1 Limited Commercial Licenses in the Pharmaceutical Industry
In 2021 the members of the Word Trade Organization signed the so-called TRIPS (Trade-Related Aspects of Intellectual Property Rights). https://www.wto.org/english/docs_e/legal_e/27-trips_01_e.htm.
This declaration gives the opportunity to governments of developing and least developed countries to issue limited commercial licenses (denoted by LCL) also know in literature with the name compulsory licenses (CL. According to its name, a limited commercial license is a government authorized non-voluntary license from a patent holder (henceforth H) to a third party, usually a generic producer (denoted by G). The government of a country C, which in what follows will be always a developing or least developed one, may issue a CL for a drug, covered by intellectual property, only under certain conditions. The most relevant one is when the sales \(S_H\) of the patent-holder H of a drug D does not reach an expected target sale denoted by \(\tilde{S}\), in the situation where H is the unique producer of the drug H in the country. Therefore, a CL is issued if in a regime of monopoly a certain drug as a selling result, that is below the expected one (i.e., \(\tilde{S}\)). Often the sale target \(\tilde{S}\) is not public, so it is considered a random variable and it is set taking into consideration, for example, the expected case of cases of the disease, for which the drug was developed.
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In the article Sarmah et al. (2020) the authors considered different scenarios, where a pharma manufacturer invests in R &D, obtaining the intellectual property of a new drug. This patent can be subjected to a limited commercial license. The author considered four different periods:
\(t_0\)
In this period the pharma company H invests in R& D for developing a new drug. At the end of this period H owns the patent for the new drug.
\(t_1\)
In this period the company H launched the drug in the market of a country C. During this period the market is a monopoly, because H is the unique company having the right to produce the drug.
\(t_2\)
In this period the government of the country C may issue a limited commercial license. This happens in the case the selling result \(S_{H_1}\) does not reach the selling target \(\tilde{S}\) in the previous phase \(t_1\). If the CL is issued, a competitor G (generic producer) also has the right to produce the drug, H must license G to produce the drug, but G has to pay a royalty fee to H. If in period \(t_1\) the selling target \(\tilde{S}\) is reached, then the CL is not issued and so H operates in a monopoly system.
\(t_3\)
The intellectual property of H expires and H must operate in a free market, where a generic producer may produce the drug.
Compulsory licensing has been the subject of interest in numerous recent studies. For example, Scherer (1977) discusses how regulators can use limited commercial licensing to restore competition in industries. Aoki and Small (2004) views compulsory licensing as a tool of anti-competitive practices and shows that it creates significant losses. Seifert (2015) documents that compulsory licensing decreases incentives for innovations, but creates benefits to consumers and total welfare. Bertran and Turner (2017) suggest that a, in duopoly, suitable royalty payments are required to improve social welfare. Bond and Saggi (2014) analyze the role of compulsory licensing in determining consumer access to a patented product sold by a patent-holder. They suggest that compulsory licensing guarantees consumer access to the patented product and increment the chances of voluntary licensing and results in the patent-holder switching from voluntary licensing to entry. In the same spirit, Stavropoulou and Valletti (2015) find that the overall welfare effects of compulsory licensing are positive even if taking into account innovation incentive. Finally, empirical studies analyze the effects in terms of incentives to innovate compulsory licensing. For instance, Baten et al. (2017); Moser and Voena (2012) suggest that compulsory licensing pushes innovators to create new patents.
In the article Sarmah et al. (2020), the authors analyze a dynamic game between the patent-holder H and a generic producer G. They considered three different scenarios, described in the picture below
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The authors in Sarmah et al. (2020) compare the different three above scenarios. The incertitude due to the risk of a compulsory license makes an investment in R &D less attractive. The issue of a compulsory license should improve the access to a new drug, but the economic-market setting should appropriately remunerate pharma companies and guarantee the profitability of the investments in R &D.
The authors in Sarmah et al. (2020) have shown that a sufficient high royalty, paid in period \(t_2\) in the case of a CL, could at the same guarantee enough good condition to H, the intellectual-property holder, and assure enough sustainable access to the drug.
