Participative democracy and local environmental issues

E CO L O G I CA L E CO N O MI CS 68 ( 20 0 8 ) 6 8–7 9

a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n

ANALYSIS

Participative democracy and local environmental issues Emmanuel Martineza , Tarik Tazdaït b,⁎, Elisabeth Tovar b,c a

EPE, 41 rue des Trois Fontanot, 92024 Nanterre cedex, France CIRED-C.N.R.S-E.H.E.S.S, Campus du Jardin Tropical 45 bis, avenue de la Belle Gabrielle, 94736 Nogent Sur Marne cedex, France c Centre d'Etudes des Politiques Economiques de l'Université d'Evry (EPEE), France b

AR TIC LE I N FO

ABS TR ACT

Article history:

The present paper deals with the management of environmental quality at the level of a city.

Received 14 November 2006

More precisely, we propose two different contractualisms among citizens that allow them to

Received in revised form

reduce the polluting emissions caused by their consumption of both a private and a collective

6 December 2007

good. In the first contractualism, the mayor proposes to citizens to establish neighbourhood

Accepted 28 January 2008

committees. They have the responsibility of defining the total amount of polluting emission

Available online 5 March 2008

by private and public transportation means in their neighbourhood. It comes out that the mediation of the neighbourhood committees brings out a lower total pollution than in the

Keywords:

case where each individual citizen acts on his own interest. However the emission level

Pollutant emissions

brought out by the neighbourhood committees is not Pareto-optimal. On the other hand,

Participative democracy

extending the first coalitional game by a supplementary stage focused on bargaining

Coalitions

between the neighbourhood committees, allows us to define a participative democracy

Bargaining

procedure. This two-stage procedure brings about a Pareto-optimal pollution level.

Transfer

© 2008 Elsevier B.V. All rights reserved.

JEL classification: C72; H41; Q53

1.

Introduction

The aim of this article is to propose a theoretical framework to study the local management of a pollution problem. Because of its decentralized nature, our model is related to the recent literature on local participative democracy. This literature hinges on two axes. On the one hand, it tries to build a common analytical framework that accounts for new forms of urban governance — among others, those prompted by the inadequacies of institutional answers to urban growth (De Carlo, 1996). On the other hand, it tries to integrate these new forms of participative democracy, in which citizens voluntarily take part in the formalisation and implementation of public policies processes (Parry et al., 1992), in the study of collective decision procedures, by confronting them in a particular to representative democracy

(McLaverty, 2002). These contemporary works are in line with Habermas (1990), who opposes procedural democracy to institutional democracy. Based on a procedural definition of democracy, a decision is said fair if each opinion is expressed by identical means and if the only factor influencing the vote is the quality of arguments, independently of the institutional mode of producing the decision. In our model, we allow citizens to be involved in the processes of decision making through neighbourhood committees, and therefore we define participative democracy as the expression of neighbourhood committees. Neighbourhood committees refer to decentralized initiatives in local development (community empowerment) that spread during the last decade, in urban zones of the Commonwealth countries but also in the rural areas of developing countries. Community empowerment can be defined as the decentralized

⁎ Corresponding author. E-mail address: [email protected] (T. Tazdaït). 0921-8009/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2008.01.025

EC O LO G I CA L E C O N O M I CS 6 8 ( 2 00 8 ) 6 8–7 9

mobilization of actors coming from a disadvantaged area (Dreier, 1996). The main characteristic of this new phenomenon is the inversion of the usual governance direction: under community empowerment, initiatives and decisions are bottom up (from decentralized agent to local authorities) rather than top down like in the usual structures of public decision. The role of public institutions is limited to mediation and, at the end of the process, to a financial support generally in complement to funds raised in a decentralized way by the community members. In North American cities, Smock (1997) notices that the emergence of a decentralized cooperation among the inhabitants of disadvantaged urban areas has become essential because of the decrease in federal subventions to local development in the 80's, and of the privatisation of local public services. We provide with a theoretical treatment based on coalition formation theory. Furthermore, our representation of participative democracy implies a redefinition of the role of public authorities, which do not intervene anymore as an active player in the game. This takes us away from the representation predominating in public economics, according to which the public authorities determine the instruments (taxes or others) that are likely to dissuade free-rider behaviours. In our approach, the public authorities do not resort to any regulation instrument. The principle of their intervention consists in proposing cooperation procedures played only by those first involved in the reduction in pollution, i.e. city citizens. Our model also moves away from the mechanism design theory (Green and Laffont, 1986, 1987) according to the designer's objective is to lead the agents to truthfully reveal their preferences through the use of various incentives taking the form of rewards or sanctions. Our conception of participative democracy does not refer to any such incentives. In our model, we determine a procedure that could be proposed by some public authority (in our example a mayor) and which allows the setting up of neighbourhood committees. With those neighbourhood committees, citizens have the possibility to reduce their pollutant emissions in an efficient way. These emissions are caused by the use of a private individual good (a car) and of a collective good (public transportation, which is assumed to be less polluting per passenger). Since the procedure of cooperation we propose is set up in a decentralized framework, our methodology is the endogenous formation of coalitions.1 This methodology was developed to understand how cooperation can happen under the threat of deviation of each agent or by a sub-set of them associated within a coalition. The stability of an agreement is then a function of its ability to avoid such behaviour. This is another way to wonder why coalitions are forming (Kurz, 1988). It implies the identification of the coalition structures for which players do not want to modify it. As noted by Greenberg (1994), the most used stability concept for games without externalities is the core of a coalition structure. A coalition structure is said to be core stable when no group of players could obtain a higher payment by forming another coalition. Unfortunately, this concept is not easily generalised to games with externalities. To be relevant, the analysis of a coalition game has to be able to incorporate the reaction of the coalition which is confronted to a deviation. Several games have been developed to study these 1

For a survey see Yi (1997) or Bloch (1997).

69

configurations, inducing the definition of several stability concepts. Aumann (1961, 1967) introduced stability concepts based on the notion of core: the α-stability and the β-stability, while Hart and Kurz (1983, 1984) proposed stability concepts based on the strong Nash equilibrium notion: the γ-stability and the δ-stability. Beside these group stability concepts, an individual stability concept was defined by d'Aspremont et al. (1983) and Donsimoni et al. (1986). Using this individual stability concept and starting from a standard framework of public economics, we show that, under specific conditions, it is possible to reach a global agreement. Such an agreement requires a stage based on the strategic analysis of negotiations. We have considered two coalitional games, a static one and a two-stage one. In the static framework, the mayor proposes to citizens to establish neighbourhood committees. They have the responsibility of defining the total amount of polluting emission by private and public transportation means in their neighbourhood. It comes out that the mediation of the neighbourhood committees brings out a lower total pollution than in the case where each individual citizen acts on his own interest. However the emission level brought out by the neighbourhood committees is not Pareto-optimal. On the other hand, extending the first coalitional game by a supplementary stage focused on bargaining between the neighbourhood committees, allows us to define a participative democracy procedure. This two-stage procedure brings about a Pareto-optimal pollution level. Its essential feature is to rest on each citizen's ability to join or leave freely the created agreement: that is what one usually calls the open membership rule. In our model, everything happens as if the public authority gives itself the objective of determining cooperation procedures with the view to proposing them to agents. Playing the games that describe the procedure, the agents are led to form a coalition to defend their interests. The difficulty is then to make sure that the coalitional game results on a burden sharing corresponding to a Pareto-optimal situation. The paper is organised as follows. Section 2 presents the assumptions of our model dealing with the management of pollutant emissions produced by each citizen (through the use of a collective and a private good). This allows us to define a coalitional game able to induce citizens to improve their situation compared to a non-cooperative context (i.e. Nash equilibrium). As this game does not allow to reach optimal allocations, Section 3 introduces a two-stage game that leads to an optimal burden sharing of emission reductions, provided some transfers between agents determined by the bargaining powers and number of members of each coalition. Section 4 generalizes the obtained results by widening the analytical framework to heterogeneous agents. Section 5 concludes with a presentation of the different highlighted results. Note that all the proofs of the different propositions are relegated to Appendix A.

