A constraint-based framework to study rationality, competition and cooperation in fisheries
aa r X i v : . [ q -f i n . E C ] J a n A constraint-based framework to study rationality, competition andcooperation in fisheries
Christian Mullon a, ∗ , Charles Mullon b a Unité de Recherche MARBEC (UMR212), IRD. France. b Department of Ecology and Evolution, University of Lausanne, Lausanne, Switzerland
Abstract
In this paper, we present a simplified framework to represent competition, coordination and bargaining infisheries when they operate under financial and technological constraints. Competition within constraintsleads to a particular type of mathematical game in which the strategy choice by one player changesstrategy set of the other. By studying the equilibria and bargaining space of this game when playersmaximize either profit or fishing capacity, we highlight that differences in financial constraints amongplayers leads to a tougher play, with a reduced bargaining space as the least constrained player canreadily exclude another from the competition. The exacerbating effects of constraints on competitionare even stronger when players maximize capacity. We discuss the significance of our results for globalocean governance in a current context characterized by financialization and technological development.We suggest that in order to maximize the chances of fruitful negociations and aim towards a fair sharingof sea resources, it would be helpful to focus on leveling current differences in the constraints facedbetween competing fishing systems by supporting local financial systems and technological control, beforeimplementing sophisticated economic tools.
Keywords:
Fisheries, competition, cooperation, constraints, financialization, global governance.
Suggested running title :
Rationality, competition and cooperation in fisheries
Keywords1. Introduction
Game theory and fisheries science have had a long intertwined history. The work of Gordon (1954) onfisheries, which anticipated Hardin (1968)’s analysis of the tragedy of the common, was an applicationof Cournot (1838)’s duopoly theory, which itself anticipated a Nash analysis of non-cooperative games(see Leonard, 1994; Myerson, 1999 for historical sketches). Since then, game theory has been widely usedfor the analysis of fishing systems and as theoretical guidance to organise the exploitation of our seas(Sumaila, 1999; Kaitala and Lindroos, 2007; Bailey et al., 2010).The close relationship between game theory and fisheries is illustrated by the use of the competitionfor a shared fish resource as an example of the prisoner’s dilemma (Feeny et al., 1996). When fishingsystems maximize their profit, competition leads to a situation in which fishing systems earn less thanwhen they cooperate. Competition in this case ends in over-exploitation and over-capacity, hence the ∗ . Corresponding author Email address:
[email protected] (Christian Mullon)
Preprint submitted to Elsevier 28 août 2018 eed to transition towards cooperation and coordination if the long term livelihood of fisheries is to bemaintained.The distinction between competition, cooperation and coordination has been a key point of game theorysince John Nash’s work (Nash, 1950a,b, 1953), which gives a theoretical framework to the basic humanpractice of bargaining. What has been referred to as Nash’s program (Binmore et al., 1986) still structuresgame theoretical considerations of fisheries systems. The bulk of the work devoted to the use of gametheory for fisheries has focused on non-cooperative situations (Arnason et al., 2000; Lindroos and Kaitala,2000) or, in the case of cooperation, on the formation of coalitions (Lindroos and Kaitala, 2000; Pintassilgo,2003; Lindroos et al., 2005).Fewer works have articulated fisheries problems in terms of cooperation, coordination and bargaining(Armstrong, 1998). However, the view that these three processes underlie the relationships among fishingsystems at different levels has been endorsed by several authors, who argue that a poor understanding ofthese process may have hindered our ability to tackle fishing crises (Fearon, 1998; Alcock, 2002; Munro,2009). For instance, Alcock (2002) uses game theoretic concepts to argue that the collapse of cod inCanada was at least partly due to a distributional conflict between small-scale and industrial fisheries.The absence of theoretical tools to understand this type of distributional conflict contributed to itsdelayed resolution and transition towards cooperation, resulting in catch diminutions coming about toolate.Alcock (2002) points at two culprits for the emergence of conflict that are evocative of current problemsfaced by many fisheries and that will guide our research. First, distributional conflict in Canada wascaused by small-scale and industrial fisheries suffering different constraints, due in particular to technicaland financial differences. Technological progress (Squires and Vestergaard, 2013) and financialization(i.e. modernization and global expansion of financial tools, Epstein, 2005) have both been playing anincreasingly important role in fisheries who increasingly rely on finance to purchase ever more advancedfleets. The two processes feedback to constrain fisheries’ scope of action as fisheries become increasinglydependent on technology to increase catch in order to repay finance.How technology and financialization influence the competitive and cooperative relationships betweenfishing systems through the constraints they generate has not been much investigated. Constraint-basedgame theory