ENEAU Sanojca Elzbieta

Ambiguities of "collaborative competences" in adult education: the case of Animacoop program

Keywords: collaborative competences, cooperation, hybrid training, community of practice, innovative organizations compétences collaboratives, formation hybride, dispositifs de formation, communautés de pratique, organisations innovantes

This paper presents two aspects of a doctoral research on collaborative competences in innovative organizations and training systems: (1) a literature review that identifies the existing work and the ambiguities in the use of these terms in French and English research; (2) an exploratory analysis of a specific training program, called "Animacoop", which intends to develop cooperation between actors both face to face and at distance. Thus, the discussion examines theoretical and empirical approaches of the terms of cooperation and collaboration and their appropriation in this training program.

INTRODUCTION

Collaborative competences are considered as being necessary to conduct complex and innovative projects in network models, these latter characterizing our contemporary era [START_REF] Choplin | Le problème de la conception collaborative des hypermédias pédagogiques[END_REF][START_REF] Blomqvist | Collaboration capability -a focal concept in knowledge creation and collaborative innovation in networks[END_REF]. Further, developing abilities to work in collaboration is also presented as a major challenge for training courses and tomorrow's organizations [START_REF] Michinov | Travail collaboratif et mémoire transactive : revue critique et perspectives de recherche[END_REF]Voogt & Pareja Roblin, 2012).

In France, there are only few references that evoke the term of "collaborative competences". Some exceptions give an idea of what they might cover, such as "learning to collaborate" and "working together" [START_REF] Simeone | Trust and credibility in an online computer-supported collaborative learning (CSCL) task[END_REF][START_REF] Simonian | Formation à distance et hybridation[END_REF]. The lack of a consensual definition suggests that this ability comes "naturally, as from the moment when people begin to work together". The illusion of this knowledge as being spontaneaous constitutes the source of our questioning on the knowledge, attitudes and skills which actors need to work collaboratively, this collaborative work being defined by [START_REF] Dejours | Coopération et construction de l'identité en situation de travail[END_REF] as "building links to voluntarily achieve a joint work".

The challenge of the thesis we are currently working on is to make visible the specific professional training related to the ability of individuals to cooperate or collaborate. This article aims to report on our advancement in this work. For this, two aspects will be stressed. First, a literature review will try to put into light the ambiguity of the term "collaborative competences". Then, an exploratory analysis of a training program will introduce a more concrete approach of the same question. For that, we will discuss the first results of a survey carried on among designers and beneficiaries of Animacoop, which is a specific training program aiming to develop cooperation between actors, both in presence and at a distance. A former French version of these analysis, initially presented in the conference "Cooperate?" (CNAM, Paris), have been enhanced and translated to be available to the English-speaking community.

I. THE USE OF THE TERMS « COOPERATION/COLLABORATION »

The complexity of the concept of "collaborative competences" is reinforced by the fact that the two composite concepts are polysemous and elusive: the term "competence", in French (sometimes translated as "competence", "ability" or "skill") can have different meanings in different approaches and disciplines (Jonnaert, 2002); the same goes for the term "collaboration" [START_REF] Thomson | Collaboration processes: inside the black box[END_REF]Henry, Lundren-Cayrol, 2001). On this semantic diversity we take the general definition of "competence" from the terminological dictionary of the Quebec Office of the French language: "an ability to act in a given context based on external and internal resources". Collaboration, will also be considered in the broadest sense of the term, in the words of Margaret Mead: "working together with a common goal" (Mead, 1937(Mead, /2002)). Furthermore, [START_REF] Henri | Apprentissage collaboratif à distance : pour comprendre et concevoir les environnements d'apprentissage virtuels[END_REF] provide an explanation entailing that collaboration is based more on equalitarian and democratic relationships than cooperation: thus it gives more power to actors in a climate of openness and shared responsibility. The implication is that we prefer the term "collaborative" to connect with "competency". However, despite this "informed choice" we take responsibility to use interchangeably the words "collaborative" and "cooperative". As a consequence, we will us the expression "collaboration/cooperation".

II. COOPERATION / COLLABORATION AS A THE OBJECT OF A RESEARCH

The difficulty in working on the cooperation/collaboration duet stems from the cultural heritage in which these terms are used and from the diversity of the scientific approaches treating them as objects of research. This chapter traces back these two sides of the problem.

II.1 Maturation of meanings

The Historical Dictionary of the French Language [START_REF] Rey | Dictionnaire historique de la langue française[END_REF] highlights the theological origin of the term cooperation, which meant in the 15th century "do something together with someone", implying participation in the "divine work". The more "secular" use of the term appeared in the 16th century when its significance was fixed on the meaning "to operate together with anyone". It acquired in the 19th century a new economical meaning. In this sense, the term "cooperation" designated "a business management method based on the distribution of profits according to one's participation". In the 1960s, the use of the word extended to "the policy by which a country contributes to another country's development, often a decolonized one".

The term "collaborate" started being used later on and immediately took on the secular meaning. It appeared in the 17th century from the term collaboratio used in medieval law, which referred to "possession acquired by spouses in joint work". In that, to collaborate meant "to work together to share benefits". A more political meaning emerged during the Second World War by introducing a negative connotation of "working with the enemy". This impairment leads to difficulty in using this verb, if not its avoidance.

The border is blurred by the multiple dimensions of the words cooperate/collaborate. In both cases, the uses cover the same fields: practical life (the action of participating and doing jointly); economical and work organization (management methods); political action (development aid program or working in partnership). The use of these terms in the practice of this ongoing research does not yet clear everything up.

II.2 Historical background on Cooperation / Collaboration research

The historical approach to cooperation and collaboration research in human and social sciences highlights the fact that the use of cooperation is preferred. In French works, it is mentioned by [START_REF] Alter | Donner et prendre: la coopération en entreprise[END_REF] among others, but we prefer to rely on [START_REF] Argyle | Cooperation (Psychology Revivals): The Basis of Sociability[END_REF] sum-up to reveal the diversity of approaches.

Argyle differentiates the pre-World War II period and the post-war period in cooperation research. During the first period, cooperation was studied by a number of psychologists, anthropologists, sociologists and economists. Cooperation was mainly dealt with as the opposite to competition. This period culminated in the publication of a piece called Competition and Cooperation published in 1937 for the American Social Science Research Council. In this study, May and Doobs gather over two hundred references connected more or less directly with cooperative/competitive behaviors.

Moreover, Argil has observed that three research axes could be noted in the field of psychology: (1) human nature: is it inherently cooperative or competitive? (2) age, and the moments in human development during which cooperation and competition are developed; (3) In educational psychology, comparative efficiency of the competitive and cooperative models on learning results.

In anthropology, Margaret Mead's book, Cooperation and Competition Among Primitive Peoples (Mead, 1937(Mead, /2002) is a significant contribution. Mead suggests that there is some form of originel cooperation in all primitive cultures. Although her approach to cooperation as a "cultural pattern" was heavily criticized, her work remains a significant reference. Similarly, the definition of cooperation as "the act of working together for a common goal" is often cited because it is broad enough to be easily agreed upon [START_REF] Candau | Pourquoi coopérer[END_REF].

After World War II, cooperation/collaboration was indirectly addressed in related topics. In many realms, such as the operation and dynamics of groups or social interactions, or helping and altruism (manque un verbe mais je trouve pas la phrase en français). In many realms, and social psychology particularly, the dominant approach for many years was the theory of the Prisoner's Dilemma Game (PDG) and its renewed version in Axelrod's "win-win" games (1984). However, Deutsch (quoted by Johnson and Johnson, 2010) criticized this theory based on experimental research. He introduced the Social Interactions Theory to understand the mechanisms of cooperative, competitive or individualistic behaviors. Following, Johnson and Johnson (2010) produced works on collaborative behaviors that constitute today a reference on the subject (see for example [START_REF] Michinov | La gestion des connaissances dans les groupes apprenants : processus, méthodes, résultats[END_REF].

Furthermore, developmental psychology continues research initiated by Piaget's studies on the age at which children develop their cooperative behavior. Despite the fact suggested by Piaget that cooperative games begin at the age of 7-8 years old, in the 80s, Marcus (cited in [START_REF] Argyle | Cooperation (Psychology Revivals): The Basis of Sociability[END_REF] evoked the age of 3 years old as the one when appear the behaviors of helping, dividing work, and achieving collective goals.

In parallel, cooperation was examined through a perspective based on evolution in studies on animal behavior. Two very different models of evolution have been developed. The first approach is that of inclusive fitness, developed by Hamilton in the 1960s, and which suggests that individuals act to promote the survival of their genes. It seems that humans cooperate better with those who share the same genes [START_REF] Candau | Pourquoi coopérer[END_REF]. The second model is that of reciprocal altruism. According to this approach, humans have an innate and exceptional ability to cooperate beyond kinship [START_REF] Tomasello | Why We Cooperate[END_REF]Nowak & Highfield, 2011).

Works on social relationships introduced new theoretical paths in 1975-1980. Among them, a theory on exchange emerged to explain the dynamics of social relationships. Following this approach, human relationships are shaped by subjective cost-benefit analysis and by comparing alternatives. Exchange and negotiation, the basic concepts of this theory, were explored for example by D'Amour (1997) or [START_REF] Kosremelli-Asmar | La collaboration interprofessionnelle : cas d'un service de pédiatrie d'un hôpital universitaire au Liban[END_REF] to explain interprofessional collaboration.