The outline of this paper is as follows. In the next Sect. 1.1 we study the problem of determining, whether the CL will be issued. In Sect. 2 we model concerning a CL by using Differential Game Theory, as presented in Dockner et al. (2000). In Sect. 3 we explain how the model can be solved. Section 4 gives some perspective for future work on the subject.
1.1 Probability of Issuing a CL
In the stochastic scenario it is extremely important to predict whether the limited commercial license will be issued or not. We consider the cumulative function \(F_{\tilde{S}}(x)=\textrm{Prob}(x\le \tilde{S})\).
By denoting \(S_{H_1}\) the selling result in phase \(t_1\), than the probability that the CL is issued is equal to \( F_{\tilde{S}}(S_{H_1})=\textrm{Prob}(S_{H_1}\le \tilde{S})\), because the CL is issued in the case the sale target result \(\tilde{S}\) is not reached in phase \(t_1\).
Therefore, we need to estimate which value can have \(\tilde{S}\), and whether the selling result \(S_{H_1}\) may reach the value \(\tilde{S}\).
In Sect. 5 in Sarmah et al. (2020), the authors model H’s belief about government’s target by using a log-logistic random variable, thus they assumefor suitable values of a and b depending on the market setting, in their article the authors have set \(a=b=2\).
$$ F_{\tilde{S}}(x)=\frac{1}{1+\left( \frac{x}{a}\right) ^{-b}}$$
Therefore, we can estimate \(\tilde{S}\) by calculating the expected value of the log-logistic random variable.
A new idea to estimate, whether the sailing result reaches the value of the estimated value for \(\tilde{S}\) is given by using a counting process on the random variable \(S_{H_1}\).
One of the easiest models of the counting process is the so-called (stationary) Poisson process. We consider a subdivision of an interval in sub-intervals with endpoints \(a_1<a_2<\ldots <a_k\). For any index \(1\le i\le k-1\) we consider a time \(b_i\in (a_i, a_{i+1})\) and an integer \(n_i\). We denote by \(N(a_i,b_i]\) the number of events of the process happening in the interval \((a_i,b_i]\). In the Poisson process, we havewhere \(\lambda \) is the parameter, which characterizes the Poisson process.
$$\begin{aligned} \textrm{Prob}(N(a_i,b_i]=n_i, i=1,\ldots , k-1)=\prod _{i=1}^{k-1}\frac{(\lambda \cdot (b_i-a_i))^{n_i}}{n_i!}e^{-\lambda \cdot (b_i-a_i)}, \end{aligned}$$
(1)
In the model, the time-points \(a_i\)’s and \(b_i\)’s can represent some marketing actions, workshops, scientific-information meetings, etc.
Note that the above formula (1) appears slightly simpler in the case \(b_i=a_{i+1}\) for all index i. The model is called stationary because the distributions are stationary, in the sense they do not depend on the numbers \(a_i\)’s and \(b_i\)’s, but on the differences \(b_i-a_i\).
Therefore, we can subdivide the period \(t_1\) in the CL-model above into several sub-intervals and estimate the corresponding selling sub-results \(n_i\) in each sub-period and with the above formula to calculate how is realizable (i.e., probable) the minimal goal \(\tilde{S}\).
As a first approximation we can consider a unique period \(t_1\) (with no subdivision), obtaining as a model a classical (univariate) Poisson distribution.
Another possible way is to consider the notion of the Hawkes process. The idea is that the selling of a new drug in a country is a self-exciting counting process. Indeed, by assuming the trivial hypothesis that the drug has a positive effect in treating the corresponding disease (for which the drug was developed), then the selling of a drug dose positively influences the selling of the next doses.
