Fabrice Lebeau
email: fabrice.lebeau@ens-lyon.fr

Corinne Touati

Eitan Altman

Nof Abuzainab
email: nof.abuzainab@inria.fr

The Social Medium Selection Game

We consider in this paper competition of content creators in routing their content through various media. The routing decisions may correspond to the selection of a social network (e.g. twitter versus facebook or linkedin) or of a group within a given social network. The utility for a player to send its content to some medium is given as the difference between the dissemination utility at this medium and some transmission cost. We model this game as a congestion game and compute the pure potential of the game. In contrast to the continuous case, we show that there may be various equilibria. We show that the potential is M-concave which allows us to characterize the equilibria and to propose an algorithm for computing it. We then give a learning mechanism which allow us to give an efficient algorithm to determine an equilibrium. We finally determine the asymptotic form of the equilibrium and discuss the implications on the social medium selection problem.

I. INTRODUCTION

Social networks involve many actors who compete over many resources. This gives rise to competitions at different levels which need to be taken into account in order to explain and predict the system behavior. In this paper, we focus on competition of individual content creators over media. A content creator has to decide which one of several media to use. The media choice may correspond to a social network that will be used for sending (and disseminating) some content. For instance, the decision can consist in choosing between twitter and facebook, or in deciding to which of several facebook groups to send the content.

The game we study in this paper is atomic and nonsplitable. We consider a decision maker (or a player) to be a single content instead. This regime can well approximate decision making where a content creator, say a blogger, occasionally sends content. Here, occasionally implies that the time intervals between generation of consecutive contents by the blogger is large enough so that the states of the system at the different times of creation of content are independent one from another. This regime is interesting not only because it is characteristic of systems with many sources of contents, but also and foremost, it turns out that it precisely characterizes bloggers that have more popularity and influence. This was established experimentally in [START_REF] May | Game in the newsroom: Greedy bloggers for picky audience[END_REF] which analyses the role of intermediate actors in dissemination of content.

A similar game as the one in this paper was already studied in [START_REF] Altman | A semi-dynamic model for competition over popularity and over advertisement space in social networks[END_REF] and [START_REF] Reiffers Masson | A time and space routing game model applied to visibility competition on online social networks[END_REF], but there the players control rates of creation of contents and/or decide how to split the rates. The resulting games are simpler than ours as they possess a single equilibrium. The model studied in this paper brings many novelties both in the system behavior as well as in the tools used to study it. The difficulty in studying the game in our atomic non-splitable game framework is due to the integrity constraint on the players as they cannot split their content between several media. This implies that the action space is discrete and thus non convex which may result in problems in the existence and/or uniqueness of the equilibrium.

Related work: Game theoretic models for competition have been proposed in a growing number of references. The authors of [START_REF] Hegde | Self-organizing flows in social networks[END_REF] focus on the competition over budget of attention of content consumers and the impact of this competition on the dynamic popularity of the content. A game model related to intermediate actors that participate in the spreading of news is considered in [START_REF] May | Game in the newsroom: Greedy bloggers for picky audience[END_REF]. The authors study how to choose the type and amount of content to send so as to be influential. In [START_REF] Lotker | A game of timing and visibility[END_REF], the authors study competition over space among content creators. The space may represent a slot (say the top one) in a timeline, and a content that arrives occupies the space pushing out the one that is already there. It then stays visible there till the next arrival of content that pushes it away. The authors study the timing game: when should a content be sent to the timeline so as to maximize the expected time it remain visible. The authors of [START_REF] Maggi | Cooperative online native advertisement: A game theoretical scheme leveraging on popularity dynamics[END_REF] study a dynamic competition model over visibility in which the rate at which creator of contents send their traffic is controlled.

Game theory has been used not only to model competition in social networks but also to design algorithms for the analysis of social networks [START_REF] Yadati | Game theoretic models for social network analysis[END_REF]: this includes community detection [START_REF] Narayanam | A game theory inspired, decentralized, local information based algorithm for community detection in social graphs[END_REF], discovery of influential nodes [START_REF]A Shapley value-based approach to discover influential nodes in social networks[END_REF] and more.

Contributions of our paper: Our first contribution is to make the observation that the game complies with the definition of congestion game introduced by Rosenthal [START_REF] Rosenthal | A class of games possessing pure-strategy Nash equilibria[END_REF]. This allows us to show that our game is a potential game, for which Nash equilibria exist. This further implies that algorithms based on best response converge to an equilibrium. We show that surprisingly, although the potential approximates a strictly concave function (in the continuous space), there may exist many equilibria. This is quite a new phenomenon in networking games, and it is due to the non convexity of the action space (due to the non splitable assumption). In order to have uniqueness, a new concept of integer concavity have to be used. We rely on the theory of M-concavity [START_REF] Murota | Discrete convex analysis[END_REF] which allows us to establish the structure of the set of equilibria for this problem. We propose a learning algorithm that converges to an equilibrium and is more efficient than the best response algorithm. We finally study the asymptotic behavior of the system as the number of players grows, both in terms of characterization of the equilibria and in price of anarchy.

II. MODEL AND NOTATIONS

We consider a set K = {1, ..., K} of seeds (content producers, bloggers, etc) that aim to publish their content in social media. We focus on the problem where each seed needs to publish in some social medium j ∈ J = {1, ..., J}. The strategy s k of seed k is the social medium it selects for disseminating its content. Define the load ℓ j on social medium j as the number of competing seeds that send their content to medium j. It can be written as ℓ j = k δ(s k , j) (δ is the Kronecker symbol, i.e. δ(a, b) equals 1 if a = b and 0 otherwise). Assume that social medium j has N j > 0 subscribers who are interested in content shipped to that medium.

The utility of a player (seed) is given as the difference between a dissemination utility and a dissemination cost. The former, that is, the value of disseminating a content at the jth social network is (i) proportional to N j and (ii) inversely proportional to ℓ j . Further, each seed pays a constant dissemination cost γ j for publishing on social medium j.

This structure of utility is very common. In the networking community, we find it in resource sharing of link capacity for flow control problems. It is also associated with the so-called Kelly mechanism (see [START_REF] Ma | On resource management for cloud users: A generalized Kelly mechanism approach[END_REF] for a similar utility in cloud computing), and models tracing back to the Tullock rent-seeking problem [START_REF] Tullock | Efficient rent-seeking[END_REF]. In the social medium context, this utility naturally arises if different seeds create similar content (say news) and thus a subscriber is not interested in receiving more than one content. This implies the structure of point (ii). The use of this type of utility in competition for resources in social networks can be found in [START_REF] Altman | A semi-dynamic model for competition over popularity and over advertisement space in social networks[END_REF].

