Sustainable coalitions in the commons

Mathematical Social Sciences 63 (2012) 57–64

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Sustainable coalitions in the commons Luc Doyen a , Jean-Christophe Péreau b,∗ a

CNRS, CERSP UMR 7204 CNRS-MNHN-UPMC, 55 rue Buffon, 75005 Paris Cedex, France

b

GREThA, UMR CNRS 5113, Université de Bordeaux, Av leon Duguit, 33608 Pessac, France

article

info

Article history: Received 28 June 2010 Received in revised form 9 October 2011 Accepted 27 October 2011 Available online 10 November 2011

abstract It is well known that the lack of cooperation among agents harvesting a renewable resource is critical for its sustainable management. The present paper gives insights into the complex balance between coalition structures, resource states or dynamics and the agent heterogeneity necessary to avoid bio-economic collapses. A model bringing together coalition games and a viability approach is proposed to focus on the compatibility between bio-economic constraints and exploited common stock dynamics. The extent to which cooperation promotes sustainability is examined. Our results suggest that the stability of the grand coalition occurs for large enough stocks. By contrast, for lower levels of resources, the most efficient user plays the role of veto player. © 2011 Elsevier B.V. All rights reserved.

1. Introduction This paper deals with the cooperation among users harvesting a renewable resource. According to recent studies (MEA, 2005), biodiversity and exploited renewable resources are under extreme pressure worldwide. For instance, three quarters of fish stocks worldwide are maximally exploited or over-exploited (FAO, 2004). Hence sustainability is nowadays a major concern for international agreements and guidelines for fishery management (ICES, 2004). In this context, exploited biodiversity management involves restoration and conservation objectives, with ecological and economic dimensions including the identification of desirable levels of stocks and profitability from catches. It inevitably raises the question of the number of active users of the resource and the ways they can cooperate. To avoid possible future collapses of stocks, catches and rents, we need to determine the conditions under which cooperation can be sustainable. Game theory modeling (Kaitala and Munro, 1995; Hannesson, 1997; Kaitala and Lindroos, 2007) provides some important insights on strategic interaction between users exploiting a common renewable resource. In particular, the relationship between the number of active agents and the sustainability of the stock has been studied in static non-cooperative games by Mesterton-Gibbons (1993) and Sandal and Steinshamn (2004) in the presence of users differing with respect to their harvesting cost. These models both show that the larger the stock, the higher the number of active players. Mesterton-Gibbons (1993) derives

∗

Corresponding author. E-mail addresses: [email protected] (L. Doyen), [email protected] (J.-C. Péreau). 0165-4896/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2011.10.006

this result by considering rent maximizing users who take into account the stock externality arising from the actions of the other users. Similarly, Sandal and Steinshamn (2004) investigate another strategic interaction between users relying on a Cournot competition. They also introduce an incumbent agent whose role is to achieve sustainability by setting the resource at its steady state. The question of coalitions has also received considerable attention in this context of renewable resources management. What distinguishes the renewable resources coalition game from many other coalition games is that major externalities occur. Particular attention has been paid to coalition issues in fishery economics (Kaitala and Lindroos, 1998; Arnason et al., 2000; Burton, 2003; Lindroos, 2004a). The need for research on cooperative fishery management arises from the current practice of international negotiations and the implementation of multi-country fishery agreements. In this context, the formation of coalitions has been analyzed in both cooperative and noncooperative games. On the one hand, the cooperative game theory literature deals with the allocation rules of the cooperation benefits between the members of a coalition using the characteristic function games (c-games). These sharing rules include Shapley value (probably the most used), nucleolus and tau-value (Lindroos, 2004b; Kronbak and Lindroos, 2007; Li, 1999). Kaitala and Lindroos (1998) show in a three-player game that there is a partial cooperative equilibrium stock level that is higher than the non-cooperative stock level (Clark, 1980), though lower than the fully cooperative stock level (Clark and Munro, 1975). Lindroos (2004a) extends these results to a four-player game. In this literature, the question raised by the stability of the coalition and especially the grand coalition is of particular interest. Hence, Arnason et al. (2000) analyze the case of Atlanto-Scandian herring and point out that the adherence of

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L. Doyen, J.-C. Péreau / Mathematical Social Sciences 63 (2012) 57–64

