Katrin Erdlenbruch

Mabel Tidball
email: tidball@supagro.inra.fr.

Daan Van Soest

Renewable resource management, user heterogeneity, and the scope for cooperation *

Keywords: JEL classification: C73, Q20 shared resource use, dynamic game, heterogeneous agents, contracts

Cooperation can increase the efficiency of commonly-owned renewable resource use. However, received knowledge is that, absent side payments, cooperative solutions are more difficult to achieve the less homogenous the agents involved. We revisit this claim by analyzing how differences in the opportunity costs of resource harvesting affect the scope for Pareto-improving contracts, where contracting is with respect to the type of technology used. We find that the scope for cooperation is largest for intermediate levels of heterogeneity.

Introduction

Absent cooperation, rates of extraction are suboptimally high if a renewable natural resource is shared by two users, or more. Achieving cooperation is typically thought to be more difficult if resource users are heterogeneous, for example with respect to the efficiency with which they can exploit the resource (cf. Cherry et al. 2005, Gardner et One reason why heterogeneity renders cooperation less likely is that it is not immediately clear what a fair allocation of the 'gains from cooperation' entails. Heterogeneity implies that the direct gains from cooperation are asymmetric and this complicates achieving Pareto improving solutions, especially if side payments are not feasible. As noted by Johnson and Libecap (1982), heterogeneity may therefore lead to (possibly prohibitively) high contracting costs. And side payments are often infeasible in practice because of practical or political constraints (Munro 1979). A second reason regards information and monitoring costs. The more dissimilar agents are with respect to skills or preferences, the more difficult it is for each individual user to infer whether their peers effectively adhere to the agreement, or not.

We contribute to the literature on the relationship between user heterogeneity and the scope for cooperation by focusing on the role of the choice of harvesting technologies herein. Technology choice is potentially able to alleviate (or even solve) both problems cited above -the monitoring problem as well as the problem of those who gain from cooperation not being able to directly compensate those who lose. Clearly, while extraction is often difficult to monitor, the type of harvesting technology used is relatively easy to observe. And because each agent can infer the other agents' privately optimal extraction given their harvesting technologies, contracting with respect to the technology type solves the monitoring problem regarding extraction itself. Second, as different technologies generate different returns to harvesting effort, technology choice can be an -albeit imperfectsubstitute for side payments.

To analyze to what extent technology choice can foster cooperation and how it interacts with the degree of heterogeneity among the various resource users, we construct a twocountry dynamic game (in spirit of Benhabib and Radner 1988, Clark 1980, Levhari and Mirman 1980). In each country there are two industries, agriculture and a resource sector. Harvesting productivity in the resource sector depends not only on the type of technology used, but also on the size of the resource stock: the larger it is, the lower the costs of harvesting an additional unit. The two countries share access to the resource, and differ only with respect to agricultural productivity -that is with respect to the instantaneous opportunity costs of resource harvesting. [START_REF] Benhabib | Joint exploitation of a productive asset: a gametheoretic approach[END_REF] Absent cooperation, the dynamic game results in overexploitation of the resource, with both countries selecting the best available technology and allocating too much labor to resource harvesting. The two countries involved can potentially increase aggregate payoffs by selecting a less-than-1 Obviously, this is just one possible source of heterogeneity. For the consequences of other sources such as differences in discount rates, see for example Munro (1979), Houba et al. (2000) and Crabbé and Van Long (1993).

perfect technology, thus accepting a decrease in technical harvesting productivity in order to increase productivity associated with having larger resource stocks.

We analyze the relationship between user heterogeneity and the scope for signing mutually beneficial technology contracts by determining the range of Pareto-improving technology sets as a function of the difference in agricultural productivity. [START_REF] Cherry | The impact of endowment heterogeneity and origin on Public Good contributions: Evidence from the lab[END_REF] We find that this range is largest at intermediate levels of heterogeneity, and hence the relationship between heterogeneity and the scope for cooperation is humpshaped. The reason is that for low levels of heterogeneity there is little opportunity for specialization whereas for high levels of heterogeneity, there is less need for cooperation. [START_REF] Clark | Restricted access to common-property fishery resources: A gametheoretic analysis[END_REF] The setup of the paper is as follows. We describe the model in section 2. In section 3, we present the solutions of the non-cooperative game and of the socially optimal program, analyze the design of Pareto-improving contracts and explore the role of heterogeneity therein. Finally, section 4 concludes. All proofs are available upon request.

The model

We consider two countries, referred to with subscripts {i, j} = {1, 2}. These countries have shared access to the same renewable resource (with size R(t) at time t), and are also able to undertake agriculture. Prices for the agricultural and resource commodities produced (denoted by P a and P r ) are determined in international markets, and are constant over time. Each country aims to maximize the net present value of the profit flow of agriculture and resource harvesting. It does so (i) by choosing a harvesting technology β i from a range of available technologies (0, β], and (ii) by deciding how much labor to allocate to resource harvesting l ri (t), taking into account the other country's decisions. Regarding the costs of adopting a harvesting technology, we impose the simplifying assumptions that 2 By focusing on the range of Pareto-improving contracts, we do not address the question which contract will actually be selected. Hence, we ignore bargaining problems (see for example Houba et al. 2000). [START_REF] Clark | Restricted access to common-property fishery resources: A gametheoretic analysis[END_REF] The only paper we are aware of that also finds that heterogeneity does not necessarily complicate obtaining efficient outcomes is that of Gaspart and Seki (2003). They consider the case where agents differ with respect to production skills, and where each agent's social status is an increasing function of his/her relative output performance. Best response implies that social status considerations tend to induce agents to mitigate differences in output performance by adjusting effort. Therefore, non-cooperative behavior may actually result in efficient production levels -but only if skill heterogeneity is not too large. Gaspart and Seki thus find that there is less need for cooperation if agents are sensitive to relative status. In contrast, in our paper the non-cooperative solution is only efficient in the trivial case where just one agent harvests (and the other prefers not to).

all technologies are equally expensive (I β = I > 0 for all β ∈ (0, β]) and that they can be discarded at zero cost. Hence, country i's maximization problem is:

Π i = max l ri ,β i ⎡ ⎣ -Iφ i + ∞ 0 (P a q ai (t) + P r q ri (t)) e -δt dt ⎤ ⎦ (1)
subject to

q ai (t) = α i l ai (t), q ri (t) = β i φ i Rl ri (t), l = l ai (t) + l ri (t) ( 2 )
Ṙ(t) = ρR(t)(K -R(t)) -q ri (t) -q rj (t), R(0) = R 0 >> 0. (3)
In objective function (1), φ i is an indicator function which takes on value 1 if country i decides to adopt a harvesting technology from the range (0, β], and 0 otherwise. Furthermore, q ai (t) and q ri (t) are the quantities produced in period t by country i, and δ is the discount rate. In (2), agricultural output is a linear function of labor input l ai (t) with constant marginal productivity α i . The harvest function q ri (t) is the standard Schaefer production function. Its arguments are labor input l ri (t) and the resource stock R(t). [START_REF] Crabbé | Entry deterrence and overexploitation of the fishery[END_REF] Furthermore, β i is the so-called catchability coefficient of the harvesting function, and reflects the share of the fish stock (q/R) that is harvested per unit of effort employed per period l r . [START_REF] Gardner | The power and limitations of proportional cutbacks in common-pool resources[END_REF] If φ i = 1, β i > 0 the country is -in principle -able to generate harvesting revenues. [START_REF] Gaspart | Cooperation, status seeking and competitive behaviour: Theory and evidence[END_REF] Therefore, decisions regarding β i embody the choice of the harvesting technology used, and given that it reflects the share harvested, this parameter must be bounded from above (i.e, β i ∈ (0, β]). Note that the Schaefer specification implies that labor's physical marginal product in harvesting is determined by two factors: the type of technology chosen (β), and the size of the resource stock (R). The last equation in ( 2) is the labor time constraint, with l the labor endowment per period and l ai (t) the amount of labor allocated to agriculture.