2.

First procedure: a coalitional game

2.1.

Payment

Let N = {1,…, n} denote the set of inhabitants in age of voting (citizens). All agents are assumed to be identical and to

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interact in a common environment threatened by a pollutant gas emitted by individual cars and the public transportation system. The economy is composed of the following commodities: – A private good, the use of a car, denoted by x = (x1,…, xn) N 0, where xi represents the level of car usage by citizen i. Assuming x ≠ 0 forbids the absence of any car usage in the economy; – a collective good, the use of public transportation, denoted by y = (y1,…, yn) N 0, where yi represents the use of public transportation by citizen i; – pollutant emissions, denoted p = (p1,…, pn) N 0, where pi is the level emitted by citizen i. We consider that p cannot be equal to zero, a reasonable assumption for most pollutions; – an environmental characteristic that is assumed measurable, z b 0. z is considered a public bad, climate change in our example. We assume that each citizen's preferences are described by a linear utility function, which satisfies2: 8iaN Ui ðxi ; yi ; zÞ ¼ xi þ yi þ z:

ð1Þ

The use of his/her own private good by citizen i induces pollutant emissions: 1=2

xi ¼ 2pi :

ð2Þ

The use of the public good by citizen i induces pollutant emissions: 1=2

yi ¼ 2bpi

with 0 b b b 1:

ð3Þ

Eqs. (2) and (3) mean that emissions resulting from the use of cars are higher than those from the use of public transport. Quantities pi are considered both as inputs for the transportation service and polluting emissions. Consequently, the use of one unit of input leads to one unit of pollution. We will admit that the ecological transfer function is determined linearly and additively by pollutant emissions, that is: Xn p: ð4Þ z¼ i¼1 i To be more rigorous, we should consider an ecological transfer function that takes into account the emissions of the considered city but also those of all other cities. However, with our focus on a local scale we represent the behaviour of agents from other cities through a constant parameter, set equal to zero for simplicity's sake. That is why we have simplified the writing of relation (4) to the only emissions emitted by the citizen of the studied city.

2.2.

The coalitional game

Let us consider the procedure by which a local public authority (in our example the mayor) establishes a form of decentralized

participative democracy by allowing the setting up of neighbourhood committees. As a consequence, within each district of the city, coalitions are forming to implement, at their own scale, the strategy guaranteeing the best management of the environmental quality, in a local context. The players of our game are then coalitions of citizens, who determine their pollutant emission volumes. Because players within each coalition only maximise their joint payoff or choose one of its efficient outcomes, we use the concept of quasi-hybrid equilibrium (Zhao, 1991, 1992), which assumes that each coalition of the partition behaves collectively by maximising its joint payoff given other players' choices. Everything happens as if citizens belonging to a coalition delegate their decisions to all coalition members in order to maximise the joint utility of the coalition, given the actions of the other coalitions. The cooperation inside any coalition is shaped by an equal sharing rule because citizens are identical. No member would accept any other sharing rule. Each SpFpN is called a coalition, and a coalition structure is a partition ε = {S1,…, St} of N, which is a collection of coalitions satisfying [tk¼1 Sk ¼ N and Sm \ Sr ¼ F for all m and r such as m ≠ r. Consequently, a coalition structure ε divides up the set of players among coalitions and/or singletons, in such a way that a citizen can belong only to one coalition (or singleton). The cardinal of a coalition Sk is: |Sk| = sk, with k = 1,…, P t and tk¼1 sk ¼ n. Definition 1. A quasi-hybrid equilibrium for each partition ε = ~ ~ ~ {S1,…, St} is a vector p = { p1,…, pn} such that for each coalition Sk a ε, ~ ~ its emissions ps = { pi / i a Sk} solves: Max

X

ðxi þ yi þ zÞ;

Subject to: X

xi V 2

iaSk

X

1=2

ð6Þ

pi

iaSk

X

yi V 2b

iaSk

X

1=2

ð7Þ

pi

iaSk

Xsk i¼1

p˜ sj ¼

pi þ z ¼ 

Xsk i V¼1;i Vpi

Xt j¼1

p˜ sj

where j p k

ð8Þ

p˜ i V:

ð9Þ

The resolution of the programme associated with Definition (1) gives: x˜ Sk ¼ 2ð1 þ bÞ; y˜ Sk ¼ z˜ ¼ 

Xsk i¼1

yi ¼ 2bð1 þ bÞ; p˜ sk ¼

ð1 þ bÞ2 sk

ð10Þ

Xt

ð1 þ bÞ2 ; k¼1 sk

and: U˜ Sk ¼ 2ð1 þ bÞ2 sk

2 We know that the utility when taking a car should be higher than taking public transportation and then we must have the following formulation: Ui(xi,yi,z) = xi + ayi + z where 0 b a b 1. As the parameter “a” does not change the results of the model, we kept formula given by Eq. (1).

ð5Þ

iaSk

Xt j¼1

ð1 þ bÞ2 : sj

ð11Þ

As already underlined, in this game each coalition maximises its joint utility given the others' best response. As a consequence, two properties characterise this game: it is both non-cooperative, as each coalition acts non-cooperatively with

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EC O LO G I CA L E C O N O M I CS 6 8 ( 2 00 8 ) 6 8–7 9

the other coalitions, and cooperative as the allocation decided by each coalition results from a cooperative process within each of them.

2.3.

