Quang-Vinh Dang
email: quang-vinh.dang@inria.fr

Claudia-Lavinia Ignat

Computational Trust Model for Repeated Trust Games

Keywords: trust management, computational trust, noncooperative games, behavioral game theory

Trust game is a money exchange game that has been widely used in behavioral economics for studying trust and collaboration between humans. In this game, exchange of money is entirely attributable to the existence of trust between users. The trust game could be one-shot, i.e. the game ends after one round of money exchange, or repeated, i.e. it lasts several rounds. Predicting user behavior in the repeated trust game is of critical importance for the next movement of the partners. However, existing behavior prediction approaches uniquely rely on players personal information such as their age, gender and income and do not consider their past behavior in the game.

In this paper, we propose a computational trust metric that is uniquely based on users past behavior and can predict the future behavior in repeated trust game. Our trust metric can distinguish between users having different behavioral profiles and is resistant to fluctuating user behavior. We validate our model by using an empirical approach against data sets collected from several trust game experiments. We show that our model is consistent with rating opinions of users, and our model can provide higher accuracy on predicting users' behavior compared with other naive models.

I. INTRODUCTION

Studying cooperation between users is a main topic of research in economy, psychology and computer science. Investment game or trust game proposed by Berg [START_REF] Berg | Trust, social history, and reciprocity[END_REF] is an important tool to study trust and collaboration [START_REF] Chakravarty | Experimental economics: a survey[END_REF]. The experiment has been repeated thousands of times all over the world 1 . In this game two users exchange money in anonymous mode. The sender sends an amount to the receiver. This amount is tripled when it arrives to the receiver. The receiver then selects an amount that should not exceed the sum received to send back to the sender. This time, the amount is not tripled when it arrives to the sender. The money exchange between users is entirely attributable to the existence of trust between them. The model of trust game can be applied for the cooperation between users that are located in different places that do not know each other. For instance, in e-commerce, buyers usually need to pay before they can receive the goods, so they need to trust the sellers.

The trust game could be one-shot, i.e. the game ends after one round of money exchange, or repeated, i.e. it lasts several rounds [START_REF] Anderhub | An experimental study of the repeated trust game with incomplete information[END_REF], [START_REF] Burks | Playing both roles in the trust game[END_REF], [START_REF] Engle-Warnick | Learning to trust in indefinitely repeated games[END_REF]. The pairs of users could be fixed [START_REF] Cochard | Trusting behavior in a repeated investment game[END_REF] or re-assigned before each round [START_REF] Dubois | Does players' identification affect trust and reciprocity in the lab?[END_REF]. The total number of rounds in the experiment can be reported to users [START_REF] Bracht | Whatever you say, your reputation precedes you: Observation and cheap talk in the trust game[END_REF], [START_REF] Bravo | Trust and partner selection in social networks: An experimentally grounded model[END_REF] or not [START_REF] Engle-Warnick | Learning to trust in indefinitely repeated games[END_REF]. These games provide different kinds of partner information to players, such as their gender, age and income [START_REF] Slonim | Increases in trust and altruism from partner selection: Experimental evidence[END_REF], or their past interaction history [START_REF] Bolton | Cooperation among strangers with limited information about reputation[END_REF], [START_REF] Dubois | Does players' identification affect trust and reciprocity in the lab?[END_REF]. Some of these games allow communication between players [START_REF] Bracht | Whatever you say, your reputation precedes you: Observation and cheap talk in the trust game[END_REF], [START_REF] Vanberg | Why do people keep their promises? an experimental test of two explanations[END_REF] or specification of user contracts [START_REF] Ben-Ner | Trust, communication and contracts: An experiment[END_REF], [START_REF] Feltovich | The role of strategic uncertainty in games: An experimental study of cheap talk and contracts in the nash demand game[END_REF].

Predicting partner behavior in the repeated trust game is of critical importance. In e-commerce, for instance, it is very important to predict the behavior of the partners before transaction completion. Game theory predicts that, in trust game, sender will send 0 and receiver will send back 0 [START_REF] Camerer | Behavioral game theory: Experiments in strategic interaction[END_REF]. However, in experimental game theory we usually do not observe this user behavior. In fact, the sending behavior of users in large-scale settings follows the normal distribution [START_REF] Johnson | Trust games: A meta-analysis[END_REF].

Existing approaches on predicting behavior of participants in one-shot trust game were proposed using additional information related to players, such as the personal information (age, gender, income, etc.) or the data collected through some tests / questionnaires. However, the only reliable available information for predicting users behavior is their behavior during previous transactions. No other work proposed prediction of user behavior based on analysis of previous interactions.

In this paper, we present a computational trust metric, which serves as a model for user behavior, to reflect and predict user's behavior in repeated trust games. We claim that, based on users past behavior in the repeated trust game, we can model and predict their next behavior. We only focus on repeated version of original anonymous trust game, i.e. the players have no information about their partners. We show that our model is robust to several types of attacks. We validate our model by using an empirical approach against data sets collected from several trust game experiments. We prove that our trust metric is consistent with users' opinion about partners trustworthiness.

The paper is structured as follows. We present related works in Section II. In Section III we present our trust metric and in section IV we describe its evaluation. Section V presents some limitations of our metric. Concluding remarks are presented in Section VI.

II. RELATED WORK

Several approaches on predicting behavior of participants in one-shot trust game were proposed using additional information related to players, such as their personal information (e.g. age, gender, income) or evaluation collected through some tests or questionnaires users needed to fill in before the experiment.

Gunnthorsdottir et al. [START_REF] Gunnthorsdottir | Using the machiavellianism instrument to predict trustworthiness in a bargaining game[END_REF] used the Mach test on users personality to predict their behavior in the game. Evans et al. [START_REF] Evans | Survey and behavioral measurements of interpersonal trust[END_REF] predicted users behavior based on the results of Propensity to Trust Survey. Using a similar idea, Yamagishi et al. [START_REF] Yamagishi | Two-component model of general trust: Predicting behavioral trust from attitudinal trust[END_REF] defined a new model called attitudinal trust to predict users' behavior in one-shot trust game. The attitudinal trust is calculated based on data collected through a questionnaire.

Yen [START_REF] Yen | An econometric analysis of household donations in the usa[END_REF] claimed that users with higher income send more to their partners than users with lower income. Falk et al. [START_REF] Falk | Did we overestimate the role of social preferences? the case of self-selected student samples[END_REF] confirmed this suggestion by showing that, students tend to send less than other social groups.

Ashraf et al. [START_REF] Ashraf | Decomposing trust and trustworthiness[END_REF] used the data collected in the dictator game to predict the behavior of the same users in the trust game.

In real world applications, sociometric information of users is usually not available [START_REF] Tang | eTrust: understanding trust evolution in an online world[END_REF]. Even if this information is available, such as a user login to a website by using Facebook account, it is not always reliable as users can declare false personal information. Our method to model and predict users' behavior does not require any additional personal data.

We are not aware of any prior work that predicted users' behavior in repeated trust game. The theoretical prediction is that in the repeated trust game participants keep exchanging an amount of 0 [START_REF] Camerer | Behavioral game theory: Experiments in strategic interaction[END_REF].

Glaeser et al. [START_REF] Glaeser | Measuring trust[END_REF] used average value of previous sending amount as the trust measurement of users in trust game. This trust metric as average of previous sending amounts was used later on by other research works [START_REF] Dubois | Does players' identification affect trust and reciprocity in the lab?[END_REF], [START_REF] Johnson | Trust games: A meta-analysis[END_REF], [START_REF] Altmann | Do the reciprocal trust less?[END_REF]. However, this average trust metric can not deal with user fluctuating behavior, as we discuss in Section IV.

