Emilie Dargaud

Maxime Menuet

Petros G Sekeris

❸ Petros

G Sekeris

Collusion and Predation Under Cournot Competition

Keywords: Collusion, Predation, Cournot competition JEL codes: D43, L13, L41 Collusion, Predation, Cournot competition. JEL: D43, L13, L41

This paper studies how predation strategies can affect the sustainability of collusion. We demonstrate that in the presence of few competitors collusion may be sustained at equilibrium for intermediate discount factors. In such instances predation implies that punishment strategies will yield low subgame perfect payoffs, thereby making collusion easier to sustain. For low discount factors collusion is not sustainable because of the high incentives to deviate to Cournot-Nash strategies. Moreover, for high discount factors it is always optimal to predate colluding firms, thus contrasting with much of the earlier literature showing that collusion is only achievable by sufficiently patient firms.

Introduction

Antitrust authorities closely scrutinize collusive practices since they allow firms to exert market power that would otherwise be absent, eventually resulting in reduced competition ✯ Univ Lyon, Université Lumière Lyon 2, GATE, UMR 5824, F-69130 Ecully, France. Email: dargaud@gate.cnrs.fr. Declarations of interest: none ❸ LEO, University of Orleans, Orleans, France. e-mail: maxime.menuet@univ-orleans.fr. Declarations of interest: none ❹ Corresponding author: TBS Business School, 1 Place A. Jourdain, 31000 Toulouse, France. E-mail: p.sekeristb@tbs-education.fr.

1 and increased prices. The literature on collusive agreements is quite extensive (see e.g. [START_REF] Choi | The Oxford Handbook of International Antitrust Economics[END_REF] for a recent review), and has mainly focused on industry characteristics that influence the sustainability of collusion (e.g. number of firms, entry barriers, market transparency, firms' asymmetry, product differentiation, demand fluctuation, . . . ). More particularly, the discount factor has been shown to play a key role: collusion arises at equilibrium only if firms are sufficiently patient not to be incentivized to reap the short run benefits of deviation.

In regards to the relationship between discount factors and cartel stability, the key connecting element is the credibility of the punishment strategies that could support collusion.

Cartels are indeed unstable in principle since firms are tempted to deviate from collusive agreements (e.g. by charging a lower price than rivals) thereby provoking a cartel breakdown. For collusive agreements to be stable, colluding firms must be able to both detect the deviating firms and to punish them. Notwithstanding deviation detections that may be hampered by market opacity, the credibility question has been shown to be truly central in explaining the stability of collusive agreements. Indeed, for collusion to be an equilibrium, firms ought to be certain that any agreed punishment will be carried out if a deviation were to take place, i.e. punishments ought to be subgame perfect. Two particular types of credible punishments have been identified: grim-trigger punishment strategies, and optimal penal codes. The former type of punishment, initially proposed by [START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF] consists in firms maximizing their competitive period payoff forever if a deviation from the collusive agreement were to occur. Optimal penal codes [START_REF] Abreu | On the theory of infinitely repeated games with discounting[END_REF]) consist in stick and carrot strategies that consider the worst possible punishments that may be history-dependent and are subgame perfect. As such, optimal penal codes allow collusion to emerge in more instances than grim-trigger strategies do.

When assessing the potential ongoing collusion in industries, competition authorities view (more) competitive settings as the benchmark against which one ought to compare ongoing practices. An implicit assumption, therefore, is that such market structure is indeed sustainable (i.e. Cournot Nash equilibria, marginal cost pricing, or any alternative competitive benchmark). Predatory practices-whereby firms predate their competitors to monopolize the market-may, however, jeopardize the very stability of the competitive benchmark, thereby necessitating a totally different approach to the study of collusion since the sustainability of collusion and deviations from collusive agreements must be reconsidered. In instances where predation strategies destabilize the competitive equilibrium, it is important to understand whether collusion may nevertheless be sustained. As such, collusion could potentially be achieved despite the threat of monopolization.

In this paper we consider a Cournot competition setup with n credit constrained firms.

Predation in our setting can emerge because firms are assumed to have (potentially mild) economies of scale, thence implying that predating firms can drive competitors to bankruptcy while still being profitable. Contrary to earlier findings, we show that with highly patient firms collusion cannot be supported at equilibrium because of the profitability of predation, i.e. because of the high discounted payoffs of monopolizing the market. Instead, collusion is shown to emerge for an intermediate range of discount factors, and provided the number of competitors is low (n ≤ 3). If firms are highly patient, predating colluding competitors will always be optimal because of the high expected flow of profits once a firm has wiped out competitors, while low discount factors imply that even when considering the harshest possible punishment strategy in case of non-compliance with the collusive agreement, the short run benefits of deviation are sufficiently high to disrupt the collusive equilibrium. For intermediate discount factors and large number of firms (i.e. n ≥ 4), the collusive profits are not sufficiently important to disincentivize predation, thus once more disrupting the collusive agreement. Last, for intermediate discount factors and few competitors (n ≤ 3) collusion can be sustained while (credibly) threatening deviating firms with the harshest available punishment which is shown to yield zero profits to the deviating firm.

The crude oil industry provides a good illustration of our theory. This industry-which constitutes the textbook example of cartel formation in Cournot competition setups-is inherently subject to increasing returns to scale, thereby opening up the space for monopolization practices. When the industry was still at its infancy in the late 1860s, Standard Oil Co. initiated aggressive predatory pricing practices, made possible through collusion with the transportation industry, i.e. the railroad industry [START_REF] Granitz | Monopolization by "raising rivals' costs": The standard oil case[END_REF], and thanks to the increasing returns to scale, enabling the company to eventually monopolize the vast majority of the U.S. market [START_REF] Mcgee | Predatory price cutting: The standard oil (n. j.) case[END_REF]. In 1911, the company was found to have violated the Sherman Antitrust Act and was subsequently broken down into 34 separate companies, only later to be consolidated again in few companies some of which became the backbone of the oil cartel known as the Seven Sisters. With the oil shocks of the 1970s the OPEC cartel made its appearance, and collusive practices were observed until the late 1980s, after which the relative abundance of oil supply de facto rendering collusion more difficult to sustain.

Although it is admittedly reductive to explain the entire evolution of the oil industry through the lenses of our model, the theory proposed in this paper contributes to understanding why monopolization was followed by collusion and thereafter by a situation more akin to Cournot competition. Indeed, our model predicts monopolization for high discount factors, collusion for intermediate ones, and Cournot-Nash equilibria for low discount factors. The discount factor of firms is a mapping of several firm-specific, industry-specific, and time-specific variables, but the available stock of oil constitutes a crucial determinant in the context of the oil industry as hypothesized and empirically supported by previous research establishing that the discount factor is a negative function of the available (known) stock of resources [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF][START_REF] O'donoghue | Doing it now or later[END_REF][START_REF] Van Veldhuizen | Renewable resources, strategic behavior and the hotelling rule: An experiment[END_REF]. In other words, as the relative availability of oil stocks has been increasing through time with the recurrent discovery of new oil fields [START_REF] Auzanneau | Or Noir: La Grande Histoire du Pétrole[END_REF], it is fair to conjecture that the firms' discount factors have been contracting though time, thence resulting in monopolization practices, followed by collusion, and eventually Cournot-Nash competition. This article contributes to the literature tying predation to market structure. Two important mechanisms that reduce future competition and increase profit by temporarily decreasing prices have been highlighted by scholars: the deep-pocket approach, and signalling mechanisms in the context of asymmetric information. Regarding the former mechanism, [START_REF] Telser | Cutthroat competition and the long purse[END_REF] shows that firms with "deep pockets" can afford the short run cost of predation in view of eventually monopolizing the market. This argument was subsequently adapted to settings where monopolists attempt retaining market power by adopting limit-pricing practices after entrants decide to challenge the incumbent [START_REF] Benoit | Financially constrained entry in a game with incomplete information[END_REF]. [START_REF] Bolton | A theory of predation based on agency problems in financial contracting[END_REF] show that mitigating agency problems between the firm and its financiers creates incentives for predatory behaviour. Second, predatory pricing can also be a rational signalling device in the context of incomplete information: low present prices can decrease future com-petition by impacting competitors' beliefs about the profitability of entry [START_REF] Milgrom | Limit pricing and entry under incomplete information: An equilibrium analysis[END_REF] or by building an aggressive reputation for incumbents [START_REF] Kreps | Reputation and imperfect information[END_REF]. As such predation has been demonstrated to be an effective tool to monopolize the market (Wiseman 2017), or to retain monopoly power.

Our work is closely related to the thin literature on predation and collusion. Earlier work has explored the role of discount factors on the sustainability of collusion in the presence of predation mechanisms. [START_REF] Kawakami | Collusion under financial constraints: Collusion or predation when the discount factor is near one?[END_REF] show that in such setups financially constrained firms can only sustain collusive agreements for intermediate discount factors.

The intuition of their result is that the firm with the deepest pockets is always able to enter in a longer war of attrition for the control of the market, thence implying that with a discount factor near one collusion will not be achievable. [START_REF] Kawakami | Collusion and predation under the condition of stochastic bankruptcy[END_REF] extends this work by considering asymmetric information on the firms' financial constraints. We complement [START_REF] Kawakami | Collusion under financial constraints: Collusion or predation when the discount factor is near one?[END_REF] and [START_REF] Kawakami | Collusion and predation under the condition of stochastic bankruptcy[END_REF] by endogenizing the firms' payoffs, in turn demonstrating that the degree of competition-as captured by the number of firms-is a crucial determinant of the sustainability of collusion. In a setup different to ours, since their work restricts the analysis to two firms and Markov perfect equilibria, [START_REF] Beviá | Oligopolistic equilibrium and financial constraints[END_REF] derive conditions for the Cournot Nash equilibrium to be sustained when firms are financially constrained. Observe that although [START_REF] Beviá | Oligopolistic equilibrium and financial constraints[END_REF] consider the possibility of firms colluding, the sole deviations from collusion that are considered are Cournot reversion strategies, thence neglecting the possibility (that we do consider) for firms to predate each other when attempting to collude. Wiseman (2017), in a different setup to ours, also shows that collusion is not sustainable for high discount factors because of monopolization practices. Our mechanism conducive to market monopolization rests on increasing returns to scale and the feature that firms have shallow pockets. Wiseman (2017) makes no specific assumptions in relation to returns to scale and it can be shown that his theory is robust to an explicit extension that would allow for returns to scale. A crucial difference, however, between Wiseman (2017) and our own theory is that, unlike us, Wiseman (2017) considers firms with deep pockets, i.e. firms that are able to experience a sequence of unfavourable outcomes, while we assume that firms are immediately driven to bankruptcy if they obtain negative profits in any given period.

