International Agreements in the Multispecies Baltic Sea Fisheries

Abstract

This study analyses a multispecies (cod, herring and sprat) partition function game among three asymmetric countries in the Baltic Sea. Through an empirical application, the study highlights the importance of dynamic multispecies models in coalition formation. We show that the grand coalition between the three countries can be stabilised when the most efficient country compensates the other countries. This solution is not feasible when using a single-species model for the cod population; thus, taking into account the multispecies nature of the Baltic Sea fisheries, the scope of cooperation increases compared with a single-species simulation. In contrast to the predominant conception, climate change may improve the likelihood of a binding agreement because with a lower cod biomass, countries will gain more from cooperation.

1 Introduction

In the early fisheries studies of Gordon (1954) and Scott (1955) with a sole owner as the single decision maker, it was demonstrated that there is an optimal solution without externalities. With most real world resource problems, however, the involvement of several decision makers is the reality. Game theory is an appropriate tool for analysing the strategic interactions of more than one rational decision maker. A fishery is one example of a common pool resource where decision makers exploit the same resource and affect the availability of the fish stock and hence the economic outcome of others by their own actions, and they thereby generate externalities. To analyse the effects of positive externalities where mergers positively affect the payoffs of non-mergers, a partition function (P-function) approach was introduced in fisheries games for the first time by Pintassilgo (2003).

Sumaila (1997) conducted one of the first multispecies studies that applied game theory. He analysed the cooperative and non-cooperative management of an age-structured predator–prey system in the Barents Sea and concluded that it is economically optimal to exploit both species under a cooperative situation if market conditions remain unchanged. A continuous time fisheries model of two ecologically interacting species was developed by Wang and Ewald (2010). Their study took ecological uncertainty into consideration and emphasised its importance due to climate change and increasing sea temperatures. According to the study’s main finding, competing species succeed more under cooperative management, whereas non-cooperation allows a predator–prey system to thrive. Fischer and Mirman (1996) studied externalities in a two-species system with two agents harvesting in a fishery and found a closed-loop Nash equilibrium. A dynamic externality occurs as two countries compete for catches, and a biological externality arises from the multispecies nature of the system. Datta and Mirman (1999) analysed similar aspects but concentrated on the effects of market power with dynamic and biological externalities. A non-cooperative game in a two-species system with biological interactions was presented by Kronbak and Lindroos (2011). The study demonstrated that a larger number of players can lead to species extinction, depending on the biological interdependencies.

Game theoretic analysis has often relied on a characteristic function (C-function) approach, which aims to find a fair distribution of cooperative benefits (see, e.g., Arnason et al. 2000a; Duarte et al. 2000; Kaitala and Lindroos 1998; Li 1998). However, C-function games usually ignore positive externalities, which follow from a lower fishing effort within a coalition, thus leading to a higher fish stock and therefore improved harvest possibilities for the non-member. Positive externalities increase the incentive to free ride. Kronbak and Lindroos (2007) applied C-function games by developing a new sharing rule that explicitly addressed the stability of cooperation when externalities are present. The authors highlighted the difficulty of sharing the payoffs among players in a C-function game.

Pintassilgo (2003) introduced a partition function, or P-function, which takes into account the whole coalition structure and not just the coalition in question. The study concluded that the eastern Atlantic bluefin tuna fishery could not achieve any cooperation with three players, not even partial cooperation. Pham and Folmer (2006) found that cooperation is not feasible due to free-rider incentives. Similarly, Pintassilgo and Lindroos (2008) demonstrated that with more than two symmetric players, the only stable coalition structure is with two singletons. Pintassilgo et al. (2010) extended this analysis to asymmetric players and found that partial and full cooperation can achieve stability depending on the level of asymmetry, the degree of efficiency and the number of players. Kulmala et al. (2013) strengthened this conclusion and found stable, partial cooperation with asymmetric players in a P-function game by allowing more flexible fishery strategies through the reallocation of benefits.