This theory, in turn, has been enriched by research on altruism, dedication, the importance of community relationships and by a renewed format of theories on reciprocity and gifts and counter-gifts. This renewal introduced a new approach to cooperation in which cooperation and collaboration were considered in terms of reciprocity and analyzed as a result of mutual gifts (Shoemaker, 1997). Since the 1990s, several authors have referred to this perspective in different fields: sociology [START_REF] Terssac (de) | Coopération et conception[END_REF][START_REF] Alter | Donner et prendre: la coopération en entreprise[END_REF][START_REF] Jobert | Exister au travail les hommes du nucléaire[END_REF], economy [START_REF] Akerlof | Labor contracts as partial gift exchange[END_REF]Eymard-Duvernet, 1994;[START_REF] Reynaud | La régulation interne des marchés du travail[END_REF]Shoemaker 1997), psychology (Enriquez, 1990) or education [START_REF] Labelle | Réciprocité éducative[END_REF][START_REF] Eneau | La part d'autrui dans la formation de soi -Autonomie, autoformation et réciprocité en contexte organisationnel[END_REF][START_REF] Eneau | From reciprocity to autonomy, or vice-versa ? The contribution of french personalism to a new perspective of self-directed learning[END_REF].

Simultaneously, several research on interactions and communications have been developed. For example, since the 1960s, the contributions of the School of Palo Alto introduced language and interpersonal communication as crucial factors to study coordination and joint actions. Later, these works inspired research which emerged with the arrival of recent technological advances and the exponential growth of Internet. Two schools of thought are particularly present in the analysis of cooperative/ collaborative works using asynchronous communication tools. The first is focused on a multidisciplinary scientific community, the Computer Supported Cooperative Work (CSCW: computer-supported cooperative work). This school addresses cooperation as a contextualized, or located, action. With this logic, the possibility of "unintentional" cooperation was evidenced. According to this view, a cooperation/collaboration could occur, without being necessarily conscious for the actors involved in the action [START_REF] Cardon | Les sciences sociales et les machines à coopérer. Une approche bibliographique du Computer Supported Cooperative Work (CSCW)[END_REF]. The second school is the approach of located or distributed cognition (Hutchins 1993). Recently, [START_REF] Michinov | Travail collaboratif et mémoire transactive : revue critique et perspectives de recherche[END_REF] reviewed and summarized the different theoretical and methodological frameworks on collective cognition. These researchers suggest that collaborative work can be explained by using a specific form of collective cognition which is Transactive Memory Systems (TMS). According to these authors, the TMS (based on the former work of Wegner) is a federative concept because it could provide elements to understand how people in groups organize and process information necessary for joint tasks.

Through this brief presentation of selective heterogeneous theoretical schools, we aim to highlight the diversity of viewpoints still active in the analysis of collaborative dynamics. What can be said of these different logics which are difficult to make converge in adult education research today?

III. DIFFERENT APPROACHES TO COLLABORATIVE COMPETENCES

In French-speaking education research, there are few references clearly presenting a definition of collaborative competences. The exploratory research in databases and search engines (e.g. Google Scholar, Eric Francis and Erudite) brings interesting statistics (Figure 1):

• since 2000, there is a constant increase in publications containing mentions of "collaborative competences"; • In educational science (e.g. ERIC) the French referrals are rare and in English, the term "collaborative skill" prevails. • the most represented fields of application are management, teaching, medicine, information and communication.

Figure 1: The coexistence of "collaborative competences" in the literature

In North American and North European literature (Scandinavian countries in particular), the examples of collaborative competences framework are more abundant [START_REF] Morse | Teaching Collaborative Governance: phases, competencies, and Case-Based Learning[END_REF]Emerson & Smutko, 2011;[START_REF] Blomqvist | Collaboration capability -a focal concept in knowledge creation and collaborative innovation in networks[END_REF]. The selection presented here explores specifically the meta-analyzes of factors for collaboration. We consider that these factors relate to the ability of actors to cooperate/collaborate.

III.1 I.1 Factors of collaborative competences

The report, produced in 1992 by Mattessich and Monsey for the Wilder Foundation, reviewed and summarized the existing research literature on factors which influence the success of collaboration. The scope of the search included the health, social science, education, and public affairs arenas. From a selection of 133 studies, 18 studies were reviewed in depth. In the end, the authors selected 19 factors facilitating collaboration, and classified them into six areas:

• factors related to the environment: history of collaboration or cooperation in the community; perception of the collaborative group as a leader in the community; favorable political and social climate • factors related to membership characteristics: mutual respect; understanding and trust; appropriate cross-section of members; perception of collaboration as members' self-interest; ability to compromise. • factors related to process and/or structure: members share a stake in both process and outcome; existence of multiple layers of decision-making ; flexibility; development of clear roles and policy guidelines; adaptability; • communication-related factors: Open and frequent communication; formal and informal communication links. • Factors related to purpose: concrete and achievable goals and objectives; shared vision; unique purpose. • Factors related to resources: sufficient financial base; skilled convener. Even though, as we have aforementioned, these cooperative and collaborative aspects were particularly studied since the advent of digital media tools, this synthesis provides an explanatory reading of collaborative events outside contexts marked by digital.

A more recent research by [START_REF] Thomson | Collaboration processes: inside the black box[END_REF] presents a similar classification. Apart from a solid theoretical foundation, these researchers collected and analyzed numerous data from empirical and longitudinal studies realized between 1995 and 2000. They developed a multidimensional model of collaboration and described the collaborative process as the Five-Dimension Collaboration Scale. Each of these dimensions provides the activities which are crucial to cooperate and all of them take commitment to process over time:

• Governance: making joint decisions on rules intended to govern the collaboration; • Administration: an efficient operating system to move from governance to action, which clearly determines roles and effective communication channels; • Organizational autonomy refers to an intrinsic tension between organizational self-interest and a collective interest. It covers the ability to address this implicit tension exhibited in collaborations between organizational self-interests and the collective interests of the group; • Mutuality: it has its roots in interdependence; it includes the ability to "work through differences to arrive at mutually beneficial relationships";

• Norms are related to the trust and reciprocity modes.

Finally, the framework of [START_REF] Morse | Teaching Collaborative Governance: phases, competencies, and Case-Based Learning[END_REF] differs from the previous two because it explicitly addresses cooperation in terms of collaborative competences. The authors advanced the idea that collaborative competences depend on the specifics of the action that is in the making. Thus, at first they identified a particular action process (e.g. collaborative governance) which they then applied to several grids created through academic and professional works. Finally they synthesized the most significant items in the collaborative competences scale adapted to the process that they analyzed (see Figure 2). The interest of this presentation is the fact that collaborative competences are dealt with as behaviors adapted to the dynamics of the process in which they operate. In other contexts (e.g. the practice of collaborative learning) observed collaborative competences would be different.

I.2 Collaborative competences and levels of collaboration.

According to another point of view, collaborative competences are considered by levels of collaborative action or group population size.

Thus, [START_REF] Blomqvist | Collaboration capability -a focal concept in knowledge creation and collaborative innovation in networks[END_REF] defend the idea that collaborative action depends on the nature of collaborative groups and identify four levels of class. For each level, they combine different items and concepts:

• At the interpersonal level: interpersonal skills with the dominance of trust, commitment and satisfaction. • At the dyad: partnership, long-term vision, ability to cooperate, relational capital.

• At the teams level: collective skills of problem solving.

• At the organizations level: integration of cooperation, networking, alliances.

The nature of the interactions, and therefore, the collaborative competences, varies in relation with the number of participants in a collaborative action [START_REF] Blomqvist | Collaboration capability -a focal concept in knowledge creation and collaborative innovation in networks[END_REF].

A similar view is proposed by D'Amour (1997) and adopted by Kosremelli-Asmar (2011). These researchers stressed the importance of organisational level-structures. The factors of collaboration are considered as an interprofessional practice on three levels:

The micro factors are associated with interpersonal relationships between team members. These factors include:

• The willingness to collaborate constitutes the most important factor in the development of collaborative practice. It hypothetically depends on education, on similar experiences from one's past, and an individual's maturity. • Mutual trust depends on individual skills. Trust is a central component of collaboration because it helps to understand and accept the different levels of expertise, roles, and individuals' maturity. • Communication influences the degree of collaboration and it is a vehicle for other factors such as mutual respect and sharing. • Respect is a prerequisite for collaboration, this means that members recognize and understand the complementarity of their expertise and roles. The meso factors related to the organizational context. Kosremelli-Asmar (2011) pinpoints:

• Flat organizational structure;

• Philosophy of the organization that promotes participation, fairness, freedom of expression and interdependence; • Administrative support, that provides the resources, the time to share and leadership based on participation and collaboration;

• Existence of coordination and communication mechanisms;

The macro factors are external to organizations and this includes: • The social system: identified as a potential obstacle to cooperation, because power relations between actors find their origins in the social system. • The cultural system: some values are deeply rooted in the minds of actors, so that negotiation and mutual adjustments become difficult, if not impossible. • The professional system, which can challenge the logic of interprofessional collaboration, especially when this system is based on power and authority. Kosremelli-Asmar's approach enables dealing with the diversity of issues that advances research in the field of collaboration.

Moreover, the three levels -micro, meso, macro -are usually employed in the field of adults' education, in France at least, and they resonate in training practices in the system Animacoop which will now be discussed.

II. THE CASE OF THE TRAINING PROGRAM ANIMACOOP

The exploratory data of this investigation on collaborative competences are based on the training program called Animacoop observed since its experimental phase and throughout two years of implementation [START_REF] Sanojca | Former aux compétences collaboratives, Actes du 7e colloque « Questions de pédagogies dans l'enseignement supérieur[END_REF].

Animacoop was prepared in response to the call for projects by the French government, by the association Outils-Réseaux in 2010. It was developed as an "education in action" program on facilitating collaborative projects. After 13 delivered sessions, 197 adults had participated in this training, many of whom were working as community organizers in local communities or with special groups such as youth or various social movements.

The program proposed an combination of periodic two-day face-to-face workshops, online support, and time and space for experimentation, and was held together with a Wiki platform, all of that over a period of 14 weeks.

The analysis of the content and learning practices observed in Animacoop, reveals three paradoxes of teaching to collaborate/cooperate.