Accordingly, Wiseman (2017) assumes some randomness in the firms' survival probability, since a deterministic setup would always map in all firms but one immediately dropping out of the market: firms have different financial strengths (their state variable) that vary stochastically over time with Markov transitions and affect the likelihood of bankruptcy. As such, Wiseman (2017) assumes a continuum of active states describing the "strength of the firm's financial position", while our own setup assumes instead complete information over that attribute, i.e. there is a unique such state (beyond bankruptcy) where for all firms the strength of the financial position is nil. Thence, besides complementing the results of Wiseman (2017) by considering firms with shallow pockets, we also view our contribution as a highly tractable model highlighting in a simple setup when collusion is sustainable in spite of the availability of predation strategies. A further important distinction between these two models is to be found in our respective predictions: given the stationarity of firms' financial constraints in our own setup, we are able to identify a non-empty parameter space-involving intermediate discount factors-where collusion is dynamically sustainable. On the contrary, in Wiseman (2017), since the firms' financial strength is a random process and this critically defines the firms' incentives to start a price war, it follows that when competing over an infinite time horizon, the likelihood that the market monopolizes converges asymptotically to one.

Predation has also been shown to be an effective tool to preserve market power in oligopolies. In [START_REF] Harrington | Collusion and predation under (almost) free-entry[END_REF], predation may be used by colluding firms to deter potential entrants, thereby preserving market power, while also facilitating collusion. Likewise, [START_REF] Marshall | Monopolization conduct by cartels[END_REF] and [START_REF] Argenton | Colluding on excluding[END_REF] demonstrate how firms may collusively predate competitors to preserve or increase their market power. [START_REF] Marshall | Monopolization conduct by cartels[END_REF] study the incentives of firms to collude and to predate potential entrants, while giving emphasis on the internal stability issue, namely, the incentives of cartel members to deviate. Last, Argenton (2020) explores the conditions under which a three-firm cartel may be predated by two of its members that subsequently collude, while demonstrating that a single firm would not predate the market because it does not have the commitment power to suffer a loss in the sacrifice phase.

In the next section we present the framework and major results. In Section 3 we discuss some extensions and implications of our results. Last Section 4 concludes.

The Model

We consider an n-firm dynamic Cournot competition setting over an infinite time horizon with time denoted by index t = 1, 2, . . .. In any time period, the inverse market demand is given by p(Q t ) = a -bQ t , where Q t designates aggregate output at time t. 1 Denote the set of n initially active firms by I. In t = 1, the n firms decide their output q i ∈ R, i = {1, . . . n}, given the following total cost function:

T C(q i ) =      F + cq i if q i > 0 0 otherwise
, where F > 0 are the fixed operating costs and c is the marginal cost. We thence assume that fixed costs are bore only by active firms. At any period t a firm can go bankrupt by obtaining (weakly) negative profits, thence reflecting the fact that firms are credit constrained/have shallow pockets in our setup. If a firm is inactive or goes bankrupt at some period τ , we assume that it leaves the market and stops being active for any t > τ , therefore implying that our state variable is the pool of potentially active firms. 2 The discount factor is assumed to be symmetric and constant, and we denote it by δ. We focus on subgame perfect Nash equilibria.

Our objective is to derive the conditions that make full collusion, i.e. collusion by all n firms in I producing individually 1 n th of the monopoly output, an equilibrium outcome. 3

Throughout this article we refer to collusion to designate full collusion, that is collusion by all active firms. We proceed in several steps. We begin by demonstrating that in our setup predation-defined as production that drives specific opponents to bankruptcy, and thus renders predated firms inactive because of the shallow pockets assumption-makes the 1 For clarity reasons we omit time subscripts in what follows.

2 Our focus is exclusively on market exit and we disregard potential entry, the absence of which could be explained by some non-modelled non-operating fixed costs.

3 Observe that one could have equally studied the incentives of a subset of firms to collude, or the incentives of firms (possibly all) to produce higher quantities than the ones maximizing their joint profits (Haag and Lagunov 2007). We briefly discuss these possibilities in Section 3. typical collusive agreements considered elsewhere in the literature collapse. Indeed, firms are shown to have incentives to predate competitors playing the Cournot-Nash strategy profile when collusion could have otherwise been supported as an equilibrium. We then demonstrate that even though firms may have incentives to predate colluding firms for high enough discount factors, there exists an intermediate range of discount factors for which collusion is an equilibrium.

Cournot-Nash equilibrium

The Cournot-Nash play is a candidate equilibrium strategy that is not conditioned on history and where all active firms maximize non-cooperatively their instantaneous profits forever. One can immediately derive the individual quantities, market prices, and individual profits compatible with the Cournot-Nash equilibrium where the firm-subscripts are omitted since quantities and profits are symmetric:

q N = a -c b[n + 1] ; p N = a + nc n + 1 ; π N = [a -c] 2 b[n + 1] 2 -F, (1)
where superscript N designates the Cournot-Nash equilibrium.

From this we deduce a necessary condition for a Cournot-Nash equilibrium to exist:

F < F N = [a -c] 2 b[n + 1] 2 . ( 2
)
Because of the fixed costs F , without any further parameter restriction our model could admit a number of Cournot-Nash equilibria involving a smaller number of active firms, i.e.

l < n. To rule out such a possibility, and to ensure therefore that the game admits a unique equilibrium as given by ( 1), we impose a stricter parameter restriction on F that ensures that if only l < n firms were active, the remaining firms would have incentives to produce a strictly positive output. This condition reads as

F < F max = [a-c] 2 4bn 2 and is derived in Appendix A.1.
Notice, moreover, that-in contrast to standard Cournot setups-in this model we consider an additional strategy: the predation strategy. Given the aggregative nature of Cournot competition games, we can express profits as a function of the focal firm's quantities (first argument) and of the competitors' aggregate output (second argument). Accordingly, the predation strategy for firm i predating firm j (for any given quantities of the remaining competitors) is defined as: q P r i→j (q 1 , . . . q j , . . . q n ) := arg min

q i    q i : π j (q j , q i + k̸ ={i,j} q k ) ≤ 0    . (3)
Since (weakly) negative profits bankrupt firms, predation implies a reduction in the number of future competitors. 4 For the n firms to be incentivized to play non-cooperatively at equilibrium, it is thus necessary that when the competitors of firm i produce q j = q N , ∀j ̸ = i, no firm i has incentives in predating the industry by producing quantities q i = q P rN such that π j (q N , q P rN + [n -2]q N ) ≤ 0, ∀j ̸ = i. 5 The no-predation condition for any firm i reads as:

6 π P rN + δ 1 -δ π M < π N 1 -δ ⇔ D = δ 1 -δ < D P rN = π N -π P rN π M -π N , (4)
where π P rN = π P rN (q P rN , [n -1]q N ) and π M denotes the monopoly profits in the event predation successfully pushes the [n -1] competitors to bankruptcy and is defined as:

max q {[a -bq]q -cq -F } , (5)
thus yielding:

q M = a -c 2b ; p M = a + c 2 ; π M = [a -c] 2 4b -F. (6)
Next, the predation strategy for firm i is such that the following condition ought to be satisfied for any firm j ̸ = i producing their symmetric Cournot-Nash equilibrium output:

π j q N , q P rN + [n -2]q N = 0 ⇔ p q P rN + [n -1]q N q N -cq N -F = 0. ( 7
)
Re-arranging terms, the predation price ought to satisfy p P rN = T C(q N ) q N

, where T C(.)

stands for total cost. We then deduce that:

p P rN = F q N + c = F b[n + 1] a -c + c.
Using the definition of the inverse market demand, and given the above predatory price, we then deduce:

q P rN = 2a + c[n -1] b[n + 1] - F [n + 1] a -c - c b . (8)
The associated instantaneous payoff of the predating firm is then equal to:

π(q P rN , [n -1]q N ) = F - bF 2 [n + 1] 2 [a -c] 2 . (9) Note that π(q P rN , [n -1]q N ) > 0 ⇔ 0 < F < F N = [a-c] 2 b[n+1] 2 .
Replacing for the derived values of π N , π M , and π P rN in (4) allows us to re-write the no-predation condition as:

D < D P rN = 4 [n + 1] 2 -4 - 8b[n + 1] 2 F [a -c] 2 [[n + 1] 2 -4] + 4b 2 [n + 1] 4 F 2 [a -c] 4 [[n + 1] 2 -4] . (10)
Thence, if (2) and ( 10) are both satisfied, the game admits a non-cooperative equilibrium defined by (1), i.e. a Cournot-Nash equilibrium. In what follows we explore existence of non-cooperative equilibria for D > D P rN .

Mixed strategy equilibrium

In the following Lemma we show that the game admits another equilibrium in the event (10) is violated: 7 a mixed strategy equilibrium.

Lemma 1 The game admits a mixed strategy subgame perfect equilibrium with zero profits for all firms if D > D P rN .

Proof in Appendix A.2.

7 Although the game admits a predation equilibrium in the event D ≥ D P rN , in D = D P rN firms are indifferent between predating the market and not, thence implying that a Cournot-Nash equilibrium exists as well. To avoid having to nuance our results for such weak inequalities because of equilibria multiplicity, we conduct the remaining analysis by inquiring whether collusion is sustainable when Cournot-Nash equilibria do not exist, i.e. when D > D P rN .