Although a wide range of game theoretic studies have been conducted in a single-species context (for some recent applications, see, e.g., Björndal and Lindroos 2012; Diekert et al. 2010; Kulmala et al. 2013), and multispecies aspects have gained increasing interest, applications using a coalition formation approach in a multispecies setting have been more or less ignored, which places our study in a central position in the literature. This paper contributes to the existing literature by combining a dynamic multispecies model and coalitional games with a partition function approach, where the multispecies nature arises from the biological interactions between the species. The model does not take into account the economic interdependency of catches, which is a quite common assumption in multispecies studies and implies that fishermen have full selectivity, i.e., they can directly target species without any by-catch of other species.

Since the 2000s, Denmark, Poland and Sweden have accounted for the majority of the total cod catches in the Baltic Sea (65–74 % of the cod catches; see Fig. 1), and this paper consequently focuses on the fisheries of these countries. The strategic interactions of the fishery are analysed by concentrating not only on cod but also on the fishery of its main prey species—herring and sprat. This analysis has several objectives: First, we examine which coalition structures are feasible regarding stability, and what the optimal catch shares of each species are. The results may be useful to consider if quota trading between individuals from different countries becomes possible in addition to the current quota exchange on the country level. Second, the effects of using a multispecies approach instead of a single-species model are assessed, and third, we explore whether the scope of cooperation is dependent on the salinity level in the Baltic Sea. Salinity conditions have a link to climate change, as there is evidence that continuing climate change may cause a decline in the salinity level of the Baltic Sea (Meier 2006; Neumann 2010), which may therefore also have a negative effect on cod recruitment. To address these questions, a dynamic multispecies bio-economic model is constructed and combined with a partition function approach. The conclusions of this paper could also be valuable for other common-pool fisheries that include a multispecies nature in two particular ways: our study develops a framework for multi-species partition function approaches, and it highlights the importance of the multispecies approach compared with the traditional single-species approach.

Fig. 1

Cod catches and total allowable catches (TACs) in the Baltic Sea from 1975–2013 (t). Source Modified from ICES (2013) and ICES (2014)

The next section describes the case area and the challenges of managing the Baltic Sea fisheries. Section 3 presents the underlying bio-economic model, while Sect. 4 introduces the P-function approach and compiles the results. Finally, Sect. 5 concludes.

2 Challenges in Baltic Sea Fisheries Management

The Baltic Sea is a fruitful case study for a game theoretic application, as the exclusive economic zones (EEZs) of the coastal countries do not reach 200 nautical miles in the narrow, semi-enclosed sea; thus, there are essentially no high seas at all. Since 2004, the EEZs of the member states have formed one common European Fishing Zone in which all members are allowed to act without restraints. This approach has created a true common pool fishery with all member nations harvesting in the same productive area. All of these coastal countries, apart from Russia, belong to the European Union (EU) and annually negotiate and agree on total allowable catches (TACs) for the most important species according to the Common Fisheries Policy (CFP) and scientific recommendations. The aim in Baltic Sea fisheries management is to obtain a joint agreement between coastal countries, where each participant cooperatively agrees on TACs and remains within the quotas. The TACs are distributed among individual countries according to a rule known as the relative stability key, and the quota distribution among fishermen on a national level is based on country-specific rules. The national quotas can be exchanged between countries, but not on an individual level.

Although EU member states are bound by the European Commission’s Common Fisheries Policy (CFP) and its relative stability principle, countries have already made strategic choices. Recently, fishing quotas have been exchanged between countries at a rate of 4 % of the total turnover in EU fisheries, although exchange has only been allowed temporarily and thus has not affected the relative stability (Andersen et al. 2009). Although the CFP reform (European Union 2013) states that ‘member states may establish a system of transferable fishing concessions’, the principle of relative stability must still hold.

However, some studies have shown that the relative stability principle does not work particularly well. Kulmala et al. (2013) showed that cooperation cannot be sustained in the Baltic salmon fishery under this principle; instead, the stable outcome is non-cooperation. The authors suggested that more flexible fishing strategies and sharing schemes could yield a higher level of cooperation. Additionally, van Hoof (2013) emphasised that because conditions (e.g., stock development, new fishing strategies, fleet evolution and demand) have changed since the introduction of the relative stability key in the early 1980s, a fixed allocation principle cannot accommodate all of the new circumstances. Hence, our simulation demonstrates what the situation would be with a more flexible management system without the restrictions of relative stability, and it is also applicable to any other fishery, even on a fleet level.

