Participative democracy and local environmental issues

E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 8 ) 6 8 – 7 9

ava i l ab l e a t www.sc i enced i r ec t . com

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ANALYSIS

Participative democracy and local environmental issues

Emmanuel Martineza, Tarik Tazdaïtb,⁎, Elisabeth Tovarb,c

aEPE, 41 rue des Trois Fontanot, 92024 Nanterre cedex, FrancebCIRED-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, FrancecCentre d'Etudes des Politiques Economiques de l'Université d'Evry (EPEE), France

A R T I C L E I N F O

⁎ Corresponding author.E-mail address: [email protected] (T.

0921-8009/$ – see front matter © 2008 Elsevidoi:10.1016/j.ecolecon.2008.01.025

A B S T R A C T

Article history:Received 14 November 2006Received in revised form6 December 2007Accepted 28 January 2008Available online 5 March 2008

The present paper deals with themanagement of environmental quality at the level of a city.More precisely, we propose two different contractualisms among citizens that allow them toreduce thepolluting emissions caused by their consumptionof both aprivate and a collectivegood. In the first contractualism, themayor proposes to citizens to establish neighbourhoodcommittees. They have the responsibility of defining the total amount of polluting emissionby private and public transportation means in their neighbourhood. It comes out that themediation of the neighbourhood committees brings out a lower total pollution than in thecase where each individual citizen acts on his own interest. However the emission levelbrought out by the neighbourhood committees is not Pareto-optimal. On the other hand,extending the first coalitional game by a supplementary stage focused on bargainingbetween the neighbourhood committees, allows us to define a participative democracyprocedure. This two-stage procedure brings about a Pareto-optimal pollution level.

© 2008 Elsevier B.V. All rights reserved.

Keywords:Pollutant emissionsParticipative democracyCoalitionsBargainingTransfer

JEL classification:C72; H41; Q53

1. Introduction

The aim of this article is to propose a theoretical framework tostudy the localmanagementof apollutionproblem.Becauseof itsdecentralized nature, ourmodel is related to the recent literatureon local participative democracy. This literature hinges on twoaxes. On the one hand, it tries to build a common analyticalframework that accounts for new forms of urban governance —among others, those prompted by the inadequacies of institu-tional answers to urban growth (De Carlo, 1996). On the otherhand, it tries to integrate these new forms of participativedemocracy, in which citizens voluntarily take part in theformalisation 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

Tazdaït).

er B.V. All rights reserved

(McLaverty, 2002). These contemporary works are in line withHabermas (1990), who opposes procedural democracy to institu-tional democracy. Basedonaprocedural definitionof democracy,a decision is said fair if each opinion is expressed by identicalmeans and if the only factor influencing the vote is the quality ofarguments, independently of the institutionalmode of producingthe decision. In ourmodel, we allow citizens to be involved in theprocesses of decision making through neighbourhood commit-tees, and therefore we define participative democracy as theexpression of neighbourhood committees.

Neighbourhood committees refer to decentralized initiativesin local development (community empowerment) that spreadduring the last decade, in urban zones of the Commonwealthcountries but also in the rural areas of developing countries.Community empowerment can be defined as the decentralized

.

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http://dx.doi.org/10.1016/j.ecolecon.2008.01.025

69E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 8 ) 6 8 – 7 9

mobilization of actors coming from a disadvantaged area (Dreier,1996). The main characteristic of this new phenomenon is theinversion of the usual governance direction: under communityempowerment, initiatives and decisions are bottom up (fromdecentralizedagent to local authorities) rather than topdown likein the usual structures of public decision. The role of publicinstitutions is limited tomediation and, at the end of the process,to a financial support generally in complement to funds raised ina decentralized way by the community members. In NorthAmerican cities, Smock (1997) notices that the emergence of adecentralized cooperation among the inhabitants of disadvan-taged urban areas has become essential because of the decreasein federal subventions to local development in the80's, andof theprivatisation of local public services.

We provide with a theoretical treatment based on coalitionformation theory. Furthermore, our representation of partici-pative democracy implies a redefinition of the role of publicauthorities, which do not intervene anymore as an activeplayer in the game. This takes us away from the representa-tion predominating in public economics, according to whichthe public authorities determine the instruments (taxes orothers) that are likely to dissuade free-rider behaviours. In ourapproach, the public authorities do not resort to any regula-tion instrument. The principle of their intervention consists inproposing cooperation procedures played only by those firstinvolved in the reduction in pollution, i.e. city citizens. Ourmodel also moves away from the mechanism design theory(Green and Laffont, 1986, 1987) according to the designer'sobjective is to lead the agents to truthfully reveal their pre-ferences through the use of various incentives taking the formof rewards or sanctions. Our conception of participative de-mocracy does not refer to any such incentives.

In our model, we determine a procedure that could beproposed by some public authority (in our example a mayor)and which allows the setting up of neighbourhood committees.With those neighbourhood committees, citizens have thepossibility to reduce their pollutant emissions in an efficientway.Theseemissionsarecausedby theuseofaprivate individualgood (a car) and of a collective good (public transportation, whichis assumed to be less polluting per passenger). Since theprocedure of cooperation we propose is set up in a decentralizedframework, our methodology is the endogenous formation ofcoalitions.1 Thismethodologywas developed to understand howcooperation can happen under the threat of deviation of eachagent or by a sub-set of them associated within a coalition. Thestability of an agreement is then a function of its ability to avoidsuchbehaviour. This is anotherway towonderwhycoalitions areforming (Kurz, 1988). It implies the identification of the coalitionstructures for which players do not want to modify it.

As noted by Greenberg (1994), the most used stabilityconcept for games without externalities is the core of acoalition structure. A coalition structure is said to be corestablewhenno group of players could obtain a higher paymentby forming another coalition. Unfortunately, this concept isnot easily generalised to games with externalities. To berelevant, the analysis of a coalition game has to be able toincorporate the reaction of the coalitionwhich is confronted toa deviation. Several games have been developed to study these

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

configurations, inducing the definition of several stabilityconcepts. Aumann (1961, 1967) introduced stability conceptsbased on the notion of core: the α-stability and the β-stability,while Hart and Kurz (1983, 1984) proposed stability conceptsbased on the strong Nash equilibrium notion: the γ-stabilityand the δ-stability. Beside these group stability concepts, anindividual stability concept was defined by d'Aspremont et al.(1983) and Donsimoni et al. (1986). Using this individualstability concept and starting from a standard framework ofpublic economics, we show that, under specific conditions, it ispossible to reach a global agreement. Such an agreementrequires a stage based on the strategic analysis of negotiations.