is an active field research that has been mostly concerned with games on networks (e.g.,Contreras et al., 2004; Li and Marden, 2010, see Krawczyk, 2005 for an application to environmentalproblems). Yet, the need for an assessment of the constraining effects of technology and finance is bolsteredby many current problems in fisheries other than distributional conflicts, such as the dangers of a rent-maximisation principle in developing countries (Béné et al., 2010), or the economic and ecological stakesassociated with deep-sea fishing (Norse et al., 2012). Here, we will show that framing economical problemsrelated to the exploitation of renewable resources within constraints offers a change of perspective onthese problems as the effect of an agent’s action on another agent translates into diminishing its scopeof action, rather than only reducing its income.The second cause of distributional conflict according to Alcock (2002) are differences in objectives betweensmall-scale and industrial fisheries (such as subsistence vs return on capital). Most of the applications ofgame theory to fisheries have been based on the idea of profit maximization, and are therefore relevantto one kind of rationality. The variety of modes of rationality at work in social-ecological systems such asfishing systems (Berkes et al., 2008) can be uncovered by addressing the simple question : who (e.g., boatskipper, firm owner, fishery, fleet, fishery manager, policy maker) maximizes what (e.g., catch, profit,capacity, well-being, pleasure, peace) ? Ignoring the variety of modes of rationality in fishing systemshas been argued to be potentially damaging for the articulation of efficient fisheries policies (Bromley,2009). In this paper, we will therefore be interested in the effects of rationality on the relationship between2shing systems and in particular, we will study the effects of capacity maximisation in addition to incomemaximisation.In view of the above considerations, we propose in this paper a constraint-based framework to studycompetition, cooperation and rationality in fisheries systems. The paper is organized as follows. (1)We define a game theoretical framework in order to consider technological and financial constraints,and games between players with different rationalities. (2) We identify a crucial criterion : the viabilitythreshold of a player, which is the maximum yield that a player faced with technological and financialconstraints can reach. (3) We compare the outcomes of games within constraints among competing fishingsystems according to whether players maximize income or capital. In particular, we analyze the games’equilibria and their bargaining space and find that the conditions that make it possible to move fromcooperation to coordination, to organize bargaining and to reach a fair distribution can be expressedin terms of the viability thresholds of competing players. (4) We discuss the relevance of our results tocurrent problems related to ocean governance.
2. Model
In order to introduce our constraint framework, we first represent in a highly stylized way how financialand technical imperatives constrain a single fishing system that exploits a fish stock and sells its yieldon a market. Such a fishing system could be a fishing country, its exclusive economic zone, its fleet andits fish market. For ease of presentation, most of corresponding mathematical relationships are assumedto be linear. Relaxing this assumption does not affect our results qualitatively. Throughout, upper- andlower-case symbols denote variables and parameters respectively.The fishing system with a capacity K for harvest expends an effort E , which cannot exceed capacity( E ≤ K ), into harvesting a fish stock that has a total biomass S . The yield Y from harvest dependson the level of stock S , as well as on effort E , according to the usual formula, Y = qSE, where q isthe fishing efficiency coefficient and represents the technological level of the fishing system. In turn, thebiomass of fish stock S decreases with yield Y , S = r − sY, where r denotes the level of biomass in theabsence of fishing, and s captures the effect of fishing on biomass.The profit of the fishing system depends on a balance of income and expenditures. Its only source ofincome is from sales. Its yield Y is sold on the market, with unitary price P decreasing with the totallevel of offer, P = a − bY, where a is the maximum unitary price, and b is the decrease in price accordingto offer. The system faces two sources of expenditures. First, fishing activity is costly. Unitary fishingcosts decrease with stock according to : f = g + h/S . Secondly, access to the market is costly, with eachunit of yield having a cost m . The net profit I of the fishing system therefore is I = Y P − Y ( f + m ) .From the above, we see that fish stock S , fish price P , income I can be expressed in terms of yield Y and fishing capacity K . Henceforth, we will consider that yield and capacity are exogenous and underliethe behavior of the fishing system. The other variables are considered endogenous.In conventional models of fisheries, the only – and often left unmentioned – constraint on a fishing systemis that profit must be non-negative : I ≥ . By contrast, in our framework, we will consider two constraintson the system. First, finance imposes an imperative rate of return upon the capital, Kp where p is theprice of a fishing capacity unit of the fishing system. Financial constraint is expressed as : I ≥ Kpk ,where k is the imperative rate of return, which implies an upper