Mariam Maki Sy

Charles Figuières

Hélène Rey-Valette

Richard B Howarth

Rutger De Wit

Valuation of Ecosystem Services and Social Choice: The Impact of Deliberation in the context of two different Aggregation Rules 1

Keywords: ecosystem services, preference elicitation, non-monetary methods, deliberation, social choice theory, coastal lagoons. JEL codes: D71, Q57

This paper describes an empiric study of aggregation and deliberation used during citizens' workshops for the preference elicitation of 20 different ecosystem services (ESs) delivered by the Palavas coastal lagoons located on the shore of the Mediterranean Sea close to Montpellier (S. France). The impact of deliberation for the preference elicitation of 20 different ecosystem services (ESs) was studied by gathering and aggregating individual preferences before deliberation that were compared to the collective aggregation after deliberation. The same aggregation rules were used before and after deliberation and we compared two different aggregation methods, i.e. Rapid Ecosystem Services Participatory Appraisal (RESPA) and Majority Judgement (MJ). RESPA had been specifically tested for ESs, while MJ evaluates the merit of each item, an ES in our case, in a predefined ordinal scale of judgment. The impact of deliberation was strongest for the RESPA method. This new information acquired from application of social choice theory is particularly useful for ecological economics studying ES, and more practically for the development of deliberative approaches for public policies.

Introduction

How can we construct a social preference for ecosystem services (ESs) based on individual preferences? The issue is particularly important for public policies focused on environmental management and spatial planning. In this context, the ambition is to provide a "means of improving the choices our societies and the public bodies make to frame our relation to nature" [START_REF] Salles | Current issues in ecosystem services valuation (ESV)[END_REF]. It is an important and recurrent practice when valuing ESs and choosing among alternative management options (e.g., designating protected areas, ecological restoration projects, spatial planning and other public policies) that lead to different outcomes [START_REF] Dendoncker | Inclusive Ecosystem Services Valuation[END_REF]. The development of participatory approaches in this area involves examining the methods of collecting and aggregating preferences. Interestingly, these real approaches often present mixtures of deliberations followed by rankings of ESs.

What can be expected from such mixtures? From a more general perspective, a wealth of potential clarifications -originating from various traditions and scientific disciplines, e.g., economics, political science, political philosophy and ecologyare helpful. Among this diversity, two approaches can be distinguished [START_REF] Dryzek | Social Choice Theory and Deliberative Democracy: A Reconciliation[END_REF].

The first approach is based on the aggregation of individual preferences. Emphasis is placed on the properties associated with the aggregation methods (e.g. [START_REF] Condorcet | Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix[END_REF][START_REF] Borda | Mémoires sur les Élections au Scrutin[END_REF]Weber, posthumous edition of 2013;[START_REF] Hare | The machinery of representation[END_REF]. A milestone of this approach is of course Arrow's famous impossibility theorem [START_REF] Arrow | Social Choice and Individual Values[END_REF], the starting point of the modern theory of social choice. In this search for a 'good' aggregation of preferences, deliberation is either absent or implicit, and to our knowledge it is not the central concern.

By contrast, the second approach relies explicitly on a deliberation process among individuals. It has been particularly promulgated by the so called 'deliberative turn' in the eighties. Nowadays this is an eminent approach in political science, which spills over into other social sciences, such as anthropology geography and sociology. It is based partly on Discourse Ethics [START_REF] Habermas | Discourse Ethics: Notes on a Program of Philosophical Justification, in Moral consciousness and communicative action[END_REF], and builds on the idea that public deliberation is the essential key of a new articulation between three democratic objectives: i) the common good, ii) justification and iii) legitimacy [START_REF] R=borda Cohen | Deliberation and Democratic Legitimacy[END_REF][START_REF] Elster | Deliberative Democracy[END_REF][START_REF] Sunstein | Group Polarization and 12 Angry Men[END_REF].

Although deliberation is defended as a prerequisite for democracy [START_REF] Dewey | The Public and its Problems[END_REF], it is not recognized as a flawless panacea. Several decades of empirical research paint a mixed picture of the merits and/or weaknesses of deliberation (e.g., [START_REF] Fishkin | Introduction[END_REF], presumably because different factors play in opposite directions. Many of these factors still remain poorly understood. This lack of knowledge is an obstacle in the quest for deliberation capable of approaching the democratic ideal. This issue, which appears of paramount importance for public policies seeking public support, appears particularly pertinent in the field of ecosystem services valuation. For our scientific analysis, we assume that any deliberative process is based, implicitly or explicitly, on a particular aggregation procedure of individual preferences. How can we hope to understand the effects of deliberation when the aggregation rule remains implicit, or when its properties are not well known? Therefore, we propose that an explicit aggregation rule should be used during deliberation, as the expectations are well known for many rules in social choice theory 6 . This approach also has the advantage that it provides a framework for assessing the impact of deliberation alone by comparing the aggregation of the individual preferences before deliberation with the subsequent outcome of the deliberation process, provided that the same aggregation rule is used during both phases. Therefore, this design requests that the individual preferences are collected at the beginning of the process and that both this collection and the deliberation process is designed according a selected aggregation rule. Hence, the impact of deliberation can be assessed in the context of the selected aggregation rule by a before/after deliberation comparison. Actually, this even suggests an entire research program, in order to assess, for each well-known aggregation rule, the potential interest of the deliberation stage.

There are several reasons to believe that adding a deliberation stage will have an effect. In many cases and particularly when dealing with ecosystem services, one can hardly consider that stakeholders' preferences are exogenous and well-informed objects for all the different ESs. Preferences are context-dependent and, to some extent at least, endogenous. Therefore, preferences must in some sense be formed during a process of consideration and/or discussion [START_REF] Spash | Deliberative monetary valuation (DMV): Issues in combining economic and political processes to value environmental change[END_REF]. This implies that deliberation facilitates information sharing among participants since they are exposed to a wide range of ideas, perspectives and viewpoints [START_REF] Lienhoop | Informing biodiversity policy: The role of economic valuation, deliberative institutions and deliberative monetary valuation[END_REF]. Deliberation explicitly gives participants the opportunity to revise their preferences after having explored the problem at hand [START_REF] Parks | What have economists learned about valuing nature? A review essay[END_REF]. From a more ethical point of view, knowing that you are going to have to defend your preferences 6 Of course there are obstacles to the 'good' properties of a deliberation other than those associated with the aggregation of preferences. Actual deliberative processes can sometimes be affected by power relations that reproduce systems of privilege and inequality. Two types of indicators can be used to assess the quality of a deliberation process. The first relates to the balance of speaking times and the transparency and traceability of the debates. The second type of indicator is related to the diversity and representativeness of the participants [START_REF] Howarth | A theoretical approach to deliberative valuation: Aggregation by mutual consent[END_REF].

publicly encourages you to go beyond your individual interest to considerations of the general interest. In one interpretation, this involves purging one's private preferences of ethically indefensible components.

Hence, backing up deliberation with explicit aggregation rules would allow one to better explore two weaknesses pointed out in the literature on deliberation. A first weakness is that deliberation can be sensitive to the details of its organization7 , including of course the aggregation rule it encompasses (in the realm of environmental issues, see for instance Smith, Chapter 4, 2003). The nature of this dependence can only remain mysterious if the properties of the aggregation rule are themselves poorly understood. Going further, this suggests choosing aggregation rules that, by construction, are consistent with the ambitions assigned to deliberation. For instance, deliberation has obviously no chance of meeting the democratic requirement if it is based on an oligarchic or dictatorial aggregation rule. A second well documented weakness is group polarization, meaning that the debates within a group tend to radicalize the opinion of the members of the group in the direction of the initially dominant opinion, regardless of the merits of this opinion [START_REF] Sunstein | Group Polarization and 12 Angry Men[END_REF]. This begs the question whether some aggregation rules are more or less sensitive to this polarization phenomenon.

Answering this question requires testing and comparing on at least two aggregation rules.

Hence, the final problem is which two aggregation rules should we choose among a wide range of possibilities?