We have considered two coalitional games, a static one anda two-stage one. In the static framework, the mayor proposesto citizens to establish neighbourhood committees. They havethe responsibility of defining the total amount of pollutingemission by private and public transportation means in theirneighbourhood. It comes out that the mediation of theneighbourhood committees brings out a lower total pollutionthan in the case where each individual citizen acts on his owninterest. However the emission level brought out by theneighbourhood committees is not Pareto-optimal. On theother hand, extending the first coalitional game by a sup-plementary stage focused on bargaining between the neigh-bourhood committees, allows us to define a participativedemocracy procedure. This two-stage procedure brings abouta Pareto-optimal pollution level. Its essential feature is to reston each citizen's ability to join or leave freely the createdagreement: that is what one usually calls the open member-ship rule. In our model, everything happens as if the publicauthority gives itself the objective of determining cooperationprocedures with the view to proposing them to agents. Playingthe games that describe the procedure, the agents are led toform a coalition to defend their interests. The difficulty is thento make sure that the coalitional game results on a burdensharing corresponding to a Pareto-optimal situation.

The paper is organised as follows. Section 2 presents theassumptions of our model dealing with the management ofpollutant emissions produced by each citizen (through the useof a collective and a private good). This allows us to define acoalitional game able to induce citizens to improve theirsituation compared to a non-cooperative context (i.e. Nashequilibrium). As this game does not allow to reach optimalallocations, Section 3 introduces a two-stage game that leadsto an optimal burden sharing of emission reductions, providedsome transfers between agents determined by the bargainingpowers and number of members of each coalition. Section 4generalizes the obtained results by widening the analyticalframework to heterogeneous agents. Section 5 concludes witha presentation of the different highlighted results. Note thatall the proofs of the different propositions are relegated toAppendix A.

2. First procedure: a coalitional game

2.1. Payment

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

70 E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 8 ) 6 8 – 7 9

interact in a common environment threatened by a pollutantgas emitted by individual cars and the public transportationsystem.

The economy is composed of the following commodities:

– A private good, the use of a car, denoted by x=(x1,…, xn)N0,where xi represents the level of car usage by citizen i.Assuming x≠0 forbids the absence of any car usage in theeconomy;

– a collective good, the use of public transportation, denotedby y=(y1,…, yn)N0, where yi represents the use of publictransportation by citizen i;

– pollutant emissions, denoted p=(p1,…, pn)N0, where pi isthe level emitted by citizen i. We consider that p cannot beequal to zero, a reasonable assumption formost pollutions;

– an environmental characteristic that is assumed measur-able, zb0. z is considered a public bad, climate change inour example.

We assume that each citizen's preferences are described bya linear utility function, which satisfies2:

8iaN Ui xi; yi; zð Þ ¼ xi þ yi þ z: ð1ÞThe use of his/her own private good by citizen i inducespollutant emissions:

xi ¼ 2p1=2i : ð2Þ

The use of the public good by citizen i induces pollutantemissions:

yi ¼ 2bp1=2i with 0 b b b 1: ð3Þ

Eqs. (2) and (3) mean that emissions resulting from the useof cars are higher than those from the use of public transport.Quantities pi are considered both as inputs for the transporta-tion service and polluting emissions. Consequently, the use ofone unit of input leads to one unit of pollution.

We will admit that the ecological transfer function isdetermined linearlyandadditivelybypollutantemissions, that is:

z ¼ �Xn

i¼1pi: ð4Þ

Tobemore rigorous,weshouldconsider anecological transferfunction that takes into account the emissions of the consideredcity but also those of all other cities. However, with our focus on alocal scale we represent the behaviour of agents from other citiesthrough a constant parameter, set equal to zero for simplicity'ssake. That is whywe have simplified the writing of relation (4) tothe only emissions emitted by the citizen of the studied city.

2.2. The coalitional game

Let us consider the procedure bywhich a local public authority(in our example themayor) establishes a form of decentralized

2 We know that the utility when taking a car should be higherthan taking public transportation and then we must have thefollowing formulation: Ui(xi,yi,z)=xi+ayi+z where 0bab1. As theparameter “a” does not change the results of the model, we keptformula given by Eq. (1).

participative democracy by allowing the setting up of neigh-bourhood committees. As a consequence, within each districtof the city, coalitions are forming to implement, at their ownscale, the strategy guaranteeing the best management of theenvironmental quality, in a local context.

The players of our game are then coalitions of citizens, whodetermine their pollutant emission volumes. Because playerswithin each coalition only maximise their joint payoff orchoose one of its efficient outcomes, we use the concept ofquasi-hybrid equilibrium (Zhao, 1991, 1992), which assumesthat each coalition of the partition behaves collectively bymaximising its joint payoff given other players' choices.Everything happens as if citizens belonging to a coalitiondelegate their decisions to all coalition members in order tomaximise the joint utility of the coalition, given the actions ofthe other coalitions. The cooperation inside any coalition isshaped 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 structureis a partition ε={S1,…, St} of N, which is a collection of coa-litions satisfying [t

k¼1Sk ¼ N and Sm \ Sr ¼ F for all m and rsuch as m≠r. Consequently, a coalition structure ε divides upthe set of players among coalitions and/or singletons, in sucha way that a citizen can belong only to one coalition (orsingleton). The cardinal of a coalition Sk is: |Sk|=sk, with k=1,…,t and

Ptk¼1 sk ¼ n.