bound on capital : K ≤ I/ ( pk ) . Second,the capacity constraint E ≤ K can be re-written as Y / ( qS ) ≤ K .3f we combine both constraints, we obtain a space of feasible states in terms of the endogenous variables,capacity K and yield Y , E = YqS ≤ K ≤ Y ( P − f − m ) kp = Ipk , (1)that is shown in figure 1. With the idea that a system’s behaviour is defined by capacity K and yield Y ,the size and shape of the space of feasible states constrain fishing behaviour. As shown in fig. 2, the spaceof feasible states is particularly sensitive to finance constraints k and fishing efficiency q (i.e., technology).The position of a fishing system within its feasible state space, and the corresponding effect on othervariables of this position, allows to infer on the underlying rationality of the system’s behaviour. As shownin Fig 3, the position of a fishing system differs according to whether it seeks to maximize profit, capacity,yield or rate of return. In particular, if a fishing system is positioned close to the capacity boundary, thenthe corresponding fishing system is overcapacited and is not optimizing its profit because this requiresbeing close to minimal capacity. If it is close to the left boundary of the feasible space, the system isovercapitalized, and again, it is not optimizing its profit. The explicit consideration of the constraints onfisheries therefore offers an intuitive framework to visualise important features of fishing system, such asexcess capacity and overcapitalization. We now extend our constraint framework to two competing fishing systems, labelled A and B , that exploitthe same stock and sell their yield on the same market. A and B could be two countries with their ownfleets that fish in the same offshore part of the ocean and sell their yield on the same international market. A and B respectively have capacities K A and K B and choose to expand an effort E A ≤ K A and E B ≤ K B into fishing. The yields of both systems depend on their respective efforts and fishing efficiencies ( q A and q B ), Y A = q A SE A and Y B = q B SE B , respectively. Natural fishing stock and unitary selling market pricedecrease with the yields of both fishing systems according to S = r − s ( Y A + Y B ) and P = a − b ( Y A + Y B ) .Fishing systems then differ in terms of fishing costs ( f A = g A + h A /S , f B = g B + h B /S ), marketingcosts ( m A , m B ), vessel values ( p A , p B ), and imperatives rates of return ( k A , k B ). The profits of A and B then are I A = Y A ( P − f A − m A ) and I B = Y B ( P − f B − m B ) , and must satisfy I A ≥ K A p A k A and I B ≥ K B p B k B , respectively.As in the preceding section, we combine the constraints of the framework to find the space of feasiblestates for each fishing system in terms of capacity and yield : Y A q A S ≤ K A ≤ Y A ( P − f A − m A ) k A p A ,Y B q B S ≤ K B ≤ Y B ( P − f B − m B ) k B p B . (2)By contrast to the single fishing system case, fish stock S and prices P depend on total yield Y A + Y B here. Therefore the spaces of feasible states for each system are inter-dependent through the yield of theother. In particular, if one fishing system increases its yield, it shrinks the feasible space of the other (fig.4). The space of feasible states entail that there is a maximum total yield ( Y A for one fishing system, Y A + Y B for two competing systems) that cannot be exceeded in order for fishing systems to survive. We definethis value as the viability threshold and write it as T (fig. 5). Its derivation and expression, when thereis a single fishing system, are given in appendix 1 (equation 4).4f there is a single fishing system, the viability threshold depends negatively on the rate of return k andpositively on the fishing efficiency q , and on a balance between price and fixed costs : a − m − g (figure2, see appendix 1, eqs. 5-7). The existence of a viability threshold is due to finance and its relationshipwith technology : there is a value of yield that an system cannot exceed because otherwise, it is not ableto meet the financial imperatives that are necessary to purchase the technology to generate that yield.If there are two fishing systems, their viability thresholds T A and T B depend on the properties of eachsystem in the same way as when there is only one fishing system (eq. 4 in appendix 1) : the viabilitythreshold of a system depends negatively on rates of return ( k A or k B ) and positively on the fishingefficiency ( q A or q B ) and on a balance of prices and costs ( a − m A − g A or a − m B − g B ). This meansthat the viability thresholds of two competing may be different. As a consequence, some amount oftotal yield may be viable for one system, but simultaneously be the death warrant for the other (e.g., if T B < Y A + Y B < T A , then only A can survive). Within these constraints, we now assume that fishing systems have a quantified objective for which theycompete. This competition can be expressed as a non-cooperative game, in which players are fishingsystems, strategies are pairs of yield and capacity, and payoffs are their quantified objective. As theset of feasible strategies of a player depends on the strategy of the other, such a game is said to be aGeneralized Nash’s equilibrium Problem (Facchinei et al., 2007). As usual, the equilibria of the gameare pairs of strategies (( Y A , K A ) , ( Y B , K B )) such that when a player unilaterally changes its strategy, itobtains a smaller payoff (Facchinei et al., 2007).We will study the game and its equilibria under two scenarios. In the first scenario, the