Anmina Murielle Djiguemde

Dimitri Dubois

Alexandre Sauquet
email: alexandre.sauquet@inrae.fr.

Mabel Tidball

Dubois, D M Djiguemde

Continuous versus Discrete Time in Dynamic Common Pool Resource Game Experiments

Keywords: Common Pool Resource, Di erential Games, Experimental Economics, Continuous Time, Discrete Time JEL Codes : C01, C73, C91, C92, Q20

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Introduction

On many issues, we have the possibility of taking decisions at any moment in time, and asynchronously with other agents: sending a message, extracting water from a groundwater table, reducing prices, etc. Many of the interactions we engage in have a real-time aspect.

How does this ability to rapidly and asynchronously adjust actions shape our behavior? This question has been of deep interest for behavioral and experimental economists over the past decade. Indeed, many questions that were initially analyzed in discrete time in laboratory experiments can today be analyzed using continuous time protocols that allow researchers to compare the behavior of agents in discrete versus continuous time.

Previous articles nd that continuous time can foster cooperation, but only under certain conditions. When presenting prisoner's dilemma games to two-person groups in three treatments, one in continuous time, one in static time (one-shot) and one in discrete time, [START_REF] Friedman | A continuous dilemma[END_REF] nd a higher median cooperation rate in continuous time. [START_REF] Bigoni | Time horizon and cooperation in continuous time[END_REF] combine elements of the design of [START_REF] Bó | Cooperation under the shadow of the future: experimental evidence from in nitely repeated games[END_REF] and of [START_REF] Friedman | A continuous dilemma[END_REF] to study cooperation in repeated prisoner's dilemma. They nd that contrary to previous results in discrete time, cooperation is easier to achieve in continuous time with a deterministic time horizon than with a stochastic time horizon. [START_REF] Oprea | Continuous time and communication in a publicgoods experiment[END_REF] study subjects' contributions in a public good game played in groups of ve people. They nd players contribute higher amounts in continuous time than in discrete time but only when a rich communication protocol among participants is included. Introducing new laboratory methods in order to eliminate inertia in a subject's decision in continuous time experiments, [START_REF] Calford | Continuity, inertia, and strategic uncertainty: A test of the theory of continuous time games[END_REF] nd strikingly di erent behaviors in continuous vs. discrete time in a simple timing game where two participants compete to enter a market. Finally, [START_REF] Leng | A minimum e ort coordination game experiment in continuous time[END_REF] study the evolution of cooperation by crossing time protocols (continuous vs. discrete time) and information feedback (group minimum e ort level vs. e ort level of each member of the group) in a minimum e ort coordination game played in groups of six people. Among the four treatments, the authors nd that the average payo increases only when continuous time is associated with the provision of information on the e ort level of each member of the group.

Although studying interactions in the prisoner's dilemma, public good, timing, or mini-mum e ort coordination games is extremely useful, these games abstract from a feature relevant to many economic applications, the presence of a state variable that makes the impact of any decision to persist through time, which is the case in common pool resource (CPR) games [START_REF] Vespa | An experimental investigation of cooperation in the dynamic common pool game[END_REF]. The vast majority of the CPR literature that combines theory and experimentation is in discrete time. A possible explanation is that discrete time is easier to implement in the lab and can be compared to a static repeated game in which the state variable evolves from one period to the other [START_REF] Herr | An experimental study of time-independent and time-dependent externalities in the commons[END_REF], Gardner et al. 1997[START_REF] Mason | Mitigating the tragedy of the commons through cooperation: an experimental evaluation[END_REF][START_REF] Hey | An experimental analysis of optimal renewable resource management: the shery[END_REF][START_REF] Suter | Behavior in a spatially explicit groundwater resource: Evidence from the lab[END_REF], for instance). Nevertheless, [START_REF] Tasneem | An experimental study of a common property renewable resource game in continuous time[END_REF] recently tested a CPR di erential game in the lab using a continuous time protocol. Focusing on Markov's perfect equilibrium strategy, they tried to determine the relevance of the nonlinear equilibria in a two-player common property resource game. [START_REF] Janssen | Lab experiments for the study of socialecological systems[END_REF] have also studied the role of communication and punishment in a CPR game in continuous time. They nd that punishment can foster cooperation only when combined with communication. The authors do not present the formal theoretical model underlying their experiment. 1In this paper we build on the previous literature to study the impact of the nature of time in a two-person common pool resource (CPR) problem. Several important di erences with previously tested games (prisoner's dilemma, public good, timing, and minimum e ort coordination games) can lead to a di erent impact of the nature of time. First, the presence of the state variable makes the impact of any decision to persist through time [START_REF] Vespa | An experimental investigation of cooperation in the dynamic common pool game[END_REF], which can, for instance, generate dynamic free riding (Battaglini et al. 2016).2 Moreover, as opposed to the prisoner's dilemma, where payo s can be directly read from a matrix, dynamic games are more di cult to handle. These two elements can make the optimal solution harder to reach in the case of CPR games. Reversely, in nite horizon can provide strategic opportunities to endogenously support cooperative outcomes (Battaglini et al. 2016). In addition, using dynamic CPR games allows us to explicitly derive equilibrium paths for three well identied types of behavior -myopic, feedback and optimal. How does the nature of time a ect strategic interactions in this context? Can continuous time still foster cooperation? Does the nature of time a ect the equilibrium path to which participants are the closest?

To analyze these questions, we consider a simple linear quadratic model, based on [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF], [START_REF] Negri | The common property aquifer as a di erential game[END_REF], and [START_REF] Rubio | Strategic behavior and e ciency in the common property extraction of groundwater[END_REF], in which agents exploit a renewable resource with an in nite horizon. The resource can be assimilated to a groundwater basin but other interpretations of CPR are possible. We use a di erential game for continuous time and propose a discretization of the CPR game so that the equilibrium paths for myopic, feedback and optimal behaviors are almost identical in the discrete and continuous time models. For the implementation in the lab we choose to lead a non-contextualized experiment in a between-subject design with four treatments. We cross the nature of time (discrete versus continuous) and the number of subjects exploiting the resource (one versus two). In the continuous time treatments, we follow the literature and mimic continuous time by allowing the agent to change his extraction level every second. In the discrete time treatments, the agent can change his extraction level every period. About one hundred subjects participated in each treatment.

Presenting subjects with the simplest setting, i.e., a single agent exploiting the resource, allows us to test whether the ability to manage a resource di ers in continuous and discrete time. Indeed, the greater number of decisions potentially taken in continuous time could facilitate a trial and error process to reach optimal management of the resource. It is important to establish this baseline because, as explained earlier, dynamic situations are complex problems to handle, and it is important to understand the impact of the nature of time without interactions. Our estimates indicating that only 37% of the agents play optimally, con rms this statement. Our results also show that in all aspects tested, a subject's ability is not affected by the nature of time in a single agent setting. This allows us to deduce that the di erences observed in the multiplayer setting are due to the impact of the nature of time on the interactions.

When running the experiment in a multiplayer setting, we nd striking di erences be-tween continuous and discrete time. For example, the average resource level is signi cantly lower in discrete time. There is a larger proportion of agents that can be classi ed as myopic and a larger proportion of agents that end up with a low resource level in discrete time, while the proportion of optimal and feedback agents are not signi cantly di erent between the discrete and continuous time. Continuous time seems to favor a more sustainable exploitation of the resource. Our underlying intuition for this result is similar to [START_REF] Friedman | A continuous dilemma[END_REF], [START_REF] Oprea | Continuous time and communication in a publicgoods experiment[END_REF] and [START_REF] Leng | A minimum e ort coordination game experiment in continuous time[END_REF]. Continuous time allows subjects to brie y switch to cooperative behavior, such as a socially optimal extraction level, in order to incite the other player to do the same, or conversely to quickly increase extraction if the other player increases their extraction too much. The fact that we observe more stable extraction levels in continuous time and that extraction levels are more homogeneous within the group is consistent with this potential explanatory mechanism. It also results in less unequally distributed payo s in continuous than in discrete time.