The profitability of a fishery varies extensively depending on the fleet segment, and some segments indeed face poor profitability. According to the Green Paper on the reform of the CFP published by the EU in 2009, management of the Baltic Sea fisheries has been economically unsuccessful due to the overexploitation of the stocks. The main reason for overfishing is assumed to be overcapacity in the fleets, which has arisen due to subsidies introduced in the 1970s that gave fishermen incentives to invest in larger vessels and new technologies. Once the fleets were highly capitalised, the decision makers had to guarantee the social and economic welfare and employment of fishermen, and TACs were thus intentionally set at higher levels than scientific recommendations. Consequently, management was driven by a short-term perspective that essentially ignored longer-term stock conservation goals (Aps and Lassen 2010; Edwards et al. 2004; Sumaila and Walters 2005). Due to their high levels, the TACs have not always restricted the yield, at least when considering reported catches. However, not all countries may have complied with the agreement; instead, some may have been free riding and exceeding the TACs, which would negatively affect potential benefits in the future because the stock would be lower than assumed. Free riding becomes profitable if a country receives higher payoffs by deviating and harvesting more fish than has been agreed.

Indeed, there is evidence of the adverse behaviour of free riding in the Baltic cod fishery. At times, even the reported cod catches have exceeded the TACs, and in addition, catches have often been misreported. It is estimated that 35–45 % more cod were landed in the Eastern Baltic from 2000–2007 than reported (ORCA 2008). In 2008–2009, the estimated misreports for Eastern Baltic cod declined to 6–7 % from the previous estimates, likely due to stricter fishing controls and decreased effort. However, this estimate was only for one year and may not accurately describe the situation. Since 2009, only a few countries have provided information about misreporting (ICES 2014). Poland is often claimed to be responsible for non-compliance, but the member states with the largest shares of TACs have also been accused. One problem in Poland is the low likelihood of inspection due to insufficient manpower and other resources in fisheries inspectorates. Therefore, inspections have not been effectively coordinated (ORCA 2008). Fisheries managers need tools and knowledge to prevent undesirable actions. One solution could be side payments, where the country that most highly values the stock makes a payment to another country if that country agrees to reduce its harvest. Thus, each country receiving such a payment would receive a free-rider payoff plus some extra payoff for cooperating.¹

Another reason for low profitability is that the quotas are not set at the economic optimum, as the CFP still follows the biological maximum sustainable yield (MSY) target when setting the TACs. However, TAC utilisation has recently been very low (Fig. 1), and in 2013, only 46 % of the Eastern Baltic cod TAC was harvested (ICES 2014). Potential reasons for this low utilisation are the reduced size of the fishing fleet due to scrapping and the low market prices of cod. The reduced cod price is probably a consequence of a decline in demand that reflects the negative images consumers still have about the cod fishery, which tended to be too aggressive. The prices are expected to decrease even further, as Norwegian cod is becoming increasingly abundant within the EU market. In addition, Baltic cod have recently suffered from malnutrition due to their improved stock size and lack of prey biomass, which is due to the relatively small geographic overlap of cod and its prey species, thus resulting in skinnier individuals and consequently in discards and high grading (European Commission 2013; ICES 2014). However, the prices of pelagic species have recently increased. Thus, with better management, higher stocks might be achieved, which would probably lead to improved economic performance due to a higher catch per unit of effort and lower costs (Blenckner et al. 2011).

3 Model

3.1 Biological Model

Here, an age-structured model is used to simulate the population dynamics of cod, herring and sprat in the Baltic Sea. This approach is chosen due to its ability to increase the accuracy of the simulated stock development. The population dynamics follow an equation in which the number of individuals N for species m in age classes² a (between 2 and 8) at time t depends on the previous period $t-1$ as follows:

$$\begin{aligned} N_{m,a} (t)=N_{m,a-1} (t-1)\times e^{-\left( {\sum _i {F_{i,m} (t-1)+M_{m,a-1} (t-1)} } \right) } \end{aligned}$$

(1)

where $N_{m,a-1} (t-1)$ is the number of individuals in age group $a-1$, e is Euler’s number, $F_{i,m} (t-1)$ represents the instantaneous fishing mortality of country i, which is assumed to be constant for each harvested age group and is the sum of all countries’ fishing mortalities, and $M_{m,a-1} (t-1)$ is the age-specific natural mortality (Hilborn and Walters 1992). For herring and sprat, the natural mortality is further divided into $M1_{m,a-1}$ and $M2_{m,a-1} (t-1)$, where the latter represents the predation mortality and the former all other natural mortality. For cod, there is only one combined natural mortality, $M_{m,a-1}$, which is constant over all periods.