Definition 1. A quasi-hybrid equilibrium for each partition ε={S1,…, St} is a vector p~={ p~1,…, p~n} such that for each coalition Skaε,its emissions p~s={ p

~i / ia Sk} solves:

MaxXiaSk

xi þ yi þ zð Þ; ð5Þ

Subject to:XiaSk

xi V 2XiaSk

p1=2i ð6Þ

XiaSk

yi V 2bXiaSk

p1=2i ð7Þ

Xski¼1

pi þ z ¼ �Xt

j¼1psj where j p k ð8Þ

psj ¼Xsk

iV¼1;iVpipiV: ð9Þ

The resolution of the programme associated with Defini-tion (1) gives:

xSk ¼ 2 1þ bð Þ; ySk ¼Xsk

i¼1yi ¼ 2b 1þ bð Þ; psk ¼

1þ bð Þ2sk

z ¼ �Xt

k¼1

1þ bð Þ2sk

;

ð10Þ

and:

USk ¼ 2 1þ bð Þ2�skXt

j¼1

1þ bð Þ2sj

: ð11Þ

As already underlined, in this game each coalition max-imises its joint utility given the others' best response. As aconsequence, two properties characterise this game: it is bothnon-cooperative, as each coalitionactsnon-cooperativelywith


the other coalitions, and cooperative as the allocation decidedby each coalition results from a cooperative process withineach of them.

2.3. Comparison of two polar cases

By supposing that each citizen is acting rationally, we candetermine the behaviour of each of them as the best responseto the other choices. This leads to a non-cooperative Nashequilibrium characterized by the finest partition {{1},…, {n}},which is obtained by setting sk=sj=1 and t=n in the previousresults.

Then, at the Nash equilibrium we get for iaN:

P

xi ¼ 2 1þ bð Þ; P

yi ¼ 2b 1þ bð Þ; P

pi ¼ 1þ bð Þ2;P

z ¼ −n 1þ bð Þ2; P

Ui ¼ 2� nð Þ 1þ bð Þ2:ð12Þ

Since the citizens are identical, they obviously emit thesame quantity of pollutant and receive the same individualutility. Nevertheless, this utility depends on the number ofcitizens included in the model, as does the quantity of am-bient pollutant.

Likewise, to get the Pareto-efficient state (or the full coop-eration) of the economy, when the coalition structure is givenby {N}, we just have to consider sk=sj=n. This results in:

pN4 ¼ 1þ bð Þ2n

; xN4 ¼ 2 1þ bð Þ; yN4 ¼ 2b 1þ bð Þ;zN4 ¼ � 1þ bð Þ2; UN

4 ¼ 1þ bð Þ2:ð13Þ

As all citizens are considered identical, determining theallocation associated to each of them amounts to adopting theegalitarian sharing rule:

pi4 ¼ 1þ bð Þ2n2 ; xi4 ¼ 2 1þ bð Þ

n; yi4 ¼ 2b 1þ bð Þ2

n;

z4 ¼ � 1þ bð Þ2n

; Ui4 ¼ 1þ bð Þ2

n:

ð14Þ

Unsurprisingly, the results associated to the Nash equili-brium do not correspond to those of the full cooperation:despite the environmental amelioration that benefits to allcitizens, everyone has interest in letting the others make aneffort 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-hybridequilibrium with the non-cooperative Nash equilibrium andthe full cooperation. From the first comparison, it appears thatthe implementation of group behaviours where divergencesare expressed at the coalition level rather than at theindividual one, contributes to an improvement of the envir-onmental quality.

Proposition 1. For the considered economy, a quasi-hybrid equili-brium is Pareto-improving compared to a non-cooperative Nashequilibrium, whatever the coalition structure formed.

Consequently, while remaining in a self-interest logic, it ispossible to envisage (via the interaction of citizen coalitions)an emission level that benefits to everyone as it provides animprovement compared with the situation where each agentdeclares for himself. The self-interest logic is now expressed

in coalitional rather than in individual terms. Favouring theformation of local coalitions contributes to improving theenvironmental quality.

Nevertheless, this coalition interest approach in a staticframework is not sufficient to reach a Pareto-optimal situa-tion, as established in Proposition (2).

Proposition 2. For the economy, full cooperation offers an utilitygain with respect to any quasi-hybrid equilibrium.

Finally, the results of Propositions (1) and (2) can be sum-marised by the following relation:

Xn

i¼1

P

Ui VXt

i¼1Ui V

Xn

i¼1Ui4: ð15Þ

The second part of this inequality i:e:Pt

i¼1 UiVPn

i¼1 Ui4

� �means that the sum obtained by playing in a coalitional way islower than or equal to the one that could be obtained, wherea single coalition gathering all citizens to emerge. In otherterms, there exists a gain to total cooperation that could beshared (in a redistributive way) in order to incite coalitions toagree on a unanimous agreement. If the inequality is satisfied,a larger agreement is then conceivable.

3. Second procedure: a two-stage game

We focus here on the definition of a game that allows for anagreement of all citizens on a management programme. Thelocal public authority now proposes a cooperation procedurebased on a two-stage game. During the first stage, citizens ofeach area form coalitions. Then, during a second stage,representatives of each area's coalition bargain to determinetheir respective effort levels (in terms of pollutant emissions),i.e. a repartition rule of the transfers that must be allocated toeach coalition. The procedure corresponds in a certain way toa form of participative, decentralised and dynamic democracy,akin to the community empowerment that is developing insome Anglo-Saxon cities.

Since we have a two-stage game with complete informa-tion, the equilibrium corresponds to a situation where:

– The defined transfer rule satisfies the Nash bargainingsolution of the second stage game.– The intermediary coalitions that are forming in the firststep must be stable.

Consequently, the model resolution is obtained by back-ward induction, starting with determining the optimal trans-fer rule in the second period. At the first period, every citizenchooses to join or to stay out of a coalition in order tomaximise his/her utility, taking as given the behaviour ofothers.

3.1. The bargaining

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


given by (U~S1,…, U

~St). This point, which corresponds to the

situation where the coalitions refuse any compromise, is suchthat each citizen prefers to belong to a coalition to defend his/her interests better. It thus coincides with the quasi-hybridequilibrium: if no solution emerges from the bargainingbetween the representatives of coalitions, then each coalitionreceives the amount it would get at the quasi-hybridequilibrium.