Through this work, we provide several contributions to the literature. We o er the rst in-lab analysis of the impact of discrete versus continuous time in the lab in the case of CPR games. We contribute to the analysis of common pool resources using di erential games, by being the rst experimental paper to consider socially optimal and myopic strategies in a continuous time setting. We also make two secondary contributions. We clearly present the experimental protocol allowing the implementation in the laboratory of a continuoustime model with an in nite horizon. Finally, to compare the behavior of subjects in the lab to theoretical projections, we combine mean-squared deviation statistics and linear regressions.

The next section of this paper presents the theoretical setting. Section 3 describes the experimental design used to test the theoretical model. Section 4 is devoted to the empirical strategy, and results are analyzed in Section 5. The nal section provides a discussion and conclusion.

The Model

We consider a simple linear quadratic model in which two agents, i, j exploit a renewable resource over an in nite horizon. The resource can be assimilated to a groundwater table.

Water pumped provides agents revenue B(w) depending only on the extraction w. Agents also incur a cost C(H, w), which depends negatively on the level of the groundwater H.

The parameters a, b, c 0 and c 1 are positive. An agent's instantaneous payo is given by the di erence between revenue and cost, as shown by equation (1):

B(w) aw - b 2 w 2 - marginal cost (c(H)) max(0, c 0 -c 1 H) w C(H,w) (1)
We take into account the positivity of the marginal or unitary cost c(H), so that it is important to adopt a piecewise marginal cost function (2) to prevent agents from perceiving the cost as a subsidy.

c(H) =      (c 0 -c 1 H) if 0 ≤ H < c 0 c 1 0 if H ≥ c 0 c 1 (2)
In the model, agents have to choose an extraction level that maximizes their instantaneous payo . The problem di ers between continuous time and discrete time. In continuous time, decisions are made at each instant t in real time and the resource evolves continuously, while in discrete time, decisions are made at each period n and the resource evolves from one period to the next. Whether in continuous or discrete time, the behavior of agents is analyzed in two settings. First, in an optimal control problem, where a sole agent exploits the groundwater, we characterize both the myopic and the optimal behaviors. Second, the behavior of agents can be analyzed in a game, where strategic interaction is introduced by considering two identical and symmetrical agents in the exploitation of the groundwater. A feedback equilibrium path can be de ned, in addition to the myopic and optimal equilibrium paths in the game.

Social optimum can be de ned as a behavior in which an agent's extraction decision allows him to maximize his discounted net payo s in order to maintain the resource at an e cient level. The social optimum is also called the "cooperative solution" in the game. In that case the resource is maintained at an e cient level by maximizing the joint discounted net payo of all agents. The myopic solution is where the agent is only interested in the maximization of his current payo (equation ( 1)), regardless of the evolution of the groundwater.

The feedback equilibrium can be seen as a scenario in which agents adopt non-cooperative behavior, maximizing their own discounted net payo s while also taking into account the evolution of the groundwater.

In continuous time, the total discounted payo (with r the discount rate) for player i is:

∞ 0 e -rt aw i (t) - b 2 w i (t) 2 -max(0, c 0 -c 1 H(t))w i (t) dt (3)
and the dynamics is given as:

                 Ḣ(t) = R -α(w i (t) + w j (t)) H(0) = H 0 and H 0 ≥ 0, H 0 given H(t) ≥ 0 w i (t) ≥ 0
In discrete time, the total discounted payo (with 1 -rτ the discount factor) and the dynamics for player i are given as:

∞ n=0 (1 -rτ ) n aw i (n) - b 2 w i (n) 2 -max (0, c 0 -c 1 H(n)) w i (n) τ (4)                  H(n + 1) = H(n) + τ (R -α(w i (n) + w j (n))) H(0) = H 0 and H 0 ≥ 0, H 0 given H(n) ≥ 0 w i (n) ≥ 0
The discretization rate τ chosen in discrete time provides a good approximation of the continuous time problem, and optimal solutions can be found by means of the Hamiltonian operator. The Nash feedback equilibrium in continuous time can be found by means of the Hamilton Jacobi Bellman (HJB) equation, by applying the guessing method to guess a quadratic value function and in discrete time by means of the Bellman equation. Finally, myopic solutions are obtained by means of a simple rst-order derivative. They can also provide a feedback representation of the solutions when considering the constraints.3 When w j is dropped from the dynamics, one is able to solve the optimal control maximization problem (the sole-agent setting).

The Experiment

The computer is naturally unable to implement "pure" continuous time. Thus, we proposed a discretization of our theoretical model in continuous time, and by varying the discretization step, we were able to implement both an experiment which approximates continuous time and one which approximates discrete time. The discretization procedure is detailed in

Appendix A.
In what follows, we present the implementation of the di erent components of our experiment: continuous time and discrete time with a sole player, continuous time and discrete time with multiple players, the in nite horizon, and the choice of the parameters.

Experimental Design

The experiment took place at the Experimental Economics Laboratory of Montpellier (LEEM).

From December 2019 to February 2020, a total of 200 students from the University of Montpellier participated in the rst part of the experiment. This part was devoted to data collection for the single agent condition. It included a total of 17 sessions, 11 for the continuous-time treatment and 6 for the discrete-time treatment. 4 From November to December 2020, a total of 190 subjects participated in the second part of the experiment, which was devoted to data collection for the two-players game. The experiment involved 20 sessions of continuous and discrete time treatments for groups of two players, so that we had 49 groups in continuous time and 46 groups in discrete time. 5 It was a non-contextualized experiment, using the oTree platform [START_REF] Chen | oTree-an open-source platform for laboratory, online, and eld experiments[END_REF], in which subjects participated in a ten-minute training phase of the game, followed by a ten-minute e ective phase of the game which counted for their remuneration. The experimental currencies (ECU) accumulated by subjects in the experiment were converted into cash payments with the conversion rate of 10 ECUs to 0.5 euro. 6 Each experimental session lasted around an hour.

Experimental Procedures

Global Description

We used a between-subject design in which participants in the sole-agent treatments were di erent from the ones in the multiple-agent treatments. In the sole-agent treatments, instructions explained the dynamics of the resource, the decision-making process and its consequences on the available resource, the cost of extraction and the payo . After an initial individual reading, an experimenter proceeded to an outloud reading of the instructions.

Next, subjects answered a digital questionnaire to make sure they understood the evolution of the resource as well as the computation of payo s. They were also invited to ask questions by raising their hands.

To familiarize subjects with the graphical interface, they participated in a 10-minute training phase before a 10-minute paid phase. At the beginning of each phase, subjects had to choose an initial extraction between 0 and 2.8 by moving their cursor on a graduated slider, However, for the sake of simpli cation, we explain the dynamics in continuous time rather than writing the di erential equation. Once the subjects chose an initial extraction level, a new screen appeared and subjects were able to see the dynamics of the resource along with their payo , which included the cumulative and continuation payo s, updated every second in the continuous time treatment and every period in the discrete time treatment.