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2 A Differential Game of Limited Commercial Licensing
In an environment in which a limited commercial license might be issued at some future point in time, the problem of drug pricing and R &D in product innovation depends on the stage of the process of issuing a CL. In each different stage of this process, the actors involved are allowed to take different actions, according to which stage is currently active. To model the different behavior of the actors involved in each stage of the limited commercial license’s life, we make use of the framework of piecewise-deterministic differential games described in Dockner et al. (2000). In this setup, a discrete set of modes, M, models have different stages of the system. In each mode, players are allowed to take some actions. Switches between modes are randomly driven by a continuous-time Markov chain with values in M. The probability of switching between two modes of the systems in general depends on the actions taken by the players and the state variables of the system. While we refer to the above-mentioned book of Dockner et al. (2000) for a detailed treatment of piecewise-deterministic differential games, in what follows we describe the details of our model of pricing and product innovation in the uncertain environment of a limited commercial license. In the following parts we will use the standard notation as the one used in the book of Dockner et al. (2000) (Chapter 8).
2.1 Preliminaries
We consider a piecewise-deterministic game with two players. The first, which we will refer to as the innovator (I), who has developed a patent for a new drug and started selling the product in a new, underdeveloped, country. The second, which we will call the generic producer (G), we want to exploit the patent of the innovator. On top of both players, there is a regulatory authority (the government or similar) in charge of evaluating the needs of the country in terms of the new drug, and eventually issuing a CL. We will assume that the issue or not of a CL depends only on sales of the new drug, leaving aside political/economical reasons.
We now introduce a set \(M=\{1,2,3\}\) of modes of the system. Broadly speaking, M identifies the stages involved during the life of a new drug with the following (this of course can be changed according on how much emphasis we want to put in each stage of development):
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1 indicates the stage in which the innovator launches a new drug on the market;
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2 models the situation in which a CL is currently active;
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3 represents the end of the process, where the patent expires.
Next, we introduce the following notation (\(i \in \{I,G\}\)): Let \(x_i = x_i(t)\) represent the cumulative sales of player i up to time t and \(p_i=p_i(t)\) be the price of the drug decided by player i at time t. Moreover, let \(A_I = A_I(t)\) model the innovator’s effort in product innovation at time t. The evolution of the cumulative sales of each player is governed by the so-called players’ instantaneous sales functions, \(f_i\), that isfor \(m\in M\). We observe that the functions that govern instantaneous sales depend on the mode of the system. This is a feature of piecewise-deterministic games that allows to take into account the different phases of the development of the limited commercial license.
$$\begin{aligned} \dot{x_i}(t) = \frac{d x_i(t)}{dt} = f_i\left( x_I,x_G, p_I, p_G, A_I,m\right) \end{aligned}$$
(2)
2.2 Profits and Sales in Each Regime
We now specify how sales evolve and what profits the players obtain in each mode of the system:
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In mode 1, since no limited commercial license exists, the generic producer is not allowed to sell the drug, and \(f_G\left( x_I,x_G, p_I, p_G, A_I,1\right) = 0\). On the other hand, the innovator operates in a situation of monopoly, as they are the only seller in the market. We assume that their instantaneous sales arewhere a is the market potential, \(b_I\) is the market elasticity with respect to the price of the patented product, and \(\theta \) is the market sensitivity to improvements in the product. In this regime, the innovator’s instantaneous profit is modeled as the difference between instantaneous market revenues and costs for production and product development, that is$$\begin{aligned} f_{I_1} = f_I\left( x_I,x_G, p_I, p_G, A_I,1\right) = a - b_I p_I + \theta A_I\text {,} \end{aligned}$$(3)being \(c_I\) the marginal production cost for production and l the coefficient for the (quadratic) cost in product development.$$\begin{aligned} \pi _I(p_I, A_I,1) = (p_I - c_I)f_{I,1} - l A_I^2\text {} \end{aligned}$$