In the study reported in this paper, we carried out an ESs social choice protocol allowing us to question the impact of deliberation, by comparing the collective rankings of ESs preferences before and after deliberation. The first aggregation rule we have selected in this study is called RESPA (for "Rapid Ecosystem Services Participatory Appraisal", see Rey-Valette et al., 2017) that has been tested for ecosystem services. Nevertheless, the impact of deliberation

has not yet been assessed for this rule. Actually, RESPA is a variant of the famous Borda's rule, preceded by a selection phase of ESs in order to arrive at a smaller subset of ESs among which stakeholders' preferences remains to be aggregated. It has interesting properties in the context of ecosystem services. In fact, when it comes to prioritizing, classifying or evaluating a large list of objects, certain methods may lead to the phenomenon of survey fatigue. With an aggregation in two nested steps, the RESPA method tries to overcome this problem. Apart from that, Borda's method is very old. Its first uses date back at least to the 2nd century AD by the Roman Senate. Its formalized study began with the Frenchman Jean Charles de Borda in the 18th century [START_REF] Borda | Mémoires sur les Élections au Scrutin[END_REF]. Closer to us, some variants of this rule have been axiomatized [START_REF] Young | An axiomatization of Borda's rule[END_REF]. In its stripped-down version, it is a simple weighted voting system. Stakeholders attribute points to each ES; the Borda score of each ES is the sum of all its points and the social ranking of ESs is then given by the order of these scores. A textbook presentation is in Mueller (Chap. 7, 2003). It has a notorious weakness: it does not abide by Arrow's Independence of Irrelevant Alternatives axiom (IIA). It is then subject to strategic manipulations, and it may also fail to rank at the top a Condorcet winner, when it exists.

The second rule we selected, the Majority Judgment ("MJ" for short; [START_REF] Balinski | A theory of measuring, electing, and ranking[END_REF][START_REF] Gasparatos | Embedded value systems in sustainability assessment tools and their implications[END_REF], 2014, 2017), has never been used in this context. The principle of MJ is that stakeholders do not rank ESs directly, but they evaluate the merit of each ES in a predefined ordinal scale of judgment, called mentions. For instance in our case: "high priority", "priority", "neutral", "low priority" and "not a priority". One then determines the median mention for each ES, and the winning ES is the one with the highest median mention.

Eventually, one not only has a winner and a ranking of medians, but also a picture of what stakeholders think about ES via the ordinal scale. It was chosen in particular because it minimizes strategic manipulation [START_REF] Balinski | A theory of measuring, electing, and ranking[END_REF]. This property suggests that it could be less subject to the phenomenon of polarization presumably associated with a deliberation.

It is worth noting that both RESPA and MJ are consistent with the democratic ideal one may expect from deliberation, at least in the specific sense that they respect the Unanimity requirement (a unanimous strict preference of ES "x" over ES "y" should aggregate into a strict social preference of ES "x" over ES "y"). Also, both have an advantage when it comes to ecosystem services: they are non-monetary methods. For good or bad reasons, monetization produces rejection phenomena when it is applied to the evaluation of nature. And we want to eliminate this noise from the equation.

Moving to practical details, our field of study is the Palavas lagoons complex located near the urban agglomeration of Montpellier (about 500,000 inhabitants) in Southern France. This lagoon complex comprises 25 km of Mediterranean coastline with seven coastal lagoons and their immediate surroundings. This area is recognized as an internationally important wetland area according the Ramsar convention and is included in the EU Natura 2000 network because of its biodiversity and habitat values, while at the same time representing cultural and recreational values for the resident population and as a holiday resort for tourist mainly during summer. More detail about the socioeconomic system is provided below together with details about the aggregation methods, with and without deliberation. The aim of the present study was to study the impact of different aggregation rules on defining collective preferences and how these preferences can change as a result of the deliberation process. Section 2 details the material and methods used. Sections 3 presents the results. Section 4 concludes with a discussion.

Material and methods

Study site

The study area (Figure 1) comprises the Palavas lagoon complex and its immediate surroundings located in South of France. It includes:

(i) Seven shallow coastal lagoons between 0.4 and 1.2 meters deep that covers a total of 3,880 ha: Ingril, Vic, Pierre-Blanche, Arnel, Prévost, Méjean and Grec lagoons, (ii) The coastal barrier of these lagoons of 25 km of which 11 is not urbanized and in a natural state, (iii) Peripheral riparian, agricultural and wetland areas, and finally, (iv) the Rhône-à-Sète canal running SW -NE through the lagoon complex. The lagoons of the complex suffered more than four decades of nutrient over-enrichment due to their proximity with the urban centers of Montpellier and Sète as well as important suburban areas (De [START_REF] Wit | Restoration ecology of coastal lagoons: new methods for the prediction of ecological trajectories and economic valuation[END_REF]. However, awareness of the risks associated with their degradation resulted in policies focusing on the improvement of water quality [START_REF] Leruste | First steps of ecological restoration in Mediterranean lagoons: Shifts in phytoplankton communities[END_REF], ecological restoration (De [START_REF] Wit | Restoration ecology of coastal lagoons: new methods for the prediction of ecological trajectories and economic valuation[END_REF][START_REF] Wit | A Multidisciplinary Approach for Restoration Ecology of Shallow Coastal Lagoons, a Case Study in South France[END_REF] and nature conservation measures [START_REF] Sy | Identifying Consensus on Coastal Lagoons Ecosystem Services and Conservation Priorities for an Effective Decision Making: A Q Approach[END_REF]. Moreover, there is a dynamic of involving stakeholders' preferences including those of local residents for a better acceptability of these restoration and conservation policies. For each citizens' workshop, after welcoming the participants, a brief introduction about the overall process of the session was realized by the co-authors of the paper. There were between 3 and 6 experts for each session, including three co-authors of the paper. The workshop session comprised lectures given by the experts using a PowerPoint support. The oral presentations, which lasted about an hour, were about ecological functioning, socio-economic dynamics and management of the Palavas lagoons complex. More precisely, the supplied information included:

(i) General information on the lagoons (definition, Mediterranean lagoons, and natural history), ecological information (salinity, hydrogeological functioning, ecological interest), issues (global warming and sea level rise related issues, eutrophication, artificialization of the coast, the costs of restoring the lagoons) and a lecture about emblematic species of the study area.

(ii) Economic value (definition of the concept of value, the distinction between use and non-use values and the total economic value), the evolution of the lagoons' management policies (the effects of the management policies, from causes at sectoral scales to ecosystem-based and concerted approaches), frameworks for analyzing interactions between nature and society: DPSIR (drivers, pressures, state, impact and response model of intervention), ecosystem services and well-being (local well-being assessment frameworks and the contributions of the lagoons to territorial well-being).

The second part of the citizens' workshop consisted of filling out individually a questionnaire focused on ES preference elicitation and questions about general sociodemographic characteristics. Preferences were elicited using the MJ and RESPA methods (see below).

There was a section in the questionnaire dedicated specifically to preference elicitation through these two methods. The preference elicitation exercise was done separately for both methods. We chose these methods because we had a long list of twenty ESs to be graded and ranked. Indeed, they were designed in order to avoid long tiresome preference elicitation exercises. The list of the twenty ESs we used, were selected from an original list comprising 31 ESs (see [START_REF] Sy | Identifying Consensus on Coastal Lagoons Ecosystem Services and Conservation Priorities for an Effective Decision Making: A Q Approach[END_REF]. These twenty ESs were judged as a priority for public policy during a focus group meeting with a diversity of stakeholders of the Palavas lagoons complex area (see Table A in Appendix A for the general definition of the considered ESs).

Groups of participants were formed for the third and fourth parts of the citizens' workshop, representing in total 8 different groups for the three workshops. These groups were asked to achieve consensus rankings for both aggregation procedures. Two of the eight groups were discarded because they did not reach such an agreement. Hence, only the remaining six groups out of eight that successfully engaged in deliberation and reached an agreement were analyzed (see Table 1), representing 31 participants in total. Each group of participants had a different set of sociodemographic characteristics. A show-up fee of fifteen euros was offered to each participant.

Table 1. Characteristics of the analyzed groups of participants

Note: The columns "Association", "Knowledge" and "Housing" stand for, respectively, member of a French environmental NGO (law association-1901), the level of knowledge of the Palavas lagoons in terms of familiarity (i.e. acquired through experience) and whether or not the participants own the house she or he is living in the Palavas lagoons area. The Rapid Ecosystem Services Participatory Appraisal (RESPA)

The preference elicitation exercise using the RESPA method included two main steps. The respondents of each workshop were first asked to select a subset of ESs they considered as important from the original list of the twenty ESs. Then, they ranked the six ESs they judged as the most important from the subset of services using a scale from one to six (1 = High priority, 6 = Not a priority), in the same manner as the Borda count. More precisely in the questionnaire, each respondent had a table (see Table 2) with a list of the considered twenty

ESs as the first column where the respondents checked the ESs they judged as important. The last column was used to rank the six most important ESs. The six ESs were ranked relative to each other. Preferences were aggregated by summing up the scores attributed to each ES.

Hence, the ranking of the ESs was done based on the associated sums of the scores.

