Aggregation[END_REF] among others). This behavioral twist, called correlation neglect is grounded on the inability of voters to understand the consequences of correlation and is considered in the literature as a plausible restriction on beliefs. Moreover, on practical terms, it simplifies the analysis as the pivot magnitudes anticipated by voters who neglect correlation just depend on the scores of the candidates and not anymore the share of the different ballots as in the classical analysis.

This result underlines that indirect equilibria exist independently of the correlation between candidates' scores.

Finally, we obtain the desired expression by observing that piv t i j = (piv i j -i) ∪ (piv i j -i j), piv t jk = (piv jk -i) ∪ (piv jk -i j) , tie t,1 abc = (piv abc -i) and tie t,2 abc = (piv abc -i j).

A.2. Proof of Theorem 1

Proof. As a direct application of the Magnitude Equivalence Theorem in [START_REF] Núñez | Condorcet Consistency of Approval Voting: A Counter Example on Large Poisson Games[END_REF], one can show that for any ballot profile τ, any type t and any pair i j of candidates we have:

µ[piv t i j ] = µ{s i = s j ≥ s k } and µ[tie t,1 abc ] = µ[tie t,2 abc ] = µ[tie abc ].
It remains to be shown that for any ballot profile τ , µ{s i = s j ≥ s k } ∈ {µ[tie i j ], µ[tie abc ]}. To do so, we first apply the Dual Magnitude Theorem of [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF]. We can write:

µ[tie abc ] = min x,y∈R f (x, y), µ{s i = s j ≥ s k } = min x∈R,y≥0 f (x, y), µ[tie i j ] = min x∈R,y=0 f (x, y)
where f (x, y) = τ i e x+y + τ ik e x + τ j e -x + τ jk e -(x+y) + τ k e -y + τ i j e y -1.

The function f is strictly convex as ∂ 2 f ∂x 2 = e x (τ i e y + τ ik ) + e -x (τ j + τ jk e -y ) > 0 and:

∂ 2 f ∂x 2 × ∂ 2 f ∂ y 2 - ∂ 2 f ∂x∂y 2 = e
x (τ i e y + τ ik ) + e -x (τ j + τ jk e -y ) × e y (τ i e x + τ i j ) + e -y (τ k + τ jk e -x ) -τ i e x+y + τ jk e -(x+y) 2 > 0.

As | f (x, y)| becomes large when either |x| or |y| is large, f admits a unique critical point (x * * , y * * ) ∈ R 2 . The value y * * solves for min y∈R f (x * (y), y), where x * (y) = arg min x∈R f (x, y) is uniquely defined. We have two cases to consider:

• Either ∂ f ∂ y (x * (0), 0) > 0.
In that case, y * * < 0, and we obtain that

µ{s i = s j ≥ s k }, µ[tie i j ] > µ[tie abc ]
. Moreover, we have:

µ i j = min x∈R,y≥0 f (x, y) = min y≥0 f (x * (y), y) = f (x * (0), 0) = min x∈R,y=0 f (x, y) = µ[tie i j ].
• Or ∂ f ∂ y (x * (0), 0) ≤ 0. In that case, y * * ≥ 0, and we obtain that

µ{s i = s j ≥ s k } = µ[tie abc ].
To conclude, the value x * (0) is obtained by setting

0 = ∂ f ∂x (x * (0), 0) = e x * (0) (τ i + τ ik ) -e -x * (0) (τ j + τ jk ),
and we get e x * (0) = τ j + τ jk τ i + τ ik . Finally, we write

∂ f ∂y x * (0), 0 = τ i e x * (0) + τ i j -τ k -τ jk e -x * (0) = δ i j (τ).
As wanted, we have shown that µ{s i = s j ≥ s k } is equal to either µ[tie i j ], if δ i j (τ) > 0, or to By application of Theorem 1, the formula for the magnitude µ ab depends on the sign of:

δ ab (τ) = τ a τ b + τ bc τ a + τ ac -τ bc τ a + τ ac τ b + τ bc .
We obtain that

δ ab (τ) > 0 ⇔ τ a (τ b + τ bc ) > τ bc (τ a + τ ac ) ⇔ τ a τ b > τ ac τ bc .
When this inequality is satisfied, we may write:

µ ab = µ[tie ab ] = min x∈R,y=0
τ a e x+y + τ ac e x + τ b e -x + τ bc e -(x+y) -

1 = -τ a + τ ac -τ b + τ bc 2 .
The formulas for the magnitudes µ ac and µ bc are obtained in a similar fashion and are hence omitted.

A.4. Proof of Theorem 2

We first prove a lemma, relating a candidates' expected scores to pivot magnitudes.

Lemma 1. Let σ be a strategy profile with associated expected score γ. We have the following: Proof. The proof relies on Theorem 1. For (i), let us assume that γ b ≥ γ a , so that we have τ b + τ bc ≥ τ a + τ ac . We write:

(i) If γ a , γ b > γ c , then µ ab > µ abc (ii) If γ a = γ b > γ c , then µ ab > µ ac = µ bc = µ abc (iii) If γ a > γ b , γ c , then µ bc = µ abc (iv) If γ a > γ b = γ c ,
δ ab (τ) = τ a τ b + τ bc τ a + τ ac + τ ab -τ bc τ a + τ ac τ b + τ bc -τ c ≥ τ a + τ ab -τ bc -τ c = γ a -γ c > 0
Hence, µ ab > µ abc , as desired.