Our proof establishes the existence of a mixed-strategy equilibrium for parameter configurations that rule out the existence of a Cournot-Nash pure strategy equilibrium. Observe, however, that this proof establishes the existence of at least one mixed strategy equilibrium, and as such does not preclude the existence of multiple equilibria. This lemma is of central importance, since it implies that when pure strategy Cournot-Nash equilibria fail to exist, the game will admit at least one equilibrium where players will mix over predation, Cournot and not producing.8

Collusion equilibrium

We next derive the conditions for the existence of a collusive equilibrium. As previously mentioned our focus is on full collusion, thus disregarding the possibility that only a subset of firms could decide to collude. If all firms collude, their individual output, associated prices and profits, are given by:

q C = a -c 2bn ; p C = a + c 2 ; π C = [a -c] 2 4bn -F. (11)
A necessary condition for collusion to be an equilibrium is that collusive profits be positive, which requires:

F < F C = [a -c] 2 4bn , (12)
a condition that is always satisfied since

F max < F C .
Moreover, for collusion to emerge at equilibrium, it should be immune to unilateral deviations. Two types of deviations ought to be considered in our setting. First, firms may unilaterally predate the industry to push to bankruptcy their competitors. Second, we inspect whether it is profitable for firms to deviate unilaterally from the collusive agreement by maximizing their instantaneous payoffs and thereafter obtaining the mixed-strategies' continuation payoff. The following proposition establishes that collusion cannot be sustained in the presence of sufficiently patient firms.

Proposition 1 Collusion can never be sustained if D > D P rC .

Proof. For collusion to be sustained at equilibrium, it is necessary that no firm has incentives to unilaterally predate the industry, which reads as:

π P rC + π M 1 -δ < π C 1 -δ , (13)
where π P rC = π q P rC , [n -1]q C . This condition can then be expressed as:

D < D P rC = π C -π P rC π M -π C . ( 14
)
To derive an explicit expression for ( 14), we re-write the predation prices as:

p P rC = T C(q C ) q C = F q C + c = 2F bn a -c + c.
We then substitute this predation price in the definition of the inverse demand for the optimal collusive output q C and set the expression equal to the above-derived term to deduce:

q P rC = an + cn + a -c 2bn - 2nF a -c - c b . (15)
Using ( 15) we obtain the instantaneous profits of the predating firm:

π q P rC , [n -1]q C = nF - 4bF 2 n 2 [a -c] 2 . ( 16
) Note that π q P rC , [n -1]q C > 0 ⇔ 0 < F < [a-c] 2 4bn , and that [a-c] 2 4bn > F max = [a-c] 2
4bn 2 , so that profits from predating colluding firms are necessarily positive in the admissible range of F parameters.

We can next replace for the optimal values of π C , π M , and π P rC in ( 14), and then rewrite the condition as:

D < D P rC = 1 n -1 - 4bn[n + 1]F [a -c] 2 [n -1] + 16b 2 n 3 F 2 [a -c] 4 [n -1] ( 17
)
Proposition 1 reverses the common finding in the litterature that collusion is easier to sustain with more patient firms. Indeed, when firms put a lot of weight on the future, their incentives to endorse the short run costs of predation-as reflected in the lower instantaneous profits obtained compared to the collusive profits-are now higher because of the high weight put on future time periods when the firm will act as a monopolist.

Since low discount factors are known to hamper collusion, and given the finding in Proposition 1, we then inspect whether collusive agreements under the threat of Cournot-Nash reversion strategies are possible. The next Corollary rules out this possibility.

Corollary 1 Collusion can never be sustained by Cournot-Nash reversion strategies.

Proof. For collusion to be sustained by Cournot-Nash reversion strategies, it is necessary that

π D + δ 1 -δ π N ≤ π C 1 -δ , ( 18
)
where π D designates the deviation payoff that obtains by optimizing the following problem:

max q i a -bq i -b[n -1] a -c 2bn q i -cq i -F .
Solving yields the following quantity, price and profits to the deviating firm i:

q D = [a -c][n + 1] 4bn ; p D = an + a + 3nc -c 4n ; π D = [[n + 1][a -c]] 2 16bn 2 -F. ( 19
)
After replacing for the values from (1), (11), and (19), inequality (18) then reads as:

δ ≥ δ C = [n + 1] 2 [n + 1] 2 + 4n ⇔ D = δ 1 -δ ≥ D C = [n + 1] 2 4n . ( 20
)
We obtain the usual result that an increase in the number of cartel firms makes collusion harder to sustain. Last, we establish that D P rC < D C , which completes the proof of the Corollary. Replacing for the expressions derived in ( 20) and ( 17) we aim at demonstrating that :

[n + 1] 2 4n > 1 n -1 - 4bn[n + 1]F [a -c] 2 [n -1] + 16b 2 n 3 F 2 [a -c] 4 [n -1]
.

The RHS of the above expressions is convex in F . Moreover when F = 0 the above inequality is true for any n ≥ 2. Lastly, since F ≤ F C as given by ( 12), if the inequality is verified for F = F C , it is verified for any F ∈ [0, F C ], and by extension for any F ∈ [0, F max ]. It is immediate to inspect that for F = F C the RHS of the inequality equals 0.

The intuition of Corollary 1 is that for any range of parameters where collusion could have been supported by Nash reversion strategies, the Cournot-Nash strategy profile is not an equilibrium since firms have incentives to predate the industry when competitors play their Cournot-Nash strategy profile. Indeed, Nash reversion strategies can only support collusion for high discount factors. Yet, high discount factors also increase the profitability of predation since the expected monopoly profits the predating firm will be able to secure will then weight more in its payoff function (Proposition 1). This in turn implies that Nash reversion strategies are not part of any subgame perfect equilibrium, and, consequently, cannot be used as a threat to deter deviations from collusion.

The interesting question then is to explore whether collusion may nevertheless be sustained using alternative punishment strategies to the Cournot-Nash equilibrium. We focus our attention on the parameter space D > D P rN where the zero-(expected) profit mixedstrategy equilibrium has been shown to exist in Lemma 1. The next Lemma opens up the possibility for a collusive equilibrium to emerge:

Lemma 2 There exists a unique threshold F ∈]0, F max ], such that for F < F , D P rN < D P rC .

Proof in Appendix A.3

Building on Lemma 2, we can now state the main finding of the paper that establishes the existence of collusive equilibria in predatory settings:

Proposition 2 For n ≤ 3, there always exist a non-empty range of fixed costs and a nonempty range of discount factors in D < D P rC such that collusion is sustained at equilibrium if the discount factor belongs to that interval.

Proof.

By Lemma (2) we know that for F < F , there always exist a non-empty range of D values, [D prN , D prC ]. To establish the proposition we thus need to show that there exists values of F ∈]0, F ] such that no deviation from full collusion exists. Bearing in mind that for D ∈ [D prN , D prC ] the Cournot-Nash strategy profile is not an equilibrium strategy profile, and firms have no incentives to predate the collusive agreement either. In view of Lemma 1, the unique alternative deviation from full collusion consists in a firm unilaterally deviating non-cooperatively from the full collusion strategy profile before obtaining zero discounted payoff in the ensuing subgame.

Denoting by D DC the D value below which such deviation is preferable to collusion, we accordingly have:

π D = π C 1 -δ DC ⇔ δ DC = π D -π C π D ⇔ D DC = π D -π C π C ,
with the values of π D and π C being given, respectively, by expressions ( 19) and ( 11).

Consequently, a non-empty range of F parameters for which collusion is sustainable at equilibrium necessarily exists if F < F and D DC < D P rC . Replacing by the appropriate values for D DC and D P rC , the expression Ξ(F ) = D P rC -D DC is given by:

Ξ(F ) = 1 n -1 - 4bn[n + 1]F [a -c] 2 [n -1] + 16b 2 n 3 F 2 [a -c] 4 [n -1] - [n+1] 2 [a-c] 2 16bn 2 -[a-c] 2 4bn [a-c] 2 4bn -F . (21)
In Appendix A.4 we show that this expression is strictly convex in F for n ≤ 4. Moreover, in F = F max , one can easily show that D P rC = 0, while D DC > 0. Consequently Ξ(F max ) < 0, which implies that there exists a range of F values where collusion is an equilibrium if and only if Ξ(0) > 0. Replacing then for F = 0 in the above expression, we obtain:

Ξ(0) = 1 n -1 - [n + 1] 2 4n + 1,
which allows us to conclude that Ξ(0) > 0 for any n ≤ 3, while for n ≥ 4, Ξ(0) < 0 Proposition 2 identifies the conditions for collusion to be sustainable at equilibrium in settings where predation needs to be considered because firms have shallow pockets and can therefore be driven to bankruptcy. Corollary 1 establishes that collusion can never be sustained by Cournot-Nash reversion strategies. This results from predation ruling out Cournot-Nash equilibria for discount factors above some threshold δ P rN -since D is a mono-tonic transformation of δ-and from the fact that this threshold is lower to the discount factor threshold above which collusion would have been an equilibrium in the absence of predation, δ C .

Collusion could potentially nevertheless be sustained by the harshest possible deviations the game can admit in case a firm deviates from the collusive agreement. Indeed, in the context of our study we demonstrate that for discount factors above the threshold δ P rN the game admits a mixed-strategy equilibrium yielding nil expected payoffs to all firms, and thus the lowest possible continuation payoffs in the context of our study. Observe, however, that even if this harshest punishment deters non-predatory deviations from the cooperative agreement, for collusion to be an equilibrium outcome firms must not have an incentive to unilaterally predate the market and subsequently obtain monopoly profits. We demonstrate that under the conditions stated in the Proposition 2, there exists a threshold on the discount factor, δ DC , below which firms have incentives to deviate from the collusive agreement (the harshest punishment does not work as a deterrent) and another threshold, δ P rC , above which firms have incentives to predate the collusive agreement, with the former threshold being smaller than the latter. Consequently collusion can only be sustained as an equilibrium outcome for an intermediate range of discount factors.

Two factors stand out as crucial in determining the possibility of observing collusive outcomes when firms have shallow pockets: the number of competitors and the magnitude of fixed costs. Lemma 2 establishes that for low fixed costs, it will not be profitable to predate the collusive equilibrium since predation involves reduced returns to scale and is therefore a costly operation. Moreover, for low fixed costs and few competitors (n ≤ 3), deviating non-cooperatively from the collusive strategy profile is not profitable either. Indeed, the incentives to deviate from collusion are a monotonically increasing function of the number of competitors since in setups with fewer competitors the expected collusive profits are higher. Second, deviations are less profitable for low fixed costs because, unlike the collusive outcome where the fixed costs will be supported at each time period, if firms deviate, fixed costs will only influence their (instantaneous) deviation profits since subsequent expected payoffs are zero for any fixed costs. Thence, lower fixed costs reduce the relative profitability of deviations.