Comparison of two polar cases

By supposing that each citizen is acting rationally, we can determine the behaviour of each of them as the best response to the other choices. This leads to a non-cooperative Nash equilibrium characterized by the finest partition {{1},…, {n}}, which is obtained by setting sk = sj = 1 and t = n in the previous results. Then, at the Nash equilibrium we get for i a N: P

P

P

xi ¼ 2ð1 þ bÞ; yi ¼ 2bð1 þ bÞ; pi ¼ ð1 þ bÞ2 ;

P

2

P

ð12Þ

2

z ¼ −nð1 þ bÞ ; Ui ¼ ð2  nÞð1 þ bÞ : Since the citizens are identical, they obviously emit the same quantity of pollutant and receive the same individual utility. Nevertheless, this utility depends on the number of citizens included in the model, as does the quantity of ambient pollutant. Likewise, to get the Pareto-efficient state (or the full cooperation) of the economy, when the coalition structure is given by {N}, we just have to consider sk = sj = n. This results in:

Proposition 2. For the economy, full cooperation offers an utility gain with respect to any quasi-hybrid equilibrium. Finally, the results of Propositions (1) and (2) can be summarised by the following relation: Xn i¼1

P

Ui V

Xt i¼1

U˜ i V

Xn i¼1

Ui4:

ð15Þ

  Pt ˜ Pn 4 The second part of this inequality i:e: i¼1 Ui V i¼1 Ui means that the sum obtained by playing in a coalitional way is lower than or equal to the one that could be obtained, where a single coalition gathering all citizens to emerge. In other terms, there exists a gain to total cooperation that could be shared (in a redistributive way) in order to incite coalitions to agree on a unanimous agreement. If the inequality is satisfied, a larger agreement is then conceivable.

3.

Second procedure: a two-stage game

2

ð1 þ bÞ ; xN4 ¼ 2ð1 þ bÞ; yN4 ¼ 2bð1 þ bÞ; n zN4 ¼ ð1 þ bÞ2 ; UN4 ¼ ð1 þ bÞ2 : pN4 ¼

in coalitional rather than in individual terms. Favouring the formation of local coalitions contributes to improving the environmental quality. Nevertheless, this coalition interest approach in a static framework is not sufficient to reach a Pareto-optimal situation, as established in Proposition (2).

ð13Þ

As all citizens are considered identical, determining the allocation associated to each of them amounts to adopting the egalitarian sharing rule: 2

2

ð1 þ bÞ 2ð1 þ bÞ 4 2bð1 þ bÞ ; yi ¼ ; ; xi4 ¼ p4i ¼ n n2 n 2

ð14Þ

2

ð1 þ bÞ ð1 þ bÞ ; U4 : z4 ¼  i ¼ n n Unsurprisingly, the results associated to the Nash equilibrium do not correspond to those of the full cooperation: despite the environmental amelioration that benefits to all citizens, everyone has interest in letting the others make an effort and benefiting from the common effort at a lesser cost — an illustration of the free-riding problem. We can now proceed to the comparison of the quasi-hybrid equilibrium with the non-cooperative Nash equilibrium and the full cooperation. From the first comparison, it appears that the implementation of group behaviours where divergences are expressed at the coalition level rather than at the individual one, contributes to an improvement of the environmental quality. Proposition 1. For the considered economy, a quasi-hybrid equilibrium is Pareto-improving compared to a non-cooperative Nash equilibrium, whatever the coalition structure formed. Consequently, while remaining in a self-interest logic, it is possible to envisage (via the interaction of citizen coalitions) an emission level that benefits to everyone as it provides an improvement compared with the situation where each agent declares for himself. The self-interest logic is now expressed

We focus here on the definition of a game that allows for an agreement of all citizens on a management programme. The local public authority now proposes a cooperation procedure based on a two-stage game. During the first stage, citizens of each area form coalitions. Then, during a second stage, representatives of each area's coalition bargain to determine their respective effort levels (in terms of pollutant emissions), i.e. a repartition rule of the transfers that must be allocated to each coalition. The procedure corresponds in a certain way to a form of participative, decentralised and dynamic democracy, akin to the community empowerment that is developing in some Anglo-Saxon cities. Since we have a two-stage game with complete information, the equilibrium corresponds to a situation where: – The defined transfer rule satisfies the Nash bargaining solution of the second stage game. – The intermediary coalitions that are forming in the first step must be stable. Consequently, the model resolution is obtained by backward induction, starting with determining the optimal transfer rule in the second period. At the first period, every citizen chooses to join or to stay out of a coalition in order to maximise his/her utility, taking as given the behaviour of others.

3.1.

The bargaining

Let us suppose that the intermediary coalitions are formed. For a fixed coalition structure ε = {S1,…, St}, we analyse the final bargaining between coalitions, with 0 b αk ≤ 1 denoting the bargaining power of coalition Sk. To do so, we use as a solution the Nash bargaining solution (1953). The status quo point is

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~

~

given by ( US1,…, USt). This point, which corresponds to the situation where the coalitions refuse any compromise, is such that each citizen prefers to belong to a coalition to defend his/ her interests better. It thus coincides with the quasi-hybrid equilibrium: if no solution emerges from the bargaining between the representatives of coalitions, then each coalition receives the amount it would get at the quasi-hybrid equilibrium. Definition 2. Given the set Ω = {(U1,…, Ut) / U1 + … + Ut ≤ UN ⁎ } and the status quo point, the bargaining rule ϖ assigns to each (t + 1)~ ~ uplet (Ω,US1,…, USt) a bargaining solution:   ϖ X; U˜ S1 ; N ; U˜ St ¼ ðuS1 ; N ; uSt Þ

ð16Þ

The components of the vector (uS1,…, uSt) represent the utilities that the coalitions get if the bargaining ends with an P agreement, given that UN4 ¼ tk¼1 usk ¼ ð1 þ bÞ2 , and knowing that UN⁎ designates the utility of the coalition that gathers all citizens. Proposition 3. The bargaining solution is such that, for a fixed coalitions structure ε = {S1,…, St}, h i Xt usk ðeÞ ¼ ð1 þ bÞ2  U˜ a þ U˜ sk j¼1 sj k

8k

ð17Þ

and the transfers τ among the coalitions of citizens take the following form:  s  s ¼ ð1 þ bÞ2 2  k þ ð1  2tÞð1 þ bÞ2 ak n X 1 þðnak  sk Þð1 þ bÞ2 j 8k: sj

ð18Þ

Following the bargaining, each coalition gets its status quo amount and a share of the gains created from the full cooperation. This supplementary share is weighted by the bargaining power αk. The transfers between coalitions are then related to the surplus resulting from the full cooperation, and their repartition depends upon two parameters, which are the coalitions bargaining power αk, and the size of the coalitions, sk. These parameters act in opposite directions. While the bargaining power increases the transfer, the number of citizen forming the coalition decreases it. Nevertheless, to understand precisely the simultaneous impact of these two effects, it is necessary to combine them. That is why we propose to study, in the next section, the relationship between them.

3.2.