Table 2. Preference elicitation table using the Rapid Ecosystem Services Participatory Appraisal (RESPA) method

Note: "NS" stands for "Not selected". It is about ESs that were not judged as important and thus not ranked during the preference elicitation process

Majority judgment (MJ)

The principle of MJ is that the respondents explicitly express their opinions on the merit of every ES on a common ordinal scale of measurement, or language of grades, which were in our case: "high priority", "priority", "neutral", "low priority" and "not a priority" [START_REF] Balinski | A theory of measuring, electing, and ranking[END_REF][START_REF] Gasparatos | Embedded value systems in sustainability assessment tools and their implications[END_REF], 2017). MJ does not require pairwise comparisons of ESs as every ES is assigned a grade independently to the others. The detailed formulation of the MJ method is presented in Box B (Appendix B). Preferences were elicited using a table (see Table 3) where the ESs were listed in the first column and the grades in the following columns. Each respondent checked the grade she or he attributed to each ES. These grades were then coded in order to facilitate the aggregation of the individually elicited preferences. The aggregation and ranking processes using MJ consisted first of computing the majority grade of each ES (see Balinski and Laraki, 2010, pp. 254-255) attributed by stakeholders. It corresponds to the middlemost or median grade, the number of observations being odd in our case (N = 31). MJ then orders ES according to their majority grade.

A potential difficulty with MJ is to deal with ex-aequo. This is simply overcome by using additional and available pieces of information. Intuitively, an ES could be ranked higher than another with the same majority grade if its proportion of grades above the majority grade is larger, or if its proportion of grades below the majority grade is smaller. More formally, the majority gauge of an ES is a triplet (p, α * , q), where: (i) p is the number or percentage of the ES's grades above the majority grade and (ii) q is the number or percentage of the ES's grades below the majority grade, (iii) α is the ES's majority grade and α * = α+ if p> q, α * = αif p ≤ q and α * = α° if p = q. Of course α+ is better than α°, which is better than α-.

Overall, considering two ESs: ES1 and ES2 with, respectively, majority gauges (p, α * , q) and In the third part of the session, the lectures were followed by a deliberation process within each group of participants. This process involved a discussion and local knowledge exchange about the relative importance of the listed ESs. Finally in the last step of the session and after the deliberation process, each subgroup of participants agreed collectively on the level of priority of each ES using both MJ and RESPA methods. The same tables filed individually were used (see Table 2 andTable 3). Groups that did not reached a consensus were discarded.

Participants were free to ask questions, during the whole process, when in doubt about a particular subject.

Data analysis

After the workshops, individual preferences issued from the MJ and RESPA methods before deliberation were aggregated both at the level of the ensemble of the 31 participants as well as for the different groups. In addition, the collective preferences were recorded for each of the six groups of respondents after the deliberation process.

The first step of the data analysis consisted of aggregating individual preferences following the MJ and RESPA methods. Thus, we computed the majority grade (i.e. the median score) associated with each ES in the case of MJ and summed up the scores attributed to each ES in the case of RESPA. Based on these aggregated scores, the ranking of the ESs according to each method was also established. In the second step of the data analysis, for each of the six groups of respondents, we compared the rankings of the ESs obtained before and after deliberation. The comparisons were made by computing the differences between the ranks of the considered ESs. It is important to note that, for each group of respondents, the collective preference generated through the RESPA method contains only six ranks associated to the six ESs that were judged collectively as the most important ones. Therefore, the before and after deliberation comparisons were only reported for these six most important ESs. Likewise, for each group of respondents, we retained only those six ESs in the case of the MJ method. The aim being, for each group of respondents, to simultaneously analyze, according to MJ and RESPA, the differences between the ranks of the retained ESs before and after deliberation. In the following step of the data analysis, we carried out correlation tests between the ranks of the retained ESs issued before and after deliberation using the Kendall Tau-B test. The more the Kendall correlation coefficients are close to 1, the more the differences between the ranks of the retained ESs issued from the before and after deliberation were small.

In the last step, the perception of the participants regarding the deliberative process and the workshops in general were examined. Five variables were used:

(i) The quality of the supplied academic information, the freedom of speech during the deliberation process.

(ii) The composition of the groups (in terms of diversity).

(iii) The complexity of the questionnaire (in terms of understanding).

(iv) The convenience related to the organization of the workshops.

(v) And the satisfaction with the show-up fee.

Results

Aggregation of individual preferences for the ranking of ESs according to MJ and

RESPA before deliberation Note: The order of presentation of the ESs followed their ranking according MJ, which is slightly different for RESPA . The two-step procedure for RESPA resulted in three ESs as "NS". This stands for "Not selected" and comprises the ESs that were never preselected as important by any of the 31 respondents in the first step during the RESPA preference elicitation process.

3.2. Differences between the rankings before and after deliberation in the different groups;

Figure 3 presents, for each of the six groups of respondents and both MJ and RESPA, the differences between the rankings of the ESs obtained before and after deliberation for the six retained ESs. These differences indicate the change in ranks when passing from before to after deliberation. Figure 3. The Ecosystem Services selected as the six most important after deliberation in the six different groups according RESPA. The radar plots indicate the differences in their rankings after deliberation with respect to their rankings before deliberation (based on the aggregation of the individual preferences of the group members) both for the MJ and RESPA aggregation rules. (Note for the radar plots that starting at the top with the ES 'Flooding regulation and protection' selected by all six groups, the selected ESs appear clockwise in the order of their MJ ranking in Table 1) Overall, we observe, for both MJ and RESPA and for all the six groups of respondents, differences between the ranks of the ESs before and after deliberation (see Figure 3). These differences were relatively smaller for MJ (i.e. closer to zero in Figure 3) than for RESPA.

More precisely, in Table 5, the percentages of change in the ranks of the two valuation practices were higher for RESPA than for MJ. Similarly, the correlation coefficients were closer to 1 for MJ than for RESPA, especially for group 3 (0.52 for MJ and -0.33 for RESPA)

and group 4 (0.67 for MJ and -0.47 for RESPA).

In addition, for both MJ and RESPA, the differences between the ranks of the ESs before and after deliberation were relatively small for regulation and maintenance services (see Figure 3).

Also, we observe that the ESs "Flooding regulation and protection" and "Banks reinforcement" are considered as a priority in terms of conservation by 5 out of the six groups of respondents. Note: the correlation coefficients were not generated for group 1 and group 2 (indicated by asterisks) because there was a perfect tie in the collective ranking of all the ESs.

In general, the results show that while the participants were satisfied with the two workshops (see figure 4), they found, however, the questionnaire moderately complex (in average).

Discussion

4.1. The impact of deliberation differs according to the rules of aggregation used for the preference elicitation

Our before/after deliberation approach allows to study the impact of deliberation on the collective ranking of preferences, but does of course not reveal how the individual opinions by each participant were impacted by the deliberation process. Hence, we clearly observed an impact of deliberation by local citizens on collective preference elicitation of ecosystem services delivered by coastal lagoons. Similar impacts of deliberation on preference elicitation have been observed in other studies (e.g. [START_REF] Howarth | A theoretical approach to deliberative valuation: Aggregation by mutual consent[END_REF][START_REF] Kaplowitz | Do focus groups and individual interviews reveal the same information for natural resource valuation?[END_REF]Kenter et al., 2016a, Kenter et al., 2016b;[START_REF] Lo | Deliberative monetary valuation: In search of a democratic and value plural approach to environmental policy[END_REF][START_REF] Mavrommati | A novel deliberative multicriteria evaluation approach to ecosystem service valuation[END_REF]. Nevertheless, while in this respect the impact of aggregation rules has been rarely studied [START_REF] Murphy | Comparing group deliberation to other forms of preference aggregation in valuing ecosystem services[END_REF]) so far, we compared two different aggregation rules, i.e., RESPA and Majority Judgement (MJ); see Methods for details. Remarkably, the differences in the ranking of the ESs before and after deliberation were generally higher for RESPA than for MJ. The differences before and after deliberation also varied among the different groups.

Hence, the strongest differences were observed for groups 3 and 4 following RESPA, while in one case the impact of deliberation was null, i.e., group 5 according MJ. For MJ, the impact of deliberation on social ranking was relatively small for the five other groups (see Figure 3 and Table 5).

To explain these remarkable differences between RESPA and MJ, we hypothesize that while MJ was designed for consensus-seeking [START_REF] Balinski | A theory of measuring, electing, and ranking[END_REF][START_REF] Gasparatos | Embedded value systems in sustainability assessment tools and their implications[END_REF], 2014, 2017) it would be less susceptible to show changes during the deliberation process. On the other hand, the two-step procedure of RESPA, while it has the advantage of preventing fatigue, also introduces an outlier group that may result in more pronounced variability both among individual preferences as well as among different groups. Hence, we could expect a larger impact of deliberation for RESPA to level out this dispersion among individuals.

Does deliberation ensure convergence and stability?