It is interesting to observe that in the context of our model the range of discount parameters for which collusion is now possible does not overlap with the ones typically identified elsewhere in the literature: collusion is possible for intermediate ranges of discount factors rather than for patient firms. Indeed if firms are highly patient, predating colluding competitors will always be optimal because of the high expected flow of profits once a firm has wiped out competitors, while low discount factors reduce the importance of continuation payoffs and incentivize firms to optimize their short run payoff. For intermediate discount factors, on the other hand, the punishment payoffs are restricted to the instantaneous profits of deviating non-cooperatively from the collusive agreement, since in the ensuing subgame we establish the existence of a mixed-strategy equilibrium where all firms derive a zero expected profit. As punishments are therefore harsher, collusion is consequently easier to sustain.

Discussion

Our focus in this article is on full collusion by all firms, but milder forms of collusion could have been considered. One could have inquired the sustainability of (full) collusion by a subset of firms alone. Such equilibria are more difficult to sustain since, on the one hand, the discount factor above which predation is optimal would now be lower than δ P rC due to the non-colluding firms producing higher output than q C , while, on the other hand, colluding firms would have higher incentives to deviate from the collusive agreement given the (now) lower instantaneous profits associated with collusion. A second approach that could have been followed would have been to consider all (or some) firms partially colluding by producing quantities lying between the fully collusive solution and the Cournot-Nash quantities. While such milder forms of cooperation are typically seen as easier to sustain (see e.g. Haag and

Lagunoff 2007), this is not necessarily true in the context of our model. Let us assume a parameter configuration such that D P rN < D P rC and low enough fixed costs so that there exists a range of discount factors for which collusion is sustainable (Proposition 2), and let us then consider milder forms of collusion. Since D P rN is independent of the degree of collusion, we are interested in inspecting how the thresholds D DC -below which firms can profitably deviate from the collusive agreement-and D P rC -above which firms predate the collusive agreement-are impacted. The former threshold is given by

D DC = π D -π C π C
and the effect of the degree of collusion on this expression is ambiguous: although both the numerator and denominator unambiguously decrease, the relative profitability of deviations (numerator) could shrink so much (for collusive agreements that are closer to the Cournot-Nash equilibrium quantities) that the threshold decreases. In other words, although milder collusive agreements generate a lower discounted expected payoff to firms, deviations from such agreements are also less profitable, thereby precluding some general prediction. Turning next to the other critical threshold, D P rC = π C -π P rC π M -π C , here too the effect of milder forms of collusion is ambiguous. Although the denominator of this expression will be higher, the sign (and magnitude) of the numerator is unclear since both π C and π P rC will be smaller.

The intuition is the following. Milder forms of collusion map into lower discounted profits for colluding firms, but predating such agreements where all firms produce higher output yields smaller instantaneous profits to the predating firm compared to the the full collusion considered elsewhere in the paper. Thence, both options (colluding or predating) become less attractive to the firm, and given the non-linearities of these expressions we cannot deduce a general rule tying milder forms of collusion to its sustainability.9

Second, we can reflect on the sustainability of collusion in the presence of predation strategies given the dynamic exit from the markets under study. Indeed, we have demonstrated that for large enough discount factors (i.e. δ > δ P rN ) the game does not admit a pure strategy Cournot Nash equilibrium, while it admits at least one mixed strategy equilibrium where firms predate their competitors with some strictly positive probability. Under such circumstances, the number of firms will dynamically become smaller, thence implying that for low enough fixed costs and intermediate discount factors (conditions of Proposition 2), a market that initially does not admit collusive equilibria will in some future time period either admit collusive equilibria or else become a monopoly.

Conclusion

In this paper we explore the possibility for firms to collude in the presence of predation. We consider a setup where firms are credit-constrained (i.e. have shallow pockets), thence implying that firms can have incentives to predate competitors if the industry admits economies of scale. In such settings, contrary to most of the literature, we demonstrate that collusion may be sustainable for intermediate discount factors alone. High discount factors increase the incentives to predate the market since collusion aims at monopolizing the market forever.

Thence, the Cournot-Nash equilibrium which would have served as a punishment threat to support cooperation collapses, and collusion is no more sustainable by such Nash-reversion strategies. Moreover, firms have then strong incentives to predate colluding firms, thereby precluding collusion from being played at equilibrium. For low discount factors, on the other hand, firms have the typical incentives identified in the literature not to collude since even when considering the harshest possible punishment, if the continuation profits weight little enough in the firm's payoff, then firms will maximize their short run profits and thereby deviate from a collusive agreement. For intermediate discount factors, however, collusion will be sustained provided fixed costs are not too high and the number of competitors is not greater than 3. Indeed, for low fixed costs, it will not be profitable to predate the collusive equilibrium since predation involves reduced returns to scale and is therefore a costly operation. Moreover, for low fixed costs and few competitors (n ≤ 3), deviating non-cooperatively from the collusive strategy profile is not profitable either since few firms imply large collusive profits.

A Appendix

A.1 Parameter restriction on F

We impose a parameter restriction such that in an n-firm setup there can be no Cournot-Nash equilibrium featuring l < n firms. Consider any such candidate Cournot-Nash equilibrium.

Accordingly, the optimal individual quantities, market price, and individual profits of any producing firm are given by:

q N (l) = a -c b[l + 1] ; p N (l) = a + lc l + 1 ; π N (l) = [a -c] 2 b[l + 1] 2 -F,
where the candidate equilibrium variables are denoted as a function of l to signal that l firms are active under this strategy profile. For this to be an equilibrium, no firm out of the pool of inactive firms should have incentives to produce positive output. The best response of any such firm if it were to produce is given by:

q(Q N (l)) = a -c 2b - Q N (l) 2 ,
where Q N (l) = lq N (l). Replacing for the value of Q N (l), we deduce the maximal profits the focal inactive firm can secure by producing equals:

π(q(Q N (l)), Q N (l)) = [a -c] 2 4b[l + 1] 2 -F, thence implying that π(q(Q N (l)), Q N (l)) ≤ 0 if F ≥ [a-c] 2 4b[l+1] 2
, in which case an l-firm Cournot-Nash equilibrium exists since no inactive firm would be incentivized to produce a positive output. Since, however, π(q

(Q N (l)), Q N (l)) is strictly decreasing in l, if F < [a-c] 2 4b[l+1] 2 for l = n -1, i.e. if F < [a-c] 2
4bn 2 = F max , then an l-firm Cournot-Nash equilibrium cannot exist since an inactive firm has incentives to produce strictly positive output. This condition thus guarantees that the game admits a unique Cournot-Nash equilibrium with n producing firms as described in (1).

A.2 Proof of Lemma 1

The proof of the existence of a mixed strategy subgame perfect equilibrium follows the steps of [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF], and can be found in the online Appendix. In what follows we demonstrate that since a mixed strategy equilibrium exists, it must necessarily involve firms earning nil expected profits. To prove that we proceed in 4 steps where we sequentially prove that for D P rN < D:

1. There is no pure strategy equilibrium.

2. If the equilibrium strategy involves a mix over a predation strategy and the Cournot-Nash strategy, then the predation strategy ought to predate the Cournot-Nash expected output.

3. If firms mix between the above-described two strategies, then they have unilateral incentives to assign a probability 1 to the predation strategy. 4. Consequently, the equilibrium mix must assign a strictly positive probability to not producing, thence implying nil expected payoffs.

1. By the definition of D P rN , the game admits no Cournot pure strategy equilibrium for D > D P rN . Second, the game admits no pure strategy predation equilibrium. Indeed, if all n firms produce such predation quantities, then if any firm produces different quantities by deviating to its Cournot Nash strategy, its intertemporal payoff is necessarily higher since in the current time period the profits would then be higher (and thus strictly positive).

For such predation strategy to be played at equilibrium then, it must be the case that it involves firms producing the exact same quantities under the predation strategy and under the Cournot Nash strategy. The predation strategy is defined as follows:

a -c -bn qPr qPr -F = 0, which implies:

qPr = [a -c] + / -[a -c] 2 -4bnF 2bn .
Considering then the [n-1] competitors of a specific (focal) firm produce these quantities, if the focal firm deviates to its Cournot-Nash strategy, its output is given by:

q N ([n -1] qPr ) = a -c -b[n -1] qPr 2b . Consequently, qPr = q N ([n -1] qPr ) if qPr = a-c b[n+1]
, or:

a -c b[n + 1] = [a -c] + / -[a -c] 2 -4bnF 2bn
⇔ [n -1][a -c] [n + 1] = +/ -[a -c] 2 -4bnF
If the right hand side term is negative, this equality cannot be satisfied, while if it is positive, it will be true if:

[n -1] 2 [a -c] 2 [n + 1] 2 = [a -c] 2 -4bnF, ⇔ 4bnF = 4n[a -c] 2 [n + 1] 2 , ⇔ F = [a -c] 2 b[n + 1] 2 = F N .
And since F max < F N , this equality can never be satisfied.

2. Having shown in the online appendix that an equilibrium exists, and since no purestrategy equilibrium exists, there must exist a mixed-strategy equilibrium. We next show that if the equilibrium strategy involves a mix over a predation strategy and the Cournot-Nash strategy, then the predation strategy ought to predate the Cournot-Nash expected output. To show that, denote by q e i the expected output of any firm i, by q e the (symmetric) expected output of any firm, and by Q e -i = [n-1]q e the aggregate expected output of firm i's competitors. The Cournot-Nash strategy being linear in the competitors' expected output, it can then be denoted for firm i by q N i ([n -1]q e ). Now, assume-contrary to what we aim to show-that the predation strategy of any firm i does not satisfy the following equality:

a -c -b q N j ([n -1]q e ) + [n -2]q e + q i q N j ([n -1]q e ) -F = 0.
Two scenarios ought to be considered.

First, if a -c -b q N j ([n -1]q e ) + [n -2]q e + q i q N j ([n -1]q e ) -F < 0, then any firm implementing its Cournot-Nash strategy would obtain negative payoffs, and firm i would then have incentives to reduce its output since q i would then neither be maximizing the period profits-an objective achieved by producing q i = q N ([n -1]q e )), nor predating rivals at the minimal cost.