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In mode 2, the limited commercial license has been issued. This means that the generic producer has entered the market in exchange for a royalty to be paid to the innovator. The sales functions are described as follows:$$\begin{aligned} f_{G_2} =&f_G\left( x_I,x_G, p_I, p_G, A_I,2\right) = a - b_G p_G + \theta A_I \end{aligned}$$(4)being \(p_G, b_G\) the price assigned by the generic producer and their price elasticity, respectively. The profit functions take into account the royalty that G needs to pay to the innovator, as described in the following$$\begin{aligned} f_{I_2} =&f_I\left( x_I,x_G, p_I, p_G, A_I,2\right) = a - b_I p_I + \theta A_I \text {} \end{aligned}$$(5)$$\begin{aligned} \pi _G(p_G,2)&= (p_G - c_G - R)f_{G_2} \end{aligned}$$(6)where R is the royalty that G needs to pay for a unit of the drug sold.$$\begin{aligned} \pi _I(p_I, A_I,2)&= (p_I - c_I)f_{I,2} - l A_I^2 + R f_{G_2}\text {,} \end{aligned}$$(7)
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In the last stage of the game, as the patent has expired, the generic producer enters the market without the need for a limited commercial license. Hence, a classical price competition takes place. Sales functions and profits are described, respectively, as$$\begin{aligned} f_{G_3} =&f_G\left( x_I,x_G, p_I, p_G, A_I,3\right) = a - b_G p_G + \theta A_I \end{aligned}$$(8)and$$\begin{aligned} f_{I_3} =&f_I\left( x_I,x_G, p_I, p_G, A_I,3\right) = a - b_I p_I + \theta A_I \text {,} \end{aligned}$$(9)$$\begin{aligned} \pi _G(p_G,3)&= (p_G - c_G )f_{G_3} \end{aligned}$$(10)Here, we are implicitly assuming that as the patent expires the innovator stops research in product innovation.$$\begin{aligned} \pi _I(p_I, 3)&= (p_I - c_I)f_{I_3} \text {.} \end{aligned}$$(11)
2.3 Switching Between Stages
In the setup we have in mind, switches between one mode of the system to another are endogenous to the problem itself. In particular, when the innovator enters a market with a new drug, they does not know if and when a limited commercial license would be issued at some future point in time. For this reason, we now introduce a continuous-time Markov chain with values in M that drives switches between modes, as it follows. Let us fix a probability space \(\left( \Omega ,\mathbb {F}, \mathscr {P}\right) \) and define a continuous-time Markov chain \(\xi = \xi (t) : \Omega \times [0, \infty ) \rightarrow M\), which generates the information flow represented by the family of \(\sigma \)-algebras \(\left\{ \mathscr {F}_t\right\} _{t\ge 0}\). This stochastic process is deterministic everywhere but at random times \(\tau _i \), \(i=1,2,\ldots \) where \(\xi \) jumps. The instant of times in which the chain jumps model the switches between regimes. The randomness of this process mimics the uncertainty of the market players about the eventual issue (and the time of issue) of the limited commercial license.
The probability law which drives the jumps of \(\xi \), and hence the switches between the stages of developments of the limited commercial license, is described by a set of function \(\lambda _{m,n}:\Omega \rightarrow \mathbb {R}_+\), for \(m,n \in M, m\ne n\), defined aswhich defines the conditional probability, measured at time t, of switching from mode m to mode n of the system in an infinitesimal amount of time dt to be proportional to dt. Such functions, usually known as hazard functions, are part of our modeling framework, as they completely characterize the Markov chain itself.
$$\begin{aligned} \lambda _{m,n}(t,x_I,X_G,p_I,P_G,A_I) = \lim _{dt\rightarrow 0}\frac{\mathbb {P}\left( \xi (t+dt) = n | \xi (t) = m\right) }{dt} \end{aligned}$$
(12)
In our setup, the system might go from mode 1 (the innovator has entered the market) to mode 2 (the limited commercial license has been issued) if the sales of drug are sufficiently low. It is thus meaningful to assume that the corresponding hazard function \(\lambda _{1,2}\), which roughly speaking defines the probability of issuing the CL, is a decreasing function of the cumulative sales in mode 1 of the system, as follows:On the other hand, the CL may never be issued if the drug reaches a considerable amount of potential patients. In such cases, the system might go directly from stage 1 to stage 3, where the patent has expired. In the same spirit of our reasoning above, the hazard function for such kinds of switches should be directly proportional to the amount of sales in mode 1, that is:To close our model, we need to define the intensity of switches between mode 2 and mode 3 of the system. Since the expiration of the patent is independent on the sales of the product, we just assume a constant hazard function, that is
$$\begin{aligned} \lambda _{1,2}= \lambda _{1,2}(t,x_I,X_G,p_I,P_G,A_I) = \frac{\delta _{1,2}}{x_I}\text {.} \end{aligned}$$
$$\begin{aligned} \lambda _{1,3}= \lambda _{1,3}(t,x_I,X_G,p_I,P_G,A_I) = \delta _{1,3}{x_I}\text {.} \end{aligned}$$
$$\begin{aligned} \lambda _{2,3}= \lambda _{2,3}(t,x_I,X_G,p_I,P_G,A_I) = \delta _{2,3}\text {.} \end{aligned}$$