Discussion about the bargaining power

Xt sk with a ¼ 1: k¼1 k n

 s  Xt ð1 þ bÞ2 s ð1 þ bÞ2 s s ¼ 2ð1 þ bÞ2 þð1 þ bÞ2 ð1  2tÞ k þ n k  sk  k j¼1 n n sj n  s  ¼ 2ð1 þ bÞ2 1  t k ; ð20Þ n

and:  8iaSk oe;

ð19Þ

ui ðeÞ ¼ ð1 þ bÞ2

 2 1 þ ð1  2tÞ ; sk n

ð21Þ

leading to the following proposition. Proposition 4. 1-A coalition gets positive transfers from other coalitions if its size is lower than the average coalition size: s N 0fsk b

n : t

ð22Þ

2-A coalition pays transfers to other coalitions if its number of citizens is greater than the average coalition size: s b 0fsk N

n : t

ð23Þ

In this case, a coalition with high bargaining power is a coalition composed of a high number of members. As shown by Proposition (4), big coalitions pay transfers to little ones. When it is composed of a high number of members, a coalition gets a strong bargaining power as it will get a significant surplus gain, then confirming the positive role of the bargaining power. However, the bigger the coalition is, the shakier is its position. In other terms, the weaker is the gap with the Pareto optimum, the more dependent is the coalition toward citizens staying outside. Actually, the incentive of citizens outside the coalition is all the more strong since, because of their free-rider position, they get important gains generated by positive externalities created by the cooperating coalition. Therefore, the number of members forming the coalition acts negatively: by creating positive externalities, the coalition decreases the incentive to cooperate for outsiders. Then, to reach a complete agreement, the obtained surplus must be allocated to external members, to compensate the loss they suffer compared to the situation where they benefit from the positive externalities. Let us now consider the second case in which all the coalitions have the same bargaining power. We obtain: s¼

ð1 þ bÞ2 ðn  tsk Þ ð1 þ bÞ2 ðn  tsk Þ Xt 1 þ ; j¼1 s nt t j

ð24Þ

and: 8iaSk oe;

In order to illustrate the influence of bargaining powers, we consider two cases: a first one where the bargaining power of a coalition is linked to the number of citizens it gathers, that is ak ¼ snk , and a second one where any coalition has the same bargaining power, that is ak ¼ 1t . For the first case, we have: ak ¼

Transfers can be rewritten:

ui ðeÞ ¼ ð1 þ bÞ2

  n  tsk Xt 1 ð1 þ bÞ2  : j¼1 s tsk tsk j

ð25Þ

It is easy to see that Proposition (4) is still correct; nevertheless the amount received by coalition members is lower than in the first case. This may be explained by interpreting the distribution of bargaining powers as a share-out of votes. Making the bargaining power depends on the size of coalition amounts to entitling each citizen with a right to vote, which gives big coalitions much more importance. Conversely, assuming that bargaining power is the same for all coalitions

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EC O LO G I CA L E C O N O M I CS 6 8 ( 2 00 8 ) 6 8–7 9

Table 1 – Case: ak ¼ Snk Coalition structure [1234] [123,4] [12,34] [12,3,4] [1,2,3,4]

Individual utilities u1

u2

u3

u4

0.56 − 0.19 0.56 − 0.56 0.56

0.56 − 0.19 0.56 − 0.56 0.56

0.56 −0.19 0.56 1.69 0.56

0.56 2.81 0.56 1.69 0.56

irrespective of their sizes amounts to entitling each coalition to only one vote, which reduces their influence insofar as they are thus on an equal footing with singletons. Tables 1 and 2 illustrate this difference. Table 1 represents the individual utilities when ak ¼ snk , n = 4 and β = 0.5 while Table 2 gives the individual utilities when ak ¼ 1t , n = 4 and β = 0.5. Notation [12,3,4] means that players 1 and 2 form a coalition while players 3 and 4 stay in a singleton position. Columns u1,…, u4 represent the respective utilities of each of the four players according to the given coalition structures. Despite their differences in gains, the two tables have several properties in common. We may indeed note that in both cases: (P1) when a coalition enlarges with the arrival of another member, the citizen who remains outside of the coalition so formed finds himself in a better position; (P2) whatever the coalition structure considered, singleton players have a higher gain than those who are members of a coalition; (P3) when a player leaves a coalition to join another of the same size, the player who remains alone sees his gain increase; (P4) when a player leaves a coalition to join another of the same size, he sees his gain decrease in comparison with his initial situation.3 Indeed, on the basis of Tables 1 and 2, we can see that: – (P1) In the coalition structure ε = [12,3,4], when the coalition {12} is joined by {3}, leading to coalition structure ε' = [123,4], player 4 sees his gain increase. – (P2) In the coalition structure ε = [12,3,4]: U3(ε) = U4(ε) N U1,2(ε). – (P3) In the coalition structure ε = [12,34], when player 3 joins the coalition {12}, then player 4 obtains a higher gain. – (P4) In the coalition structure ε = [12,34], when player 3 joins the coalition {12}, then player 3 obtains a lower gain. More generally, theses different properties confirm the idea that the free-riding behaviour is fairly dominant which hampers the formation of a stable coalition.

3.3.

of the stability concept to be used. Several definitions of stability have been developed in the literature depending on the stability concept discussed: individual or group stability. The group stability concepts have the particularity to protect the coalition formed by all citizens against both individual and group deviations. Yet, to prevent these deviations from happening (particularly group deviations) group stability implies assuming that the benchmark situation is that of a coalition gathering all citizens formed under the collective interest logic. Only a central planner (e.g. the mayor of a city), as the legitimate depositary of citizen sovereignty, is in a position to defend the common interest of all community members. However, in our model, once the public authority has proposed the procedure, it is the citizens who play the game and the formation of a coalition is not the result of researching collective interest but of citizens acting in their own interests (i.e. comparing the gain they get whether they join the coalition or not). Furthermore, some stability concepts are based on the potential resort to deter deviations. This is not compatible with a participative democracy framework, where choices are done from consent, not from obligation. That is why we retain the individual stability concept. This implies the introduction of supplementary notions. ~ Let Ui,sk be the utility obtained by citizen i when he decides to join the coalition formed by sk − 1 citizens and ~ U− i,sk his utility when he stays out of the coalition formed by sk cooperating citizens. Let L(sk) denote the function giving the citizen's incentive to defect from a coalition formed by ~ ~ sk citizens. It follows that L(sk) = U− i,sk− 1 − Ui,sk. Moreover, let ~ ~ − L(sk + 1) = Ui,sk + 1 − U− i,sk be the incentive for a non-cooperative citizen to join a sk-coalition (that therefore becomes a sk + 1-coalition). We can now propose the following definition (d'Aspremont et al., 1983). Definition 3. A coalition formed by sk citizens is stable if it satisfies the conditions of internal stability (i.e. that a citizen member of the coalition has no incentive to leave it: L(sk) b 0) and external stability (i.e. that no citizen outside the coalition have any incentive to join the sk-coalition: − L(sk + 1) b 0). This concept of individual stability is based on the open membership rule introduced by d'Aspremont et al. (1983): each citizen gets the possibility to join or leave the coalition freely. The open membership rule must be separated from the exclusive membership rule one can find in some group stability concepts and where the membership of a coalition implies the consent of each member of the coalition (Hart and Kurz, 1983; Yi and Shin, 1995). In the case where ak ¼ snk , we obtain the following proposition.

Intermediary coalition formation Table 2 – Case: ak ¼ 1t ak ¼ 1t

Studying the stability of coalition is equivalent to determining intermediary coalition structures that would form at the first stage of the game. Then, an immediate problem is the choice 3

As Yi (1997) shows, these properties are valid for all coalition games with positive externalities provided that the agents are identical. This is no longer true in the case of heterogeneous agents.

