Guillaume Blin

Marie Gasparoux

Sebastian Ordyniak

Alexandru Popa

Sobra

SOBRA -Shielding Optimization for BRAchytherapy

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Introduction

In France, every year, almost 200.000 patients are treated by radiotherapy as part of their cancer treatment. This kind of therapy is using ionizing radiation aiming at controlling or killing malignant cells as a curative procedure or as part of adjuvant therapy and is widely used (in 2/3 of the cancer treatments). While internal radiotherapy treatments are currently widespread and considered as routine, there is still room for related innovative developments. The aim is to concentrate the radiation beams as precisely as possible towards the tumor site while sparing as much as possible normal tissues such as skin or organs which radiation must pass through to treat the tumor (the so-called organs at risk).

Brachytherapy -also sometimes referred to as Curietherapy -refers to a short distance (brachys in Greek) treatment of cancer with radiation from small, encapsulated radionuclide sources. Brachytherapy uses encapsulated radioactive sources (also called seeds) to deliver a high dose to tissues near the source. It is characterized by strong dose gradients, i.e., the dose becomes negligible in a very short distance from the source (about 10% per mm) [START_REF] Potter | Recommendations from gynaecological (gyn) {GEC} {ESTRO} working group (ii): Concepts and terms in 3d image-based treatment planning in cervix cancer brachytherapyâĂŤ3d dose volume parameters and aspects of 3d image-based anatomy, radiation physics, radiobiology[END_REF]. This type of treatment is given by placing sources directly into or near the volume to be treated. The dose is then delivered continuously, either over a short period of time (temporary implants) or over the lifetime of the source to a complete decay (permanent implants). There are many different techniques and sources available.

In this contribution, we focus on High Dose Rate implants. High dose rate or HDR brachytherapy is a form of internal radiation which temporarily expose abnormal tissue to a high amount of radiation. Under Computed Tomography (CT) and Fluoroscopy guidance, a bronchoscope or a needle is used to deliver a catheter into a position at the tumor site. The other end of this catheter is connected to a computerized machine. This machine passes a small radioactive metal seed through the catheter. The catheter guides the seed to the tumor site. The seed moves step by step through the catheter in order to cover the whole tumor site. The time spent at each position -also known as dwell time -is used to control the radiation dose in different regions of the tumor. The overall effect of HDR brachytherapy is to deliver short and precise amounts of high-dose radiation to a tumor while minimizing healthy tissue exposure. After a series of treatments the catheter is removed leaving no radioactive seeds in the body.

One of the main drawback of this technique comes from the lack of precise modulation of the irradiation field and thus the conformation to the shape of the tumor site. In this paper, we aim at studying the benefit of an innovative modulation technique in brachytherapy using tunable shields (as done in external radiotherapy). This approach will allow accumulating both the temporal modulation currently used and the shielding modulation. The aim is to provide treatment of better accuracy by adapting more precisely to the tumor shape. Indeed, currently, the modulation of the radiation source is done by controlling the time spent at each position by the source along the catheter. The main problem is that, at any position, the irradiation is uniform and can be represented as a cylinder surrounding the catheter. This shape does not always conform to the relative placement of the tumor and the organs at risks (i.e., in the radiation field). In this contribution, we consider the technic which aim at modulating a unique radioactive source using a gear inspired by external radiotherapy.

The use of the shield will allow to preserve, for a given position along the catheter, some part of the surrounding area. The so-called rotating shield brachytherapy (RSBT) was conceptually proposed by Ebert in 2002 [START_REF] Ebert | Possibilities for intensity-modulated brachytherapy: technical limitations on the use of non-isotropic sources[END_REF]. In RSBT the dose is delivered through a partially shielded radiation source in an optimized step-shot fashion (as done in classical brachytherapy treatment) to improve tumor dose conformity. The intensity of radiation is modulated by the amount of time the shield is pointed in a given direction. RSBT [START_REF] Liu | Rapid emission angle selection for rotating-shield brachytherapy[END_REF][START_REF] Liu | Dynamic rotating-shield brachytherapy[END_REF][START_REF] Yang | Rotating-shield brachytherapy for cervical cancer[END_REF] and other intensitymodulated brachytherapy techniques such as dynamic modulated brachytherapy (DMBT) [START_REF] Webster | Dynamic modulated brachytherapy (dmbt): Concept, design, and system development[END_REF][START_REF] Webster | Dynamic modulated brachytherapy (dmbt) for rectal cancer[END_REF][START_REF] Webster | Su-f-bra-11: Dynamic modulated brachytherapy (dmbt): Concept, design, and application[END_REF] were further studied with the aim of improving intracavitary brachytherapy dose distributions for rectal and cervical cancer. We will first focus on a peculiar type of shield which have been briefly described in the patent [START_REF] Smith | Radiation therapy apparatus with selective shielding capability[END_REF] and studied in [START_REF] Liu | Paddle-based rotating-shield brachytherapy[END_REF]. It corresponds to a set of shield segments forming a cylinder that can be individually retracted to produce circumferentially limited radiation output, directed radially. According to the way the sources are introduced in the patient body, and the physical constraints of the material, it is not possible to build sector of size as small, and thus as high resolution, as wanted. Therefore, using the possible rotation of the equipment, the aim is to find a sequence of sectors configurations that allows delivering a dose as near as possible to the prescribed dose. The corresponding algorithmic aspects are unexplored and the goal of this paper is to conduct an algorithmic study which will guide the final development of the equipment. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints that we formaly introduced in Section 2.

Preliminaries

Considering each dwell position of the irradiation source (denoted I), our main objective is to deliver to each part of the surrounding volume its proper irradiation dose. For this purpose, we will use a paddle-based shielding equipment P of K paddles (also referred as sectors for ease) that can stop the radiation going through when they are not retracted. We will consider the surrounding volume to be treated as a circular volume of interest divided in N subvolumes. In the following, a treatment plan for a given dwell position will be defined as a sequence of T shield configurations (P 1 , τ 1 ), (P 2 , τ 2 ) . . . (P T , τ T ) where P t , 1 ≤ t ≤ T , is a paddle configuration and τ t is its dwell time. Each paddle configuration is represented as a binary string P t = p t 0 p t 1 . . . p t K-1 where p t k represents the state (open or closed) of the sector k of P t . An open sector of the shield (paddle retracted allowing radiation going through) is represented by a 1, while a closed one (paddle is out and radiation is stopped) is represented by a 0.

For each given step (P t , τ t ) in the treatment plan, a corresponding received doses D t by the surrounding volume is defined as a string of integers

D t = d t 0 d t 1 . . . d t N -1
where d t n corresponds to the total irradiation time the subvolume n was exposed to during this step. Roughly, it corresponds to the contribution of the corresponding treatment step to the whole treatment plan. For ease, without loss of generality, when parameters P t and τ t are not informative, we will write D. Regarding the entire treatment plan, we will denote the prescribed doses as a string of nonnegative integers D = d0 d1 . . . dN-1 where dn corresponds to the total irradiation time needed to achieve the right dose for the subvolume n. We will moreover denote the total received doses as a string of integers D = d 0 d 1 . . . d N -1 , such that for all d n ∈ D, d n = 1≤t≤T d t n . For ease and without loss of generality, we assume that each shield sector is associated to w = N/K consecutive patient volumes, and, for simplicity, that K divides N (so w is an integer). By default, each shield sector p k will be associated to 1a). We can remark that D = D 0 D 1 . . . D K . Informally, one may see P and D as circular strings, P placed inside D and representing a mask that can stop the radiation from going through (see Figure 1b, with a counterclockwise indexation). In future work, planned for the full version of this paper, we plan to explore a variant of the problem where one will be able to change this association by applying a rotation of the shield represented by P .

D k = D[k • w, (k + 1) • w -1] = d k•w d k•w+1 . . . d k•w+w-1 of length w (see example Figure
Let us consider the practical case where one is applying a given shielded configuration (represented by P ) on a patient (represented by D) for a given amount of time τ (expressed in a given unit of time). Let us denote D(P, τ ) = One may consider several variants of the problem, depending on constraints and parameters. First of all, the shield configuration can be considered as fixed or dynamic (one fixed mask or a minimal number of chosen masks) and provided with or without rotation capabilities (this last property is not considered here). This properties are related to manufacturing purposes and constraints. We moreover consider allowing or not irradiation overdoses (d n > dn ). Indeed, in practice, it is convenient to overdose a tumor region while one should try to not overdose organ at risks regions. From a combinatorial point of view, there are two parameters that alter the overall treatment time; namely, the sum of the irradiation times and the number of configurations (as a transition between two configurations will require some time). In the following, we will consider variants of the problem based on the previous observations. In the first two variants, the input consists of only one shield configuration that is given and fixed. The goal is to decide what is the optimum amount of radiation that can be applied when allowing or not overdoses. As proven in Section 3, these variants of the problem are polynomial time solvable.

d 0 d 1 . . . d N -

Problem 1 (FixMask).

Given a prescribed dose represented as a string of nonnegative integers D = d0 d1 . . . dN-1 and a fixed shield configuration represented as a binary string P = p 0 p 1 . . . p K-1 , find the dwell time τ minimizing

N -1 n=0 | dn -d n | with D = D(P, τ ) s.t. ∀n < N .
While in FixMask variant of the problem, dn -d n can be negative -that is overdoses are allowed -in FixMask + variant, we moreover impose that ∀n < N , d n ≤ dn -thus, forbidding overdoses. We now consider variants of the problem where multiple shield configurations are allowed. As mentioned previously, two different criteria can be optimized in such a treatment plan. One would like to either achieve the optimal difference between the prescribed dose and the actual total delivered dose using a minimal number of shield configurations or given an upper bound on the number of shield configurations, achieving the minimum reachable difference. From a computational point of view, there is no evidence that these problems are equivalent. Formally, the problems are defined as follows.

Problem 2 (MinFixMasks opt ). Given two nonnegative integers K and diff and a string of integers D = d0 d1 . . . dN-1 (with N being a multiple of K), find a treatment plan (P 1 , τ 1 ), (P 2 , τ 2 ) . . .

(P T , τ T ) minimizing T such that N -1 n=0 dn -d n < diff , where ∀d n ∈ D, d n = 1≤t≤T d t n .
Problem 3 (MinFixMasks bound ). Given two nonnegative integers K and T max and a string of integers D = d0 d1 . . . dN-1 (with N being a multiple of K), find a treatment plan (P 1 , τ 1 ), (P 2 , τ 2 ) . . . (P T , τ T ) where T < T max minimizing

N -1 n=0 dn -d n , where ∀d n ∈ D, d n = 1≤t≤T d t n .
Similarly to in FixMask + , in MinFixMasks + opt and MinFixMasks + bound variants of the problem, we moreover impose that ∀n < N , d n ≤ dn -thus, forbidding overdoses.

Polynomial Results

In this section, we show that the variants of the problem where the shield configuration is given and fixed are solvable in polynomial time. Clearly, for a fixed masked, the doses associated to closed paddles cannot be brought closer to the corresponding prescribed doses and will thus not be considered.

Theorem 1. FixMask + can be solved in O(N ).

Proof. Obviously, the maximal irradiation time τ max that can be applied is the minimum dj of D associated to a 1 bit of P , since any greater value would decrease dj -d j below 0. Therefore, τ max = min{ dj ∈ D : p j/N = 1 } which leads to an overall difference of

N -1 n=0 ( dn -d n ) = N -1 n=0 dn -p k ∈P (p k • w • τ max ) (
where w = N/K). τ max can be obtained by browsing D in linear complexity.