Norway is crucial to the stability of any coalition. This result has connections with Lindroos (2004a) who considers the possibility of veto countries. Kronbak and Lindroos (2007) predict a stable grand coalition while Lindroos (2004b) points out the connections between safe minimum biological levels and stability of full cooperation. On the other hand, the non-cooperative game theory literature focuses on the endogenous formation of coalitions and coalition structures using a partition function game taking into account externalities across the coalitions (Bloch, 1996; Yi, 1997; Finus, 2001). Pintassilgo (2003) derives general results regarding the stability of the coalition structures in straddling stock fisheries. In the presence of positive externalities, the grand coalition is not stable. Using the static Gordon–Schaefer bio-economic model, Pintassilgo and Lindroos (2008) show that complete noncooperation is the only equilibrium when at least three symmetric players are concerned. Our paper questions the shape and size of coalitions with respect to sustainable requirements for the exploited renewable resource. Although our work is in a direct line with the models of the literature, it gives new insights by considering a dynamic game focusing on viability constraints. First our approach is dynamic in the sense that the analysis is not restricted to steady states and equilibrium yield. Second, a viability viewpoint is adopted to cope with sustainability. The viability (or viable control) approach does not strive to identify optimal or steady state paths, but rather aims at identifying the conditions that allow desirable objectives or constraints to be fulfilled over time, considering both present and future states (Pezzey, 1997; Baumgärtner and Quaas, 2009; Béné et al., 2001). As emphasized in DeLara and Doyen (2008) or Martinet and Doyen (2007), viability is closely related to the maximin (Rawlsian) approach with respect to intergenerational equity. Viability may also allow for the satisfaction of both economic and environmental constraints and is, in this respect, a multi-criteria approach. Viability analysis has been applied to renewable resources management and especially to fisheries (Béné et al., 2001; Eisenack et al., 2006; Martinet et al., 2007), but also to broader (eco)-system dynamics (Cury et al., 2005; Doyen et al., 2007; Béné and Doyen, 2008). Relationships between sustainable management objectives and reference points as adopted in the ICES1 precautionary approach are discussed in DeLara et al. (2007). Here the viability framework makes it possible to determine the conditions under which coalitions can fulfill positive profitability and conservation objectives along time, considering both present and future states of the renewable resource system. The paper is organized as follows. Section 2 is devoted to the description of the dynamic bio-economic model together with the profitability constraints. Section 3 provides the results related to the shape of the viable coalitions with respect to the level of the resource. The contribution of the agents to the viability is also analyzed using the minimum number of active players. The last section concludes.

We consider that the resource is exploited by n agents, implying an amount of harvest h( t ) =

ei (t )x(t ),

(2)

i=1

with ei ∈ [0, 1] standing for the effort (or harvesting mortality) of each agent i ∈ {1, . . . , n}. Since take-off cannot exceed resource stock, a scarcity constraint holds 0 ≤ h(t ) ≤ x(t ),

(3)

implying a constraint on the effort n −

ei (t ) ≤ 1.

i=1

Following Clark (1990), the rent of each agent i is defined by

πi (x(t ), ei (t )) = pei (t )x(t ) − ci ei (t ),

(4)

where ci measures the cost per unit of effort and the price p of the resource is assumed to be constant. The rent is positive for each agent i when the stock is larger than the zero-rent level (or open access stock) namely x(t ) > xoa i =

ci p

.

Eqs. (1)–(4) refer to the Gordon–Schaefer model which is the main reference in theoretical approaches to the bio-economics of fisheries. This aggregated model has shown a significant potential to demonstrate fundamental economic principles in fishery management (Hannesson, 1997; Clark, 1990; Pintassilgo and Lindroos, 2008). Following Mesterton-Gibbons (1993), we assume that the agents are heterogeneous in the sense that they differ in their cost c1 < c2 < · · · < cn−1 < cn , or equivalently in their open access stocks oa oa oa xoa 1 < x2 < · · · < xn−1 < xn .

It is worth noting that two kinds of externalities occur in this game as every agent may alter both the current catches (and rents) of others agents by the scarcity constraint (3) and also the future catches of agents through harvesting and dynamics (1) which impact the stock for the next period. Sustainability problem. Following Pezzey (1997); Béné et al. (2001); Baumgärtner and Quaas (2009), the sustainability of the system is approached through constraints to satisfy along time, especially profitability goals. In our multi-agent framework, the dynamic problem that we handle is to determine coalitions S ⊂ {1, . . . , n}, harvesting strategies among the coalition ei (t ), i ∈ S and a stock path x(t ) which ensure that the aggregated rent of the agents i belonging to the coalition S remains strictly positive

−

2. The dynamic model

n −

πi (x(t ), ei (t )) > 0,

t = 0, 1, . . . , T .

(5)

i∈S

Following Conrad and Clark (1987) and DeLara and Doyen (2008), the dynamics of a renewable resource stock x(t ) ∈ R+ is given by x(t + 1) = f (x(t ) − h(t )),

t = 0, 1, . . . , T ,

(1)

where catches h(t ) occur at the beginning of the period. The natural resource productivity is represented by function f , possibly nonlinear. We denote by K the capacity charge of the resource K = sup(x ≥ 0, f (x) ≥ x).