II.1 First paradox: the collaborative competences are invisible but very present

It is interesting to note here that the objectives don't explicitly name the collaborative competences. The operational goals refer to initiation and deliberation in network groups, to use of collaborative tools and teleworking.

In 2012, a survey conducted among 71 participants showed that only 10% of them read the course throughout the training course [START_REF] Sanojca | Former aux compétences collaboratives, Actes du 7e colloque « Questions de pédagogies dans l'enseignement supérieur[END_REF]). Yet, the satisfaction of these participants at the end of the program is very positive and persons evoked a real learning of collaboration.

The analysis of the training structures and contents can explain the understanding of this learning despite that this goal was not clearly displayed and in spite of the lack of a real appropriation of the contents of the training course.

Thus three modalities of learning to collaborate were implemented through the course:

• Individual learning modality: each learner has to take in the uploaded content every week. • Collective learning modality: activities are structured so as to bring the participants to progressively write a paper to be published on the Internet. • learning through the project modality: one of the prerequisites to participate in the program was to have a specific project in mind that would serve as a test bed to apply the course content Although in most cases, the learners only put their mind to the modality of "collective learning", the engineering of a three-ways learning remains interesting. It is conceivable that the "learning to collaborate" was realized in the combination of these three modalities.

II.2 Second paradox: method which in spite of being informal are nevertheless very structured

In terms of the content's structure, the course was designed so that participants would experience all the stages in the life cycle of a network as they themselves worked together over 14 weeks (see Figure 3). Structuring group activities is another way of understanding the modality of learning to collaborate /cooperate. Indeed, the activities are conducted with groups where the number of participants was increased gradually (Figure 4).

Figure 4: Modality of "collective learning" (from Sanojca (2013))

Despite the fact that the training objectives for developing collaborative competences are not clearly put into words, the fact is that the pedagogical structures were designed so that participants would learn about cooperation and collaboration.

II.3 Third paradox: a combination of pedagogical expertise and "do-it-yourself"

The term used by Animacoop's designers to describe their practice is "actioneducation-action" ant that reflects their point of view on their profession. Indeed they experiment practices and tools without imposing a particular result. Also, they work from a "logic of attention" rather than a "logic of intention": Animacoop's trainers propose conceptual and technical tools in ways to encourage starting small and simple, encouraging their learners to reflect on their practices and on the needs of their individual projects. This attention conduct to innovate both of them: the trainers experiment new attentional ways and participants became designers in their own right.

In this idea of "action-education-action" Animacoop's designers have aimed to name the collaborative competences to be developed in their program. They associated the 12 action verbs to 12 key-concepts proposed during the courses (see Figure 5). This classification proposes the distinction of the three levels where collaborative competences could be practiced. However, we note some ambiguities: first, individuallevel skills relate to the leader's skills and group dynamics. Then, the environment-level skills -e.g. "collective production of resources and methods" relate to process group. However, this effort of formalization of collaborative competences, although imperfect, is relatively rare in professional training. Thus, that demonstrates a real interest by the practitioners to formalize how to learn to collaborate/cooperate.

Figure 5: key-concepts and collaborative action verbs

III. DISCUSSION AND DIRECTIONS FOR FUTURE RESEARCH

In a society operating in open networks, the development of collaborative competences becomes a major challenge for training, especially in programs that use technical tools, and hybrid forms of teaching methods.

As shown in the first part of this article, there are several models on collaborative competences and the consequence is that it is difficult to prefer one model. Nevertheless, it can be suggested that this model could be derived from a crossdisciplinary approach and a systematic analysis of multiple dimensions.

III.1 Literature lighting for the concept of "collaborative competences"

Several researches revealed two logics that could facilitate the understanding of collaborative competences. Firstly and foremost, the factors of collaboration constitute several indicators for naming skills and abilities required to cooperate/collaborate. Despite the fact that they remain highly variable, they refer to three components of the collaborative process: antecedent, i.e. the favorable conditions; the process, related to collective action; and the results, which include the common productions.

On these grounds, this structure of collaborative process can be compared to a "systemic model of work activity" [START_REF] Aubé | Les interventions régulatrices groupales et le rendement des équipes de travail : un modèle théorique[END_REF][START_REF] Thomson | Collaboration processes: inside the black box[END_REF] particularly relevant to design training devices to be taken into account and to develop collaborative competences.

Secondly, we have stressed these competences in relation to the particular situation and action; so that, they are doubly located: because the nature of competences themselves can be revealed only in specific situations, the nature of cooperation/collaboration is to "make whole". As we have suggested, the collaborative competences are mobilized differently according to their action (e.g. designing, learning or governance). Thus, before going on to develop the collaborative competences in training, it is more appropriate to discern the process of collective action in which the persons are implicated and only after that adopt the most convenient competences to be consolidated. The example of the work of Morse & Stephens (2009) stresses this point of view.

Finally, those contextualized and identified competences could be observed in three different contexts in which they operate: (1) in the interactions between individuals (micro); (2) in the organizational structures (meso), correlated with the capacity to organize the non-hierarchical models, to manage and share resources; (3) in the external environment (macro), connected with the political and cultural context.

Thus, we suggest combining three logics to analyze the collaborative competences: sequential analyses: (input-process-output), dynamics analyses (linked to action) and gradual (by levels).

III.2 From the formation project to the development of collaborative competences

The program Animacoop can be analyzed as an attempt to develop collaborative competences. Although as we have seen, they are not explicitly included in the training program, factly, they are developped by learners through group work. Progress of learning to cooperate/collaborate through the five stages of a network life is comparable to the "process approach" described by Morse & Stephens (2009). It seems to indicate that this process is artificial in the Animacoop program: the participants refer to five stages of a network life during the collective presentations yet they did not adopt that in their work. Moreover this structure of courses in a framnetwork life, is not connected with two other modalities of learning: "collective learning" and "learning in the project". This pragmatic approach to conceptualizing training programs should be explored in future works.

The exercises "group size scaling extended" could be underlined as another form of learning cooperation/collaboration. Thus, staff experimented the idea of character "aggregative" social networks. As Proulx suggests (2008), the use of digital technologies strongly influence identity development of virtual communities; they are more an "ephemeral aggregations of individuals," formed around similar interests, sparking partial participation. The ability of people to aggregate/disaggregate seems to develop through interactions in groups of different sizes, constituted gradually: two, four or eight participants. However this point needs to be examined in greater depth in future research because a number of other issues remain unresolved: "Ability to aggregate", is a particular goal really fixed by instructors? Would the participants benefit from teaching in this exercise? Is this specific ability really significative to develop collaborative competences? If this was the case, could we appoint specific competences according to the level on which a cooperation/collaboration is realized, as suggested by [START_REF] Blomqvist | Collaboration capability -a focal concept in knowledge creation and collaborative innovation in networks[END_REF]?

Thus, to continue the ongoing research, we will first check the adequacy of available models and, in particular, the collaborative competences scale of Morse and Stephens. A questionnaire distributed to 197 participants of Animcoop with twenty-five indicators proposed by these authors is currently analyzed.

The next phase of the survey is to interview in greater depth, the training programed methods that facilitate to work in common. This point is addressed currently by a series of semi-structured interviews, with a sample of thirty trainees.

As we interest ourselves in the working contexts of Animacoop's participants after their training, we want to know, amongst others, how they have reinvested or not these collaborative competences. The first results suggest that adult learners, prior to cooperating/ collaborating actually, once they return to their workplace, begin to transmit to their colleagues what they learned during the training. It seems that, as well as collaborative competences being acquired in the training-action, they reinvest in a particular process of action-training. This perspective reveals the attitude of "improvisation" and "creativity" as much in the training process as in the professional activity. We will examine this in depth.

The expected results will allow us to validate a collaborative skill scale built from the work of Morse and Stephens (2009).

However, the challenge of this work lies not in the design of repository of collaborative competences. We prefer to apprehend these competences like "know to act" to facilitate the necessary adaptation to a changing environment (Voogt & Pareja Roblin, 2012;[START_REF] Sennett | Ensemble : pour une éthique de la coopération[END_REF]. In other words, the ability to cooperate/collaborate seems like the forefront of competences to be acquired, in contemporary professional work.

Figure

  Figure 2: Collaborative competences (from Morse & Stephens (2012, p. 572)) Assessment Initiation Deliberation Implementation

Figure 3 :

 3 Figure 3: Structure of content according to the stages of community development


Guillaume Cheikbossian
email: guillaume.cheikbossian@umontpellier.fr

Group Cooperation against an Incumbent

Keywords: Collective Action, Group Cooperation, Repeated Game, Trigger Strategies, Renegotiation JEL Classi…cation: C73, D72, D74

In this paper, I study the ability of group members to cooperate against an incumbent in a repeated rent-seeking game and where group members and the incumbent have di¤erent valuations of the prize. I …rst consider that group members use Nash Reversion Strategies (NRS) to support cooperative behavior and show that full cooperation within the group is more easily sustained as a Stationary Subgame Perfect (Nash) Equilibrium (SSPE) as either group size, or the heterogeneity in the valuation of the prize, increases. In turn, I show that full cooperation within the challenger group can also be sustained as a Weakly Renegotiation-Proof Equilibrium (WRPE). Yet, an increase in group size makes it more di¢ cult to sustain within-group cooperation but an increase in the relative valuation of the prize by group members still facilitates group cooperation.