It has often been alleged that deliberation produces a convergence of opinion. Several ideas have been forwarded to explain such a convergence of opinions. First of all, it might be explained by a better sharing of the local knowledge of the study site among the participants. Indeed, we observed that during the debates within the groups more knowledgeable participants shared their local ecological knowledge (see e.g., [START_REF] Narchi | Marine ethnobiology a rather neglected area, which can provide an important contribution to ocean and coastal management[END_REF] with the other participants (based on notes without using systematic recording). Such a process can lead to creation of so-called collective wisdom, which as such reduces the diversity of opinions as shown by [START_REF] Navajas | Aggregated knowledge from a small number of debates outperforms the wisdom of large crowds[END_REF]. In addition, during the deliberation process, there is generally a preliminary phase of information sharing that is as objective as possible with experts contrasting arguments. During our citizens' workshops the participants received information from expert of ecological and socio-economic issues, respectively, through small lectures in the first part (Fig. 2) and further exchanged with these experts during the deliberation if they requested more specific information. Moreover, in the specific case of citizen juries, there is the possibility of self-referral among the participants about any lacking information on the subject. Furthermore, for deliberation to be successful it has been underscored that the choice of tools for deliberation processes is of paramount importance [START_REF] Gasparatos | Embedded value systems in sustainability assessment tools and their implications[END_REF] and some more ludic approaches can stimulate the participants as they should engage in a collective learning process. Hence, the participants need to possess the specific capabilities, feel comfortable and adapt their tools and methods for such an approach. This is not always the case as one of the groups adopted a voting system for the collective preferences stating that they wanted to go faster than possible by deliberation (one of the two groups not taken into account in our analysis, see Methods section).

These above-mentioned information inputs play an important role in the convergence of positions and constitutes a benchmark for the participants to argue their positions during the debates (Randhur and Shriver, 2009). This multiple information inputs (external and internal to the group) corresponds to the spirit of the contribution of Habermas' deliberation which gives a large place to information sharing with, nevertheless, the risk of a polarization of the exchanges [START_REF] Hargittai | Cross-Ideological Discussions among Conservative and Liberal Bloggers[END_REF][START_REF] Lawrence | Self-Segregation or Deliberation? Blog Readership, Participation and Polarization in American Politics[END_REF][START_REF] Wilhelm | Democracy in the Digital Age[END_REF].

4.3 Which type of deliberation we need for scientific studies and practical cases?

While this empiric study was based on comparing the collective rankings before and after deliberation with the deliberation backed up by the same explicit aggregation rule, this does often not correspond to the procedures used in practical governance and court cases. For the United States there is an abundant social choice literature focused on court procedures [START_REF] Iaryczower | Can words get in the way? The effect of deliberation in collective decision making[END_REF], while in France it is mainly linked to a strong interest for participatory approaches for public policies. In both cases, it is more common to organize the deliberation prior to the pronouncement of individual or collective preferences. As mentioned in the introduction it is assumed that preferences are often constructed during discussions [START_REF] Spash | Deliberative monetary valuation (DMV): Issues in combining economic and political processes to value environmental change[END_REF] as it relies on information sharing among participants [START_REF] Lienhoop | Informing biodiversity policy: The role of economic valuation, deliberative institutions and deliberative monetary valuation[END_REF].

Hence, the popular juries in court cases typically represents the case where deliberation precedes individual pronouncements, while the final decision of the court is then based on voting. If the objective is studying how the individual preferences are influenced by deliberation, it is needed to complement our approach with an additional gathering of individual preferences after the deliberation. Participative approaches for public policies often use deliberation prior to seeking a consensus that should represent a collective preference elicitation or ranking. The theory of public choice is thus very useful to study the value of argued and balanced debates beyond simple votes [START_REF] Davis | The Web of Politics[END_REF][START_REF] Carpini | Public participation, Discursive Participation and Engagement: A Review of the Empirical Literature[END_REF]Talpin, 2013).

These results are encouraging for an interdisciplinary rapprochement of ecological economics based on social choice both with sociology and political sciences, with the aim to study participatory approaches in public policies. These public policies are very much dependent on the local context and many of the problems related to the management of ecosystems and their associated ESs have to be dealt with at the local scale by decentralized governance.

Nevertheless, to the best of our knowledge the currently used participatory approaches in

France have not yet directly addressed the ESs, but rather focus on spatial planning and hydrological measures. Hence, the implementation of participatory approaches for public policies would be better accepted by increased understanding of the deliberative process and the impact of the different aggregation rules, e.g. as those studied here (RESPA, MJ).

Following our observation of a smaller impact of deliberation for MJ, one could argue that adoption of MJ aggregation of individual preferences would allow to pursue the participative process without deliberation. However, MJ shows the problem of ex-aequo and is more susceptible to fatigue then RESPA, which, in addition, has the advantage to produce highly standardized results that can be more easily compared among groups (see e.g. Fig. 3).

Furthermore, the idea of participative approaches is to improve the quality and transparency of the decision process with the aim to achieve negotiated solutions [START_REF] Madani | Intellectual Capital: Investigating the Role of Customer Citizenship Behavior and Employee Citizenship Behavior in Banking Industry in Iran[END_REF].

Finally, the important role of information supply during participatory approaches needs to be highlighted as this may result in raising awareness and willingness to participate in discussions not only for the highly-involved stakeholders. Restricting the participatory approach to the latter group should be prevented as this creates a group of new experts with a restricted diversity of points of view.

Figure 1 .

 1 Figure 1. The Palavas lagoon complex in S. France on the Mediterranean Sea with its coastal barrier (25 km long running SW-NE) and its fringing wetlands. (Coastal lagoon area retrieved from Oxsol data base, which is a regional refinement of Corine Land Cover; background OpenStreetMap).

  Figure 2. The overall steps of the data collection during the citizens' workshops.

  (r, β * , s). The MJ ranking process places ES1 ahead of ES2when: (i) α*≻β* or, (ii) α*=β*= α+ and p>r or, (iii) α*=β*=αand q<s or, (iv) α*=β*=α° and p<r.

Figure 4 :

 4 Figure 4: workshops valuation by the participants (averaged).

Table 3 .

 3 Preference elicitation table using Majority judgement method

Table 4

 4 presents the individual preferences aggregation and the ranking of the twenty ESs according to RESPA and MJ. The individual preferences were aggregated based on the scores attributed to the ESs by the 31 respondents retained for this study (see Methods). The resultsshow differences between the rankings of the ESs issued from MJ and RESPA. However, these differences were small. Moreover, we observed a general pattern in the ranking of the ESs. More precisely, for both MJ and RESPA, the top five ESs were all regulation and maintenance services. Likewise, ESs related to relaxation (sentiment of relaxation), cognitive

(environmental education and research opportunity) and contemplative activities (recreational hiking and walking, aesthetic value of landscapes; bird watching and aesthetic value of habitats or species) were ranked next in the top twelve, both for MJ and RESPA. Next ranked ESs related to patrimonial (historical sites), symbolic (local identity) and provisioning services (shellfish farming, biomass for grazing and fish resources), again both according MJ and RESPA. Finally, the ESs that were ranked last are those associated with sports (nonmotorized water sports) and nature activities (recreational fishing and waterfowl hunting).

Table 4 .

 4 Aggregation of individual preferences according to MJ and RESPA before deliberation for the whole set of participants(N= 31)

Table 5 .

 5 Correlation coefficient and percentages of change in the ranking of ESs before and after deliberation in the different groups.

For example, an unstructured process might be dominated by the powerful participants, particularly if they are in agreement. In contrast, a facilitated process might amplify the voices of people in the minority, forcing engagement and social learning on matters of disagreement[START_REF] Howarth | A theoretical approach to deliberative valuation: Aggregation by mutual consent[END_REF].

Acknowledgements

This study has been financed by the DRIIHM LabEx (ANR-11-LABX-0010_DRIIHM), "Device for Interdisciplinary Research on human-environments Interactions" within the framework of the Human-environment observatory "Mediterranean coastline. Special thanks are due to Nicole Lautredou and Mylène Farge for their help during the citizen workshops.

We acknowledge the support of the municipalities of Lattes, Villeneuve-lès-Maguelone and Mireval for providing conference rooms for the workshops. The local managers of the former Syndicat Mixte des Etangs Littoraux (SIEL) and Ms Nicole Plank (municipal council of Lattes) are thanked for their support and useful suggestions. We are most thankful to the 42 participants at the citizens' workshops and particularly to those 31 who fully engaged in the deliberative process.