1 International Council for the Exploration of the Sea.

Cooperation among a group of players implicitly relies on the establishment of an organization with the purpose of jointly managing the catches and protecting the resource stock. In our cooperative game, the objective of the coalition is not to optimize the global rent but to sustain this rent throughout time. In such a viability perspective, it is possible to highlight sub-optimal but sustainable solutions and performances in which some less efficient users may have negative profits compensated by positive rents for other users among the coalition. It turns out that internal transfer payments between the members of the coalition are required to fulfill the profitability condition (5) which holds for the whole coalition. However, the problem of the repartition of the

L. Doyen, J.-C. Péreau / Mathematical Social Sciences 63 (2012) 57–64

profit between the members of the coalition is beyond the scope of this paper. The profitability constraint (5) entails the following resource viability requirement2 captured by the critical bio-economic level xS x(t ) > xS = min xoa i .

(6)

i∈S

Such a stock constraint points out that the conservation of the resource above a safe threshold is a necessary condition for sustaining the rent among the coalition S. Eq. (6) also shows that the most efficient agent of coalition S which has the lowest floor stock level xoa i plays a major role. The present paper intends to give insights into the shape and size of coalition S regarding the initial value of the stock x0 , the resource dynamics f , the economic context (c and p) and these sustainability goals (5) and (6). Viable states and viability kernel. To achieve this, we define the viability kernels Viab(S , t ) for a given coalition S through backward induction inspired by dynamic programming. First, at the terminal date T , we set Viab(S , T ) = x | x > xS .





For any time t = 0, 1, . . . , T − 1, we compute the viability kernel Viab(S , t ) at time t from the viability kernel Viab(S , t + 1) at time t + 1 as follows Viab(S , t )

   ∃(ei )i∈S ∈ A  S (x),         − − s ei − ej ∈ Viab(S , t + 1), , = x > x f x 1 −      i∈S j̸∈S       ∀(ej )j̸∈S ∈ BN \S ((ei ), x)     

(7)

  −  π (x, ei ) > 0 , AS (x) = e ∈ ∆S   i∈S i

(8)

and



j̸∈S

(9)

The previous definition of AS stresses the fact that the agents among the coalition cooperate for sustainable profitability goals both in the present and the future. By contrast, the definition of BN \S for outsiders of the coalition captures the fact that they act as singletons and are myopic regarding goals of profitability. This may occur in practice for renewable resource management because agents (fishermen, hunters, etc.) can have a strongly myopic and non virtuous behavior and harvest the maximum of natural resource available in one shot because of the open-access feature of the resource. The myopic behavior of the outsiders can encompass several strategies including optimizing or inertial strategies which are potentially dangerous and risky for the resource. The viability kernel Viab(S , t ) for coalitions can be defined in terms of trajectories and strategies. However, for the sake of

2 The proof is presented in Appendix A.1.

(10)

Such a viable feedback effort e∗S (t , x) exists as long as the resource state x belongs to the viability kernel Viab(S , t ). Of course the design of such viable efforts requires knowledge of the whole sequence of viability kernels. 3. Results Hereafter, the stock productivity f : R+ → R+ is assumed to be continuously increasing f ′ > 0 and to satisfy f (0) = 0. We also assume that the open-access stocks xoa i lie in the part where the resource grows in the following sense



(11)

3.1. Viable coalitions and states Let us first identify the viable stocks through the computation of the viability kernels for every coalition.

• • • •

  − |S |  e ≤1 , ∆S = (ei ) ∈ R+   i∈S i 

 

 ∗ eS (t ,x) ∈ AS (x),     − − ∗ ej ∈ Viab(S , t + 1), f x 1− ei (t , x) −   i∈S j̸∈S  ∀(ej ) ∈ BN \S ((e∗i )(t , x), x).

Theorem 1. The viability kernels at time t (t < T ) are

with ∆S denoting the simplex of coalition S defined by



Viable feedbacks. The viable harvests in terms of feedback of coalition S denoted by e∗S (t , x) = (e∗i (t , x))i∈S are deduced from the backward recursive definition (7) of the viability kernel Viab(S , t ). Given a resource state x at time t, they are characterized by

oa xoa ⊂]0, K [= {x ≥ 0, f (x) > x}. 1 , xn



BN \S ((ei ), x) = (ej ) (ei , ej ) ∈ ∆N , min πj (x, ej ) ≥ 0 .

simplicity, we preferred to restrict the definition to its dynamic programming characterization which is convenient and useful for discrete time dynamic systems under constraints as emphasized in DeLara and Doyen (2008). In such a perspective, a direct natural strategy for decisions among the coalitions are feedback controls; the strategy for outsiders should rely on non-anticipative strategies (Cardaliaguet et al., 2007) as it can depend on insiders decisions along time. This all however requires the introduction of important notations that we wanted to avoid here.



where AS (x) is the set of feasible decisions for the coalition S while BN \S ((ei ), x) stands for the set of admissible decisions for the outsiders in N \ S. These feasibility sets are defined by

59

Viab(S , t ) = ∅ if 1 ̸∈ S.   Viab({1, . . . , n}, t ) = xoa . 1 , +∞  oa  Viab({1, . . . , j} , t ) = x1 , xoa for j < n. j +1 Viab(S , t ) = Viab( S , t ) for  S = ∪i ({1, . . . , i} ⊂ S ).