Introduction

Most of economic, social and political activities involve groups (countries, political parties, …rms, administrative units, etc....) with shared common interests within groups and, potentially, con ‡icting interests across groups. The e¤ectiveness of a group in carrying out its objective then crucially depends on how it deals with its collective action -or free-rider -problem. [START_REF] Olson | The Logic of Collective Action[END_REF] celebrated theory argues that this problem makes larger groups less e¤ective than smaller groups and that, ultimately, collective action must break down when group size becomes very large. 1 The di¢ culties facing large groups seem to be also persistent in a dynamic setting. Despite the potential of trigger strategies to induce mutual cooperation in a repeated game [START_REF] Axelrod | The Emergence of Cooperation among Egoists[END_REF], the common wisdom is that cooperation becomes more di¢ cult to sustain as the size of the collectivity increases (e.g., [START_REF] Hardin | Collective Action[END_REF][START_REF] Olson | The Rise and Decline of Nations[END_REF][START_REF] Sandler | Collective Action: Theory and Applications[END_REF][START_REF] Taylor | Community, Anarchy and Liberty[END_REF].

A limitation of the traditional collective action theory, is that it analyses the collective action problem within a single group isolated form the rest of society. However, in many situations, collective action is often undertaken to counter similar actions by competing groups. Collective action might also be necessary for defending collective resources against an external entity or may also be undertaken for challenging the dominant position of an incumbent. In this paper, I analyze the logic of ongoing collective action in a repeated game setting in which an interest group -called the challenger group -invests some e¤orts to take over an incumbent for the award of a collective prize. The incumbent is referred to as such in that it does not have any collective action problem to solve. Also, the collective prize has the characteristics of a (pure) public good and all members of the challenger group have the same valuation of the prize, which can be di¤erent from that of the incumbent. One might think for example of several green NGOs making lobbying e¤orts in order to induce governments to pass a law on environmental protection in a speci…c production sector. The coveted public good could also be something more political as in the case of a parliamentary coalition challenging the power of the incumbent party.

In this paper, I will use a simple model where the challenger group's probability of winning the prize is given by the classic function introduced by [START_REF] Tullock | E¢ cient Rent-seeking[END_REF]. In this context, I …rst investigate the e¤ectiveness of Nash Reversion Strategies (NRS), à la [START_REF] Friedman | A Non-Cooperative Outcome for Supergames[END_REF], to support full cooperation within the group. These strategies prescribe that any deviation from the cooperative level of e¤ort is met with permanent reversion to the non-cooperative outcome. The di¢ culty of supporting cooperation within the group is then measured by the lowest discount factor supporting the optimal level of group e¤ort as a subgame perfect outcome. I show that cooperation within the group is more easily sustained as either the relative valuation of the prize by group members or group size increases. This last result is in contrast with the common wisdom that group-size is detrimental to the e¤ectiveness of trigger strategies to support cooperation. The intuition is that the Nash punishment threat becomes relatively more e¤ective as group size increases. Next, I investigate whether full cooperation within the group can be sustained as a Weakly Renegotiation-Proof Equilibrium (WRPE), in the sense of [START_REF] Farrell | Renegotiation in Repeated Games[END_REF]. With perfect and frictionless renegotiation, full cooperation can indeed be sustained as a WPRE even in very large groups although it becomes more di¢ cult to sustain as group size increases. However, an increase in the relative valuation of the prize by group members still makes it easier to sustain within-group cooperation. Thus, overall, I …nd only partial support for the Olson's theory.

My analysis is related to the literature on tacit cooperation in repeated games applied to public economics. [START_REF] Mcmillan | Individual Incentives in the Supply of Public Inputs[END_REF] was the …rst to apply NRS to a supergame model of private provision of a public good but without investigating the impact of an increase in the number of agents. In a prisoner's dilemma game with n players, [START_REF] Bendor | Institutional Structure and the Logic of Ongoing Collective Action[END_REF] show that punishment strategies are not very valuable to sustain an e¢ cient outcome in large groups when there is imperfect monitoring. [START_REF] Pecorino | The E¤ect of Group Size on Public Good Provision in A Repeated Game Setting[END_REF] focuses on the e¤ect of an increase in the number of contributors on their ability -as measured by the critical value of the discount parameter -to overcome free-riding in a model of private provision of a pure public good.

He cannot derive monotonicity results but shows that cooperation can be maintained when the number of contributors goes to in…nity for several speci…cations of the individual payo¤ function. [START_REF] Haag | On the Size and Structure of Group Cooperation[END_REF] analyze a collective action game played by a collection of heterogeneous individuals in terms of time preferences. They show that homogeneous and larger groups are more cooperative on average for a large class of collective action games.

All these works analyze the collective action problem within a group isolated from the rest of society. In contrast, in the present analysis, group action aims at contesting similar action by a unitary opponent. In another paper [START_REF] Cheikbossian | The Collective Action Problem: Within-group Cooperation and BetIengroup Competition in a Repeated Rent-Seeking Game[END_REF], I analyze the ability of group members to cooperate in rent-seeking for a private prize -fully divisible among group members -in a context of competition between two groups of unequal size. The two groups face a collective action problem and all individuals have the same valuation of the prize. I then show that the set of parameters for which cooperation can be sustained within the larger group as a subgame perfect outcome is as large as that for which cooperation can be sustained in the smaller group. Again, in the present analysis, I consider that one group only has a collective action problem to solve but the two entities have di¤erent valuations for the public prize. More importantly, I consider that group members can renegotiate in case within-group cooperation breaks down.

The present analysis is thus also related to the literature on the concept of 'weak-renegotiation proofness' applied to oligopoly models and to the issue of international agreements. Using Farrell and Maskin (1989)'s solution concept of WRPE for two-players games, [START_REF] Farrell | Renegotiation in Repeated Oligopoly Interaction[END_REF] shows that full collusion as a WRPE is impossible in repeated Bertrand competition when there are more than three …rms, or more than nine …rms in repeated Cournot competition (with linear demand and cost functions). [START_REF] Barrett | A Theory of Full International Cooperation[END_REF] develops a model of international cooperation where each country has a binary choice between cooperating or defecting and also shows that cooperation can only be supported by a 'small'number of countries. In a similar model, [START_REF] Asheim | Regional versus Global Cooperation on Climate Control[END_REF] show that two international agreements can do better than a single one for involving a larger number of countries in an agreement. Still in a model where players have a binary choice, [START_REF] Froyn | A climate Agreement with Full Participation[END_REF] show that full participation can be sustained as WRPE provided that only a limited number of countries are permitted to punish a defection. Finally, [START_REF] Asheim | Renegotiation-Proof Climate Agreements with Full Participation: Conditions for Pareto-E¢ ciency[END_REF] show that this result carries over a more general model where players or countries have a continuum of (emission) choices.

Yet, they consider a very simple model since the Nash equilibrium of the stage game is an equilibrium in dominant strategies. In other words, the optimal choice of one player does not depend on the choices made by the other players. Furthermore, all players can participate to the cooperative agreement. In my analysis, there exist strategic interactions in that a change in a group member's action induces a change in the equilibrium actions not only of the other group members but also of the opponent.

The model 2.1 The stage game

I start by specifying the details of the stage game G. There are n + 1 risk-neutral agents, indexed by i = 0; 1; 2; ::::; n who compete for a public prize. Agent 0 is a unitary entity -called the incumbentand has a valuation V I for the prize. Agents i = 1; 2; :::; n are grouped into a common entity -called the challenger group -and join their e¤orts to take over from the incumbent in the award of the prize.

All group members have the same valuation V G of the prize, which is supposed to be higher than that of the incumbent, i.e. V G V I . These valuations are publicly known. :

X=(X + Y ) if (X; Y ) 6 = (0; 0); 1=2 if (X; Y ) = (0; 0): (1)
The preferences of all players are represented by an additively separable utility function, i.e.

v i = p (X; Y ) V G x i ; (2)
for member i = 1; 2; :::; n of the challenger group, and

v 0 = (1 p (X; Y ))V I Y; (3)
for the incumbent.

I …rst analyze the one-shot equilibrium outcome in which group members do not cooperate in the contest with the opponent. Let X i = P j6 =i x j be the sum of rent-seeking e¤orts of the group members excluding that of member i. Maximizing v i with respect to x i subject to the non-negativity constraint x i 0 yields group member i's best response to X i and to Y , i.e. r i (X i ; Y ), which is given by the following …rst order-condition 2Y

(x i + X i + Y ) 2 V G = 1; (4)
with equality for x i > 0:

Similarly, maximizing v 0 with respect to Y subject to the non-negativity constraint Y 0, yields the best-response function of the incumbent to the collective e¤ort X of the challenger group, i.e. R 0 (X), which is given by the following …rst order-condition

X (X + Y ) 2 V I 1; (5)
with equality for Y > 0.

Due to the public-good nature of collective rent-seeking, …rst-order conditions only determine group e¤ort. I look, however, for a symmetric equilibrium such that all group members make the same level of e¤ort. The symmetric equilibrium is thus a couple x N ; Y N such that R 0 (X N ) = Y N and r i (X N i ; Y N ) = x N for all i = 1; 2; :::; n (and where X N i + x N = X N = nx N ). Solving the system given by ( 4) and [START_REF] Bendor | Institutional Structure and the Logic of Ongoing Collective Action[END_REF] and letting be the ratio of the valuations of the prize, i.e. = V G =V I 1, we have that

x N = n [1 + ] 2 V G ; (6)
while

Y N = X N = .
The equilibrium winning probability for the challenger group is thus given by

p(X N ; Y N ) = 1 + ; (7)
which is independent of the size of the challenger group. Indeed, with the lottery contest success function and linear cost functions, the increased free-rider problem induced by a larger group size is exactly compensated by the larger number of contributors (see [START_REF] Katz | Rent-seeking for Pure Public Goods[END_REF][START_REF] Ursprung | Public Goods, Rent Dissipation and Candidate Competition[END_REF].

Under non-cooperation, each group member has thus the following utility

v N = [n(1 + ) 1] n [1 + ] 2 V G ; (8)
while that of the incumbent is given by

v N 0 = V I [1 + ] 2 : (9)

The optimal solution

Consider now a situation where there can be full cooperation within the challenger group. In other words, group members can jointly choose individual e¤orts so as to maximize the aggregate welfare of their group, i.e., they choose X so as to maximize

P i v i = np (X; Y ) X, given Y .
The collective best response of the challenger group to Y , i.e. R(Y ), is given by the following …rst-order condition nY

(X + Y ) 2 V G 1; (10)
with equality for X > 0.