Definition 2. Given the set Ω={(U1,…, Ut) /U1+…+Ut≤UN⁎ } andthe status quo point, the bargaining rule ϖ assigns to each (t+1)-uplet (Ω,U

~S1,…, U

~St) a bargaining solution:

ϖ X; US1 ; N ; USt

� �¼ uS1 ; N ;uStð Þ ð16Þ

The components of the vector (uS1,…, uSt) represent the

utilities that the coalitions get if the bargaining ends with anagreement, given that UN4 ¼ Pt

k¼1 usk ¼ 1þ bð Þ2, and knowingthat UN⁎ designates the utility of the coalition that gathers allcitizens.

Proposition 3. The bargaining solution is such that, for a fixedcoalitions structure ε={S1,…, St},

usk eð Þ ¼ 1þ bð Þ2�Xt

j¼1Usj

h iak þ Usk 8k ð17Þ

and the transfers τ among the coalitions of citizens take the followingform:

s ¼ 1þ bð Þ2 2� skn

� �þ 1� 2tð Þ 1þ bð Þ2ak

þ nak � skð Þ 1þ bð Þ2X

j

1sj

8k:

ð18Þ

Following the bargaining, each coalition gets its status quoamount and a share of the gains created from the fullcooperation. This supplementary share is weighted by thebargaining power αk. The transfers between coalitions arethen related to the surplus resulting from the full cooperation,and their repartition depends upon two parameters, which arethe coalitions bargaining power αk, and the size of thecoalitions, sk. These parameters act in opposite directions.While the bargaining power increases the transfer, thenumber of citizen forming the coalition decreases it. Never-theless, to understand precisely the simultaneous impact ofthese two effects, it is necessary to combine them. That is whywe propose to study, in the next section, the relationshipbetween them.

3.2. Discussion about the bargaining power

In order to illustrate the influence of bargaining powers, weconsider two cases: a first onewhere the bargaining power of acoalition is linked to the number of citizens it gathers, that isak ¼ sk

n , and a second one where any coalition has the samebargaining power, that is ak ¼ 1

t.For the first case, we have:

ak ¼ skn

withXt

k¼1ak ¼ 1: ð19Þ

Transfers can be rewritten:

s ¼ 2 1þ bð Þ2þ 1þ bð Þ2 1� 2tð Þ sknþ n

skn� sk

� �Xt

j¼1

1þ bð Þ2sj

� sk 1þ bð Þ2n

¼ 2 1þ bð Þ2 1� tskn

� �; ð20Þ

and:

8iaSkoe; ui eð Þ ¼ 1þ bð Þ2 2sk

þ 1n

1� 2tð Þ� �

; 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 bnt: ð22Þ

2-A coalition pays transfers to other coalitions if its number ofcitizens is greater than the average coalition size:

s b 0fsk Nnt: ð23Þ

In this case, a coalition with high bargaining power is acoalition composed of a high number of members. As shownby Proposition (4), big coalitions pay transfers to little ones.When it is composed of a high number ofmembers, a coalitiongets a strong bargaining power as it will get a significantsurplus gain, then confirming the positive role of the bargain-ing power. However, the bigger the coalition is, the shakier isits position. In other terms, the weaker is the gap with thePareto optimum, the more dependent is the coalition towardcitizens staying outside. Actually, the incentive of citizensoutside the coalition is all the more strong since, because oftheir free-rider position, they get important gains generated bypositive externalities created by the cooperating coalition.Therefore, the number of members forming the coalition actsnegatively: by creating positive externalities, the coalitiondecreases the incentive to cooperate for outsiders. Then, toreach a complete agreement, the obtained surplus must beallocated to external members, to compensate the loss theysuffer compared to the situation where they benefit from thepositive externalities.

Let us now consider the second case in which all thecoalitions have the same bargaining power. We obtain:

s ¼ 1þ bð Þ2 n� tskð Þnt

þ 1þ bð Þ2 n� tskð Þt

Xt

j¼1

1sj; ð24Þ

and:

8iaSkoe; ui eð Þ ¼ 1þ bð Þ2 n� tsktsk

� �Xt

j¼1

1sj� 1þ bð Þ2

tsk: ð25Þ

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

Table 1 – Case: ak ¼ Skn

Coalitionstructure

Individual utilities

u1 u2 u3 u4

[1234] 0.56 0.56 0.56 0.56[123,4] −0.19 −0.19 −0.19 2.81[12,34] 0.56 0.56 0.56 0.56[12,3,4] −0.56 −0.56 1.69 1.69[1,2,3,4] 0.56 0.56 0.56 0.56

Table 2 – Case: ak ¼ 1tak ¼ 1

t

Coalitionstructure


u1 u2 u3 u4


irrespective of their sizes amounts to entitling each coalitionto only one vote, which reduces their influence insofar as theyare thus on an equal footing with singletons.

Tables 1 and 2 illustrate this difference. Table 1 representsthe individual utilities when ak ¼ sk

n , n=4 and β=0.5 whileTable 2 gives the individual utilities when ak ¼ 1

t, n=4 andβ=0.5.

Notation [12,3,4] means that players 1 and 2 form acoalition while players 3 and 4 stay in a singleton position.Columns u1,…, u4 represent the respective utilities of each ofthe four players according to the given coalition structures.

Despite their differences in gains, the two tables haveseveral properties in common. We may indeed note that inboth cases: (P1) when a coalition enlarges with the arrival ofanother member, the citizen who remains outside of thecoalition so formed finds himself in a better position; (P2)whatever the coalition structure considered, singleton playershave a higher gain than those who are members of a coalition;(P3) when a player leaves a coalition to join another of thesame size, the player who remains alone sees his gainincrease; (P4) when a player leaves a coalition to join anotherof the same size, he sees his gain decrease in comparison withhis 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 thecoalition {12} is joined by {3}, leading to coalition structureε'=[123,4], player 4 sees his gain increase.– (P2) In the coalition structure ε=[12,3,4]: U3(ε)=U4(ε)NU1,2(ε).– (P3) In the coalition structure ε=[12,34], when player 3joins the coalition {12}, then player 4 obtains a higher gain.– (P4) In the coalition structure ε=[12,34], when player 3joins the coalition {12}, then player 3 obtains a lower gain.