2.4 The Problem
The problem is defined as a differential game, in which both players choose their strategies so as to maximize the discounted expected value of future profits. Let r be the rate used by both players to discount their profits. Define the objective functionals of both players, in each mode of the system m, as follows:Then, both players choose their pricing and product development strategies so as to maximize the objective functional above under the dynamic constraints given by (12) and
$$\begin{aligned} \begin{aligned} J_G(x,y,p_I,p_G, A_I, m)&= \mathbb {E}\left( \int _0^\infty e^{-r s}\pi _G\left( p_G,\xi (s)\right) |x_I(0) = x; x_G(0)=y; \xi (0) = 1\right) \\ J_I(x,y,p_I,p_G,A_I, m)&= \mathbb {E}\left( \int _0^\infty e^{-r s}\pi _I\left( p_I,A_I,\xi (s)\right) |x_I(0) = x; x_G(0)=y; \xi (0) = 1\right) \end{aligned} \end{aligned}$$
$$\begin{aligned} \dot{x_i}(t) = \frac{d x_i(t)}{dt} = f_i\left( x_I,x_G, p_I, p_G, A_I,m \right)&\quad i \in {I,G}; m \in M\text {.} \end{aligned}$$
(13)
3 Solving the Model
To solve the problem, one must first choose the type of strategies available to the players. As it is standard in the theory of stochastic (in this case piecewise deterministic) differential games, two classes of strategies are available. Open-loop strategies correspond to an entire path chosen for the control variable, while with closed-loop strategies (also known as feedback or Markov strategies) the control variable are a function of the observed level of the state space. The latter class of strategies is more appealing as it appears to be more realistic than open-loop strategies. On the other hand, solving for feedback strategies is, in general, much more difficult. In the case of the problem at hand, we do believe that it can be solved in closed-loop strategies, with the help of the paradigm of backward induction and numerical techniques.
Let us begin with feedback strategies, also known as Markovian strategies. In this setup, the optimal controls are expressed in terms of the state variables, that is \(p_i(t) = \psi _i\left( x_I(t),x_G(t)\right) \). This means that at each time players perform an action based on the observed state of the system. In this case, the solution of the problem can be expressed in terms of a system of Hamilton-Jacobi-Bellman equations. Let \(V_i(x,m)\) denote the value function of player \(i \in \{I,G\}\) in regime \(m \in \{1,2,3\}\), with \(x = (x_I,x_G)\). Then, for \(V_i(x,m)\) to be the solution of the piecewise-deterministic differential game, they must solve the following system:Let us now focus on open-loop strategies. In this case, the actions of the players are expressed in terms of time, the regime of the system and the state of the system at the last observed switch, that is \(p_i(t) = \phi _i\left( m\left( s(t)\right) , x\left( s(t)\right) ,t-s(t)\right) \), where s(t) denotes the time in which the last switching of regime occurred before time t. In such case, the value functions depend explicitly on time. Thus, for \(V_i(x,m,t)\) to be the solution to the problem under piecewise open-loop strategies, they must solve the following system of Hamilton-Jacobi-Bellman equation:The systems of equation in (14) and (15) describe sufficient conditions for the solution of the problem sketched in Sect. 2. However, both conditions do not admit closed-form solution, so that we must use a mix of analytical and numerical techniques to derive the optimal policies followed by the players. One approach is to discretize the system of Hamilton-Jacobi-Bellman equation by means of a semi-lagrangian approach (Falcone and Ferretti 2013).1 This entails splitting the time horizon into a sequence of equidistant steps. We then approximate the variable x(t) by means of the sequence \(x^h_n\). We will make use of the conditional probability that the process will jump from mode i to mode j in an analogous time step, which we approximate asThe continuous-time optimal control problem is thus replaced by the following first-order discrete-time approximationwhere we set the discount factor \(\beta = e^{-\omega h}\). Camilli (1997) shows that \(V^h(x,i)\) satisfies the following dynamic programming equationFinally, (18) gives the discrete-time infinite dimensional system of equations satisfied by the value functions \(V^h(x)\overset{def}{=}\ \left\{ V^h(x,i):\; i \in I \right\} \)where the dynamic programming operators \(\mathscr {N}_i(\cdot )\) are defined byProblem (20) is still infinite dimensional in the state variable. However, we can convert it into a set of finite-dimensional equations by partitioning the state space into a grid \(\Gamma = \{x_k : k=1,\ldots , K\}\) and solve (20) only for \(x \in \Gamma \). To make the scheme operative, we need to reconstruct the values \(V^h(x_k+h f(x_k,\alpha ,i),i)\) since in general the points \(x_k+h f(x_k,\alpha ,i)\) do not coincide with any point of \(\Gamma \).