III. TRUST CALCULATION

We define our trust metric as a measure of how well a user behaved in the past. We claim that we can predict users behavior based on the computed trust metric. For instance, a user with high trust score tends to behave well in the future. As we discuss in Section IV-C2, some users try to behave well at the beginning and then suddenly deviate. Our trust calculation will take into account this strategy that we call fluctuate strategy. In general, for trust games, trustworthiness of a user depends on the amount sent to her partners [START_REF] Dubois | Does players' identification affect trust and reciprocity in the lab?[END_REF], [START_REF] Glaeser | Measuring trust[END_REF]. A higher sending amount should lead to higher trustworthiness.

The trust score formula needs to satisfy the following requirements:

1) The trust value is higher if the sending amount is higher.

2) The trust value can distinguish between different types of users.

3) The trust value considers user behavior over time. 4) The trust value encourages a stable behavior rather than a fluctuating one. 5) The trust value is robust against attacks.

A. Parameters initial values

The values of the parameters used for the trust metric computation are displayed in Table I

B. Current trust

As we described in Section I, in each round, two users interact by sending a non-negative amount. For senders, the maximum amount they can send is set to 10, and for receivers, the maximum amount they can send is the amount they received from the sender (i.e. three times of what the sender sent). For both roles, we normalize the send proportion t as the sending proportion of a user at round t, with t ≥ 1:

send proportion t = sending amount t maximum sending amount t (1)
It is obvious that ∀t, 0 ≤ send proportion t ≤ 1.

We define next the trust metric for a single interaction between users that we call current trust. current trust t is a function of send proportion t , meaning that the trustworthiness of a user in a single interaction depends on how much she sends to her partner in round t. We define current trust t as a value between 0 and 1 inclusive. This function should satisfy the following properties (for convenience, we use the notation f (x), f : [0, 1] → [0, 1] for the function of current trust t , with x being send proportion t ):

• f (x) is continuous in [0, 1]. • f (0) = 0, meaning that current trust is 0 if the user sends nothing. • f (1) = 1, meaning that current trust is 1 if the user sends the maximum possible amount. • f (x) > 0 with x ∈ [0, 1],
meaning that current trust is strictly increasing when send proportion increases from 0 to 1. f (x) denotes the derivative of function f (x).

• f (x) ≤ 0 with ∈ [0, 1] meaning that the function is concave, i.e. the closer to 1 the value of current trust is, the harder is to increment it.

• f (x -) = f (x + ), ∀x ∈ [0, 1],
meaning that the function is smooth, i.e. there is no reason that at some point the current trust increases roughly less than previously. We proposed the following function that satisfies the above mentioned conditions:

current trust t = log(send proportion t × (e -1) + 1) (2)
where current trust t is the current trust function at round t and send proportion t is the value of send proportion at round t.

Explanation about the selection of the formula 2 will be provided in Section IV-A2.

C. Aggregate Trust

current trust t uniquely computes the value of trust based on the current interaction t. However, the previous interactions between two users have to be taken into account. The calculation of trust for multiple interactions is inspired by the trust model SecuredTrust [START_REF] Das | SecuredTrust: A dynamic trust computation model for secured communication in multiagent systems[END_REF]. The main drawback of SecuredTrust is that the metric assumes the existence of current trust t value. However, as shown in the previous subsection, computing current trust is not an easy task as it has to satisfy certain requirements. Moreover, SecuredTrust was mainly designed for peer-to-peer network systems, where computation of the trust in a peer node relies on information provided by the neighbours in the network. In this way, the trust value in one peer is in fact the reputation of that peer computed as an aggregation of the neighbor trust values on that peer. Nevertheless, in collaborative environments different users have different experiences with a certain user and therefore their trust values on that user are different.

Furthermore, SecuredTrust uses a constant value of α as forgetting factor. If this property can be valid in the peer-topeer network field, it does not hold for human users. Based on psychological peak-end rule [START_REF] Fredrickson | Duration neglect in retrospective evaluations of affective episodes[END_REF] we applied a dynamic α. The peak-end rule claims that, in a series of experiences, humans remember the extreme and the last experience, and forget the other ones.

We calculate aggregate trust as follows:

δ t = |current trust t -current trust t-1 | (3) β t = c × δ t + (1 -c) × β t-1 (4)
α t = threshold + c × δ t 1 + β t ( 5
)
aggregate trust t = α t × current trust t (6) + (1 -α t ) × aggregate trust t-1
As we described in the section III-A, current trust 0 = 0. The δ t is the change of current trust value by two sequential interactions t and t -1 between two users. We calculated δ t to see how much a person changes her behavior since her last activity. It is easy to prove that, α t is bigger if δ t is bigger, and vice versa. It means that, if the trust of the current interaction is much different from accumulated trust of all previous interactions, the current interaction will play a more important role in the final trust value.

D. Dealing with fluctuating behavior

Some users may try to collaborate in the beginning and then suddenly betray. We added a change rate t variable into our model to punish this kind of activity.

First, we calculate the trend factor t at round t representing the recent trend of user behavior, with higher value meaning that users improved lately their behavior:

trend factor t =                trend factor t-1 + φ if current trust t -aggregate trust t > trend factor t-1 -φ if aggregate trust t -current trust t > trend factor t-1 otherwise ( 7
)
adj atf t = atft 2 if atf t > MAX ATF atf t otherwise ( 8
)
atf t =                  adj atf t-1 + (current trustt -aggregate trustt ) 2 if current trust t -aggregate trust t > φ adj atf t-1 + (aggregate trust t -current trust t ) if aggregate trust t -current trust t > φ adj atf t-1 otherwise (9) change rate t = 0 if atf t > MAX ATF cos ( π 2 × atft MAX ATF ) otherwise (10)
In formula 9, we present the accumulated trust fluctuation (atf) function. Both kinds of fluctuate behaviors are punished: the latest sending amount is suddenly higher or lower than usual behavior. However, it is obviously that the latter case is more critical than the former one. Therefore, the punishment in the latter case should be stronger.

The accumulated trust fluctuation is a non-decreasing function. The increase depends on the change over time of user's behavior. If the behavior is stable or it changes within the allowed range (defined by the constant φ), atf t will not change.

When atf t reaches the threshold value MAX ATF, it means that accumulated change in user behavior over time reaches the level of betrayal and therefore change rate t drops to 0. Otherwise, as shown by Equation 10, change rate t decreases if atf t increases.

The cosine function is used in formula 10 because the cos function has a low degradation rate in the initial stage, and a high degradation rate in the case of repeated fluctuating behavior [START_REF] Das | SecuredTrust: A dynamic trust computation model for secured communication in multiagent systems[END_REF]. It means that, if a user starts adopting a fluctuating behavior the punishment is low, but it increases fast while fluctuating behavior persists. the trust value after round t is calculated by: trust value t = expect trust t × change rate t [START_REF] Slonim | Increases in trust and altruism from partner selection: Experimental evidence[END_REF] where, expect trust t = trend factor t × current trust t + (1 -trend factor t ) × aggregate trust t

The trust value is updated on every round.