The proofs of Theorem 1 are presented in Appendix (Appendices A.2–A.6).3 Theorem 1 identifies what the size and the composition of the viable coalition of users are with respect to the stock of the resource. To illustrate this, consider Table 1 where we can distinguish the viability kernels in the case of three players n = 3. Table 1 reads as follows:  any initial stock level lying for instance in oa the interval xoa , x belongs to the viability kernel Viab({1, 2}, t ) 1 3 of the coalition of players 1 and 2. The tragedy of open-access revisited. It turns out that cooperation promotes viability as the higher the cooperation between users, the larger the viability domain. In particular, the grand coalition N = {1, 2, 3} (social  viability) corresponds to the largest viability kernel xoa 1 , +∞ . By contrast, the smallest viable coalition occurs with singletons. In particular, viability vanishes for singletons S = {2} or S = {3} since both viability kernels Viab({2}, t ) and Viab({3}, t ) are empty. Another significant viable coalition is

3 Note that all the results could be extended to the infinite horizon case since it turns out that the viability kernels do not depend on time.

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L. Doyen, J.-C. Péreau / Mathematical Social Sciences 63 (2012) 57–64

Table 1 Viability kernels (in black) with three players n = 3. Kernel \ Stock x

xoa 1

0

xoa 2

xoa 3

K

Viab({1, 2, 3}) Viab({1, 2}) Viab({1}) = Viab({1, 3}) Viab({2}) = Viab({3}) = ∅

formed by agents 1 and 2. However viability is reduced in this partial cooperation case as Viab({1, 2}, t ) is strictly contained in Viab({1, 2, 3}, t ). Of interest is the fact that the coalition formed by players 1 and 3 is equivalent to singleton {1}, emphasizing that the role of player 3 is minor in this case. Agent 1 is a veto player. As already pointed out in Eq. (6), the agent with the lowest cost plays a major role for the sustainability of the resource. The viability kernels given by Theorem 1 show that agent 1 is a veto player as his presence is always required for cooperation to be viable. In other words, if the most efficient user 1 who has the lowest floor stock level is not a member of the coalition, the associated viability kernels are empty. In particular, the only viable singleton is S = {1}. 3.2. A focus on the grand coalition (social viability) Coordination promotes the viability as the grand coalition N =   {1, . . . , n} corresponds to the largest viability kernel xoa 1 , +∞ . Such a configuration makes it possible to recover several wellknown cases of bio-economic states. In particular it turns out that the maximum economic state xmey belongs to the viability kernel Viab(N , t ) of the grand coalition.

(xmey , emey ) ∈ Arg 

max

(x∑ ,e) f (x(1− ei ))=x

n −

πi (x, ei ).

i =1

Corollary 1. The maximum economic state xmey > 0 is viable for the grand coalition: xmey ∈ Viab(N , t ). The proof, detailed in Appendix A.7, relies on the idea that the maximum economic equilibrium is always larger than the openaccess equilibrium. Let us point out that the only clear result regarding the maximum economic yield xmey holds with respect to the grand coalition. There is no reason for such a MEY to belong to other viability kernels. The case with two close (with similar costs) players is illustrative at this stage because there are few chances for the reference point mey to belong to the thin viability kernel oa oa Viab({1}, t ) =]xoa 1 , x2 [ since the two open-access levels xi are very close. Similarly the non cooperative equilibrium xnc belongs to the viability kernel Viab(N , t ) of the grand coalition. To prove this point, introduce the notation x(ei , e−i ) as the solution of the equilibrium equation f (x(1 − ei − e−i )) = x. The Nash equilibrium nc xnc is the solution of the problem with xnc = x(enc i , e−i ) nc πi (xnc , enc i ) = max πi (x , ei ), ei ≥0

∀i = 1, . . . , n.

Corollary 2. Non cooperative equilibrium xnc > 0 is viable for the grand coalition: xnc ∈ Viab(N , t ). Again the proof relies on the idea that the non cooperative equilibrium xnc is always larger than the open-access equilibria xoa and consequently xoa i 1 . However this assertion may sound counterintuitive as a non-cooperative equilibrium is not a good candidate for full cooperation. Actually, this non-cooperative equilibrium xnc has some cooperative features because it is a steady state for which the agents implicitly agree on setting the resource at some stationary (and thus sustainable or viable) level.

Fig. 1. An example of minimal players n∗ (x) for a viable coalition.