$$\begin{aligned} \begin{aligned} r V_i(x,m) = \max _{p_i} \left\{ \pi _i(p_i,m) +\frac{d}{dx_I} V_I(x,m) f_I(x, p_I,p_G,h) +\right. \\ \frac{d}{dx_G} V_G(x,m) f_G(x, p_I,p_G,h) + \left. \sum _{n\ne m} \lambda _{n,m}\left( V_i(x,n)-V_i(x,m)\right) \right\} \end{aligned} \end{aligned}$$
(14)
$$\begin{aligned} \begin{aligned} r V_i(x,m,t ) - \frac{d}{d_t}V_i(x,m,t ) = \max _{p_i} \left\{ \pi _i(p_i,m) +\frac{d}{dx_I} V_I(x,m,t) f_I(x, p_I,p_G,h) +\right. \\ \frac{d}{dx_G} V_G(x,m,t) f_G(x, p_I,p_G,h) + \left. \sum _{n\ne m} \lambda _{n,m}\left( V_i(x,n,0)-V_i(x,m,t)\right) \right\} \end{aligned} \end{aligned}$$
(15)
$$\begin{aligned} P^h_{x,i,j}(q) = 1-e^{-h \lambda _{i,j}(x,q)}. \end{aligned}$$
(16)
$$\begin{aligned} V^h(x,i) = \max _{q_1,q_2,\ldots } E_{x,i}\left\{ \sum _{l=0}^{\infty }\sum _{n=N_l}^{N_{l+1}-1}h \beta ^n\pi ^{\xi _l}(x^h_n,q_{n-N_l})\right\} , \quad i\in I\text {,} \end{aligned}$$
(17)
$$\begin{aligned} V^h(x,i) = \max _{q} E_{x,i}\left\{ h \pi ^i(x,q) + \beta V^h(x^h_1,i)\right\} . \end{aligned}$$
(18)
$$\begin{aligned} V^h(x,i) = \mathscr {N}_i\bigl ( V^h(x)\bigr ) \quad i \in I, \end{aligned}$$
(19)
$$\begin{aligned} \begin{aligned} \mathscr {N}_i\bigl (V^h(x)\bigr ) \overset{def}{=}\ \max _{q} \left\{ h \pi ^i(x,q) + \right. \beta P^h_{x,i}(q)V^h(x+h G(x,q,i),i)+ \\ \sum _{j \ne i } P^h_{x,i,j}(q)V^h(x,j).\} \end{aligned} \end{aligned}$$
(20)
4 Perspective
We have outlined a novel framework for the analysis of investments of a large company under the risk of limited commercial licensing. The framework captures all the key features on the subject and allows researchers to tackle important research questions such as
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How does the risk-limited commercial licensing affect the investment patterns of innovators in research and development?
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How does limited commercial licensing impact in the pricing of new, licensed, products?
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Can a regulator devise a licensing mechanism that boosts investments in research and developments and protect public welfare at the same time?
In future research, we plan to exploit the setup produced in this paper with the aim of tackling the research questions above.
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