IV. EVALUATION OF TRUST METRIC

In this section, we evaluate the performance of our trust metric according to the following three aspects:

1) Evaluation with simulated data. We evaluate our trust metric with simulated data. More specifically, we analyse whether our trust metric can distinguish between user types and cope with a fluctuating strategy. 2) Consistency with human opinions. We study how our trust metric can be validated with real user data. More specifically, given the same data set, we analyse whether our trust metric provides the same ratings of user behavior as the ones manually assigned by humans. 3) Evaluation of prediction with real data. We study whether our trust metric can predict users future behavior. In other words, we analyse whether the trust score assigned by the trust metric to a user reflects her future behavior.

A. Evaluation with simulated data

Our trust metric should punish fluctuating user behaviors. Moreover, it should detect user behavior patterns, i.e. it should be able to distinguish different types of user profiles: low, medium and high. In this subsection, we verify that our trust metric satisfies these criteria.

1) Fluctuating user behaviors: We define three types of user profiles according to the values of send proportion: low, medium and high. Similar to [START_REF] Buntain | Trust transfer between contexts[END_REF], we define that a user with a low profile sends in average 20% of the maximum possible amount, while for a medium profile user the send proportion is 50% and for a high profile user it is 80%. We also define a fluctuate profile user who first tries to behave well and then deviates.

By means of simulations for the above user profiles, we compare the behavior of our trust metric with the average trust metric where the trust score is calculated by an average of previous sending amounts [START_REF] Anderhub | An experimental study of the repeated trust game with incomplete information[END_REF], [START_REF] Ben-Ner | Trust, communication and contracts: An experiment[END_REF], [START_REF] Cochard | Trusting behavior in a repeated investment game[END_REF], [START_REF] Dubois | Does players' identification affect trust and reciprocity in the lab?[END_REF], [START_REF] Engle-Warnick | Learning to trust in indefinitely repeated games[END_REF], [START_REF] Glaeser | Measuring trust[END_REF], [START_REF] Johnson | Trust games: A meta-analysis[END_REF]. The behavior of our trust metric in first 10 rounds is displayed in Fig. 1 and the behavior of the simple average trust metric is displayed in Fig. 2.

We can easily see that, our trust metric can cope and punish the fluctuating behavior very well, as it reduces the trust score of fluctuating user to the same as of a low profile user. On the other side, the simple average metric cannot distinguish between fluctuating and high profile users. 2) Distinction between user types: In subsection IV-A1, we analyzed the behavior of our trust metric on constant sending behavior versus fluctuating behavior. However, the constant sending behavior is not realistic, and in this section, we relax our user profiles by allowing them to vary their behavior around the average value. In particular, we defined the behavior of low profile, medium profile and high profile as normal distributions with means of 0.2, 0.5 and 0.8 respectively, with standard deviation of 0.15 (this standard deviation value has been approximated from [START_REF] Johnson | Trust games: A meta-analysis[END_REF]). In what follows we analyze whether our trust metric can distinguish between different user types. Hence, after a large number of rounds, trust scores of different users will follow a distribution. In order to distinguish between different profiles, these distributions must satisfy the following properties:

• The trust values assigned to fluctuating users should be similar with the trust values assigned to low profile users, and should not overlap with the trust values assigned to medium profile users. profile respectively, and by std low , std medium and std high the corresponding standard deviations, then:

mean low + std low ≤ mean medium + std medium (12)
mean medium + std medium ≤ mean high + std high [START_REF] Vanberg | Why do people keep their promises? an experimental test of two explanations[END_REF] • The ratio of any two variances of these distributions should not be larger than 3, as suggested by Keppel [START_REF] Keppel | Design and analysis: A researcher's handbook[END_REF]. It is not easy to find a current trust function which can satisfy these above requirements. After an empirical process, the formula presented in Equation 2 is the only function we found so far that can satisfy these requirements.

For instance, if we replace our current trust formula by a new formula such as current trust = send proportion, this trust metric will not be able to distinguish between medium profile and fluctuating users. As we show in Fig. 3, after ten rounds, the new trust metric will assign overlapping trust scores to medium profile and fluctuating users, but our metric still can distinguish between these two user profiles as displayed in Fig. 4.

B. Consistency with human opinions

In this section, we evaluate our trust metric according to user ratings obtained by an existing experimental study of the repeated trust game [START_REF] Keser | Trust and reputation building in e-commerce[END_REF].

Keser [START_REF] Keser | Trust and reputation building in e-commerce[END_REF] organized a repeated trust game experiment where users could rate in each round their partners' sending behavior. The three levels proposed were: negative, neutral or positive. Based on the data published in this study, we created three virtual users called positive user, neutral user and negative user respectively corresponding to the levels of possible ratings. These virtual users follow the average behavior of real users who have the corresponding rating. In what follows we analyse the results obtained by our trust metric applied for the behavior of these virtual users. Since we are using a continuous rating score and Keser was using a discrete rating score, the two rating scores do not match completely. However, we should expect that our trust metric does not conflict with Keser's results, i.e. for any two behaviors A and B, if A was rated higher than B (for instance, positive versus neutral or positive versus negative), our trust metric should assign a higher trust score to A than B.

The analysis is displayed in Fig. 5. As expected, our trust metric assigns in all cases higher trust values to positive user than neutral user, and higher trust values to neutral user than negative user.

The conclusion is that our trust metric and people's opinion about trustworthiness of behavior in repeated trust games do not contradict each other.

C. Evaluation with real data

We showed that our trust metric matches real people's opinions about partner's behavior in the past. In this section we address the issue whether it can predict the future behavior of users. For instance, if our trust metric assigns a high trust value for a user, we are interested whether this particular user behaves well or badly in the future.

We note that, a low R 2 value is usual in predicting human behavior, but in many cases, it does not mean that the prediction is useless [START_REF] Gunnthorsdottir | Using the machiavellianism instrument to predict trustworthiness in a bargaining game[END_REF]. For instance, Ashraf et al. [START_REF] Ashraf | Decomposing trust and trustworthiness[END_REF] used a list of ten factors to predict users' behavior in one-shot trust game, and achieved the average R 2 of 0.25.

1) Data sets:

We analyze the performance of our trust metric on three data sets.

The first data set was collected by an experiment we conducted in our laboratory. We recruited 30 participants through a public announcement and we ran five sessions, each with six people. Before a session started, participants were asked to read user instructions and sign a consent form. During the experiment, users were asked to play a repeated anonymous trust game that we implemented by using zTree [START_REF] Fischbacher | z-Tree: Zurich toolbox for ready-made economic experiments[END_REF]. However, participants were not told the total number of rounds in order to avoid them behaving differently at the end of the session. In each round, the six participants were paired randomly, and for each pair the roles were assigned randomly. We ensured that during the session each user interacted exactly five times with each other user.

The two other data sets were obtained by the experiments described in [START_REF] Dubois | Does players' identification affect trust and reciprocity in the lab?[END_REF] involving 36 participants and in [START_REF] Bravo | Trust and partner selection in social networks: An experimentally grounded model[END_REF] involving 108 participants. Both of these experiments are done anonymously, and the data is provided under the form of a behavior log of participants.

The total data set comprises behavior of 174 participants in repeated trust games.

2) Observation on data: First, we show that our models on user profiles (low, medium, high and fluctuate profiles) are consistent with data collected throughout experiments. Next, we show that real data proves the existence of different user types such as participants who send in average a high amount and those who send in average a low amount. We also show that real data proves the existence of users with a fluctuating behavior.