For (ii), it suffices to show that µ ac = µ abc . As γ c ≤ γ a , we obtain as before δ ac (τ) ≤ γ a -γ b = 0.

Hence, µ ac = µ abc , as desired.

For (iii), we use the fact that γ c ≤ γ b . We obtain δ bc (τ) ≤ γ b -γ a < 0. Hence, µ bc = µ abc , as desired.

For (iv), it will be easier to show the claim with the ordering γ b > γ a = γ c . 14 As γ b > γ a , we obtain δ ab (τ) ≥ γ a -γ c = 0, with a strict inequality only if τ a > 0 or τ bc > 0. We thus have two possibilities: either µ ab > µ abc (if δ ab (τ) > 0) or τ a = τ bc = 0. The argument for µ ac is the same.

We now prove Theorem 2.

Proof. We now consider each of the two possible cases of non-discriminatory equilibrium.

Equilibrium with:

µ ab = µ ac > µ bc .
Assume that σ is an equilibrium with µ ab = µ ac > µ bc . As µ ab , µ ac > µ bc , we get from Theorem 1 that

µ ab = -( τ a + τ ac -τ b + τ bc ) 2 , µ ac = -( τ a + τ ab -τ c + τ bc ) 2
and that these formulas remain valid in a neighborhood of τ. Therefore, there is a ballot profile τ arbitrarily close to τ such that µ ab > µ ac > µ bc , and a ballot profile τ for which µ ac > µ ab > µ bc . If σ was stable, σ t would be a best response to both τ and τ . We obtain a contradiction by considering type t = ca for which BR(ca | nτ ) = ac and BR(ca | nτ ) = c for n large enough.

Equilibrium with : µ ab = µ ac = µ bc .

Assume that σ is an equilibrium with µ ab = µ ac = µ bc . By application of Lemma 1, we have two possibilities: either γ a = γ b = γ c or (γ a > γ b = γ c and τ b = τ ac = τ c = τ ab = 0). The second case can be ruled out: we have by assumption r ba > 0, so that τ b > 0 or τ ab > 0.

We can find τ and τ arbitrarily close to τ such that γ a = γ b > γ c and γ a = γ c > γ b . In the first case, pivot ab dominates, while pivot ac dominates in the second case, by application of (ii) in Lemma 1. We get a contradiction as before by considering t = ca.

14 The discriminant δ ab (τ) is defined as the difference between the expected scores of a and c conditional on tie ab .

One could also prove the claim directly, by defining the symmetric discriminant defined as the difference between the expected scores of b and c conditional on tie ab .

A.5. Proof of Theorem 3

Proof. Let σ be a direct equilibrium satisfying µ ab > µ ac > µ bc . It follows from condition (*) that γ a = τ a + τ ac , γ b = τ b + τ bc and γ c = τ ac + τ bc . We observe that τ ac τ bc < τ a τ b since we have We have:

r a r -a ≤ 1/3 + ε 1/3 -ε and r ca r ba ≥ 1/3 -ε ε .
For ε small enough, we have that r ca /r ba > r a /r -a . Thus, no direct equilibrium exists with µ ab > µ ac > µ bc . Moreover, as r ca /r ba < 1, no direct equilibrium exists with µ ac > µ ab > µ bc .

A symmetric argument applies for direct equilibria where b or c belongs to the two first pivots. We conclude that there exists a neighborhood of r for which no direct equilibrium exists. This set of preference profiles is of positive measure, which concludes the proof.

A.7. Proof of Theorem 4

Proof. Let σ be a stable equilibrium with µ ab > µ ac = µ bc and let t ∈ {ac, bc}.

1. The proof relies on the offset theorem introduced in [START_REF] Myerson | Large poisson games[END_REF]. The theorem establishes that events differing from each other by a single translation have a probability ratio that can be computed using the offset ratios.

As σ is an equilibrium satisfying µ ab > µ ac = µ bc , we must have τ ab = τ c = 0 (as µ ab dominates, ab and c are never best responses), τ a ≥ τ bc (as µ bc = µ abc ) and τ b ≥ τ ac (as µ ac = µ abc ).

Claim:

The offset ratios are identical in events piv ac , piv bc and tie abc .

Let us prove the claim for piv ac and tie abc . By slightly abusing notation, we can write piv ac = {Z a = Z bc } ∩ {Z ac > Z b } and tie abc = {Z a = Z bc } ∩ {Z ac = Z b }. We first observe that the offset ratios of ballots a and bc will be the same for the two events: they are each equal to the corresponding offset ratio in the event {Z a = Z bc }. Moreover, as τ b ≥ τ ac , we have that for any n, the most likely outcome in the event {Z ac > Z b } belongs in fact to the event {Z ac = Z b + 1}, which implies that the two events admit the same offset ratios (from the definition of offset ratios). In turn, the offset ratios of ballots ac and b are the same for events {Z ac = Z b + 1} and {Z ac = Z b }, which proves the claim.