(i) Appendix

1 This work was supported by French National Research Agency Grants ANR-17-EURE-0020.

Appendix A: the list of the ecosystem services used in the study Table A. The set of the twenty ecosystem services (ESs) used in this study. The ESs have been selected following [START_REF] Sy | Identifying Consensus on Coastal Lagoons Ecosystem Services and Conservation Priorities for an Effective Decision Making: A Q Approach[END_REF] and categorized according to the classification designed for coastal and marine ESs by Liquete et al. (2013) and currently included in CICES version 5.1 (Haines-Young and Potschin, 2018).

ES

Environmental education

Appendix B Box B: Formulation of the Majority judgement method

Consider a set of a finite number of ecosystem services 𝑆 = {𝑆𝐸 1 , … , 𝑆𝐸 𝑚 }; a finite number of voters 𝑉 = {1, … , 𝑛}; and a common language of grades Ʌ = {𝛼, 𝛽, 𝛾, … } which is a totally ordered set. The grades or words are "absolute" [START_REF] Balinski | Judge: Don't Vote ! Oper[END_REF] in the sense that every voter uses them to measure the level of priority of each ES independently.

The matrix of inputs is formulated as:

Where 𝛼 𝑖𝑗 = 𝜑(𝐸𝑆 𝑖 , 𝑣) ∈ Ʌ is the grade assigned by voter 𝑣 ∈ 𝑉 to 𝐸𝑆 𝑖 ∈ 𝑆.

The majority grade attributed to an ES by all the voters correspond to its middlemost or median grade when 𝑛 is odd and its lower middlemost when 𝑛 is even [START_REF] Balinski | Judge: Don't Vote ! Oper[END_REF].

Suppose an ES majority grade is 𝛼 * , and that 𝑝% of his grades are higher than 𝛼 * and 𝑞% are lower. Then its majority gauge is (𝑝, 𝛼 * , 𝑞), where 𝑝 > 𝑞 implies 𝛼 * is endowed with a "+", and otherwise it is endowed with a "-" (Balinski andLaraki, 2010, 2014). It is formulated as follow:

The majority gauge (𝑝, 𝛼 * , 𝑞) determine the majority-gauge-ranking of ESs.

Consider two ESs 𝐸𝑆 1 and 𝐸𝑆 2 with majority gauges (𝑝, 𝛼 * , 𝑞) and (𝑟, 𝛽 * , 𝑠), respectively.

The majority-gauge-ranking "≻ 𝑚𝑔 " places 𝐸𝑆 1 ahead of 𝐸𝑆 2 , 𝐸𝑆 1 ≻ 𝑚𝑔 𝐸𝑆 2 , or (𝑝, 𝛼 * , 𝑞) ahead of (𝑟, 𝛽 * , 𝑠), (𝑝, 𝛼 * , 𝑞) ≻ 𝑚𝑔 (𝑟, 𝛽 * , 𝑠) when: (ii) References


Kaouther Messaoud

Nachiket Deo

Mohan M Trivedi

Fawzi Nashashibi

Trajectory Prediction for Autonomous Driving based on Multi-Head Attention with Joint Agent-Map Representation

come   L'archive ouverte pluridisciplinaire

Trajectory Prediction for Autonomous Driving based on Multi-Head Attention with Joint Agent-Map Representation

Kaouther Messaoud 1,2 , Nachiket Deo 2 , Mohan M. Trivedi 2 and Fawzi Nashashibi 1

Abstract-Predicting the trajectories of surrounding agents is an essential ability for autonomous vehicles navigating through complex traffic scenes. The future trajectories of agents can be inferred using two important cues: the locations and past motion of agents, and the static scene structure. Due to the high variability in scene structure and agent configurations, prior work has employed the attention mechanism, applied separately to the scene and agent configuration to learn the most salient parts of both cues. However, the two cues are tightly linked. The agent configuration can inform what part of the scene is most relevant to prediction. The static scene in turn can help determine the relative influence of agents on each others motion. Moreover, the distribution of future trajectories is multimodal, with modes corresponding to the agent's intent. The agent's intent also informs what part of the scene and agent configuration is relevant to prediction. We thus propose a novel approach applying multi-head attention by considering a joint representation of the static scene and surrounding agents. We use each attention head to generate a distinct future trajectory to address multimodality of future trajectories. Our model achieves state of the art results on the nuScenes prediction benchmark and generates diverse future trajectories compliant with scene structure and agent configuration.

I. INTRODUCTION

Autonomous vehicles navigate in highly-uncertain and interactive environments shared with other dynamic agents. In order to plan safe and comfortable maneuvers, they need to predict future trajectories of surrounding vehicles. The inherent uncertainty of the future makes trajectory prediction a challenging task. However, there is structure to vehicle motion. Two cues in particular provide useful context to predict the future trajectories of vehicles: [START_REF] Alahi | Social LSTM: Human trajectory prediction in crowded spaces[END_REF] The past motion of the vehicle of interest, and the motion of its neighbouring agents and (2) the static scene structure including road and lane structure, sidewalks and crosswalks.

A major challenge in trajectory prediction is the high variability in both context cues. Static scene elements and agents in a traffic scene can have various configurations. Several existing machine learning models for trajectory prediction have employed the attention mechanism with single [START_REF] Vemula | Social attention: Modeling attention in human crowds[END_REF], [START_REF] Sadeghian | Sophie: An attentive gan for predicting paths compliant to social and physical constraints[END_REF], [START_REF] Park | Diverse and admissible trajectory forecasting through multimodal context understanding[END_REF] or multiple heads [START_REF] Messaoud | Non-local social pooling for vehicle trajectory prediction[END_REF], [START_REF] Messaoud | Relational recurrent neural networks for vehicle trajectory prediction[END_REF], [START_REF] Messaoud | Attention Based Vehicle Trajectory Prediction[END_REF] to learn the most salient subsets of the static scene and agent configuration, to make both cues tractable. However, a limitation of existing approaches is that they either consider just one of the two cues [START_REF] Vemula | Social attention: Modeling attention in human crowds[END_REF], [START_REF] Messaoud | Non-local social pooling for vehicle trajectory prediction[END_REF], [START_REF] Messaoud | Relational recurrent neural networks for vehicle trajectory prediction[END_REF], or apply attention modules separately to representations of the agent configuration and static scene [START_REF] Sadeghian | Sophie: An attentive gan for predicting paths compliant to social and physical constraints[END_REF]. The two cues are tightly linked; each can inform the most salient parts of the other. For example, the presence 1 INRIA Paris, {kaouther.messaoud,fawzi.nashashibi}@inria.fr 2 LISA, UCSD, {mkaouther,ndeo,mtrivedi}@ucsd.edu of a nearby pedestrian could make a crosswalk in the path of the vehicle of interest a salient part of the static scene, as opposed to if there were no nearby pedestrians. On the other hand, the presence of a crosswalk would make a nearby pedestrian on the sidewalk a more salient part of the input context as opposed to if there was no crosswalk.

Another challenge in trajectory prediction is the multimodality of the distribution of future trajectories. The vehicle of interest could have one of several plausible intents, each of which would correspond to a distinct future trajectory. To address multimodality most existing approaches build one latent representation of the context [START_REF] Deo | Convolutional social pooling for vehicle trajectory prediction[END_REF], [START_REF] Deo | Multi-modal trajectory prediction of surrounding vehicles with maneuver based LSTMs[END_REF], [START_REF] Gupta | Social gan: Socially acceptable trajectories with generative adversarial networks[END_REF], [START_REF] Zhao | Multi-agent tensor fusion for contextual trajectory prediction[END_REF] and then generate multiple possible trajectories based on this representation. However, we believe that each possible future trajectory is conditioned on a specific subset of surrounding agents' behaviors and scene context features. For each possible intent, a different partial context is important to understand the future behavior.

To address the above challenges, we propose a model for multimodal trajectory prediction of vehicles, utilizing multihead attention as proposed in [START_REF] Vaswani | Attention is all you need[END_REF]. In particular, our model has the following characteristics:

1) Joint agent-map representation: Unlike prior approaches, we use a joint representation of the agents and static scene (represented as an HD map) to generate keys and values for attention heads. This allows us to better model the inter-dependency of both cues. Our experiments show that attention heads applied to a joint agent-map representation outperform those applied to separate representations of agents and the map.

2) Attention heads specializing in prediction modes:

We model the predictive distribution as a mixture model, with each attention head specializing in one mixture component. This allows each attention head to weight different parts of the scene and agent configuration conditioned on agent intent. Our experiments show that attention heads specializing in modes of the predictive distribution outperform an ensemble of attention heads used for predicting every mode.

We evaluate our model on the publicly available NuScenes dataset [START_REF] Caesar | nuscenes: A multimodal dataset for autonomous driving[END_REF]. Our model achieves state of the art results on the NuScenes prediction benchmark, outperforming all entries on 9 of the 11 evaluation metrics. Our model generates diverse future trajectories, that conform to the static scene and agent configuration. The code for the proposed model will be made available at: https://github.com/KaoutherMessaoud/MHA-JAM.