Hence, the utility of seed k is given by

u k ( s) = N s k ℓ s k -γ s k , (1)
where s is the vector of strategies of the seeds s = (s k ) k∈K For notational convenience, in the following we will denote by S = J K the set of strategy profiles, and by Γ = (K, (N j , γ j ) j∈J ) the game setting with K players (the seeds), the set J of social media of parameters ((N j , γ j ) j∈J ) and the utility (u k ) k∈K defined in Eq. [START_REF] May | Game in the newsroom: Greedy bloggers for picky audience[END_REF].

Since the utility of each player (the seeds) only depends on the number of users choosing the same action (i.e. the same social media), the game is equivalent to a congestion game in the sense of Rosenthal [START_REF] Rosenthal | A class of games possessing pure-strategy Nash equilibria[END_REF], where the resources are the social media. It therefore is a potential game [START_REF] Monderer | Potential Games[END_REF], that is to say that there exists a function Pot such that:

∀k ∈ K, ∀ s ∈ S, ∀s k , u k (s 1 , ..., s k-1 , s ′ k , s k+1 , ..., s K ) -u k ( s) = Pot(s 1 , ..., s k-1 , s ′ k , s k+1 , ..., s K ) -Pot( s).

Let us introduce the Harmonic number:

H n = n j=1 1/j if n ≥ 1 and H n = 0 if n ≤ 0.
Then, one can readily check that a suitable potential of the game is:

Pot( s) = j N j H ℓj -γ j ℓ j .
Let ℓ be the vector of loads induced by s and D the set of possible vector loads: D = (ℓ 1 , ..., ℓ J ) ∈ N J | J j=1 ℓ j = K . By using the equivalence with the congestion game of [START_REF] Rosenthal | A class of games possessing pure-strategy Nash equilibria[END_REF] we get the potential as a function of the loads ℓ ∈ D of the social media:

Pot( ℓ) = j N j H ℓj -γ j ℓ j .
Let us finally introduce the following notations that will come in handy in the rest of the paper:

• For x ∈ R n , supp + ( x) = {j | x j > 0} , • . | . is the euclidian scalar product over R J : x | y = j x j y j , • . ∞ is the uniform norm: x ∞ = max j |x j |,
• ( e j ) j is the Euclidean base of R J .

III. DISCRETE POTENTIAL ANALYSIS

A. Nash equilibria

Potential games have received a lot of attention in the past years as they draw a natural bridge between the theory of games and optimization. Indeed, the definition of a potential implies the following. A strategy profile s is a Nash equilibrium iff it is a maximizer of the potential, that is,

∀ s, ∀k ∈ K ∀s ′ k ∈ J, Pot(s 1 , ..., s k-1 , s ′ k , s k+1 , ..., s K ) ≤ Pot( s). (2)
Note that a change of strategy of a seeds from social media i to social media j amount in reducing the load ℓ i of network i and increase that of j by one unit. Hence, in the space of load vectors, Equation (2) becomes:

∀i, j : ℓ j > 0 ⇒ Pot( ℓ + e i -e j ) ≤ Pot( ℓ). (3)
For a given load vector ℓ, let V( ℓ) be the set of possible load vectors obtained after the deviation of a single player:

V( ℓ) = ℓ + e i -e j | i, j ∈ J st ℓ j > 0 .
Then, the Nash equilibria are all the strategy profiles s for which the vector of loads is a local (in the sense of V) maximum of Pot : D → R.

B. M-concavity

The potential is defined over a discrete set, and therefore classical convexity properties do not hold. In order to understand the structural and uniqueness properties of the Nash equilibrium, we study the properties of the potential function in terms of M-concavity 1 .

Definition 1. A function f : Z J → R is M-concave if for all x, y in D and for all u ∈ supp + ( x -y): ∃v ∈ supp + ( y -x), f ( x) + f ( y) ≤ f ( x -e u + e v ) + f ( y -e v + e u ).
We have the fundamental property: Theorem 2. The function f : Z J → R defined by:

f ( x) = Pot( x) if x ∈ D, -∞ otherwise is M-concave. Proof. Let x, y ∈ D. First, if supp + ( x-y) = ∅,
then the property is trivially true. Otherwise, assume that there is some i in supp + ( xy). If we had supp + ( yx) = ∅ then we would also have

j y j = y i + j =i y j < x i + j =i x j = j x j = K.
This is absurd since y ∈ D. Hence supp + ( yx) = ∅. 1 For more information about M-convexity, see [11, sec. 4.2].

Then, let u ∈ supp + ( xy) and v ∈ supp + ( yx). Since x u > y u ≥ 0 and y v > x v ≥ 0 we have that xe u + e v and ye v + e u are in D. Then:

f ( x -e u + e v ) + f ( y -e v + e u ) -f ( x) -f ( y) = N v 1 x v + 1 - 1 y v ≥0 +N u 1 y u + 1 - 1 x u ≥0 .
Note that since we did not choose a particular v in the preceding proof, we have actually shown a much stronger property, that is, that the inequality holds for any v, which is decisive for the rest of the analysis:

∀ x, y ∈ D, ∀u ∈ supp + ( x -y), ∀ ∀ ∀v ∈ supp + ( y -x), f ( x) + f ( y) ≤ f ( x -e u + e v ) + f ( y -e v + e u ). (4)

C. Properties of the Nash equilibria

In this section, we show properties of the Nash equilibria of this game using the M-concavity of f . Theorem 3. Let s ∈ S and let ℓ be the vector of loads of s. Then: s is a Nash equilibrium for the game ⇔ ℓ maximizes globally the potential over D.

Proof. The sufficient condition is a direct consequence of Eq. (3). Conversely, assume that s is a Nash equilibrium. Then, by (3), we know that ℓ is a local maximum of Pot (over V( ℓ)). Let u, v ∈ J:

If ℓ-e u + e v / ∈ D then f ( ℓ) > f ( ℓ-e u + e v ).
Otherwise, we have ℓe u + e v ∈ V( ℓ). Hence ℓ satisfies the property:

∀u, v ∈ J, f ( ℓ) ≥ f ( ℓ -e u + e v ).
We then apply [START_REF] Murota | Discrete convex analysis[END_REF]Thm 4.6] with the M-concave function f on ℓ, given that ℓ is a global maximum of f . Therefore ℓ is a global maximum of Pot on D.