More generally, theses different properties confirm the ideathat the free-riding behaviour is fairly dominant whichhampers the formation of a stable coalition.

3.3. Intermediary coalition formation

Studying the stability of coalition is equivalent to determiningintermediary coalition structures that would form at the firststage of the game. Then, an immediate problem is the choice

3 As Yi (1997) shows, these properties are valid for all coalitiongames with positive externalities provided that the agents areidentical. This is no longer true in the case of heterogeneousagents.

of the stability concept to be used. Several definitions ofstability have been developed in the literature depending onthe stability concept discussed: individual or group stability.

The group stability concepts have the particularity toprotect the coalition formed by all citizens against bothindividual and group deviations. Yet, to prevent these devia-tions from happening (particularly group deviations) groupstability implies assuming that the benchmark situation isthat of a coalition gathering all citizens formed under thecollective interest logic. Only a central planner (e.g. the mayorof a city), as the legitimate depositary of citizen sovereignty, isin a position to defend the common interest of all communitymembers. However, in our model, once the public authorityhas proposed the procedure, it is the citizens who play thegame and the formation of a coalition is not the result ofresearching collective interest but of citizens acting in theirown interests (i.e. comparing the gain they get whether theyjoin the coalition or not). Furthermore, some stability conceptsare based on the potential resort to deter deviations. This isnot compatible with a participative democracy framework,where choices are done from consent, not from obligation.That is why we retain the individual stability concept. Thisimplies the introduction of supplementary notions.

Let U~i,sk be the utility obtained by citizen i when he

decides to join the coalition formed by sk−1 citizens andU~− i,sk his utility when he stays out of the coalition formed by

sk cooperating citizens. Let L(sk) denote the function givingthe citizen's incentive to defect from a coalition formed bysk citizens. It follows that L(sk)=U

~− i,sk−1−U

~i,sk. Moreover, let

−L(sk+1)=U~i,sk+ 1−U

~− i,sk be the incentive for a non-coopera-

tive citizen to join a sk-coalition (that therefore becomes ask+1-coalition). We can now propose the following defini-tion (d'Aspremont et al., 1983).

Definition 3. A coalition formed by sk citizens is stable if it satisfiesthe conditions of internal stability (i.e. that a citizen member of thecoalition has no incentive to leave it: L(sk)b0) and external stability(i.e. that no citizen outside the coalition have any incentive to join thesk-coalition: −L(sk+1)b0).

This concept of individual stability is based on the openmembership rule introduced by d'Aspremont et al. (1983):each citizen gets the possibility to join or leave the coalitionfreely. The openmembership rule must be separated from theexclusive membership rule one can find in some groupstability concepts and where the membership of a coalitionimplies the consent of each member of the coalition (Hart andKurz, 1983; Yi and Shin, 1995). In the case where ak ¼ sk

n , weobtain the following proposition.

[1234] −0.56 −0.56 −0.56 −0.56[123,4] −1.37 −1.37 −1.37 1.87[12,34] −0.56 −0.56 −0.56 −0.56[12,3,4] −2.25 −2.25 1.12 1.12[1,2,3,4] −0.56 −0.56 −0.56 −0.56


Proposition 54. Given ak ¼ skn , the unique coalition structure

satisfying internal and external stability is the singletons one.

This result can be explained by reverting to the notion offree-riding. If the bargaining on transfers fails, the grandcoalition {N} will not be formed and the efficient policy ofemission reductions will not be implemented. Nevertheless,intermediary coalitions will remain formed since the pastagreements are steady. Then, each coalition will set up apolicy defined in accordance with the quasi-hybrid equili-brium terms. Anticipating this evolution of the game, citizenswill tend to stay outside any coalition to get a credible threat,notably by sending the signal that every dissatisfaction wouldlead to an even worse situation than the quasi-hybrid equi-librium, namely to the Nash equilibrium. This threat is all themore credible as it provides incentives to citizens to take partin the definition of a transfer rule that can guarantee theforming of a global agreement during the bargaining stage. Tosum up, having the choice between two status quo points,citizens will not favour the more advantageous one because itimplies forming coalitions that provide positive externalities.The presence of these externalities would increase the in-centives provided to isolated citizens to ruin the bargainingbecause of the highness of the benefits of the coalitions'efforts. Because of this free-riding, the only stable coalitionstructure is the singletons one.

However, it is possible to limit this behaviour by accordinga significant advantage to big coalitions in comparison tosmaller ones. To do so, it is sufficient to increase the bar-gaining power of the biggest coalitions, thereby inducing anincrease of their members' utility while making that ofsingletons tend towards its status quo level. This is illustratedby considering the following bargaining power ak ¼ sk

n

� �2. Eachcitizen's utility is then expressed as follows:

8iaSkoe ui eð Þ ¼ skn� 1

h iXt

j¼1

1þ bð Þ2sj

þ 2 1þ bð Þ2sk

þ 1þ bð Þ2sk 1� 2tð Þn2 : 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 onlystable structure is [12,34]. In other words, increasing thebargaining power of big coalitions in relation to small onesresults in a higher number of stable coalitions. The existenceof several stable coalitions of the same size implies theinterchangeability of certain members of a stable coalitionwith other members on the fringe. Indeed, insofar as thecitizens are identical, changing from one stable coalition toanother of the same size only requires permuting members ofthe coalition with members of another coalition.

The wider a coalition is, the stronger is its bargainingpower. Nevertheless, cooperation between all citizens cannotconstitute a stable situation. Indeed, the more citizens in thecoalition, the greater the incentive for certain citizens to defectin an attempt to benefit from the reductions made by othercitizenswithout suffering the disadvantages. Anticipating this

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

behaviour, the citizens have very little incentive to join acoalition, all the more so as the coordination of behaviourscause considerable reductions on the part of those cooperat-ing. Consequently the maximum number of two citizens canbe understood as the number of citizens which facilitates thesustainability of behaviour coordination without any incen-tive for defection being possible. The coalition is then self-enforcing.