The dynamics of the resource stock are given by (3), with ρ the intrinsic growth rate and K the carrying capacity. Because of their joint impact on the size of the resource stock, the two countries are tied in a game, where the initial resource stock equals R 0 .

The two countries only differ with respect to agricultural productivity, α i , which is an inherent characteristic of the country's location (climate, soils, etc.).

The last three assumptions are the following. First, country 1 is assumed to have a natural advantage with respect to agricultural production relative to country 2;

α 1 > α 2 .
Second, each country is able to drive the resource stock to extinction if it adopts the most efficient harvesting technology β, and a sufficient condition for this to hold is that βl > ρK. Third, the costs of purchasing a harvesting technology (I) are sufficiently low and the initial resource stock (R 0 ) is sufficiently large such that harvesting is a profitable enterprise for both countries involved. Hence, φ i = 1 for i = {1, 2}.

Pareto-improving contracts

Our model is very similar to that of Clark (1980), from which we can derive the noncooperative solution.

Proposition 1 The Nash equilibrium feedback solution is that both countries adopt the best harvesting technology ( β), and the allocation of harvesting effort is as follows:

l N * r1 (t) = l if R(t) > R1 0 if R(t) ≤ R1 , ( 4
)
l N * r2 (t) = ⎧ ⎨ ⎩ l if R(t) > min(R * 2 , R1 ) (ρ/ β)(K -R(t)) ≡ lN * r2 if R(t) = min(R * 2 , R1 ) 0 if R(t) < min(R * 2 , R1 ) . ( 5
)
Here, R * i is defined as country i's privately optimal steady state resource stock,

R * i = K 4 - δ 4ρ + α i P a 4 βP r + K 4 - δ 4ρ + α i P a 4 βP r 2 + α i P a δ 2 βP r ρ , ( 6
)
and Ri is the resource stock at which country i is indifferent between allocating another unit of labor to resource harvesting or to agriculture -given that it is tied in a non-cooperative game with country j:

Ri = α i P a βP r . ( 7
)
In other words, Ri is the level of the resource stock where country i's marginal benefits of labor allocated to agriculture (α i P a ) are equal to the instantaneous (or gross-) marginal benefits of resource harvesting (P r βR, that is ignoring the user cost of harvesting).

To investigate whether contracts can be designed that are Pareto-improving compared to the non-cooperative outcome, we first identify the solution which maximizes joint profits.

Proposition 2 The program that maximizes aggregate payoffs consists of both countries

adopting the best technology β, and allocating labor as follows:

l S * r1 (t) = l if R(t) > R 0 if R(t) ≤ R , ( 8
)
l S * r2 (t) = ⎧ ⎨ ⎩ l if R(t) > R * 2 ρ/ β K -R * 2 ≡ l S * r2 if R(t) = R * 2 0 if R(t) < R * 2 , ( 9
)
where R ( R > R1 ) is defined as the stock level at which country 1's marginal benefits of agriculture equal the net marginal benefits of resource harvesting. [START_REF] Hämäläinen | Bargaining on whales: A differential game model with Pareto optimal equilibria[END_REF] Comparing Propositions 1 and 2, both the Nash equilibrium feedback solution and the socially optimal harvesting program imply each country selecting the best technology, β. Also, the allocation of labor over agriculture and harvesting is identical with and without cooperation in the early periods, but not in the later periods. [START_REF] Hannesson | Fishing as a supergame[END_REF] Contracting with respect to the allocation of labor is typically difficult because of imperfect observability, and therefore we focus on the case in which countries can contract with respect to the harvesting technologies used. From ( 8) and ( 9) we know the following:

Proposition 3 The contract that maximizes aggregate payoffs is given by:

β 1 = β if R(t) > R 0 if R(t) R , ( 10
)
β 2 = β for all 0 R(t) . ( 11
)
In words, country 2 should always harvest using technology β, whereas country 1 should discard it when the resource stock is depleted down to R. This contract is obviously both incentive-compatible and payoff-enhancing for country 2, but it is not for country 1. Indeed, if R1 R * 2 , country 2's best response function as described by Clark (see Proposition 1 above) coincides with the way in which country 2 should employ its labor in order to maximize aggregate payoffs. And if R1 < R * 2 , country 2 can credibly commit to continue allocating l to resource harvesting in case country 1 reneges on the contract. [START_REF] Houba | Can negotiations prevent fish wars?[END_REF] But country 1 is not willing to sign this contract. To maximize aggregate payoffs, it should cease harvesting at a level of the resource stock ( R) where the instantaneous (or gross-) marginal benefits of harvesting exceed its marginal benefits of agriculture (as R > R1 ).

Hence, absent side payments, country 1's discounted payoff falls if it abides with the contract that maximizes joint payoffs.

Let us therefore consider whether country 2 can induce country 1 to sign a Paretoimproving contract. Given that both countries select β in the Nash equilibrium, we can state that aggregate payoffs increase if β 1 = β 2 = βε (ε > 0), substituting natural resource productivity (larger R) for technical productivity (higher β). But contracts specifying the adoption of equally inefficient technologies by the two countries involved are not necessarily Pareto-improving. For country 1, the use of an inefficient technology implies lower harvesting productivity which is not likely to be compensated by the productivity increase associated with higher resource stocks at every point in time along the depletion path. Country 2, on the other hand, benefits from larger resource stocks not only during the transition phase, but also in the long run. Therefore, perhaps surprisingly, if side payments are not feasible, the only way in which the low opportunity cost country can induce the high opportunity cost country to sign a Pareto-improving contract is by selecting an even more inefficient technology than does the high opportunity cost country.

Let us use ( β1 , β2 ) to denote sets of technologies specified in Pareto-improving contracts. Absent side payments, country 1 is only willing to sign the contract if β2 < β1 .

By limiting its own harvesting potential, country 2 allows country 1 to benefit from lower search costs (due to larger resource stocks) during the transition phase. But country 2 is only willing to sign the contract if β1 < β; if not, aggregate payoff does not increase and country 2 would necessarily lose. Hence, Pareto-improving contracts are characterized by combinations of technologies such that β2 < β1 < β.