We show next that the set of loads corresponding to the different Nash equilibria of the game, are all neighbors of each other: Theorem 4. Let E Γ be the set of the loads of the Nash equilibria of the game. Then:

∀ x, y ∈ E Γ , x -y = u∈supp + ( x-y) e u - v∈supp + ( y-x) e v .
In other words, all Nash equilibria x, y ∈ E Γ satisfy xy ∞ ≤ 1.

Proof. First, if E Γ = { x}, then the theorem is trivially true. Otherwise, assume that there exists x, y ∈ E Γ such that xy ∞ > 0. Since x and y are in D, we can write xy as xy = e ue v + z for some u, v ∈ J and z ∈ Z J satisfying u = v, z | e u ≥ 0 and z | e v ≤ 0. Therefore, we have u ∈ supp + ( xy) and v ∈ supp + ( yx).

Let a = xe u + e v and b = ye v + e u . By Eq. ( 4) we have:

f ( x) + f ( y) = 2f ( x) ≤ f ( a) + f ( b). Hence we get f ( a) = f ( b) = f ( x) by global maximality of f ( x). Then f ( x) -f ( a) + f ( y) -f ( b) = 0, which in turns implies that N u x u + N v y v = N u y u + 1 + N v x v + 1 .
Since x u ≥ y u + 1 and y v ≥ x v + 1, the last equation implies that x u = y u + 1 and x v = y v -1. Therefore:

x -y = u∈supp + ( x-y) e u - v∈supp + ( y-x) e v .
Using this result, we can find a bound over the number of Nash equilibria: Proposition 5. For any setting Γ, the number of Nash equilibria is upper bounded. More precisely, let E Γ be the set of the loads of the Nash equilibria of Γ. Then:

|E Γ | ≤ J J 2 .
Further, this bound is tight. Indeed, let J ≥ 2, m ∈ N * and γ ∈ R + . We define the game Γ by K = J 2 and ∀j ∈ J, N j = m, γ j = γ.

Then |E Γ | = J ⌊ J 2 ⌋ . The proof of Proposition 5 is given in Appendix A. Note that the bound is in O 2 J √ J
and that it is independent of the number of seeds K. As the number of social media is typically small, then there is a limited number of equilibria.

IV. ALGORITHMIC DETERMINATION OF AN

EQUILIBRIUM

In this section we see how to compute a Nash equilibrium. Note that, from Theorem 5, computing all the loads of the Nash equilibria would require Ω 2 J √ J operations. Further, for each load vector ℓ maximizing Pot, computing all the corresponding Nash equilibria s ∈ S would require up to O(K!) operations because of the symmetry of the game.

A. Maximization of the potential

Consider the following optimization mechanism:

Step 1) Start with some ℓ ∈ D

Step 2) Find ℓ * the argmax of Pot on V( ℓ)

Step 3) If ℓ * = ℓ then stop

Step 4) Let ℓ = ℓ * and repeat from step 2) Thanks to Theorem 3, we know that this mechanism converges to a vector of loads of a Nash equilibrium. From the point of view of the seeds, it is similar to a guided best-response mechanism where at each step the seed which could increase the most the potential by changing its strategy is selected.

The problem is that, in the worst cases, this algorithm visits all the load vectors of the domain D, which leads to O K J steps to find a maximum. However, we can exploit the M-concavity of function f to compute a Nash equilibrium in a far more efficient way. To that end, we adapt the algorithm MODI-FIED STEEPEST DESCENT given in [15, p.8] to our problem, which is presented in Algorithm 1 below.

Algorithm 1: SD MAX

Input: Γ = (K, (N j , γ j ) j∈J ) Proof. We implemented the active domain B of the algorithm used in [15, p.8] by a vector b satisfying:

Output: A vector in E Γ 1 Let ℓ = K e 1 and b = 0 ∈ Z J 2 while ∃u, ℓ u -1 ≥ b u do 3 Compute v ∈ arg max t∈J Nt ℓt+1 -γ t 4 b v ← ℓ v + 1 5 ℓ ← ℓ -e u + e v
B =    x | u∈J x u = K and ∀u ∈ J, x u ≥ b u    .
We then remarked that we do not need to compute the potential since v = arg max t∈J f ( ℓe u + e t ) is equivalent to:

∀t ∈ J, f ( ℓ -e u + e v ) -f ( ℓ -e u + e t ) ≥ 0 ⇔ ∀t ∈ J, N v ℓ v + 1 -γ v - N t ℓ t + 1 + γ t ≥ 0 ⇔ v = arg max t∈J N t ℓ t + 1 -γ t .
We can then apply the same analysis as in [START_REF] Muriguchi | Scaling Algorithms for M-convex Function Minimization[END_REF] for the correctness of the algorithm.

The quantity 0 ≤ u∈J (Kb u ) ≤ KJ decreases by at least one at each step of the algorithm. Therefore, the algorithm terminates with at most KJ iterations. Moreover, finding a u satisfying the loop condition and computing the value of v can be done in O(J). Hence, this algorithms has a time complexity in O KJ 2 .

We have also designed a refinement of this algorithm thanks to some scaling properties of function f , which gives a complexity in O J 3 log K/J . It is omitted due to length requirements.

B. An efficient learning mechanism

Note that the previous algorithm starts with some arbitrary load vector in D and then iteratively finds the best improvement until reaching a global maximum.

Instead, we propose a novel approach in which the seeds arrive one by one. We then show that at each arrival of seed k, the strategy s k can be computed in such way that after all arrivals, the resulting vector s is a Nash equilibrium. This approach relies on the following theorem: Theorem 7. Let Γ = (K, (N j , γ j ) j∈J ) be a setting of the game and Γ ′ = (K + 1, (N j , γ j ) j∈J ) be the setting obtained by adding an extra seed on Γ. Let s be a Nash equilibrium for Γ and σ the strategy profile of Γ ′ in which the K first seeds choose the same strategy as in s (i.e. s k = σ k for all k ≤ K) and the additional seed chooses one of the social media which maximizes its payoff. Then σ is a Nash equilibrium for Γ ′ .

Formally, let ℓ ∈ E Γ be the load of some Nash equilibrium of Γ and w ∈ J. Then

ℓ + e w ∈ E Γ ′ ⇔ w ∈ arg max t∈J N t ℓ t + 1 -γ t .
Proof. Let w ∈ arg max t∈J Nt ℓt+1γ t . We proceed to show that ℓ + e w is in E Γ ′ .