4. Citizen heterogeneity

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

4.1. Model modification

We use the two-stage model, making the same assumptionsas before, excepted that we now have:

8iaN xi ¼ mip1=mii where mi N 1; ð27Þ

with:

dxidpi

¼ p1�mimii N 0 and

d2xidp2i

¼ 1�mi

mip

1�2mimi

i b 0: ð28Þ

The emission function is concave and increasing.The coefficient 1

mirepresents the degree of homogeneity of

the emission function. Since 1mib1 implies decreasing returns

to scale, the larger themi is, the less polluting the private goodowned by citizen i:

dxidmi

¼ p1mii þ m2

ilnpi

N 0 and limpiY0dxidmi

¼ 1: ð29Þ

In the same way, we consider that citizens are charac-terised by heterogeneity in the production of emissions by thecollective good. We will suppose that we have:

yi ¼ bmip1=mii where mi N 1: ð30Þ

4.2. The status quo point

Before determining the game solution, we have to characterisethe status quo point in order to get a benchmark point duringbargaining. As we have already underlined, it correspondsto the quasi-hybrid equilibrium of the coalitional game. Theprogramme that must be solved is then:

MaxXiaSk

xi þ yi þ zð Þ; 8Skae;8iaSk; ð31Þ


Under the following constraints:

XiaSk

xi VXiaSk

mip1=mii ð32Þ

XiaSk

yi V bXiaSk

mip1=mii ð33Þ

Xski¼1

pi þ z ¼ �Xt

j¼1psj where j p k ð34Þ

psj ¼Xsj

iV¼1;iVpipiV: ð35Þ

That is:

pi ¼1þ bsk

� mimi�1

Z psk ¼Xsk

i¼1pi ¼

Xski¼1

1þ bsk

� mimi�1

;

xsk ¼Xsk

i¼1xi ¼

Xski¼1

mi1þ bsk

� 1mi�1

ð36Þ

ysk ¼Xsk

i¼1yi ¼ b

Xski¼1

mi1þ bsk

� 1mi�1

;

z ¼ �Xt

k¼1psk ¼ �

Xt

k¼1

Xski¼1

1þ bsk

� mimi�1

:

ð37Þ

Usk ¼ 1þ bð ÞXsk

i¼1mi

1þ bsk

� 1mi�1

�skXt

k¼1

Xski¼1

1þ bsk

� mimi�1

:

ð38Þ

With differentiated citizens we cannot use the egali-tarian rule to determine the repartition of gains withincoalitions. It thus becomes harder to define the individualutilities of the members of a given coalition. Several ruleshave been developed in the literature, such as the Nashsolution (1953) or the Shapley value (1953). In order to stayclose enough to the previous model, we use a rule accord-ing to which the members of a coalition share its gains infunction of each citizen's ability to use a less pollutingprivate good:

8iaSk Ui ¼miPiaSk mi

USk : ð39Þ

From these results, we can compute the Nash equilibriumand the Pareto optimum in the presence of heterogeneouscitizens. It comes down to characterising both the extremecases of the finest partition {{1},…, {n}} and the grand coalition{N}.

For ε={{1},…, {n}}, we have:

Ppi ¼ 1þ bð Þ

mimi�1;

Pxi ¼ mi 1þ bð Þ 1mi�1;

Pyi ¼ bmi 1þ bð Þ 1mi�1;

Pzi ¼ �n 1þ bð Þmi

mi�1

ð40Þ

P

Ui ¼ mi 1þ bð Þ 1þ bð Þ 1mi�1�n 1þ bð Þ

mimi�1: ð41Þ

On the other hand, for ε={N}, we get:

pN4 ¼X

iaNpi ¼

XiaN

1þ bn

� mimi�1

;

xN4 ¼X

iaNxi ¼

XiaN

mi1þ bn

� 1mi�1

;

ð42Þ

yN4 ¼X

iaNyi ¼ b

XiaN

mi1þ bn

� 1mi�1

;

z4 ¼ �Xn

i¼1

1þ bn

� mimi�1

ð43Þ

UN4 ¼ 1þ bð Þ

XiaN

mi1þ bn

� 1mi�1

�Xn

i¼1

Xn

i¼1

1þ bn

� mimi�1

ð44Þ

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

Ui4 ¼ miPn

i¼1 miUN4 ¼ miPn

i¼1 mi1þ bð Þ

XiaN

mi1þ bn

� 1mi�1

�Xn

i¼1

Xn

i¼1

1þ bn

� mimi�1

:

ð45Þ

4.3. Bargaining

To resolve the two-stage game, we adopt the backwardinduction method supposing that citizens have alreadyformed intermediary coalitions. We can then focus on thebargaining about the repartition of full cooperation gainsbetween the representatives of the intermediary coalitions.

Proposition 6. The solution of the bargaining is such that, for a fixedcoalition structure ε:

usk eð Þ ¼ UN4�

Xt

j¼1Usj

h iak þ Usk 8k; ð46Þ

and transfers among the coalitions of citizens take the following form:

sk ¼ 1þ bð ÞPski¼1 mi

1þ bsk

� mi

mi � 1þ nak � skð ÞPtk¼1

Pski¼1

1þ bsk

� mi

mi � 1

þak½ 1þ bð ÞPni¼1 mi

1þ bn

� 1mi � 1�Pn

i¼1Pn

i¼11þ bn

� mi

mi � 1

� 1þ bð ÞPtk¼1

Pski¼1 mi

1þ bsk

� 1mi � 1�

�Psk

i¼1 miPni¼1 mi

½ 1þ bð ÞXn

i¼1

1þ bn

� 1mi � 1 �

Xn

i¼1

Xn

i¼1mi

1þ bn

� mi

mi � 1�8k:ð47Þ

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

4.4. Stability

The theoretical treatment of stability with heterogeneousagents is made impossible by a particular difficulty. Comparedto the previous model, every citizen permutations must nowbe studied. In order to simplify the calculations, we study the