We notice that changes in user behavior in repeated trust games are very usual. Fig. 7 illustrates the average and standard deviation of sending amount proportions of each user in the three datasets previously mentioned. The standard deviations of user sending proportions are large compared with their average sending proportions, meaning that users often change their sending behavior during the experiments. For instance, Fig. 6 illustrates a selected user behavior from our dataset: this player cooperates very well at beginning then deviates and never cooperates again. We observed that in all data sets, only few players send a constant amount throughout a session. Fig. 7 shows that for all three datasets, the average sending proportions of participants vary from 0 to 1, matching with our defined profiles: low, medium and high corresponding to a sending proportion of 0.2, 0.5 and 0.8 respectively.

We can conclude that, fluctuating behavior is a fact in all three data sets, and for this reason, it is important to design a trust metric that copes with this behavior.

3) Predicting users' behavior: Based on the behavior log we applied our trust metric on users' behavior at a certain round, then used the output trust score as the independent variable to predict the user's behavior in the next round. For all rounds starting with round five, we found a high correlation between the output trust scores and user behavior in the next round. However, due to space limitation, we present the results of our analysis only for rounds five and ten.

In our analysis the independent variables are the trust values for each user after fourth and ninth interaction and the dependent variables are the sending proportions of users in the fifth and tenth round. For the data in [START_REF] Dubois | Does players' identification affect trust and reciprocity in the lab?[END_REF], we tested the relationship between our trust metric and the user behavior at round five and ten. However, because of the design of the experiment in [START_REF] Bravo | Trust and partner selection in social networks: An experimentally grounded model[END_REF], we could only test the relationship between our trust metric and user behavior at round five. Fig. 8 displays the prediction of user sending behavior at round ten by using the data set from our experiment. Fig. 9 displays the prediction of user sending behavior at round five by using the Bravo dataset.

The summary of all linear regressions previously mentioned is displayed in Table II, where the independent variable (x-axis in Fig. 8 and Fig. 9) is the trust value our metric assigned to each user before a particular round, and the dependent variable is the behavior of this user in this round (y-axis in Fig. 8 and Fig. 9). We can notice that the slopes of all regressions are significant, meaning that our trust metric predicts well user's behavior. Similar results were obtained for the same analysis in other rounds (i.e. a significant slope value and a positive r-value), but, as previously mentioned, due to lack of space, we do not present them in detail. 4) Performance comparison: As previously mentioned, there is no prior work in predicting users behavior in repeated trust game. For this reason, in this section, we compare our model with two other baseline models: average model and null model.

Average model predicts that, the next sending amount of a user is equal to the average of her previous sending amounts. On the other hand, the null model predicts that, the next sending amount of a user is equal to her previous sending amount.

In order to compare the performance of these three models, we calculate the predicting values of each of these models. We compute the adjusted R 2 value for each model from round five to round ten and then calculate the average of adjusted R 2 . The higher average R 2 a model achieves, the better this model is in predicting users behavior.

The comparison of performance of different models is displayed in Table III. For our data and data of Dubois, we calculated an average adjusted R 2 values in predicting users behavior from round five to ten. As Bravo's dataset contains only five rounds, we computed the average adjusted R 2 values We can see that, our model outperforms the other two baseline models in predicting users behavior in repeated trust games.

D. Robustness of the trust model

In this section, we consider the robustness of our model against some common attacks to the reputation systems as addressed in [START_REF] Hoffman | A survey of attack and defense techniques for reputation systems[END_REF] with an adaptation to trust games context. We show that our trust metric is quite robust against some common kinds of attacks to reputation systems.

• Self-promoting attack implies that "attackers manipulate their own reputation by falsely increasing it". In the context of trust games, we can define self-promoting strategy as fluctuating strategy. Our trust metric can punish the fluctuating strategy by assigning people who use this strategy same score with low-profile users. • Whitewashing attack implies that "attackers try to repair and restore their reputation". In trust game context, the whitewashing strategy corresponds to users that try to repair their reputation by behaving better than in the past. Users adopting this strategy will be classified as having a fluctuating strategy and therefore they need a long period to repair their reputation. • Slandering attack implies that "attackers try to report false data". There are two types of slandering attacks: the attackers can report falsely bad information about honest users, or can report falsely good information about malicious users. The first type of slandering attack is not possible to appear in our method as data is collected by the system and not by user report. Our metric is robust against the second type of slandering attack. In order to falsely increase the reputation level, malicious users can use fake accounts to increase the trust score of their main account by providing positive reviews about it. However, our trust metric is based on personal experience of a user with another particular partner, and does not rely on the relationship of this partner with other users.

V. DISCUSSION

Nowadays reputation-based systems are widely used and trust scores assigned to users could be observed in many popular websites, such as Wikipedia, eBay, Amazon and online discussion forums (e.g. Stack Overflow). A good trust metric can suggest honest users to select the right partners and reduce the probability that they are cheated in on-line environments.

Our model has a limitation: the cold-start problem. The model requires the data for several interactions between users in order to calculate and predict their next behavior. This drawback generally holds for behavior modelling approaches: without data on previous behavior no prediction can be done [START_REF] Tang | eTrust: understanding trust evolution in an online world[END_REF]. Several potential solutions for this problem exist, such as querying the information from common friends or computing trust path through network [START_REF] Liu | Computational Trust Models and Machine Learning[END_REF].

On the other hand, our model has several advantages compared with other approaches presented in Section II. Our trust metric only requires the information about the interaction between the user and the partner in the past. This information is available on users side, without querying any central server or other nodes in the network. The trust metric also does not require any personal information such as gender, age, income or Mach score, which is not always available in collaborative systems today. These advantages facilitate the application of our trust metric to real systems.

VI. CONCLUSION

In this paper, we presented a trust metric to measure the trustworthiness of users in repeated trust games. We prove that our trust metric can deal with fluctuating user behavior and can distinguish different types of participants. Moreover, we show that our trust metric is consistent with human opinions. Last but not least, we show that our trust metric achieves a better performance in predicting user behavior in repeated trust games in comparison with other baseline approaches.

As repeated trust game is a general model for studying human cooperation, we expect that our trust metric could be applied to collaborative systems [START_REF] Truong | A contract-extended push-pullclone model for multi-synchronous collaboration[END_REF] where users need to trust and collaborate with other users such as Wikipedia or open source software development projects.

Fig. 1 .

 1 Fig. 1. Our trust metric on different user types in the first 10 interactions.

Fig. 2 .

 2 Fig. 2. The average trust metric on different user types.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Distribution of the trust current trust = send proportion after ten rounds. The trust scores assigned to fluctuating users overlap with trust scores assigned to medium profile users.

Fig. 5 .

 5 Fig. 5. Validating trust metric with real users' ratings.

Fig. 6 .

 6 Fig. 6. An observation of fluctuating behavior from our data set.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Average and standard deviation of sending proportions in datasets.

Fig. 9 .

 9 Fig. 9. Relationship between trust metric and user behavior at round five in the Bravo dataset.

  . The left side of the table contains the initial values of the corresponding parameters, while the right side of the table contains the constant values of the corresponding parameters.

TABLE II REGRESSION

 II BETWEEN TRUST METRIC AND FUTURE USERS' BEHAVIOR.

		Intercept	Slope Adj.R 2
	Our dataset (round 5)	0.071 0.701***	0.319
	Our dataset (round 10)	-0.022 0.913***	0.542
	Bravo dataset (round 5)	-0.006 0.715***	0.362
	Dubois dataset (round 5)	0.072 0.848***	0.356
	Dubois dataset (round 10)	0.027 0.855***	0.357
	We denote '***' as significant level of 99.9%.

TABLE III COMPARISON

 III OF R 2 VALUES OF DIFFERENT PREDICTING MODELS.