Coalition structure [1234] [123,4] [12,34] [12,3,4] [1,2,3,4]

Individual utilities u1

u2

u3

u4

−0.56 −1.37 −0.56 −2.25 −0.56

− 0.56 − 1.37 − 0.56 − 2.25 − 0.56

− 0.56 − 1.37 − 0.56 1.12 − 0.56

−0.56 1.87 −0.56 1.12 −0.56

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E CO L O G I CA L E CO N O MI CS 68 ( 20 0 8 ) 6 8–7 9

Proposition 54. Given ak ¼ snk , the unique coalition structure satisfying internal and external stability is the singletons one. This result can be explained by reverting to the notion of free-riding. If the bargaining on transfers fails, the grand coalition {N} will not be formed and the efficient policy of emission reductions will not be implemented. Nevertheless, intermediary coalitions will remain formed since the past agreements are steady. Then, each coalition will set up a policy defined in accordance with the quasi-hybrid equilibrium terms. Anticipating this evolution of the game, citizens will tend to stay outside any coalition to get a credible threat, notably by sending the signal that every dissatisfaction would lead to an even worse situation than the quasi-hybrid equilibrium, namely to the Nash equilibrium. This threat is all the more credible as it provides incentives to citizens to take part in the definition of a transfer rule that can guarantee the forming of a global agreement during the bargaining stage. To sum up, having the choice between two status quo points, citizens will not favour the more advantageous one because it implies forming coalitions that provide positive externalities. The presence of these externalities would increase the incentives provided to isolated citizens to ruin the bargaining because of the highness of the benefits of the coalitions' efforts. Because of this free-riding, the only stable coalition structure is the singletons one. However, it is possible to limit this behaviour by according a significant advantage to big coalitions in comparison to smaller ones. To do so, it is sufficient to increase the bargaining power of the biggest coalitions, thereby inducing an increase of their members' utility while making that of singletons tend towards its status quo level. This is illustrated  2 by considering the following bargaining power ak ¼ snk . Each citizen's utility is then expressed as follows: 8iaSk oe

ui ðeÞ ¼

hs

k

n þ

1

i Xt j¼1

ð1 þ bÞ2 2ð1 þ bÞ2 þ sj sk

ð1 þ bÞ2 sk ð1  2tÞ : n2

ð26Þ

It is easy to see that in the case where b ¼ 12 and n = 4, the structure of singletons is no longer stable. In fact, the only stable structure is [12,34]. In other words, increasing the bargaining power of big coalitions in relation to small ones results in a higher number of stable coalitions. The existence of several stable coalitions of the same size implies the interchangeability of certain members of a stable coalition with other members on the fringe. Indeed, insofar as the citizens are identical, changing from one stable coalition to another of the same size only requires permuting members of the coalition with members of another coalition. The wider a coalition is, the stronger is its bargaining power. Nevertheless, cooperation between all citizens cannot constitute a stable situation. Indeed, the more citizens in the coalition, the greater the incentive for certain citizens to defect in an attempt to benefit from the reductions made by other citizens without suffering the disadvantages. Anticipating this

behaviour, the citizens have very little incentive to join a coalition, all the more so as the coordination of behaviours cause considerable reductions on the part of those cooperating. Consequently the maximum number of two citizens can be understood as the number of citizens which facilitates the sustainability of behaviour coordination without any incentive for defection being possible. The coalition is then selfenforcing.

4.

In this section, we remove the most restrictive assumption of our model so far, i.e. citizens homogeneity, to consider a more general and realistic framework of heterogeneous citizens. The differentiation will concern here the production of public and private goods. We consider that the first heterogeneity source is produced by the fact that some citizens have a private good “cleaner” than the others' (greener cars, for example). The second source of heterogeneity stems from allowing a differentiated use of the collective good by agents.

4.1.

For ak ¼ 1t , we obtain a similar result.

Model modification

We use the two-stage model, making the same assumptions as before, excepted that we now have: 1=mi

8iaN xi ¼ mi pi

where mi N 1;

ð27Þ

with: i dxi d2 xi 1  mi mi i m ¼ pi i N 0 and ¼ pi b 0: dpi mi dp2i 1m

12m

ð28Þ

The emission function is concave and increasing. The coefficient m1i represents the degree of homogeneity of the emission function. Since m1i b1 implies decreasing returns to scale, the larger the mi is, the less polluting the private good owned by citizen i: 1 m2 dxi dxi m ¼ pi i þ i N 0 and limpi Y0 ¼ 1: dmi lnpi dmi

ð29Þ

In the same way, we consider that citizens are characterised by heterogeneity in the production of emissions by the collective good. We will suppose that we have: 1=mi

yi ¼ bmi pi

4.2.

where mi N 1:

ð30Þ

The status quo point

Before determining the game solution, we have to characterise the status quo point in order to get a benchmark point during bargaining. As we have already underlined, it corresponds to the quasi-hybrid equilibrium of the coalitional game. The programme that must be solved is then: Max

4

Citizen heterogeneity

X iaSk

ðxi þ yi þ zÞ;

8Sk ae; 8iaSk ;

ð31Þ

75

EC O LO G I CA L E C O N O M I CS 6 8 ( 2 00 8 ) 6 8–7 9

On the other hand, for ε = {N}, we get:

Under the following constraints: X

X

xi V

iaSk

1=m mi pi i

ð32Þ

iaSk

X

X

yi V b

iaSk

1=mi

ð33Þ

mi pi

 mi 1 þ b mi 1 p ¼ ; iaN i iaN n  1 X X 1 þ b mi 1 x ¼ m ; xN4 ¼ i i iaN iaN n p4N ¼

iaSk

Xsk i¼1

p˜ sj ¼

pi þ z ¼ 

Xsj i V¼1;i Vpi

yN4 ¼

Xt j¼1

p˜ sj

where j p k

ð34Þ

p˜ i V:

ð35Þ

X

X

X

z4 ¼ 

y ¼b iaN i

X

mi iaN

ð42Þ

 1 1 þ b mi 1 ; n

ð43Þ

mi Xn 1 þ b mi 1 i¼1 n

UN4 ¼ ð1 þ bÞ

 1  mi 1 þ b mi 1 Xn Xn 1 þ b mi 1 mi  iaN i¼1 i¼1 n n

X

ð44Þ

That is: mi  mi Xsk Xsk 1 þ b mi 1 1 þ b mi 1 ˜ Z psk ¼ p ¼ ; pi ¼ i¼1 i i¼1 sk sk