Table 4

Coalitionstructure

Individual monetary transfers

τ1 τ2 τ3 τ4

mi m1 ¼ 54

m2 ¼ 74

m3 ¼ 94

m4 ¼ 114

mi

mi � 1m1

m1 � 1¼ 5 m2

m2 � 1¼ 7

3m3

m3 � 1¼ 9

5m4

m4 � 1¼ 11

7

1mi � 1

1m1 � 1

¼ 41

m2 � 1¼ 4

31

m3 � 1¼ 4

51

m4 � 1¼ 4

7

[1234] 0 0 0 0[123,4] −0.428 −0.599 −0.77 1.797[124,3] −0.27 −0.378 1.244 −0.595[134,2] −0.135 0.673 −0.242 −0.296[234,1] 0.094 −0.024 −0.031 −0.038[12,34] −0.468 −0.655 0.505 0.617[13,24] −0.199 0.216 −0.357 0.34[14,23] −0.009 0.013 0.016 −0.02[12,3,4] −1.045 −1.462 0.905 1.602[13,2,4] −0.635 0.176 −1.143 1.601[14,2,3] −0.324 0.157 0.879 −0.712[23,1,4] −0.618 −0.44 −0.566 1.624[24,1,3] −0.645 −0.086 0.867 −0.135[34,1,2] −0.694 0.078 0.277 0.338[1,2,3,4] −1.5 −0.5 0.5 1.5


stability based on a 4-citizens example, using a bargainingpower defined as:

ak ¼Psk

i¼1 miPni¼1 mi

: ð48Þ

Each citizen i utility then becomes:

ui eð Þ ¼ miPni¼1 mi ½ 1þ bð Þ

Xn

i¼1mi

1þ bn

� 1mi � 1

�Xn

i¼1

Xn

i¼1

1þ bn

� mi

mi � 1� 1þ bð ÞXt

k¼1

Xski¼1

mi1þ bsk

� 1mi � 1

þnXt

k¼1

Xski¼1

1þ bsk

� mi

mi � 1�þ miPsk

i¼1 mi1þ bð Þ

Xski¼1

mi1þ bsk

� mi

mi � 1�skXt

k¼1

Xski¼1

1þ bsk

� mi

mi � 1" #

:

ð49Þ

We consider the case where N={1,2,3,4}, β→0 and m1 ¼ 54 ;

m2 ¼ 74 ; m3 ¼ 9

4 ; m4 ¼ 114 . In other terms, citizen 4 has the clean-

est technology and citizen 1 has the dirtiest one. Table 3presents the utility of each citizen according to the differentconceivable coalition structures.

It appears that all the coalition structures satisfy the ex-ternal stability condition, but the singleton structure [1,2,3,4] isthe only one to satisfy the internal stability as well. We cannotclaim, based on our 4-citizens example, that no coalitionstructure can be stable but the singleton regardless of thenumber of citizens. Given its complexity, the model does notallow any generalisation.

From this example, we can deduce some behaviouralspecificities from citizens. We can observe notably that

Table 3

Coalitionstructure


u1 u2 u3 u4

mi m1 ¼ 54

m2 ¼ 74

m3 ¼ 94

m4 ¼ 114

mi

mi � 1m1

m1 � 1¼ 5 m2

m2 � 1¼ 7

3m3

m3 � 1¼ 9

5m4

m4 � 1¼ 11

7

1mi � 1

1m1 � 1

¼ 41

m2 � 1¼ 4

31

m3 � 1¼ 4

51

m4 � 1¼ 4

7

[1234] 0.207 0.29 0.372 0.455[123,4] −0.221 −0.309 −0.398 2.252[124,3] −0.063 −0.089 1.616 −0.14[134,2] 0.072 0.963 0.13 0.159[234,1] 0.3 0.265 0.341 0.417[12,34] −0.261 −0.365 0.877 1.072[13,24] 0.008 0.506 0.015 0.795[14,23] 0.198 0.302 0.389 0.435[12,3,4] −0.838 −1.173 1.278 2.057[13,2,4] −0.428 0.466 −0.77 2.057[14,2,3] −0.117 0.447 1.251 −0.257[23,1,4] −0.411 −0.151 −0.194 2.079[24,1,3] −0.438 0.203 1.239 0.32[34,1,2] −0.487 0.368 0.649 0.794[1,2,3,4] −1.293 −0.21 0.872 1.955

those possessing a strongly polluting good get incentives tosearch alliances before negotiating, while those possessing aweakly polluting good do not. The reason is that, for the lattercitizens the necessary effort in case of implementation ofsome pollution reduction policy is weaker. These citizens arethus in a strong position during the bargaining and theirstrategy of bargaining alone allows us to increase the pressureon transfers they will get from other citizens. This pressurewill be all the more important as other citizens will regroup incoalition. Being alone at the bargaining stage is the moreadvantageous as individual players get higher gains (com-paratively to those forming a coalition).

Table 4 gives the individual monetary transfers betweencitizens in our four players game.

We can notice that negotiating as an individual player isbeneficial 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 areother singletons; in that case he should transfer funds to thoseother citizens. In our example, this is the case for citizen 1 forthe 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 ofcitizens) that transfer funds to the other, the citizen with thecleaner technologywill always pay themore. That comes fromshare rules inside the coalition.

5. Conclusion

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


corresponds to a possible form of participative democracy.Through a two-stage game, we showed that transfers in aglobal agreement depend not only on the capacity of a citizento reduce pollutant emissions, but also on the formationof negotiating coalitions. This allows us to point out theimportance of intermediary coalition formation in the reparti-tion of cooperation gains.We have also clarified how transfersset up among citizens are dependent on multiple variables,both of environmental and economic nature, and on thebargaining power of coalitions that participate to the negotia-tion. This last variable is very important because it can leadto very different results in terms of stability of the coalitionstructure.

Beyond these analytical results, the main contribution ofthis model is the importance it gives to a decentralisedcontractualism which leads to a Pareto-optimal allocation.This representation tends to offer a favourable argument tothe local experiences of participative democracy set up inQuebec, in some Commonwealth countries and in someNorthAmerican cities. The interest of this form of regulation lies inthe redefinition of institutions it proposes. The role ofinstitutions is no longer limited “to punish or reward” (asit is in the agency theory framework) but extended to thedevelopment and proposition of cooperation procedures tocitizens. These procedures will not be controlled by anyonebut citizens themselves as the onus is on them to play pro-cedures while defending their own interests. These pro-cedures have to be developed in such a way that citizenshave an interest in cooperating, without deriving it from anysanction strategy.