From now on, we can thus write φ m for the offset ratio of ballot m. We know from the formulas derived in the main text that φ m > 0 for any m ∈ {a, b, ac, bc}. Applying the offset theorem, we obtain that:

lim n→∞ P[piv t,1 abc | nτ] P[tie abc | nτ] = φ t 1 and lim n→∞ P[[piv t,2 abc | nτ] P[tie abc | nτ] = φ t 1 t 2 .
For the pivots with two candidates, let us focus on pivot ac. With the notations in the statement of the theorem, this corresponds to i = a (the pivot under consideration is ac) and j = b (b is the candidate not appearing in the pivot). Applying the offset theorem, we obtain:

lim n→∞ P[piv t ac | nτ] P[piv ac | nτ] = lim n→∞ P[piv ac -t 1 | nτ] P[piv ac | nτ] + P[piv ac -t 1 t 2 | nτ] P[piv ac | nτ] = φ t 1 + φ t 1 t 2 .
What remains to be done is to compare the likelihood of events piv ac and tie abc . We rely on two observations: first, piv ac = (piv ac ∪ tie abc ) -b,

as {Z ac > Z b } = {Z ac ≥ Z b } -b.
Second, the offset ratio of b in piv ac ∪ tie abc is the same as in any of these two events, precisely because φ b is the same in the two events. Applying the offset theorem, we can write:

| nτ] = φ b 1 -φ b (φ t 1 + φ t 1 t 2 ).
Finally, by symmetry of the problem, we obtain a symmetric formula for the pivot bc. Note that we have implicitly assumed in the proof that all offset ratios are different from 1. One can check that the formulas remain valid if some offset ratios reach the value of 1, concluding the proof of 1. Thus, for n large, the sign of ∆(t | nτ) is the same as the sign of g(u t ). The function g is decreasing and takes value 0 for u * t ∈ (0, 1), defined by:

Applying

u * t = φ t 1 + φ t 1 t 2 3 φ t 1 t 2 -1 + 1 + φ t 1 φ t 1 + φ t 1 t 2 3 φ t 1 t 2 -1 + 3 1/φ t 1 -1 + 2 + 2φ t 1 .
As σ is an equilibrium, we conclude using Proposition 1 that u t < u * t ⇒ σ t = t 1 and u t > u * t ⇒ σ t = t 1 t 2 .

Note again that, while we have assumed that the offset ratios are different from 1, the result remains valid (defining u * t as a limit), when some offset ratios reach this value.

APPENDIX B. ALTERNATIVE MODEL WITH CORRELATION NEGLECT

In the benchmark model, voters condition on the ballot profile τ to determine their best response (that is, they compute the sign of ∆(t | nτ) for each type t). If voters neglect this correlation, they compute pivot probabilities assuming that the scores of the different candidates are independent random variables. In other words, rather that conditioning on the ballot profile τ to compute pivot probabilities, a voter conditions on the score distribution γ assuming that each pair S i , S j is a pair of independent random variables with respective expected scores γ i and γ j . As a simple corollary of Theorem 1, one can write that for any score distribution γ, any type t ∈ T and any pair ab of candidates:

1. In this section, we provide further justifications for our focus on stable equilibria. Starting from a weaker definition of a (large) equilibrium, we first show that discriminatory and stable equilibria coincide. Then, we prove that, for generic preference profiles, any equilibrium is stable. Finally, we demonstrate that discriminatory equilibria, as defined by Myerson (with a condition on relative pivot probabilities), coincide with the stricter definition of discriminatory equilibria we use (with a condition on pivot magnitudes).

We start by stating a definition of a large equilibrium, considering the limit behaviour when n → ∞. Definition 3. The pure strategy profile σ = (σ t ) t∈T is a large equilibrium if there is some n 0 such that for any n ≥ n 0 , for each t ∈ T, σ t ∈ BR(t | nτ).

This definition is more practical, even if formally more restrictive, than that of a large equilibrium in [START_REF] Myerson | Large poisson games[END_REF], which is based the convergence of a sequence of finite-n-equilibria.

C.1. Equivalence between stable and discriminatory equilibria

We first provide a result reciprocal to Theorem 2: any discriminatory large equilibrium is stable.

Proposition 5. Let σ be a discriminatory large equilibrium (µ ab > µ ac , µ bc ). Then, σ is a stable equilibrium.

Proof. There are two kinds of discriminatory large equilibria: direct (µ ab > µ ac > µ bc ) and indirect ones (µ ab > µ ac = µ bc ).

For a direct equilibrium, the conditions (*) and (**) identified in Section 4 must be satisfied.

These two conditions are strict inequalities on the ballot profile τ. Then, in any neighborhood of τ, the magnitude ordering remains, and the best responses are identical: the equilibrium is indeed stable.

For an indirect equilibrium with µ ab > µ ac = µ bc , the conditions of Theorem 4 must be satisfied. As the profile is in pure strategies, and r is assumed to exhibit no tie, we have that (i) all the offset ratios lie strictly between 0 and 1, so that u * ac , u * bc ∈ (0, 1); and (ii) the inequalities As we have assumed µ ab = µ ac , we obtain: This contradicts the assumption that pivot ab becomes infinitely more likely than the other pivots at the limit.