3.3. Minimum number of players Given a stock level x, we define by n∗ (x) the minimum number of players in a viable coalition by n∗ (x) = min (|S | | x ∈ Viab(S , 0)) , where |S | stands for the cardinal of the coalition S. Using Theorem 1, we derive the following condition oa n∗ (x) = min j | xoa 1 < x < x j +1 ,





which is illustrated by the stepwise increasing function displayed in Fig. 1. This minimum number of active players in a coalition in our dynamic framework is a generalization of the steady state participation condition of Mesterton-Gibbons (1993). It is worth noting that our approach extends this result since steady states are particular cases of viability (DeLara and Doyen, 2008). Let us emphasize that the number of viable players increases with stock. In particular, this suggests that the grand coalition is stable whenever the stock is large enough. 3.4. Viable efforts for the viable coalitions The next step of the analysis is to exhibit the viable effort of the members of the coalition. Applying the ‘‘dynamic programming’’ characterization of viable feedbacks displayed in (10), we deduce a characterization of the viable feedbacks e∗S (t , x) for a given coalition S. To encompass the grand coalition case in the following theorem, we need to introduce the notation xoa n+1 = +∞. As shown in proof in Appendix A.4 from Eq. (12), we obtain: Theorem 2. Consider a viable coalition S = {1, . . . , j} and any  oa viable stock x in Viab(S , t ) = xoa 1 , xj+1 . The viable feedbacks e∗S (t , x) = (e∗i (t , x))i∈S are solutions of the linear constraints

− ∗ ei (t , x)(px − ci ) > 0,     i∈S       f −1 xoa − ∗ 1 ei (t , x) = α 1 − x  i ∈ S       f −1 xoa  j +1   , + (1 − α) max 0, 1 −  x

where 0 < α < 1. Note that several catch efforts can satisfy previous linear inequalities and consequently there is some flexibility in the decision process. Among these viable choices, one can favor efficient or conservative rules or different trade-offs between ecological or economic performance. Low values of α refer to an

L. Doyen, J.-C. Péreau / Mathematical Social Sciences 63 (2012) 57–64

61

ecological viewpoint since they favor the resource. When α tends to zero, the effort of the coalition S = {1, . . . , j} is such that the stock is close to xoa j+1 . By contrast, high values of α promote current efforts and profitability and lessen the resource close to xoa 1 while warranting the viability. In fact, to prevent the outsiders from collapsing the resource and the rents, the coalition has to manage the resource in a way that the outsiders become passive. The coalition S = {1, . . . , j} oa has to maintain the resource in its viability kernel xoa 1 , xj+1 to guarantee its sustainability. Actually, the coalition achieves this by neutralizing the outsiders in a sustainable way. This neutralization occurs by preventing every profitability for outsiders and more specifically by maintaining the resource below open-access levels xoa < · · · < xoa j+1 n+1 for outsiders. A similar mechanism of neutralization is applied in Clark (1990, p. 155). This means that the stock falls below the open access level of outsiders but is still above the open access level of coalition members. In such a context, since all the outsiders of the coalition are obliged to be passive, the coalition does not have to take into account that it can play against either a coalition formed by the outsiders or against individual outsiders (Pintassilgo and Lindroos, 2008).

Table 2 Sensibility analysis of viability assessed by the length |Viab| of the viability kernel with respect to price p and costs c for three players n = 3.

3.5. Sensitivity analysis

4. Conclusion

This paragraph provides a qualitative analysis of the role played by major parameters of the model on the viability outcomes. More specifically we pay attention to the sensibility of the viability kernel Viab with respect to price p and unit costs c = (ci )i=1,...,n . To achieve this, we differentiate the size of the viability kernel with respect to these parameters and use the characterizations = cpi . To of Theorem 1 involving the open-access stocks xoa i avoid infinite values for the size of the viability kernel (especially in the case of the grand coalition), we restrict the kernel to its informative part below the capacity charge K . The viability performance I (S , p, c ) is thus represented by

This paper has analyzed the conditions under which cooperation of active heterogeneous users within a coalition is required to promote the bio-economic viability of a renewable resource. We have proposed a dynamic model bringing together coalition games and a viability approach to focus on the compatibility between bio-economic constraints and an exploited common stock dynamic. The model makes it possible first to revisit the tragedy of open-access and the seminal work of Hardin as it shows to what extent lack of cooperation reduces or jeopardizes the viability of the whole bio-economic system. Focusing on the grand coalition, it has been shown how the usual ‘‘sustainable’’ (steady) states including the maximum economic yield (MEY) are particular cases of viability. We have also determined the minimum number of viable players expanding the equilibrium approach of Mesterton-Gibbons (1993) and Sandal and Steinshamn (2004) to a more dynamic context. Such a study stresses the fact that diversification in technologies (ratio costs-catchability) is relevant for high levels of stock while specialization, rationalization and dictatorship situations as in Arnason et al. (2000) or Lindroos (2004a) are well-suited for low resources. This suggests how the grand coalition is stable for large resource levels which reinforces assertions of Pintassilgo (2003); Kronbak and Lindroos (2007) and Lindroos (2004b). More generally, the present work gives new insights into the mix between games and viability. Up to now, the viability approach has rather focused on centralized management without accounting for strategic interactions and coordination between users. Exceptions include works on differential games (Cardaliaguet et al., 2007) focusing on two players within a continuous time context. However, to our knowledge, coalition issues have not been studied in such a viability framework in the context of a game with n asymmetric users.