quantified objectiveof the systems is profit. When A and B have equal viability thresholds ( T A = T B ), both systems stablyco-exist (fig. 6, top). When A and B have different viability thresholds ( T A = T B ), the situation is morecomplicated (fig. 6, middle and bottom). For the sake of argument, suppose the viability threshold of A is greater than B ’s ( T A > T B ), i.e., A is in a dominant position and can exclude B from the competitionby choosing a yield greater than B ’s viability threshold ( T A > Y A > T B ). We find two possible outcomes,either A ’s optimal (i.e., which maximises profit) yield is greater than T B in which case B is excluded anddies out, or A ’s optimal yield is lower than T B in which case B survives (this is what is plotted in fig. 6,right). The greater the difference between the viability thresholds of two competing systems, the morelikely one system will vanish (fig. 6, bottom).In the second scenario, the quantified objective of the systems is capacity. Under this scenario, wefind that when A and B have equal viability thresholds, the game has an infinite number of equilibria(fig. 7, top). Every sharing of the total yield, which equals the viability threshold of both systems( Y A + Y B = T A = T B ), is a possible equilibrium. This process can be characterized by a Nash demandgame (Nash, 1953). In order for them to stably co-exist, it is therefore for them to agree on how toshare the total yield. Cooperation in this case is a requirement. When A and B have different viabilitythresholds, the game has a single equilibrium under which only the system with the greatest viabilitythreshold survives (fig. 7, bottom). This is because by maximizing its capacity, the system with thegreatest viability threshold always pushes the total yield above the threshold of the other system, whichcan no longer satisfy its financial constraints and thus goes out of business. In some cases, there exists pairs (( Y A , K A ) , ( Y B , K B )) of strategies that are away from the Nash equili-brium and that provide a better reward to both players. Such strategies constitute the bargaining space5f the system as both players have an interest to bargain in order to reach these strategies. Bargainingspaces are plotted in the right parts of figs 6 and 7. Inside the bargaining there is what is referred to asthe Pareto frontier. It is made of pairs of strategies such that an unilateral change of strategy results in asmaller reward to one one player (see (Muthoo, 1999) for definitions). The goal of bargaining is to reacha point on the Pareto frontier.As the top and centre right plots of fig. 6 illustrate, when players maximize profit and have equal ormarginally different viability thresholds, a bargaining space exists. Hence, competing players may reachan agreement through bargaining. This situation parallels the prisoner’s dilemma problem : both playersmay reap greater payoffs if they coordinate to cooperate. When have important differences in viabilitythresholds, the bargaining space is empty (bottom right of fig. 6) is because the dominant player has nointerest in bargaining : it can simply eliminate the other from the competition. Differences in viabilitythresholds therefore reduce the possibilities of coordination.When players maximize their capacity and have equal viability thresholds, the bargaining space is large(top right of fig. 7) because all pairs of possible strategies other than the Nash equilibrium provide abetter reward. However, the situation is different to the case when players maximise profit because here,the threat values of a player (i.e., the maximum payoff it obtains in the absence of negotiation and withthe other player’s worst strategy) is zero. Bargaining is therefore imperative and the situation is akinto a pure demand game in which players have to agree on the sharing of a pie (Nash, 1953). Whenplayers maximize capacity, any difference in viability thresholds leads to the collapse of the bargainingspace (centre and bottom right of fig. 7). As before, this is because the dominant player can exclude theother from the competition and thus has no need to bargain. The sensitivity of the bargaining space todifference in viability thresholds when players maximize capacity highlights that when fishing systemsoperate with modes of rationality other than profit maximization, possibilities for bargaining can besignificantly destabilized.Our analyses show that differences in viability thresholds compromise the co-existence of multiple systems,first by exacerbating the effects of competition and second by shrinking the possibilities of bargaining.This is particularly so when capacity is maximized. When systems have different viability thresholdsand are maximizing capacity, two steps are necessary to ensure their long-term survival. First, fisheriesmanagement must aim at equalizing viability thresholds among competing systems. Recall that viabilitythresholds T depend positively on cost related function ( a − m − f ) and fishing efficiency q , and negativelyon rates of returns k . There are therefore three ways to equalize viability thresholds. For the sake ofargument, suppose that A is dominant over B , i.e., T B < T A . Then, to reach T B = T A , it is possible tomodulate costs (i.e., increase ( a − m B − f B ) and/or decrease ( a − m A − f A ) ), to change fishing efficiencies(i.e., increase B ’s q b and/or decrease A ’s q A ), or change the rates of return (i.e., increase A ’s k A and/ordecrease B ’s k B ). Second, players must find a fair division of the yield between them.