Second, if a -c -b q N j ([n -1]q e ) + [n -2]q e + q i q N j ([n -1]q e ) -F > 0, then when firm i plays its predation strategy, either it drives the profits of rivals playing their predation strategy to zero, or not. In the former case a firm playing the predation pure strategy would either obtain negative profits when facing at least one other firm equally predating the market, or else positive profits if facing firms having played their Cournot Nash strategy, and yet lower profits than if the same firm had played its Cournot-Nash strategy. Deviating to the Cournot-Nash strategy is therefore necessarily profitable. We therefore conclude that if the equilibrium strategy involves a mix over a predation strategy and the Cournot-Nash strategy, then the predation strategy ought to predate the Cournot-Nash expected output.

3. We next characterize the mixed strategy profile described in 2. and establish that there can be no equilibrium where firms mix over these two strategies alone. The proof is conducted for n = 2 firms, and can be extended to n > 2 firms. Define by λ the probability with which firms play their Cournot-Nash strategy, and by [1 -λ] the probability they implement their predation strategy. Moreover, denote for brevity by q N (q e ) the Cournot-Nash strategy when expecting the competitors to produce q e , and by q P r (q e ) = q P r (q N (q e )) the predation strategy that sets the profits of firms implementing their Cournot-Nash strategy to zero. Imposing symmetry and setting then q e = λq N (q e ) + [1 -λ]q P r (q e ), we must have at a mixed-strategy equilibrium: λ 1 -λδ π(q N (q e ), q N (q e )) = λπ(q P r (q N (q e )), q N (q e )) + δ

1 -δ π M . (22)
The left-hand side describes the discounted expected payoff from playing the Cournot Nash strategy in every subgame: if at any time period the firm obtains the same payoff from implementing any of the pure strategies over which it mixes at equilibrium, then the firm can be seen as playing any pure strategy in any subgame. On the right hand side, we have the expected profit of the firm when it plays its predation pure strategy, since in such a scenario with probability [1 -λ] the competitor equally plays his predation strategy and the firms' expected payoff is then nil.

We first focus on the LHS of the expression. The Cournot best response function for an output q e of the competitor is given by:

q N (q e ) = a -c 2b - q e 2 ,
and replacing for the value of q e , q N (q e ) = a -c 2b -λq N (q e ) + [1 -λ]q P r (q e ) 2 .

We thence deduce:

q N (q e ) = a -c [2 + λ]b - [1 -λ]q P r (q e ) [2 + λ] .
Next, the predation quantities are defined as:

q P r (q N (q e )) : a -c -b[q P r (q N (q e )) + q N (q e )] q N (q e ) -F = 0.

Or, q P r (q N (q e )) = a -c b -q N (q e ) -F bq N (q e ) .

Replacing next q P r (q N (q e )) in the above-computed expression describing q N (q e ) and dropping q e to simplify notations we obtain:

q N [2 + λ -b(1 -λ)] - λ b [a -c] - [1 -λ]F b 2 q N = 0.
We next compute the roots of this expression to find equilibrium value of q N , and since the lower root involves negative output, q N is given by:

q N = λ b [a -c] + / -λ 2 b 2 [a -c] 2 + 4[2+λ-(1-λ)b](1-λ)F b 2 2[2 + λ -(1 -λ)b] . (23)
Using last these values of q N and of q P r (q N ) in ( 22), it can be shown using numerical simulations that for D = D P rN ⇔ δ = D P rN 1+D P rN , for any λ ∈ [0, 1], λ 1-λδ π(q N (q e ), q N (q e )) < λπ(q P r (q N (q e )), q N (q e )) + δ 1-δ π M . Moreover, we can equally show that increasing the value of δ makes this inequality more likely to hold, thence ruling out the existence of a mixed strategy equilibrium where firms mix over their Cournot and predation strategies alone. 4. Last, having proved in Lemma 1 that the game always admits an equilibrium, and since the game admits no pure strategy equilibrium for the parameter configuration we are considering, it must be the case that firms play an "idle" strategy where they produce zero quantities with strictly positive probability. Consequently, since any pure strategy ought to secure same expected profits at the mixed strategy equilibrium, and given that producing zero quantities implies nil payoffs, it must be the case that the firms' expected profits at the mixed strategy equilibrium is zero.

A.3 Proof of Lemma 2

Using expressions (10) and ( 17), we can write Ψ(F ) = D P rC -D P rN as:

Ψ(F ) = 1 n -1 - 4bnF [n + 1] [a -c] 2 [n -1] + 16b 2 n 3 F 2 [a -c] 4 [n -1] - 4 [n + 1] 2 -4 + 8b[n + 1] 2 F [a -c] 2 [[n + 1] 2 -4] - 4b 2 [n + 1] 2 F 2 [a -c] 4 [[n + 1] 2 -4] Simplifying yields: Ψ(F ) = n -1 [n + 1] 2 -4 - 4bF [n + 1] [a -c] 2 [n 3 -3n + 2] [n -1][[n + 1] 2 -4] + 4b 2 F 2 [a -c] 4 4n 3 n -1 - [n + 1] 4 [n + 1] 2 -4 Observe that Ψ(F ) is a convex function, with Ψ(F ) |F →0 > 0. Moreover, we have imposed that F < F max = [a-c] 2 4b[n+1]
2 . We substitute for F max in Ψ to obtain: Wiseman, T. ( 2017), 'When does predation dominate collusion?', Econometrica 85(2), 555-584.

On-line Appendix: Proof of Lemma 1

In what follows we first demonstrate the existence of a stationary equilibrium for n = 2 using the theorem of [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF], and we then extend the reasoning to n > 2, thereby allowing us to conclude that this game always admits an equilibrium.

Observe first that with 1 firm the game admits a unique equilibrium where the firm produces q M and earns profits equal to π M . In what follows, we therefore abstain from detailing the continuation payoffs of the monopolist, and we instead use throughout the proof π M in such instances.

B Case with n = 2 firms

We focus on stationary equilibria, namely equilibria where firms' quantities are time-invariant.

Accordingly, we can write firm 1's discounted payoff, as:

Π 1 (q 1 , q 2 ) = [a -c -b(q 1 + q 2 )]q 1 -F + +∞ i=1 δ i µ 1 [a -c -b(q 1 + q 2 )]q 1 -F ] + +∞ i=1 δ i ν 1 π M ,
where, µ 1 = 1 {[a-c-b(q 1 +q 2 )]q 1 -F >0, [a-c-b(q 1 +q 2 )]q 2 -F >0}, and,

ν 1 = 1 {[a-c-b(q 1 +q 2 )]q 1 -F >0, [a-c-b(q 1 +q 2 )]q 2 -F ≤0} ,
with 1 the indicator function.

The above expression can then be written as:

Π 1 (q 1 , q 2 ) = [a -c -b(q 1 + q 2 )]q 1 -F

+ δ 1 -δ µ 1 ([a -c -b(q 1 + q 2 )]q 1 -F ) + δ 1 -δ ν 1 π M . ( 24
)
Firm 2's discounted payoff can be defined likewise.

We next define the following sets:

E 1 := {(q 1 , q 2 ) ∈ (R + ) 2 |[a -c -b(q 1 + q 2 )]q 1 -F > 0},
E 2 := {(q 1 , q 2 ) ∈ (R + ) 2 |[a -c -b(q 1 + q 2 )]q 2 -F > 0}.

From ( 24), denoting by E c 1 the complementary set to set E 1 , we have the following cases:

1. If (q 1 , q 2 ) ∈ E c 1 , then:

Π 1 (q 1 , q 2 ) = Π 1 (q 1 , q 2 ) = [a -c -b(q 1 + q 2 )]q 1 -F.

2. If (q 1 , q 2 ) ∈ E 1 ∩ E 2 , then:

Π 1 (q 1 , q 2 ) = Π 1 (q 1 , q 2 ) = 1 1 -δ ([a -c -b(q 1 + q 2 )]q 1 -F ).

3. If (q 1 , q 2 ) ∈ E 1 ∩ E c 2 , then:

Π 1 (q 1 , q 2 ) = [a -c -b(q 1 + q 2 )]q 1 -F + δ 1 -δ π M .

B.1 Preliminaries: Characterizing the discontinuities

We denote the set of discontinuities of Π 1 by D. From (24), this set is just defined as:

D = ∂E c 1 ∪ {E 1 ∩ ∂E c 2 },
where ∂ designates the boundary of the set, namely

∂E c 1 = {(q 1 , q 2 ) ∈ (R + ) 2 |[a -c -b(q 1 + q 2 )]q 1 -F = 0},
and, E 1 ∩ ∂E c 2 = {(q 1 , q 2 ) ∈ (R + ) 2 |[a -c -b(q 1 + q 2 )]q 1 -F > 0, [a -c -b(q 1 + q 2 )]q 2 -F = 0}.

At this stage, we want to represent graphically on the (q 1 , q 2 )-place the space of discontinuity.

First, let us focus on ∂E c 1 . For any q 1 > 0, we can write:

[a -c -b(q 1 + q 2 )]q 1 -F = 0 ⇔ q 2 = 1 b a -c -bq 1 -F q 1 =: f (q 1 ).

f (q 1 ) is described by an inverted-U reaching a maximum forf (q 1 ) = f ( F/b) = a-c b -2 F/b, as depicted in Figure 1. It follows that f (q 1 ) is a one-to-one mapping on (0, q1 ] and on [q 1 , +∞), respectively. For the range of admissible parameters (F < F max = [a-c] 2 4bn 2 ), we necessarily have that f (q 1 ) ≥ 0. Consequently, ∂E c 1 can be written as the sum of two subsets where q 2 is a continuous and one-ton-one function of q 1 : ∂E c 1 = {(q 1 , q 2 ) ∈ (0, q1 ] × R + |q 2 = f (q 1 )} ∪ {(q 1 , q 2 ) ∈ [q 1 , +∞) × R + |q 2 = f (q 1 )}.

Second, let us focus on ∂E c 2 . We know that the following equality, [a -c -b(q 1 + q 2 )]q 2 -F = 0, characterizes an implicit function q 2 = ψ(q 1 ). According to the above reasoning, we can

q 2 q 1 D E 2 E 1
q 2 = ψ(q 1 ) q 2 = f (q 1 ) Figure 1: Graphical representation of D write the set E 1 ∩ ∂E c 2 as the sum of two subsets in which q 2 is a continuous and one-to-one function of q 1 . By denoting q1 the maximum of ψ and q2 = ψ(q 1 ), we have E 1 ∩∂E c 2 = {(q 1 , q 2 ) ∈ E 1 ∩[0, q1 ]×[0, q2 ]|q 2 = ψ(q 1 )}∪{(q 1 , q 2 ) ∈ E 1 ∩[0, q1 ]×[q 2 , ∞)|q 2 = ψ(q 1 )}.