 

I (S , p, c ) = Viab

 ∫   [0 , K ] =

K

1Viab(S ) (x)dx, 0

where 1Viab(S ,t ) (.) is the indicator function defined by

1Viab(S ,t ) (x) =



1 0

if x ∈ Viab(S , t ), if x ∈ ̸ Viab(S , t ).

Straightforward computations stemming from the values of the boundaries xoa = cpi of the viability kernels yield the following i sensibility results for price p: Proposition 1. At any time t > T and for any p, c > 0, we have

• • •

∂I (S , p, c ) = 0 if 1 ∂p ∂I (N , p, c ) > 0. ∂p ∂I ({1, . . . , j} , p, c ) ∂p

̸∈ S. < 0 for j < n.

To prove these assertions, we basically use that I ({1, . . . , j} , p, c ) =

K

∫ 0

=

oa oa 1]xoa ,xoa [ (x)dx = xj+1 − x1 1 j+1

cj+1 − c1 p

.

Similar computations entail the following sensibility results for unit costs of effort ci : Proposition 2. At any time t > T and for any p, c > 0, we have

•

∂I ( S , p, c ) ∂ ci

= 0 if 1 ̸∈ S.

Coalition \ Parameters

p

c1

c2

c3

Viab({1, 2, 3}) Viab({1, 2}) Viab({1}) = Viab({1, 3}) Viab({2}) = Viab({3}) = ∅

+ − −

− − −

0 0

+

0

0

0

• •

+

∂I (N , p, c ) < 0 and ∂∂cIi (N , p, c ) = 0 for i ≥ 2. ∂ c1 for j < n, ∂∂cI ({1, . . . , j} , p, c ) < 0, ∂∂cI ({1, . . . , j} , p, c ) i 1 i ≥ 2 and ∂ c∂ I ({1, . . . , j} , p, c ) > 0. j+1

0 0 0

= 0 for

Table 2 illustrates such comparative analysis for n = 3 players. It stresses the fact that the unit cost c1 of veto agent 1 plays a detrimental role (− or 0) for the viability while other costs c2 or c3 improve it (+ or 0). In other words, the efficiency of player 1 favors viability while the efficiency of other players can alter it. The role of price p is more ambiguous with a positive impact for the grand coalition and a negative or nil impact in other cases.

Acknowledgments This work was supported by the FRB (Foundation for the Research on Biodiversity) through the project entitled BIOMER and the ANR (French National Agency for Research) through the project entitled ADHOC. We wish to thank the editor and two anonymous referees, and the participants at the EAERE2009 conference in Amsterdam (Netherlands) and ASSET2010 in Alicante (Spain) for their comments and suggestions.

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Appendix. The proofs

A.3. The grand coalition S = N At date t = T , we have

A.1. State constraint We are interested in the existence of efforts ei (t ) such that profitability constraint (5) holds true. In other words, we want

−

 sup

(ei (t ))i∈S ∑ei (t )≥0 ei (t )≤1 i∈S

πi (x(t ), ei (t )) > 0.

Viab(N , t ) = {x > xN = xoa 1 }. From the very recursive definition (7), the viability kernel at time T − 1 reads

 Viab(N , T − 1) =

i∈S

     n  −  × f x 1 − ei ∈ Viab(N , T ) .  i=1

This is a linear optimality problem since

−

π i ( x, e i ) =

i∈S

−

ei (px − ci ).

Hence

i∈S



It can be proved that

−

 sup

(ei )i∈S ∑ei ≥0 ei ≤1 i∈S

x > xoa 1 , ∃e ∈ An (x)

Viab(N , T − 1) =

ei (px − ci ) = max(px − ci ), i∈S

i∈S

 

x>

xoa 1

, sup f e∈An (x)

x 1−

n −



 >

ei

xoa 1

.

i=1

Since f is increasing, we have, using Lemma 1

  sup f

= px − min ci ,

e∈An (x)

i∈S

x 1−

n −

 ei

  =f

x 1−

i =1

= px − ci∗ ,

n −

inf

e∈An (x)

 ,

ei

i =1

= f (x).

and the optimal effort e (i ) = 1 and e∗ (i) = 0 for i ̸= i∗ . Consequently the viability constraint reads ∗ ∗

px(t ) − min ci > 0,

−1 oa and therefore Viab(N , T − 1) = x > xoa x1 . 1 , x > f From the properties of function f depicted in (11), we have   −1 oa max xoa x1 = xoa 1 ,f 1 . It yields







Viab(N , T − 1) =]xoa 1 , +∞[.