Obviously, private payoffs of country 1 (2) increases if β2 ( β1 ) falls. Hence we can determine the scope for Pareto-improving contracts by determining the combination of β 1 and β 2 that maximizes country i's payoff subject to the constraint that the other country is indifferent between signing that contract, or not. [START_REF] Johnson | Contracting problems and regulation: The case of the fishery[END_REF] Let us use ( βi 1 , βi 2 ), i = 1, 2, to denote the sets of technologies chosen by an own-payoff maximizing country i when it has the right to propose a contract. That means that ( βi 1 , βi 2 ) is the solution of max

β 1 ,β 2 Π i (β 1 , β 2 ) s.t. ( 12
) Π j (β 1 , β 2 ) = Π Nash j . (13)
In figure 1, we depict the combinations of βi 1 and βi 2 (i = 1, 2) as a function of heterogeneity (α 1α 2 ). We find that independent of which country can draft the proposed contract (i = 1 or 2), the Pareto-improving technology level of the high opportunity cost country (country 1), βi 1 , is close to the maximum technology level for both low and high levels of heterogeneity, and is lowest for intermediate levels of heterogeneity. The intuition behind these results is as follows.

For low levels of heterogeneity, the two countries are too similar for country 2 to specify a contract with β 2 much smaller than β 1 ; its own payoff would fall. Because β2 cannot be much lower than β1 , country 2 is not able to provide much compensation along the depletion path to induce country 1 to select a sufficiently inefficient technology. Hence, 10 Country 1's commitment to using β1 < β is not time-consistent unless the discounted flow of benefits of switching from β1 to β fall short of the investment costs (I) associated with this switch -taking into account that country 2 can credibly commit to its trigger strategy. We assume that this is the case as designing a time-consistent contract complicates the analysis without adding additional insights. For a proper analysis of the importance of switching costs, see Nostbakken (2006). An alternative solution to the potential time-inconsistency problem would be to assume that countries can adopt a harvesting technology just once. β1 cannot be too low. That means that, whichever country has the right to draw up the contract (country 1 or 2), the technologies specified in the contract are close to β. Therefore, the net gain associated with signing the contracts are small.

For very high levels of heterogeneity, the possibilities for gains via cooperation are also limited, as the costs of non-cooperation are smaller. The higher α 1 , the shorter the time period in which country 1 engages in resource harvesting (as R1 is then larger). Obviously, if α 1 is such that country 1 does not harvest at all (not even in period 1, and a sufficient condition for this is that α 1 > βP r R 0 /P a ; see (7)), it does not pay for country 2 to choose an inefficient technology (i.e., β2 = β). For slightly lower levels of α 1 , country 2 can gain from country 1 choosing a less than efficient technology, but only a little. Therefore, it is willing to accept only a very small difference in technological efficiency levels, and hence β 1 should be close to the maximum efficiency level. Again, that means that, whichever country has the right to draw up the contract (country 1 or 2), it can not gain much from that contract. [START_REF] Laukkanen | Cooperative and non-cooperative harvesting in a stochastic sequential fishery[END_REF] 11 Note that for α 1 -α 2 < 30, country 2 assigns itself an even more inferior technology than what would be assigned by country 1; that is, β2

2 < β1 2 . This may seem surprising, but it can be explained by noting that β2

1 << β1 1 . In words, if country 2 has the right to make a proposal, it assigns country 1 a technology which is very much inferior to what that country would assign itself -if it had the right to draw up the contract. The only way country 2 can get away with this is by assigning itself an even worse technology,

We can now establish the scope for Pareto-improving contracts by plotting the difference between β1 j and β2 j , j = 1, 2; see figure 2. Picking any technology level β 1 ∈ ( β2

Conclusions

Absent property rights, rates of exploitation are suboptimally high if a resource is shared by several countries. Cooperation can enhance aggregate payoffs but previous research suggests that, unless side payments are feasible, cooperation may be difficult to achieve when the countries involved differ with respect to important characteristics such as the (opportunity) costs of resource harvesting. The reason is that socially optimal contracts are generally not Pareto-improving if agents are heterogeneous.

In this paper, we revisit the claim that, absent side payments, achieving cooperation is more difficult the more heterogeneous the resource users are. If side payments are not feasible, the only means countries have to achieve cooperation is by establishing so that country 1 is at least as well off with the contract as without.

Pareto-improving contracts with respect to easily observable decision variables, such as technology choice. In our two-country linear dynamic game, efficiency of resource use can be increased by contracting on a relatively inefficient technology. Harvesting productivity is a function of technological efficiency as well as of the size of the resource stock, and hence the adoption of relatively inefficient harvesting technologies can improve aggregate payoffs because they result in larger resource stocks. Each country can only induce the other country to adopt a relatively inefficient technology (that increases aggregate payoffs) if it adopts a relatively inefficient technology itself.

The scope for Pareto-improving contracts is limited, though, for both low and high levels of heterogeneity. When countries are fairly similar, the low opportunity cost country cannot afford to have a much less efficient technology than its counterpart, and hence is not able to compensate the high opportunity cost country for the adoption of a sufficiently inefficient technology. Alternatively, when countries are very dissimilar to start with, there is less necessity for the low opportunity cost country to select a technology that is much more inferior than that of its counterpart. The more dissimilar the countries involved, the smaller is the time period where the high opportunity cost country is involved in resource harvesting, and hence the smaller the scope for Pareto-improving contracts.

In summary, we find that the range of Pareto-improving technology sets is a humpshaped function of the difference in agricultural productivity. Hence we find that the scope for cooperation is largest for intermediate levels of heterogeneity.

  al. 2000, Hannesson 1997, Johnson and Libecap 1982, Ledyard 1995, Munro 1979, Varughese and Ostrom 2001).

Figure 1 :

 1 Figure 1: Combinations of technologies ( βi 1 , βi 2 ) that maximize country i's payoff such that the other country's payoff does not fall, as a function of heterogeneity (α 1α 2 ). Parameter values: α 2 = 5, β = 20, δ = 0.05, K = 100, ρ = 0.03, R 0 = 100, P a = 100, P r = 10, and l = 1.

Figure 2 :

 2 Figure 2: Ranges of technologies for countries 1 and 2 that are Pareto-improving. Parameter values: see figure 1.

  ), we can find a β 2 or β 1 that increases private payoffs of both countries involved, and the range is largest for intermediate levels of heterogeneity.

	1 , β1 1 )
	or β 2 ∈ ( β1 2 , β2 2

Note that the Schaefer harvesting function is special as it assumes increasing returns to scale. If both inputs (l r and R) are doubled, the amount of resource harvested is quadrupled. Indeed, this is a key characteristic of the Schaefer specification, and well-noted in the fisheries literature. The main reason why it is often used is its mathematical convenience, but it can be defended on the basis that it captures search costs. Take the example of a fishery where fish tend to spread evenly over the available waters. Then, the larger the size of the stock, the higher the fish density and the more fish will be caught per sweep of the net.

The parameter can be interpreted in several ways. For example, in the case of a fishery it may reflect the type of fishing gear used (fishing rods, or nets), the mesh size of the nets used, the number of nets used simultaneously, the size of the boat used, the average speed of the boat used, etc. Some of these inputs can be viewed as capital that can be accumulated over time, others not. Specifically including capital would result in a model with two state variables, which are very difficult to solve. We follow the bulk of the renewable resource literature and think of β as a harvesting technology (a boat, or a net) that can be chosen freely and which comes in different sizes with different levels of productivity.

Clearly, the assumption that all technologies are equally expensive (I β = I ∀ β ∈ (0, β]) is less offending in case β reflects mesh size (i.e., nets with finer mesh size are not likely to be much more expensive than nets with larger mesh size) than if it is thought to reflect boat size or speed; cf. footnote 5. Note, however, that our results are even reinforced if we were to assume investment costs to be an increasing function of technological productivity.