II. RELATED RESEARCH

Cross-agent interaction: Drivers and pedestrians operate in a shared space, co-operating with each other to perform safe maneuvers and reach their goals. Thus, most state of the art trajectory prediction approaches model cross-agent interaction. Alahi et al., [START_REF] Alahi | Social LSTM: Human trajectory prediction in crowded spaces[END_REF] propose social pooling, where the LSTM states of neighboring agents were pooled based on their locations in a 2-D grid to form social tensors as inputs to the prediction model. Deo et al. [START_REF] Deo | Convolutional social pooling for vehicle trajectory prediction[END_REF] extend this concept to model more distant interactions using successive convolutional layers. Messaoud et al., [START_REF] Messaoud | Non-local social pooling for vehicle trajectory prediction[END_REF] instead apply a multihead attention mechanism to the social tensor to directly relate distant vehicles and extract a context representation. Our approach is closest to [START_REF] Messaoud | Non-local social pooling for vehicle trajectory prediction[END_REF], however, we additionally consider an encoding of the static scene as an input to the multi-head attention modules.

Agent-scene modeling: Static scene context in the bird's eye view is also an important cue that has been exploited in prior work for trajectory prediction. Zhao et al. [START_REF] Zhao | Multi-agent tensor fusion for contextual trajectory prediction[END_REF] concatenate social tensors and an encoding of the scene in the bird's eye view and apply convolutional layers to extract a joint representation of the scene and surrounding agent motion. Sadeghian et al. [START_REF] Sadeghian | Sophie: An attentive gan for predicting paths compliant to social and physical constraints[END_REF] deployed two parallel attention blocks; a social attention for vehicle-vehicle interactions and a physical attention for vehicle-map interactions modeling. Yuan et al. [START_REF] Park | Diverse and admissible trajectory forecasting through multimodal context understanding[END_REF] use two attention modules as well but deploy them sequentially by feeding the output of the cross-agent attention module as a query to the scene attention module. Unlike [START_REF] Park | Diverse and admissible trajectory forecasting through multimodal context understanding[END_REF], [START_REF] Sadeghian | Sophie: An attentive gan for predicting paths compliant to social and physical constraints[END_REF], our model generates a joint representation of the scene and social tensor similar to [START_REF] Zhao | Multi-agent tensor fusion for contextual trajectory prediction[END_REF], which we use as an input to multi-head attention modules to model the most salient parts of the scene and agent configuration.

Multimodal trajectory prediction:

The future motion of agents is inherently multimodal, with multiple plausible trajectories conditioned on the agents' intent. Thus recent approaches learn one to many mappings from input context to multiple future trajectories. The most approach is sampling generative models such as conditional variational autoencoder (CVAE) [START_REF] Lee | Desire: Distant future prediction in dynamic scenes with interacting agents[END_REF] and Generative Adversarial Networks (GANs) [START_REF] Zhao | Multi-agent tensor fusion for contextual trajectory prediction[END_REF], [START_REF] Gupta | Social gan: Socially acceptable trajectories with generative adversarial networks[END_REF], [START_REF] Sadeghian | Sophie: An attentive gan for predicting paths compliant to social and physical constraints[END_REF]. Other methods sample a stochastic policies learnt by imitation or inverse reinforcement learning [START_REF] Li | Which way are you going? imitative decision learning for path forecasting in dynamic scenes[END_REF], [START_REF] Deo | Trajectory forecasts in unknown environments conditioned on grid-based plans[END_REF]. Ridel et al. [START_REF] Ridel | Scene compliant trajectory forecast with agent-centric spatio-temporal grids[END_REF] predict the probability distributions over grids and generates multiple trajectory samples.

In this paper, we utilise a mixture model similar to [START_REF] Cui | Multimodal trajectory predictions for autonomous driving using deep convolutional networks[END_REF], [START_REF] Chai | Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction[END_REF], generating a fixed number L of plausible trajectories for an agent, where each trajectory is represented by a sequence of two-dimensional Gaussians and the probability associated with each of the L trajectories.

III. MULTI-HEAD ATTENTION WITH JOINT AGENT-MAP REPRESENTATION

A. Input Representation

Interaction space: We first define the interaction space of a target vehicle T as the area centered on its position at the prediction instant t pred and oriented in its direction of motion, and denote it as A T . We consider all agents present in A T and the static scene elements in it as inputs to our model. This representation enables us to consider a varying number of interacting agents based on their occupancy in this area. We consider A T to extend ±25m from the target vehicle T in the lateral direction, 40m in the longitudinal direction ahead of T and 10m behind it.

Trajectory representation: Each agent i in the interaction space is represented by a sequence of its states, for t h past time steps between t pred -t h and t pred ,

S i = [S t pred -t h i , . . . , S t pred i ]. (1)
Here, superscripts denote time, while subscripts denote agent index. Absence of subscripts implies all agents in A T , and absence of superscripts implies all time steps from t pred -t h to t pred . Each state is composed of a sequence of the agent relative coordinates x t i and y t i , velocity v t i , acceleration a t i and yaw rate θt i ,

S t i = (x t i , y t i , v t i , a t i , θt i ). ( 2
)
The positions are expressed in a stationary frame of reference where the origin is the position of the target vehicle at the prediction time t pred . The y-axis is oriented toward the target vehicle's direction of motion and x-axis points to the direction perpendicular to it.

Map representation: We use a rasterized bird's eye view map similar to [START_REF] Cui | Multimodal trajectory predictions for autonomous driving using deep convolutional networks[END_REF], to represent the static scene elements in A T . The map includes the road geometry, drivable area, lane structure and direction of motion along each lane, locations of sidewalks and crosswalks. We denote the map as M.

B. Multimodal Output Representation

We wish to estimate the probability distribution P (Y |S, M) over the future locations Y of the target vehicle, conditioned on the past trajectories S of agents in A T , and the map M. To account for multimodality of P (Y |S, M), we model it as a mixture distribution with L mixture components. Each mixture component consists of predicted location co-ordinates at discrete time-steps over a prediction horizon t f .

Y l = [Y t pred +1 l , . . . , Y t pred +t f l ], l = 1, . . . , L. (3)
Here, superscripts denote time, while subscripts denote the mixture component. Each predicted location Y t l is modeled as a bivariate Gaussian distribution. Our model outputs the means and variances Θ t l = (µ t l , Σ t l ) of the Gaussian distributions for each mixture component at each time step. Additionally it outputs the probabilities P l associated with each mixture component.

C. Encoding layers

The encoding layers generate feature representations of the past trajectories S, and the map M. They comprise:

Trajectory encoder: The state vector S t i of each agent is embedded using a fully connected layer to a vector e t i and encoded using an LSTM encoder from t pred -t h to t pred ,

h t i = LST M (h t-1 i , e t i ; W enc ). (4)
Here, h t i and h t T are the hidden states vector of the i th surrounding agent and the target vehicle respectively at time t. All the LSTM encoders share the same weights W enc .

Map encoder: We use a CNN to extract high level features from the map M. The CNN outputs a map feature representation F m of size (M, N, C m ), where (M, N ) represent spatial dimensions of the feature maps and C m the number of feature channels.

D. Joint Agent-Map Attention

The first step in modeling vehicle-agents and vehicle-map interactions is to build a combined representation of the global context. To do so, we first build a social tensor as proposed in [START_REF] Alahi | Social LSTM: Human trajectory prediction in crowded spaces[END_REF]. We divide A T into a spatial grid of size (M, N ). The trajectory encoder states of the surrounding agents h t pred i at the prediction instance are placed at their corresponding positions in the 2D spatial grid, giving us a tensor F s of size (M, N, C h ), where C h is the size of the trajectory encoder state. Figure 1 shows an example of a social tensor. We then concatenate the social tensor F s and map features F m along the channel dimension to generate a combined representation F of agent trajectories and the map,

F = Concat(F s , F m ). (5)
We use the multi-head attention mechanism [START_REF] Vaswani | Attention is all you need[END_REF] to extract the salient parts of the joint agent-map representation F as shown in figure 2, with an attention head assigned to each of the L mixture components. For each attention head, the hidden state of the target vehicle h

t pred T
is processed by a fully connected layer θ l to give the query

Q l = θ l (h t pred T ; W θ l ). (6)
The agent-map representation F is processed by 1 × 1 conv. layers φ l and ρ l to give keys K l and values V l ,

K l = φ l (F; W φ l ), (7)
V l = ρ l (F; W ρ l ). (8)
The output A l of each attention head is then calculated as a weighted sum of value vectors V l (m, n, :),

A l = M m=1 N n=1 α l (m, n)V l (m, n, :). (9)
Here, α l (m, n) weights the effect of each value vector

V l (m, n), α l (m, n) = Sof tmax( Q l K T l (m, n, :) √ d ), (10)
where Q l K T l (m, n, :) is the dot product of Q l and K l (m, n, :) and d is the dimension of Q l and K l (m, n, :).