i∈S

or equivalently ci x(t ) > min . i∈S p

Now let us assume the viability domain for the grand coalition S = N at time t + 1 is Viab(N , t + 1) = {x > xoa 1 }. From the recursive formulation we have Viab(N , t )



A.2. Notations and lemmas

= x|x>

For the sake of clarity, we introduce the following notations and lemmas

• The set of feasible decisions A{1,...,j} (x) for coalition Sj = {1, . . . , j} is denoted by Aj (x). • The set of feasible decisions B{j+1,...,n} (x) for outsiders N \ {1, . . . , j} is denoted by Bj (x). ∑j Lemma 1. If x > xoa i=1 ei = 0. 1 then m = infe∈Aj (x) > xoa ̸= ∅. Then pick some e = 1 then Aj (x) ∑ (e1 , . . . , ej ) ∈ Aj (x). First, since ei ≥ 0, it is clear that ji=1 ei ≥ 0 and thus m ≥ 0. Now assume for a while that m > 0. This means that for any integer k ∈ N there is a sequence of vector ek ∈ Aj (x) Proof. If x

such that j

m≤

−

eki ≤ m +

i =1

1 k

′

i =1

j −

. ek , 2

′

we have ek ∈ Aj (x) and

eki ≤ 0.5 ∗ (m + 1/k).

i=1

′

ei

 >

xoa 1

.

i =1

Using a similar reasoning, we obtain oa Viab(N , t ) = x > xoa =]xoa 1 , f (x) > x1 1 , +∞[.





A.4. The coalition S = {1, . . . , j} Let us consider the case where coalition S is formed by agents 1 to j, i.e. S = {1, . . . , j}. At date t = T , we have Viab({1, . . . , j} , t ) = {x > x{1,...,j} = xoa 1 }. At time t = T − 1, we have Viab({1, . . . , j} , T − 1)

eki < m and a contradiction occurs.

At this stage we have to specify the behavior of agent j + 1 who is outside the coalition. We distinguish two cases according to whether player j + 1 is passive or remains active. oa • If x ∈]xoa 1 , xj+1 [ then players l = j + 1, . . . , n become passive

i=1

For large enough k, we claim that 0.5 ∗ (m +

∑j

x 1−



   ∃(ei ) ∈ Aj (x), ∀(el ) ∈ Bj ((ei ), x),        − − = x > xoa . oa 1 f ei − el > x1    x 1−    i∈S l̸∈S ′

eki = 0.5 ∗

, ∃e ∈ An (x), f

n −

  

But taking the sequence ek = j −

  xoa 1

1 k

) < m. Hence



Lemma 2. ∀x ∈ ]0, K [, we have f (x) > x and x > f

because if they aim at having a non-negative rent their efforts are null el = 0. Then, using Lemma 1, the viability condition becomes

 −1

(x).

Proof. The first part comes from assumptions given in Eq. (11). Since function f is continuously increasing, it is a bijection and the second part of the lemma is proved. 

sup f

e∈Aj (x)

1−

j −

  ei

x

oa > xoa 1 ⇒ f (x) > x1 .

i=1

−1 oa From Lemma 2, we have xoa x1 . 1 < x which implies x > f oa oa The viability domain is thus ]x1 , xj+1 [⊂ Viab({1, . . . , j} , T − 1).





L. Doyen, J.-C. Péreau / Mathematical Social Sciences 63 (2012) 57–64

• If x ≥ xoa j+1 , the stock of the resource is too high to ensure a positive rent for the coalition S = {1, . . . , j} whatever the effort played by player j + 1, . . . , n. Indeed, in that case a feasible reply of player j + 1 against the coalition S = {1, . . . , j} is to harvest ∑j ej+1 = 1 − i=1 ei ≥ 0 since it induces πj+1 = ej+1 (px − cj+1 ) ≥ 0. Such a catch strategy entails the extinction of the resource as ∑j f (x(1 − i=1 ei ) − ej+1 ) = f (0) = 0. Therefore, in that case [xoa Viab({1, . . . , j} , T − 1) = ∅. j+1 , ∞[ oa Hence ]xoa 1 , xj+1 [= Viab({1, . . . , j} , T − 1). Now let us assume the viability domain for the coalition S = oa {1, . . . , j} at time t + 1 is Viab({1, . . . , j} , t + 1) =]xoa 1 , xj+1 [. From the recursive formulation we have

Viab({1, . . . , j} , t )

   ∃e ∈ Aj (x), ∀(el ) ∈ Bj ((ei ), x),       − = x > xoa . oa oa 1 f ei ∈]x1 , xj+1 [   x 1−      