  FIGURE 1. Ordering Conditions.

FIGURE 2 .

 2 FIGURE 2. Graphical representation of the Approval game.

FIGURE 4 .

 4 FIGURE 4. A direct equilibrium with µ ab > µ ac > µ bc .

FIGURE 5 .

 5 FIGURE 5. Best responses (for n large) in an indirect equilibrium with µ ab > µ ac = µ bc .

FIGURE 7 .

 7 FIGURE 7. Distribution of the number of possible winners at a stable equilibrium when a Condorcet winner exists

FIGURE 8 .

 8 FIGURE 8. Distributions of the probabilities of existence of a stable equilibrium electing the Condorcet winner (resp. electing a non-Condorcet winner), when a Condorcet winner exists

FIGURE 10 .

 10 FIGURE 10. Distribution of the number of possible winners at a stable equilibrium when a Condorcet cycle exists

FIGURE 11 .

 11 FIGURE 11. An indirect equilibrium with µ ab > µ ac = µ bc

  µ[tie abc ] otherwise. A.3. Proof of Proposition 2 Proof. By application of the Dual Magnitude Theorem, as in the proof of Theorem 1, we have µ abc = min x,y∈R τ a e x+y + τ ac e x + τ b e -x + τ bc e -(x+y) -1, which implies that µ abc = -τ a -τ bc 2 -τ b -τ ac 2 .

  then µ bc = µ abc and (µ ab > µ abc or τ b = τ ac = 0) and (µ ac > µ abc or τ c = τ ab = 0)

  lim n→∞ P[piv ac | nτ] P[piv ac | nτ] + P[tie abc | nτ] = lim n→∞ P[(piv ac ∪ tie abc ) -b | nτ] P[piv ac ∪ tie abc | nτ] = φ bFrom this we obtain that lim n→∞ P[piv ac | nτ]P[tie abc | nτ] = φ b 1 -φ b ,and we conclude that lim n→∞ P[piv t ac | nτ] P[tie abc

  µ[tie ab ] = -( γ a -γ b ) 2 , 2. µ[tie abc ] = 3 γ a γ b γ c 1 3 -γ a -γ b -γ c , 3. µ[piv t ab ] = µ[tie ab ] if γ 2 c < γ a γ b , µ[tie abc ] if γ 2 c ≤ γ a γ b , 4. µ[tie t,1 abc ] = µ[tie t,2 abc ] = µ[tie abc ].Following the methods developed in the article, we obtain Theorem 5.APPENDIX C. TECHNICAL DISCUSSION

  and ln (P[Za + Z ac = Z b + Z bc | nτ]) n = -1 + 2 (τ a + τ ac )(τ b + τ bc ) τ a + τ ac )(τ b + τ bc ) + o n→∞ (1)= µ ab -1 2n ln 4πn (τ a + τ ac )(τ b + τ bc ) + o n→∞ (1).

  Definition 1. A strategy profile σ * = (σ * t ) t∈T is a stable equilibrium if ∃ ε > 0 and n 0 ≥ 0 such that for any ballot profile τ in the simplex over M with |τ -τ * | < ε and n ≥ n 0 , we have σ * t ∈ BR(t | nτ), where τ * denotes the ballot profile associated to σ * . For any voter with type t and i ∈ {1, 2, 3}, we denote by t i the candidate in i-th position in his preference ranking (for example, t 1 is his most liked candidate). By a slight abuse of notation, t 1 denotes a ballot in favor of his best candidate and t 1 t 2 a ballot in favor of his two best candidates. Thus, depending on the context, a may denote the ballot a or the candidate a, and ab may denote the ballot ab or the type ab. It can be shown that, 5 for each ballot profile τ, each n and each type t, U t [t 1 | nτ], U t [t 1 t 2 | nτ] > U t [m | nτ] for any m ∈ M \ {t 1 , t 1 t 2 }. It follows that w.l.o.g. the strategy set of each type t can be restricted from M to M t = {t 1 , t 1 t 2 }, simplifying the strategic analysis.

	1.3. Best Responses

  µ ab > µ abc and τ b < τ ac since µ ac > µ abc . Combining the pair of previous inequalities implies that τ bc < τ a , which, in turn, implies that γ a > γ c . Moreover, the inequality τ b < τ ac implies on its own thatγ c > γ b . It follows that γ a > γ c > γ b .Let σ be an indirect equilibrium. By a direct application of Lemma 1, it cannot be that γ c > γ a (considering all the possible values for γ b ). Therefore γ a ≥ γ c , and similarly γ b ≥ γ c . A.6. Proof of Proposition 3 Proof. Consider a preference profile r in a neighborhood of r (the symmetric Condorcet cycle), say for each t ∈ T , |r t -r t | ≤ ε.

  the previous result, we obtain lim n→∞ ∆(t | nτ)/P[tie abc | nτ] = g(u t ) withg(u t ) = 1/φ t 1 t 2 1 -1/φ t 1 t 2 (φ t 1 + φ t 1 t 2 ) 1 -u t 2 + φ t 1 1 -φ t 1 (φ t 1 + φ t 1 t 2 ) -u t 2 + (2φ t 1 + φ t 1 t 2 ) 1 -2u t 6 .