For each attention head we concatenate the output A l with the target vehicle trajectory encoder state h t pred T to give a context representation z l for each mixture component l = 1, . . . , L, z l = Concat(h

t pred T , A l ). (11)
E. Decoding layer Each context vector z l , representing the selected information about the target vehicle's interactions with the surrounding agents and the scene, and its motion encoding are fed to l LSTM Decoders. The decoders generate the predicted parameters of the distributions over the target vehicle's estimated future positions of each possible trajectory for next t f time steps,

Θ t l = Λ(LST M (h t-1 l , z l ; W dec )). (12)
All the LSTM decoders share the same weights W dec and Λ is a fully connected layer. Similar to [START_REF] Cui | Multimodal trajectory predictions for autonomous driving using deep convolutional networks[END_REF], we also output the probability P l associated with each mixture component.

To do so, we concatenate all the scene representation vectors z l , feed them to two successive fully connected layers and apply the softmax activation to obtain L probability values.

F. Loss Functions

Regression loss: While the model outputs a multimodal predictive distribution corresponding to L distinct futures, we only have access to 1 ground truth trajectory for training the model. In order to not penalize plausible trajectories generated by the model that do not correspond to the ground truth, we use a variant of the best of L regression loss for training our model, as has been previously done in [START_REF] Gupta | Social gan: Socially acceptable trajectories with generative adversarial networks[END_REF]. This encourages the model to generate a diverse set of predicted trajectories. Since we output the parameters of a bivariate Gaussian distribution at each time step for the L trajectories, we compute the negative log-likelihood (NLL) of the ground truth trajectory under each of the L modes output by the model, and consider the minimum of the L NLL values as the regression loss. The regression loss is given by

L reg = min l t pred +t f t=t pred+1 -log(P Θ t l (Y t l |S, M)). (13)
Classification loss : In addition to the regression loss, we consider the cross entropy as used in [START_REF] Cui | Multimodal trajectory predictions for autonomous driving using deep convolutional networks[END_REF], [START_REF] Chai | Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction[END_REF],

L cl = - L l=1 δ l * (l)log(P l ), ( 14
)
where δ is a function equal to 1 if l = l * and 0 otherwise. Here l * is the mode corresponding to the minimum NLL in equation 13. Y l * is the predicted trajectory corresponding to l * and P * l its predicted probability. Off-road loss: While the loss given by equation 13 encourages the model to generate a diverse set of trajectories, we wish to generate trajectories that conform to the road structure. Since the regression loss only affects the trajectory closest to the ground-truth, we consider the auxiliary loss function proposed in [START_REF] Niedoba | Improving movement prediction of traffic actors using off-road loss and bias mitigation[END_REF], [START_REF] Park | Diverse and admissible trajectory forecasting through multimodal context understanding[END_REF] that penalizes points in any of the L trajectories that lie off the road. The off-road loss L or for each predicted location is the minimum distance of that location from the drivable area. The overall loss for training the model is given by

L = L reg + λ cl .L cl + λ or .L or , (15)
where the the weights λ cl and λ or are empirically determined hyperparameters.

G. Implementation details

The input states are embedded in a space of dimension 32. We use an image representation of the scene map of size of (500, 500) with a resolution of 0.1 meters per pixel. Similar to [START_REF] Phan-Minh | Covernet: Multimodal behavior prediction using trajectory sets[END_REF] representation, our input image extents are 40 m ahead of the target vehicle, 10 m behind and 25 m on each side. We use ResNet-50 pretrained on ImageNet to extract map features. This CNN outputs a map features of size (28, 28, 512) on top of them we place the trajectories encodings. The deployed LSTM encoder and decoder are of 64 and 128 randomly initialized units respectively. We use L = 16 parallel attention operations applied on the vectors projected on different spaces of size d=64. We use a batch size of 32 and Adam optimizer [START_REF] Kingma | A method for stochastic optimization[END_REF]. The model is implemented using PyTorch [START_REF] Paszke | Automatic differentiation in pytorch[END_REF].

IV. EXPERIMENTAL ANALYSIS AND EVALUATIONS

A. Dataset

We train and evaluate our model using the publicly available nuScenes [START_REF] Caesar | nuscenes: A multimodal dataset for autonomous driving[END_REF] dataset. The dataset was captured using vehicle mounted camera and lidar sensors driving through Boston and Singapore. It comprises 1000 scenes, each of which is a 20 second record, capturing complex inner city traffic. Each scene includes agent detection boxes and tracks hand-annotated at 2 Hz, as well as high definition maps of the scenes. We train and evaluate our model using the official benchmark split for the nuScenes prediction challenge, with 32,186 prediction instances in the train set, 8,560 instances in the validation set, and 9,041 instances in the test set.

B. Baselines

We compare our model to six baselines, two physics based approaches, and four recently proposed models that represent the state of the art for multimodal trajectory prediction. All deep learning based models generate up to L = 25 trajectories (L = 16 for all implemented methods) and their likelihoods. We report the results considering the k most probable trajectories generated by each model.

Constant velocity and yaw :

Our simplest baseline is a physics based model that computes the future trajectory while maintaining constant velocity and yaw of the current state of the vehicle.

Physics oracle : An extension of the physics based model introduced in [START_REF] Phan-Minh | Covernet: Multimodal behavior prediction using trajectory sets[END_REF]. Based on the current state of the vehicle (velocity, acceleration and yaw), it computes the minimum average point-wise Euclidean distance over the predictions generated by four models: (i) constant velocity and yaw, (ii) [START_REF] Phan-Minh | Covernet: Multimodal behavior prediction using trajectory sets[END_REF].

Multipath [START_REF] Chai | Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction[END_REF]: Similar to MTP, the Multipath model uses a CNN with same input. However, unlike MTP, it uses fixed anchors obtained from the train set to represent the modes, and outputs residuals with respect to anchors in its regression heads. We implement MultiPath as described in [START_REF] Chai | Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction[END_REF].

CoverNet [START_REF] Phan-Minh | Covernet: Multimodal behavior prediction using trajectory sets[END_REF]: The CoverNet model formulates multimodal trajectory prediction purely as a classification problem. The model predicts the likelihood of a fixed trajectory set, conditioned on the target vehicle state.

Trajectron++ [START_REF] Salzmann | Trajec-tron++: Multi-agent generative trajectory forecasting with heterogeneous data for control[END_REF]: is a graph-structured recurrent model that predicts the agents trajectories while considering agent motions and heterogeneous scene data.

C. Metrics

MinADE k and MinFDE k : We report the minimum average and final displacement errors over k most probable trajectories similar to prior approaches for multimodal trajectory prediction [START_REF] Gupta | Social gan: Socially acceptable trajectories with generative adversarial networks[END_REF], [START_REF] Lee | Desire: Distant future prediction in dynamic scenes with interacting agents[END_REF], [START_REF] Cui | Multimodal trajectory predictions for autonomous driving using deep convolutional networks[END_REF], [START_REF] Chai | Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction[END_REF], [START_REF] Deo | Trajectory forecasts in unknown environments conditioned on grid-based plans[END_REF]. The minimum over k avoids penalizing the model for generating plausible future trajectories that don't correspond to the ground truth.

Miss rate: For a given distance d, and the k most probable predictions generated by the model, the set of k predictions is considered a miss based on,

Miss k,d = 1 if min ŷ∈P k max t=t f t=t pred y t -ŷ t d. 0 otherwise (16)
The miss rate MissRate k,d computes the fraction of missed predictions over the test set.

Off-road rate: Similar to [START_REF] Deo | Trajectory forecasts in unknown environments conditioned on grid-based plans[END_REF], we consider the off-road rate, which measures the fraction of predicted trajectories that fall outside the drivable area of the map. * Results reported in the nuScenes challenge Leaderboard: https://evalai.cloudcv.org/web/challenges/ challenge-page/591/leaderboard/1659

D. Quantitative Results

We compare our model MHA-JAM (MHA with joint agent map representation trained with off road loss) to various baselines in table I. Our model outperforms all baselines on 9 of the 11 reported metrics, while being second on the remaining two, representing the state of the art on the nuScenes benchmark as of writing this paper.

For the MinADE k and MinFDE k metrics, our model achieves the best results for k ∈ {1, 10, 15} and second best to Multipath [START_REF] Chai | Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction[END_REF] when k = 5. Having the best performance for k ∈ {10, 15} shows that our method generates a diverse set of plausible trajectories that match the ground truth. However, for k = 5, our classifier doesn't seem to succeed in selecting the closest trajectories to the ground truth among the 5 most probable ones, while the Multipath classifier does.