We graphically represent these boundaries as depicted on Figure 1.

B.2 [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF] theorem.

We now first recall the theorem of [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF], before applying it to our setup.

Theorem 5 of Dagupsta and Maskin (1986). Let A i ⊆ R (i = 1, ..., N ) be a closed interval and let U i : A → R + (i = 1 . . . , N ) be continuous except on a subset A * * (i) of A * (i),

where A * (i) is defined as:

and Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ) = Π 1 (q 10 , q 20 ) = [a -c -b(q 10 + q 20 )]q 10 -F + δ 1-δ π M .

As before, wow consider the four following subcases: (i) {q 1m , q 2m } m ⊆ E 1 ∩ E c 2 ; (ii) {q 1m , q 2m } m ⊆ E 1 ∩ E 2 ; (iii) {q 1m , q 2m } m ⊆ E c

1 ∩ E 2 , and (iv) {q 1m , q 2m } m ⊆ E c 1 ∩ E c 2 .

(i) We have: [a -c -b(q 1m + q 2m )]q 1m -F > 0, [a -c -b(q 1m + q 2m )]q 2m -F ≤ 0.

Therefore, Π 1 (q 1m , q 2m ) + Π 2 (q 1m , q 2m ) =

[a -c -b(q 1m + q 2m )]q 1m -F + δ 1 -δ π M + [a -c -b(q 1m + q 2m )]q 2m -F → [a -c -b(q 10 + q 20 )]q 10 -F + δ 1 -δ π M

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

(ii) We have: [a -c -b(q 1m + q 2m )]q 1m -F > 0, [a -c -b(q 1m + q 2m )]q 2m -F > 0.

Therefore, Π 1 (q 1m , q 2m ) + Π 2 (q 1m , q 2m ) = 1 1 -δ ([a -c -b(q 1m + q 2m )]q 1m -F ) + 1 1 -δ ([a -c -b(q 1m + q 2m )]q 2m -F ) → 1 1 -δ ([a -c -b(q 10 + q 20 )]q 10 -F )

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

(iii) We have: [a -c -b(q 1m + q 2m )]q 1m -F ≤ 0, [a -c -b(q 1m + q 2m )]q 2m -F > 0.

Therefore, Π 1 (q 1m , q 2m ) + Π 2 (q 1m , q 2m ) = [a -c -b(q 1m + q 2m )]q 1m -F + [a -c -b(q 1m + q 2m )]q 2m -F + δ 1 -δ π M

→ [a -c -b(q 10 + q 20 )]q 10 -F + δ 1 -δ π M

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

The assumption that firms go bankrupt with zero profit is made for expositional clarity. The results would remain qualitatively and quantitatively unaffected if bankruptcy required strictly negative profits.

Notice that these predation quantities are the same for any predating firm given the model's symmetry, thence allowing us to drop the index i → j to simplify notation.

Throughout the article we use the transformation D = δ 1-δ rather than δ to simplify the presentation of the results.

Observe that the logic of the argument, and the main 0-profit result are reminiscent of[START_REF] Tasnádi | Production in advance versus production to order[END_REF] and van den Berg and Bos (2017), although our respective setups differ in fundamental ways since these authors consider firms (sequentially) deciding both quantities and prices.

We are able to show with numerical simulations that D P rC is increasing in the "mildness" of the collusive outcome, unless the agreement is very mild, i.e. firms produce quantities that are close enough to their Cournot-Nash equilibrium output.

In the words of[START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF], the assumption that functions f d ij need to be one-to-one aims to "exclude f d ij 's that are vertical or horizontal on some interval " (p. 7-8). This is clearly not the case when looking at the Figure1.

The sign of Ψ(F max ) is then given by:

⇔ sign{Ψ(F max )} = sign -n 5 + n 4 -6n 3 + 10n 2 -n -3 And this expression is negative for any n ≥ 0.

We thus conclude that there exists a unique root F of Ψ(F ) on ]0, F max ] such that Ψ(F ) ⋛ 0 ⇔ F ⋚ F .

A.4 Convexity of Expression (21) for n ≤ 4

We differentiate twice Ξ(F ) as given by ( 21) so as to obtain:

Re-arranging terms and simplifying, this expression is positive if the following inequality is verified:

4bn 2 , the above expression is then necessarily verified in the admissible range of F parameters if:

is upper semi-continuous and U i (a i , a -i ) is bounded and weakly lower semi-continuous in a i . Then the game [(A i , U i ); i = 1, . . . , N ] possesses a mixed-strategy equilibrium. [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF] First, let us recall how our notations relate to [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF]. We have N = 2,

B.3 Applying

Second, we suppose that for each i, the discontinuities of Π i are confined to a subset of a continuous manifold of dimension less than N (N = 2 in this specific instance). To be precise, following Dasgupta and Maskin, for each pair of agents i, j ∈ 1, 2, let D(i) be a positive integer, and for each integer d, with 1 ≤ d ≤ D(i), let f d ij : R → R be a one-to-one, continuous function. Finally, for each i ∈ 1, 2 define

We shall suppose that the discontinuities of Π i are confined to a subset Q * * (i) of Q * (i). In other words, Q * * (i) is the set of discontinuities of Π i . As argued by Dasgupta and Maskin (p. 7), "the requirement that the function f d ij be one-to-one does not rule out a non-monotonic curve of discontinuities; since each monotonic subcurve can be made to correspond to a different f d ij ."

To apply the Theorem, we shall show that it applies to our setup for fi rm 1, and given the symmetry of the game, this holds true for player 2 as well. We need to ensure four conditions, which are checked in the following subsections, respectively.

B.3.1 Discontinuities

This point is evident, because the set of discontinuities of Π i (D = Q * * (1)) can be written by the union of subset, in which q 2 is in one-to-one continuous link with respect to q 1 (see

Observe that if one firm goes bankrupt when the number of initial competitors is such that n = 2, the subgame admits an equilibrium where the surviving firm produces q M forever after.

We can then distinguish four cases:

Recall that the definition of an upper semi-continuous mapping: f (x) is upper semicontinuous mapping in x 0 iff:

As usual, to ensure the upper semi-continuity we introduce a sequence {q 1m , q 2m } m≥1 ⊆ (R + ) 2 that is convergent to {q 10 , q 20 }. First of all, it is obvious that -by constructionoutside the set of discontinuities D, Π 1 is continuous, and therefore upper semi-continuous.

We just need to focus on the set D. There are two cases (see Figure B.1).

First: (q 10 , q 20 ) ∈ ∂E c 1 , i.e.,

[a -c -b(q 10 + q 20 )]q 10 -F = 0, and Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ) = Π 2 (q 10 , q 20

Now consider the four following subcases: (i)

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

(ii) We have:

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

(iii) We have:

Therefore,

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

(iv) We have: [a -c -b(q 1m + q 2m )]q 1m -F ≤ 0, [a -c -b(q 1m + q 2m )]q 2m -F ≤ 0.

Therefore,

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

Partial conclusion: the mapping Π 1 (q 1 , q 2 ) + Π 2 (q 1 , q 2 ) is upper semi-continuous on ∂E 1 .

Second:

≤ Π 1 (q 10 , q 20 ) + Π 2 (q 10 , q 20 ).

Conclusion: the mapping Π 1 (q 1 , q 2 ) + Π 2 (q 1 , q 2 ) is upper semi-continuous on E 1 ∩ ∂E c 2 , so does on the set of the discontinuity points D, and thus on (R + ) 2 .

Note that the mapping ζ : q 1 → [a -c -b(q 1 + q 2 )]q 1 -F is continuous on R + , and described an inverted U-shaped curve, with ζ(0) = 0, a maximum at q1 = (a -c -bq 2 )/2b, a root at q1 > q1 , i.e., ζ(q 1 ) = 0, and ζ(+∞) = -∞.

Let us introduce an upper bound on q 1 denoted by q1 , where q1 > q1 . We can establish the following:

1. If (q 1 , q 2 ) ∈ E c 1 , then Π 1 (q, q 2 ) ≤ Π 1 (q 1 , q 2 ) ≤ 0.

2. If (q 1 , q 2 ) ∈ E 1 ∩ E 2 , then 0 < Π 1 (q 1 , q 2 ) ≤ Π 1 (q 1 , q 2 ).

3. If (q 1 , q 2 ) ∈ E 1 ∩ E c 2 , then δ 1-δ π M < Π 1 (q 1 , q 2 ) ≤ Π 1 (q 1 , q 2 ).

Conclusion: that the mapping q 1 → Π 1 (q 1 , q 2 ) is bounded on [0, q1 ].

B.3.4 Π i (q i , q -i ) weakly lower semi-continuous in q i

As [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF] claim, if a function is lower semi-continuous, this function is weakly lower semi-continuous (footnote 14, p. 25).

By definition, Π 1 (q 1 , q 2 ) is lower semi-continuous in q 1 if and only if, for any α ∈ R, the set {q 1 |Π(q 1 , q 2 ) ≤ α} is a closed interval. This condition holds when we restrict the values of q 1 to the closed interval [0, q1 ]. For example, if α ≥ ζ(q 1 ) + δ 1-δ π M , then {q 1 |Π(q 1 , q 2 ) ≤ α} = [0, q1 ], and if α ≤ 0, then {q 1 |Π(q 1 , q 2 ) ≤ α} = [q 1 , q1 ].

B.3.5 General conclusion

Since firms are symmetric in our setup, the exact same reasoning applies for player 2, therefore allowing us to conclude that given [START_REF] Dasgupta | The existence of equilibrium in discontinuous games, i: Theory[END_REF]'s Theorem 5, the game possesses a mixed-strategy stationary equilibrium on the space [0, q1 ] × [0, q2 ], where q2 is defined in a similar way to q1 .