The main observation required to show that FixMask can also be solved in polynomial time is given in the following lemma.

Lemma 1. For a sequence S of natural numbers and a natural number x consider the function f (x) with f (x) = s∈S |s -x|. Then f (x) has a unique minimum, which is only reached by any number x in between the at most two medians of S. Moreover, for any x not between the at most two medians of S, the function f (x) decreases with the distance of x to a median of S.

The above lemma implies that an optimum dwell time for an instance of FixMask is a median of the subsequence of D containing all prescribed doses for which the paddles are open.

Quasi-polynomial Algorithms for MinFixMasks

In this section, we present exact algorithms for all variants of the MinFixMasks problem.The presented algorithms run in quasi-polynomial time if the values of the precribed patient doses are bounded by a polynomial in the number of precribed doses. As a by product we show that the problems MinFixMasks opt and MinFixMasks + opt can be approximated in polynomial-time within a factor of log dmax of the optimum where dmax is the maximum prescribed dose to a subvolume of the patient, i.e., dmax := max dn∈ D dn . We first show that it is sufficient to consider treatment plans where the applied dwell times are pairwise distinct.

Lemma 2. For any instance of MinFixMasks opt , MinFixMasks +

opt , MinFixMasks bound , and MinFixMasks + bound there is an optimal solution (P 1 , τ 1 ), (P 2 , τ 2 ) . . . (P T , τ T ) satisfying τ i = τ j for every i and j with 1 ≤ i = j ≤ T .

Proof. Let P = (P 1 , τ 1 ), . . . (P T , τ T ) be an optimal solution of an instance I of any of the variants of MinFixMasks problem. Let i and j with 1 ≤ i = j ≤ T be such that τ i = τ j . Let (i) τ j * = 2τ j , (ii) the binary string P i * be obtained by the XOR of binary strings P i and P j , and (iii) the binary string P j * be obtained by the AND of binary strings P i and P j . Then the treatment plan obtained from P by replacing (P i , τ i ) with (P i * , τ i * ) and (P j , τ j ) with (P j * , τ j * ) is also an optimal solution of I. Moreover, by applying this procedure iteratively we eventually obtain an optimal solution of I such that all dwell times are pairwise distinct.

Let S be a set of dwell times. We say that S is complete if it contains a subset S for every number 1 ≤ i ≤ dmax such that i = s∈S s. We say that a treatment plan is S-restricted if it uses only dwell times from S and each of them at most once. Because any non-trivial instance of MinFixMasks opt and MinFixMasks + opt require at least one step, we obtain the following corollary from the above lemma.

Lemma 3. Let S be a set of dwell times. Then an S-restricted treatment plan minimizing

Corollary 1. MinFixMasks opt and MinFixMasks + opt can be approximated in polynomial time within a factor of log dmax of the optimum.

We are now ready to show our main theorem of this section. Proof. The algorithm goes over all sets S containing at most log dmax + 1 (respectively at most min{ log dmax + 1, T max in the case of MinFixMasks bound and MinFixMasks + bound ) dwell times between 1 and dmax . For every such set S, the algorithm then uses Lemma 3 to compute the optimal (the meaning of optimal here depends on the considered problem) S-restricted treatment plan. Finally, the algorithm outputs the best treatment plan that satisfies

Corollary 2. MinFixMasks +

opt can be solved in quasi-polynomial time if dmax is bounded by a polynomial in the size of the input instance.

Hardness of MinFixMasks opt and MinFixMasks bound

In this section, we provide the first hardness results on the MinFixMasks opt and MinFixMasks bound problems (i.e., when allowing overdoses). We prove our hardness results via a reduction from the Monotone 1-3 SAT problem (proven to be NP-complete in [START_REF] Schaefer | The complexity of satisfiability problems[END_REF]): Given a boolean formula φ = {c 1 , c 2 , . . .} in 3-CNF of |φ| clauses built on a set V = {v 1 , v 2 , . . .} of |V | variables, such that its clauses contain only unnegated literals, does there exist a truth assignment on V satisfying φ such that each clause is satisfied by exactly one of its three literals?

We first prove that both the MinFixMasks opt and MinFixMasks bound problems are NP-hard even when each shield sector is associated to two consecutive patient volumes (i.e., w = 2). Given any instance (φ, V ) of Monotone 1-3 SAT problem, we build an instance of MinFixMasks as follows. For all i ∈ [1, |V |], let q i be an integer value computed using the following recurrence formula:

q i = 1 + 2 × i-1 j=1 (1 + q j ) with q 1 = |V |.
For each variable v i ∈ V , we build the sequence V i = (q i , 1 + q i ). For each clause c m = (v a , v b , v c ) ∈ φ, we build the sequence C m composed of two copies of (q a + q b + q c + 2). For each pair (v i , v j ), i < j ≤ |V |, we build the sequence V i,j = (q i + q j , q i + q j + 2). Let V * ,j be the concatenation of V 1,j , V 2,j , . . . V j-1,j . The sequence D is obtained by concatenating in order

V 1 V 2 . . . V |V | C 1 C 2 . . . C |φ| V * ,2 V * ,3 . . . V * ,|V | . We finally set K = |V | + |φ| + |V |•(|V |-1)

2

, N = 2 • K (i.e., w = 2). For the MinFixMasks bound variant, we further set T max = |V |. An illustration is given in Figure 2.

φ = (v 1 , v 2 , v 3 ) ∧ (v 3 , v 4 , v 5 ) ∧ (v 2 , v 4 , v 5 ) ∧ (v 1 , v 2 , v 4
) which only admits one optimal solution (v 1 = v 5 = true and v 2 = v 3 = v 4 = false) where N = 18 and m = 5. For ease of notation, v x will denote x occurrences of the element v (thus 61 2 corresponds to 61 61) and most elements V i,j have been ommited.

One can first state some interesting properties of this construction (the proof is omitted here and provided in Appendix). For all 1 ≤ n |V |, τ n is defined accordingly to the boolean value v n as τ n = q n if v n = true; τ n = 1 + q n otherwise. Each P n is obtained by concatenating three substrings corresponding to the V i 's, C m 's and V i,j 's :

P n = P n v P n c P n V * where P n v = 0 n-1 10 |V |-n , P n c = In(n, 1)In(n, 2) . . . In(n, |φ|) (In(n, m) is 1 if v n ∈ c m , 0 otherwise) and P n V * = P n V * ,1 P n V * ,2 . . . P n V * ,|V | . Each P n V * ,i is defined accordingly to i and n as P n V * ,i = 0 i-1 if i < n ; P n V * ,i = 1 n-1 if i = n and P n V * ,i = 0 n-1 10 i-1-n otherwise.
By construction, with the obtained treatment plan, a total dwell time τ n (either q n or 1 + q n ) is applied to each V n . Moreover, any C m corresponding to a clause (v a , v b , v c ) receives a total dwell time of q a + q b + q c + 2, since by hypothesis exactly one of {v a , v b , v c } is true in our assignment: that is either q a + (1 + q b ) + (1 + q c ) or (1 + q a ) + q b + (1 + q c ) or (1 + q a ) + (1 + q b ) + q c . Finally, a total dwell time τ i + τ j such that q i + q j ≤ τ i + τ j ≤ q i + q j + 2 has been applied to each V i,j , lowering its cost to 2. Thus, any Monotone 1-3 SAT solution over φ give us an optimal solution for our instance of MinFixMasks using |V | shield configurations.

(⇒) Let (P 1 , τ 1 ), (P 2 , τ 2 ) . . . (P T , τ T ) be an optimal solution (that is such that diff = |V | 2 ). By Lemma 5, we know that T ≥ |V |. Suppose w.l.o.g. that T = |V |.

In the following, for ease, we say that a step t contributes to a sequence V i if the block V i is irradiated at step t (the i th bit of its mask P t is set to 1) and t minimizes a sequence V i if the step t is the last one of the treatment plan contributing to V i : at step t -1 and before, V i did not reach its minimum yet, at step t + 1 and after, V i cannot be lowered. Note that a sequence is minimized at exactly one step. We first show that in one step we cannot minimize more than one sequence V i , which is a corollary of the following proposition. Proposition 2. If a sequence V j is minimized at step t, then this step cannot contribute to any V i , i < j.

Proof. Let V j be the greatest sequence minimized at a step, t, also contributing to a V i , i < j. Then τ t ≤ 1 + q i < q j , so other contributing steps to V j exist. As V j is the highest sequence sharing a dwell time with a lower sequence, the missing contributions can only be from steps minimizing lower sequences. Thus, the sum of these other contributions is bounded by

j =j-1 j =1 (1 + q j ).
This leads to an upper bound for the sum of all contributions received by V j :

j-1 j =1 (1 + q j ) + τ t ≤ j-1 j =1 (1 + q j ) + 1 + q i ≤ j j =1
(1 + q j ) < q j . Then, the total irradiation time received by V j is not enough for it to be minimized at step t, which contradicts our starting hypothesis. Corollary 3. There exists no step minimizing more than one sequence.

Proof. According to Proposition 2, if two sequences V i and V j , i < j are minimized at the same step, then this step is both minimizing V j and contributing to V i , which is impossible.

Since we have |V | sequences and |V | steps, exactly one V i is minimized at each step of the treatment plan. Without loss of generality, let us order the steps of our treatment plan such that each sequence V i is minimized at step i. In the next proposition, we prove that the dwell time at step i can take one of two possible values, thus corresponding to a true/false assignment of variable i. Proposition 3. For any i, τ i ∈ {q i , 1 + q i }.

Proof. We prove the result by induction. From Proposition 2, it follows that q i -i-1 j=1 τ j ≤ τ i ≤ 1 + q i for each step i. Thus, for the first induction step, it holds that q 1 ≤ τ 1 ≤ 1 + q 1 , so τ 1 ∈ {q 1 , 1 + q 1 }.

Considering the step j, and the sequence V j-1,j = (q j-1 + q j , q j-1 + q j + 1), no step after j can contribute to V j-1,j since

τ j+1 ≥ q j+1 - j i=1 (1 + q i ) ≥ 1 + 2 × j i=1 (1 + q i ) - j i=1 (1 + q i ) > q j-1 + q j + 2 Moreover, j-1 i=1 τ i ≤ j-1 i=1 (1 + q i ) < q j ,
so the contribution of step j is mandatory to minimize V j-1,j which induces that

τ j ≥ q j-1 + q j - j-1 i=1 τ i ≥ q j-1 + q j - j-1 i=1 (1 + q i ) ≥ q j -1 - j-2 i=1 (1 + q i ) Suppose now that there exists k ≥ 2 such that τ j ≥ q j-1 -1 - j-k i=1 (1 + q i ).
Consider then V j-k,j = (q j-k + q j , q j-k + q j + 2). Applying a similar reasoning as before, we conclude that the contribution of step j is mandatory, and, with our last lower bound over τ j that q j-k + q j -τ j ≤ q j-k + q j -q j -1 -

j-k i=1 (1 + q i ) ≤ q j-k + 1 + j-k i=1 (1 + q i ) < q j-k+1
Thus, steps strictly between j -k and j cannot contribute. Therefore, the only steps able to contribute are j and 1 to j -k:

τ j ≥ q j-k + q j - j-k i=1 τ i ≥ q j-k + q j - j-k i=1 (1 + q i ) ≥ q j -1 - j-(k+1) i=1 (1 + q i )
We obtain a greater lower bound for τ j . This reasoning can be applied as long as V j-k,j exists, that is as long as j -k ≥ 1. The last application (k = j -1, k + 1 = j) leads to

τ j ≥ q j -1 - j-j i=1 (1 + q i ) ≥ q j -1
On the whole, we obtain τ j ∈ {q j -1, q j , 1 + q j }. Moreover, if τ j = q j -1, then step j is not enough to minimize V j = (q j , 1 + q j ) (an amount of 1 or 2 is missing). But we can only use the dwell times of the treatment plan, and the lowest one is

τ 1 ∈ {|V |, |V | + 1}
, where |V | is the number of variables in φ (so |V | ≥ 3). Thus, τ j = q j -1 is impossible. This leads to τ j ∈ {q j , 1 + q j } for any step j.