Let u, v ∈ J such that ℓ u +δ(u, w) > 0 and u = v. We need to show that Pot( ℓ + e w ) ≥ Pot( ℓ + e we u + e v ). There are three cases detailed below.

First, consider that u = w and v = w. Then Pot( ℓ + e w )-Pot( ℓ + e we u + e v ) = Pot( ℓ)-Pot( ℓe u + e v ) so it is proven in this case.

Second, consider that v = w. We have

Pot( ℓ + e w ) -Pot( ℓ + e w -e u + e v ) = Pot( ℓ + e w ) -Pot( ℓ + 2 e w -e u ) = N u ℓ u - N w ℓ w + 2 + γ w -γ u ≥ N u ℓ u - N w ℓ w + 1 + γ w -γ u ≥ 0 since ℓ ∈ E Γ .
Third, consider that u = w. We have

Pot( ℓ + e w ) -Pot( ℓ + e w -e u + e v ) = Pot( ℓ + e w ) -Pot( ℓ + e v ) = N w ℓ w + 1 - N v ℓ v + 1 + γ w -γ v ≥ 0
by definition of w.

The reciprocal follows from the last formula: if Nw ℓw+1γ w was not maximal, then there would be some v such that Pot( ℓ + e w ) < Pot( ℓ + e v ). Hence ℓ + e w would not be in E Γ ′ from Theorem 3.

We use Theorem 7 to build an efficient algorithm finding a vector of loads of a Nash equilibrium (Algorithm 2): it begins with no seed and simulates K times the arrival of a seed maximizing its payoff.

Algorithm 2: ORDER LEARNING

Input: Γ = (K, (N j , γ j ) j∈J ) Proof. Let Γ k = (k, (N j , γ j ) j∈J ) for k ∈ {0, ..., K}. Since 0 ∈ Z J is a vector of loads of a Nash equilibrium of Γ 0 , then, from Theorem 7, at the end of the kth iteration of the loop, ℓ is a vector of loads of a Nash equilibrium of Γ k , hence the correctness of Algorithm 2.

Output: A vector in E Γ 1 Let ℓ = 0 ∈ Z J and k = 1 2 while k ≤ K do
Since we can compute w in O(J) and there are K iterations of the loop, then the time complexity of Algorithm 2 is in O(KJ).

V. ASYMPTOTIC BEHAVIOR

In this section, we discuss the form of the Nash equilibria when we have a lot more seeds than social media, so K ≫ J. We are interested in this case in practice as the activity in the Internet tend to be concentrated in a restricted number of famous websites.

A. Intuition

First, we can make a hypothesis about the asymptotic behavior of this game when K → ∞ according to the form of the potential. Recall that H n ∼ n→∞ ln(n) + µ, where µ is the Euler-Mascheroni constant. Then in order to find approximate Nash equilibria, we can study the function

P ( ℓ) = j, ℓj >0 (N j ln(ℓ j ) -γ j ℓ j ) .
We can see that, for large values of ℓ j , the linear part in γ j ℓ j is determinant compared to the logarithm part in N j ln(ℓ j ). Therefore we can make the hypothesis that when the quantity j ℓ j = K is large enough, then the only ℓ j that continue to increase are the ones with minimal cost. Then, it seems natural that all social media with minimal cost would behave as if they were in a subgame where new seeds would only choose them.

B. Asymptotic Analysis

Following our hypothesis, we define γ m the minimal cost and G the set of social media with minimal costs:

γ m = min j γ j and G = arg min j γ j .
We know, thanks to Theorem 7, that when K increases, the coordinates of the loads of the Nash equilibrium we consider can only increase. We proceed to show our intuition. In the following, we note Γ K = (K, (N j , γ j ) j∈J ) and E K = E ΓK . We study the vectors in E K obtained with the mechanism implemented in Algorithm 2. Let ℓ (K) be the vector in E K obtained after the Kth iteration of the loop in the algorithm. Theorem 9. When K goes to infinity, at the Nash equilibria, the social media are divided into two groups:

• The loads of the social media with non-minimal cost stop increasing when they reach a constant. Formally:

∀j ∈ J \ G, ℓ (K) j -→ K→∞ N j γ j -γ m -1.
• The loads of the social media with minimal cost goes to infinity, and the proportion of seeds a social medium get among the one with minimal cost is equal to its market share. Formally:

∀w ∈ G, ℓ (K) w t∈G ℓ (K) t -→ K→∞ N w t∈G N t .
The proof of Theorem 9 is given in Appendix B.

VI. NUMERICAL RESULTS

Figure 1 shows the convergence of the equilibrium when the number of seeds K grows large. Note that there may be up to 3 equilibria and that the plots of the figure correspond to the outputs of Algorithm 2. The asymptotes obtained in Theorem 9 are represented in dashed lines with colors matching those of the loads of their associated social media (SM). The SM 2 and 3, which have minimal cost, have loads growing to infinity with the number of seeds. The asymptote of SM 2 has a higher slope than that of 3 because it has a higher number of subscribers (N 2 > N 3 ). Finally, while for large values of seeds the cost of the social media is predominant, in contrast, for low values of seeds, the number of customers M plays the larger role in determining the loads of the different social media. Figure 2 shows the evolution of the social welfare, that is, the sum of total utilities, k u k ( s), at the Nash equilibrium and at the social optimum. The asymptotic behavior at the Nash equilibria is given by -γ m K from Theorem 9. Further, let L = {j, N j ≥ γ j }. Recall that the social optimum is the strategy vector maximizing the social welfare. Then, for K large enough, the social optimum satisfies max ℓ j,ℓj >0 (N jγ j ℓ j ) = j∈L (N jγ j )-γ m (K -|L|) ∼ -γ m K. Hence, as the number of seeds grows to infinity, the price of anarchy converges to 1.

Finally, Figures 3 and4 show the sensitivity of the equilibria with respect to N and γ for a case with J = 2 social media.

We observe that the load of a social medium is increasing with its number of customers, as expected (Fig. 3). Further, if the dissemination cost of SM 2 is higher or equal to that of SM 1 and if it has no customer, then its load is zero, as exhibited in the red and blue plots. Otherwise, even though it has no customer, if its cost is minimal, it will receive some seeds (green plot). Finally, note that as the cost of SM 1 decreases, the number of customers in the SM 2 has lower effect of the evolution of the load ℓ 2 .