3. Discussion
We have framed a game between fishing systems within the technological and financial constraints theyface. These constraints are reflected in the existence of viability thresholds , which correspond to themaximum total fish resource that a fishing system can catch. A fishing system cannot exceed its viabilitythreshold because otherwise, it is not able to meet the financial imperatives that are necessary to purchasethe technology to generate that yield. Different fishing systems exploiting the same resources influenceone another by constraining their scope of action, and when actors face different constraints, one canpotentially “kill" the other by pushing the system above the viability threshold of its competitor.6hen we analysed games’ equilibria and bargaining spaces, we found that they depend critically onviability thresholds and how they differ among players. When fishing systems have different viabilitythresholds, competition is exaggerated, and the transition towards coordination through bargaining thatis needed to avoid over-exploitation and over-capacity is compromised. This is because when a dominantcan simply eliminate the other from the competition, it has no motivation in finding a solution throughbargaining.
We found that the effects of differences in viability thresholds are particularly important when fishingsystems maximize capacity. This result leads to the question of the pertinence of such a behaviour, whichmay appear incongruous at first. The difference between profit maximisation and capacity maximisationas a goal can be framed as a question : at national level, do national fishing policies encourage investmentin boats for insuring a bigger income to a country, or do they target a minimal profit to fishing companiesfor insuring the development of their national fishing fleet ? Or, at another level, do fishing entities acquirebig boats to catch fish and thus earn money, or do they sell fish in order to earn money and thus financebig boats ? The answer to such questions is not obvious, at least to us.Capacity maximisation is perhaps the most parsimonious explanation to overinvestment and overcapa-city (Bell et al., 2016). A more explicit illustration of fishing capacity maximisation can be found indeliberate national policies. In India for example, there have been 12 national plans since 1950 that allsupport capacity development (Bhathal, 2014), which has led to significant excess capacity, a productionthat exceeds the needs of the country (Bhathal and Pauly, 2008; World-Bank, 2010; Sathyapalan et al.,2011; Bhathal, 2014) and a national opposition to any international agreement for banning subsidies(Campling and Havice, 2013). This nationwide increase of capacity was initially motivated by the neces-sity to develop and to eradicate poverty (Thorpe et al., 2005), but seems today to be motivated by theanticipation of competition with fleets from developed countries (Campling and Havice, 2013)..The notion that fishing entities favour an increase in capacity also provides a straightforward explanationto (1) the failures of decommissioning schemes (i.e., plans for decreasing fleets size through buybacks) evenwhen they come at the expense of profit (Holland et al., 1999), (2) the observation that most subsidies,whatever their intentions, are redirected towards new equipment (Sumaila et al., 2010), (3) the hightechnical turnover (i.e., capital stuffing Townsend, 1985) that has been observed in many fishing entities(Pauly et al., 2002). Capacity maximisation in fisheries behaviour may be motivated by the anticipationof future capacity limitations (Guyader and Thébaud, 2001), of individual quotas distributed accordingto capacity (Squires et al., 1995), or of subsidies of buybacks programs (Clark et al., 2005), or by thedesire to diminish the risk of exploiting a highly variable resource (Branch et al., 2006).In addition, the standard hypothesis that fishing systems behave as to maximise profit does not alwaysfit well with the empirical evidence. Fisheries anthropology (Acheson, 1981; Palsson, 1988; Olson, 1997),studies on fishermen behavior (Jentoft, 1989; Salas and Charles, 2007; Holland, 2008; Rijnsdorp et al.,2008) and on their communities (Jentoft, 2000; Pomeroy et al., 2006; Béné et al., 2010) describe fishermenwho are mostly committed to their occupation, even if that means remaining poor, and who co-existby relying on mutualistic rules (Jentoft, 1986; Allison and Ellis, 2001). They constitute closely-knit,sometimes closed, societies, within which competition is often symbolical : for the biggest fish, the biggestcatch, the most productive fishing plots, the most efficient gear, the biggest boat or the best skipper(Palsson, 1988).Altogether, these observations lend support to the idea the some fishing systems are not only concernedwith profit, but may also seek to increase capacity as a goal. The significant effects of capacity maxi-7isation on constrained fisheries that we observed on our results call for future assessments of whethersuch behaviour is at play, and more in-depth analysis of its effects on competition and bargaining.