To complete our proof, it remains to show that a sequence C m = (q a +q b +q c + 2, q a + q b + q c + 2) cannot be minimized by other steps, except those corresponding to an assignment to the variables a, b and c. Proposition 4. Minimizing a sequence C m = (q a + q b + q c + 2, q a + q b + q c + 2) implies the contribution of exactly the steps a, b, and c. Proof. W.l.o.g. let a < b < c. To minimize C m , we need to apply a total amount of exactly τ = q a + q b + q c + 2. Since τ c+1 ≥ q c+1 ≥ 1 + 2 × c i=1 (1 + q i ) > τ , step c + 1 or higher cannot contribute to C m . Thus the contribution of step c is mandatory since

c-1 i=1 τ i ≤ c-1 i=1 (1 + q i ) < 1 + 2 × c-1 i=1 (1 + q i ) < q c .
Similarly, τ b+1 ≥ q b+1 > τ -τ c , so steps strictly between b and c cannot contribute, and b-1 i=1 τ i < q b ≤ τ -τ c inducing that the contribution of step b is mandatory. Finally, τ a+1 ≥ q a+1 > τ -τ c -τ b , so steps strictly between a and b cannot contribute, implying that the contribution of step a is mandatory since

a-1 i=1 τ i < q a ≤ τ -τ c -τ b .
Gathering the previous results, we have an optimal solution to our Min-FixMasks instance if and only if each sequence C m corresponding to a clause (v a , v b , v c ) receives exactly the dwell times received by the sequences V a , V b and V c . Moreover, each of theses V i receives either q i or 1 + q i as a (total) dwell time. Finally, minimizing C m implies that exactly one of the three V i receives the lowest of its two possible values. This corresponds to a truth assignment over φ such that each of its clauses contains exactly one true variable.

Appendix

umor coverage under 3D im-e due to the use of the interber of catheters, the location ell times are computed in an ng dose distributions are still metric dose distribution, source therapy techniques such as (RSBT) 11-13 and dynamic BT) 14-16 were introduced as a ary brachytherapy dose distril cancer. When used as proworks by using a shield that c brachytherapy (eBT) source ashua, NH, USA), which roradiation away from healthy cerous tissue. 11-13 Due to the n in cervical cancer patients, sion angles would need to be r to ensure the dose conformity Small emission angles with atment times. A rapid emistechnique with single-shield ed to strike the best balance ose distribution quality, which and OAR sparing. 12 Dynamicthe use of di↵erent azimuthal livery via a layered shielding pendently rotatable to flexibly ws. 17 With a delivery time of ction (fx), D-RSBT can pron S-RSBT, while S-RSBT may ry time is limited (<20 min/fx) ingle large emission angle. 17 SBT lies in the limit of the ngle that can be formed by the a paddle-based rotating-shield ce, a novel conformal brachy-translation of the whole applicator, shown in Fig. 1. As this is a conceptual study for assessing the P-RSBT technique, the exact diameter of the applicator is not finalized. It is expected that an applicator with a diameter of less than 10 mm could be constructed, which would necessitate appropriate anesthesia techniques for the brachytherapy procedure. As shown in Fig. 1, an integer, K, number of shield paddles, is arranged to form a cylindrical tube with each paddle shielding a sector of 360 /K angularly. The maximum number of paddles (K) achievable will depend upon the method used to drive the paddles. A candidate motor available today is the Faulhaber microdrive, of 1.9 mm diameter and 9.5 mm length. With a 5.4 mm radiation source, up to four microdrives could be used Proof. Let m be a number inbetween the at most two medians of S (if S has only one median, then m is equal to that median) and let x be any natural number that is not inbetween the at most two medians of S. We denote by S ≤x the set of natural numbers s in S with s ≤ x and similarily with S >x the set of natural numbers s in S with s > x. We first show that f (x) > f (m). We distinguish two cases: [START_REF] Cormen | Introduction to Algorithms[END_REF] x is larger than the largest median of S and (2) x is smaller than the smallest median of S.

≤ m ≤ m 2 . f (m 1 ) = s∈S |s -m 1 | = s∈S∧s≤m1 |s -m 1 | + s∈S∧s≥m2 |s -m 1 | = s∈S∧s≤m1 |s -m| + |m -m 1 | + s∈S∧s≥m2 |s -m| -|m -m 1 | = s∈S∧s≤m1 |s -m| + |S ≤m1 ||m -m 1 | -|S ≥m2 ||m -m 1 | = s∈S |s -m| = f (m)
This shows that f (x) has a unique minimum which is only reached by any number x inbetween the at most two medians of S.

Lemma 3 Let S be a set of dwell times. Then an S-restricted treatment plan minimizing

N -1 n=0 dn -d n can be found in time O(( dmax ) 2 |S| + w 2 log w + K dmax ). Moreover, an S-restricted treatment plan minimizing N -1 n=0 dn -d n under the additional constraint dn -d n ≥ 0 for every 0 ≤ n ≤ N -1 can be found in O(( dmax ) 2 |S| + Kw + K dmax ) time. Moreover, if S is complete then the S-restricted treatment plans returned by the above algorithms are optimal among all (not necessarily S-restricted) treatment plans.

(P 1 , s 1 ), . . . (P T , s |S| ) where s 1 , . . . , s |S| is an arbitrary ordering of the numbers in S and the i-th bit of the binary string P j is 1 if and only if s j ∈ S(d i ). This completes the description of the algorithm. The running time of the algorithm is at most O(( dmax ) 2 |S| + w 2 log w + K dmax ) (in the case that overdoses are allowed) and O(( dmax ) 2 |S| + Kw + K dmax ) (in the case that no overdoses are allowed). Finally, if S is complete then the S-restricted treatment plan returned by the algorithm minimizes Proof. The best solution in terms of irradiation differences is to apply to each V i a total dwell time τ i corresponding to the median of q i and 1 + q i , which is either τ i = q i or τ i = 1 + q i , leading to a distance of 1. Therefore, the minimal overall distance one can achieve for all the V i sequences is |V |.

Let us remark that, by construction, the V i s are increasingly ordered in D (i.e., V i < V i+1 , 1 ≤ i < |V |). Let (P 1 , τ 1 ), (P 2 , τ 2 ) . . . (P T , τ T ) be a treatment plan reaching the minimal overall distance |V |. Let us prove that T ≥ |V |. Suppose that using t < T different configurations of the treatment plan, one was able to apply a total dwell time τ i corresponding to the median of q i and 1 + q i to each V i for 1 ≤ i ≤ j. In other words, one has acheived the minimal overall distance for all V i for 1 ≤ i ≤ j. Regarding V j+1 , the maximal overall contribution related to the use of these t different configurations is lower or equal to j i=1 1 + q i . Moreover, none of these t configurations can be applied again since any of these contributes to at least one of {V 1 , V 2 , V j } and would increase the overall distance. Therefore, in order to reach the minimal distance for V j+1 , at least an extra configuration is needed. By induction, one cannot achieve the minimal overall distance of |V | on the V i sequences without using at least |V | different configurations.

Moreover, let us consider the V * ,i sequences. Each V i,j = (q i + q j , q i + q j + 2) has a minimum cost of 2. Thus, the total cost of the V * ,i sequences cannot be lower than 2

• |V |•(|V |-1)

2

. By adding the minimal cost for all the V i and V * ,i sequences, one obtain an optimal diff

= |V | + |V | • (|V | -1) = |V | 2 .

Figure 1 :

 1 Figure 1: Relation between P and D (K = 3, N = 6)

Theorem 2 .

 2 FixMask can be solved in O(N log N ). Proof. Because of Lemma 1 the best possible value that we can achieve for N -1 n=0 | dn -d n | is obtained by setting the dwell time τ 0 to any median of the subsequence of D containing only the prescriped doses for which the paddles are open (in the given mask). A median of this sequence can be found by sorting the corresponding set in time O(N log N ).

N - 1 n=0Lemma 4 .

 14 dn -d n can be found in time O(( dmax )2 |S| + w 2 log w + K dmax ). Moreover, an S-restricted treatment plan minimizingN -1 n=0 dn -d n under the additional constraint dn -d n ≥ 0 for every 0 ≤ n ≤ N -1 can be found in O(( dmax ) 2 |S| + Kw + K dmax ) time.Moreover, if S is complete then the S-restricted treatment plans returned by the above algorithms are optimal among all (not necessarily S-restricted) treatment plans. There is a treatment plan minimizing N -1 n=0 dn -d n using at most log dmax + 1 steps. Moreover, such a treatment plan can be found in polynomial time. The same holds for a treatment plan minimizingN -1 n=0 dn -d n under the additional constraint that dn -d n ≥ 0 for every n with 0 ≤ n ≤ N -1.Proof. Because the set S = { 2 i : 0 ≤ i ≤ log dmax } is complete and has size log dmax + 1, this follows immediately from Lemma 3.

Theorem 3 .

 3 MinFixMasks opt and MinFixMasks bound can be solved in time O( d log dmax +1 max ( dmax ) 2 ( log dmax + 1) + w 2 log w + K dmax )) . Moreover, MinFixMasks + opt and MinFixMasks + bound can be solved in time O( d log dmax +1 max ( dmax ) 2 ( log dmax + 1) + Kw + K dmax )).

N - 1 n=0

 1 dn -d n < diff found for any considered set S, or the algorithm returns failure if no treatment plan exists with N -1 n=0 dn -d n < diff . The stated running of the algorithm follows because there are at most d log dmax +1 max such sets S and for each of them, we require time at most O( dmax ) 2 |S| + w 2 log w + K dmax ) respectively time at most O(( dmax ) 2 |S|+Kw +K dmax ) (in the case of the problems MinFixMasks + opt and MinFixMasks + bound ). The correctness of the algorithm follows from Lemmas 2, 3 and 4.

5 6 Figure 2 :

 62 Figure 2: Example of an instance of MinFixMasks considering the boolean formulaφ = (v 1 , v 2 , v 3 ) ∧ (v 3 , v 4 , v 5 ) ∧ (v 2 , v 4 , v 5 ) ∧ (v 1 , v 2 , v 4) which only admits one optimal solution (v 1 = v 5 = true and v 2 = v 3 = v 4 = false) where N = 18 and m = 5. For ease of notation, v x will denote x occurrences of the element v (thus 61 2 corresponds to 61 61) and most elements V i,j have been ommited.

Lemma 5 .Proposition 1 .