We also observe that the load of a social medium is decreasing with its dissemination cost, as expected (Fig. 4). Further, numerical results show that the load decreases more abruptly for lower number of users N 1 , but that the drop occurs for larger values of γ 2 .

VII. CONCLUSION

In this paper, we have studied competition for popularity of seeds among several social media. We have shown that the game is equivalent to a congestion game and hence has a potential. We then studied the properties of the potential in terms of M -concavity. We have shown that there may exist several Nash equilibria, all belonging to a single neighborhood and provided examples where the number of equilibria is maximal. We have provided a novel efficient learning algorithm based on a remarkable property of the Nash equilibria in some subgames. We also investigated the asymptotic behavior of the equilibria of the game and the price of anarchy. As future work, we will study the underlying competition among the social media in the Stackelberg setting for a discrete number of seeds: according to their number of subscribers (who consume content), how could they appropriately set up their prices? We further plan to extend our model to the case where seeds have different dissemination utilities for sending to the various media. The game is no more equivalent to a congestion game but turns out to be equivalent to crowding games [START_REF] Milchtaich | Congestion games with player-specic payoff functions[END_REF]. This allows to show existence of (pure) equilibria but best response policies need not converge, as there need not be a potential anymore. Thus, designing learning algorithms for this extension is yet an open problem.

Lemma 11. Let α ∈ J. Then min(α,J-α) k=0 α k J -α k = J α .
Proof. We show this result using a combinatorial argument. First, note that since J α = J Jα , then one can restrict the analysis to the case where α ≤ Jα.

We want to select α elements in J. To do that, we partition the set J into two subset A and B such that |A| = α (so |B| = Jα).

Selecting α elements in J amounts to choosing k the number of elements we select in B, then select these k elements and finally select αk elements in A. Therefore

J α = |A| k=0 |B| k |A| α -k = α k=0 α α -k J -α k = min(α,J-α) k=0 α k J -α k .
We can now proceed to the proof of Theorem 5:

Proof. Let x ∈ E Γ and: We conclude the proof by applying Lemma 11, using the increasing property of function J . over {0, ..., ⌊J/2⌋} and the fact that J p = J J-p for all p:

U = {u | ∃ y ∈ E Γ , x u > y u } = y∈EΓ supp + ( x -
|E Γ | ≤ J α ≤ J J 2
.

B. A Tight Class of Settings

Let J ≥ 2, m ∈ N * and γ ∈ R + . We define the game Γ by K = J Proof. Assume that there exists x ∈ E Γ and u ∈ J such that x u > 1. Since K < J, there exists v ∈ J such that x v = 0. Consider the vector y = xx u e u + x u e v . Since all the N j and γ j are equal, the potential of y is equal to the potential of x. Therefore y ∈ E Γ . But we have y vx v = x u > 1 which contradicts Theorem 4 and concludes the proof.

Since E Γ = ∅, let x ∈ E Γ . By Lemma 12, we can note x = u∈A e u for some A ⊂ J. Let B ⊂ J verifying |B| = J 2 and y = v∈B e v . Then, we have

Pot( x) = u∈A (m -γ) = J 2 (m -γ) = v∈B (m -γ) = Pot( y).
Therefore,

|E Γ | = A ⊂ J | |A| = J 2 = J J 2
.

arbitrarily high with K large enough, so we determine an approximation of a load of a Nash equilibrium for the social media in G by solving Since P is concave, we apply a Lagrangian maximization method. Let L be the Lagrangian for this problem:

L( x, λ) = P ( x) -λ t∈G x t -K G ,
where λ and the x t are nonnegative.

Since P is concave, the unique maximum x * verifies ∀t ∈ G, ∂L ∂x t ( x * ) = 0. Therefore, we get that for any t:

N t x * t -γ m -λ = 0 ⇔ x * t = N t γ m + λ .
Now we determine the value of λ:

t∈G x * t = K G ⇒ t∈G N t γ m + λ = K G ⇒ λ = 1 K G t∈G N t -γ m .
Hence ∀w ∈ G, x * w = K G Nw t∈G Nt . Thanks to [START_REF] Yadati | Game theoretic models for social network analysis[END_REF]

6 return ℓ Proposition 6 .

 66 Algorithm 1 terminates, returns a vector in E Γ with a time complexity in O KJ 2 .

3-γ t 4 ℓ ← ℓ + e w 5 k ← k + 1 6 return ℓ Proposition 8 .

 4518 Compute w ∈ arg max t∈J Nt ℓt+1 Algorithm 2 terminates and returns a vector in E Γ with a time complexity in O(KJ).

3 Fig. 1 .

 31 Fig. 1.Convergence to the asymptotic behavior. Case with: (N 1 , γ 1 ) = (100, 2), (N 2 , γ 2 ) = (25, 1) and (N 3 , γ 3 ) = (20, 1).

Fig. 2 .

 2 Fig.2. Sum of utilities at equilibrium compared with the social optimum (same setting that Figure1).

5 Fig. 3 .

 53 Fig. 3. Influence of the number of users. Case with 2 social media: N 1 = 250, γ 2 = 10 and K = 100.

Fig. 4 .

 4 Fig. 4. Influence of the dissemination cost. Case with 2 social media: γ 1 = 30, N 2 = 250 and K = 100.

.

  y) and V = y∈EΓ supp + ( yx). By Lemma 10, we have U ∩ V = ∅ and |U| + |V| ≤ J. We then define the set A = { x} ∪ min(|U |,|V|) We know by Proposition 5 and Lemma 10 that all vectors in E Γ are of the form given in the previous expression, hence E Γ ⊂ A. Let α = |U|. We have |V| ≤ Jα. Then |E Γ | ≤ |A| = 1 + min(|U |,|V|)

2 and

 2 ∀j ∈ J, N j = m, γ j = γ. Lemma 12. The Nash equilibria of game Γ satisfy the property:ℓ ∈ E Γ ⇒ ∃A ⊂ J, |A| = J 2 and ℓ = u∈A e u .

  ln(x t )γ m x t ) s.t. x ∈ D G .

  and since H nµ ∼

	get that ∀w ∈ G,	ℓ t∈G ℓ (K) w	(K) t	ln n, we finally N w t∈G N t n→∞ -→ K→∞ .