Adapted instructions were provided to subjects in the multiple-agent treatments. Environments remained the same as in the sole agent treatments, except that subjects extracted the resource in groups of two. The layout of the user interface was slightly di erent from that of the sole agent treatments, with an additional curve showing the pair's total extraction. Complete instructions for the four treatments can be found in the online supplementary materials.

Parameters

Table 1 reports the parameters used. To get comparable results, parameters were the same in continuous time and discrete time for both the sole-and multiple-agent treatments. We also chose these parameters so that the steady state level of the resource in the socially optimal case is strongly separated from other cases. The socially optimal behavior leads to a high level of the groundwater, while the myopic behavior results in low groundwater levels (see the right sides of gures 1 and 2).

Both the natural recharge R and the return ow coe cient α were designated at a small enough size to capture the renewable nature of the resource, simulate real life conditions and avoid oods in the model. 7In situations where a subject's extraction is higher than the available resource, the rule was to set the extraction to zero until she changed her decision or until the amount of the resource increased enough to allow for a new extraction. This rule was chosen because it is easy to implement in the lab and because setting an allocation rule for the extraction in proportion to the available resource would have led to a multiplicity of equilibria, which would have greatly complicated the empirical strategy needed to compare lab results to equilibrium paths without revealing any (particularly) interesting information on the behavior of agents.

Decision Timing in Continuous and Discrete Time

In the sole agent continuous time treatment, subjects were able to change their extraction rate at any moment by simply moving the graduated slider displayed on their computer.

Every second, the computer transmitted the slider value to the server, which then performed the computations (resource and payo ) and updated the values displayed on the computer's graph and text interfaces.

In the two-player continuous-time treatment, player 2's computer sent the cursor value to the server as soon as it changed, while player 1's computer transmitted the cursor value to the server every second, which triggered the server to continuously broadcast the updated values to both players. Thus, every second, the server took player 1's current extraction and player 2's most recent extraction (i.e. the last one transmitted by his computer). In this way, the time was synchronized between the two members of the group, since only one player was triggering the continuous updating of the information.8

Since we have tried to provide an experiment that is as close as possible to continuous time, we have chosen a discretization step that is as small as possible, τ = 0.1, to capture the speci c characteristic of continuous time, i.e., its uninterrupted evolution. This means that in our continuous time treatment, one second of real time corresponds to 0.1 instant in the model. Thus, 10 minutes of experiment are equal to 600 seconds and equivalent to 60 instants.

In the discrete time treatment, we have chosen a larger but reasonable discretization rate, τ = 1. With this rate, 1 period equals 1 instant in the model. Therefore, subjects participated in a 60-period dynamic environment. In addition, in order to ensure a similar duration in both treatments, we gave the subject exactly 10 seconds in each period to take her decision, which means that the play time was also 10 minutes in discrete time.

The graphical user interface was divided into four areas. On the top left, a graph showed the evolution of the player's extraction. At the top right, a graph displayed the evolution of the resource, and at the bottom left there was a graph showing the evolution of the payo .

Finally, at the bottom right, a text box presented the same information as the graphs but in text form.

In nite Horizon

In both continuous and discrete time, the in nite horizon is implemented through the payo , as in [START_REF] Tasneem | An experimental study of a common property renewable resource game in continuous time[END_REF] and [START_REF] Tasneem | Sustainable management of renewable resources: an experimental investigation in continuous time[END_REF]. The payo is composed of two elements:

(i) a cumulative payo from the rst instant of play (t = 0) to the present instant (t = p), and (ii) a continuation payo , which is computed as an integral of payo s from the present instant (t = p) to in nity (t = ∞), assuming that the player's extraction remains unchanged.

In the two-player game, the continuation payo was calculated assuming that both players' extraction remained unchanged.

The cumulative payo in continuous time corresponds to the discounted integral of the instantaneous payo s from the beginning of the experiment up to the present instant. Thus, the discount rate is r = 0.5% and means that the payo of instant t is multiplied by e -0.005×t .

The discounting principle allows subjects to understand that the same instantaneous payo has a di erent discounted value according to the instant. In other words, as time goes on, the payo s of the last instants have a lesser impact on the subject's total payo for the experiment. Similarly, the cumulative payo in discrete time corresponds to the discounted sum of each period's payo from the beginning of the experiment up to the present period.

Thus, the discount factor is β = 0.995 and means that the payo of period n is multiplied by 0.995 n . The discounting principle allows subjects to understand that the same payo has a di erent discounted value according to the period. In other words, in the experiment, the same instantaneous payo contributes less to the total nal payo when it occurs in the later periods rather than in the earlier periods.

Empirical Strategy

Two hundred subjects participated in the sole-agent (optimal control) experiment and 190 in the multiple-player (game) experiments. They took (paid) extraction decisions for 600 seconds during each session. We use these extraction decisions data to understand whether agents take di erent decisions in continuous vs. discrete time, and in the control vs. in the game. Through the empirical analysis, we use standard tests such as the Mann-Whitney and the Fisher exact proportion tests to compare our indicators among the di erent treatments.

Furthermore, to determine whether agents demonstrated myopic or optimal behavior (or feedback behavior in the game), we use the empirical strategy presented in this section. For ease of understanding, the empirical strategy is rst explained in detail for the sole-agent setting.

To identify which theoretical extraction pattern an agent's extraction comes closest to, a widely used statistics is the mean squared deviations (MSD, e.g., [START_REF] Herr | An experimental study of time-independent and time-dependent externalities in the commons[END_REF]). The minimum MSD gives the agent type. The MSDs are calculated for each agent such that:

M SD th my = T t=1 w(t) -w(t) th my 2 T M SD th op = T t=1 w(t) -w(t) th op 2 T (5)
where w(t) is the extraction of the agent at time t, w(t) th my is the constrained myopic theoretical extraction at time t, and w(t) th op is the optimal theoretical extraction at time t. Agents can be classi ed as myopic or optimal, depending on which MSD, M SD th my or M SD th op is the smallest. Comparing extractions of the agent to the theoretical constrained myopic and optimal extraction in this way is imperfect since an agent can make mistakes and begin adopting an optimal path after, say, 30 seconds, which will not be captured correctly by the method.

For instance, if an agent under-extracts for the rst 30 seconds, the optimal extraction at time 31, given the observed groundwater level H (called conditional, w(31) c op ) will be greater than the optimal extraction at time 31 if the agent behaved perfectly optimally from time 0 (w(31) th op ). Thus, in order to correctly identify an agent's behavior type -myopic or optimalwe compare observed extraction to conditional extractions throughout the remainder of the paper. Conditional extractions are computed with respect to the t -1 actual groundwater level. Thus, we compute the following MSDs :

M SD c my = T t=1 w(t) -w(t) c my 2 T M SD c op = T t=1 w(t) -w(t) c op 2 T , (6)
where w(t) c my is the conditional constrained myopic extraction of the agent at each second (every ten seconds for discrete time), and w(t) c op is the conditional optimal extraction of the agent. Agents are classi ed as myopic or optimal depending on which MSD, M SD c my or M SD c op is the smallest. The inconvenient of a classi cation of agents based on the MSD alone is that an agent will always be classi ed, even if he doesn't follow the theoretical patterns studied at all. 9 To overcome this aw, we add a second criteria based on a regression analysis. Supposing that for a given agent, we have:

w(t) c my < w(t) c op , or w(t) c my > w(t) c op , (7)
then we run the following regression:

w(t) = β 0 + β 1 w(t) c my + ε t , or w(t) = β 0 + β 1 w(t) c op + ε t . (8)
We consider an agent to be signi cantly myopic (or optimal) if β 1 is positive and signicantly di erent from 0. This allows us to categorize the agents as: myopic, optimal, or undetermined. 10 Regarding the econometric time series treatments, we implement an augmented Dickey-Fuller test to detect the presence of unit roots in the series. In case of non-stationarity of the variables, we run our regressions on a di erentiated series. Serial correlation of the error terms is dealt with using Newey-West standard errors, and sensitivity tests using 1, 5, and 10 lags are implemented. 11

We follow exactly the same strategy to analyze experimental data for the game, but this time for three instead of two predicted behaviors, namely: myopic, optimal and feedback.