The number of cod recruits in the first age class follows Heikinheimo’s (2011) application of a function by Hilborn and Walters (1992):

$$\begin{aligned} R_c (t)={\textit{SSB}}_c (t)\times e^{\alpha _c -\beta _c \times {\textit{SSB}}_c (t)+\gamma \times \varepsilon } \end{aligned}$$

(2)

where ${\textit{SSB}}_c$ ³ is the cod spawning stock biomass, $\alpha _c$, $\beta _c$, and $\gamma$ are recruitment parameters, and $\varepsilon$ represents a salinity parameter. The salinity parameter illustrates the salinity conditions for cod recruitment and is the deviation from the average salinity level in the Baltic Sea corresponding to the scenarios in ICES (2005). The deviation was approximately 0.8 from 1974–1986, which is used here as a proxy for ‘high’ salinity conditions. A period of poor salinity and low cod recruitment occurred from 1987–2004, thus, $-0.05$ is used as a proxy for ‘low’ salinity conditions, which are assumed to represent the current situation (Heikinheimo 2011).

The recruitment for herring and sprat follows Ricker’s function according to ICES (2005):

$$\begin{aligned} R_m (t)=\alpha _m \times {\textit{SSB}}_m (t)\times e^{-\beta _m \times {\textit{SSB}}_m (t)} \end{aligned}$$

(3)

where $R_m (t)$ is the number of recruits, and $\alpha _m$ and $\beta _m$ are species-specific recruitment parameters.

The spawning stock biomass, ${\textit{SSB}}_m (t)$, is the sum of the biomasses of mature fish over all age groups:

$$\begin{aligned} {\textit{SSB}}_m (t)=\sum _{a=1}^8 {N_{m,a} (t)\times e^{-\left( {\sum _i {F_{i,m} (t-1)+M_{m,a} (t-1)} } \right) }\times MO_{m,a} \times WA_{m,a} } \end{aligned}$$

(4)

where $MO_{m,a}$ is the proportion of mature individuals in age group a, and $WA_{m,a}$ is the age-specific mean weight of an individual.

Figure 2 illustrates the species interactions that are taken into account in the biological model. The main interaction is the total cod predation on herring and sprat, $M2_m$, which is known as predation mortality. This dependence is modelled by using predation mortality functions based on functional responses according to Heikinheimo (2011). The functional response $P_a\left( t\right)$ illustrates the rate of predation as a function of prey density, i.e., the number of herring and sprat consumed by one cod in age group a in one year t:

$$\begin{aligned} P_a (t)=\frac{G_a \times \left( {N_{h+s} \left( t \right) } \right) ^{n}}{(N_{h+s} (t))^{n}+\left( {D_{h+s} } \right) ^{n}} \end{aligned}$$

(5)

where $G_a$ is the maximum annual consumption of prey individuals (sum of herring and sprat) by one cod within an age group a. The values for $G_a$ are derived according to the key run of the Study Group on Multispecies Assessment in the Baltic (SGMAB) (ICES 2005). A half-saturation constant $D_{h+s}$ ⁴ represents the size of the combined herring and sprat stock when cod consume half of the maximum consumption. This approach follows Heikinheimo (2011) and is applied to the entire Baltic Sea. $N_{h+s} (t)$ is the total size of the herring and sprat stocks in numbers, and n is a constant determining the type of functional response. In this paper, the type III functional response is used, which is a logistic curve. It is commonly applied to predators that have several prey species; according to Heikinheimo (2011), it is able to produce realistic results within this system. This type of response function allows the predation on prey to increase with a decelerating rate as the prey density grows. With a lower density level of prey, the predator is assumed to switch to consuming other species.