Given (i) φ 1 = φ 2 = 1 and (ii) β 1 = β 2 = β, the current-value Hamiltonian of the intertemporal harvesting decision reads as H = P a α 1 l -l r1 + P r βRl r1 + P a α 2 l -l r2 + P r βRl r2 + λ ρR(K -R) -βR(l r1 + l r2 ) , where λ is the shadow price of the resource stock. Then, R is implicitly defined by ∂H/∂l r1 = 0, that is by P a α 1 = (P r -λ) βR.

[START_REF] Hannesson | Fishing as a supergame[END_REF] Note that we impose the assumption about the initial abundance of the resource (given I) to rule out situations where there is no conflict of interests. One example of such a case would be the one in which both R 0 and R * 2 are strictly below R1 . Here, country 1 never undertakes any harvesting and country 2 either harvests (ifR * 2 ≤ R 0 < R1 ) or does not harvest (if R 0 < R * 2 < R1), but the resource always ends up in the same steady state, R * 2 . This would actually be socially optimal; private and social objectives coincide. To rule out such cases, we assume that the resource is sufficiently large initially. A sufficient condition is that R 0 > R, so that R(t) < R 0 for all t > 0.

Hence if country 2 uses a trigger strategy it is incentive-compatible and credible; cf.[START_REF] Hämäläinen | Bargaining on whales: A differential game model with Pareto optimal equilibria[END_REF] or[START_REF] Laukkanen | Cooperative and non-cooperative harvesting in a stochastic sequential fishery[END_REF].

The authors gratefully acknowledge the European Union's financial support via the Fifth European Framework's BIOECON project, and Daan van Soest also thanks the Netherlands Organisation for Scientific Research for their support as part of the program on 'Evolution and Behavior'.


Huet S Cemagref

Advances in Complex Systems c World Scientific

. Individuals are characterised by twodimensional continuous attitudes, each associated with an uncertainty u, supposed constant in this first study. Individuals interact by random pairs. If their attitudes are closer than u on both dimensions, or further than u on both dimensions, or closer than u on one dimension and not further than u + δu on the other dimension, then the rules of the BC model apply. But if their attitudes are closer than u on one dimension and further than u + δu on the other dimension, then the individuals are in a dissonant state. They tend to solve it by shifting away their close attitudes. The model shows metastable clusters, which maintain themselves through opposite influences of competitor clusters. Our analysis and first experiments support the hypothesis that, for a large range of uncertainty values, the number of clusters grows linearly with the inverse of the uncertainty, whereas this growth is quadratic in the BC model.

Introduction

Many behaviours, especially in conditions of a high involvement, can be understood as originating from underlying attitudes. For instance, one may vote for an extreme nationalist party because of a negative attitude towards immigrants, or consume organic food because of a positive attitude towards environmentally friendly agriculture. Hence attitudes motivate behaviour and exert selective effects at various stages of information processing [START_REF] Eagly | The Psychology of Attitudes[END_REF]. Consequently, to understand behavioural change it is essential to understand the underlying attitudinal dynamics that give rise to such a change. Attitude is here understood in its psychological meaning as a tendency to evaluate a particular entity with some degree of favour or disfavour. The dynamics of attitudes are closely related to social influence, which includes individual influence on feelings, beliefs and behaviours of others [START_REF] Mason | Situating Social Influence Processes: Dynamic, Multidimensional Flows of Influence Within Social Networks[END_REF]. These dynamics are studied, by experiments in laboratory on individuals and small groups, and are the subject of a variety of theories and assumptions. The most common assumption is a tendency of attitudes to get closer to already similar ones (attraction). A less usual assumption is a tendency to reject the other's attitude if it is psychologically uncomfortable (rejection).

Whereas an abundance of studies have been published in social psychology on the processes leading towards attitude change, relative little attention has been devoted to the interactions between multiple attitudes in social interactions. Yet, the issue of interactions between attitudes in a social interaction context seems to be highly relevant. People often discuss different (unrelated) issues, and shifts on one attitude dimension may have an impact on other dimensions. For example, if a friend, who is having attitudes similar to yours on different issues, is speaking favourably about organic food, about which you have a negative attitude, the resulting dissonance may be resolved by either developing a more positive attitude on organic food as well, or by shifting away from the attitude of your friend on other issues. In contrast, if a person you disagree with on many issues also advocates for organic food, your attitudes are not likely to change as no dissonance is experienced.

This paper proposes a simple model, implementing individuals with both these opposite tendencies (attraction and rejection in some conditions), and studies through agent based computer simulations how a population initially uniformly distributed in the attitude space evolves towards different global patterns. Our main result is that we observe fewer clusters than in the case of dynamics only based on attraction for a large range of uncertainty values. Before going through this result in more detail, we briefly present related research in social simulation and social psychology.

To begin, we consider the assumption of homophily. It assumes that people, especially if they are uncertain about their capacity and knowledge to evaluate a particular object, are more likely to adopt opinions and attitudes of similar others. For example, [START_REF] Festinger | A Theory of Cognitive Dissonance[END_REF] shows that people like to have opinions similar to the ones of people they interact with. Similarity between receiver and source has a strong impact on the influence level of Word of Mouth [START_REF] Moschis | Social Comparison and Informal Group Influence[END_REF]. Additionally, [START_REF] Price | The role of interpersonal sources in external search: an informational perspective[END_REF] suggest that homophily facilitates the flow of information between people because of perceived ease of communication. Secondly, besides a perspective on what drives people's attitudes towards each other, some experiments and theories focus on the forces that may drive people's attitudes apart. At the individual level, the reactance theory [START_REF] Brehm | A theory of psychological reactance[END_REF], the balance theory [START_REF] Heider | Attitudes and Cognitive Organization[END_REF], the motivation to protect oneself [START_REF] Rogers | Protection motivation theory[END_REF], and the social judge-ment theory [START_REF] Sherif | Social Judgement[END_REF] indicate that a persuasive effort can induce a rejection reaction: the behaviour, and/or the attitude changes in the direction opposite to the persuasion effort. In groups, the social identity theory [START_REF] Tajfel | Differentiation between social groups: Studies in the social psychology of intergroup relations[END_REF], the self-categorization theory [START_REF] Turner | Rediscovering the social groups: A self-categorization theory (Cambridge[END_REF] and the optimal distinction theory [START_REF] Brewer | The Social Self: On Being the Same and Different at the Same Time[END_REF] consider a capacity to differentiate from the individuals who are members of the same group by rejecting their opinions. This rejection is usually called the 'boomerang effect'. The conditions of its occurrence vary from one theory to another. Furthermore, some social psychologists admit that the boomerang effect remains poorly understood [START_REF] Eagly | Process Theories of Attitude and Change: Attribution Approaches and Social Judgment Theory[END_REF]. The social judgement theory states that uncertainty plays an important role in both attitude attraction and rejection. The social identity theory stresses that attitude rejection is linked to the salience, at a given time, of the individual social identity. At the individual level, the theories link attitude rejection to loss of control or freedom, or a negative relation with others. From these theories, we retain that attitude rejection occurs when several attitudes are implicitly or explicitly activated. Moreover, it is favoured by a 'dissonant' situation, such as agreement on some attitudes and disagreement on others. As an example, [START_REF] Wood | Self-definition, defensive processing, and influence: the normative impact of majority and minority groups[END_REF] reports about students who, informed that their attitudes regarding a particular issue are close to the one of the Ku Klux Klan, decide to reinterpret this issue and adopt an attitude further away from the one of the Ku Klux Klan.