Similarly, the best-response function of the incumbent to the collective e¤ort X of the challenger group, i.e. R 0 (X), is given by the following …rst-order condition

X (X + Y ) 2 V I 1; (11)
with equality for Y > 0.

An interior equilibrium is a couple of e¤orts

(X C ; Y C ) such that R(Y C ) = X C and R 0 (X C ) = Y C .
Assuming that the members of the challenger group share equally the collective e¤ort, and denoting by

x C = X C =n this common individual e¤ort under within-group cooperation, I obtain

x C = n [1 + n] 2 V G ; (12)
and

Y C = x C = .
As a result the equilibrium winning probability for the challenger group is given by

p(X C ; Y C ) = n 1 + n : (13)
Under cooperation within the challenger group, each group member has thus the following utility

v C = n 1 + n 2 V G ; (14)
while that of the incumbent is given by

v C 0 = V I [1 + n] 2 : (15)
3 The repeated game

Preliminaries

The n + 1 agents play an in…nitely repeated game with discounting. Time is discrete and dates are denoted by t = 0; 1; 2:::Let G 1 ( ) be the repeated game obtained by repeating G in…nitely often, and where 2 (0; 1) is the discount parameter per period for each player. I assume that the e¤ort level produced by all group members and the incumbent are perfectly observed by all agents. Let Y (t) 2 < + and x(t) (x 1 (t); x 2 (t); :::; x n (t)) 2 < n + be, respectively, the e¤ort of the incumbent and the vector of individual e¤orts within the challenger group in period t. A public history in period t 1 is thus h(t) = ((x(0); Y (0)) ; (x(1); Y (1)) ; :::;

(x(t 1); Y (t 1))) or h(t) = f(x(s); Y (s))g t 1 s=0
. The initial history is the null set, H 0 f?g, and H t the set of t-period histories. A pure strategy for group member i in G 1 ( ), for i = 1; 2; :::; n, is an in…nite sequence of functions i = f t i g 1 t=1 , where t i : H t ! < + is a mapping from the set of all possible public histories up to t 1 (included) into the set of e¤ort levels. Similarly, a pure strategy in G 1 ( ) for the incumbent is de…ned on the same sets, i.e. 0 = f t 0 g 1 t=1 , where t 0 : H t ! < + . The strategy pro…le is denoted by ( 0 ; 1 ; 2 ; :::; n ).

Any strategy pro…le generates an outcome path S = fx(t); Y (t)g 1 t=0 , de…ned inductively by fx(0); Y (0)g = (?) and (x(t); Y (t)) = ( t 0 ; t 1 ; :::; t n ) for all t 1. A path of rent-seeking e¤orts S thus implies an in…nite stream of stage-game payo¤s fv i (x(t); Y (t))g 1 t=0 for group member i = 1; 2; :::; n and fv 0 (x(t); Y (t))g

1 t=0 for the incumbent. Let v i (t) v i (x(t); Y (t)
) for i = 0; 1; 2; :::n -with i = 0 for the incumbent and i = 1; 2; :::; n for the members of the challenger group. The average discounted payo¤ of player i for the outcome path

S = fx(t); Y (t)g 1 t=0 is given by V i (S) = (1 ) X 1 t=0 t v i (t) (16)
Thus, the average discounted payo¤ of player i, for i = 0; 1; 2; :::; n in G 1 ( ) obtained with the strategy

pro…le is V i ( ) = V i (S ( )).
is a Nash equilibrium if 0 is a best response to ( 1 ; 2 ; :::; n ) and if i , for i = 1; :::; n, is a best response to i = ( 0 ; 1 ; :::; i 1 ; i+1 ; :::; n ). And it is a subgame perfect equilibrium in G 1 ( ) if after every history, the continuation of is a Nash equilibrium in the corresponding subgame. I will restrict attention to Stationary Subgame Perfect (Nash) Equilibria (SSPE), i.e., equilibria in which after any history, a stationary pro…le of strategies is played thereafter, and which also satisfy the additional requirement of symmetry within the challenger group, in the sense that all group members make the same level of e¤ort at every history.

Nash Reversion Strategies

I …rst consider that the members of the challenger group use Nash Reversion Strategies (NRSs) à la [START_REF] Friedman | A Non-Cooperative Outcome for Supergames[END_REF] in order to support cooperative behavior. NRSs prescribe that group members cooperate in the …rst period and in all subsequent periods if all members cooperated in the previous period. If any member deviated in the previous period, then all group members revert to the single-shot Nash equilibrium forever, independently of the behavior of the incumbent. Yet, group members' actions must be mutual best responses and must also be individual best-responses to the action of the incumbent -in case of non-cooperation within the group -or must constitute a collective best-response function to the action of the incumbent -in case of cooperation within the group. Furthermore, it is supposed that the incumbent plays in every period his static best-response to the group members'strategies -whether they cooperate or not -regardless of the history (i.e. t Y = R 0 (x(t)) for all t). To put this informally, there is always non-cooperation between the challenger group and the incumbent.

In general, in in…nitely repeated games, the set of SSPE is very large. I thus focus on the "best" SSPE from the viewpoint of group members, that is the one sustaining the joint-payo¤ maximizing level of rentseeking e¤ort x C . 3 Formally, I thus focus on the strategy pro…le characterized by: t i = x C for i = 1; 2; :::; n

and t 0 = R 0 (X C ) = Y C if t = 0 or if h(t) = (x C (s); Y C (s))
t 1 s=0 for t 1; t i = x N for i = 1; 2; :::; n and

t 0 = R 0 (X N ) = Y N otherwise.
This strategy pro…le is denoted by ~ . Let x D be the e¤ort level of a contributing member who considers defecting from the cooperative phase while other play ~ . The payo¤ of the deviator would thus be given by

v D = (n 1)x C + x D (n 1)x C + x D + Y C V G x D , ( 17
)
where x C is given by ( 14) and where x C = Y C . The deviator chooses x D so as to maximize v D . The …rst-order condition to this problem implies that

@v D @x D = Y C [(n 1)x C + x D + Y C ] 2 V G 1 0. (18)
In any period in which all group members contribute the joint-maximizing level of e¤ort x C , each member's best possible deviation from ~ is given by:

x D = 8 > > < > > : = p n(1 + n) n [1 + (n 1)] (1 + n) 2 V G if n = 2; 0 if n 3: (19)
Thus each member's best possible deviation from ~ is to cut her contribution to 0 for any n 3, independently of . In contrast, x D > 0 for n = 2, independently of . 4 Substituting ( 21) into [START_REF] Salant | Losses from Horizontal Merger: The E¤ects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium[END_REF] and using [START_REF] Mcmillan | Individual Incentives in the Supply of Public Inputs[END_REF], I obtain the (best) deviation payo¤ for any group member, 3 I must insist that this SSPE is Pareto-e¢ cient only from the point of group members. This is reason why I use the term "best" SSPE. 4 Indeed, x D 0 if and only if

[(1 + n)=(1 + (n 1))] p n.
On the one hand, the left-hand term of this inequality is increasing in and thus reaches a maximum when is going to in…nity, in which case the inequality reduces to n=(n 1) p n or p n (n 1), which does not hold for any n 3. On the other hadn, the left-hand term of the above-mentioned inequality reaches a minimum in = 1, in which case it reduces to

(n + 1)=n p n, that is 3 > 2 p 2 (when n = 2).
that is

v D = 8 > > > > < > > > > : = 1 + (1 + n) [n(1 + ) 2 p n] (1 + n) 2 V G if n = 2; (n 1) 1 + (n 1) V G if n 3: (20)
The one-period net gain to deviating from the cooperative agreement within the group is v D v C , while the per-period net bene…t to maintaining cooperation within the group is v C v N . Thus, the discounted value of avoiding a (permanent) breakdown of cooperation within the group is given by

[ = (1 )] v C v N .
Therefore, no group members have an incentive to deviate from the cooperative agreement if and only if

v D v C 1 v C v N : (21)
Focusing on situations where self-enforcement is a binding constraint on the abilities of group members to cooperate, the critical value of the discount parameter above which cooperation can be sustained as a SSPE through NRSs is then 10), ( 16) and ( 22) into this expression, I obtain

= v D v C = v D v N . Substituting (
N (2) = 2(3 2 p 2)(1 + 2 )(1 + ) 2 8(2 p 2) 3 + 4(8 5 p 2) 2 + (23 16 p 2) + 2(3 2 p 2) ; (22)
if n = 2, and

N (n) j n 3 = n(1 + ) 2 [ n(n 2) + (n 1)] (1 + n) 2 [ (n 2 n 1) + (n 1) 2 ] (23)
if n 3.