		Average model Null model Our model
	Our data	0.42	0.43	0.55
	Dubois's data	0.28	0.34	0.40
	Bravo's data	0.3	0.32	0.36
	in predicting users behavior at round five.

According to Google Scholar, the paper has been cited 3,810 times, as on

27-Jun-2016. An incomplete study counted that there are more than 23,000 participants involved in trust game experiments[START_REF] Johnson | Trust games: A meta-analysis[END_REF].

VII. ACKNOWLEDGEMENTS

This research was partially supported by USCoast Inria associate team and PSPC OpenPaaS::NG project funded by the Investissements d'Avenir French government program managed by the Commissariat général à l'investissement (CGI). We are thankful to Prof. Valerie L. Shalin, Wright State University, Department of Psychology, for assistance with distribution analysis.


E Merle-Lucotte

D Heuer

C Le Brun

L Mathieu

R Brissot

E Liatard

O Meplan

A Nuttin

Fast Thorium Molten Salt Reactors started with Plutonium

One of the pending questions concerning Molten Salt Reactors based on the 232 Th/ 233 U fuel cycle is the supply of the fissile matter, and as a consequence the deployment possibilities of a fleet of Molten Salt Reactors, since 233 U does not exist on earth and is not yet produced in the current operating reactors. A solution may consist in producing 233 U in special devices containing Thorium, in Pressurized Water or Fast Neutrons Reactors. Two alternatives to produce 233 U are examined here: directly in standard Molten Salt Reactors started with Plutonium as fissile matter and then operated in the Th/ 233 U cycle; or in dedicated Molten Salt Reactors started and fed with Plutonium as fissile matter and Thorium as fertile matter. The idea is to design a critical reactor able to burn the Plutonium and the minor actinides presently produced in PWRs, and consequently to convert this Plutonium into 233 U. A particular reactor configuration is used, called 'unique channel' configuration in which there is no moderator in the core, leading to a quasi fast neutron spectrum, allowing Plutonium to be used as fissile matter. The conversion capacities of such Molten Salt Reactors are excellent. For Molten Salt Reactors only started with Plutonium, the assets of the Thorium fuel cycle turn out to be quickly recovered and the reactor's characteristics turn out to be equivalent to Molten Salt Reactors operated with 233 U only. Using a combination of Molten Salt Reactors started or operated with Plutonium and of Molten Salt Reactors started with 233 U, the deployment capabilities of these reactors fully satisfy the condition of sustainability.

I. INTRODUCTION

It has been demonstrated recently [START_REF] Mathieu | Proposition for a Very Simple Thorium Molten Salt Reactor[END_REF][START_REF] Mathieu | The Thorium Molten Salt Reactor: Moving on from the MSBR[END_REF][START_REF] Mathieu | Cycle Thorium et Réacteurs à Sel Fondu : Exploration du champ des Paramètres et des Contraintes définissant le Thorium Molten Salt Reactor[END_REF] that the Molten Salt Reactors (MSRs), which are one of the systems selected by the Generation IV forum, may be operated in simplified and safe conditions in the Th/ 233 U fuel cycle and with fluoride salts, whether in a moderated or a fast neutron spectrum. These MSRs are called Thorium Molten Salt Reactors (TMSRs). Their main specific advantages come from their liquid fuel. The amounts of fissile and fertile matter can be adjusted without unloading the core, avoiding any initial reactivity reserve. Similarly, the Fission Products which poison the core can be taken out efficiently. This is true for instance for gases such as Xenon, which are easily removed with helium bubbling.

In Section II, we first present the motivations for a fast neutron TMSR. We show that such a reactor can operate with the relevant properties, in terms of breeding ratio, feedback coefficients, trans-uranian production and material steadiness to irradiation. The last paragraph of this section is devoted to studies addressing the problem of fissile inventory, the only drawback of this reactor type. Two methods are presented to minimize this inventory.

One of the pending questions concerning MSRs is thus the supply of the fissile matter, and as a consequence the deployment possibilities of a fleet of MSRs, since 233 U does not exist on earth and is not yet produced in the current operating reactors. In Sections III and IV, we detail two ways to produce 233 U: directly in standard MSRs started with Plutonium as fissile matter and then operated in the Th/ 233 U cycle; or in dedicated MSRs started and fed with Plutonium as fissile matter and Thorium as fertile matter. Our idea is to design a reactor able to burn the Plutonium and the minor actinides produced in currently operating reactors, and consequently to convert this Plutonium into 233 U. We analyze the characteristics of such reactors, in terms of deterministic safety parameters, fissile matter inventory, salt reprocessing scheme, trans-uranian production, and overall conversion capabilities This work is based on the coupling of a neutron transport code called MCNP [START_REF] Briesmeister | MCNP4B-A General Monte Carlo N Particle Transport Code[END_REF] with a materials evolution code. The former calculates the neutron flux and the reaction rates in all the cells while the latter solves the Bateman equations for the evolution of the materials composition within the cells. These calculations take into account the input parameters (power released, criticality level, chemistry ...), by adjusting the neutron flux or the materials composition of the core on a regular basis. Our calculations rest on a precise description of the geometry and consider several hundreds of nuclei with their interactions and radioactive decay; they allow a thorough interpretation of the results.

II. FAST NEUTRON MOLTEN SALT REACTOR BASED ON THE 232 Th/ 233 U FUEL CYCLE

II.A. Description of the General TMSR

The general concept of the Thorium Molten Salt Reactor (TMSR) is a 2500 MWth (1 GWe) graphite moderated reactor based on the 232 Th/ 233 U fuel cycle. Its operating temperature is 630 °C and its thermodynamic efficiency is 40 %. The graphite matrix comprises a lattice of hexagonal elements with 15 cm sides. The density of this nuclear grade graphite is set to 1.86. The salt runs through the middle of each of the elements. One third of the 20 m 3 of fuel salt circulates in external circuits and, as a consequence, outside of the neutron flux. The salt used is a binary salt, LiF -(Heavy Nuclei)F 4 , whose (HN)F 4 proportion is set at 22 mole % (eutectic point), corresponding to a melting temperature of 565°C. The 233 U proportion in HN is about 3 %. The salt density at 630°C is set at 4.3 with a dilatation coefficient 5 of 10 -3 /°C.

A graphite radial blanket containing a fertile salt surrounds the core so as to improve the system's regeneration capability. The properties of the blanket are such that it stops approximately 80 % of the neutrons, thus protecting external structures from irradiation while improving regeneration. We assume that helium bubbling in the salt circuit is able to extract the gaseous Fission Products (FP) and the noble metals within 30 seconds. For the first presented study, we consider a delayed reprocessing of the total salt volume over a 6 month period with a complete extraction of the FPs and of the Trans-Uranians (TRU). We assume that the 233 U produced in the blanket is also extracted within a 6 month period.

The moderation ratio can be altered by changing the channel radius. This modifies the neutron spectrum of the core, placing it anywhere between a very thermalized neutron spectrum and a relatively fast spectrum. The core size is adjusted to keep the whole salt volume constant. Fig. 1 shows the influence of the channel radius on the neutronic behavior. In order to evaluate the performance of a reactor configuration, we check a number of parameters: total reactivity feedback coefficient, breeding ratio, graphite life span and initial fissile inventory.