Another group of interesting results for our purpose comes from the social influence paradigm which has exhibited two important group behaviours: the average consensus [START_REF] Allport | Social Psychology[END_REF][START_REF] Sherif | A study of some social factors in perception[END_REF] and the polarized consensus [START_REF] Moscovici | The Group as a Polarizer Of Attitudes[END_REF]. The average consensus occurs when the value of an object given by a group after discussion, is close to the average of the values given by individuals before discussion. The polarized consensus takes place when the value given by the group after discussion is significantly more extreme than the average of individual opinions before discussion. Following these studies, Nowak [START_REF] Nowak | Dynamical Social Psychology[END_REF], in the social simulation domain, has recommended to investigate the tendency of individual attitudes to become more extreme (polarisation) as well as the tendency of individual to aggregate themselves in groups (clustering).

A large number of computer models are based on homophily. They postulate the existence of an attractive force between agents having close attitudes, which can be formulated using thresholds that determine when agents move towards each other's position [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF][20][START_REF] Hegselmann | Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation[END_REF][START_REF] Latané | Dynamic Social Impact: The Creation of Culture by Communication[END_REF][START_REF] Weisbuch | Meet, discuss and segregate ! Complexity[END_REF] (see [START_REF] Castellano | Statistical physics of social dynamics[END_REF] for an interesting review on opinion dynamics). This attraction threshold, also called uncertainty, can be fixed or dynamic [START_REF] Deffuant | How can extremism prevail ? A study based on the relative agreement interaction model[END_REF][START_REF] Weisbuch | Persuasion dynamics[END_REF].

Other models, less numerous and more recent, also include a rejection mechanism in addition to assimilation. In formalising the Social Judgement Theory [START_REF] Jager | A dynamical perspective on attitude change[END_REF][START_REF] Jager | Multiple Attitude Dynamics in Large Populations[END_REF][START_REF] Jager | Uniformity, Bipolarization and Pluriformity Captured as Generic Stylized Behavior with an Agent-Based Simlation Model of Attitude Change[END_REF], an individual has two thresholds on an attitude dimension: a first for assimilation and a second one for rejection (the second is assumed higher than the first). In [START_REF] Salzarulo | A continuous Opinion Dynamics Model Based on the Principle of Meta-Contrast[END_REF], based on the theory of self-categorisation and the meta-contrast principle, an individual tends to minimise the distance to a prototypical opinion which defines his own group and, at the same time, he maximises the distance to an external group. Moreover, a rejection effect appears in [START_REF] Urbig | Dynamics of structured attitudes and opinions[END_REF][START_REF] Urbig | Drifting to more extreme but balanced atttiudes: Multidimensionnal attitudes andselective exposure[END_REF] as an emerging effect of homophilic individual interactions. This effect is due to the fact that getting closer in the 2dimensional attitude space may in some cases result in a shift away on the global attitude (which is a weighted sum of the attitudes).

Another form of rejection mechanism can be found in the 'contrarians' of Galam [START_REF] Galam | Contrarian deterministic effects on opinion dynamics: 'the hung elections scenario[END_REF][START_REF] Borghesi | Chaotic, staggered and polarized dynamics in opinion forming: the contrarian effect[END_REF] who tend to adopt an attitude which is opposite to the one of the majority (attitudes are supposed binary). The stochastic Sznajd model [START_REF] De La Lama | Spontaneous emergence of contrarianlike behaviour in an opinion spreading model[END_REF][START_REF] Wio | Contrarian-like behavior and system size stochastic resonance in an opinion spreading model[END_REF] also includes individuals who oppose to the majority. In this model, a social temperature implies with a probability p the application of the appropriate Sznajd rule for the opinion choice of an agent, and the application of the opposite rule with the probability 1 -p. Both of these models consider one-dimensional binary attitudes and tend to a particular final state, due to the "contrarians" effect, for which 50% of the population adopts one opinion, and the other 50%, the other opinion.

The attitude dynamic model we propose postulates multidimensional attitudes, like in [27, 29-34, 36, 43-44]. Considering two dimensional attitudes with an equal importance, our main assumption is that, if you strongly disagree with someone on attitude x 1 , and are close on attitude x 2 , you tend to solve the dissonance by shifting away on attitude x 2 . More precisely, when attitudes are both far or both close from each other, we follow the hypotheses of bounded confidence (BC) models [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF][START_REF] Hegselmann | Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation[END_REF][START_REF] Weisbuch | Meet, discuss and segregate ! Complexity[END_REF][START_REF] Deffuant | How can extremism prevail ? A study based on the relative agreement interaction model[END_REF][START_REF] Weisbuch | Persuasion dynamics[END_REF] ( [START_REF] Lorenz | Continuous Opinion Dynamics Under Bounded Confidence: A Survey[END_REF] for a review): when both are close, the attitudes tend to get closer, when both are far apart, there is no influence. Two models are usually identified as bounded confidence models: the Deffuant-Weisbuch model [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF] and the Hegselmann-Krause model [START_REF] Hegselmann | Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation[END_REF]. These two models differ regarding their communication regime. Agents of the Hegselmann-Krause model adopt the average opinion of all agents which lie in her area of confidence. Agents of Deffuant-Weisbuch model meet in random pairwise encounters after which they comprise or not. The model presented in this paper follows the Deffuant-Weisbuch communication regime. Therefore, our model is similar to a multi-dimensional bounded confidence model, except that we added the rejection mechanism when people are close on one attitude and far apart on the other.

The next part of this paper describes the model in a simplified version of the ODD framework [START_REF] Grimm | A standard protocol for describing individual-based and agentbased models[END_REF] which is a protocol for describing individual and agent based models in three blocks (Overview, Design concepts, and Details). Following that, we present examples of simulation runs for different parameters, which lead to the hypothesis that the number of clusters grows linearly with the inverse of the uncertainty for a large range of uncertainty values. Other example show that higher uncertainty values tend to less consensus than the classical two dimensional bounded confidence model. Then, we show results of a systematic exploration of the parameter space which support these hypothesis. Finally we will discuss the results and conclude.

State variables and scales

We consider a population of N individuals, each having a 2-dimensional attitude or two different attitudes x 1 and x 2 , represented by real numbers between -1 and +1, and the related uncertainties u 1 and u 2 . Uncertainty is a term used for convenience, because this variable may represent confidence in one's own attitude position as well as the motivation to comply with other's attitude positions (social susceptibility). It corresponds also to the latitude of acceptance of the Social Judgement Theory and represents the level of ego-involvement in the value of the attitude. In the following experiments, all individuals have the same uncertainties U on both attitudes u 1 = u 2 = U ).

Process overview and scheduling

At each time step, we choose a pair of individuals A and B at random, and they may influence each other. More precisely, at each time step, the algorithm is as follows:

N times repeat: -choose couple of individuals (A,B) at random; -B may influence A.

The influence depends on the conditions describing the values of attitudes and uncertainties. Suppose A has attitudes a 1 and a 2 with uncertainties u 1 and u 2 , and B has attitudes b 1 and b 2 with uncertainties v 1 and v 2 . We only studied the case where all individuals have the same uncertainty for all their attitudes. Thus, for sake of simplicity, we used U instead of u 1 and u 2 in the following since u 1 = u 2 = U . Then, A compares its attitudes with the ones of B. Three cases arise.