Xsk

x˜ sk ¼

y˜ sk ¼

i¼1

xi ¼

m Ui4 ¼ Pn i

 1 1 þ b mi 1 mi i¼1 sk

y ¼b i¼1 i

Xt

Xsk

mi i¼1

p˜ ¼  k¼1 sk

U˜ sk ¼ ð1 þ bÞ

ð36Þ

Xsk

Xsk

z˜ ¼ 

Given the sharing rule we adopted, each citizen gets a utility level:

i¼1

 1 1 þ b mi 1 ; sk

ð37Þ

mi Xsk 1 þ b mi 1 : i¼1 k¼1 sk

Xt

Xsk

mi i¼1

mi  1 Xt Xsk 1 þ b mi 1 1 þ b mi 1 sk : k¼1 i¼1 sk sk

With differentiated citizens we cannot use the egalitarian rule to determine the repartition of gains within coalitions. It thus becomes harder to define the individual utilities of the members of a given coalition. Several rules have been developed in the literature, such as the Nash solution (1953) or the Shapley value (1953). In order to stay close enough to the previous model, we use a rule according to which the members of a coalition share its gains in function of each citizen's ability to use a less polluting private good: 8iaSk

iaSk

mi

U˜ Sk :

ð39Þ

From these results, we can compute the Nash equilibrium and the Pareto optimum in the presence of heterogeneous citizens. It comes down to characterising both the extreme cases of the finest partition {{1},…, {n}} and the grand coalition {N}. For ε = {{1},…, {n}}, we have: P

mi

1

P

P

1

pi ¼ ð1 þ bÞmi 1 ; xi ¼ mi ð1 þ bÞmi 1 ; yi ¼ bmi ð1 þ bÞmi 1 ;

P

P

mi

mi iaN

 1 1 þ b mi 1 n

ð45Þ

mi Xn Xn 1 þ b mi 1  : i¼1 i¼1 n

Bargaining

Proposition 6. The solution of the bargaining is such that, for a fixed coalition structure ε: h i Xt ˜ s ak þ U˜ U usk ðeÞ ¼ UN4  j sk j¼1

8k;

ð46Þ

and transfers among the coalitions of citizens take the following form: sk ¼ ð1 þ bÞ

Psk

mi i¼1

½

þak ð1 þ bÞ

ð1 þ bÞ

 mi  mi P P k 1 þ b mi  1 1 þ b mi  1 þðnak  sk Þ tk¼1 si¼1 sk sk

Pn

Pt k¼1

i¼1

mi

Psk

½

 1  mi 1 þ b mi  1 Pn Pn 1 þ b mi  1  i¼1 i¼1 n n

mi i¼1

 1 1 þ b mi  1 sk

Psk Xn 1 þ b mi  Pi¼1 ð1 þ bÞ i¼1 n n m i i¼1



1 mi  1



Xn Xn i¼1

mi i¼1



 mi 1 þ b mi  1 8k: n

ð47Þ We can note that the results presented in Proposition (6) are a generalisation of those we obtained in a homogeneous citizens framework. Indeed, setting mi = 2 leads back to the result of Proposition (3).

4.4.

ð41Þ

The theoretical treatment of stability with heterogeneous agents is made impossible by a particular difficulty. Compared to the previous model, every citizen permutations must now be studied. In order to simplify the calculations, we study the

mi

1

mi

X

ð40Þ

zi ¼ nð1 þ bÞmi 1 Ui ¼ mi ð1 þ bÞð1 þ bÞmi 1 nð1 þ bÞmi 1 :

i¼1

ð1 þ bÞ

To resolve the two-stage game, we adopt the backward induction method supposing that citizens have already formed intermediary coalitions. We can then focus on the bargaining about the repartition of full cooperation gains between the representatives of the intermediary coalitions. ð38Þ

m U˜ i ¼ P i

4.3.

mi

m UN4 ¼ Pn i

Stability

76

E CO L O G I CA L E CO N O MI CS 68 ( 20 0 8 ) 6 8–7 9

stability based on a 4-citizens example, using a bargaining power defined as: Psk mi : ak ¼ Pi¼1 n i¼1 mi

Table 4 Coalition structure

ð48Þ mi

Each citizen i utility then becomes: m ui ðeÞ ¼ Pn i i¼1

mi

½

ð1 þ bÞ

Xn i¼1

 mi

1 1 þ b mi  1 n

mi  1 Xn Xn 1 þ b mi  1 Xt Xsk 1 þ b mi  1 ð1 þ bÞ mi i¼1 i¼1 i¼1 k¼1 n sk mi Xt Xsk 1 þ b mi  1 þn k¼1 i¼1 sk " mi #  mi Xt Xsk 1 þ b mi  1 Xsk mi 1 þ b mi  1 þ Psk ð1 þ bÞ m s : i k k¼1 i¼1 i¼1 sk sk mi i¼1





ð49Þ We consider the case where N = {1,2,3,4}, β→0 and m1 ¼ 54 ; 7 9 11 ¼ m2 4 ; m3 ¼ 4 ; m4 ¼ 4 . In other terms, citizen 4 has the cleanest technology and citizen 1 has the dirtiest one. Table 3 presents the utility of each citizen according to the different conceivable coalition structures. It appears that all the coalition structures satisfy the external stability condition, but the singleton structure [1,2,3,4] is the only one to satisfy the internal stability as well. We cannot claim, based on our 4-citizens example, that no coalition structure can be stable but the singleton regardless of the number of citizens. Given its complexity, the model does not allow any generalisation. From this example, we can deduce some behavioural specificities from citizens. We can observe notably that

Table 3 Coalition structure mi

Individual utilities u1 m1 ¼

5 4

u2 m2 ¼

7 4

u3 m3 ¼

9 4

u4 m4 ¼

11 4

mi mi  1

m1 ¼5 m1  1

m2 7 ¼ m2  1 3

m3 9 ¼ m3  1 5

m4 11 ¼ 7 m4  1

1 mi  1

1 ¼4 m1  1

1 4 ¼ m2  1 3

1 4 ¼ m3  1 5

1 4 ¼ m4  1 7

[1234] [123,4] [124,3] [134,2] [234,1] [12,34] [13,24] [14,23] [12,3,4] [13,2,4] [14,2,3] [23,1,4] [24,1,3] [34,1,2] [1,2,3,4]

0.207 −0.221 −0.063 0.072 0.3 −0.261 0.008 0.198 −0.838 −0.428 −0.117 −0.411 −0.438 −0.487 −1.293

0.29 − 0.309 − 0.089 0.963 0.265 − 0.365 0.506 0.302 − 1.173 0.466 0.447 − 0.151 0.203 0.368 − 0.21

0.372 −0.398 1.616 0.13 0.341 0.877 0.015 0.389 1.278 −0.77 1.251 −0.194 1.239 0.649 0.872

0.455 2.252 − 0.14 0.159 0.417 1.072 0.795 0.435 2.057 2.057 − 0.257 2.079 0.32 0.794 1.955

Individual monetary transfers τ1 m1 ¼

5 4

τ2 m2 ¼

7 4

τ3 m3 ¼

9 4

τ4 m4 ¼

11 4

mi mi  1

m1 ¼5 m1  1

m2 7 ¼ m2  1 3

m3 9 ¼ m3  1 5

m4 11 ¼ 7 m4  1

1 mi  1

1 ¼4 m1  1

1 4 ¼ m2  1 3

1 4 ¼ m3  1 5

1 4 ¼ m4  1 7

[1234] [123,4] [124,3] [134,2] [234,1] [12,34] [13,24] [14,23] [12,3,4] [13,2,4] [14,2,3] [23,1,4] [24,1,3] [34,1,2] [1,2,3,4]