A wide variety of neutronic behaviors is available by changing the moderation ratio. We define three types of configurations: thermal, epithermal and fast spectrum. Each one has advantages and drawbacks: a thermal spectrum leads to a low fissile inventory but positive feedback coefficient, while a fast spectrum implies a high breeding ratio but large fissile inventory. Studies have been carried out on these three fields of research to improve these configurations and to find relevant solutions [START_REF] Mathieu | The Thorium Molten Salt Reactor: Moving on from the MSBR[END_REF][START_REF] Mathieu | Cycle Thorium et Réacteurs à Sel Fondu : Exploration du champ des Paramètres et des Contraintes définissant le Thorium Molten Salt Reactor[END_REF] . In this paper we will focus only on the fast neutron spectrum since it appears to be the most promising and simple path of investigation. Moreover, a fast neutron spectrum opens the possibility of using fissile matter such as Plutonium, when is it hardly achievable with a thermal neutron spectrum [START_REF] Nuttin | Potential of Thorium Molten Salt Reactors[END_REF] . When the hexagons are fully filled with fuel salt, there is no more graphite within the core. The reactor is then composed of a single big salt channel. In such a configuration there is no graphite irradiation problem, since there is none of it inside the high neutron flux. The graphite blanket structure is much less irradiated and, as a result, has a longer life span. We now present the characteristics of this configuration. The feedback coefficient can be broken down into three sub-coefficients related to the different components of the core presented above:

heating graphite dilatation salt heating salt total dT dk dT dk dT dk dT dk _ _ _ ⎟ ⎠ ⎞ + ⎟ ⎠ ⎞ + ⎟ ⎠ ⎞ = ⎟ ⎠ ⎞
The third sub-coefficient is negligible in our case since there is no graphite moderator in core. The contributions of the heating of the salt and of its dilatation are of the same order of magnitude, as shown in Table 1.

The total feedback coefficient for this TMSR configuration is equal to -5.37 pcm/°C. The uncertainties on the coefficients come first from statistical errors which are precisely estimated and also from systematic errors that are not quantified, like the evaluation of the cross-sections for example. Only the statistical uncertainties are given in Table 2. The systematic uncertainties are not precisely known but are quite large, around 1 to 2 pcm/ °C. The safety level of such a MSR is excellent, as presented in reference [START_REF] Merle-Lucotte | Molten Salt Reactor: Deterministic Safety Evaluation[END_REF] . Fig. 4 shows the evolution of the uranium inventory during 100 years of operation. The sequential formation of 234 U and heavier uranium isotopes is clearly visible. The initial fissile inventory, 5.5 metric tons, necessary to start such a reactor is comparatively large. Even if this system is overbreeder, with a breeding ratio equal to 1.12 (thanks to the thorium blanket and the 6 month reprocessing of the FPs), this large initial inventory is the main drawback of this very interesting TMSR configuration, leading to deployment capabilities of such reactors that are not satisfactory in terms of sustainability. Solutions for the minimization of this initial fissile inventory are explored in the next paragraph.

II.C. Minimization of the initial fissile inventory of fast TMSRs

The idea is to de-correlate the fuel function and the coolant function of the salt, leading to a reduced proportion of heavy nuclei in the salt.

This can be achieved in two ways:

-By introducing Beryllium in the fuel salt to lower its eutectic point. The salt considered is then LiF-BeF 2 -(HN)F 4 at the same operating temperature as before. -By operating at a higher temperature (around 1000°C), still using a binary fluoride salt LiF-(HN)F 4 .

We present in the following of this section the results of the first solution, the Beryllium addition in the fuel salt.

Fig. 5. Breeding ratio and initial fissile inventory of the fast TMSR as a function of the heavy nuclei proportion Fig. 5 shows the variation of the breeding ratio and of the fissile inventory of the fast TMSR for different proportions of heavy nuclei in the fuel salt. We want a reference reactor configuration that ensures breeding (i.e. with a breeding ratio greater or equal to 1) and that can be started with as small an initial fissile inventory as possible. With less than 7% of heavy nuclei, breeding is out of reach and, with more than 10%, the fissile inventory is too large.

The reference configuration we have chosen is the TMSR with an 8.5% proportion of heavy nuclei in the salt. The heavy nuclei mole percentage being fixed, the remaining 91,5% are divided in Lithium (around 4/5) and Beryllium (around 1/5). We have checked that the Beryllium mole percentage in the salt has no influence on the neutronic behavior of the reactor.

To keep the same heavy nuclei flux in the reprocessing unit as presented in paragraph II.A, we consider here a delayed reprocessing of the total salt volume over a 3 month period.

The open question is now to determine the best way to produce the necessary initial 233 U inventory of these TMSRs in order to verify their deployment capabilities, and more generally to study how to launch a fleet of molten salt reactors.

To produce 233 U, a solution may consist in the use of special devices containing Thorium 8 , in Pressurized Water (PWR) or Fast Neutron (FNR) Reactors. As already mentioned, we have considered here the production of 233 U directly in MSRs using an already available fissile matter such as 235 U or Plutonium. The use of 235 U to start/operate a MSR requires enrichment levels higher than some 30%, which is not realistic in terms of proliferation resistance. Consequently it is not discussed here. We have analyzed This has two consequences for this MSR configuration:

• It has to be started and operated with Plutonium. In fact, to be more realistic, we have started and fed this MSR with a mix of Pu, Np, Am and Cm (Table 2) corresponding to the trans-uranians of UOX fuel after one use in a standard PWR and five years of storage.

• Consequently, this Pu-operated MSR has to be based on a fast neutron spectrum, and thus must contain a large percentage of heavy nuclei in the salt. We have chosen 28% of heavy nuclei, which is the second eutectic point of this LiF-(HN)F 4 salt as shown on Fig. 6.

III.B. Characteristics of the configuration

The initial fissile (Pu) inventory necessary to start such a reactor is 13 metric tons, and the Thorium initial inventory is 43 metric tons. The corresponding 233 U produced during the reactor's operation is displayed on Fig. 7 (red curve). To minimize the wastes rejected during the reprocessing, we have studied the effect on the 233 U production of a slower reprocessing of the fission products and transuranians (Fig. 7) , 233 U being still extracted in a six month period. With a full core reprocessing in 16 years instead of six months, the 233 U production decreases by around 14% only, while the waste rejection is reduced by a factor 32. We thus considered a 16-year reprocessing for these MSRs fed with Plutonium.

Concerning safety, the total feedback coefficient for this MSR configuration is equal to -2.5 pcm/°C, it does not vary with the reprocessing time. As systematic uncertainties lie around 1 to 2 pcm/°C, this value is not negative enough to ensure a high safety level [START_REF] Merle-Lucotte | Molten Salt Reactor: Deterministic Safety Evaluation[END_REF] .

III.C. Evaluation and minimization of the wastes produced

Concerning the waste production, the amount of leakage occurring during the reprocessing is negligible compared to the final inventory, thanks to the 16-years reprocessing described in paragraph III. B. This trans-uranian inventory after 50 years of operation is detailed in Table 3 (4 th column), the trans-uranians being not extracted during the reactor operation. The corresponding trans-uranian inventory of the reference TMSR (paragraph II. C) is also listed in Table 3 (first column).

The final inventory in trans-uranians of the MSR fed with Plutonium is large enough to cause trouble in terms of radiotoxicity. A simple solution is illustrated in the last column of Table 3: after 50 years of operation as Pu to 233 U converter, the Pu supply is stopped and the MSR continues operating with a 232 Th supply only, acting as a transuranian incinerator. This helps to reduce significantly the trans-uranian inventory, even if the inventories of the heaviest trans-uranians continue to increase. Thanks to the low quantities implied, this does not pose a huge problem.