Case 1: B is close to A on both attitudes:

a t 1 -b t 1 ≤ U and a t 2 -b t 2 ≤ U (1)
Then both attitudes of A get closer to the ones of B:

a t+1 1 = a t 1 + µ b t 1 -a t 1 and a t+1 2 = a t 2 + µ b t 2 -a t 2 (2)
Here µ is a kinetic parameter of the model, representing the velocity of the attraction or the rejection. In our following study, µ has the same value for all individuals. 2.3.2. Case 2: B is far from A on both attitudes:

a t 1 -b t 1 > U and a t 2 -b t 2 > U (3)
Then, there is no influence of B on A.

2.3.3.

Case 3: B is far from A on one attitude and close to A on the other.

a t 1 -b t 1 ≤ U and a t 2 -b t 2 > U (4)
We only describe the case where people are close to each other on the attitude 1 and far from each other on the attitude 2 because the case where people are close on the attitude 2 and far on the attitude 1 is analog with a 1 and b 1 interchanged with a 2 and b 2 . Then two cases arise, depending on whether A and B differ strongly on attitude 2. We introduce the positive parameter δ, ruling the intolerance threshold which globally depends on the uncertainty: Case 3.1: A and B do not differ strongly on attitude 2

a t 2 -b t 2 ≤ (1 + δ)U (5)
Then, the disagreement is not strong enough to trigger the rejection. A approaches B on attitude 1 and ignores B on attitude 2:

a t+1 1 = a t 1 + µ b t 1 -a t 1 ( 6
)
Case 3.2: A and B differ strongly on attitude 2

a t 2 -b t 2 > (1 + δ)U (7)
Then, A shifts away from B on attitude 1. The movement is proportional to the distance needed to get b 1 out of A's range of uncertainty around a 1 . The specific form of the equation express that people move their own average attitude in order to put the average attitude of the unacceptable other out their own attitude segment. This means people try to adopt a new attitude in such a way that they don't judge themselves similar to the unacceptable other.

a t+1 1 = a t 1 -µ psign b t 1 -a t 1 U -b t 1 -a t 1 (8)
Where psign() is a particular sign function, which returns -1 if its argument is strictly negative, or +1 otherwise. The particularity compared with the standard sign function is that when the argument is 0 psign returns +1. Moreover, we confine the attitude within the bounds (-1, +1) of the attitude space:

If a t+1 1 > 1 then a t+1 1 := sign(a t+1 1 ) (9)
The following figures illustrate the different types of interactions (attraction, rejection or indifference). Figure 1 shows on the left the case where A is not influenced by B : they are far from each other on both dimensions. On the right, Figure 1 shows the case where A is attracted by B and vice-versa because they are close to each other. This means each one has his attitude in the other's acceptance zone.

Initialisation

When we do not vary the population size, we consider a population of 1000 individuals with two attitudes. On each dimension, the attitude is randomly initialised following a uniform distribution comprised between -1 and +1. Uncertainty U is constant and identical on each dimension.

Analysis of several examples

In this section, we observe several simulation examples. Their analysis leads in particular to formulate the hypothesis that the number of clusters is a linear function of 1/U for weak to average values of uncertainty. Higher uncertainties globally exhibit close final states from those of the 2-dimensional bounded confidence model. is progressively organized into several clusters. The clusters are not regularly organised on horizontal and vertical lines, as observed with the classical bounded confidence model. They rather tend to be located on oblique lines, which are not strictly regular. Moreover, the individuals fluctuate in the clusters, with constant amplitude of fluctuation, leading to a permanent diversity within the cluster. The reason is that individuals are pushed away from the cluster by other clusters, located close on one dimension and far enough on the other. These movements compensate each other because generally there are several neighbouring clusters that reject the cluster in opposite directions. Moreover, individuals are attracted by the cluster itself, especially if the cluster includes many individuals. Therefore one can say that the clusters are metastable, because if there is a strong perturbation (deletion of a neighbouring cluster) this may dramatically modify the equilibrium. This is a big difference with the classical bounded confidence model, in which after a while, clusters keep concentrating with time, each independently from the others. A second important difference with the classical BC model is that we get clusters on the border of the attitude domain. With the BC model, the first clusters are always inside the attitude domain, on a distance which is about the uncertainty U. On the contrary, with this model, it appears that there are always some clusters which have an extremer attitude position than any individual at the initialisation. These border clusters are flat, because their neighbouring clusters tend to push them away outside the attitude domain. This is a polarisation phenomenon in the sense of Nowak: a part of the individuals gets more extreme. If we removed the constraint to remain within the bounds of the attitudes, the global range of attitude would grow, and we would finally end up with stable clusters, not disturbing each other, in a significantly larger attitude domain.

Evolution with uncertainty U=0.2 and intolerance threshold with δ=0

Figure 3 shows an example of evolution for uncertainties U =0.2, and intolerance parameter δ=0, and the kinetic parameter µ=0.3. The number of time steps t appears on the top of each picture.

Spatial organisation of the clusters and hypothesis of linearity of their number with 1/U

The spatial organisation of the clusters can be further analysed. In this particular case where δ = 0, we note that there is only one cluster on a horizontal or vertical line. Indeed, two clusters on the same horizontal or vertical line is an unstable situation. If the clusters are far, they tend to push each other from the line. If the clusters are close, they tend to merge. This can be checked by considering the histogram of presence of the individuals on each axis on Figure 4. We note that 13 clusters appear on the projection of both axes. Moreover, the distance between the clusters is small enough to trigger rejection (11 clusters is the maximum, to provide a distance of at least U between two consecutive clusters), which explains why the individuals fluctuate in the clusters. In this case, the number of clusters can be analysed on a single axis: there should be a minimum interval between the clusters on each axis which is about the value of U. As we have seen, because of the metastability, it is possible to get slightly smaller intervals. Nevertheless, one can expect a number of clusters varying linearly with 1/U. When the intolerance threshold gets higher, the conditions for rejection are more restricted: the disagreement on one attitude must be higher. We observe that, for these values of δ, it becomes possible to get two clusters on the same horizontal or vertical line, when they are not too far apart (they remain in the tolerance zone). This explains why there are more clusters. Nevertheless, we can hypothesise that this number should still vary linearly with 1/U, but with a higher coefficient.

Influence of intolerance threshold δ > 0

Moreover, for δ = 1.5, we observe flat clusters inside the attitude domain, whereas this did not take place for δ = 1. Such a flat cluster appears when all the neighbour clusters are on the same line in the tolerance zone, or far on both attitudes. The rejection interactions are therefore only in one direction. U. This first exploration suggests that the number of clusters decreases with U, like with the BC model. The observations made on our first simulation extend to these cases: Oscillations of individuals remain, with higher oscillations when U increases, and spatially organised to avoid two clusters on the same horizontal or vertical line. In each case, we get flat clusters with the maximum value for one attitude (polarisation).

Different values of uncertainty U with intolerance threshold δ=0

For U = 0.6, we observe that the clusters become very concentrated, like in the simple BC model even if, for the same uncertainty value, the simple BC model has only one cluster for a population of 1000 individuals. The reason is that with 4 clusters, the intervals between the clusters on a same horizontal or vertical line can easily be higher than U, and therefore avoid to generate a competition between the attraction in the cluster and the rejection from the neighbouring clusters.