  P[piv ac | nτ] P[piv ab | nτ] ≥ (τ a + τ ac )(τ b + τ bc ) τ a τ bc + o n→∞ (1).

Note that[START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF] defines a discriminatory equilibrium as one in which a pivot (say ab) is infinitely more likely than the other ones, and this does not imply that pivot magnitudes are different a priori. However, we show in the appendix (Section C.3) that the two definitions coincide for stable equilibria.

Note that direct equilibria are indeed stable, since the magnitude ordering remains the same in a neighborhood of τ, by virtue of condition ( * * ).

Such candidates must exist since for any subsets of types T, T ⊆ T , with T T , t∈T r t t∈T r t by assumption.

 12 Note that the magnitude ordering is preserved in a neighborhood of τ, and that each u * t is a continuous function of τ. Thus, the strategy profile σ is indeed a stable equilibrium.

Tasos Kalandrakis, Yukio Koriyama, Jérôme Lang, Klaus Nehring and Yulliy Sannikov for useful comments. Financial support from the ANR-14-CE24-0007-01 (CoCoRICoCoDEC) project and the ANR-17-EURE-001 (a French government subsidy managed by the Agence Nationale de la Recherche under the framework of the Investissements d'avenir programme) is acknowledged. a NOKIA-BELL LABS FRANCE. This author carried out the work presented in this paper at LINCS (www.lincs. fr).

Proof. We describe in-depth the decision problem of a voter with type i j (with k denoting the remaining candidate). As the type t is fixed in this proof, we will write u for u t . By definition, we have

(4)

Like we did for the particular case of two candidates, we define for each K ⊆ K the event piv K = z ∈ Z | s i (z) = s j (z) > s k (z), ∀i, j ∈ K, ∀k ∉ K , where all candidates in K are tied and have larger scores than the other candidates. Note that piv abc = tie abc . The expected utility of a voter with type t approving of i equals:

whereas the expected utility of this voter approving of both i and j at event z equals:

Thus, the difference between both expected utilities,

The previous expression can be simplified due to the following observations. Note first that (piv i -i) = (piv i -i j) ∪ (piv i j -i j). Hence,

Similarly, observe that (piv j -i j) = (piv j -i) ∪ (piv i j -i) ∪ (piv jk -i) ∪ (piv abc -i). Thus,

Finally, following a similar reasoning, one can see that (piv ik -i) = (piv ik -i j) ∪ (piv abc -i j)

and therefore,

Combining equalities ( 5) to ( 7), ( 1) can be rewritten as :

on τ justifying the magnitude ordering are strict. Therefore, in a neighborhood of τ, the magnitude ordering is unchanged, and the values u * ac , u * bc move continuously. It follows that the equilibrium is indeed stable.

C.2. On the generic non-existence of non-stable equilibria

We establish that pure-strategy equilibria that are not stable only exist for a 0-measure set of ordinal preference profiles. Proposition 6. The set of preference profiles r for which a non-stable large equilibrium exists (µ ab = µ ac > µ bc or µ ab = µ ac = µ bc ) is of measure 0.

Proof. We consider each of the two possible cases of non-discriminatory large equilibrium.

Assume that σ is an equilibrium with µ ab = µ ac > µ bc . As µ ab , µ ac > µ bc , we get from Theorem 1 that

and that these formulas remain valid in a neighborhood of τ. Therefore, such an equilibrium cannot be found generically in a neighborhood of r (as strategies are pure).

Assume that σ is an equilibrium with µ ab = µ ac = µ bc . As shown in the proof of Theorem 2, the expected scores of two candidates must be equal. This cannot remain true generically in a neighborhood of r.

C.3. On the notion of a discriminatory equilibrium [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF] introduces the notion of discriminatory equilibrium in which one pivot event between some pair of candidates is infinitely more likely than any other pivots event. For any pair of candidates i j, the event piv i j is serious if lim n→∞ P[piv i j |nτ] K⊆K P[piv K |nτ] > 0. That is, the pivot between i and j is serious if, in the event that a close race exists in the election, the conditional probability that i and j are in this close race has a positive limit as the expected population gets large. A candidate i is said to be serious if there is some other candidate j such that piv i j is serious. According to [START_REF] Myerson | Comparison of Scoring Rules in Poisson Voting Games[END_REF], a large equilibrium is discriminatory if and only if there is a candidate in K who is not serious whereas we have assumed that a discriminatory equilibrium is characterized by the ordering of the magnitudes being µ ab > µ ac , µ bc up to a relabeling of the candidates. The next theorem shows that both notions are identical. Theorem 6. If a large equilibrium is not discriminatory: µ ab = µ ac ≥ µ abc , then the relative probabilities of pivots ab and ac do not diverge: P[piv ab | nτ]/P[piv ac | nτ] is bounded above and bounded away from 0.