I have the following result. 5Result 1: (i) N (n) is decreasing in n for any n 3 independently of ; (ii) N (n) is decreasing in for any n 2 independently of n:

The sustainability of within-group cooperation depends on the impact of increasing n or on the one-period net gain of deviating from the cooperative agreement -given by v D v C -relative to the net bene…t of maintaining within-group cooperation -given by v C v N . I focus on the situation where n 3. I have

v C v N = (n 1) n 2 (1 + ) (1 + n) n(1 + ) 2 (1 + n) 2 V G ; (24)
and

v D v C = [ n(n 2) + (n 1)] [1 + (n 1)] (1 + n) 2 V G : (25)
As shown in the Proof of Result 1 (in the Appendix), the net bene…t of cooperation -given by v C v Nis increasing in n. This re ‡ects the fact that an increase in group size exacerbates the free-rider problem, which in turn makes the punishment threat relatively more e¤ective. As regards the impact of group size on the incentive to deviate, that is on the net bene…t of defection, -given by v D v C -it is decreasing in n for any n 4. The incentive to deviate from the cooperative outcome is determined by the bene…t of withdrawing his/her individual contribution and by the resulting negative impact on the probability of success of the group. But this last is concave in aggregate e¤ort. Hence, the decrease in the probability of success due to an individual defection becomes less pronounced as n increases, which would reinforce the incentive to defect. However, the concavity of the probability of success also implies that, when n increases, it increases the optimal level of aggregate e¤ort but by a lower extent than the size of the group. In turn, an individual contribution and thus the bene…t of withdrawing his/her contribution also decreases with n, which would relax the incentive to deviate. It turns out that the decrease in the bene…t of withdrawing his/her individual contribution dominates that of the associated cost in terms of probability of success. As a result, the net bene…t of defection decreases with group size and this e¤ect goes in the same direction than the increase in the punishment threat for facilitating the sustainability of group cooperation. This is in contrast with the general presumption that decentralized strategies of reciprocity may fail to enforce cooperation in large groups. In the present analysis, where group members face an external entity, I have that an increase in group size unambiguously facilitates group cooperation. Furthermore, cooperation remains an equilibrium in very large groups if their members place any weight at all on the future since, then, the critical discount factor N (n) j n 3 converges to 0 as n goes to in…nity.

Also, an increase in the valuation of the prize by group members compared to that of the incumbenti.e. an increase in -makes within-group cooperation less di¢ cult to sustain. 6 As shown in the appendix, an increase in increases the net bene…t of cooperation and thus the e¤ectiveness of the punishment threat as does an increase in group size. However, it also increases the net bene…t of deviating from the cooperative outcome. That is, as increases, the bene…t of withdrawing his/her individual contribution becomes more important than the cost in terms of probability of success. As a result the net bene…t of defection increases but this e¤ect is always dominated by the increased punishment threat as rises, which in turn makes within-group cooperation easier to sustain.

Renegotiation

Although in…nite Nash reversion is subgame perfect, it is not very realistic because it is unforgiving and, furthermore, it hurts the punishers just as the deviator. Indeed, it is hard to believe that agents remain forever in a Pareto dominated equilibrium without renegotiating back to a cooperative outcome. I thus now consider another strategy pro…le which is robust to renegotiation. Speci…cally, I analyze whether full cooperation can be sustained as a Weakly Renegotiation-Proof Equilibrium (WRPE), in the sense of Farrell and Maskin (1989). To be WPRE, a strategy pro…le must satisfy two requirements: (i) The …rst is that of subgame perfection which requires that no player can gain by a one-period deviation after any history;

(ii) The second requirement is that, after any history, there must not exist two continuation equilibria such that all players are better o¤ in one continuation equilibrium than in the other. Otherwise, the players would renegotiate from one to the other.

A strategy pro…le satisfying these two requirements is Penance-p, which has been proposed by [START_REF] Asheim | Regional versus Global Cooperation on Climate Control[END_REF] and [START_REF] Froyn | A climate Agreement with Full Participation[END_REF] in the context of a prisoner's dilemma game. Our contribution here is to extend the use of this concept to a game of group cooperation with continuous payo¤s and strategic interactions between players and, more importantly, with one player not participating to the agreement.

Speci…cally, the Penance-p subgame perfect strategy pro…le, denoted by ~ (p), speci…es (i) that a group member contributes the joint-maximizing level of e¤ort except if another group member has been the sole deviator from Penance-p in the previous period and (ii) that if a deviation occurs then p group members -the punishing members -stop contributing to collective action, whereas the n p other group members continue to act cooperatively. This means that they produce the joint-maximizing level of group e¤ort of a group consisting of n p members, while the p punishing members do not contribute at all to collective action but still bene…t from the collective action of their group. Punishments last only one period and so this strategy pro…le is robust to renegotiation if not all group members strictly gain by restarted full cooperation instead of carrying out the punishment.

Again, I focus on the "best" WRPE from the viewpoint of group members, that is the one sustaining the joint-payo¤ maximizing level of rent-seeking e¤ort x C . Recall that x(t) (x 1 (t); x 2 (t); :::; x n (t)) 2 < n + . Thus, the Penance-p strategy pro…le can be formally described as follows

t i = 8 > > < > > : x C if t = 0 or if (x(t 1); Y (t 1)) = (x C ; Y C );
x P if x i (t 1) 6 = x C or if x j (t 1) 6 = x C for j 6 = i and i = 2 P j ; 0 if x j (t 1) 6 = x C for j 6 = i and i 2 P j ;

(26) for i = 1; 2; :::; n.

x P is joint-payo¤ maximizing level of rent-seeking e¤ort of a group of size (n p), and where P j is the set of punishing members when member j deviates from ~ (p). As mentioned above, the number of punishing members is the same for all i = 1; 2; :::; n and is denoted by p. Finally, as in the previous section, the incumbent is assumed to play in every period his static best-response to the group members'strategies regardless of the history, that is t 0 = R 0 (x(t)) for all t.

The subgame-perfection requirement

Consider …rst the requirement of subgame perfection. There are two kinds of histories to check: When no deviation has taken place in the previous round t 1 and when a deviation did occur in t 1.

No deviation in period t 1:

In this history, Penance-p prescribes that all group members continue to cooperate. If however, one member deviates in period t and reverts to Penance-p in t + 1 then his/her deviation payo¤ is v D given by [START_REF] Tullock | E¢ cient Rent-seeking[END_REF] in period t, while in period t + 1 he/she obtains the payo¤ of group cooperation with n p contributing members. Substituting n by n p into (16), I obtain

v P = [ (n p)] 2 [1 + (n p)] 2 V G : (27)
In period t + 2 onward, full cooperation is restored so that member i obtains v C , given by ( 16) in that period and in any future period. Therefore, any group member has no incentives to unilaterally deviate from the cooperative outcome unless he/she can obtain a larger discounted sum of payo¤s in period t and t + 1 by deviating, that is if and only if

v D + v P (1 + )v C or P with P v D v C = v C v P .
Recall that the best deviation payo¤ v D from the cooperative phase depends on whether n = 2 or n 3.

If n = 2, then p = 1. Indeed p must always be strictly positive otherwise defection would never be punished and so each member would always defect. And it must also be lower than 2, otherwise the strategy pro…le would correspond to Nash reversion strategies and the renegotiation-proofness requirement would not be satis…ed. Then, using ( 16), the …rst expression in ( 22) and ( 29), I obtain for n = 2 and p = 1;

P (2; 1) = (3 2 p 2)(1 + 2 )(1 + ) 2 2 (3 + 4 ) : (28)
Now consider that n 3. In this case, the member who defects from the cooperative phase cuts her contribution to 0 (see [START_REF] Taylor | Community, Anarchy and Liberty[END_REF], so that the best deviation payo¤ is given by the second expression in [START_REF] Tullock | E¢ cient Rent-seeking[END_REF]. Also using ( 16) and ( 29), I then obtain the following threshold value of the discount parameter

P (n; p) j n 3 = [1 + (n p)] 2 [ n(n 2) + (n 1)] p [1 + (n 1)] [2 n(n p) + 2n p] : (29)
We now have to verify the subgame-perfect requirement, when a single deviation has taken place in period

t 1.
A deviation in period t 1:

In this history, Penance-p prescribes that in period t, there are p punishing members who do not contribute to the collective action of the group. I …rst determine under which condition the n p group members (including the member who deviated in t 1) that are prescribed to cooperate to the bene…t of the whole group by contributing x P have no incentive to deviate from Penance-p in period t.

If n = 2 (and thus p = 1) or n p = 1, the best deviation payo¤ from the punishment phase, denoted v DP is equal to v P because v P is the optimal payo¤ of the unique member who makes rent-seeking e¤orts in t (because he/she deviated in t 1). Thus, the incentive compatibility constraint on the punishment path is always satis…ed if n p = 1.

Now suppose that (n p) > 1 and let x DP be the e¤ort level of a contributing member who defects from the punishment in period t. The best deviation payo¤ of a contributing member in the punishment phase who decides to deviate depends on whether n p = 2 or n p 3 and is given by (??) with n replaced by n p, that is

2 v DP = 8 > > > > < > > > > : = 4 2 + (3 2 p 2)(1 + 2 ) (1 + 2 ) 2 V G if n p = 2;
(n p 1) 1 + (n p 1)

V G if n p 3:

: (30)
Therefore, each of the n p contributors in period t has no incentive to unilaterally deviate from Penance-p unless he/she can obtain a larger discounted sum of payo¤s in period t and t + 1 by deviating, that is if

and only if v DP + v P v P + v C or DP with DP v DP v P = v C v P .
Thus, using ( 16), ( 29) and ( 35), the minimum discount factor above which no contributing member in the punishment phase has an incentive to deviate is given by

DP (n) j n p=2 = (3 2 p 2)(1 + 2 )(1 + n) 2 (n 2) [n + 2 + 4 n] 2 ;
(31)

if n p = 2 and DP (n; p)

j n p 3 = (1 + n) 2 [ (n p)(n p 2) + (n p 1)] p [ (n p 1) + 1] [2 n(n p) + 2n p] ; (32)
if n p 3.

I now establish the following result that will prove useful in the rest of the analysis.

Result 2:

The minimum discount factors P (n; p) j n 3 and DP (n; p) j n p 3 above which the strategy pro-…le ~ (p) satis…es the subgame-perfect requirements are both decreasing in the number of punishing members p:

I also need to verify that the punishing members who do not contribute at all to collective action in the punishment period have no incentives to deviate by producing a positive level of e¤ort. If the p punishing members adhere to the punishment in period t and do not participate to collective action, then the payo¤ of each of them is given by the probability of success (multiplied by V G ) of a cooperative group consisting of n p members, that is from (15),

v P A = (n p) 1 + (n p) V G : (33)
If n p 3, then each group member's best possible deviation is to cut her contribution to 0, which corresponds exactly to what is prescribed by penance-p for the p punishing members. So, in this case, the punishing members do not have any incentives to deviate in the punishment phase, all the more it will trigger the punishment again in t + 1.