C Extension to n > 2

Having shown that an equilibrium exists for any subgame involving n ≤ 2 firms on restricting the space of q i to closed interval, we will now apply a recursive reasoning to generalize the existence result to any finite n. Indeed, the problem is structurally equivalent for any finite number of firms, so that with n = 3 firms the only difference for proving the upper semicontinuity of Π i (q) and the weakly lower semi-continuity of Π i (q i , q -i ) in q i is the increased dimensionality of the discontinuity set. The existence proof can be extended to n = 3, and recurrently to any finite n, with the same constraint on q i .


I Silva

F Gentils

P Rain

EFFECT OF ASYMMETRIC ELECTRODES ON DC CURRENT IN SILICONE RUBBER

INTRODUCTION

After decades of developments in HVDC technologies for power transmission, attention has been devoted to new studies for MVDC applications in order to turn DC power distribution a reality. Among them, the validation or improvements of insulating systems is a challenging one because materials and designs change considerably between HV and MV systems, and between AC and DC. In any case, one particular aspect that is systematic on real HV/MV equipment is the use of asymmetric electrodes, i.e., electrodes that differs in nature and design on each side of the insulation. Into bushings, for example, the insulating material is associated with the conductor (usually copper or aluminum) in one side and a screen on the other side (potentially some other metal, as steel). In laboratory scale, dielectric samples are often sputter-coated either for the application of an external voltage or as a technique to improve the contact with the sample and reduce roughness effects. Contrary to the application, the same metal is usually sputtered on both sides, normally gold (Au) or platinum (Pt), since they are chemically inert and shouldn't oxidate nor interfere with migration effects. As consequence, there is a difference between measures performed at the laboratory and the application design that can make validation procedures difficult. It has been reported in literature that the nature of electrodes can interfere on injection and distribution of charges [START_REF] Ko | Contact-injected space charge in PET[END_REF][START_REF] Hirsch | Contact-injected space charge in PET[END_REF][START_REF] Thielen | Heat treatment effects on the metal-polymer interface and space-charge formation in thin metallized polyethylene terephthalate films[END_REF][START_REF] Viertel | Electrode Influence on DC Conductivity Measurements of Low Density Poly Ethylene[END_REF]. This has been attributed to energy barriers associated to each metal. Polarity should also play a major role since the energy barriers varies for electron and hole injection. For a positive polarity, an impact on current measurements up to one order of magnitude has been reported for low-density polyethylene (LDPE) [START_REF] Viertel | Electrode Influence on DC Conductivity Measurements of Low Density Poly Ethylene[END_REF] and high-temperature vulcanized silicone rubber (HTV SiR) [START_REF] Silva | Investigating the I-V characteristics of an HTV silicone rubber for MVDC electrical insulation[END_REF] just because of the change of the metal that coats the sample. For liquid silicone rubber [START_REF] Freye | DC Conductivity Measurements of Liquid Silicone Rubber: Influence Analysis and Repeatability[END_REF], the electrode system also interferes on repeatability of results. For additive-free LDPE, space charge measurements with samples sputter-coated with an asymmetric design (gold on one side and aluminum on the other side) have shown an increase of charges in the bulk when compared to symmetric electrodes [START_REF] Chen | Electrodes and Charge Injection in Low-density Polyethylene Using the Pulsed Electroacoustic Technique[END_REF]. Similar studies were carried out for cross-linked polyethylene (XLPE) [START_REF] Hao | The effect of electrode material and semicon bonding on space charge dynamics of XLPE[END_REF], Polyethylene terephthalate (PET) [START_REF] Rain | Influence of the electrode metal on the injection of space charges in PET and epoxy resin[END_REF], and Polyimide (PI) [START_REF] Kaneko | Effects of Electrode Material on Injection and Space Charge Formation in Polyimide[END_REF]. For XLPE, injection has occurred with dominated charge depending on the electrode nature. For PI, also injection has been identified, but no dependence of electrode material has been found. In the case of PET with gold electrodes, negative and positive injections were observed, the former being larger than the latter. This work aims to associate both approaches with a focus on an High Temperature Vulcanized Silicone Rubber (HTV SiR), which is a material mostly applied in MV applications. In the first part, the current mechanism of both symmetric and asymmetric configurations is presented. Then, measures of space charge will be presented. They should support the comprehension about the mechanisms responsible for conduction and the impact of asymmetric systems on conductivity. At the end, the influence of polarity is discussed.

MATERIALS AND METHODS

Material of the Study

An HTV SiR filled with silica has been shaped in 1 mm thickness disks of 70 mm in diameter (for current measurements) and 110 mm (for space charge measurements). As any real industrial material, the presence of all kind of additives, such as antioxidants, plasticizers, demolding and pigments has to be taken into account.

Sample Conditioning and Preparation

All disks were dried into an oven at 70°C and under vacuum conditions for at least 2 days. Two main reasons supports the drying step: (1) it helps us to control the water content, to which conductivity seems to be very sensitive [START_REF] Hirsch | Contact-injected space charge in PET[END_REF]; (2) it removes some volatiles and byproducts that can biase the analysis. Only dry samples are tested in this work. Then, samples are sputter-coated with a turbomolecular pumped coater at 10 -3 mbar. Four different configurations were applied, as shown in Figure 1. A sputter-coat metallization consisted of a 200 nm thick layer of gold and/or copper. In a first moment, a positive high-voltage source will be connected to the bottom electrode for all measurements. Then the bottom electrode becomes the anode (A) and the upper one the cathode (K) when sample are under voltage. One can notice that inverting copper and gold arrangements as in designs (c) and (d) corresponds to a polarity invertion. Upper and bottom coats have different sizes in order to follow the design of electrodes of current and space charge measurement cells, sizes are available in TABLE 1. For current measurements, the guard electrodes surfaces have not been coated. Current-Voltage (I-V) Set-up and Protocol I-V measurements were conducted by simply connecting sample to an HV source from one side and to an electrometer on the other side. A vessel was to hold the samples. It can be hermetically closed and dried air can be injected, enabling the increase of voltage without risk of discharges. As it can seen in Figure 2, conductive silicone tapes have been installed as cell electrodes so that the electrodes could deform and ensure a good contact with the material. By sputtercoating the sample, the effect of these electrodes should be eliminated.

The vessel is closed and, after vacuum application, dry air is pumped in under 1.5 -2 bars. The vessel is located inside a climatic chamber with relative humidity (RH) control. Current measurements have been executed at 40°C with environment inside the vessel controlled within 1-3% RH, so that the sample could not absorb moisture during the test. After 4 hours of discharging and temperature stabilization, tests were performed from 1 kV to 5 kV by steps of 1 kV. For each voltage level, the method consisted of 1 hour of polarization and 1 hour of depolarization (so called 1h/1h method).

Figure 2 Scheme of I-V test arrangement (a); external view of test cell (b); view of electrodes arrangement (c).

The conduction current (Ic) is obtained by the sum of currents at the end of polarization (Ip) and depolarization (Id) steps. No relevant offset has been identified. The current density J is calculated by dividing Ic by the effective surface of the sample.

Space Charge Set-up and Protocol

Space charges were measured using the laser induced pressure pulse (LIPP) method which consists of a pressure pulse excited by a laser source that propagates through the sample thickness, generating a signal in voltage or current proportional to the charge or field distribution [START_REF] Sessler | Measurement of Charge Distribution in Polymer Electrets by a New Pressure-Pulse Method[END_REF]. Details about the main steps (calibration, volt-on and volt-off measures) as well as all the data treatment strategy can be found in [START_REF] Rain | Field and temperature dependence of space charge injection in a silicone rubber[END_REF]. The measures were conducted at similar conditions as I-V measures, i.e. dried samples at 40°C with volt-on and volt-off measures.

The voltage of 5 kV has been applied for 1 hour. The entire set-up can be seen in Figure 3.

RESULTS

Current-Voltage (I-V) Measurements

The current density as a function of the electric field for Au/Au, Cu/Cu, Au/Cu and Cu/Au are presented in Figure 4. Note that the values of J are divided by 𝐽 0 , that is the lowest value among all the results. Each curve can be divided in two regions: one linear or ohmic, in which the current density is proportional to the field, and another non-linear or non-ohmic. In Figure 4, the linear domain is suggested by the gray line in which a slope equals to 1 has been extrapolated from data at 1kV/mm. Some additional datapoints for Au/Au samples have been measured and added to the graph in order to verify this transition. The non-linear region is underlined by a slope higher than the unity, with their values listed in the table.

In general, current density values of symmetric gold electrode samples are one order of magnitude lower than all the other systems of the study. The non-linear region has a slope equals to 3.43. The current density of symmetric copper electrode samples has a slope in the non-linear region equals to 2.47. The systems with two different electrodes presented a non-linear mechanism with a slope close to 2 (slope = 1.86 and 1.87). Both symmetric copper sputter-coated and asymmetric samples presented close values of J.

It is worth to note that the threshold (represented by the circle) seems to be the same for symmetric samples (E = 1.64 kV/mm), and also the same for asymmetric ones (E = 1.88 kV/mm).

One may ask if a slope of 2 is systematic for asymmetrical systems. Additional measurements have been executed for new samples in which just one side has been sputter-coated. The other side stayed in contact with the conductive silicone tape (SiR-C, i.e. filled with carbon black, so as the conductor is made of carbon) that has always present (Figure 2). The result, that can be seen in Figure 5, shows that all the profiles in the non-linear region can be fit with a slope very close to 2 as well, including symmetric SiR-C electrodes (i.e. without any coating).

In Figure 5, all the asymmetric systems exhibits higher currents than the symmetric one. In addition, the polarity applied to the conductive silicone separates regions of current amplitude with close values for Au/C and Cu/C, or for C/Au and C/Cu. Again, a slope equal to 1 has been extrapolated from 1 kV/mm and presented as the gray line. In this case, the threshold seems to be higher when gold was applied as sputter-coat.

Because the insulating materal is the same, it leads to conclude that the measure of the current density (so as the volumetric conductivity itself) cannot be dissociated from the electrode system. In other words, conductivity seems to be a property of the electric insulated system, which includes the metallic contacts.

Space Charge Measurements

The charge density profile through the symmetric goldcoated sample under voltage (Volt-on) and immediately after short-circuit (Volt-off) is showed in Figure 6. On both curves, the charge distribution seems homogeneous inside the bulk, and the volt-off curve reveals that the material has been slightly negatively charged. The presence of negative charge is confirmed by positive induced charges at the electrodes on both sides of Voltoff measure.