Note that the continuous time framework provides us with 600 decisions per agent, while 9 To take a concrete example, instead of comparing the agent's extraction w(t) to the conditional constrained myopic and conditional optimal extraction, w(t) c my and w(t) c op , we could compare it to the temperature in Moscow and Istanbul from day 1 to day 600, and we would nd that our agent's extraction is closer to the temperature either in Moscow or in Istanbul, because one MSD will always be smaller than the other, even if completely irrelevant.

10 An alternative is proposed by [START_REF] Suter | Behavior in a spatially explicit groundwater resource: Evidence from the lab[END_REF], who run a similar regression (without the constant term) and consider that an agent follows a given behavior if the coe cient is not signi cantly di erent from 1. A natural way to do this is to implement a Wald test with:

H 0 : β 1 = 1 H A : β 1 = 1, and W = ( β1 -1) 2 var( β1 ) → F (1,300)
In this case, a very imprecisely estimated coe cient β 1 (very large var( β1 )) will lead us to reject H A and classify the agent as myopic or optimal, although he follows neither an optimal or myopic path. This is the reason why we propose the aforementioned alternative rule for classi cation. 11 We present regression results using 1 lags. Results using 5 and 10 lags are available upon request.

the discrete time framework provides us with only 60. This greatly impacts our empirical strategy as β-coe cients would have more chances to be signi cant in continuous time -a greater number of observations leading to a lower minimum e ect size. To avoid this issue, we keep only one observation every ten seconds when running the regressions in continuous time.

Results

Figure 3 presents an overview of our results. We plotted the mean resource by treatment along with the 95% con dence interval around the estimated mean. It seems we have close average resource levels in the two time treatments in the control, but di erent ones in the game. Also, the average resource level increases in the control and decreases in the game.

Figure 3 -Evolution of mean resource level by treatment

In the rest of the section we take a closer look at what happens within each treatment. We rst compare the agents in the control setting. Second, we compare the average behaviors in the control and in the game. Third, we thoroughly study behaviors in the game. Finally, we build speci c indicators to examine the potential mechanism at play. Note that through the rest of the paper, the term 'agents' is used to refer to subjects in the control, the term 'players' to subjects in the game, and the term 'groups' to groups of two subjects that were paired in the game. perhaps more surprisingly, the number of times the players change their extraction level is not signi cantly di erent between the continuous and discrete time treatments, while in theory they had the possibility to change it 61 times in discrete time and 601 times in continuous time. The fact that we observe a substantial share of agents reaching a resource level above 20 and very few ending up with a resource level below ten is consistent with the fact that the average resource level in the control observed in Figure 3 is closer to the optimal than to the myopic path. This is con rmed by the MSDs map Figure 4, which presents the location of agents with respect to the M SD c op on the y axis and the M SD c my on the x axis. Agents located above the bisector can be considered as more myopic (M SD c op > M SD c my ) and vice versa. Very few agents have a greater M SD c op than the M SD c my , i.e., 19 over 200. This proportion is slightly lower in discrete than in continuous time (see the last test in Table 2).

Analysis of the Optimal Control

Figure 4 -Map of conditional MSDs in the control

As we explained in Section 4, using the MSD alone is unsatisfactory, because we want to know if agents are signi cantly optimal or myopic. Applying the regression lter presented in the previous section leads us to nd that in discrete time 33 agents can be classi ed as signi cantly optimal and one as myopic, and 41 can be considered optimal and four as myopic in continuous time. Proportions of optimal and myopic agents are not signi cantly di erent between the two treatments. As expected, average payo s are not signi cantly di erent either (see Table 3). The proportion of optimal agents seems comparable to the experiment of [START_REF] Tasneem | Sustainable management of renewable resources: an experimental investigation in continuous time[END_REF] who found that extraction behavior results in a steady state of the resource 56% of the time, with the mode of the distribution being optimal.12 Also, the average e ciency ratio (individual payo over the optimal payo , here 220 ECUs) is 83% in Tasneem et al. ( 2019)'s study while it is 88% in ours. [START_REF] Suter | Behavior in a spatially explicit groundwater resource: Evidence from the lab[END_REF] found a slightly higher e ciency ratio in the optimal control in a discrete time experiment, about 95%. To summarize, in a control setting, both continuous and discrete times lead to similar choices by participants. Having made this rst observation we now study how the nature of time a ects strategic interactions between players.

The Control Versus the Game

The rst observation that can be made by looking at Figure 3 is that the average level of the resource is lower in the game than in the control and decreases over time, whereas the resource level was increasing over time in the control. Mann-Whitney tests reported in Table 4 con rm that, compared to the control, the average resource level in the game is signi cantly lower and the average extraction level signi cantly higher. This is consistent with what one To summarize, even if we cannot prove the mechanism at play, the fact that extractions are more stable and that within-group di erences in nal payo s are lower in continuous time is consistent with the fact that continuous time o ers a less costly opportunity to in uence the other player's decisions. As a result, continuous time seems to reduce inequality in payo distribution, in addition to favoring more sustainable resource exploitation.

Discussion and conclusion

In this paper, we intended to determine the impact of the nature of time, discrete or continuous, on the behavior of agents in the context of a dynamic CPR game. To this end, we considered a simple linear quadratic model in which agents exploit a renewable resource over an in nite time horizon. Starting from a di erential game, we proposed a discretization such that the equilibrium paths for the myopic, feedback and optimal behaviors are almost identical in discrete and continuous time. We then took on the challenge of implementing continuous time and in nite horizon in the lab, allowing participants to make extraction decisions every second, and adding continuation payo s to cumulative payo s to simulate an in nite horizon.

To determine whether the nature of time has an impact on the ability of agents to manage a resource, we rst looked at the situation where the resource is owned by a single agent.

Observations showed no di erence between discrete and continuous time, based on a battery of indicators, including the average level of the resource, the average level of extraction, the proportion of myopic agents, and the proportion of optimal agents. Furthermore, about 35% of the subjects could be classi ed as signi cantly optimal and the average resource level increased over time, as is the case with the optimal solution.

In the context of a two-player game, the results were dramatically di erent. First, unlike what we observed with a single agent, the average resource level decreased over time, as is the case with the myopic and feedback equilibrium paths. Furthermore, only 2% of the groups behaved according to the optimal (cooperative) path. The competitive nature of the game when multiple players simultaneously extract on the same resource explains the di culty in adopting a sustainable path. Second, we observed signi cant di erences between discrete and continuous time settings. In particular, the discrete time setting led to the observation of a larger number of agents exhibiting myopic behavior, thus leading to a much lower average resource level than that observed in the continuous time setting. The continuous time environment seems to allow for better cooperation within groups and thus greater resource sustainability. Although our experimental design does not allow us to prove the exact mechanism at play, our intuition is consistent with [START_REF] Friedman | A continuous dilemma[END_REF], [START_REF] Oprea | Continuous time and communication in a publicgoods experiment[END_REF] or [START_REF] Leng | A minimum e ort coordination game experiment in continuous time[END_REF]: compared to discrete time, continuous time allows for rapid and adaptive strategic choices that promote the emergence of cooperation, either by attempting to in uence the other or by retaliating against their tendency to over-exploit the resource. The observed greater stability of continuous-time extraction, as well as the greater homogeneity within groups in this environment, is consistent with this explanatory mechanism.