If n p 2, then the best deviation for a contributing member is to make a positive level of e¤ort.

However, a non-contributing member still cannot do better than not contributing at all to collective action independently of n and p. Indeed, let x A be the e¤ort level of a punishing member who considers defecting in the punishment period t. His/her payo¤ would be given by

v A = (n p)x P + x A (n p)x P + x A + Y P V G x A ; (34)
where x P = Y P . @v A =@x A is decreasing in x A and thus a punishing member does not have any incentive to deviate from penance-p by producing a positive level of e¤ort if and only if

@v A =@x A j x A =0 0, that is if V G = [1 + (n p)] 2 Y P . Since Y P = x P = = h (n p)= [1 + (n p)] 2 i
V G , this inequality is always satis…ed for n p 1.

The renegotiation-proofness requirement

I now turn to the renegotiation-proofness requirement. It implies an upper bound on the number of punishing countries. Indeed, full cooperation is weakly renegotiation-proof if not all group members are strictly worse o¤ with the punishment than with renegotiation (i.e. returning immediately to cooperation without punishment). Suppose member i deviates in t 1. Clearly, in period t the n p members (including the defecting member of period t 1) who contribute to group e¤ort are worse o¤ with the punishment than with renegotiation. Thus, weak-renegotiation proofness requires that the p punishing members are at least as well o¤ with punishment as with renegotiation.

If the p punishing members adhere to the punishment in period t and do not participate to collective action, then the payo¤ of each of them is given by v P A . If however, full cooperation is restored immediately, then all members obtain v C given by [START_REF] Olson | The Rise and Decline of Nations[END_REF]. In either cases, all group members obtain v C in t + 1 onward.

Thus, weak renegotiation-proofness requires that v P A v C or p p, with

p n(1 + n) 1 + 2 n : (35)
Intuitively, the number of punishing members must not be too large, otherwise the e¤ective level of collective action in the punishment path is too low to prevent renegotiating back to the fully cooperative outcome.

One can also easily verify that p 2 (n=2; (n + 1)=2) for any > 1. Thus, the maximal number of punishing members satisfying the renegotiation-proofness requirement will be given by:

p = n=2 if n is even or p = (n 1)=2 if n is odd. ( 36
)
From Result 2, I have that P (n; p) j n 3 is decreasing in the number of punishing members. Thus, substituting (42) into (31), the lowest values of the discount factor satisfying the incentive compatibility constraint on the cooperative phase are given by

P (n) = [2 + (n + 1)] 2 [ n(n 2) + (n 1)] (n 1) [1 + (n 1)] [1 + 3n + 2 n(n + 1)]
if n = 3; 5; 7; :::

[2 + n] 2 [ n(n 2) + (n 1)] n 2 (3 + 2 n) [1 + (n 1)]
if n = 4; 6; 8; ::::

DP (n; p) j n p 3 is also decreasing in the number of punishing members provided that n p 3. It can be easily veri…ed that this inequality reduces to n 5 if n with p = (n 1)=2 if n is odd; while it reduces to n 6 if n is even with p = n=2, if n is even. Thus, under these conditions and substituting (42) into (37), the lowest values of the discount factor satisfying the incentive compatibility constraint on the punishment phase are given by

DP (n) = 2 [1 + n] 2 [ (n + 1)(n 3) + 2(n 1)]
(n 1) [2 + (n 1)] [1 + 3n + 2 n(n + 1)] if n = 5; 7; 9:::

2 [1 + n] 2 [ n(n 4) + 2(n 2)] n 2 (3 + 2 n) [2 + (n 2)] if n = 6; 8; 10: (38)
Thus, provided that n 5, Penance-p strategies with full cooperation within the challenger group is a WPRE if and only if Max

n P (n); DP (n) o :
Let us now consider that n 4. Again, with a group of two members, subgame-perfection only requires that each member has no incentive to deviate from the cooperative path, i.e. P (2; 1), where P (2; 1) is given by (30). The renegotiation-proofness requirement is also satis…ed since p = 1. Now, for a group of three members, the renegotiation-proofness requirement also implies that p = 1. And subgame-perfection requires that M ax (i) P (2; 1) when n = 2;

n P (
(ii) P (n) when n = f3; 4g;

(iii) P (5) when n = 5 and ~ or DP (5) when n = 5 and ~ , with ~ ' 1:13;

(iv) DP (n) when n 6:

As mentioned above, [START_REF] Froyn | A climate Agreement with Full Participation[END_REF] in a model where players have a binary choice between cooperating and defecting, and Asheim and Holstmark (2009) in a model with continuous choices -but without strategic interactions between players -…nd that full cooperation can be sustained a WPRE through penance strategies. I show that this result extends to a more general model with strategic interactions between players and in a context where full cooperation within a group is undertaken to counteract the action of a unitary opponent. Furthermore, contrary to the above mentioned analysis, the number of players determines which incentive compatibility constraint is more stringent than the other. Observe that the inequality DP (n)

P (n) reduces to v DP v P v D v C since DP (n) v DP v P = v C v P and P (n) v D v C = v C v P .
v DP v P represents the net bene…t of deviation for a contributing member on the punishment path, while v D v C represents the net bene…t of deviation for each member on the cooperative path. When group size is larger than 5, the best possible deviation from either the cooperative phase or the punishment phase is to cut his/her contribution to 0. In this case, as shown and explained in the previous subsection, the net bene…t of defection decreases with group size -that is with the number of contributing members. It follows that there are more incentives to deviate from the punishment path than from the fully cooperative outcome (and hence DP (n) P (n)). This result is reversed when the challenger group has few members (less than 5). In this case, the best possible deviation from the punishment path is to make a positive level of e¤ort since contributing members are not enough to make it pro…table full defection. In turns, it makes defection from the punishment path less attractive than defection from the equilibrium path.

I now characterize the impact of an increase in group size n, or in the relative valuation of the prize by group members given by , on the di¢ culty of sustaining within-group cooperation, as measured by the lowest discount factor supporting the optimal level of group e¤ort as a WRPE. For n 5, I have the following result.

Result 4: Suppose that n 5, then:

(i) P (2; 1) < P (4) < P (3) ;

(ii) P (5) > P (4) and DP (5) > P (4);

(iii) P (2; 1), and P (n) for n = f3; 4; 5g and DP (5) are decreasing in .

Thus, for a group of a small size, increasing the number of group members generally makes full cooperation more di¢ cult to sustain as a WRPE except when group size increases from 3 to 4 members. Also, whether a group of 5 members has more or less di¢ culty to sustain within-group cooperation as a WRPE compared to a group of 3 members is indeterminate and depends on the exact value of . Intuitively, a group of 3 members is that for which the punishment is relatively the least severe since, in that case, the number of punishing members in proportion of group size -that is p=n = 1=3 -is the smallest compared to a group of 2, 4 or 5 members. Yet, an increase in the valuation of the prize by group members compared to that of the incumbent -i.e. an increase in -unambiguously makes within-group cooperation less di¢ cult to sustain as a WRPE. The question now is whether these results remain valid for larger groups with an arbitrary number of members. For n 5, I have the following result.

Result 5: Suppose that n 5, then:

(i) DP (n + 1) < DP (n) if n is odd ; (ii) DP (n) is increasing in n independently of ; (iii) DP (n) is decreasing in independently of n:
Point (i) of this result echoes the previous one for a group of less than …ve members and where the incentive compatibility constraint along the cooperative path is more stringent than that of the punishment path.

For a group of more than 5 members, the binding constraint is that corresponding to the punishment path. But the intuition remains the same. If the challenger group has an odd number of members, then integrating one more member facilitates within-group cooperation because this allows increasing the number of punishing members also by 1, thus increasing the proportion of group members punishing a deviation.

Nevertheless, an increase in group size generally raises the threshold value of the minimum discount factor above which within-group cooperation on the e¢ cient level of group e¤ort can be sustained as a WRPE.

As just mentioned above, the net bene…t of defection from the punishment path, that is v DP v P , is decreasing in the number of contributing members in the same way as does the net bene…t of defection from cooperation with group size. However, and in contrast to the situation where group members use Nash Reversion Strategies, the punishment threat also becomes less e¤ective as group size increases due to the renegotiation proofness requirement. In fact, Result 5 shows that v C v P is decreasing even more with group size than v DP v P , thus increasing DP (n) and making full cooperation more di¢ cult to sustain as a WRPE as group size increases. In other words, the constraint of renegotiation-proofness brings us back to the general presumption that maintaining cooperation becomes more di¢ cult as the size of the collectivity increases (see, e.g., [START_REF] Hardin | Collective Action[END_REF][START_REF] Olson | The Rise and Decline of Nations[END_REF][START_REF] Sandler | Collective Action: Theory and Applications[END_REF][START_REF] Taylor | Community, Anarchy and Liberty[END_REF]). Yet, in contrast to [START_REF] Olson | The Logic of Collective Action[END_REF]'s conjecture full cooperation can still be maintained in arbitrarily large groups provided the discount factor of group members is su¢ ciently high.

As for the impact of the heterogeneity in the valuation of the prize on the di¢ culty of sustaining full cooperation as a WRPE, I obtain the same result than that obtained when group members use simple Nash Reversion Strategies without any possibility of renegotiation. The cooperative level of individual e¤ortwhether there are n or only (n p) contributing members -increases with and, hence, withdrawing his/her individual contribution becomes more and more tempting even though a defection also decreases the probability of winning the public prize. However, the net bene…t of maintaining full cooperation relative to the punishment phase also increases with , thus making the punishment threat more e¤ective. And, as

shown by Result 5, the renegotiation-proofness requirement with a limiting number of punishing members does not alter the increased e¤ectiveness of the punishment threat for making within-group cooperation easier to sustain, as the relative valuation of the prize by group members increases.