 51 To achieve a solution with diff = |V | 2 (which is the optimal solution one can get), one has to use at least |V | different configurations. The MinFixMasks opt and MinFixMasks bound problems are NP-hard when w = 2. Proof. Let us prove that, if w = 2, the MinFixMasks opt and MinFixMasks bound problems are equivalent to the Monotone 1-3 SAT problem. (⇐) Given any truth assignment satisfying φ such that each clause is satisfied by exactly one of its literals, let us build a treatment plan consisting of a set of |V | binary strings {P 1 , P 2 , . . . P |V | } with the corresponding irradiation times {τ 1 , τ 2 , . . . τ |V | } as follows. We build a a set of |V | binary strings {P 1 , P 2 , . . . P |V | } with the corresponding irradiation times {τ 1 , τ 2 , . . . τ |V | }, using a given truth assignment over φ.

Fig. 1 .Figure 3 :Lemma 1

 131 Fig. 1. A conceptual design of a P-RSBT applicator (a) 3D view. For the purpose of legibility, not all paddle shafts are drawn. (b) Cross-sectional view.

N - 1 n=0Lemma 5

 15 dn -d n under all possible treatment plans. The same applies when considering the additional constraint dn -d n ≥ 0 for every n with 0 ≤ n ≤ N -1. To achieve a solution with diff = |V | 2 (which is the optimal solution one can get), one has to use at least |V | different configurations.

  1 the string of integers obtained by applying radiation for a time τ to D through the mask P . We consider that p k = 1 (resp. p k = 0) denotes applying radiation (resp. no radiation applied) to the area D k . Moreover, d n = τ (resp. d n = 0) if radiation is applied to the volume n for a time τ (resp. if no radiation is applied there). In other words, each patient volume associated to an open sector (represented by a 1) is irradiated τ units of time, while volume associated to a closed sector (represented by a 0) is left in its previous state.

  In the first case |s -x| ≥ |s -m| + |x -m| for every s ≤ x and |s -m| ≤ |s -x| + |x -m| for every s > x. Moreover, because x is larger than any median of S, we obtain that |S ≤x | > |S >x |. HenceThe proof for the second case is symmetric. This shows that a minimum of f (x) can only be reached if x is a number inbetween the two median of S and that if x is not inbetween the at most two medians of S, then f (x) decreases with the distance of x to a median of S. Hence in the case that S has exactly one median there is nothing else to show. Now consider the case where S has exactly two medians say m 1 and m 2 . In this case |S ≤m1 | = |S ≥m2 | and hence we obtain the following for any number m with m 1

	f (x) =	|s -x|
	s∈S
	≥	|s -m| + |x -m| +	|s -m| -|x -m|
	s∈S∧s≤x	s∈S∧s>x
	=	|s -m| + |S ≤x ||x -m| -|S >x ||x -m|
	s∈S
	>	|s -m|
	s∈S
	= f (m)

This research was partially supported by PoPRA project funded by Conseil Régional d'Aquitaine and European FEDER and IdEx Bordeaux. We thank Maxence Ronzié for initial assistance in the very first step of the project.

Proof. Because every treatment plan has to apply the same dose to every patient volume under the same paddle, we obtain that d iw = d iw+1 = • • • = d (i+1)w-1 for every 0 ≤ i ≤ K -1. Hence, finding a treatment plan minimizing N -1 n=0 dn -d n is equivalent to finding K natural numbers d 0 , . . . , d K-1 minimizing N -1 n=0 dn -d n/w such that each number d i can be realized by a treatment plan. Since we are only considering S-restricted treatment plans, it holds that the set of realizable numbers (for any such treatment plan) is equal to the set of numbers that can be written as a sum of a subset of S. Moreover, because we can choose the masks arbitrarily it holds that for every K numbers d 0 , . . . , d K-1 , which can be written as the sum of subsets of S, there is an S-restricted treatment plan realizing these numbers. To see this let d 0 , . . . , d K-1 be any numbers such that for each d i there is a subset S(d i ) of S with d i = s∈S(di) s, then (P 1 , s 1 ), . . . (P T , s |S| ) where s 1 , . . . , s |S| is an arbitrary ordering of the numbers in S and the i-th bit of the binary string P j is 1 if and only if s j ∈ S(d i ) is an S-restricted treatment plan realizing the numbers d 0 , . . . , d K-1 . It follows that given the set of allowed dwell times S, we can minimize each of the sums

is to a median of the sequence diw , . . . , d(i+1)w-1 . Moreover, if we consider the case where we have the additional constraint that dn -d n ≥ 0 for every 0 ≤ n ≤ N -1, then the optimal value for d n/w is the one that is closest to the minimum of the sequence diw , . . . , d(i+1)w-1 . These considerations naturally lead to the following algorithm to find an S-restricted treatment plan minimizing

The algorithm first computes a table that contains for every number i between 1 and dmax either a subset S of S such that i = s∈S s or Nil if no such subset S exists for i. Using the standard text-book algorithm for the Subset Sum problem [START_REF] Cormen | Introduction to Algorithms[END_REF] (running in O(sN ) time, where s is the sum that needs to be obtained and N is the size of the set of integers), this can be achieved in O(( dmax ) 2 |S|) time. In the case that we have no additional constraint the algorithm then computes the (at most two medians) m 1 i and m 2 i of the sequence diw , . . . , d(i+1)w-1 for every 0 ≤ i ≤ K -1. This can be achieved (by sorting each of these sequences separately) in at most O(w 2 log w) time. In the case that we have the additional constraint dn -d n ≥ 0 for every 0 ≤ n ≤ N -1, the algorithm computes the minimum min i for each of these sequences in O(Kw) time. Finally, the algorithm computes for every 0 ≤ i ≤ K -1, the number d i as the number between 1 and dmax that has a non-nil entry in the table and either: (1) is closest to one of the medians m 1 i and m 2 i (in the case with no additional constraints) or ( 2) is closest to the minimum min i (in the case that overdoses are not allowed). Using the table and the medians m 1 i and m 2 i respectively the minima min i this can be achieved in time O(K dmax ). Considering an arbitrary ordering s 1 , . . . , s |S| of the numbers in S and denoting by S(d i ) the subset of S contained in the table for the number d i , the algorithm then outputs the treatment plan


Poels Alix
email: alix1.poels@etu.u-pec.fr

Alix Poels

Gratuité sur Internet : l'émergence d'

Keywords: Internet, Gratuité, Pratiques de consommation, Téléchargement, Culture

Est

La gratuité est une notion essentielle et inhérente à notre société, son étude a cependant été relativement éludée en marketing. Cet article, de nature conceptuelle cherche dans un premier temps à circonscrire l'idée de gratuité, en en dressant une typologie, et en la reliant aux concepts de don, de partage et de valeur. Puis nous montrerons en quoi la culture à l'origine du Réseau des réseaux et son architecture permettent aux individus d'exercer ce que l'on nomme le « consumer power » et comment, sur internet, l'accession gratuite à des biens culturels est désormais ancrée dans les pratiques de consommation des individus malgré la mise en place de dispositifs dissuasifs et répressifs. D'où cette question, la gratuité sur internet signe t'elle l'émergence d'une nouvelle culture de consommation ancrée dans les pratiques ?

Introduction

« Nous entrons dans une ère où la gratuité sera considérée comme la norme et non pas comme une anomalie » [START_REF] Anderson | Free !entrez dans l'économie du gratuit[END_REF].

« Tout se vend sur le Net, sauf les biens immatériels qui touchent la culture et l'information. Ceux-ci sont sanctuarisés sous la bannière du gratuit et personne n'envisage vraiment qu'une épreuve de vérité puisse un jour balayer brutalement ce miracle » Monique Dagnaud (2010a).

Ces deux citations, prophétiques et antinomiques sont la preuve de la polémique suscitée par la gratuité, tour à tour revendiquée ou vilipendée. Certains, tels qu [START_REF] Anderson | Free !entrez dans l'économie du gratuit[END_REF], le rédacteur en chef de la revue Wired en font le pilier de l'économie du numérique, d'autres comme [START_REF] Bomsel | Gratuit ! Du déploiement de l'économie numérique[END_REF], économiste, la dénoncent et prédisent sa fin. Des sociologues célèbrent l'avènement des logiciels libres (Goldenberg et Proulx, 2010), ou pleurent la mort de la Culture (Dagnaud, 2010b). Mais nous débattons trop souvent de la gratuité sans vraiment chercher à la définir. Nous nous intéressons à ses conséquences, ses externalités, du téléchargement en P2P1 aux logiciels libres en passant par les wiki2 sans jamais vraiment la placer au coeur du débat et lui assigner sa position charnière, sans vraiment montrer comment et pourquoi elle est peut-être aujourd'hui ancrée dans les modes de consommation. Cet article, de nature conceptuelle cherche à montrer en quoi nous pourrions faire face aujourd'hui à une « culture de la gratuité » sur internet profondément enracinée dans les mentalités et les pratiques des consommateurs. Au sein de cet article nous nous pencherons sur ce concept aux contours flous, pour tenter de le délimiter et nous montrerons en quoi nous pouvons aujourd'hui parler, dans le cadre de la gratuité, d'une culture de consommation.

1. Pourquoi s'interroger sur le concept de gratuité ?

Un concept multiforme et complexe

Par gratuité, nous pouvons entendre « sans prix », « un acheté, un offert », « école publique », c'est pourquoi il nous semblait nécessaire d'en dresser une typologie en nous appuyant sur les travaux de [START_REF] Grassineau | Rationalité économique et gratuité sur Internet : cas du projet Wikipédia[END_REF].

Tableau 1. -Typologie de la gratuité d'après Grassineau (2009) Il existe quatre types de gratuité : une gratuité que l'on pourrait qualifier « d'imposée », dont l'objectif est la redistribution et le partage des richesses dans un but égalitaire et constitutive du « sens de vivre en société » [START_REF] Sagot-Duvauroux | La gratuité, une utopie réalitaire[END_REF], une gratuité « communautaire » (Dang Nguyen et Pénard, 2001) dont l'exemple type serait celui du forum sur Internet, une gratuité « naturelle », comme celle de l'air que nous respirons, « qui ne résulte pas de l'activité humaine » [START_REF] Grassineau | Rationalité économique et gratuité sur Internet : cas du projet Wikipédia[END_REF] et une gratuité « marchande », modèle économique dans lequel le consommateur n'a pas à fournir d'effort monétaire direct pour acquérir un bien ou un service. Au centre de ses différents types de gratuité, une définition commune : « est considérée comme gratuite une action effectuée (librement) sans contrepartie monétaire directe » (Demazière, Horn et Zune, 2008).

Gratuité naturelle

Gratuité contrainte

Gratuité marchande

Gratuité en réseau

Cette typologie de la notion de gratuité nous en permet un éclaircissement, cependant sa limite tient à ce qu'elle décrit plus qu'elle n'explique les phénomènes à l'oeuvre. La complexité à rendre compte du concept de gratuité tient également à sa proximité avec d'autres concepts, tels que le don notamment (on parle souvent de don « gratuit ») même si elle ne s'y réduit pas.