We voluntarily used a very simple design, as to our knowledge we are the rst paper to test the impact of the nature of time in dynamic CPR games. Consequently, many extensions are possible. We hope our work can o er a basis for future works examining, for instance, whether continuous time can still foster cooperation when increasing the group size, as the continuous time frame by itself was able to induce cooperation compared to the discrete time frame in a two-person prisoner's dilemma in [START_REF] Friedman | A continuous dilemma[END_REF], but not in a ve-person public good game as in [START_REF] Oprea | Continuous time and communication in a publicgoods experiment[END_REF] or a six-person minimum e ort game as in [START_REF] Leng | A minimum e ort coordination game experiment in continuous time[END_REF]. Also, many re nements of the underlying theoretical model and of the game setting are possible. In particular, the role of major mechanisms such as rewards, punishments and communication settings in the continuous versus the discrete time frame remain to be examined.

Appendices A The Discretization of the Continuous Time Model

This section presents the procedure adopted to discretize the continuous time model. Let's consider the following continuous time model:

max w(t)
∞ 0 e -rt f (w(t), H(t))dt (9)

s.t        Ḣ(t) = R -αw(t) H(0) = H 0 ≥ 0, H 0 given H(t) ≥ 0 w(t) ≥ 0
For the discretization of the model above, let's consider τ as the discretization step and n as a period. Time is discretized into intervals of length τ , such that the di erential equation and the payo are approximated in each interval nτ , (n + 1)τ . Thus, the discretization of the objective function gives:

  which displayed values up to two decimal points. Due to the quadratic nature of our revenue function, any extraction level led to a positive revenue. Figure B.1 in the Appendix B shows a concave revenue curve with a maximum revenue reached for an extraction of 1.4. Figure B.2 in the Appendix B also shows the unitary cost function, which decreases as the available resource increases and vanishes when the level of the available resource is above 20. In the continuous time instructions, the extraction refers to an extraction rate, while in the discrete time instructions it refers to an extraction level. In addition, a distinction is made between the di erential equation representing the dynamics of the resource in continuous time and the di erence equation representing the dynamics of the resource in discrete time.

Figure 1

 1 Figure 1 and 2 below show the theoretical time paths for the extraction and resource levels in continuous time for 100 seconds. As the theoretical time paths in discrete time are almost identical to those in continuous time, we do not show them.

Figure 1 -

 1 Figure 1 -Extraction behaviors and resource levels in sole-agent continuous time

  Figure B.3 in Appendix B shows a screenshot of the user interface for the sole agent treatment in continuous time. In the multiple agent treatments, the user interface was identical except that an additional curve in the upper left graph showed the evolution of the group's total extraction.

Figure 5 -

 5 Figure 5 -Map of conditional group MSDs in the game

Figure 6 -

 6 Figure 6 -Cumulative density functions of the resource levels by type

  Figure B.3 -Decision-making screen shot. We follow a hypothetical subject who chooses his extraction rate at random

Table 1 -

 1 Parameters for the experiment

	Variable	Description	Value
	a	Linear parameter in the revenue function	2.5
	b	Quadratic parameter in the revenue function	1.8
	c 0	Maximum average cost	2
	c 1	Variable cost	0.1
	c 0 -c 1 H	Marginal or unitary cost	2 -0.1H
	r	Discount rate in continuous time	0.005
	β = (1 -rτ )	Discount factor in discrete time	0.995
	R	Natural recharge (rain)	0.56
	α	Return ow coe cient	1
	H 0	Initial resource level	15
	τ	Discretization step	0.1 & 1

  Table2compares continuous and discrete time over various indicators. The average resource level is not signi cantly di erent between the two treatments. About 40% of the players reach a resource level greater than 20 in each treatment (the optimal steady state resource level) and at approximately the same time. Only three agents in each treatment end up with a resource level below ten. Finally, the average extraction level is around 0.50 in both treatments and,

Table 2 -

 2 Continuous versus discrete time in the control

			Average agent's resource level	Mann-Whitney test
		Mean S.D.	N	Z-stat	Exact prob
	Discrete time	17.572 2.639	98	-0.98	0.328
	Continuous time 17.144 3.297	102	-	-
			Agents reaching R=20	Fisher exact test
		Yes	No	N	Odds ratio Exact prob
	Discrete time	39	59	98	0.983	0.535
	Continuous time	41	61	102	-	-
			Time agents reach R=20	Mann-Whitney test
		Mean S.D.	N	Z-stat	Exact prob
	Discrete time	23.795 13.546	39	-0.563	0.577
	Continuous time 23.115 15.460	41	-	-
			Agents ending up with R<10	Fisher exact test
		Yes	No	N	Odds ratio Exact prob
	Discrete time	3	95	98	1.042	0.640
	Continuous time	3	99	102	-	-
			Average agents extraction	Mann-Whitney test
		Mean S.D.	N	Z-stat	Exact prob
	Discrete time	0.497 0.064	98	0.992	0.322
	Continuous time 0.501 0.075	102	-	-
		Number of agents extraction change	Mann-Whitney test
		Mean S.D.	N	Z-stat	Exact prob
	Discrete time	34.122 17.603	98	-0.304	0.762
	Continuous time 44.902 47.515	102	-	-
		Agents with smaller M SD c my than M SD c op	Fisher exact test
		Yes	No	N	Odds ratio Exact prob
	Discrete time	6	92	98	0.446	0.087
	Continuous time	13	89	102	-	-

Table 3 -

 3 Classi cation and payo s in the control

		Proportion of optimal agents	Fisher exact test
		Yes	No	N	Odds ratio Exact prob
	Discrete time	33	65	98	0.755	0.209
	Continuous time	41	61	102	-	-
		Proportion of myopic agents	Fisher exact test
		Yes	No	N	Odds ratio Exact prob
	Discrete time	1	97	98	0.253	0.198
	Continuous time	4	98	102	-	-
		Average agents payo s	Mann-Whitney test
		Mean S.D.	N	z-stat	Exact prob
	Discrete time	191.370 38.497	98	-0.755	0.452
	Continuous time 196.605 16.878	102	-	-

Note also that some authors such as[START_REF] Noussair | Cooperation in a dynamic shing game: A framed eld experiment[END_REF] conduct their experiments in discrete time, while their theoretical model is in continuous time, which poses the question of to what theoretical predictions should we compare lab results: those from discrete or those from continuous time models? Moreover,[START_REF] Tasneem | Sustainable management of renewable resources: an experimental investigation in continuous time[END_REF] study the ability of a single economic agent to exploit a renewable resource e ciently. To do that they test in the laboratory an optimal control problem with an in nite horizon in continuous time and show that extraction behavior results in a steady state of the resource only 56% of the time.

Battaglini et al. (2016) de ne dynamic free-riding this way: "an increase in current investment by one agent[which] typically triggers a reduction in future investment by all agents". In the context of a CPR, a decrease in extraction level can be seen as an investment to obtain a higher resource level.

The feedback representation is obtained when the solution is written according to the state variable, instead of according to time.

Since the continuous time condition involves higher network tra c, we limited the number of participants per session to a maximum of 14, which explains the greater number of sessions for this treatment.