Systematic analysis of the number of clusters

We are interested in comparing the final number of attitude clusters with the one generated by the standard BC model proposed by [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF]. First we describe how we compute the final number of clusters. Then, we analyse this final number of cluster regarding two different behaviour zones of the BC model (see [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF][START_REF] Ben-Naim | Bifurcations and Patterns in Compromise Processes[END_REF] for more details). The first zone is a zone for which the population is organized in several clusters; it is the object of our second point. The second zone is a zone for which the wide majority of people go in one cluster; it is the object of our third point.

Computing the number of clusters

From the individual-based simulations, we collect the average, minimum and maximum final number of clusters. To compute the number of clusters, we define a minimum distance between attitudes, below which we consider that they belong to the same cluster. We compute the clusters as groups of agents such that between any couple of agents of opinions x and x in the group, there is a list of agents in the group of opinions (x 1 , x 2 , . . ., x k ) making a chain of couples distanced from each other of at most an Euclidian distance lower than . The following pseudo-code can be used to compute the clusters; necessaryToLookAt is a table containing In practice, we chose =0.2U and we neglected the clusters of size lower or equal to 3 individuals. The simulations are stopped after 1,000,000 iterations. They can be stopped before if the number of clusters has not changed after 100,000 iterations. Even though [START_REF] Ben-Naim | Bifurcations and Patterns in Compromise Processes[END_REF] have demonstrated the interest of minor clusters in wide populations and for high value of µ [START_REF] Laguna | Minorities in a Model for Opinion Formation[END_REF], we focus only on major clusters in this first study.

Final number of clusters on the 'multiple clusters zone' of the BC model

The BC model, in one dimension, yields a final number of clusters n c in a population initialised with a uniform law on an attitude space of width 2M , with all the same uncertainty U, which can be approximated by:

n c ≈ M U (10)
In the 2-dimensional case, when both attitude axes are adjusted independently and all have the same uncertainty U on both attitude dimensions, this rule is repeated on all lines of the space, therefore we get: This result is confirmed by Figure 7 which presents on abscissa 1/U 2 and on y-axis, the average number of clusters obtained on 30 replicas.

n c ≈ M U 2 (11)
We focus on the zone where BC model exhibits a final state including several clusters. For our attitudinal domain (attitudes beteween -1 and +1), it goes from U = 0 to U = 0.54. Figure 8 shows the number of clusters obtained with rejection dynamics, for different values of U and δ. These results confirm the hypothesis of linearity of the number of clusters with 1/U for δ = 0 and δ = 0.5 (left) on this zone. For δ = 1, 1.5, 2 and 3, there is a non-linearity for U larger than 1 (only 1 and 2 are presented on the figure). When U is larger than 0.3, and δ is large, the conditions for rejection are much constrained by the size of the domain: two individuals must be at both sides of the domain. Most of the interactions correspond therefore to the standard BC, and the curve is therefore quadratic. When U decreases (1/U grows), the rejection becomes more common and the curve becomes linear.

Let's now verify if these results are robust when the population size varies (100, 1000 and 5000 individuals). Figure 9 shows the number of clusters obtained with rejection dynamics for δ = 0 and different values of U and population size N . To be able to compare the different population sizes, we count all the clusters (no threshold). From Figure 9, we note that the population size does not change our conclusion: the final number of clusters tends to be linear with 1/U.

Final number of cluster on the "one major cluster" zone of the BC model

We now focus on the zone where the BC model exhibits one single major cluster. For our attitudinal domain, it begins for U > 0.54. Since for U > 1, the rejection mechanism cannot work (all attitudes are at a distance which is within the attraction range), we only study the U value range from 0.54 to 1 (indeed, for U > 1, all people go in one unique central cluster, exactly as in the BC model). Figure 10 shows the results in this zone for different population sizes. We immediately see that the final number of clusters is not linear with 1/U. In the standard BC model proposed by [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF], the final state for this zone is one major cluster containing a large majority of the population with, in some cases, several very minor clusters when the population is very large [START_REF] Castellano | Statistical physics of social dynamics[END_REF]. Figure 11 shows that our model has also, on the zone of U values, one major cluster containing a majority of the population (from a part of half to the whole population depending on the parameter value). In our model with a rejection mechanism, we finally obtain between two and six final clusters as shown on Figure 10. Are the non-major clusters the same as those of the BC model ? From [START_REF] Laguna | Minorities in a Model for Opinion Formation[END_REF], we know the very small clusters of the standard BC model are very numerous and do not exist for low values of µ. In our attraction-rejection model, minor clusters are not numerous, from one to five on average, and larger than those of the classical BC model. Moreover, they remain when we run simulation with a value of µ equal to 0.01. Finally, our population size is not large enough to really observe 'minor clusters' in the sense used by [START_REF] Ben-Naim | Bifurcations and Patterns in Compromise Processes[END_REF].

Discussion and conclusion

In the model of 2-dimensional attitude dynamics we propose, an agent shifts away from a close attitude on one axis when the interlocutor is far on the other axis. We assume that this is a way to solve the dissonance between the attitude axes. The distance threshold to trigger rejection depends on the intolerance parameter δ and on the uncertainty U, which may define a non-commitment zone, in which the dissonance is tolerated. When the conditions of rejection are not met, that is when we exclude the case where two individuals differ strongly on one attitude and are similar on the other attitude, the model behaves exactly like the 2D bounded confidence (BC) model.

The first explorations of this model, in the simple case where all uncertainties are the same, show several striking results, in comparison with the 2D BC model:

• When the uncertainty is lower than 1, allowing the rejection to occur, the dynamics leads to several metastable clusters, which are generally in competition and tend to reject each other. The stability is due to contradictory rejections from neighbouring clusters, which compensate each other. If one of its neighbouring clusters is removed, the position of a cluster changes significantly, and it may even disappear. Moreover, individuals belonging to a cluster are in constant fluctuation around the cluster center, with amplitudes depending on the cluster size and on the proximity of competing clusters. In this respect, the configuration is very different from the one obtained with simple BC model where, after a while, clusters keep concentrating with time, each independently from the other.

• Several clusters are moving towards the limits of the attitude domain. This may be interpreted as a radicalisation of a part of the population, which reaches the maximum absolute value of one of the attitudes. This never happens with the 2D BC model. • In the case where the intolerance threshold δ = 0, two clusters cannot be maintained on the same horizontal or vertical line. Therefore, the clusters tend to occupy points of the space where they are as far as possible from other clusters on each axis. This analysis suggests a number of clusters growing linearly with 1/U for values of U for which the 2D BC model exhibits several clusters called 'major' and 'central' clusters by [START_REF] Ben-Naim | Bifurcations and Patterns in Compromise Processes[END_REF]. However, for the 2D BC model, for this same range of U values, the cluster number grows quadratically with 1/U in the 2D BC model. When δ grows, configurations with more than one cluster on a line may be stabilised, but this number is limited by the size of the tolerance zone. Therefore, the growth of the cluster number should still be linear, but with a factor growing with δ. First systematic experiments support this statement, for different population sizes. • For values of U for which the 2D BC model exhibits only one cluster called 'central' in [START_REF] Ben-Naim | Bifurcations and Patterns in Compromise Processes[END_REF], our model does not follow the same law and tends to have less consensus than the 2D BC model. Indeed, depending on the parameter value, it exhibits from two to six clusters on average, with one major cluster containing a majority of people. The other clusters are generally on the limits of the attitude domain. [START_REF] Ben-Naim | Bifurcations and Patterns in Compromise Processes[END_REF] and [START_REF] Laguna | Minorities in a Model for Opinion Formation[END_REF] show that the 2D BC model has, for a subpart of this zone of U values, numerous very minor clusters when µ is high and when the population size is wide. However these very minor clusters, even if they are located close to the bound of the attitudinal space, are different from the minor clusters of extremists of our model.