1.1.1. Le don et le partage, concepts limitrophes de la notion de gratuité Le concept de don s'entend comme « toute prestation de bien ou de service effectuée, sans garantie de retour, en vue de créer, nourrir, ou recréer le lien social entre les personnes » (Caillé, 1994) [START_REF] Cheal | The gift economy[END_REF]. Pour [START_REF] Mauss | Essai sur le don : forme et raison de l'échange dans les sociétés archaïques[END_REF] le don s'inscrit dans une dimension de sociabilité que sous-tend un cercle d'infinie réciprocité, d'où cette tension entre « la gratuité du don » et « l'exigence de l'échange » [START_REF] Boltanski | L'amour et la justice comme compétences : trois essais de sociologie de l'action[END_REF]. « Le « don échangé » enferme bien un « paradoxe », et l'expression même « échange de don » constitue une contradiction dans les termes. En effet, soit l'insistance est mise sur le « don », c'est à dire sur le caractère gratuit du cadeau, et l'on perd de vue l'échange, soit l'accent est mis sur l'échange et la gratuité du don ne peut plus apparaître que comme une illusion voire une tromperie ». [START_REF] Boltanski | L'amour et la justice comme compétences : trois essais de sociologie de l'action[END_REF]. Ce que souligne [START_REF] Derrida | Donner le temps. 1. La fausse monnaie[END_REF] « pour qu'il y ait don, il faut que le donataire ne rende pas, n'amortisse pas, ne rembourse pas, ne s'acquitte pas, n'entre pas dans le contrat, n'ait jamais contracté de dette ». Cependant pour [START_REF] Bourdieu | Esquisse d'une théorie de la pratique ; précédé de Trois études d'ethnologie kabyle[END_REF] ce qui vient distinguer le don de l'échange est la notion de temporalité. Le don suppose qu'une temporalité le sépare du contredon, alors que dans l'échange nous sommes dans le « donnant-donnant ». Enfin, une autre distinction entre le don et l'échange tient au fait que : « Le don existe aussitôt qu'est acceptée la possibilité d'un défaut de la réciprocité, et que cette acceptation constitue le signe suffisamment dénué d'équivoque de la générosité et du désintéressement » (Caillé, 1994). La gratuité dans le prisme du don apparaît comme la quintessence de ce concept. Le terme de gratuit vient ici signifier le « défaut de réciprocité » et l'absence d'attente de retour.

Dans notre discipline, les sciences de gestion et plus particulièrement en marketing, des auteurs tels que [START_REF] Belk | Gift Giving as Agapic Love: An Alternative to the Exchange Paradigm Based on Dating Experiences[END_REF][START_REF] Giesler | Consumer Gift Systems[END_REF] ont remis en cause la dimension dyadique du don. Ces auteurs, Belk en particulier, soulignent le caractère gratuit de certains de nos actes, contre les présupposées idéologiques réduisant les individus à des êtres motivés uniquement par la maximisation de leur profits. Ainsi [START_REF] Belk | Gift Giving as Agapic Love: An Alternative to the Exchange Paradigm Based on Dating Experiences[END_REF] montrent comment le don peut être sous tendu par le concept d'Agapé3 qui n'induit pas de réciprocité. [START_REF] Giesler | Consumer Gift Systems[END_REF] quant à lui montre comment au sein de la communauté Napster4 , certains membres ouvrent au partage un grand nombre de fichiers sans pour autant attendre de recevoir en retour. D'autres études enfin, ont permis soit d'affiner les motivations des donneurs et d'en dresser des profils (Bergadaà, 2006) ou de montrer en quoi le don pouvait avoir une dimension émancipatrice, ainsi Kozinets ( 2002) montre comment il peut être une alternative au marché dans son étude sur le festival du Burning Man.

Le partage est aussi un concept limitrophe de la notion de gratuité. Nous partageons au quotidien, et cet acte est si naturel qu'il nous paraît quasiinsignifiant. Rares d'ailleurs sont les études à avoir porté sur ce sujet.

Il existe plusieurs acceptations du terme de « partage ». Le terme peut d'abord désigner l'action de diviser en plusieurs éléments un ensemble, on partage alors « entre » les individus (répartition/distribution). On peut aussi entendre le terme « partage » comme « le fait d'avoir ou de faire en commun quelque chose avec quelqu'un » [START_REF] Autant | Le partage : un nouveau paradigme ?[END_REF], on partage alors « avec ». [START_REF] Belk | Why Not Share Rather Than Own?[END_REF] le définit comme « l'acte ou le processus de recevoir ou de prendre quelque chose des autres pour son propre usage », le partage est naturel et non réciproque (au contraire du don). Dans le partage la distinction entre « ce qui est mien » et « ce qui est tien » disparaît, le partage définit « ce qui est nôtre » [START_REF] Belk | Sharing[END_REF]. On pensera alors aux systèmes de P2P notamment, sanctionnés par l'Hadopi et contre laquelle des voix s'élèvent comme celle d'Aigrain (2012) qui dans son essai, Sharing : Culture and the economy in the Internet Age, défend le droit au partage des oeuvres culturelles entre les individus. Un autre exemple est celui du Book Crossing dont le principe est de « libérer » des livres dans la nature pour qu'ils puissent être retrouvés et lus par d'autres personnes qui les « libéreront » à leur tour. Certains y voient un glissement du paradigme de la propriété à celui de l'accès [START_REF] Giesler | Consumer Gift Systems[END_REF], ce qu'annonce notamment les travaux de [START_REF] Rifkin | L'âge de l'accès : la révolution de la nouvelle économie[END_REF] «On peut penser que d'ici à vingt-cinq ans l'idée même de propriété paraîtra singulièrement limitée, voire complètement démodée, à un nombre croissant d'entreprises et de consommateurs. En tant qu'institution, la propriété fonctionne à un rythme beaucoup trop lent pour la nouvelle économie, qui opère en termes de nanosecondes. L'idée de propriété se fonde sur la valeur attribuée à la possession d'un patrimoine matériel ou foncier sur une durée plus ou moins définie. « Avoir », « posséder », « accumuler » ont toujours été des notions fort prisées. Mais de nos jours, l'enchaînement ultra-rapide des innovations technologiques et le rythme vertigineux de l'activité économique rendent souvent la notion de propriété fort problématique». Belk dans son article « Why not share rather than own » (2007) montre en quoi « l'accès » seul aux biens et services serait bénéfique à la société de consommation, il donne notamment l'exemple du covoiturage et d'internet tout en soulignant les avantages du partage par rapport à la propriété.

La lecture « brève » de ces deux concepts nous permet de voir que la gratuité comme les notions de don et de partage ont en commun de tisser des formes de sociabilité entre les individus. En sciences de gestion, l'étude de la notion de gratuité s'est avant tout portée sur la notion de « gratuité ponctuelle », de type promotionnelle dont l'on comprend bien les effets d'aubaine notamment. A contrario, nous nous intéressons dans cet article à une « gratuité permanente » dans le contexte numérique.

Une notion peu étudiée en marketing

Au sein de notre discipline, le marketing, l'étude de la gratuité se décline autour de deux grands axes : ses impacts sur le comportement des individus, et sa corrélation à la notion de valeur. En ce qui concerne les impacts de la gratuité, celles ci ont étés principalement étudiées dans le contexte muséal (Le [START_REF] Gall-Ely | Une étude exploratoire des représentations de la gratuité et de ses effets sur le comportement des publics des musées et des monuments[END_REF]2008), [START_REF] Bourgeon- | Les effets positifs de la gratuité sur le processus de décision du consommateur : le cas de la visite des musées et des monuments[END_REF]2009 ;2010) ou de spectacles sportifs (Derbaix et al., 2010). Les résultats montrent que "la gratuité permet l'essai et peut provoquer l'attachement [des publics] à l'objet gratuit, mais aussi à la gratuité" (Le [START_REF] Gall-Ely | Une étude exploratoire des représentations de la gratuité et de ses effets sur le comportement des publics des musées et des monuments[END_REF]. Cependant la gratuité n'est pas le principal déterminant de décision, elle va seulement en modifier le processus, en autorisant l'erreur. Il n'y a ni geste de paiement, ni effort monétaire. Ces éliminations sont les conséquences les plus immédiatement perçues de la gratuité [START_REF] Gombault | La gratuité des musées et des monuments qu'en pensent les publics en France ?[END_REF]. Faire payer un prix, n'importe quel prix, instaure donc une barrière psychologique que la plupart des gens s'abstiendront de franchir. La gratuité, en revanche, coupe court à cette décision, accroissant ainsi le nombre de personnes qui essaieront ce qu'on leur propose, il n'y aura pas ou peu de conséquences négatives d'un mauvais choix (Gombault & al,. 2008). Les risques financiers sont inexistants et à la recherche d'informations se substitue un essai [START_REF] Gombault | La gratuité des musées et des monuments qu'en pensent les publics en France ?[END_REF]). Elle crée un effet promotionnel, et peut même engendrer une certaine forme d'irrationalité que [START_REF] Shampanier | Zero as a special price: the true value of free products[END_REF] associent à l'effet prix zéro. Ils montrent de même que lorsqu'un prix est mentionné, les individus appliquent les normes du marché alors que lorsque le prix est de zéro, ils vont appliquer des normes sociales (celles de l'échange social) pour déterminer leur choix.

Gratuité et valeur

Baudrillard étudie le rapport entre gratuité et valeur et suggère que, dans le cadre de la gratuité, valeur d'échange et valeur d'usage disparaissent : « une fois neutralisée dans un processus de don, de gratuité, de prodigalité, de dépense, la valeur d'usage elle-même devient insaisissable » pour être remplacée par la relation valeur signe/valeur d'échange symbolique [START_REF] Baudrillard | Pour une critique de l'économie politique du signe[END_REF].

Dans le cadre de visites gratuites des musées, Bourgeon-Renault et ses collègues suggèrent que la demande des individus devient davantage sensible à la valeur d'échange symbolique [START_REF] Bourgeon- | Gratuité et valeur attachée à l'objet de consommation culturelle : le cas des musées et des monuments[END_REF]. Si la gratuité permet la démocratisation de l'accès aux biens culturels [START_REF] Bourgeon- | Facteurs explicatifs de l'attitude favorable à la gratuité des musées[END_REF], elle peut également induire une dévalorisation de l'offre, comme cela a été montré dans le secteur médical [START_REF]The cost of free services : organisation impediments of access to public services[END_REF][START_REF] Gorn | The impact of free and exaggerated prices on perceived quality of services[END_REF].

Ce paradoxe de la valeur de la gratuité tient à ce que celle ci peut être envisagée selon trois niveaux différenciés. En effet, d'un point individuel la gratuité va être dévalorisée, elle va évoquer la mauvaise qualité notamment. D'un point de vue collectif, elle va, pour les individus, permettre la création de lien social et enfin, d'un point de vu sociétal, elle va être rattachée (toujours par les individus) aux grandes valeurs que sont la démocratie, la justice, l'égalité [START_REF] Poels | Valeurs et pratiques associées au gratuit : émergence d'une nouvelle culture de consommation[END_REF].

La gratuité offerte par les offreurs (musées, consultations médicales) a donc des effets contrastés sur les publics concernés (visiteurs, patients) et selon le niveau (individuel, collectif, ou sociétal) dont elle est envisagée. Mais qu'en est-il d'une gratuité qui découle des changements de distribution de l'offre ? Dans le cadre d'internet, de nombreux biens/ services sont accessibles de façon gratuite sans que les offreurs/ayantsdroits en autorisent toujours l'accès. Cela résulte souvent des possibilités de l'environnement technologique (offre numérisée notamment), de la culture à l'origine du réseau et des pratiques de plus en plus usitées par les consommateurs qui exercent leurs capacités à prendre le pouvoir par leur consommation, une tendance récemment étudiée sous le terme de consumer power.

2. Internet « culture » ou « contre-culture » de la gratuité ?

De la culture à l'origine du réseau….