ORSEE[START_REF] Greiner | Subject pool recruitment procedures: Organizing experiments with orsee[END_REF] is the platform used by the LEEM to manage the subject pool.

ECU means Experimental Currency Unit.

The return ow coe cient is the quantity of water returning to the groundwater after each extraction.

This also reduced network tra c because as long as the second player did not change his extraction, his computer did not transmit a new value.

A more precise comparison of the results is not possible since the authors use a di erent empirical strategy.

The maximum group payo is 240 ECUs, so we computed the individual e ciency ratio by halving this value. Nevertheless, it is possible to get "more than your own share". Obviously, if one of the two members of the pair extracts a very small amount of groundwater, the other member can obtain more than 50% of the total maximum payo .

To make continuous and discrete time comparable, we take the di erence between two decisions separated by ten seconds in continuous time.

In Figure7.a we also see an increase in stability over time for both treatments. Note, however, that the greater instability in the beginning of the play time can be explained by the game setting. Indeed, players need rst to either let the resource grow or deplete it before reaching a steady state, depending on their preferred equilibrium.

To make continuous and discrete time comparable, we use only one decision every ten seconds in continuous time.

 17 The c.d.f. are statistically di erent according to the Kolmogorov-Smirnov test (p-value < 0.

05).18 The average di erence in extraction between players of the same group at the end of the game equals 0.18, while the average player's extraction level equals 0.27.

Concentration (Gini) indexes are signi cantly di erent whether we use the standard, Erreygers or Wagsta indexes (O'Donnell et al.

2016).

of the LEEM working group and Nicolas Querou for comments, the ANR GREEN-Econ [grant number: ANR-16-CE03-0005] for nancial support, and the Experimental Economic Laboratory of Montpellier for technical support. Online supplementary materials available here.

would expect if agents had unlimited rationality, since they would play optimal in the control and feedback in the game. In addition, we observe that agents change their extraction levels more often in the game than in the control. Finally, the MSDs map reported in Figure 5 shows that, compared to Figure 4, signi cantly more agents have a smaller M SD c my than M SD c op in the game than in the control (32 groups over 95, see Fisher test in Table 4). change in their behavior. For example, one player can temporarily lower his extraction level to see if the other player will do the same. This type of test is less expensive in continuous time than in discrete time. Indeed, in discrete time, the player can only make one decision per period and this corresponds to one instant, whereas in continuous time, the player can make one decision per second and this corresponds to only 0.1 of an instant. In other words, the opportunity cost of testing a strategy, in terms of payo , is much lower in continuous time, because only a fraction of the payo is given up during the temporary test strategy. This mechanism through which continuous time can foster cooperation was also advanced by [START_REF] Friedman | A continuous dilemma[END_REF], [START_REF] Oprea | Continuous time and communication in a publicgoods experiment[END_REF] and [START_REF] Leng | A minimum e ort coordination game experiment in continuous time[END_REF]. [START_REF] Oprea | Continuous time and communication in a publicgoods experiment[END_REF] calls this "pulse behavior" and sees it as a non-verbal form of communication. It can be used as a way to incite the other player to decrease extraction up to the optimal level or to retaliate if the other players increase their extraction level too much. Applying the regression lter presented in Section 4 leads us to nd that 14 groups (28 players) can be classi ed as signi cantly myopic in discrete time versus three groups in continuous time, making the proportion of myopic behavior signi cantly larger in discrete time.

Six groups are classi ed as feedback in the two treatments, and we nd only two optimal in discrete time and one in continuous time. Proportion of optimal and feedback agents are not signi cantly di erent between discrete and continuous time. Note that the presence of optimal groups is consistent with Battaglini et al. (2016)'s argument that in nite horizon can provide strategic opportunities to endogenously support cooperative outcomes.

As a result, we observe signi cantly higher average individual payo s in continuous time than in discrete time. E ciency ratios in the game are lower than in the control, and lower in discrete time (48%) than in continuous time (64%). 13 Finally, Figure 6 provides an overview of the results of the classi cation by type by plotting the cumulative density functions (c.d.f.) of the resource levels. The distribution of the observed resource levels rank as expected, with the myopic groups experiencing the lowest resource levels, followed by the feedback and optimal groups. The undetermined group

The discretization of the dynamics gives:

The discrete time problem can be de ned as:

The discrete time model therefore converges towards the continuous time model when the discretization step τ tends toward zero.

B Figures from Experimental Instructions


M De Angeli
email: marco.deangeli@istp.cnr.it

D Ripamonti

F Ghezzi

P Tolias

C Conti

C Arnas

M Jerab

D L Rudakov

C P Chrobak

J Irby

B Labombard

B Lipschultz

G Maddaluno

Cross machine investigation of magnetic tokamak dust: Morphological and elemental analysis

Keywords:

The presence of magnetic dust can be an important issue for future fusion reactors where plasma breakdown is critical. Magnetic dust has been collected from contemporary fusion devices (FTU, Alcator C-Mod, COMPASS and DIII-D) that feature different plasma facing components. The results of morphological and elemental analysis are presented. Magnetic dust is based on steel or nickel alloys and its magnetism is generated by intense plasma-material interactions. In spite of the strong similarities in terms of morphology and composition, X-ray diffraction analysis revealed differences in the structural evolution that leads to non-trivial magnetic responses.

Introduction

The presence of dust in fusion devices constitutes an important issue with safety and operational implications [1][2][3][4]. Despite the fact that ferromagnetic dust has been detected in a number of machines [5][6][START_REF] Novokhatsky | Proc. 38th EPS Conf[END_REF][START_REF] Angeli | [END_REF], magnetic dust has never been systematically investigated probably owing to the obvious absence of bulk magnetic materials in the vessel components. In particular, comprehensive dust collection activities in most tokamaks have never attempted a characterization of the magnetic properties of the sampled dust population [9][10][11][12][13][14][15].

It has been suggested that ferromagnetic dust can be generated in tokamaks by a non-equilibrium austenite-to-ferrite phase transformation that is induced by rapid temperature quenching in the presence of strong external magnetic fields [START_REF] Angeli | [END_REF]. Moreover, there is strong experimental evidence of pre-plasma remobilization of ferromagnetic dust in FTU that has been further supported by theoretical estimates [16]. This provided motivation for a re-evaluation of the old hypothesis concerning the possible impact of prematurely re-mobilized dust on the toka mak start-up [5,16]. Accurate predictions of the extent of such impact require reliable input concerning the overall amount and the size distribution of the magnetic dust.

The present work reports the first cross-machine dust collection study that focuses on the magnetic properties of tokamak dust and constitutes the first attempt to map the magnetic dust inventory of contemporary fusion devices. A morphological and elemental analysis of magnetic dust sampled from FTU, Alcator C-Mod, COMPASS and DIII-D is presented. These characteristics are compared to draw general conclusions concerning magnetic dust in tokamaks.

Magnetic dust collection and vessel composition

In the present work, the term magnetic dust refers to any particulate that can be isolated from a specimen under the effect of a strong permanent magnet, i.e. discussion is not confined to ferromagnetic dust but extends to strongly paramagnetic dust. The dust collection activities were carried out with different methods and in different locations for the tokamaks investigated. Magnetic dust from FTU, Alcator C-Mod and COMPASS was separated, by means of a strong magnet, from an entire dust batch collected by vacuuming. Magnetic dust from DIII-D was collected in-situ by means of a permanent magnet dragged around the vessel during a scheduled machine shut-down. The above difference naturally introduces limitations in the comparison between DIII-D and the other tokamaks.