APPENDIX A PROOF OF PROPOSITION 5

A. Proof of the upper bound |E Γ | ≤ J ⌊ J 2 ⌋ . Lemma 10. Let x be a load vector at a Nash equilibrium, x ∈ E Γ , and u a social medium, u ∈ J. Then (∃ y ∈ E Γ , x u > y u ) ⇒ (∀ z ∈ E Γ , x u ≥ z u ) . Proof. Assume that there exists y and z in E Γ such that x u > y u and x u < z u . Then, by Theorem 4, y u = x u -1 and z u = x u + 1. Hence z uy u = 2 which contradicts Theorem 4. PROOF OF THE ASYMPTOTIC BEHAVIOR Note that, by definition of the learning mechanism implemented in Algorithm 2, for all K ∈ N and j ∈ J we have

A. Social media with non minimal cost

We want to prove that

We begin by proving the following two lemmas.

Lemma 13. The quantity

Proof. First, this quantity is decreasing. Moreover, by definition of the ℓ (K) , we have that for all K,

Therefore, there exists some u ∈ J such that

2 We denote by K "large enough" the fact that there exists some K 0 such that the property is verified for all K > K 0 .

Hence M (Kn) is arbitrarily close to -γ u for n large enough. We conclude by noticing that -γ u ≤ -γ m . Lemma 14. Let K > 0 and u ∈ J. Then N u ℓ

Proof. First, assume that Nu ℓ (K) u

+1

γ u ≤ -γ m . Then for some w ∈ G and for all K ′ ≥ K we have

Therefore, for all K ′ > K, (5) leads to ℓ

which concludes the proof.

We can now proceed with the proof of ( 6). Let j ∈ J \ G. We know by Lemma 14 that ℓ (K) j increases with K as long as

Therefore, for K large enough we have

Then, let p ∈ N. Since j ∈ J \ G, we have γ j > γ m .

We solve

which concludes the proof.

B. Social media with minimal cost

We can directly conclude from Lemma 14 that the load of any social medium having a minimal cost goes to infinity as K increases. Formally:

Now we proceed to find the values of ℓ (K) w

for the social media with minimal cost. Let K be large enough so that (6) is verified. Let K G = K - Consider the game Γ G = (K G , (N t , γ m ) t∈G ). From [START_REF] Yadati | Game theoretic models for social network analysis[END_REF], the loads of the social media in G can be


Marie Ménoret

'Don't think, just Pink': Knowledge and Ignorance in Public Health Policies

Breast cancer is the most frequent cancer in human populations, particularly in industrialized countries 1 . Touching a woman out of twelve, its importance drove to the establishment of many public policies for its prevention, its detection and treatment. Based on the accumulation of knowledge in clinical and epidemiological areas, these public policies, are often presented as canonical examples of the Evidence-based public health policies (EBPH), although they have been largely built before the formation of the concept, derived from that of Evidence-Based Medicine (EBM) 2 .

Based on scientific literature and on the recommendations for mammogram screening in many countries, this paper analyzes the ways in which the systematic mammographic monitoring of women, established in the 1980s and 1990s from previous knowledge, address new knowledge, whether produced by clinical trials, observational studies or knowledge of patients. It shows how certain knowledge are deliberately ignored or minimized, leading to very minor changes in these policies monitoring, despite growing criticism of their effectiveness or their drawbacks. Finally, we conclude by asking the question of the historicity of this type of public health policy design and the scientific knowledge they carry.

1/ Systematic mammography: an evidence-based public policy

The issue of monitoring of breasts for cancer detection purposes has been discussed for a century, framing and being framed by battle against cancer as a fight against delays 3 . In the interwar period, different activists promoted self-examination techniques to detect tumors at the earliest, for the purpose of medical examination and intervention, including mastectomy 4 . Despite the lack of evidence of their effectiveness, these self-examination techniques were recommended from the post-war by many learned societies and became the subject of public policies, mostly in terms of health education. In the 1970s, monthly self-breast examinationlike clinical examination by a doctor -were widely considered as practices to promote, in order to reduce mortality from breast cancer. From the 1980s on, criticisms have been arising: its proponents responded by saying practices had to be overseen, women trained, right protocols adopted 5 . Clinical trials were launched worldwide, sometimes very large, like the one shaped by the USSR and the WHO 6 or the one sponsored by the US National Cancer Institute in Shanghai 7 . Indeed, the techniques were considered relatively simple and mostly inexpensive, therefore likely to be adopted by a large part of the world population. Trials did not respond to the preventive hopes widely supported by WHO. That gradually led to recommendations not to promote self-examination from the early 2000s, particularly because of the significant number of consecutive biopsies of "detections" by patients, even as mortality was not affected by the self-examination. 8 Conversely, the use of mammography in the prevention of breast cancer mortality has very early been the object of investigations intended to prove its efficacy. Truly developed from the 1940s based on X-rays technology, it gradually brung out a medical segment with its own specialists and equipment, developed by private industries in the mid 1960s. This segment was very active in the production of scientific evidence, especially in two large randomized controlled trials (RCTs): one began in 1967 in New York 9 and one in two Swedish counties in 1977 10 , and both showed a 30% reduction in mortality. This led many public authorities to promote mammography monitoring and to pay for its implementation management from the mid-1980s on. This political agenda has been supported by the results of RCTs in the 1990s, showing once again the importance of reducing of mortality 11 .

While some critics of this political enthusiasm appeared very early 12 , they had no efficacy in opposing to the implementation of screening programs in Europe. By the late 1980s, systematic screenings policieswere launches in the Netherlands, Finland and Great Britain, soon followed by most Spanish regions, Sweden, Denmark and France. Meanwhile, in the United States, very active social movements led to the extension of local programs and public funding 13 , while Canada developed its national program. In the early 2000s, all highly industrialized countries -including Korea and Japan -had their mammography surveillance program: while the age groups involved vary (from 40 to 50 years -64 years to death), in every cas it is the whole female population that is targeted through time, regardless of their individual or group risk factors 14 .

Obviously, the implementation of such policies is not solely due to a knowledgetransfer process from scientific arenas to policy makers who would then apply a rational evidencebased policy 15 . The construction of a public policy remains a political process in each country, alliances between medical specialists, screening companies and sometimes women's movements have produced slogans, campaigns, funding. Moreover, national or local institutions in charge of these policies are allied in the International Breast Cancer Screening Network, which published comparative studies and lobbied for screening. But at the heart of this public policy, there is always a reference to the decline in mortality, proven by controlled trials to introduce the screening as a true knowledge-based policy, made to save lives.