Fig. 2

The interactions of the species taken into account in the model

The species-specific functional responses are:

$$\begin{aligned} P_{m,a} (t)=\frac{w_m \times N_m (t)\times P_a (t)}{\mathop \sum _m w_m \times N_m \left( t \right) } \end{aligned}$$

(6)

where $N_m\left( t\right)$ is the total number of individuals in the stock. As cod prefer sprat to herring, a preference coefficient $w_m$ is used that is higher for sprat than for herring (Heikinheimo 2011). The total annual predation mortalities are

$$\begin{aligned} M2_m (t)=\sum _{a=1}^8 {\frac{N_{c,a} \left( t \right) \times P_{m,a} (t)}{N_m (t)}} \end{aligned}$$

(7)

where $N_{c,a}\left( t\right)$ is the number of cod in age group a. Because younger individuals are preferred as food, the predation mortality of herring in the first age group is assumed to be three times higher (i.e., $M2_{h,1} (t)=3\times \sum _{a=1}^8 {\frac{N_{c,a} (t)\times P_{m,a} \left( t \right) }{N_m (t)}}$) than in the other age groups based on the ICES data.

This type of simple predation function is easier to apply in bio-economic modelling, and the use of more complex and data-demanding ecosystem or multispecies models is therefore avoided. In addition, simplified models are effective for understanding basic interactions. One example of an alternative multispecies model is the stochastic multispecies model (SMS) used by ICES. Another popular way to examine effects on the whole marine ecosystem is a mass-balanced ecosystem model. However, these models demand a great deal of data, such as data concerning the catch-at-age, stomach contents and annual feeding rate for the predators. Complicated multispecies models are also very sensitive to structural uncertainty (ICES 2005). A simpler alternative could be the Lotka–Volterra predator–prey interaction model. Unfortunately, it is not considered to be a very realistic model, as it ignores the age structure and assumes identical individuals (Rockwood 2006).

The interrelations between prey species are also included in the model. It is assumed that herring benefit from a lower sprat stock, as the abundance of food is consequently higher, and the mean individual weight in the herring spawning stock increases under these conditions. However, as the sprat stock size is very sensitive to fluctuations, whereas cod stock shows more stability, we modelled this dependence according to Heikinheimo (2011) through cod density, which has a connection to sprat stock. Additionally, the mean individual weight in the sprat spawning stock (age groups 2–8) is assumed to be higher with lower sprat density. Cod cannibalism is taken into consideration by assuming a higher natural mortality rate for young age groups when cod are abundant.

All biological parameters, as well as the initial values of the population dynamics (which are the means between 2011 and 2013), are compiled in the “Appendix” and are in accordance with Heikinheimo (2011) and ICES (2014).

3.2 Economic Model

In the economic part of the model, the species-specific prices, $p_{i,m}$, are constant but asymmetrical for each country i. The prices are in accordance with the European Commission (2013) and its supplementary data,⁵ and they represent the means between 2010 and 2012. The revenue function is linear:

$$\begin{aligned} Q_{i,m} (t)=p_{i,m} \times H_{i,m} (t) \end{aligned}$$

(8)

where $H_{i,m}\left( t\right)$ is the country-specific catches in biomass applied to all species according to the following function:

$$\begin{aligned} H_{i,m} (t)= & {} \sum _{a=1}^8 \frac{F_{i,m} \left( t \right) }{\sum _{i=1}^3 {F_{i,m} (t)+M_{m,a} \left( t \right) } }\times N_{m,a} (t)\nonumber \\&\times \left( {1-e^{-\left( {\sum _i {F_{i,m} \left( {t-1} \right) +M_{m,a} \left( {t-1} \right) } } \right) }} \right) \times WA_{m,a} (t) \end{aligned}$$

(9)

The harvesting costs of cod follow a quadratic function of Arnason et al. (2000b):

$$\begin{aligned} C_{i,c} (t)=c_{i,c} \times \frac{H_{i,c} \left( t \right) ^{2}}{B_c (t)} \end{aligned}$$

(10)

where $c_{i,c}$ is a country-specific cost parameter, $H_{i,c} \left( t\right)$ is the cod catch and $B_c\left( t\right)$ is the total cod biomass according to:

$$\begin{aligned} B_c\left( t\right) =\sum _{a=1}^8 {N_{c,a} (t)\times WA_{c,a}} \end{aligned}$$