Proof. Let σ be a discriminatory equilibrium with µ ab = µ ac = µ bc and assume that pivot ab is infinitely more likely than the other pivots. We can apply the formula in Proposition 2 for the magnitudes. The equality of all magnitudes imply that τ a = τ bc and τ b = τ ac . Without loss of generality, we may assume that τ a ≥ τ b . Let us show that, in this case, pivot ac is no less likely than pivot ab. While the argument is done for the events piv K it extends to the events piv t K .

We have Z a , Z bc ∼ P (τ a n) and Z b , Z ac ∼ P (τ b n). We write:

We obtain the desired contradiction: pivot ab cannot be infinitely more likely than pivot ac at the limit.

Finally, let us consider an equilibrium σ with µ ab = µ ac > µ bc and assume that pivot ab is infinitely more likely than the other pivots. Applying the formula in Proposition 2, we must have:

We can show that:


R Gantois

A A⇤

Cantarel

G Dusserre

J.-N Félices

F Schmidt

NUMERICAL SIMULATION OF RESIN TRANSFER MOLDING USING BEM AND LEVEL SET METHOD

Keywords: Resin Transfer Molding (RTM), Level Set, Boundary Element Method (BEM)

Resin Transfer Molding is widely used to produce fiber-reinforced materials. In the process, the resin enters a close mold containing the dry fiber preform. For mold designer, numerical simulation is a useful tool to optimize the mold filling, in particular to identify the best positions of the ports and the vents. An issue in mold filling simulation is the front tracking, because the shape of the resin front changes during the flow. In particular, topological changes can appear resulting from internal obstacles dividing the front or multi-injection. A previous approach [1] using the Boundary Element Method (BEM) in a moving mesh framework shows the capability of the method to compute accuratlely the front propagation at low CPU time. The present paper describes a method developed to handle complex shapes, using BEM together with a Level Set approach. Numerical results in two dimensions are presented, assuming a Newtonian non-reactive fluid, and an homogeneous and not-deformable reinforcement. The resin flow in the fibrous reinforcement is modeled using Darcy's law and mass conservation. The resulting equation reduces to Laplace's equation considering an isotropic equivalent mold. Laplaces equation is solved at each time step using a constant Boundary Element Method to compute the normal velocity at the flow front. It is extended to the fixed grid and next used to feed a Level Set solver computing the signed distance to the front. Our model includes a boundary element mesher and a Narrow Band method to speed up CPU time. The numerical model is compared with an analytical solution, a FEM/VOF-based simulation and experimental measurements for more realistic cases involving multiple injection ports and internal obstacles.

INTRODUCTION

Liquid Composite Molding (LCM) processes are widely used in industry. Among them, Resin Transfer Molding (RTM) is one of the most popular. It consists in injecting the liquid resin in the dry preform held in position in a mold. Recent aerospace programms focus on forming structural parts using LCM. As mechanical performances strongly depend on filling conditions, resin flow prediction is important in mold design. In particular gates' locations are adjusted so that the resin impregnates correctly the entire preform. In that task, numerical simulation can be usefull. As the resin flows, a tracking technique is employed to follow the moving front. It can be performed using moving mesh methods [START_REF] Schmidt | Numerical simulation of resin transfer molding using linear boundary element method[END_REF][START_REF] Um | A study on the mold filling process in resin transfer molding[END_REF][START_REF] García | Meshless methods with application to liquid composite molding simulation[END_REF], Volume Of Fluid methods [START_REF] Trochu | Numerical analysis of the resin transfer molding process by the finite element method[END_REF][START_REF] Hirt | Volume of fluid (vof) method for the dynamics of free boundaries[END_REF] or Level Set methods [START_REF] Sethian | Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry[END_REF][START_REF] Soukane | Application of the level set method to the simulation of resin transfer molding[END_REF]. The main advantage of the last one is that the front is accurately captured in an Eulerian ⇤ Corresponding author: Postal address: Ecole des Mines Albi, Campus Jarlard, F-81013 Albi, France. Phone : (+33) 5.63.49.30.00. Email address : renaud.gantois@mines-albi.fr framework. The present paper focus on a technique combining a Boundary Element Method(BEM) [START_REF] Schmidt | Numerical simulation of resin transfer molding using linear boundary element method[END_REF][START_REF] Um | A study on the mold filling process in resin transfer molding[END_REF][START_REF] Soukane | Application of the level set method to the simulation of resin transfer molding[END_REF][START_REF] Yoo | Numerical simulation of the resin transfer mold filling process using the boundary element method[END_REF] and a Level Set method.The first part of this paper considers the governing equations. The second part describes the implemented model. The last part covers some applications.