63

Consider x > xS > xoa 1 and assume for a while that there exists a relevant e∑ = (ei )i∈S ∈ AS (x). Set the effort of player 1 to e1 = 1 − i∈S ei ≥ 0 and the effort of other players l outside the coalition to zero el = 0. The strategy of outsiders l ̸= 1 is admissible as their profit πl (x, el ) = el (px − cl ) = 0 is zero. The c1 strategy of outsider 1 is also admissible as x > xoa 1 = p and its profit satisfies π1 (x, e1 ) = e1 (px − c1 ) ≥ 0. However with such a strategy, the dynamic of the stock collapses as

  f



x 1 − e1 −

−

−

ei −

= f (0) = 0 < xS ,

el

l̸=1, l̸∈S

i∈S

which leads to a contradiction. Consequently Viab(S , T − 1) = ∅. Therefore by backward induction Viab(S , t ) = ∅ for t < T . A.6. The coalition  S such that Viab(S , t ) = Viab( S , t ) for  S = ∪i ({1, . . . , i} ⊂ S )

i

• If x ≥ xoa j+1 , the stock of the resource is too high to ensure a positive rent for the coalition S = {1, . . . , j} whatever the effort played by player j + 1. Indeed, in that case a feasible reply of player j + 1 against the coalition S = {1, . . . , j} is ∑j to harvest e∗j+1 = 1 − i=1 ei for any e ∈ Aj (x). Such a catch strategy entails the extinction of the resource as f (x(1 − ∑j oa oa i=1 ei ))= f (0) = 0 ̸∈]x1 , xj+1 [. Therefore in that case oa [xj+1 , ∞[ Viab({1, . . . , j} , t ) = ∅. • If x < xoa j+1 , player j + 1 becomes passive because if he aims at having a positive rent its effort is null ej+1 = 0. Then using the viable feedbacks displayed in (10), the global viability effort of the coalition is such that



−

 


∗

−

e∗i (t , x) = α

i∈S

1−

f

  f



x 1−

−

ei −

i∈S

−

∈ Viab( S , T ).

el

l̸∈S

Let us derive the viable strategy e′S for the coalition S from the



ei 0

if i ∈  S, if i ∈ S + = S \  S.

It turns out that

• e′S ∈ AS (x). • For every choice (el )l̸∈S ∈ BN \S (e′S , x) ensuring for all outsider l ̸∈ S a positive rent πl ≥ 0, we have    − − ′ f x 1− ei − el i∈S

l̸∈S

 

 oa

x1

≥f

x 1−

 −

x

i∈S





f −1 xoa j +1 + (1 − α) max 0, 1 − x

ei −

−

el

∈ Viab( S, T )

l̸∈S

= Viab(S , T ).

 ,

(12)

with 0 < α < 1. A.5. The coalition S such that 1 ̸∈ S At date t = T , we have Viab(S , T ) = {x > xS } with xS > xoa 1 . At time t = T − 1, we deduce Viab(S , T − 1)

   ∃e ∈ AS (x), ∀(el ) ∈ Bj ((ei ), x),      − − S  = x > x f x 1 − e − . S ei − el >x 1      i∈S l̸=1, l̸∈S   

First pick some x ∈ Viab( S , t ). There then exists a strategy eS ∈ AS (x) such that for all outsider l ∈ N \  S for every choice el ensuring positive rent πl ≥ 0, we have

e′i =

The RHS of the inequality is since from Lemma 2  positive  we have x > xoa > f −1 xoa 1 1 . However we have to take into account that the LHS can be negative, which implies that viability effort takes the form

 −1

oa At time t = T − 1, we know that Viab ( S , T − 1) =]xoa 1 , xj∗ +1 [.

S

x



 Viab(S , T ) = {x > xoa 1 } = Viab(S , T ).

S as follows: viable strategy e′ for the coalition 

xoa j +1

x

We assume that 1 ∈ S. Let us write S =  S S + where  S = ∗ + S. At date t = T , the equality holds true {1, . . . , j } and S = S \  since



Once again we have to distinguish between two cases:

We deduce that x ∈ Viab(S , T − 1). Now pick some x ∈ Viab(S , T − 1) and let us prove that x ∈ oa oa ]xoa 1 , xj∗ +1 [. Assume for a while that x ≥ xj∗ +1 . Then for whatever feasible strategy of the coalition eS ∈ AS (x), a feasible reply of player j∗ + 1 against the coalition S is to harvest ej∗ +1 = 1 − ∑ ∗ ∗ ∗ e i∈S i ≥ 0 since it induces πj +1 = ej +1 (px − cj +1 ) ≥ 0. Such a catch strategy entails the extinction of the resource as

∑ j∗

S ∗ f (x(1 − i=1 ei − ej +1 )) = f (0) = 0 < x . Consequently we derive a contradiction. We therefore obtain the equality Viab(S , T − 1) = Viab( S, T − oa 1) =]xoa , x [ . ∗ 1 j +1 Similar backward induction reasoning makes it possible to generalize the previous result at any time t < T .

64

L. Doyen, J.-C. Péreau / Mathematical Social Sciences 63 (2012) 57–64

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