The tokamaks under investigation are characterized by different plasma-facing material composition and different working toroidal magnetic field strengths. As summarized in Table 1, FTU and COMPASS feature plasma-facing materials that can directly seed magnetic dust (namely stainless steel and Inconel respectively), while Alcator C-Mod and DIII-D can only indirectly seed magnetic dust through diagnostic components or structural materials. Moreover, FTU and Alcator are compact machines with high toroidal magnetic field up to 8 T, whereas COMPASS and DIII-D have moderate field up to 2.2 T, implying a variable ability of magnetic dust production. Even though interest on plasma-facing materials is currently focused on the slightly paramagnetic tungsten and the slightly diamagnetic beryllium, as we shall discuss, the machine combination studied herein can be considered as representative of existing and future devices with respect to magnetic dust generation.

The dust batches that were selected for analysis are listed in the following. (a) FTU. The dust was collected by gross vacuuming around the vessel floor and was meshed out into two different size groups [START_REF] Angeli | [END_REF]. The batch with sizes m was considered in the present investigation. (b) Alcator C-Mod. The dust was collected by vacuuming the floor, the upper areas of the lower outer divertor, the horizontal areas and beneath the "FG" sector of the outer divertor in 2007. (c) COMPASS. Three dust batches were collected by vacuuming in different locations: the batch Co2 was collected inside the east port duct, the batch Co3 was collected inside the south port duct and the batch Co4 was collected in the divertor at the Langmuir probe feed-through position. (d) DIII-D. Magnetic dust was collected by a permanent magnet dragged around bottom surfaces of an outboard midplane and bottom ports during an entry vent. In this case, the overall amount was minuscule.

Table 2 catalogues the dust batch names, weights and the analysis performed. It should be noted that dust from all the tokamak devices could be covered by boron as a result of boronisation activities. In addition, only for FTU, dust could be covered by lithium compounds courtesy of the liquid lithium limiter [START_REF] Angeli | [END_REF]17]. Both elements are highly reactive with oxygen (O) and are difficult to detect by EDX analysis either due to the low sensitivity at low energies or due to partial overlapping with other elements.

Table 1

Vessel material composition at the time of the dust collection together with the maximum strength of the toroidal magnetic field. TZM is a molybdenum dominated alloy (Ti 0.5 wt%, Zr 0.1 wt%, Mo balance); SS stands for stainless steel; Inconel is a Ni based super-alloy. Note also that, during the 2007 Alcator C-Mod experimental campaign, a toroidal row of W tiles was inserted in the strike point region [13].

Analysis of magnetic dust

The magnetic dust collected from the four aforementioned tokamaks is based either on stainless steel (SS) or on Nickel (Ni) alloys that originate from different plasma facing components (PFCs) such as walls, divertor or limiter tiles, diagnostics or antennas. In terms of shape, the dust can be roughly categorized as flakes, splashes and spheroids.

Dust flakes. The majority of dust grains in all magnetic batches are flakes based on SS. Characteristic examples of flake-shaped dust that was collected from the four tokamaks are illustrated in Fig. 1. Their maximum dimension ranges from few tens up to m depending on the fusion device of origin. EDX analysis revealed that the flakes have low oxygen content ( at% on average) with the exception of FTU flakes which feature a much higher O concentration, probably due to the presence of Li. The average Fe/Cr ratio is , estimated on the grains analyzed from each device batch, to be compared with that of the standard SS AISI304/316 grade (3.3-4.0). It should be noted that many debris in Alcator and COMPASS definitely originate from machining, performed either in-vessel or transported from outside. In fact, some debris have "accordion-like" structures typical of mechanical shavings, see Fig. 2(a,b). This has been verified by a morphological comparison with SS shaves obtained from workshop milling and lathing activities.

Splashed droplets. Splash-shaped magnetic dust has been found in all devices except DIII-D. This is expected owing to the absence of vacuuming in the present DIII-D collection as well as the enhanced adhesion of splashes [18] or of immobile dust with high temperature pre-history [19]. Splashes are based on SS or Ni alloys, see Fig. 3. Their maximum dimension ranges from 100 up to m and they have a low O concentration (10 at% on average). The SS-based splashes possess a Fe/Cr ratio of 3.5-3.9.

Spheroids. Sphere-like magnetic grains have been collected in all tokamaks under investigation as either SS-or Ni-based grains with diameters ranging from up to m. Sphere-like dust can be divided into two classes; those with clear dendrite-like surface texture (see Fig. 4) and those without dendrite-like surface texture (see Fig. 5). The O content of the un-textured SS spheroids is lower than that of the textured SS spheroids (25 at% vs. 45 at% on average), probably due to their higher surface porosity. The average Fe/Cr ratio is 2.5-4 in both classes. The un-textured grains from DIII-D have a highly irregular surface but feature high oxygen content. The presence of a clear surface texture depends on the dynamic cooling rate after melting [START_REF] Kou | Welding Metallurgy[END_REF], which is not possible to estimate solely based on post-mortem evidence. The presence of both textured and un-textured grains in the magnetic dust batches lead to the conclusion that the grain cooling rate is not a critical parameter for the austenite-to-ferrite phase transformation. Note that Ni-based sphere-like dust has been detected in the FTU, COMPASS Co2 and DIII-D batches. Aiming to establish whether spheroidal magnetic dust originate from in-vessel welding activities (a common practice inside the DIII-D vessel), dust has been collected after in-lab welding of Inconel 625 studs on Inconel 625 or SS AISI316 plates used in DIII-D. The analysis of the produced spheroidal grains revealed that only welding involving SS materials could generate magnetic grains, while pure Ni-based grains are not magnetic. Furthermore, the welding-produced magnetic SS dust seems to have been exposed to much higher heat loads and presents some irregular surface texture structure (though not well defined) compared to dust collected in tokamaks. Examples of magnetic and non-magnetic SS grains produced by in-lab welding activity are illustrated in Fig. 2c andd. Note the high O content and low Fe/Cr ratio (

). X-ray Diffraction Analysis. XRD analysis was carried out on some magnetic dust batches (see Table 2). All batches analyzed were con firmed to be composed of SS and Ni alloys. Some of them, namely the ones from FTU and COMPASS, feature an increased ferrite peak compared to the peak of the standard untreated AISI304/316 SS material, see Fig. 6. This can be explained by the austenite-to-ferrite phase transformation that occurs in re-solidified spherical droplets and splashes when embedded in a strong magnetic field environment [START_REF] Angeli | [END_REF]. In addition, the FTU spectrum reveals the probable presence of martensite, typical of rapid cooling down of molten grains, a peculiarity that can be attributed to the fact that FTU is a cryogenic device [START_REF] Pizzuto | [END_REF]22].

On the other hand, the XRD spectrum of Alcator magnetic dust (Fig. 6, red line) is similar to the standard spectrum of untreated AISI316 steel (dashed line in Fig. 6), where the austenitic Fe-peak is much higher than the ferrite Fe-one; Fe-about 10% of Fe-. This can be explained by the fact that most Alcator magnetic grains were flakes, which as discussed above are believed to primarily originate from either in-vessel or ex-vessel machining. A more detailed analysis of the XRD results will be reported in a separate work.

Discussion and conclusions

This investigation catalogues the magnetic dust inventory of fusion devices that are equipped with different plasma-facing materials. Concerning its surface morphology, magnetic dust is encountered in the shape of flakes, splashes or spheroids with dimensions that range from few microns up to several hundreds of microns.