GOVERNING EQUATIONS

PRELIMINARY TRANSFORMATIONS

Our model assumes that the liquid resin is a Newtonian non-reactive fluid, flowing in isothermal conditions through an homogeneous and not deformable fibrous reinforcement. In the impregnated area ⌦, the macroscopic resin motion is governed by the modified Darcy's law [START_REF] Simacek | Role of acceleration forces in numerical simulation of mold filling processes in fibrous porous media[END_REF] and incompressibility equation

< ! v >= [K] µ ! rp 0 (1) ! r. < ! v >= 0 (2) 8 > > < > > :
where < ! v > is the macroscopic velocity, [K] the permeability tensor, µ the liquid resin viscosity and p 0 the acting pressure. That pressure given by p 0 = p + ⇢gz includes a gravity term, where ⇢ is the resin specific mass and g gravity. Combining the previous equations and transforming the coordinates in the isotropic equivalent domain ⌦ e of permeability K e , we obtain

! v = K e µ✏ ! rp 0 (3)
4p 0 = 0 (4) 8 > < > :
where ! v is the resin velocity, computed from the macroscopic velocity using ✏ (medium porosity), and K e is the equivalent isotropic permeability. The isotropic equivalent transformation [START_REF] Schmidt | Numerical simulation of resin transfer molding using linear boundary element method[END_REF][START_REF] Adams | Radial penetration of viscous liquid into a planar anisotropic porous medium[END_REF] is

" x e y e # = p K e " 1 p K 1 0 0 1 p K 2 #" x y # (5)
where K 1 and K 2 are the principal permeabilities of the reinforcement. The isotropic equivalent permeability, is given as

K e = p K 1 K 2 .

BOUNDARY CONDITIONS

Let us consider e the isotropic equivalent front bounding ⌦ e . Boundary conditions are assigned according to the location of the point M 2 e under consideration. Dirichlet conditions (imposed pressure) are prescribed on the gate ep 0 and on the free edge

ep f p 0 = ( p 0 + ⇢gz 0 8M 2 ep 0 p f + ⇢gz f 8M 2 ep f (6)
and Neuman conditions (imposed normal pressure gradient) are prescribed on the mold wall eq 0 for a nonpenetration condition

! rp 0 . ! n = 0 8M 2 eq 0 (7)
where ! n is the unit outwards vector at point M .

NUMERICAL METHOD

OUTLINE

Our programm is implemented using Matlab environement. At the beginning, pre-processing imports a standard mesh file, performs the isotropic equivalent transformation, assigns material and processing data, and locates the injection ports. Next, the filling routine based on a Level Set formulation advances the front. At the end, the post-processing plots the results once the real domain is recovered.

The filling stage is divided into a finite number of quasi steady states. At each step time, governing equations are solved using a constant Boundary Element Method. More details can be found in section 3.2. The front is advanced by feeding a Level Set solver with the BEM-computed velocities. How it is done is covered in section 3.3. It is repeated until the mold is completely filled.

BOUNDARY ELEMENT METHOD

For clarity, the subscript e referring to the isotropic equivalent domain is omitted by the next. As mentioned earlier, BEM confines the calculation on the front which is a close curve bounding the calculation domain ⌦. A known value of pressure p or normal pressure gradient q is prescribed on the edges p or q . Laplace's equation is multiplied by the Green function p ? . Integrated twice by parts over the calculation domain and using Green's theorem leads to Somigliana's equation [START_REF] Schmidt | Numerical simulation of resin transfer molding using linear boundary element method[END_REF][START_REF] Brebbia | Boundary elements: an introductory course[END_REF][START_REF] Mathey | Optimisation numérique du refroidissement des moules dinjection de thermoplastiques basée sur la simulation des transferts thermiques par la méthode des éléments frontières[END_REF]

c i p i + ˆ pq ? d = ˆ p ? qd with c i = ✓ 2⇡ (8)
where p i is the value of the pressure at a point M i on the boundary, q ? the pressure gradient associated with p ? and ✓ the internal angle of the corner in radians. For a twodimensional domain, p ? and q ? are given as

p ? = 1 2⇡ ln ✓ 1 r ◆ and q ? = 1 2⇡ ! r . ! n r 2 (9)
where r is the distance from the point M i of application of the Dirac delta function to any point under consideration.

Meshing the boundary into N constant boundary elements and applying Equation 8 leads to

c i p i + N X j=1 p j ˆ j q ? d = N X j=1 q j ˆ j p ? d (10)
Rewritten in a matrix form using H and G, Equation 10 is then transformed into

N X j=1 p j H ij = N X j=1 q j G ij (11)
where

H ij = 1 2 ij +
´ j q ? d and G ij =

´ j p ? d are N 2 matrix. Finally, the previous equation is reordered to take the form of a linear system AX = F , where X is a vector of N unknowns.

LEVEL SET METHOD

The moving front is captured using the signed-distance function , defined so that the zero level set corresponds to the interface [START_REF] Sethian | Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry[END_REF][START_REF] Sethian | Level set methods for fluid interfaces[END_REF]

(t) = (x, y) 2 R 2 / (x, y, t) = 0 (12)
For each point under consideration, distance is signed negative if located in the impregnated area and positive otherwise. We consider two meshes : a fixed grid made of unstructured triangle elements and a moving mesh made of beam elements. Level Set equations [START_REF] Sethian | Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry[END_REF][START_REF] Sethian | Level set methods for fluid interfaces[END_REF] govern the evolution of the signed-distance function. To combine with BEM, we use a formulation involving a propagating interface with a velocity in its normal direction, given as follows

@ @t + F | ! r | = 0 (13) (x, y, t = 0) = 0 (14) 8 > < > :
where F is the extended normal velocity, built to coincide with the velocity of the front (it is extrapolated outside whithout physical meaning). The filling stage is initialized by choosing 0 as the signed-distance to the inlet gates.