As economic development and population growth increase demand for seafood, technological progress isopening up new fishing areas in high- and deep-seas, which according to the 1982 United Nations Conven-tion on the Law of the Sea (UNCLOS) are governed by a principle of freedom of fishing (Oda, 1983). Forreasons that are biological, ethical (Moore and Squires, 2016; Barbier et al., 2014; Brooks et al., 2013) oreconomic (Rogers et al., 2014), a growing number of calls have been made to control access and organisenegotiations on how to distribute fishing pressure in these new areas (Hayashi, 2004; Molenaar, 2005;Moore and Squires, 2016; Barbier et al., 2014). Future issues faced by such a global ocean governanceand how they relate to our results can be understood by first considering the context in which they wouldtake place.The institutional context is one in which governance depends on multilateral negotiations (Barrett, 2005),which rely on a principle of efficiency (negotiation must succeed) and of rationality (players must have aninterest in the success of the negotiation). Meanwhile, the economic context is one of globalization andfinancialization (Whalen, 2001), which in fishing systems generates highly competitive, highly efficient,excessively capitalized fishing entities (Squires and Vestergaard, 2013; Eigaard et al., 2014) due to amutual reinforcement of technical progress, financial modernization and competitiveness. Finally, theinternational political context is underlain by a requirement for equity (Ostrom et al., 1999; Kaul et al.,1999; Rao, 1999; Sen, 1999), which emphasizes that how the access to fish is shared should be fair,especially for developing countries (Allison, 2001).In view of this context and our results, we may anticipate several possible futures for ocean governance.First, a politically stressed international context leads to the abandonment of the principle of equityand all obstacles to sea grabbing are lifted. The re-enforcement of technology, finance and competitioncontinues and leads to a world in which only a small number of large fishing companies survive byproviding revenues to the globalized finance system. Second, following what has been done for climatechange negotiations, countries agree on an institutional frame but do not consider any kind of rationalityother than profit maximization, and thus proceed to use tools such as side payments to ensure equity,which probably fail for the same reasons buybacks programs fail. Third, the re-enforcement of technology,finance and competition is halted by fisheries policies that (1) maintain fragile differentiated independentlocal finance systems, such as local cooperative credit systems, in order to emancipate fishing entitiesfrom globalized finance ; and (2) stabilise fishing efficiency at a reasonable level, instead of systematicallypromoting competition by encouraging efficiency. As a result, viability thresholds are levelled, bargainingspaces are enlarged and the equity issue can be appropriately addressed.
To conclude, our model highlights that because the competition of fishing systems occurs within constraints,it is inherently unfair as differences in viability thresholds can lead to the survival of only the most viablefishing systems. This unfairness is exaggerated when fishing systems operate with modes of rationa-lity other than profit maximization, such as capacity maximisation, that destabilize the possibilities forbargaining. Altogether, our results lead us to question the relevance of creating economical tools, suchproperty rights, fishing rights or side payments, that aim towards enlarging the space of bargaining inthe framework of the prisoner’s dilemma, adding to the growing number of voices that question thisframework (Feeny et al., 1990; Alcock, 2002; Richerson et al., 2002). In order to fulfill the demands ofequity on future global governance, we suggest that it would be more relevant to focus on equalizing the8onstraints on fishing systems (i.e., their viability thresholds) by paying special attention to their localfinancial systems and by limiting technological development.