Si l'on peut parler d'une « culture » de la gratuité sur Internet, c'est d'abord parce qu'elle est partie intégrante de l'ADN du Réseau des réseaux. A l'origine, ses concepteurs issus de la communauté scientifique, puis, quelques années plus tard les « hackers » nés de la contre culture Californienne « rêvent d'un monde où les hommes pourraient échanger de l'information librement d'un bout à l'autre de la planète, où les communautés en ligne renforceraient ou remplaceraient les communautés locales, où les conférences informatiques permettraient d'exercer une intelligence collective, de construire un nouvel espace public » [START_REF] Flichy | La place de l'imaginaire dans l'action technique[END_REF]. Il s'affirme alors une culture commune « une sensibilité aux valeurs du don qui suppose la confiance dans les autres, la "foi" dans le lien social » [START_REF] Coris | La culture du don dans la modernité[END_REF]citée par Dagnaud, 2010b). L'informatique libre repose sur la culture hacker, ou hackerisme, définie par quelques principes-clés [START_REF] Coris | La culture du don dans la modernité[END_REF] : l'accès aux ordinateurs devrait être illimité et total ; l'information devrait être libre et gratuite ; il faut se méfier de l'autorité et promouvoir la décentralisation ; les gens devraient être jugés en fonction de la valeur de ce qu'ils créent, et non en fonction de leur position sociale ou des diplômes qu'ils détiennent. Aujourd'hui et malgré l'ouverture du réseau au grand public, le monde virtuel est toujours considéré comme l'espace de liberté par excellence [START_REF] Odou | Téléchargement illégal, déviance et gestion individuelle du conflit moral: l'apport de la théorie de la neutralisation[END_REF] et autonome vis-à-vis du monde réel. La valeur de la gratuité sous-tend les mythes fondateurs d'Internet, elle est présente au coeur même des modèles qui l'ont dessiné. Cette « culture de la gratuité » peut apparaître sous deux angles différents : soit une recherche consumériste de biens et de services aux meilleurs prix, voire à prix nul soit une revendication, militante, conscientisée, de rapports sociaux libres et égalitaires (reconnaissance d'une possibilité de contribution citoyenne) (Proulx et [START_REF] Goldenberg | Internet et la culture de la gratuité[END_REF].

Pour [START_REF] Bomsel | Gratuit ! Du déploiement de l'économie numérique[END_REF], la gratuité sur Internet est une illusion. Elle repose, en grande partie sur ce que l'on appelle les subventions croisées : ainsi une tierce personne va « payer » un échange gratuit entre deux autres parties (cas de la presse gratuite, des moteurs de recherche notamment). L'économiste décrit les mécanismes des marchés à deux versants (à l'image de Google notamment). Le gratuit marchand chez Bomsel est décrit comme une « arme économique redoutable », destiné à accroître le nombre d'utilisateurs d'un bien/service du fait des effets de réseaux, en effet, les biens immatériels sont soumis à des effets de réseaux, en effet l'utilité d'un bien va dépendre du nombre d'utilisateurs : être abonné à Internet par exemple ne vaut que si d'autres individus le sont aussi. [START_REF] Anderson | Free !entrez dans l'économie du gratuit[END_REF] quant à lui soutient que le gratuit, notamment en matière numérique s'impose aujourd'hui comme un nouveau modèle économique, « reposant sur les bits et non plus les atomes », il permet d'abaisser le coût des biens et des services aux environs de zéro. Cependant, Bomsel comme Anderson, n'abordent pas dans leurs analyses tout une catégorie de biens culturels dont l'importance est croissante (publications bénévoles, musique gratuite, films, livres etc..) qui sont autant de manifestations de « prises de paroles » par les individus.

2.1.1. Internet et la notion de «consumer power » L'architecture du réseau permet un exercice du pouvoir par les individus, cette notion, avec Internet, semble n'avoir jamais été aussi prégnante comme l'illustre cette citation de Kucuk et Krishnamurthy (2007) : «La révolution industrielle fut pour les fabricants ce que la révolution d'Internet est aux consommateurs. Nous assistons aujourd'hui à une renégociation des relations [de pouvoir] entre les entreprises et les consommateurs ». En effet, le Réseau des Réseaux permet aujourd'hui aux individus un rééquilibrage des rapports de pouvoir avec les entreprises, à travers quatre dimensions : technologique, économique, sociale et légale [START_REF] Kucuk | An analysis of consumer power on the Internet[END_REF] [START_REF] Becker | Outsiders, études de sociologie de la déviance[END_REF]. Cette requalification tend à marginaliser des comportements propres à internet. Elle fait d'une « culture » de la gratuité, une « contre-culture » déviante, et anormale.

Voies de recherche

Nous posons ici un cadre d'analyse, qu'il semble nécessaire de prolonger par une étude empirique. Ainsi comment les internautes consommant de manière gratuite des contenus vont-ils qualifier leurs pratiques ? Quels vont être les différents types d'argumentaires utilisés ?

Les pratiques permettant l'accès aux contenus gratuits découlent-ils de la recherche consumériste du prix nul, ou d'une démarche idéologique ?

Nous avons vu que la gratuité est liée à la notion de valeur, comment les individus vont-ils hiérarchiser les contenus auxquels ils accèdent ?

L'accès gratuit, enfin modifie t-il le rapport à l'oeuvre culturelle?

Bibliographie

  Sommaire Introduction ............................................................................................... 1. Pourquoi s'interroger sur le concept de gratuité ? ................................ 1.1. Un concept multiforme et complexe .............................................. 1.1.1. Le don et le partage, concepts limitrophes de la notion de gratuité ............................................................................................... 1.2. Une notion peu étudiée en marketing ............................................. 1.2.1. Gratuité et valeur ..................................................................... 2. Internet et le phénoméne de gratuité .................................................... 2.1 De la culture à l'origine du réseau…. ............................................. 2.1.1. Internet et la notion de «consumer power » ............................ 2.2. A une culture de la gratuité ? ....................................................... 3. Voies de recherche .......................................................................... 5. Bibliographie ...................................................................................

  .Peut-on réellement parler de « culture » dans le cas de la gratuité ? Le concept de « culture » étant intrinsèquement lié à la notion de normes, de valeurs et d'institutions. Or « dans le cas du web, la définition de normes comportementales claires est loin d'être établie »[START_REF] Wynn | Hyperbole over cyberspace: Self-Presentation and social boundaries in Internet home pages and discourse[END_REF]. Ces résultats montrent que nous sommes bien ici dans un ancrage de la gratuité dans les pratiques de consommation. Sur le net, elle apparaît comme une norme, intériorisée par les individus. Ce fait est appuyé par deux recherches qui se sont attachées à l'étude du téléchargement au prisme de la théorie de la neutralisation. Ainsi les individus auraient un comportement volontairement déviant : « on peut imaginer, vu la diffusion de la pratique du téléchargement, que celle-ci concerne des individus qui reconnaissent le bien fondé du droit à la propriété intellectuelle mais agissent en non-conformité avec cette règle sociale lorsqu'ils sont sur Internet »[START_REF] Odou | Téléchargement illégal, déviance et gestion individuelle du conflit moral: l'apport de la théorie de la neutralisation[END_REF]. Et « on assisterait donc à une forme d'acculturation du « gratuit musical », s'appuyant sur le discours et le comportement dominant ainsi que sur la technologie existante, et tendant à faire du téléchargement gratuit d'oeuvres musicales un comportement normatif, du moins chez les jeunes »[START_REF] Divard | Comprendre les comportements volontairement déviants en termes de paiement : application de la théorie de la neutralisation au téléchargement illégal de fichiers musicaux[END_REF]. Ces études se placent sous le paradigme de la déviance, hors le partage est-il une déviance ? Nous partageons des livres entre amis, des connaissances au sein des bibliothèques. Le principe, sur le fond est le même que celui du partage de fichiers. Dans son article sur les « pirates » du net, Denegri-Knott, (2004) montre comment peu à peu les pouvoirs publics, les organisations ont cherché à « normaliser » les comportements sur Internet, en requalifiant certaines pratiques de « déviantes », (là ou elles n'étaient pas considérées comme telles) et comment peu à peu les « hackers » se sont vus « étiquetés » du terme de « pirates » par les « entrepreneurs de morale »

	reconnaissance croissante du Réseau des Réseaux comme plateforme de des attaques sur plusieurs sites gouvernementaux (comme le FBI aux
	revendications (Zureik et Mowshowitz, 2005). Les individus disposent Etats Unis, mais aussi celui d'Hadopi 8 , en France, en raison du soutien du
	alors d'un pouvoir d'opposition et de contestation que Glickman (2009) chef de l'état à la fermeture du site 9 ). Cette action relève d'une démarche
	relie à la notion de citoyenneté. Cet activisme, peut être rapproché de la conscientisée et militante. Dans le même temps, on a aussi assisté à une Tableau 2. -Sources de pouvoir des consommateurs sur internet, d'après notion de résistance des consommateurs que Peñaloza et Price (1993) recrudescence du téléchargement en P2P 10 en France (pourtant surveillé Kucuk et Krishnamurthy (2007) définissent en reprenant la définition proposé par Poster (1992, p1) par Hadopi) par les internautes, ce qui nous montre bien en quoi l'accès
	Dimensions comme « la manière dont les individus ou les groupes pratiquent des Avantages gratuit à des contenus est ancré dans les pratiques de consommation, Description du changement stratégies d'appropriation en réponse à des structures de domination ». ainsi comme le notent Goldenberg et Proulx, (2010) : « Pour les
	Technologique Praticité internautes habitués à utiliser les ressources sociales, techniques et Mobilité Possibilité de faire, suivre, et contrôler des transactions à tout moment Egalité face à la technologie Ainsi la culture à l'origine d'Internet ainsi que son architecture cognitives du Web, le rapport entre Internet, création et culture de la engendrent des normes comportementales différentes de celles du monde gratuité ne renvoie pas seulement à un défi, mais à des pratiques et des réel. La notion de gratuité sous tend les modèles de l'internet, le partage réflexions amorcées depuis un certain temps ». Ainsi, certaines études en est une valeur fondamentale et enfin, il apparaît comme un espace de montrent que malgré la mise en place de procédures dissuasives (avec liberté que l'on doit défendre (on pensera notamment aux manifestations Hadopi notamment pour la France) sont loin d'avoir l'effet escompté sur anti-ACTA 5 ) les pratiques des individus. Selon Dejean, Pénard et Suire (2010) : « Le
	Economique piratage semble fortement ancré dans les habitudes de consommation de Valeur Accès aux marchés les plus attractifs 2.2. A une culture de la gratuité ? contenus vidéos et audio sur Internet ». Les changements de
	Possibilité de construire de la valeur comportement des individus « portent plus sur une évolution des
	économique (marchés C2C) pratiques ou techniques de piratage numérique, que sur un arrêt pur et
	simple du piratage ». Ces auteurs retracent alors les trajectoires
	Sociale d'internautes (2000 individus) suite au vote de cette loi : Hadopi a su Connectivité Accès aux réseaux sociaux
	On voit ainsi s'opposer sur le réseau les consommateurs d'une part, et les dissuader certains « pirates » de télécharger sur les réseaux peer-to-peer, Accès aux communautés institutions (les pouvoirs publics entre autres) dans une sorte de bataille dans le même temps « la réduction du nombre d'internautes qui utilisent
	opposant les tenants de la gratuité à ceux pour qui l'accès aux contenus les réseaux peer-to-peer s'est [...] accompagnée d'une hausse des autres Accès aux experts culturels a un prix (l'argument phare étant la rémunération des artistes). formes de piratage non prises en compte par la loi Hadopi (+27%). Cet
	Légale Or certains producteurs et artistes ont su prendre en compte et s'adapter Protection Accès à des sources accroissement fait plus que compenser la diminution du nombre
	gouvernementales aux spécificités du web (on pensera au free culture movement d'utilisateurs des réseaux Peer-to-Peer. Le bilan de la loi Hadopi, 3 mois
	Accès aux associations de notamment, tout un pan de la création contemporaine est ainsi sous à peine après son adoption, fait ressortir une légère augmentation du licences libres). La fermeture du site Megaupload 6 , le 19 janvier dernier nombre de pirates sur Internet (+3%) ».
	consommateurs notamment a entraîné des réactions assez vives de la part de groupes
	Accès aux réseaux activistes agissant, par exemple sous la bannière Anonymous 7 qui ont alors lancé
	La possibilité d'accéder à une grande variété d'informations renforce le
	pouvoir des consommateurs, il accroît la transparence, permet une
	communication horizontale (many-to-many) (Barrutia et Echebarria,
	2004). Internet apparaît comme un instrument d'expression puissant, le
	récent terme « cyberactivisme » est d'ailleurs un indicateur de la

Peer to Peer (Pair à Pair/P2P) : ce système permet à plusieurs ordinateurs de communiquer via un réseau, de partager des objets (fichiers), mais également des flux multimédias continus (streaming) sur internet.