Moreover, our method presents significant improvements compared to others when considering miss rate and offroad rate metrics. Having the lowest miss rate suggests that our predicted trajectories are less likely to deviate from the ground truth over a threshold of d = 2m. In addition, our model achieves significantly lower off-road rates especially when trained with the off-road loss that penalizes predictions outside of the drivable area. Therefore, it generates scene compliant trajectories.

E. Ablation Experiments:

To get a deeper insight on the relative contributions of various cues and modules affecting the overall performance, we perform the following ablation experiments.

Importance of input cues: Our model relies on two main inputs, past motion of surrounding agents and scene context captured with maps. To investigate the importance of each input, we compare our model MHA-JAM to two models: (1) MHA with purely agent inputs (MHA-A), and (2) MHA with purely map inputs (MHA-M). When considering only the surrounding agents without any information about the scene structure (MHA-A), the model shows poor results according to the three metrics MinADE k , missRate k,2 , and off-road rate (cf. Figure 3a). This highlights the importance of the map information to make more accurate and scene compliant predictions. Moreover, since MHA-JAM has the best performance, we infer that considering the surrounding agents also helps our model make a better prediction. shows that all the metrics decrease when we increase the number of the attention heads. This proves the usefulness of using different attention heads to generate multimodal predictions.

Effectiveness of joint context representation:

To prove the effectiveness of using joint context representation, we compare our model to MHA-SAM (MHA with Separate-Agent-Map representation). MHA-SAM is composed of two separate MHA blocks (cf. Figure 4): MHA on surrounding agents (similar to MHA-A) and on map (similar to MHA-M). We concatenate their outputs to feed them to the decoders.

The main difference between MHA-SAM and MHA-JAM is that MHA-JAM generates keys and values in MHA using a joint representation of the map and agents while MHA-SAM computes keys and values of the map and agents features separately. Figure 3b shows that MHA-JAM performs better compared to MHA-SAM especially according to the off-road rate metric. This proves the benefit of applying attention on a joint spatio-temporal context representation composed of map and surrounding agents motion, over using separate attention blocks to model vehicle-map and vehicle-agents interaction independently. Effectiveness of a specialized attention heads: We also compare our method to MHA-JAM (with Joint-Attention-Heads JAH). While MHA-JAM uses each attention head head l to generate a possible trajectory, MHA-JAM with joint attention heads uses a fully connected layer to combine the outputs of all attention heads head l , l = 1 . . . L. It generates each possible trajectory using a learnt combination of all the attention heads.

Comparing MHA-JAM and MHA-JAM (JAH) reveals that We notice that the off-road loss helps generating trajectories more compliant to the scene by reducing the off-road rate while maintaining good prediction precision.

F. Qualitative Results

Figure 5 presents two examples of vehicle trajectory prediction, their corresponding 5 most probable generated trajectories and their associated attention maps. We notice that our proposed model MHA-JAM (off-road) successfully predicts diverse possible maneuvers; straight and left for the first Example 5a and straight, left and right for the second Example 5b. In addition, it produces different attention maps which implies that it learnt to create specific context features for each predicted trajectories. For instance, the attention maps of the going straight trajectories, assign high weights to the drivable area in the straight direction and to the leading vehicles (the dark red cells). Moreover, They show focus on relatively close features when performed with low speed and further ones with high speed (cf. Example 5a). For the left and right turns, in both examples, the corresponding attention maps seem to assign high weights to surrounding agents that could interact with the target vehicle while performing those maneuvers. For instance, in the left turn (cf. Example 5b), the attention map assigns high weights to vehicles in the opposite lane turning right. For the left turn of the first example and for the right turn of the second example, the attention maps assign high weights to pedestrians standing on both sides of the crosswalks. However, for the right turn, the model fails to take into account the traffic direction.

Figure 6 shows the average attention maps, for 4 generated possible maneuvers (going straight with low and high speed, left and right), over all samples in the test set. We note that each attention map assigns high weights, on average, to the leading vehicles, to surrounding agents and to the map cells in the direction of the performed maneuvers. This consolidates the previous observations in Figure 5. We conclude that our model generates attention maps that focus on specific surrounding agents and scene features depending on the future possible trajectory.

V. CONCLUDING REMARKS

This work tackled the task of vehicle trajectory prediction in an urban environment while considering interactions between the target vehicle, its surrounding agents and scene. To this end, we deployed a MHA-based method on a joint agents and map based global context representation. The model enabled each attention head to explicitly extract specific agents and scene features that help infer the driver's diverse possible behaviors. Furthermore, the visualisation of the attention maps reveals the importance of joint agents and map features and the interactions occurring during the execution of each possible maneuver. Experiments showed that our proposed approaches outperform the existing methods according to most of the metrics considered, especially the off-road metric. This highlights that the predicted trajectories comply with the scene structure.

Fig. 1 :

 1 Fig. 1: MHA-JAM (MHA with Joint Agent Map representation): Each LSTM encoder generates an encoding vector of one of the surrounding agent recent motion. The CNN backbone transforms the input map image to a 3D tensor of scene features. A combined representation of the context is build by concatenating the surrounding agents motion encodings and the scene features. Each attention head models a possible way of interaction between the target (green car) and the combined context features. Each LSTM decoder receives an context vector and the target vehicle encoding and generates a possible distribution over a possible predicted trajectory conditioned on each context.

Fig. 2 :

 2 Fig. 2: Attention modules in MHA-JAM: We generate keys and values by applying 1x1 convolutional layers to a joint representation of the map and surrounding agents, while the trajectory encoding of the target agent serves as the query.

  Advantage of using multiple attention heads: We train our model with different numbers of attention heads (L) and we compare the MinADE L , MinFDE L and MissRate L,2 . TableII

Fig. 3 :

 3 Fig. 3: Ablation experiments: We evaluate through ablation experiments, the importance of input cues (top), the effectiveness of a joint agent map representation for generating keys and values for attention heads (middle), the effectiveness of attention heads specialized for particular modes of the multimodal predictive distribution (middle), and finally the effectiveness of the auxiliary off-road loss (bottom). For each experiment we plot the metrics MinADE k (left), MissRate k,2 (middle) and off-road rate (right) for the k likeliest trajectories output by the models.

Fig. 4 :

 4 Fig. 4: MHA with separate agent-map representation: We compare our model to a baseline where attention weights are separately generated for the map and agent features

2 Fig. 5 :Fig. 6 :

 256 Fig. 5: Examples of produced attention maps and trajectories with MHA-JAM (off-road) model

TABLE I :

 I Results of comparative analysis on nuScenes dataset, over a prediction horizon of 6-seconds

		MinADE1	MinADE5	MinADE10	MinADE15	MinFDE1	MinFDE5	MinFDE10	MinFDE15	MissRate5,2	MissRate10,2	Off-Road Rate
	Const vel and yaw Physics oracle MTP [4] Multipath [3] CoverNet * [18] Trajectron++ * [21]	4.61 3.69 4.42 4.43 --	4.61 3.69 2.22 1.78 2.62 1.88	4.61 3.69 1.74 1.55 1.92 1.51	4.61 3.69 1.55 1.52 --	11.21 9.06 10.36 10.16 11.36 9.52	11.21 9.06 4.83 3.62 --	11.21 9.06 3.54 2.93 --	11.21 9.06 3.05 2.89 --	0.91 0.88 0.74 0.78 0.76 0.70	0.91 0.88 0.67 0.76 0.64 0.57	0.14 0.12 0.25 0.36 0.13 0.25
	MHA-JAM MHA-JAM (off-road) *	3.77 3.69	1.85 1.81	1.24 1.24	1.03 1.03	8.65 8.57	3.85 3.72	2.23 2.21	1.67 1.70	0.60 0.59	0.46 0.45	0.10 0.07
	constant velocity and yaw rate, (iii) constant acceleration and yaw, and (iv) constant acceleration and yaw rate.
	Multiple-Trajectory Prediction (MTP) [4]: The MTP model uses a CNN over a rasterized representation of the scene and the target vehicle state to generate a fixed number of trajectories (modes) and their associated probabilities. It uses a weighted sum of regression (cf. Equation13) and classification (cf. Equation14) losses for training. We use the implementation of this model by

TABLE II :

 II MinADE and MinFDE with different numbers of attention heads (L)

	L	1	4	8	12	16	20
	MinADE L MinFDE L MissRate L,2	3.48 1.72 1.26 1.13 1.02 1.00 8.01 3.54 2.29 1.91 1.64 1.60 0.91 0.76 0.59 0.50 0.40 0.40