0 −0.428 −0.27 −0.135 0.094 −0.468 −0.199 −0.009 −1.045 −0.635 −0.324 −0.618 −0.645 −0.694 −1.5

0 −0.599 −0.378 0.673 −0.024 −0.655 0.216 0.013 −1.462 0.176 0.157 −0.44 −0.086 0.078 −0.5

0 − 0.77 1.244 − 0.242 − 0.031 0.505 − 0.357 0.016 0.905 − 1.143 0.879 − 0.566 0.867 0.277 0.5

0 1.797 −0.595 −0.296 −0.038 0.617 0.34 −0.02 1.602 1.601 −0.712 1.624 −0.135 0.338 1.5

those possessing a strongly polluting good get incentives to search alliances before negotiating, while those possessing a weakly polluting good do not. The reason is that, for the latter citizens the necessary effort in case of implementation of some pollution reduction policy is weaker. These citizens are thus in a strong position during the bargaining and their strategy of bargaining alone allows us to increase the pressure on transfers they will get from other citizens. This pressure will be all the more important as other citizens will regroup in coalition. Being alone at the bargaining stage is the more advantageous as individual players get higher gains (comparatively to those forming a coalition). Table 4 gives the individual monetary transfers between citizens in our four players game. We can notice that negotiating as an individual player is beneficial for all citizens except the one with the less advance (and therefore more polluting) technology. For this individual, negotiating alone is the worst strategy as soon as there are other singletons; in that case he should transfer funds to those other citizens. In our example, this is the case for citizen 1 for the following coalition structures: [23,1,4], [24,1,3], [34,1,2] and [1,2,3,4]. We can also notice that in a coalition (or a group of citizens) that transfer funds to the other, the citizen with the cleaner technology will always pay the more. That comes from share rules inside the coalition.

5.

Conclusion

In this paper we have proposed a decentralised mechanism of environmental quality management at a local scale which

77

EC O LO G I CA L E C O N O M I CS 6 8 ( 2 00 8 ) 6 8–7 9

corresponds to a possible form of participative democracy. Through a two-stage game, we showed that transfers in a global agreement depend not only on the capacity of a citizen to reduce pollutant emissions, but also on the formation of negotiating coalitions. This allows us to point out the importance of intermediary coalition formation in the repartition of cooperation gains. We have also clarified how transfers set up among citizens are dependent on multiple variables, both of environmental and economic nature, and on the bargaining power of coalitions that participate to the negotiation. This last variable is very important because it can lead to very different results in terms of stability of the coalition structure. Beyond these analytical results, the main contribution of this model is the importance it gives to a decentralised contractualism which leads to a Pareto-optimal allocation. This representation tends to offer a favourable argument to the local experiences of participative democracy set up in Quebec, in some Commonwealth countries and in some North American cities. The interest of this form of regulation lies in the redefinition of institutions it proposes. The role of institutions is no longer limited “to punish or reward” (as it is in the agency theory framework) but extended to the development and proposition of cooperation procedures to citizens. These procedures will not be controlled by anyone but citizens themselves as the onus is on them to play procedures while defending their own interests. These procedures have to be developed in such a way that citizens have an interest in cooperating, without deriving it from any sanction strategy.

Acknowledgements We would like to thank Frédéric Ghersi, Rabia Nessah and Antoine Saglio for their helpful comments on previous versions of this paper. The participants of the “Journées de l'AFSE 2004” (Rennes, May 2004) and the “First International Seminar on the Environment” (Bejaia, June 2005), together with three anonymous referees are also gratefully acknowledged.

Appendix A Proof of proposition (1). We have seen that, at the noncooperative Nash equilibrium, each citizen gets a satisfaction level equal to: U¯ i = (2 − n)(1 + β)2, while at the quasi-hybrid 2 2 P ˜ i ¼ 2ð1þbÞ  tj¼1 ð1þbÞ . equilibrium, we have: U sk sj To get the terms of Proposition (1), it is sufficient to show that: P

˜ i z Ui ; 8iaN U Which is equivalent to: n2z

Xt j¼1

Xt Xr1 1 1 Xt 1 2 1  f sk  2 z  þ : k¼1 j¼1 s j¼rþ1 s sj sk sk j j

Adding sk to each side, we get: Xt

s j¼1;j pk j

2z

Xr1 1 Xt 1 1 þ   sk : j¼1 s j¼kþ1 s sk j j

P P P Pk1 1 Pt 1 As: tj¼1 sj z tj¼1 s1j and tj¼1;j pk sj z j¼1 j¼kþ1 sj , we just sj þ have to check that: 2 z 

1  sk ; sk

that is: s1k þ sk z2, which is always satisfied since sk ≥ 1. Proof of proposition (2). Let us note that: " # Xt ð1 þ bÞ2 Xt X 2 2 ð 1 þ b Þ s Usk ¼ k k¼1 j¼1 sj 2 X t ð1 þ bÞ : ¼ 2tð1 þ bÞ2 n j¼1 sj Proposition (2) validity is then guarantied if: 2tð1 þ bÞ2 n

Xt j¼1

ð1 þ bÞ2 V ð1 þ bÞ2 ; sj

that is to say: Xt j¼1

1 2t  1 ; z sj n

ð1Þ

with: t b n, t N 1 and n N 2. Pt 1 We will show that if the minimum of j¼1 sj subject to Pt j¼1 sj ¼ n satisfies condition (1), then it is always satisfied. The Lagrangian associated to the minimisation programme can be written: L¼

Xt j¼1

hXt i 1 þk s n ; j¼1 j sj

It results that: AL 1 ¼ 2 þk¼0 Asi sj

ð2Þ

AL Xt ¼ s  n ¼ 0: j¼1 j Ak

ð3Þ

Consequently, the first order condition (2) implies: k ¼ ¼ N ¼ s12 , hence: s1 = … = st. Substituting this condition in (3), t we get: sj ¼ nt. The function's minimum is then given by: 1 s21

t

t t2 ¼ : n n

Since the objective function is convex and the constraint is linear, L is convex and the minimum determined by the Lagrangian corresponds to a global minimum. Let us demonstrate that this minimum satisfies condition (1): t2 2t  1 : N n n This inequality can be rewritten: (t − 1)2 N 0. As t N 1, the inequality is always satisfied. Proof of proposition (3). The solution of the bargaining, (uS1,…, uSt), is given by: t   ˜ st ak ; Max j Uk U k¼1

under the following constraints: Xt k¼1

Uk ¼ ð1 þ bÞ2 ¼ U4 N

and

Xt k¼1

ak ¼ 1;

78

E CO L O G I CA L E CO N O MI CS 68 ( 20 0 8 ) 6 8–7 9

where 0 b αk ≤ 1 denotes the bargaining power of coalition Sk. To ~ ~ determine the solution, we write: Vj = Uj − USj, with Uj ≠ USj. The programme can be rewritten: t

a

Max j Vj j

Xt

s:t:

j¼1

j¼1

Vj ¼ ð1 þ bÞ2 

Xt j¼1

U˜ sj ¼ D N 0:

The Lagrangian is written: Xt  t a V  D ; L ¼ j Vj j  k j j¼1 j¼1

Proof of proposition (6). The principle of this proof is the same as the one used for Proposition (3).