2) Forbidden knowledge, ignored knowledge and minimized knowledge in an EBPP

Disagreeing with these policies, many scholar critics voiced out in the early 2000s, being widely published in biomedical literature; promoters of screening policies partly addressed them in commentaries, answers and new articles. The development of this controversy, which has been lasting for fifteen years, shows that the knowledge-based policy does not take into account certain knowledge, by various mechanisms, which analysis would feed the literature on the social production of ignorance. We will not discuss here all the criticisms of mammography programs based on the absences 16 they produce as they fulfill the whole public space of "cancer prevention", for example in the US the movement focused on environmental causes of cancer17 or the gradual commercialization of a the social movement against breast cancer [START_REF] King | Pink Ribbons Inc: breast cancer activism and the politics of philanthropy[END_REF] . In this communication, we are not focusing on the contrasting aspects of the different national implementations, but rather on the common and invariant characteristics that homogenize the treatment of ignorance in these screening policies. a/ The ethical forbidding: the establishment of a consensus on the basis of clinical trials prevented the repetition of RCTs with new screening machines on behalf of the loss of opportunity for patients that would not have access to mammography. So here we have to deal with an assumed ignorance which produces undone science [START_REF] Frickel | Undone science: charting social movement and civil society challenges to research agenda setting[END_REF] , not because certain fractions of the public or patientsasked for a certain science, but in the name of the health of these populations.

b/ Minimization of risks: certain "tumors" identified by mammogram devices, more and more sophisticated and « specific », would never become morbid and even less deadly, as their natural history would be longer than the life expectancy of tested women. So, for its critics, the "benefit" of detection corresponds in fact to a "therapeutic risk" given the severity of the therapeutic measures (biopsy, surgery, chemotherapy) or their induced pathological effects. The issue of overdiagnosis, common to all cancer screening programs has been little publicized and understood by patients until the early 2000s [START_REF] Black | Overdiagnosis: an underrecognized cause of confusion and harm in cancer screening[END_REF] . Then it became more public and discussed in some screening programs, it is always discussed in relation to the decline in mortality proven by clinical trials, thus allowing developers to describe a positive benefits/risk ratio 21 in favor in the scientific literature and insist more on profits than on the risks in their public discourse [START_REF] Chen | Stakeholders' positions in the breast screening debate, and media coverage of the debate: a qualitative study[END_REF] . c/ The endogenisation of uncertainties: the reading of mammograms is a complex process that requires significant expertise. If semi-automated devices have been developed to cope with the influx of clichés produced by screening programs, they have little decreased the variability of interpretations produced by radiologists, particularly in the USA [START_REF] Beam | Variability in the interpretation of screening mammograms by US radiologists: findings from a national sample[END_REF] . Most public programs have adopted a double reading system, publicly touted as additional security for tested women but that poses new problems in the many cases of discrepancy [START_REF] Duijm | Independent Double Reading of Screening Mammograms in the Netherlands: Effect of Arbitration Following Reader Disagreements 1[END_REF] . d/ The theoretical hierarchy of knowledge: many contemporary observational studies have shown a slighter decrease of mortality for women screened, or even a lack of effect. Through comparative studies between countries having implemented or not screening policies, they have shown a lack of mortality difference. In every country, breast cancer mortality diminishes through time, this evolution being mainly attributed toprogress in therapeutic management 25 . These results are often not discussed or are minimized due to their level of evidence lower than controlled clinical trials in the dominant version of EBM 26 . e/ The lack of recognition of women's knowledge and experience subject to the public health imperative. Despite strong incentives to submit to screening programs, participation rates remain insufficient in the eyes of its proponents, with the notable exception of Scandinavian countries 27 . This relative lack of membership is typically read as much resistance or cognitive biases. The literature produced by medical anthropology 28 , medical sociology 29 or women's studies 30 on the subject is almost never cited in the biomedical world and therefore not taken into account.

3) Indirect effects of ignored knowledge

These criticisms have very limited effects on screening policies, as no country has decided to stop its program on behalf of these complimentary or alternativeknowledge. Nevertheless, there are three types of consequences of the publication and dissemination of this knowledge that do not support routine mammogram, much less its expansion and, in certain cases, aims at stopping screening programs. board, recommend stopping screening program in Switzerland 33 . However, these recommendations do not have direct normative effects, and they had a very limited effect on women's practices at least in the US 34 . b/ Marginal changes in public policies: two separate one can be observed, mainly in the USA and Canada, and at least partly in the UK. First, as we have just told, debates about overdiagnosis and reassessment of the benefits/risks led to postpone the recommendation of a screening for women aged 40-49 years, and even not to recommend monitoring. Second, in three countries there appears a much more centered on full disclosure of women and the principle of informed consent in program participation [START_REF]Following a tradition that has been legally settled by US activists on breast surgery in the 1970s[END_REF] . It should be noted that other countries are not at all affected by these changes: thus in France, a mammogram out of two is still performed out of the surveillance program, primarily to women from 40 to 49 years old, without effect despite the fact that these mammograms are publicly funded [START_REF] Rogel | Évaluation du programme national de dépistage organisé du cancer du sein en France: période 2004-2009[END_REF] . c/ Debates on the limitations of screening programs reached non-academic areas, including broadcast media,women press, and electronic forums [START_REF] Barker | Dead by 50: Lay expertise and breast cancer screening[END_REF] . This is especially true of the works of Peter Götzsche which besides its abundant scientific production and comments, published a book for doctors and a brochure for the general public from the Nordic Cochrane Center 38 .

Conclusion: current state of science and public policies

Screening policies, including mammography discussed here, are clearly the result of Foucault's biopolitics, where the State defends the health of the governed. But they rely on composite social forces, combining doctors, manufacturers and patients, in order to be defined and implemented, and it is these same actors who later defended them. We have shown that many types of knowledge were excluded, ignored or even forbidden to produce, but the screening defenders apply a symmetrical treatment to their critics: they would produce bad science, would base their conclusion on incomplete studies or would be subject to bias interpretation [START_REF] Hendrick | United States preventive services task force screening mammography recommendations: science ignored[END_REF] .