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We show here that there exists a maximal value for yield and give its expression. We start from thetechnological and finance constraints on yield that have been obtained in above equation 1 : E = YqS ( Y ) ≤ K ≤ Y ( P ( Y ) − f ( Y ) − m ) kp Using definitions : S ( Y ) = r − sY , P ( Y ) = a − bY and f ( Y ) = g + h/S ( Y ) , they can be developed as : q ( r − sY ) ≤ kp × (( a − bY ) − ( g + hr − sY ) − m ) That is : kp ≤ (( a − bY ) − ( g + hr − sY ) − m )( q ( r − sY )) To simplify, we put l = a − g − m . Obviously situations with l < are meaningless. It is routine to showthat previous inequality can be re-written as : AY + BY + C ≥ (3)With A = bqsB = ( − bqr − aqs + gqs + mqs ) = − q ( br + sl ) ≤ C = ( − kp − hq + qrl ) = q ( rl − h ) − kp We look for values of Y satisfying inequality 3. We solve the corresponding second degree equation.Possible yields if they exist are outside its roots. Its discriminant is : ∆ = q ( br + sl ) − qbs ( qlr − ( qh + kp ))= q ( br − sl ) + 4 qbs ( qh + kp ) > Solutions are : T = q ( br + sl ) − √ ∆2 bqsT = q ( br + sl ) + √ ∆2 bqs Recall that S = r − sY . Thus we are interested only in values of Y that are outside roots and such that Y ≤ r/s . We remark that we have : √ ∆ = p ( br − sl ) + 4 qbs ( qh + kp ) > | br − sl | We deduce : rs − T = br − sl bs + √ ∆2 bs > T − rs = − br + sl bs + √ ∆2 bs > Therefore, values of Y greater than T are not valid. Possible values for Y are thus such that Y ≤ T .13alues of parameters ( a, b, s, r, g, h, etc. ) resulting in a system with T < are not realistic.Finally the viability threshold is : T = q ( br + sl ) − √ ∆2 bqs (4)It is easy to derive the dependency of the viability threshold according to systems parameters. We have : ∂T∂q = kpq √ ∆ > (5) ∂T∂k = − p √ ∆ < (6) ∂T∂l = 12 b √ ∆ + q ( br − ls ) √ ∆ ≥ b √ ∆ − | q ( br − ls ) |√ ∆ > (7)14 ables Maximizing profit Maximizing capacityEqual viability thresholds One equilibrium with co-existence of bothfishing systems.An explorable bargaining space Every share of the total yield, whichequals the viability threshold, is a Nashequilibrium.The bargaining space is made of all pos-sible pairs of strategies. The Pareto fron-tier is made of a sharing of the yield at thecommon value of viability thresholds.Sligthly different viabilitythresholds One equilibrium which allows the co-existence of both systems.An explorable, but complicated, bargai-ning space. One equilibrium with the survival of onlythe dominant system.Bargaining space is empty.Different viability thre-sholds One equilibrium with the survival of onlythe dominant system.Bargaining space is empty. One equilibrium with the survival of onlythe dominant system.Bargaining space is empty.
Table Competition between fishing systems. Nash equilibria and bargaining space.15 igures captions
Figure The space of feasible states for a fishing system. A fishing system is defined by its yield Y (x-axis) and itscapacity K (y-axis). The lower bound reflects the technical constraint : a minimal capacity is required to achieve a certainyield. The upper bound, meanwhile, is the financial constraint. Together, the constraints define a feasible state space wherecapacity allows yield and yield allows to finance capacity. Feasible states are shown in green. Figure Effects of fishing efficiency and rate of return on feasible space. Left : Decreasing fishing efficiency q increasesthe minimal capacity and shrinks the feasible space. Right : Increasing rate of return k decreases the maximum capacityand and shrinks the feasible space Figure Constraints and rationality. The space of feasible state is colored (increasing from green to yellow) accordingto the values of variables (endogenous yield and capacity or exogenous profit and rate of return. igure Competition between fishing systems. The feasible set of a fishing system depends on the yield of the otherfishing system
Figure Viability thresholds of two competing fishing systems igure Competitive equilibrium and bargaining space with profit maximisation. From top to bottom row : (a) equalviability thresholds :