Site web dont le contenu peut être modifié par tous. Les wikis permettent entre autre de communiquer et de diffuser rapidement des informations.

l'Agapè, est une notion fondée sur le don, mais ne reposant pas sur la notion de réciprocité, elle est indépendante du désir et ne se pose pas la question de la valeur de l'objet auquel elle s'adresse, elle est indifférente au mérite, et gratuite : « en effet à la différence de la philia, fondée [...] sur la notion de réciprocité, qui constitue l'un des concepts fondamentaux des sciences sociales modernes, et plus particulièrement de l'anthropologie , l'agapè, définie par le don , n'attend pas de retour, ni sous la forme d'objets , ni même sous l'espèce immatérielle d'amour en retour ».[START_REF] Boltanski | L'amour et la justice comme compétences : trois essais de sociologie de l'action[END_REF]

Site d'échange de fichiers musicaux en P2P jusqu'en 2001

http://obsession.nouvelobs.com/high-tech/20120210.OBS1061/de-nombreusesmanifestations-anti-acta-prevues-ce-samedi.html

 6 Megaupload est un site web, fermé en 2012, qui proposait un service d'hébergement de fichiers en un clic. Il permettait aux internautes de mettre en ligne n'importe quel type de fichier, pour les autres

internautes.7 Anonymous est un groupe d'hacktivistes, regroupant des membres de communautés d'internautes agissant de manière anonyme dans un but particulier, ils se présentent comme des défenseurs absolus de la liberté d'expression. Leur devise: We are Anonymous. We are Legion. We do not forgive. We do not forget. Expect us.

Hadopi : Haute Autorité pour la diffusion des oeuvres et la protection des droits sur Internet, cette loi sanctionne entre autres le téléchargement en Peer-to-Peer

http://www.lemonde.fr/web/recherche_breve/1,13-0,37-1180773,0.html

http://www.numerama.com/magazine/21393-le-p2p-explose-apres-la-fermeture-demegaupload.html


CAN": ILLEGAL DOWNLOADING IN FRANCE SINCE 2009

Keywords: Illegal downloading, illegal practices, market system

How are some illegal practices sustained? Although prior research has been focusing on the domestication or integration of illegal practices by the market, little attention has been paid to their sustainability. We propose to study, through the case of illegal downloading in France since 2009, 1° How consumers get organized after the enactment of a new regulation 2° How these practices could form a parallel system of supply and demand chain. The main results show that consumers pay for illegal content and to be "protected". Moreover, downloaders are certain that even their favorite platform shut down they will be able to find alternatives. This research's intended contribution to the literature on illegal practices as well as market system dynamics will be discussed.

Introduction

"You can make it illegal, but you can't make it unpopular" Martin Behrman Why are some practices easily regulated and others not? In 2008, in France, smoking in shared spaces (cafes, restaurants, stations) was made illegal. This new regulation has been quickly integrated 1 and is today widely respected. In contrast, although downloading has been considered an illegal practice since 2009 2 , 49% of the French population is consuming through this way 3 . The French legal apparatus appears to be inefficient to reduce these behaviors, even in providing a discourse on consumer's morality 4 or in punishing the downloaders.

Illegal behavior, illicit practices and deviant forms of consumption [START_REF] Robertson | Illegal Downloading, Ethical Concern, and Illegal Behavior[END_REF][START_REF] Moschis | Deviant Consumer Behavior[END_REF][START_REF] Fullerton | What is Consumer Misbehavior?[END_REF][START_REF] Rose | An Attributional Analysis of Resistance to Group Pressure Regarding Illicit Drug and Alcohol Consumption[END_REF], to quote some examples, have been widely studied in consumer research from a consumer-centric analytical mode. Prior research has also been focusing on the domestication or integration [START_REF] Goulding | The Marketplace Management of Illicit Pleasure[END_REF] by the market of illegal practices. Some studies not directly addressing illegal practices suggest that the status of a practice could change through market co-optation [START_REF] Bengtsson | Prisoners in Paradise: Subcultural Resistance to the Marketization of Tattooing, Consumption[END_REF][START_REF] Kozinets | Utopian Enterprise: Articulating the Meaning of Star Trek's Culture of Consumption[END_REF][START_REF] Schouten | Subcultures of Consumption: An Ethnography of the New Bikers[END_REF][START_REF] Thompson | Countervailing Market Responses to Corporate Co-optation and the Ideological Recruitment of Consumption Communities[END_REF]. A marginal practice can become popular when the market appropriates and commercializes it (Sandicki et Ger 2010). In a similar way, Humphrey (2010a) in her paper dedicated to the creation and sustainability of a new industry shows how a practice historically associated with crime may become legitimate under the influence of multiple stakeholders. "The author finds that frames such as crime, business, and regulation change over time and that these frames are used by multiple stakeholders to structure normative conceptions about the practice of casino gambling "(p.1).

But what about uncontained / undomesticated practices? Literature remains scarce on the particular networks in which these practices are embedded, allowing their existence. Our goal in this research is to question the processes through which illegal practices could be sustained. Specifically we explore two key questions: How do consumers get organized after a law enforcement prohibiting downloading? How could these practices form a parallel set of supply-demand relations?

Illegality and market systems

Market systems, a pretty much recent body of research [START_REF] Press | Legitimating community supported agriculture through American pastoralist ideology[END_REF][START_REF] Thompson | Countervailing Market Responses to Corporate Co-optation and the Ideological Recruitment of Consumption Communities[END_REF]Giesler 2008Giesler , 2012;;Humphreys 2010aHumphreys , 2010b;;[START_REF] Karababa | Early Modern Ottoman Coffeehouse Culture and the Formation of the Consumer Subject[END_REF] explores the creation, emergence, and dynamics of markets: "This (stream of) research places markets as social systems at the center of their analysis and examines how they are created and/or change through the discursive negotiations and the practices of various market stakeholders" (Siebert and Thyroff 2012, p.1). For instance, [START_REF] Karababa | Early Modern Ottoman Coffeehouse Culture and the Formation of the Consumer Subject[END_REF] portray the construction of coffee house culture and an active consumer subject in the early modern Ottoman Age through the struggles between religious morality and pursuit of pleasure. However, little attention has been paid to illegal markets' dynamics, despite their importance in terms of diversity (illegal markets cover a wide range of consumption fields from prostitution to drugs to media piracy) and economy (world illegal market value represents $1,810,052.83 Million)5 . [START_REF] Goulding | The Marketplace Management of Illicit Pleasure[END_REF] in their study on illicit pleasure are paving the way, depicting the processes of regulation of the rave phenomenon6 and take into account some actors like the club promoters, the State Authorities and the police. They show how clubbing could appear as the "marketization" of rave, supporting an illegality that is easier to control and manage and confined to permanent spaces.

Clubbing is a form of containment of illicit practices with the complicit knowing wink of the police and state authorities. The authors suggest that "an effect of this marketization process has been the subsequent reemergence of rave"(p.769), interpreted as a resistance and reject of the clubbing culture.

However, is resistance [START_REF] Penaloza | Consumer resistance: a conceptual overview[END_REF][START_REF] Roux | Consumer resistance: proposal for an integrative framework[END_REF] the "sine qua none" condition to the resurgence of such practices? Most of the studies focusing on the key role of consumers in market evolution "conceptualize consumers role as some form of rebellion or resistance to prevailing market logics or market-induced stigma" (Schouten and Martin, 2013, p.855). Giesler (2008), in his study based on anthropology of performance [START_REF] Turner | The Anthropology of Performance[END_REF], characterizes four steps of drama in marketplace evolution: breach, crisis, redress, and reintegration through the historical analysis of the war on music downloading. He concludes that the downloaders are reintegrated over a consensus. If downloaders' narratives are exposed, less attention is paid to their practices. Is the reintegration implying a disappearance of the illicit downloading phenomena? Giesler (2008, p.749) notes nevertheless that « the music market seems likely to be pushed into another cycle of discord and conflict ». However, this analysis of discord and conflict cycle does not say much about consumers' practices and their evolution.

Downloading illegally

Downloading has been regulated in many countries [START_REF] Bishop | Building International Empires of Sound: Concentrations of Power and Property in the[END_REF][START_REF] Easley | Ethical Issues in the Music Industry, Response to Innovation and Piracy[END_REF][START_REF] Lessig | Free Culture: The Nature and Future of Creativity[END_REF]). "But despite the attempts to encourage consumers in countries including the United Kingdom, France and United States to download music from legal sources, many not only continue to download from unauthorized sources but reject the very idea that what they are doing is wrong" (Cluley 2013, p.263).

Why are these practices persisting? Literature, even not addressing directly the sustainability of these practices, brings some answers. For some, the practices of illegal downloading are anchored in consumers' behaviors [START_REF] Proulx | Internet et la culture de la gratuité[END_REF]. It could also be explained by the difficulty to normalize consumers' behaviors on the web. Denegry-knott (2006) suggests that the balance of power on the Internet is likely to shift from producers to consumers. The architecture of the Internet allows that power be exercised by the individuals: « the industrial revolution was to manufacturers what the digital revolution is to consumers » (Kucuk et Krishnamurthy, 2007, p.47).

Literature has also been conceptualizing the war on downloading as a quest for legitimacy (Denegry- [START_REF] Denegri-Knott | Sinking the Online "Music Pirates:" Foucault, Power and Deviance on the Web[END_REF][START_REF] Denegri-Knott | Consumers behaving badly: deviation or innovation? Power struggles on the web[END_REF]Giesler, 2008;[START_REF] Cluley | Downloading deviance: Symbolic interactionism and unauthorised file-sharing[END_REF]. Illegal downloading appears as an illustration of power struggles on the web, a battle over a discourse determining what the norm is and what a deviance is through the processes of labeling the downloaders (Denegri- [START_REF] Denegri-Knott | Sinking the Online "Music Pirates:" Foucault, Power and Deviance on the Web[END_REF]2006). However, in this battle, some of the actors, like the cultural industries, can suffer from a lack of legitimacy. [START_REF] Cluley | Downloading deviance: Symbolic interactionism and unauthorised file-sharing[END_REF] suggests that proponents and opponents to downloading are following the same International Conference on ICT in Organizations, Paris 2015 economic imperative (e.g. creators should be rewarded), and "are locked in a battle over who can define behavioural standards for consuming music"(p.263).