What is at stake in this controversy is the opposition between two visions of EBM: the first is aimed at the purification of biomedical evidence, only produced in RCTs and, facing their limits, by the generalization of meta-studies 40 ; the second, which recognizes the need to embrace as much knowledge as possible, and that RCT should be criticized as much in their methodological foundations and their implementation than other devices of knowledge production [START_REF] Victora | Evidence-based public health: moving beyond randomized trials[END_REF] . In other words, the first fully assumes ignorance it co-produces, because it isthe condition of the formation of good knowledge for EBPH in a linear model of diffusion from science to politics 42 ; the second, in a form that is similar to the defense of the many forms of "evidence" produced by the social sciences [START_REF] Head | Three lenses of Evidence-Based policy[END_REF] , defends a model of aggregation of different types of knowledge.

Faced with a growing crystallized controversy and their relative failure in changing screening policies, critics have recently produced an explanation that would account for the divergence of perspectives: clinical trials, so dear to the advocates of screening, would adequately represent breast cancers before 1990, while the observational data would be adequate to the current cancers [START_REF] Welch | Why mammography screening has not lived up to expectations from the randomised trials[END_REF] . Patients consult more often and earlier, tumors would be better managed, and the most appropriate treatments with fewer side effects, all kinds of factors would make obsolete the decline in mortality. But accounting for this type of explanation is potentially devastating for EBPH: if the "all things being equal" can not be ascertained and biomedical and epidemiological conditions keeps on varying, how to base a policy on knowledge that is by nature outdated?

11

  Nyström, L., et al. "Breast cancer screening with mammography: overview of Swedish randomised trials." The Lancet 341.8851 (1993): 973-978. 12 See for example Skrabanek, Petr. "False premises and false promises of breast cancer screening." The Lancet 326.8450 (1985): 316-320. 13 Kolker, Emily S. "Framing as a cultural resource in health social movements: funding activism and the breast cancer movement in the US 1990-1993." Sociology of Health & Illness 26.6 (2004): 820-844. 14 Klabunde, Carrie N., Rachel Ballard-Barbash, and International Breast Cancer Screening Network. "Evaluating population-based screening mammography programs internationally." Seminars in breast disease. Vol. 10. No. 2. WB Saunders, 2007. 15 For a discussion on this point, see for example Marston, Greg, and Rob Watts. "Tampering with the evidence: A critical appraisal of evidence-based policy-making." The drawing board: An Australian review of public affairs 3.3 (2003): 143-163. 16 Croissant, Jennifer L. "Agnotology: Ignorance and Absence or Towards a Sociology of Things That Aren't There." Social Epistemology 28.1 (2014): 4-2

  a / Official recommendations of screening reduction: the political work made by critics of screening mammography is crystallized by groups of experts, doing literature reviews, that declineate from their previously more extensivepolicy recommendations. In 2009, in the case of the United States Preventive Services Task Force 31 , in 2011 the Canadian Task Force on Preventive Health Care 32 . In 2013, for the first time, such a committee, the Swiss medical 25 Autier, Philippe, et al. "Breast cancer mortality in neighbouring European countries with different levels of screening but similar access to treatment: trend analysis of WHO mortality database." Bmj 343 (2011): d4411. 26 Independent UK Panel on Breast Cancer Screening. "The benefits and harms of breast cancer screening: an independent review." The Lancet 380.9855 (2012): 1778-1786. 27 Youlden, Danny R., et al. "The descriptive epidemiology of female breast cancer: an international comparison of screening, incidence, survival and mortality." Cancer epidemiology 36.3 (2012): 237-248. 28 For example Wardlow, Holly, and Robert H. Curry. ""Sympathy for my body": Breast cancer and mammography at two Atlanta clinics." Medical anthropology 16.1-4 (1994): 319-340. 29 Barker, Kristin K., and Tasha R. Galardi. "Dead by 50: Lay expertise and breast cancer screening." Social science & medicine 72.8 (2011): 1351-1358. 30 Klawiter, Maren. "Breast cancer in two regimes: the impact of social movements on illness experience." Sociology of Health & Illness 26.6 (2004): 845-874. 31 Woolf, Steven H. "The 2009 breast cancer screening recommendations of the US Preventive Services Task Force." Jama 303.2 (2010): 162-163. 32 Canadian Task Force on Preventive Health Care. "Recommendations on screening for breast cancer in average-risk women aged 40-74 years." Canadian Medical Association Journal 183.17 (2011): 1991-2001.

  The Breast Cancer Wars: Hope, Fear, and the Pursuit of a Cure in Twentieth-Century America: Hope, Fear, and the Pursuit of a Cure in Twentieth-Century America. Oxford University Press, 2001. 5 See for example Kegeles, S. Stephen. "Education for breast self-examination: Why, who, what, and how?"

	Preventive medicine 14.6 (1985): 702-720 et Janz, N. K., et al. "Interventions to enhance breast self-examination
	practice: a review." Public health reviews 17.2-3 (1988): 89-163.
	6 Semiglazov, V. F., et al. "The role of breast self-examination in early breast cancer detection (results of the 5-
	years USSR/WHO randomized study in Leningrad)." European journal of epidemiology 8.4 (1992): 498-502
	7 Thomas, David B., et al. "Randomized trial of breast self-examination in Shanghai: methodology and
	preliminary results." Journal of the National Cancer Institute 89.5 (1997): 355-365.
	8 That didn't prevent clinical breast examiniation to be performed by specialized or general health care
	professionals.
	9 Shapiro, Sam, Philip Strax, and Louis Venet. "Periodic breast cancer screening in reducing mortality from
	breast cancer." Jama 215.11 (1971): 1777-1785.
	10 Tabar, Lazio, et al. "Reduction in mortality from breast cancer after mass screening with mammography:
	randomised trial from the Breast Cancer Screening Working Group of the Swedish National Board of Health and
	Welfare." The Lancet 325.8433 (1985): 829-832.

3 Aronowitz, Robert A. "Do not delay: breast cancer and time, 1900-1970." The Milbank Quarterly 79.3 (2001): 355. 4 H. Lerner.

Jemal, Ahmedin, et al. "Global cancer statistics." CA: a cancer journal for clinicians 61.2 (2011): 69-90.

The concept was not really coined by a single paper, but rather discussed in many public health or epidemiology journals from around 1997 on. See for example Jenicek, Milos. "Epidemiology, evidencedbased medicine, and evidence-based public health." Journal of epidemiology 7.4 (1997): 187-197 and Brownson, Ross C., James G. Gurney, and Garland H. Land. "Evidence-based decision making in public health." Journal of Public Health Management and Practice 5.5 (1999): 86-97.

McCormick, Sabrina, Phil Brown, and Stephen Zavestoski. "The personal is scientific, the scientific is