Acknowledgements

We would like to thank Frédéric Ghersi, Rabia Nessah andAntoine Saglio for their helpful comments on previous ver-sions of this paper. The participants of the “Journées de l'AFSE2004” (Rennes, May 2004) and the “First International Seminaron the Environment” (Bejaia, June 2005), together with threeanonymous referees are also gratefully acknowledged.

k¼1 N k¼1

Appendix A

Proof of proposition (1). We have seen that, at the non-cooperative Nash equilibrium, each citizen gets a satisfactionlevel equal to: Ui=(2−n)(1+β)2, while at the quasi-hybridequilibrium, we have:Ui ¼ 2 1þbð Þ2

sk�Pt

j¼11þbð Þ2sj

.To get the terms of Proposition (1), it is sufficient to show

that:

8iaN Ui zP

Ui;

Which is equivalent to:

n� 2zXt

j¼1

1sj� 2sk

fXt

k¼1sk � 2z

Xr�1

j¼1

1sj� 1sk

þXt

j¼rþ1

1sj:

Adding �sk to each side, we get:

Xt

j¼1;jpksj � 2z

Xr�1

j¼1

1sjþXt

j¼kþ1

1sj� 1sk

� sk:

As:Pt

j¼1 sjzPt

j¼11sjand

Ptj¼1;jpk sjz

Pk�1j¼1

1sjþPt

j¼kþ11sj, we just

have to check that:

�2 z� 1sk

� sk;

that is: 1skþ skz2, which is always satisfied since sk≥1.

Proof of proposition (2). Let us note that:

XUsk ¼

Xt

k¼12 1þ bð Þ2�sk

Xt

j¼1

1þ bð Þ2sj

" #

¼ 2t 1þ bð Þ2�nXt

j¼1

1þ bð Þ2sj

:

Proposition (2) validity is then guarantied if:

2t 1þ bð Þ2�nXt

j¼1

1þ bð Þ2sj

V 1þ bð Þ2;

that is to say:

Xt

j¼1

1sj

z2t� 1

n; ð1Þ

with: tbn, tN1 and nN2.We will show that if the minimum of

Ptj¼1

1sj

subject toPtj¼1 sj ¼ n satisfies condition (1), then it is always satisfied.

The Lagrangian associated to the minimisation programmecan be written:

L ¼Xt

j¼1

1sjþ k

Xt

j¼1sj � n

h i;

It results that:

ALAsi

¼ � 1s2j

þ k ¼ 0 ð2Þ

ALAk

¼Xt

j¼1sj � n ¼ 0: ð3Þ

Consequently, the first order condition (2) implies: k ¼1s21¼ N ¼ 1

s2t, hence: s1=…=st. Substituting this condition in (3),

we get: sj ¼ nt. The function's minimum is then given by:

ttn¼ t2

n:

Since the objective function is convex and the constraint islinear, L is convex and the minimum determined by theLagrangian corresponds to a global minimum. Let us demon-strate that this minimum satisfies condition (1):

t2

nN2t� 1

n:

This inequality can be rewritten: (t−1)2N0. As tN1, theinequality is always satisfied.

Proof of proposition (3). The solution of the bargaining, (uS1,…,

uSt), is given by:

Max jt

k¼1Uk �Ust� �ak

;

under the following constraints:

XtUk ¼ 1þ bð Þ2¼ U4 and

Xtak ¼ 1;


where 0bαk≤1 denotes the bargaining power of coalition Sk. Todetermine the solution, we write: Vj=Uj−U

~Sj, with Uj≠U

~Sj. The

programme can be rewritten:

Max jt

j¼1Vajj s:t:

Xt

j¼1Vj ¼ 1þ bð Þ2�

Xt

j¼1Usj ¼ D N 0:

The Lagrangian is written:

L ¼ jt

j¼1Vajj � k

Xt

j¼1Vj � D

� �;

Consequently:

ALAVk

¼ akVak�1k j

t

j p kVajj � k ¼ 0 ð1Þ

Xt

j¼1Vj ¼ D: ð2Þ

From (1), we have: akjtj¼1 Vaj

j ¼ kVk, that is:

Xkak j

t

j¼1Vajj ¼ k

XkVk:

Using (2), we get:jtj¼1 Vaj

j ¼ kD, and then (1) could berewritten: Vk=αkD. This yields:

usk � Usk ¼ ak 1þ bð Þ2�X

Usj

h i:

It follows:

usk ¼ ak 1þ bð Þ2�X

Usj

h iþ Usk:

As we know that: Usk ¼ 2 1þ bð Þ2�skPt

j¼11þbð Þ2sj

, we get:

usk ¼ 2 1þ bð Þ2þ 1þ bð Þ2 1� 2tð Þak þ nak ¼ skð ÞXt

j¼1

1þ bð Þ2sj

:

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

usk4 ¼X

iaskUi4 ¼

Xiask

1þ bð Þ2n

¼ sk 1þ bð Þ2n

;

then we get the relation: usk⁎+ τ=usk, that is:

s ¼ usk � usk4 ¼ 2 1þ bð Þ2þ 1þ bð Þ2 1� 2tð Þak

þ nak � skð ÞXt

j¼1

1þ bð Þ2sj

� sk 1þ bð Þ2n

:

Proof of proposition (5). We have seen that:

Ui;sk ¼ 1þ bð Þ2 2sk

þ 1n

1� 2tð Þ� �

Now, if citizen i decides to leave the sk-coalition in order toact as a singleton, his utility is:

U�i;sk�1 ¼ 1þ bð Þ2 2� 1n

1þ 2tð Þ� �

:

The difference gives us:

U�i;sk�1 � Ui;sk ¼ 1þ bð Þ2 2 sk � 1ð Þsk

� 2n

� �:

This difference is always positive when nN2 and skN2. Inother terms, the internal stability condition is satisfied onlyfor skb2: a citizen i always gets incentives to leave a sk-coalition he is member of. With this reasoning valid for all i ofSk, the result is that no coalition gathering more than onemember will appear. The development of calculations asso-ciated to the external stability condition allows us to confirmresults following from the internal stability. The only stablecoalition structure is the finest partition {{1},…, {n}}.

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

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Participative democracy and local environmental issues

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