Concluding remarks

In a repeated contest game, I analyze the ability of group members to cooperate in order to challenge the position of an incumbent for the award of a public prize. The results show that group cooperation can be more easily sustained through simple Nash Reversion Strategies (NRS) as the relative valuation of the prize by group members, or group size, increases. The intuition is that bad outcomes under a static setting make for e¤ective punishments in a repeated game setting. Next, I consider that group

CEE-M Working Papers

  Let x i 2 < + be the rent-seeking expenditure/e¤ort expended at unit cost by group member i, and X =

	n P i=1	x i 2 < + be the sum of rent-
	seeking e¤orts of group members. The rent-seeking expenditure/e¤ort of the external entity is denoted
	Y 2 < + . Following much of the contest literature, I assume that the probability of winning the prize p : < 2 + ! [0; 1] for group members is given by a contest success function, which has the Logit form, i.e. 8 <
	p (X; Y ) =

  3); DP (3) j n p=2 o, where P (3) and DP (3) j n p=2 are given by (42) and (36), respectively. For n = 4, the renegotiation proofness requirement is such that p 2.

	Since P (n; p) j n 3 is decreasing in p, P (4) most relaxes the incentive compatibility constraint on the cooperative phase. In this case, Penance-p strategies with full cooperation within the challenger group is a WPRE if and only if Max n P (4); DP (4) j n p=2 ; DP (4; 1) j n p 3 o , where P (4) j ne 4 , DP (4) j n p=2 , and DP (4; 1) j n p 3 are given by (41), (36) and (37), respectively. I have the following Result.

Result 3: The strategy pro…le ~ ( p) with full cooperation within the challenger group is a WPRE if and only if:

For a review of the literature on collective action theory à la[START_REF] Olson | The Logic of Collective Action[END_REF], see[START_REF] Pecorino | Olson's Logic of Collective Action at Fifty[END_REF].

Note that the marginal return to an additional unit of individual rent-seeking e¤ort (as well as to an additional unit of group rent-seeking e¤ort) is decreasing in e¤ort. Hence, each player's problem is strictly concave and the …rst-order conditions are both necessary and su¢ cient for characterizing the best-response functions of the players.

All the Proofs of the results are given in the Appendix.

Both v C v N and v D v C depend on and on V G = V I . We consider that V I is …xed and thus the increase in is only driven by the increase in V G :

 ----------? I thank participants of the Annual Conference of the Association for Public Economic Theory in Rio de Janeiro (APET), the conference in "Contests: Theory and Evidence" at the University of East Anglia (Norwich), and seminar participants in Montpellier and Toulouse School of Economics. This research received funding from the French Agence Nationale de la Recherche (ANR) as part of the research project GREENGO -New Tools for Environmental Governance: The Role of NGO -Contract ANR-15-CE05-0008. 1

members can renegotiate, without costs, returning back to the cooperative outcome in case of a defection. I show that group cooperation can still be maintained independently of the size of the group as a Weakly Renegotiation-Proof Equilibrium (WRPE). Furthermore, the renegotiation-proofness requirement does not change the result that a greater heterogeneity in the valuation of the prize makes it easier within-group cooperation. However, the comparative static result with respect to group size is reversed. In particular, the critical discount factor above which full cooperation can be maintained converges to 0 under NRS while, under the constraint of renegotiation-proofness, it converges to 1 as the size of the collectivity goes to in…nity. Therefore, it would be necessary to investigate other strategies of cooperation against against an opponent that are in between the impossibility of renegotiation and perfect or frictionless renegotiation.

And the (im)perfectness of renegotiation should depend, among other features, on group size. Thus, the next step should be to introduce a cost of renegotiation as a function of group size in this simple framework, before considering a generalization of the model.

Appendix

Proof of Result 1

I …rst evaluate the impact of group size n and of the relative valuation of the prize by group members on the threshold value of the discount factor N (n). Regarding the impact of group size, I have

which is strictly negative for any n 3.

The impact of the ratio of valuations on N (n) is given by

)

which is also strictly negative for any n 3. I now evaluate the impact of group size n and of the relative valuation of the prize by group members on the net bene…t to maintaining within-group cooperation v C v N and on the net bene…t of deviating from the cooperative outcome v D v C . Calculating the derivative of v C v N with respect to n, I obtain

which is clearly positive for any n 3.

The derivative of v D v C with respect to n is given by

One can see that this derivative is negative for any n 4 and 1. Now, I consider that V I remains unchanged and I calculate the derivative of v C v N with respect to = V G =V I .

which is clearly positive.

The derivative of v D v C with respect to is given by @ v D v C @ = n(2n 2 6n + 3) 2 + (4n 2 8n + 1) + 2(n 1)

which is also clearly positive for any n 3.

Calculating the derivative of N (2), with respect to , I have

I now compare N (2) and N (3). Recall that in contrast to n 3, the group member who defects from the cooperative outcome chooses a positive level of e¤ort when n = 2. Using (25), [START_REF] Axelrod | The Emergence of Cooperation among Egoists[END_REF] is given by

Calculating the di¤erence between N (2) and N (3), I have

which can be positive or negative depending on .

Proof of Result 2

Observe that P v D v C = v C v P depends on p only thought v P , which is itself decreasing in p for any n 2 and p 1. Indeed, the derivative of v P with respect to p is given by

which is negative. It follows that P (n; p) j n 3 is decreasing in the number of punishing members p. I now turn to the impact of the number of punishing members p on the threshold value of the discount factor DP (n; p) j n p 3 . I have

Clearly, 0 (n; p) and 1 (n; p) are strictly positive for any n p 2. Calculating the derivative of the term [:] in 2 (n; p) with respect to p, I have @ 2 (n; p)=@p= 4 2(n p); which is negative for any n p 2. Thus, the term [:] in 2 (n; p) reaches a maximum in p = 1, in which case, I have 2 (n; 1) = n 2 5n + 7, which is positive for any n 3. As a result, 2 (n; p) is positive for any n 3. In turn, @ DP (n; p) j n p 3 =@p is negative and thus DP (n; p) j n p 3 is a decreasing function of the number p of punishing members for any n p 3 (implying n 4).

Proof of Result 3

Let …rst consider a group of three members, i.e. n = 3 so that p = 1. Using (42), P (3) is given by

while using (36) DP (3) j n p=2 is given by

Calculating the di¤erence, we have

which is clearly positive. Now consider a group of four members. For n = 4, the renegotiation proofness requirement is such that p 2. Since P (n; p) j n 3 is decreasing in p, the optimal number of punishing members that most relaxes the incentive compatibility constraint on the cooperative phase is given by P (4). Using (42), we have

I now determine whether p = 1 or p = 2 most relaxes the incentive compatibility constraint on the punishment path. Using (36) and (37), we have

Also, the derivative of (n) with respect to n is given by

where

1)(3n + 1):

One can observe that i (n) 0 for i = f1; 2; 3; 4; 5g so that the sign of this derivative is always positive. Hence, (n) reaches a minimum in n = 5. In this case, one obtain

(5) = 15 4 + 4

Also, the derivative of ( p) with respect to n is given by

where

One can observe that i (n) 0 for i = f1; 2; 3; 4; 5g so that the sign of this derivative is always positive. Hence, (n) reaches a minimum in n = 6. In this case, I have

which is strictly positive for any 1. As a result, I have that DP (n) > P (n) for any n odd and larger than 6 independently of .

Proof of Result 4

(i) Using (42), we have

and thus P (3) > P (4).

Using (30) and (42), we also have P (4) P (2; 1) = (1 + 2 ) 5:94 3 + 5:20 2 0:43 0:69

and thus P (3) > P (4)> P (2; 1).

We also have > 0 (A31) and thus DP (5) > P (4).

I now analyze the impact of increasing on these thresholds values of the discount parameter. I obtain, @ P (2; 1)

and @ P (3) @ = 2 27 2 + 24 + 5 2 (5 + 12 ) 2 < 0 (A33) and @ P (4) @ = 8 2 + 6 + 1 4 2 (1 + 3 ) 2 < 0 (A34) and @ P (5) @ = 15 2 + 8 + 1 4 2 (1 + 4 ) 2 < 0 (A35)

Proof of Result 5

I now consider that n 5 and odd values of n: I have

and thus DP (n + 1) < DP (n) when n is odd.

I now analyze the impact of increasing n on DP (n). When n is odd and greater than 5, we have

where 0 (n) = 6(n 1) 2 ;

The term in the [:] in the numerator of (A37) is positive if

We have that i (n) 0 for i = 0; 1; 3 while 2 (n) is negative for n 9 and positive for n 11. Since i (n) 0 for i = 0; 1; 3 and 1, a su¢ cient condition for the above inequality to be satis…ed is then that 3

n) 0; which is indeed the case since it is equal to (n + 1) n(n 2 + 9n + 11) 5 > 0.

When n is even and greater than 6, we have

where 0 (n) = 6(n 4);

We have that i (n) 0 for i = 0; 3. 1 (n) is negative for n 8 and positive for n 10 while 2 (n) is negative for n 22 and positive for n 24. If n 8, a su¢ cient condition for the above inequality to be satis…ed is 3 (n) 3 0 (n), which is indeed the case since 3 (n) 0 (n) and 1. If n 10, a su¢ cient condition for the inequality to be satis…ed is

n) 0, which is indeed the case since it is equal to n(3n 2 + 19n + 18) > 0.

I now analyze the impact of increasing on DP (n). and thus @ DP (n) =@ is negative.

When n is even, I have and thus @ DP (n) =@ is also negative when n is even.