As far as the chemical composition is concerned, magnetic dust is based on steel or nickel alloys. The low surface content of oxygen on dust splashes and some spheroidal grains suggests that they originated from plasma-material interactions and not from in-vessel welding activities that can be a common practice. The latter possibility can be excluded for the majority of grains because of the morphological and chemical differences between plasma-generated and welding-generated dust, namely the high oxygen concentration and the Fe/Cr ratio in dust that is produced by in-lab welding. This is the so-called Cr surface segregation phenomenon that typically accompanies very high temperature treatment [23], as in welding, and leads to a higher O content owing to the formation of Cr oxides. The collection of Ni-based spheroidal dust (in DIII-D) further supports the claim that such grains do not originate from welding activities, as discussed in Section 3.

X-ray Diffraction analysis has demonstrated the different nature of dust magnetism depending on the fusion device. In particular, in magnetic dust collected from FTU and COMPASS, the presence of ferrite and/or martensite structures is revealed that constitutes evidence of SS dust magnetic phase transition under intense plasma heat loads [START_REF] Angeli | [END_REF]. The presence of an ambient magnetic field plays a role in magnetic phase retention during dust resolidification, as observed by some authors [24,25], but the magnetic field strength does not appear to be a key point, since magnetic phase transitions were detected in devices with high and moderate toroidal magnetic field. Probably, this stems from the fact that magnetic fields of the order of few Tesla suffice to saturate the magnetic domains of ferritic grains. On the other hand, the large quantity of SS flakes collected in Alcator C-Mod is revealed to be in the native austenite state.

In the case of ferromagnetic spherical dust, it has already been demonstrated that the magnetic moment force owing to the toroidal field can overcome both the gravitational and the adhesion force leading to pre-plasma remobilization [16]. In the case of strongly paramagnetic dust, a simple evaluation of the magnetic force that is exerted on a spherical grain in presence of toroidal field gradients suggests that it is several times larger than gravity. This indicates that, in principle, any SS grain could be classified as magnetic dust. The complications due to the non-spherical morphology of the flakes and a comparison with adhesion forces [26,27] will be addressed in future work.

Overall, a non-negligible amount of magnetic dust has been collected from the tokamaks considered in this work. This holds for FTU and COMPASS that can directly seed magnetic dust from the first wall, but also holds for Alcator C-Mod and DIII-D that can only indirectly seed magnetic dust from their diagnostics and antennas. Taking into ac

U N C O R R E C T E D P R O O F
count that the ITER diagnostic first wall is composed of stainless steel [28], it is possible that great amounts of magnetic dust will be produced in ITER, where plasma breakdown is critical. This is confirmed by the latest projections for dust generation in ITER, which anticipate comparable amounts of stainless steel and tungsten dust by the end of the fusion power operation phase [START_REF] De Temmerman | Progress in analysis of PWI issues for ITER and research needs[END_REF]. In particular, the lower estimate is 8 kg W and 9 kg SS dust, whereas the upper estimate is 37 kg W dust and 23 kg SS dust [START_REF] De Temmerman | Progress in analysis of PWI issues for ITER and research needs[END_REF]. In the prospective of the possible extended utilization of reduced-activation ferritic-martensitic steels in DEMO [START_REF] Roth | [END_REF], it is possible that large amounts of magnetic dust will also be generated beyond ITER. The understanding of the nature and origin of magnetic dust is an essential step towards the reliable modelling of its pre-plasma remobilization and of its interference with the start-up of future fusion devices.

Disclaimer

This report was prepared as an account of work sponsored in part by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Fig. 1 .

 1 Fig. 1. Dust flakes and their EDX analysis (at%). (a) SS-based dust from FTU (C , O , Cr , Fe ); (b) Ni (1) and SS (2) based dust from Alcator (O 10.1-0, Cr 0-19.6, Fe 0.6-68.5, Ni 88.5-7.7, respectively); (c) SS-based dust from the COMPASS Co3 batch (C , O , Cr , Fe , Ni ); (d) SS-based dust from DIII-D (C 23.4, O 8.3, Cr 11.3, Fe 46.6, Ni 6.3).

Fig. 2 .

 2 Fig. 2. (a and b) Chips in the Alcator magnetic batch probably due to machining. EDX analysis (at%): O 0.8-2.6, Cr 18.8-19, Fe 69.1-66.3 respectively. (c and d) Spheroidal dust from in-lab Inconel studs welded on SS plates and EDX (at%); (c) magnetic dust (O 63.7, Cr 18.6, Fe 1.4, Ni 0); (d) non-magnetic dust (O 58.7, Cr 28.3, Fe 1.2, Ni 0.5).

Fig. 3 .

 3 Fig. 3. Dust splashes and their EDX analysis (at%). (a) SS dust from FTU (O 10.1, Cr 14.2, Fe 61.3, Ni 8.0); (b) SS dust from Alcator (O 1.1, Cr 18.8, Fe 72.2, Ni 6.6); (c) Inconel dust from the COMPASS Co2 batch (B 24.6, O 4.8, Cr 14.3, Fe 0.9, Ni 37.7), no splashes in the Co3,4 batches; d) SS dust from Alcator (O 15.9, Cr 16.3, Fe 51.7, Ni 7.2).

Fig. 4 .

 4 Fig. 4. Spheroidal dust with clear dendrite-like textured surface and their EDX analysis (at%). (a) Ni-based dust from FTU (O 15.4, Cr 12.7, Fe 10.5, Ni 51.24); (b) SS dust from Alcator (O 53.7, Cr 8.0, Fe 32.8, Ni 4.8); (c) SS dust from the COMPASS Co4 batch (O 48.3, Cr 8.7, Fe 32.1, Ni 3.3); (d) SS dust from DIII-D (C 26.9, O 41.6, Cr 7.2, Fe 21.7, Ni 1.9).

Fig. 5 .

 5 Fig. 5. Spheroidal dust without any clear dendrite-like textured surface and their EDX analysis (at%). (a) SS dust from FTU (C 19.6, O 15.7, Cr 9.6, Fe 47.2, Ni 7.6); (b) SS dust from Alcator (C 0, O 0, Cr 21.2, Fe 70.0 Ni 8.0); (c) SS dust from the COMPASS Co4 batch (C 50.4, O 9.1, Cr 7.9, Fe 27.4, Ni 4.4); (d) SS dust from DIII-D (C 53.4, O 23, Cr 4.7, Fe 15.8, Ni 2.3).

Fig. 6 .

 6 Fig. 6. XRD spectra of the Alcator C-Mod (red), FTU (blue) and COMPASS Co4 (green) magnetic dust batches. The spectrum of untreated AISI316 steel is also shown with the black dashed line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2

 2 Dust batches investigated. "N/A" suggests that the amount of dust is too scarce to be weighted, "XRD" stands for X-ray diffraction, "SEM" for scanning electron microscopy and "EDX" for energy-dispersive X-ray analysis. The fractions are in weight percentage.

	Tokamak device	Year of dust collection	Wall material	Machine configuration	Limiter or divertor material	Maximum toroidal magnetic field
	FTU Alcator C-Mod COMPASS DIII-D		304LN Mo Inconel 625 Graphite	Limiter Divertor Divertor Divertor	TZM Mo Graphite Graphite	8.0 T 8.0 T 2.1 T 2.2 T

Acknowledgments

This work is supported in part by the U.S. Department of Energy under DE-FC02-04ER54698. DIII-D data shown in this paper can be ob tained in digital format by following the links at https://fusion.gat.com/ global/D3D_DMP.

Conflict of interest

The authors declare no conflict of interest.

Declaration of Competing Interest

The authors report no declarations of interest.