IV.B. Characteristics of the configuration

The initial Plutonium inventory of this MSR is equal to 7010 kg. The 233 U production is displayed on Fig. 8 (solid lines), together with the corresponding 233 U production in the reference TMSR (dashed lines). This production is far better in the Pu-started MSR, improved by a factor 1.4.

Fig. 8. 233 U production for the TMSR (dashed lines) and for the MSR started with Plutonium (solid lines)

Concerning safety, the total feedback coefficient for this MSR configuration lies around -9 pcm/°C. The contributions of the heating of the salt and of its dilatation are of the same order of magnitude. This very negative feedback coefficient ensures an excellent deterministic safety level [START_REF] Merle-Lucotte | Molten Salt Reactor: Deterministic Safety Evaluation[END_REF] .

IV.C Evaluation of the wastes produced

In terms of trans-uranian inventory, the MSRs started with Plutonium turn out to be nearly equivalent to Thorium Molten Salt Reactors operated with 233 U only after around forty years, as shown in Table 4 and Fig. 9.

As for the Pu-fed MSR (paragraph III.C), the inventories of the heaviest trans-uranians are higher for the MSR started with Plutonium than for the TMSR. Again, thanks to the low quantities implied, this does not pose a huge problem.

Fig. 9. Heavy nuclei inventory for the TMSR (dashed lines) and for the MSR started with Plutonium (solid lines) We have simulated the deployment of several reactor technologies and examined how well they satisfy the anticipated energy demand:

-The first scenario involves light water reactors [START_REF] Merle-Lucotte | Scenarios for a Worldwide Deployment of Nuclear Power -Full Report[END_REF] (LWRs) and fast neutron breeder reactors (FNRs) (see Table 5). In this view, we first pay particular attention to the availability of uranium 235, the only natural fissile element, which is, as a consequence, the major constraining factor in the frame of sustainable development. Secondly, we evaluate the possibility of eventually shutting down the reactor fleets started, by taking in consideration the heavy nuclei produced, whose handling is tricky.

In terms of natural resources availability, as shown on Fig. 10 which represents the integrated amount of natural uranium a required by the nuclear industry, the best scenario for a sustainable use of the natural uranium reserves is the third one, based on LWRs, TMSRs and MSRs operated with Plutonium. The second scenario is also better than the first one (with FNRs) in terms of natural resource sustainability. U-Pu fuel cycle and the amounts of plutonium and minor actinides produced are significantly smaller than in the scenario with FNRs. As a result, waste management is made simpler and easier to implement. Nuclear power deployment in this case is sustainable and efficient, the use of fissile matter and the production of wastes are optimized.

IV.E Further improvements

An interesting path of investigation to improve the MSRs consists in a reactor operating temperature increase. We have considered the same MSR configurations based on a LiF-BeF 2 -(HN)F 4 salt (with HN equal to 8.5%) operating at 1030 °C, both for a TMSR and for a MSR started with Plutonium. At this temperature, the thermodynamic efficiency is assumed to increase from 40 % to 60 %. This has an incidence on the power of the reactor: a 2500 MWth reactor operating at 1030 °C produce 1500 MWe instead of 1000 MWe when operating at 630 °C. The characteristics (feedback coefficient, inventories) of these reactors are comparable to the results presented in paragraphs II.C and IV.B, except for the power production which is 1.5 times higher and for the breeding ratio which is slightly smaller. As a consequence, a worldwide deployment scenario based on these TMSRs and MSRs operated with Plutonium, and PWRs, is equivalent to the very good third scenario detailed in paragraph IV.D in terms of natural resource sustainability, but this time with reactors having an excellent level of deterministic safety.

The major problem of this solution is that common structure materials cannot withstand such a temperature increase. However, new promising solutions based on carbon (carbon-carbon, carbon fiber, carbides...) could help solve this problem. If these technologies are not implemented, then this solution will have to be ignored.

V. CONCLUSIONS

We have first presented a very interesting Molten Salt Reactor configuration called fast Thorium Molten Salt Reactor (TMSR); it is based on the 232 Th/ 233 U fuel cycle and a fast neutron spectrum. We have detailed the characteristics of this reference TMSR configuration. A solution has been proposed to minimize the initial fissile inventory of a fast TMSR down to 2.8 metric tons only. We have then tried to give an answer to a major question on MSRs, concerning the supply of the fissile matter ( 233 U), and as a consequence the deployment possibilities of a fleet of MSRs.

We have examined two new ways of producing 233 U: directly in standard MSRs started with Plutonium as fissile matter and then operated in the Th/ 233 U cycle; or in dedicated MSRs started and fed with Plutonium as fissile matter and Thorium as fertile matter. Our idea was to design a critical reactor able to burn the Plutonium presently produced in PWRs, and consequently to convert this Plutonium into 233 U.

We have analyzed the characteristics of such reactors, in terms of deterministic safety parameters, fissile matter inventory, salt reprocessing scheme, waste production… We have demonstrated that the conversion capacities of both MSR types are excellent.

In the case of MSRs started and fed with Plutonium, we have shown that the 233 U production is very large and allows the more sustainable worldwide deployment. We have also studied the large radiotoxicity and trans-uranian production, which seem to be the main drawbacks of this concept. We have proposed a solution that consists in extending the reactor's operation as trans-uranian incinerator. The last negative point for these reactors is their rather low deterministic safety level, which could result in lower economical advantages.

For MSRs only started with Plutonium, we have checked that the assets of the Thorium fuel cycle are quickly recovered and that the reactor's characteristics turn out to be equivalent to MSRs operated with 233 U only, except for the 233 U production capacity which is much better. Moreover, by combining the reference TMSR configuration presented at the beginning of this paper with this MSR started or operated with Plutonium, we have shown that the worldwide deployment of such a fleet of MSRs is possible and sustainable.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Influence of the channel radius on the TMSR's behavior II.B. Description of the fast neutron TMSR

Fig. 3 .

 3 Fig. 3. Horizontal section of the fast TMSR core Vertical and horizontal sections of this configuration are presented on Figs. 2 and 3. The salt channel, 1.25 m radius and 2.60 m height, is surrounded by a thorium blanket and by two axial reflectors. These reflectors are made of ZrO 2 or ZrC (same neutronic behavior) in order to avoid the use of a moderator material. An efficient bubbling system extracts gazeous FPs and noble metals in 30 s, and a reprocessing unit slowly removes other FPs and TRUs of the fuel salt in 6 months.

Fig. 4 .

 4 Fig. 4. Inventory evolution of uranium and its isotopes for the single channel configuration.

  two types of MSR using Plutonium as fissile matter: MSRs operated with Plutonium and Thorium (section III) and MSRs only started with Plutonium and then based on the 232 Th/ 233 U fuel cycle (section IV). In both cases, we have combined fast reference TMSRs with the MSRs producing 233 U, to evaluate the deployment abilities of a fleet of Molten Salt Reactors. III. FAST NEUTRON MOLTEN SALT REACTOR OPERATED WITH PLUTONIUMIII.A. Description of the reactorWe first describe the best way to quickly produce large amounts of 233 U. This could be done by using specific MSRs. The idea is to spare all 233 U produced and thus not to use it as fissile matter in the MSR, but to extract it as soon as possible. The use of a liquid fuel makes this quick Uranium extraction possible during reactor operation.

Fig. 7 .