These results suggest several points to discuss.

• The metastability of the clusters is due to the bounds we impose on the attitude values. Indeed, without these bounds, the attitudes grow until the distance between the clusters is higher than the uncertainty in all directions. Then, the clusters do not influence each other, and they keep concentrating as in the BC model. First simulations performed on the same model with an unbounded attitude domain indicate that the final number of clusters is close to the one obtained with the bounded domain. However, the unbounded case should be the object of a particular study. In any case, the metastability of the clusters is an particular feature of this model, which better fits real group dynamics than the perfect similarity obtained without a bound (or by a standard BC model). • Even without bounds, we obtain a global result which shows strong simi-larities with social identity and self-categorization theories. Our individuals tend to minimise their in-group distance and maximise their out-group distance (to competing groups). We also get some polarized groups (which have more extreme opinions than all the individuals initially). This reminds of the results from Moscovici and Zavalloni [START_REF] Moscovici | The Group as a Polarizer Of Attitudes[END_REF]. Therefore, with a model considering only paired interactions, we get group dynamics which seem to make sense in a social psychology perspective.

• However, the model remains highly simplified, and a challenge that remains is checking if these interesting properties last when adding more sophisticated hypotheses. In particular, in our model, all attitudes are considered to have the same weight on the behaviour, whereas one expects that only disagreements on attitudes deeply related to social identity can lead to rejection. To take this aspect into account, we should consider attitudes of different types. • We have chosen the particular communication regime of the Deffuant-Weisbuch model. Considering the communication regime as a parameter of the bounded confidence model as in the formulation proposed by [START_REF] Urbig | Communication regimes in opinion dynamics: Changing the number of communicating agents[END_REF], would be worth investigating. Moreover, it would be a logical extension to relate the chance of interacting to the attitude similarity between the agents, thus reflecting principles of preferential attachment.

In future research, we plan to continue to explore the properties of this model. In particular, introducing extremists like in [START_REF] Deffuant | How can extremism prevail ? A study based on the relative agreement interaction model[END_REF] could produce unexpected effects.

Acknowledgement

We thanks the anonymous reviewers for their suggestions of improvements and their relevant remarks.

1 .

 1 Purpose of the modelThe purpose of the model is to test the collective effects of a particular rejection mechanism in 2-dimensional bounded confidence models which are based on individual attraction mechanisms. The rejection takes place when individuals are close on one attitude and far on the other.

6

  Huet S., Deffuant G., Jager W.

  Fig. 1. A and B in a situation of no influence on both dimensions (left) and in situation of attraction (right)

Fig. 2 .

 2 Fig. 2. Left: A and B in a situation of attraction on one dimension (on attitude 1 dimension here) and indifference on the other dimension. Right: A and B in a situation of rejection on one dimension (on attitude 1 dimension here) and indifference on the other dimension

Figure 2 8

 2 Figure 2 left shows another case where people are close to each other on only one dimension. People are far from each other on one dimension but not far enough to consider the proximity on the other dimension as unacceptable. Thus, they attract each other on the dimension where they are close. On the contrary, Figure 2 right shows the cases where people are far enough from each other on one dimension. The proximity on the other dimension is perceived as unacceptable. Thus, they move away from each other on this dimension.

Fig. 3 .

 3 Fig. 3. Initial population uniformly distributed in 2D attitude space. U =0.2, µ=0.3, δ=0. We observe the emergence of metastable clusters, with remaining fluctuations of individuals within the clusters. Moreover, some flat clusters are located on the borders of the attitude domain, containing radicalised individuals

Fig. 4 .

 4 Fig. 4. Kernel density estimator on horizontal axis (left) and vertical axis (right), for the final situation of Figure 3 (U =0.2, t=20000). One notes that the 13 final clusters are regularly distributed on each axis.

Fig. 5 .

 5 Fig. 5. Example of final configuration for U = 0.2, µ=0.3, N = 1000, δ=1 (left), δ=1.5 (right). It is possible to get 2 clusters on the same horizontal and vertical line, which is unstable when δ=0. Moreover, for δ=1.5, some clusters are flat inside the attitude domain.

Figure 5 Rejection

 5 Mechanism in 2D BC Provides More Conformity 11 examples of final attractors, for U =0.2, δ=1 (left) and δ=1.5 (right). The number of clusters appears to increase with δ .

Fig. 6 .

 6 Fig. 6. Examples of attractor configurations for different values of uncertainty U and intolerance parameter δ = 0, µ = 0.3. Population size N = 1000.

Figure 6 12

 6 Figure 6 shows several attractor configurations for different values of uncertainty

Fig. 7 .14

 7 Fig. 7. Average final number of clusters of the 2D bounded confidence model as a function of 1/U 2 . Error bars indicates minimum and maximum obtained on 30 replicas.

Fig. 8 .

 8 Fig. 8. Mean final number of clusters for the model with rejection as a function of 1/U, for various values of δ. N =1000 and δ=0.3. The error bars are the minimum and maximum numbers met in 30 replicas. On the left, for δ=0 and δ=0.5, the number of clusters seems linear with 1/U. On the right, the behaviour is not linear for large U.

Rejection 15 Fig. 9 .

 159 Fig. 9. Mean final number of clusters for the model with rejection as a function of 1/U, for δ = 0 and various value of population size N (100, 1000, 5000) and U (from U=0.09 to U=0.5). The error bars are the minimum and maximum numbers met in 30 replicas. The number of clusters seems also linear with 1/U.

Fig. 10 .

 10 Fig. 10. Mean final number of clusters for the model with rejection as a function of 1/U, for δ = 0 and various value of N (100, 1000, 5000) and U (from U=0.55 to U=0.95). The error bars are the minimum and maximum numbers met in 30 replicas. The number of clusters is not linear with 1/U.

16

  Huet S., Deffuant G., Jager W.

Fig. 11 .

 11 Fig. 11. Average mass of the final larger cluster for δ = 0 and various value of N (100, 1000, 5000) and U (from U =0.55 to U =0.95).

  the identification number of each individual for all the population:

	March	27,	2008 19:3 WSPC/INSTRUCTION	FILE
	ACS2008HuetDeffuantJager
			Rejection Mechanism in 2D BC Provides More Conformity 13
			necessaryToLookAt[j] = 0
			currentCluster.add(j)
			compt++
		currentCluster.remove(0)
		nbClusters++
		if compt = populationSize then i = populationSize
		for all i of the population
		if necessaryToLookAt[i] > 0
		currentCluster.add(i)
		compt++;
		necessaryToLookAt[i] = 0
		while currentCluster.isNotEmpty()
		for all j of the population
		if necessaryToLookAt[j] > 0
		if distance(pop[currentCluster.get(0)],pop[j]<epsilon)