Consequently: t AL a ¼ ak Vkak 1 j Vj j  k ¼ 0 AVk jpk

Xt j¼1

Vj ¼ D:

ð1Þ

REFERENCES

ð2Þ

Aumann, R.J., 1961. The core of a cooperative game without side payments. Trans. of the AMS 98, 539–552. Aumann, R.J., 1967. A survey of cooperative games without side payments. In: Shubik, M. (Ed.), Essays in Mathematical Economics in Honor of O. Morgenstern. Princeton University Press, Princeton, pp. 3–27. Bloch, F., 1997. Noncooperative models of coalition formation in games with spillovers. In: Carraro, C., Siniscalco, D. (Eds.), New Directions in the Economic Theory of the Environment. Cambridge University press, Cambridge, pp. 311–352. d'Aspremont, C., Jacquemin, A., Gabszewicz, J.J., Weymark, J., 1983. On the stability of collusive price leadership. Can. J. Econ. 16, 17–25. De Carlo, L., 1996. Gestion de la Ville et Démocratie Locale. L'Harmattan, collection Villes et Entreprises, Paris. (284 pp.). Donsimoni, M.P., Economides, N.S., Polemarchakis, H.M., 1986. Stable cartels. Int. Econ. Rev. 27, 317–327. Dreier, P., 1996. Community empowerment strategies: the limits and potential of community organizing in urban neighborhoods. Cityscape 2, 121–159. Green, J., Laffont, J.-J., 1986. Incentive theory with data compression. In: Heller, W., Starr, R., Starrett, D. (Eds.), Uncertainty, Information and Communication (Essays in Honor of K.J. Arrow, vol. 3). Cambridge University Press, Cambridge, pp. 239–253. Green, J., Laffont, J.-J., 1987. Limited communication and incentive compatibility. In: Groves, T., Radner, R., Reiter, S. (Eds.), Information, Incentives, and Economic Mechanisms. University of Minnesota Press, Minneapolis, pp. 308–329. Greenberg, J., 1994. Coalition structures. In: Aumann, R., Hart, S. (Eds.), Handbook of Game Theory, vol. 2. North Holland, Amsterdam, pp. 1305–1337. Habermas, J., 1990. Moral Consciousness and Communicative Action. MIT Press, Cambridge, MA. (225 pp.). Hart, S., Kurz, M., 1983. Endogenous formation of coalitions. Econometrica 51, 1047–1067. Hart, S., Kurz, M., 1984. Stable coalition structures. In: Holler, M.J. (Ed.), Coalitions and Collective Action. Physica Verlag, Vienna, pp. 235–258. Kurz, M., 1988. Coalitional value. In: Roth, A. (Ed.), The Shapley Value: Essays in Honor of Lloyd Shapley. Cambridge University Press, Cambridge, pp. 155–173. McLaverty, P., 2002. Is public participation a good thing? In: McLaverty, P. (Ed.), Public Participation and Innovations in Community Governance. Ashgate Publishing, Burlington, pp. 185–197. Nash, J., 1953. Two-person cooperative games. Econometrica 21, 128–140. Parry, G., Day, N., Moyser, G., 1992. Political Participation and Democracy in Britain. Cambridge University Press, Cambridge. (527 pp.).

a

From (1), we have: ak jtj¼1 Vj j ¼ kVk , that is: X

t

k

a

ak j Vj j ¼ k j¼1

X k

Vk : a

Using (2), we get: jtj¼1 Vj j ¼ kD, and then (1) could be rewritten: Vk = αkD. This yields: h i X ˜ s ¼ a ð1 þ bÞ2  U˜ sj : usk  U k k It follows: h i X U˜ sj þ U˜ sk : usk ¼ ak ð1 þ bÞ2  P Þ As we know that: U˜ sk ¼ 2ð1 þ bÞ2 sk tj¼1 ð1þb sj , we get: 2

usk ¼ 2ð1 þ bÞ2 þð1 þ bÞ2 ð1  2tÞak þ ðnak ¼ sk Þ

Xt j¼1

ð1 þ bÞ2 : sj

Transfers among coalitions will implement in function of the joint utility, that is to say: UN⁎ = (1 + β)2. Yet, we have: us4k ¼

X iask

Ui4 ¼

X

ð1 þ bÞ2 sk ð1 þ bÞ2 ¼ ; iask n n

then we get the relation: u⁎sk+ τ = usk, that is: 2 2 s ¼ usk  u4 sk ¼ 2ð1 þ bÞ þð1 þ bÞ ð1  2tÞak

þðnak  sk Þ

Xt j¼1

This difference is always positive when n N 2 and sk N 2. In other terms, the internal stability condition is satisfied only for sk b 2: a citizen i always gets incentives to leave a skcoalition he is member of. With this reasoning valid for all i of Sk, the result is that no coalition gathering more than one member will appear. The development of calculations associated to the external stability condition allows us to confirm results following from the internal stability. The only stable coalition structure is the finest partition {{1},…, {n}}.

ð1 þ bÞ2 sk ð1 þ bÞ2 :  sj n

Proof of proposition (5). We have seen that:   2 1 U˜ i;sk ¼ ð1 þ bÞ2 þ ð1  2tÞ sk n Now, if citizen i decides to leave the sk-coalition in order to act as a singleton, his utility is:   1 U˜ i;sk 1 ¼ ð1 þ bÞ2 2  ð1 þ 2tÞ : n The difference gives us:   ˜ i;s ¼ ð1 þ bÞ2 2ðsk  1Þ  2 : U˜ i;sk 1  U k sk n

EC O LO G I CA L E C O N O M I CS 6 8 ( 2 00 8 ) 6 8–7 9

Shapley, L., 1953. A value for N-person game. In: Kuhn, H.W., Tucker, A.W. (Eds.), Contribution to the Theory of Games, vol. II. Princeton University Press, Princeton, pp. 307–317. Smock, K., 1997. Comprehensive community initiatives: a new generation of urban revitalization strategies. The On-Line Conference On Community Organizing and Development Working Papers, http://comm-org.utoledo.edu. Yi, S., 1997. Stable coalition structures with externalities. Games Econ. Behav. 20, 201–237.

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Yi, S., Shin, H., 1995. Endogenous formation of coalitions in oligopoly. Working Paper 95-2, Dartmouth College, Department of Economics. Zhao, J., 1991. The equilibria of multiple objective games. Int. J. Game Theory 20, 171–182. Zhao, J., 1992. The hybrid solution of a n-person game. Games Econ. Behav. 4, 145–160.