 7 Fig. 7. 233 U production for various Pu-operated MSR fuel reprocessing times and for the MSR-incinerator

  -The second scenario involves light water reactors, MSRs producing 233 U and started with Pu (paragraph IV.A), and reference TMSRs (paragraph II.C). -The third involves light water reactors, reference MSRs producing 233 U and operated with Pu (paragraph III.A), and reference TMSRs.

Fig. 10 .

 10 Fig. 10. Integrated natural uranium resources required by the three deployment scenariosOur aim in this paragraph is to explore the potential for worldwide nuclear power deployment and its limitations.In this view, we first pay particular attention to the availability of uranium 235, the only natural fissile element, which is, as a consequence, the major constraining factor in the frame of sustainable development. Secondly, we evaluate the possibility of

Fig. 11 .

 11 Fig. 11. Inventories in Plutonium and 233 U for two deployment scenarios: with LWRs and FNRs, and with LWRs, MSRs started with Pu and TMSRs In terms of fissile matter inventories (Fig. 11), only the first two scenarios (the scenario with LWRs and FNRs, and the scenario with LWRs, MSRs started with Pu and TMSRs) are displayed, since the results of the third scenario are equivalent to the results presented here for the second scenario. A small number of MSRs operated with Pu are indeed necessary since they produce around twenty times more 233 U than the MSRs only started with Pu (Fig. 7 and Fig. 8). For a worldwide deployment, this compensates their individual larger trans-uranian inventory. The scenarios based on MSRs allow a significant reduction of Plutonium inventories, down to around 1700 metric tons in 2100, instead of 45000 tons for the scenario based on FNRs, as displayed in Fig 11. The 233 U inventory in the scenarios with MSRs, replacing the Pu inventory for FNRs, lies around 10000 tons only, corresponding to a total amount of Uranium of 17000 tons. The role of the MSRs in the second and third scenarios is thus to close the

TABLE 1

 1 Total feedback coefficient for the TMSR and break down in sub-coefficients

	Total Coefficient	Salt Heating	Salt Dilatation
		(Doppler)
	-5.37 ± 0.04	-3.14 ± 0.04	-2.02 ± 0.04
	pcm/°C	pcm/°C	pcm/°C

TABLE 2

 2 Proportions of trans-uranians in UOX fuel after one use in PWR without multi-recycling (burnup of 60 GWd/ton) and after five years of storage[START_REF] De Saint Jean | Scénarios CNE : réacteurs classiques, caractérisation à l'équilibre[END_REF]

	Element	Proportion in the mix
	Np 237	6.3 %
	Pu 238	2.7 %
	Pu 239	45.9 %
	Pu 240	21.5 %
	Pu 241	10.7 %
	Pu 242	6.7 %
	Am 241	3.4 %
	Am 243	1.9 %
	Cm 244	0.8 %
	Cm 245	0.1 %
	Fig. 6. LiF -ThF 4 phase diagram

TABLE 3

 3 Trans-Uranian inventory/feeding (in kg) in the reference TMSR with 8.5% of heavy nuclei (at 60 years) and in the MSR with 28% of heavy nuclei fed with PlutoniumThe idea in this section is to have a reactor able to burn Pu but without loosing the advantages of the 232 Th/ 233 U fuel cycle. We have thus simulated MSRs started only with Plutonium and then operated in the 232 Th/ 233 U fuel cycle. More precisely, as stated in paragraph III.A, these MSRs are started with the same mix of Pu, Np, Am and Cm listed in Table2. They have a low heavy nuclei proportion in their fuel, as discussed in paragraph II.C. More precisely, the fuel salt considered is composed of LiF-BeF 2 -(HN)F 4 with 8.5% of heavy nuclei.

		TMSR	MSR-	MSR-	MSR-	MSR-
			Pu	Pu	Pu	Inciner
		Invent.	Initial	Feeding	Invent	Invent.
		60 yrs	invent	during	.	+ 50 yrs
			.	50 yrs	50 yrs
	237 Np	108.2	955.9	3430	614	48.84
	238 Pu	149.3	423.6	1520	1751	351
	239 Pu	57.14	7257	26040	4958	215.7
	240 Pu	43.95	3400	12200	9619	2258
	241 Pu	15.72	1703	6113	1940	421.3
	242 Pu	10.73	1074	3854	3341	1206
	241 Am	2.425	523.4	1878	1112	242.8
	242m Am 0.093	0	0	61.1	16.51
	243 Am	3.700	296.8	1965	1083	449.8
	242 Cm	0.312	0	0	52.5	12.11
	243 Cm	0.063	0.97	3.46	7.68	2.164
	244 Cm	4.045	126.6	454.3	962	506.5
	245 Cm	1.500	11.73	42.11	292.1	185.8
	246 Cm	0.610	1.67	5.981	108.2	161.9
	247 Cm	0.154	0	0	12.91	28.61
	248 Cm	0.038	0	0	3.06	12.94
	249 Bk	0.003	0	0	0.052	0.243
	249 Cf	0.006	0	0	0.197	1.125
	250 Cf	0.004	0	0	0.059	0.417
	251 Cf	0.002	0	0	0.008	0.090
	252 Cf	0.0002 0	0	0.0004 0.005
	IV. FAST NEUTRON MOLTEN SALT REACTOR
		STARTED WITH PLUTONIUM
		IV.A. Description of the reactor

TABLE 4 Trans

 4 -Uranian inventory (kg) in the TMSR with 8.5% of heavy nuclei and in the MSR started with Plutonium, initially and after 60 years of operationThe deployment scenarios described below rest on the following nuclear power progression 10 : starting at zero in 1970, nuclear power production rises to 340 GWe.y (GigaWatt electric-year) in 1985, to 400 GWe.y in 2000. Nuclear power remains stable from 2000 to 2015, then increases at the rate of 6.2% per year until 2050, achieving an eightfold increase by 2050; it then slowly increases by 1.1% per year until 2100. Extrapolating up to 2100 allows us to verify that the deployment scenarios are lasting.

		TMSR	MSR-Pu	MSR-Pu
		Invent. after	Initial	Invent. after
		60 yrs (kg)	invent. (kg)	60 yrs (kg)
	237 Np	108.2	537.5	101.8
	238 Pu	149.3	238.2	139.1
	239 Pu	57.14	4080	52.85
	240 Pu	43.95	1911	51.10
	241 Pu	15.72	957.9	18.97
	242 Pu	10.73	604.0	33.00
	241 Am	2.425	294.3	3.237
	242m Am 0.093	0	0.129
	243 Am	3.700	166.9	15.14
	242 Cm	0.312	0	0.425
	243 Cm	0.063	0.546	0.094
	244 Cm	4.045	71.19	27.78
	245 Cm	1.500	6.61	12.80
	246 Cm	0.610	0.932	23.60
	247 Cm	0.154	0	10.53
	248 Cm	0.038	0	5.120
	249 Bk	0.003	0	0.498
	249 Cf	0.006	0	1.080
	250 Cf	0.004	0	0.886
	251 Cf	0.002	0	0.511
	252 Cf	0	0	0.088
		IV.D Worldwide deployment scenarios

TABLE 5

 5 Characteristics of the FNRs considered

		Liquid metal coolant
		FNR
	Output capacity	1.0 GWe
	First operating date	2025
	Reactor lifespan	50 yrs
	Pu amount (per load)	6 tons
	Loading periodicity	5 yrs
	Number of loads	2
	Breeding (per reactor-yr):	300 kg of Pu

ACKNOWLEDGMENTS

We are thankful to Elisabeth Huffer for her help during the translation of this paper.