Next, values for each points are updated using an Euler scheme

(x, y, t + t) = (x, y, t) F | ! r | t ( 15
)
where t is the time step, adjusted to match its upper limit (CFL conditions).

For the sake of efficiency, is not updated on the entire grid, but only on few nodes around the front in a "narrow band" [START_REF] Sethian | Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry[END_REF]. Next, front is rebuilt using a boundary element mesher, based on interpolating the zero level set on the grid.

The method directly handles topological changes, involving merging or dividing fronts, but does not ensure that the resin remains inside the mold. Contact with mold is implemented using a fixed level set describing the mold walls. It acts by correcting as follows

= max( , mold ) ( 16
)
where mold is the signed-distance to the mold.

APPLICATIONS 4.1 ISOTROPIC RADIAL INJECTION

We consider here the standard case of an isotropic radial injection, for a 0.25 m 2 square plate. The resin enters the part from the center and flows through the preform describing circular patterns. Four vents located on the corners ensure the complete filling. Material and processing parameters are summerized in Tables 1 and2. In the first case, gravity is neglected for a comparison with an analytical solution [START_REF] Schmidt | Numerical simulation of resin transfer molding using linear boundary element method[END_REF][START_REF] Adams | Radial penetration of viscous liquid into a planar anisotropic porous medium[END_REF]. It is given as

Table 1: Material data K[m 2 ] µ[P a.s 1 ] ⇢[kg.m 3 ] ✏[ ] 1.10 9 0.1 1150 0.5
p 0 [P a] p f [P a] r 0 [m] g[m.s 2 ] 2.
 r f r 0 2 .  2 ln ✓ r f r 0 ◆ 1 + 1 = 4K (p 0 p f ) t ✏µr 2 0 ( 17
)
where r f and r 0 are the radii of the moving front and the inlet gate, p 0 and p f the inlet and outlet pressures. The mold is meshed using 2500 triangle elements. CPU time is around 20 s on a 2.26 GHz / 1.93 Go of RAM laptop. The predicted filling time is 98 s. Our model is assessed by fitting the numerical fronts using the analytically predicted circles. Figure 1 shows the comparison between our numerical results (left) and the analytical results (right). Analytical results (in bold lines) are overlaid on our results, for 10.7 s and 63.3 s elapsed time. Using L1 norm leads to the relative error of 2.1%, which shows the accuracy of our numerical model. The second case takes into account gravity effects. On Figure 2 we compare our results (left) with a FEM/VOFbased code (PAM-RTM TM ) (right) using the same mesh.

As expected, the flow is slightly deported in the lower part of the plate. The predicted filling time is 101 s for our simulation and 89 s for PAM-RTM TM , which shows a good qualitative accordance. The difference is due to the contact implementation which is different in Level Set (see Equation 16) and VOF. Further developements will improve the contact accuracy by refining the mesh on the mold wall.

ANISOTROPIC INJECTION INVOLVING COMPLEX SHAPES

We compared our numerical results with experimental data, performing an infusion experiment involving a more realistic case. It consisted in impregnating an anisotropic knitted glass preform (1x1 rib knit fabric manufactured by Textile Aero Tarn) using a canola oil. The reinforcement (one layer) was placed under a transparent flexi-ble bag and filled using two inlet gates and two vaccum lines (on the upper and lower sides). The dimensions of the mold are 0.3 m per 0.35 m. Some internal obstacles were placed to simulate contacts with mold walls. Material data (see reference [START_REF] Dusserre | Effect of deformation on knitted glass preform in-plane permeability[END_REF] for measurement procedure) and processing parameters are given in Tables 3 and4. CPU time is around 200 s for a model involving 3693 elements. Figure 3 shows the comparison between our results (at the top) and experimental data (at the bottom) at different times.The agreement is fair at any time. In particular, fronts merging (at 3 s) and dividing on the obstacles (at 7 s) are accurately predicted.

CONCLUSION

We developed a software to predict a two-dimensional resin impregnation for both isotropic and anisotropic cases. Results were compared with an analytical solution, a FEM/VOF-based simulation and experimental data, with a fair agreement. The implemented model includes a gravity term, but it can easily be modified to take into account other body forces. Further developements will include a 3D approach (technique remains unchanged), and a high permeability layer to simulate infusion process.

Figure 1 :

 1 Figure 1: Comparison with analytical solution

Figure 2 :

 2 Figure 2: Comparison with FEM/VOF (PAM-RTM TM )

.10 3 Figure 3 :

 33 Figure 3: Comparison with infusion experiment

Table 2 :

 2 Processing data

Table 3 :

 3 Material data

	K 1 [m 2	]	K 2 [m 2	]	µ[P a.s 1	]	✏
	1.50.10 9	7.75.10 10	0.067		0.705

Table 4 :

 4 Processing data

ACKNOWLEDGEMENT

This work was supported by DAHER-Socata and CRCC. The authors are grateful to B. Cosson (ENSM Douai) and M. Bordival (ICA-Albi) for their contributions.