The cultural industries' resistance is also invoked, and could be understood as a cause of the sustainability of the illegal downloading. Garcia- [START_REF] Garcia-Bardidia | La consommation illégale de musique numérique : y résister ou se l'approprier ?[END_REF] show how the cultural industries have been reacting to the downloading phenomenon: they tried to "re-moralize" the consumers (e.g. Hadopi Law in France), to "re-materialize cultural goods" (e.g. Vinyls) and to "reintermediate" in the market (creation of distribution platforms e.g. Spotify). Defending their position through discourse, the cultural industries are trying to reproduce the existing order (Blanc and Huault, 2012). This static state is a key enabling a better comprehension of the sustainability of the downloading practices. Indeed, Garcia-Bardidia et al., (2011) suggest that downloaders are resisting to the labelization game, adopting other consumption alternatives. The coercive approaches of the downloading phenomena (e.g. Hadopi law) are susceptible to reinforce the position of the downloaders, pushing them from a mainstream position to an activist position (Garcia- [START_REF] Garcia-Bardidia | Consumer resistance and anticonsumption: Insights from the deviant careers of French illegal downloaders[END_REF].

Studies paying attention to consumers' discourses regarding their practices of illegal downloading are likely to bring some answers too. Indeed, downloaders are mobilizing neutralization techniques [START_REF] Sykes | Techniques of neutralization: a theory of delinquency[END_REF] to manage their moral conflict [START_REF] Odou | Téléchargement illégal, déviance et gestion individuelle du conflit moral: l'apport de la théorie de la neutralisation[END_REF][START_REF] Divard | Les comportements volontairement déviants en termes de paiement[END_REF][START_REF] Garcia-Bardidia | Consumer resistance and anticonsumption: Insights from the deviant careers of French illegal downloaders[END_REF]). These strategies are defensive and imply 1) a responsibility neutralization (transfer, dilution and victimization), 2) a pragmatic neutralization (euphemization, retrospective compensation and ethical re-appropriation) and 3) an ideological neutralization (claim, stigmatization) [START_REF] Odou | Téléchargement illégal, déviance et gestion individuelle du conflit moral: l'apport de la théorie de la neutralisation[END_REF]. Finally, practices of illegal downloading, even condemned by the law, seem stimulated by the liberty and the absence of price [START_REF] Divard | Les comportements volontairement déviants en termes de paiement[END_REF]. However, this consumption is also implying other forms of alienation and constraints (addiction phenomenon). The downloader is a free agent, experiencing the torments of freedom (Garcia- [START_REF] Renaud | Consommer sans limites ou les affres de la liberté: le téléchargement de produits culturels[END_REF].

Research method

Although research has focused on why are some illegal practices sustained, literature remains scarce on how these practices are sustained. Our main goal in this research is to understand how the consumers have been dealing with law enforcement and how a parallel system of supply and demand chain have emerged.

We focus our research on the case of illegal downloading in France since the enactment of the law (2009). According to [START_REF] Yin | Case study research: Design and methods[END_REF] a case study design should be considered when: (a) the focus of the study is to answer "how" and "why" questions; (b) you cannot manipulate the behavior of those involved in the study; and (c) you want to cover contextual conditions because you believe they are relevant to the phenomenon under study [START_REF] Baxter | Qualitative Case Study Methodology: Study design and implementation for novice researchers[END_REF].

Data collection

We conducted nineteen in-depth interviews [START_REF] Mccracken | The long interview[END_REF] with informants involved in illegal downloading. Our respondents varied in terms of age (16-51), family status, education and occupation. The consumers' interviews approximately added up to 350 pages of single-spaced text. Interviews were conducted in private homes and, in selected few cases, in cafes and by Skype. Participants were informed that the interview pertained to « a study about illegal downloading » and were insured that their identity would remain confidential. The interviews were structured as follows: 1) Story of the practice of illegal downloading 2) Perception and experience of the practice (illegal downloading) 3) International Conference on ICT in Organizations, Paris 2015

Perception of the Hadopi law. The duration varied between 35-minute length to 270-minute. Six other in-depth interviews were conducted with some of our informants after a year and a half (selected for several reasons: some have been "caught" by the Hadopi Law, some are "followers" and needed to be informed by others downloaders, some are skillful downloaders having fall back solutions), to dig on the topic " How was it to download before 2009?" These interviews were conducted to understand if a shift (in terms of practices) really happened between a fuzzy period and a period of prohibition. Consumers' interviews were entirely transcribed and coded. A content analysis was done to allow the emergence of the categories (Strauss and Corbin, 2004).

Results

A constant adaptation to the imposed constraints

Illegal downloading or l'Hydra of Lerna

Respondents evoke the phenomenon of the resurgence of platforms. A lot of them have been confronted to the shutdown of their favourite platforms (as Megaupload, for example), but they are certain to find alternatives: "The day this website will shutdown, we will find some others. 15 000 will appear, so, we can't stop the downloading phenomena or you have to stop the entire Internet!!!"(Sam, 24-year-old, graduate student).

The downloaders that we qualify as amateurs (in the sense that they are experiencing a dependence to the others to "be furnished" in cultural contents and "tip-off" (Maxime, 25-year-old, manager) are convinced and have the assurance that they will find solutions as Fred: "There must be individuals who are aware of others websites where I will be able to continue to download" (Fred, 21-year-old, undergraduate student).

" Catch me if you can "

It seems that the respondents are not afraid by being detected in their practices by the Hadopi Law. Thus, the "repressive" law (Caro, 51-year-old, accountant) seems to be obsolete for them: " It was easy to be caught on Emule, Hadopi just came when the torrent technology was already used…Hadopi was tracking on Emule…(laughs), we were already having largely a step ahead" (Raphaël, 41-yearold, psychologist).

Consumers are adapting theirs practices to continue to download. For example, Clemence (28-yearold, nurse) has been "caught": she has received a second warning from Hadopi. This warning just slowed down her practice for several months. She decided after this period to 1) get some information from more experienced downloaders and 2) to equip herself to continue to download: "I'am actually installing a VPN software to protect my laptop, to have a secured connection, which makes my IP address invisible. When I will have find self-confidence again, I will download again and be proud of myself. This software is installed since a while, I just have to re-install my downloading software and I will continue to download on T411".

Finally, consumers are using alternative forms of consumption as streaming, which is not punished by the law: "Hadopi...it was not positive because I just go back to streaming"(Lola, 25-year-old, without employment).

5.2

Constitution of a parallel system of supply-demand chain

Paying for illegal content

International Conference on ICT in Organizations, Paris 2015

Our respondents evoked the fact to pay for illegal contents. This type of solution seems easier for a vast majority of our respondents: "In fact, I was thinking that when you want to see movies, it's fragmented in different parts and after, when you unzip, your file could not be entirely read, I was thinking that I have to read 20 min per 20 min, so I just subscribe a month ago, for the first time to Rapidshare. And now I have a Rapidshare account, I'm paying 10 euros per month and I have everything, and its gooood!!"(Léo, 24-year-old, undergraduate student). These platforms, as Rapishare, are declined in free versions (more restrictive in terms of access) and paying versions. They answer to consumers needs in terms of facility and variety of contents, but they are not getting the "battle-hardened" downloaders: "I don't want to pay for something that is illegal…"(Romain, 21 years old, undergraduate student) or: «I don't think it's right, most of the time if you don't know internet, you have to pay" (Léandre, 16-year-old, waiter).

Paying to be protected

Some services allow the consumers to protect their connections: "And even if I'm caught by the Hadopi law, some solutions exist, paying solutions but…you have Pirate bay, who created a service, IPredator which allows you to…You have to pay 5 Swedish krona per month, and they provide you a VPN, after you can also take your neighbor connection…but it's more risky…"(Sam, 24-year-old, graduate student).

The results show that downloaders are mobilizing some adaptation strategies to the legislative constraints (as protecting theirs connections, going to others types of illegal consumption). It seems interesting to be aware that consumers are ready to pay to continue to download illegally.

On the influence of non-state actors

Downloaders evolve in an illegal ecosystem. This ecosystem, far from being constructed on the notions of gift and sharing, is a "business" (Caro,accountant). Individuals are conscious regarding the model they are nourishing as Sam, 24-year-old, graduate student: "I think that the business owner of Emule was earning 600 000 Euros per year (…), he was driving a Porsche".

However these businessmen seem legitimate for the downloaders: "He was doing a lot of money (…) but maybe he will be the leader of a certain progress. Even he doesn't care about network neutrality" (Jules, 24-year-old, informatician) Some of the respondents describe this ecosystem as a market, an illegal market:

-Respondent: "The day after the shutdown of Megaupload, you had… it's like drug traffic, one is caught, five are appearing. They are taking back the market". (Sam, 24-year-old, graduate student).

DISCUSSION

Our main goal is to bring a comprehension of the sustainability of illegal practices. We chose the context of illegal downloading for several reasons: these practices are mainstream (Hadopi, in a recent study, shows that a half of respondents are consuming cultural goods on the internet through illegal ways), and despite the regulation, in France, these practices are still enduring. Another point retains our attention: contrary to the findings of other scholars [START_REF] Denegri-Knott | Sinking the Online "Music Pirates:" Foucault, Power and Deviance on the Web[END_REF][START_REF] Giesler | Consumer Gift System[END_REF]Giesler, , 2008)), downloading in France does not seem to put in evidence a subculture founded on a libertarian ethos [START_REF] Odou | Téléchargement illégal, déviance et gestion individuelle du conflit moral: l'apport de la théorie de la neutralisation[END_REF]. This phenomenon of subculture could be a reason for the sustainability of such practices, but it does not seem that, in France, downloading illegally is fed by such a subculture [START_REF] Divard | Les comportements volontairement déviants en termes de paiement[END_REF]. International Conference on ICT in Organizations, Paris 2015

Our research has resonances with the works of [START_REF] Garcia-Bardidia | Consumer resistance and anticonsumption: Insights from the deviant careers of French illegal downloaders[END_REF], 2012), showing that these repressive measures are pushing the downloaders to alternatives forms of consumption.

Contributions

As it is a work in progress, we envision several contributions to the ongoing marketing debate by bringing a critical perspective on:

The sustainability of illegality -Market systems have been studied so far in terms of changes, ruptures, creation, innovation, emergence-. Giesler (2008) in his study on downloading conceptualizes markets' dynamics in the cultural domain as characterized by four phases: breach, crisis, rupture and reintegration. Reintegration of the downloaders on the one hand, and adaptation of the offer on the other hand. Our research completes this model, showing that: 1) Downloaders have not been reintegrated 2) a parallel market has been constituted, embedded in the legal market and centered on the distribution of contents.

Our research also contributes to the literature on illicit practices. [START_REF] Goulding | The Marketplace Management of Illicit Pleasure[END_REF] show how illegal practices are contained (in terms of space and time) with the complicit knowing wink of the authorities. This approach helps us to understand the processes of domestication of illicit practices but say not much about the resurgence of such practices. Finally, we will contribute to the notion of legitimacy, and more particularly on the sources of legitimacy and on how the very presence of an illicit market can erode the belief that certain illegal activities are illegitimate.

Overall, those possibilities represent an attempt to shed light on illegality as part of market dynamics, which we believe, is being under-accounted by extant literature so far.

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A rave is a large party, or festival illegally organized featuring performances by DJs and occasionally live performers playing electronic music