When the sample is copper-coated on both sides, this profile is altered, as depicted in Figure 7. Induced peaks in Volt-on curves are lower but wider. An accumulation of charges is evidenced by the negative charge-packet located on the cathode side both on Volt-off and Volton curve. Either under volt-on or volt-off, the peaks at the electrodes are broader than for Au/Au sample. Under volt-on, these broad peaks can be attributed to homocharges injected under both polarities. Under voltoff, positive homocharges are still observed but negative homocharges are not. Conversely, a slight concentration of positive heterocharges is detected under volt-off at the cathode. In the case of the asymmetric electrode system with gold as cathode and copper as anode, qualitative observations are similar as for Figure 7 with slight differences. The negative charge-packet is centred deeper inside the sample, as portrayed in Figure 8. This logically increased the peak induced under volt-off at the cathode which is larger than in Figure 7. The increase of the induced peak at the anode reveals a larger concentration of these negative space charges as a whole with the gold cathode. The Au/Cu sample has been discharged and tested again at the same voltage level, but with an opposite polarity. Figure 9 shows a similar Volt-on profile as in Figure 8, as if the curve were reflected by the y axis, i.e. an accumulation of positive charges into the bulk instead of negative charges. It can be noted that the peak at the copper side in Figure 9 (representing an injection of electrons) is broader than its equivalent in Figure 8 (injection of holes). It suggests that the copper injects easily electrons than holes. These electrons are not found in the volume, but near the anode, as the Volt-off curve suggests. Even if profiles looks similar, the quantity of charge is not. More charge was accumulated when a positive polarity was applied, as if gold had an easier facility to inject electrons than holes (or the opposite, as if copper has a facility to inject holes instead of electrons). Considering these two figures, the dominant polarity of the space charges inside the gap is imposed by the polarity of the gold electrode: with a gold cathode, space charges are thoroughly negatives whereas with a gold anode, space charges are thoroughly positives.

DISCUSSION

Space Charge Profiles and current mechanisms

In a previous work [START_REF] Hirsch | Contact-injected space charge in PET[END_REF], the effects of electrode nature on current measurements in this HTV SiR as well as the linear caracteristics of lnJ vs square(E) suggested a mechanism limited by injection of charges (particularly Schottky injection). Nevertheless, an ionic conduction should not be entirely excluded since the sample is not free of additives. The Au/Au current, with a slope of 3.43, could fit with a model of SCLC mechanism with different trap levels.

However, the flat profile of the low density space charges pictured in Figure 6 as well as the low amplitude of the current density lead rather to the hypothesis of a current limited by a Schottky injection as proposed in [START_REF] Silva | Investigating the I-V characteristics of an HTV silicone rubber for MVDC electrical insulation[END_REF].

As for gold electrodes, current measurements with copper electrodes could also fit a Schottky injection as seen in [START_REF] Silva | Investigating the I-V characteristics of an HTV silicone rubber for MVDC electrical insulation[END_REF]. However, with a slope of 2.47, it may also correspond to a SCLC mechanism with different levels of traps. The profile seen in Figure 7 for Cu/Cu looks like a smoothed version of Figure 8, with a chargepacket closer to the cathode side. A mechanism of SCLC would be in better accordance with this charge accumulation. Space charge limited current (SCLC) mechanism with no traps, or with traps of a single level of energy, is characterized by a slope of 2 in the lnJ vs lnV curves.

The investigated asymmetric electrode systems all exhibit a slope of 2, suggesting a SCLC regime. The accumulation of space charges presented Figure 8 for Au/Cu sample is consistent with a model of SCLC. The higher attenuation expected at the cathode should be compensated by a higher enhancement at the anode. Higher enhancement induces higher stresses, and higher stresses increase the injection of charges. The current is no longer limited by injection but by charge transport.

Polarity of the space charges with respect to the metal of the electrodes Figures 6 to 9 exhibit space charge distributions with a dominant polarity. A first attempt of interpretation is to compare injection at both sides. For example in Figure 6, dominant negative space charges within the gap suggest a larger injection of negative charge at the cathode than of positive charges at the anode. Figure 7 leads to the same deduction for Cu/Cu system. As concerns asymetric electrodes, Figure 8 suggests a larger injection of negative charges at the Au cathode than of positive charges at the Cu anode and Figure 9 suggests a larger injection of positive charges at the Au anode than of positive charges at the Cu cathode. Therefore, a scale of easier injection can be deduced at this field of 5 kV/mm in the silicone rubber: Au cathode > Au anode > Cu cathode > Cu anode. However this classification does not explain the observed quantitative differences in the space charge concentration. Further explanations are needed.

One Electrode is Likely to Impact What's Happening with Another

If we have the same cathode for Au/Au (Figure 6) and Au/Cu (Figure 8), why much more negative charges are observed in the last case? Two hypothesis can be raised. First, copper can simply injects more easily (both holes and electrons) than gold. For the case of the positive polarity applied to Au/Cu, holes would cross the sample and accumulate at the cathode, reinforcing the field there. The higher the field, the higher the stress, the higher the injection. When a negative polarity is applied for Au/Cu sample, as the gold will inject less holes, copper will inject less electrons, so the magnitude of the charge density should be lower in comparison with the positive polarity. Consequently, a current should be limited by injection when no other mechanism is involved and when the material doesn't inject so much. If so, the current will rapidly turn into a SCLC. Second, the energy barriers of both sides are somewhat modified and ultimately related since the beginning. The modification of the enegy barriers is possible because many events, such as reactions and rearrangements at the interface, can change their nature, creating a density of states that possess its own energy barrier 𝜙 𝐼𝑆 [START_REF] Viertel | Electrode Influence on DC Conductivity Measurements of Low Density Poly Ethylene[END_REF]. Nevertheless, the proof that the energy barrier of the cathode is related to the energy barrier of the anode since the beginning is more difficult to achieve. An approach based on the Double Schottky effect sounds promising. There, the Schottky injection is considered to occur on both sides, with each contact seen as resistances in parallel to the insulating. As the electrical current through all resistances is the same, and as the equationing of Schottky injection take into acount the energy barrier, a relationship between energy barriers is possible [START_REF] Grillo | A currentvoltage model for double Schottky barrier devices[END_REF].

Limitations of the Study

As mentioned before, as the sample is not additive-free, there is no guarantee that ionisation, dissociation, and migration won't occur and have an impact on space charge results. On the contrary, these phenomena would partially justify a density of states that deviates the energy barrier heights from theoretical ones. It is known that copper can easily oxidize, and this should change the electrode properties. In addition, it was not excluded that oxidized copper can migrate within the sample. However, an analysis of Scanning Electron Microscopy with EDS have shown no trace of copper into the bulk or close to the surface. It is worth to mention that the mechanical pressure during the electrical tests was not controlled. It is also supposed that the difference of the electrode's surfaces would not play a major impact on the mechanisms studied.

CONCLUSIONS

In this study, I-V tests and Space Charge Measurements via the LIPP method have been performed with 1 mm thick samples of High-Temperature Vulcanized Silicone Rubber (HTV SiR). Samples have been coated by sputtering and divided in two categories: symmetrics (when anode and cathode were made of the same material) and assymetrics (when the material of the anode differed from the one of the cathode). Gold, copper and conductive SiR have been used as electrode materials.

As main conclusions, it can be set that:

-Au/Au samples presented a non-linear current mechanism attributed to injection of Schottky type, whereas Cu/Cu samples behaviour suggested a SCLC mechanism.

Proceeding of the International Electrical Insulation Conference -INSUCON 2023 / IET Centre, Birmingham, UK, 18-20 April 2023 -Assymetric samples presented systematically a SCLC in the non-linear region, no matter the nature of the anode and cathode.

-Accumulation of charges was considerably higher in assymetric samples or when copper was used as electrode material than with the Au/Au system. Space charge observations are thus consistent with current fitting laws.

-I-V characteristics of silicone rubber are related to the entire system, which includes the electrode system, and not just to the insulating material. The impact of mechanical pressure has to be evaluated, since silicone rubbers are elastomers and their densities can change under mechanical pressure.

Figure 1

 1 Figure 1 Electrode designs for symmetrical (a and b) and asymmetrical (c and d) configurations.

Figure 3 .

 3 Figure 3. Scheme of the LIPP method.

Figure 5

 5 Figure 5 Current density as a function of electric field for samples with electrode systems made of carbon (C) on both sides, or by gold (Au) or copper (Cu) replacing one of the carbon electrodes.

Figure 6 - 1 Figure 7

 617 Figure 6 Charge distribution under Volt-on and Volt-off. Au electrodes. Applied voltage: +5000V for 1 hour.

Figure 8 Figure 9

 89 Figure 8 Charge distribution of SiR. Cathode Au and anode Cu. Applied voltage: +5000V for 1 hour.

TABLE 1 Electrode diameters for current and space charge measurements Electrode Current meas. (mm) Space charge meas. (mm)

 1

	Upper electrode	50	40
	Botton electrode	54	33

Figure 4 Current density as a function of electric field for silicone rubber samples associated with different electrode systems: Au or Cu on both sides, and Au/Cu at both polarities.

						Au	(anode)	and	C	(cathode)
						Au	(cathode)	and	C	(anode)
						Cu	(anode)	and	C	(cathode)
						Cu	(cathode)	and	C	(anode)
						C	(anode	and	cathode)
					J/J 0	1E-7	s lo p e = 1
						Sample	Slope
						Au/C	1.93
						C/Au	1.95
						Cu/C	1.85
						C/Cu	2.1
						C	1.98
						1E-8
						1	2	3	4	5
						E	(kV/mm)
	1E-6	Au (Anode and Cathode) Cu (Anode and Cathode)
		Au (Anode) and Cu (Cathode)
		Cu (Anode) and Au (Cathode)
		Threshold
	1E-7 J/J 0	slo pe = 1
	1E-8
				Sample Slope
				Au/Au 3.43
				Cu/Cu 2.47
	1E-9			Au/Cu 1.87 Cu/Au 1.86
	1	2	3	4	5
		E (kV/mm)

ACKNOWLEDGEMENTS

This study is supported by Schneider Electric (France). We express our sincere thanks to Mr. Flury, Mr. Pollet, and Mr. Cuilla for the technical assistance.