(11)

The cost parameter for the Danish cod fishery is derived from the total variable cost data of the Danish Baltic Sea fishery according to the FishSTERN report (Blenckner et al. 2011) and applied to consider only the cod fishery. To calibrate the parameter, a root-mean-square error (RMSE) is used for the cost data from 2002–2007. For Poland and Sweden, the costs are scaled to the Danish costs by comparing the fishing efficiencies of the countries according to the data from the European Commission (2013). Therefore, in Poland and Sweden, the costs for cod harvesting are assumed to be 39 % lower and 16 % higher than the Danish costs, respectively.⁶ The harvesting costs of herring and sprat are combined and derived from a linear cost function according to Gordon (1954) and applied in Kulmala et al. (2007):

$$\begin{aligned} C(t)=c\times E(t) \end{aligned}$$

(12)

where c is the harvesting cost per effort day (the same for all countries) and $E\left( t\right)$ is the number of effort days. As it is known that $E\left( t\right) =\frac{F\left( t\right) }{q}$, the cost function can be rewritten as $C\left( t\right) =c\times \frac{F(t)}{q}$ and further split to separately apply to sprat and industrial herring as well as human-consumed herring. The country-specific total cost for combined herring and sprat harvesting becomes:

$$\begin{aligned} C_{i,h+s} (t)=X_i (t)\times c_f \times \frac{F_{i,h} (t)+F_{i,s} (t)}{q_f }+\left( {1-X_i (t)} \right) \times c_d \times \frac{F_{i,h} (t)}{q_d } \end{aligned}$$

(13)

where parameter $X_i\left( t\right)$ illustrates the proportion of industrial herring and sprat catches in the total herring and sprat catches. The catchability coefficient for industrial herring and sprat, $q_f$, differs from that for human-consumed herring, $q_d$. Furthermore, the cost parameter $c_d$ is higher for human-consumed herring than for sprat and industrial herring $(c_f)$, as the vessels are larger, have more crew, face higher fuel costs, and have higher quality demands related to hygienic requirements (Kulmala et al. 2007). The cost parameters of herring and sprat are in compliance with Kulmala et al. (2007) and are symmetric among the countries.

In this paper, it is assumed that the players are asymmetric, as they face different harvesting costs for cod, fish prices and discount rates. The country-specific social discount rates $r_i$ are based on Florio et al. (2008). These differences affect the optimal harvesting strategies of the countries. Three possible game structures are analysed: the grand coalition, non-cooperation (three singletons) and partial cooperation (a two-player coalition and a singleton). In the grand coalition, all countries have negotiated a binding agreement and maximise their joint payoff or net present value (NPV) over the chosen 50-year time period (2014–2063) by annually adjusting the country-specific choice variable $F_{i,m} \left( t\right)$, fishing mortality, for each species under biological constraints. The objective function in the grand coalition (GC) is:

$$\begin{aligned}&\textit{NPV}_{GC}=\max \limits _{F_{i,m}\left( t\right) }\sum _{t=1}^{50} {\sum _{i=1}^3 {\frac{\sum _m Q_{i,m}\left( t\right) -C_{i,c} \left( t\right) -C_{i,h+s}\left( t\right) }{\left( {1+r_i}\right) ^{t-1}}}}\end{aligned}$$

(14)

$$\begin{aligned}&\hbox {s.t.}\nonumber \\ N_{m,a}\left( t\right)= & {} N_{m,a-1} \left( {t-1} \right) \times e^{-\sum _{i=1}^3 {F_{i,m} \left( {t-1} \right) -M_{m,a-1} \left( {t-1} \right) } } \end{aligned}$$

(15)

$$\begin{aligned} H_{i,m} (t)= & {} \sum _{a=1}^8 {\frac{F_{i,m} \left( t \right) }{\sum _{i=1}^3 {F_{i,m} (t)+M_{m,a} \left( t \right) } }} \times N_{m,a} (t)\nonumber \\&\quad \times \left( {1-e^{-\left( {\sum _{i=1}^3 {F_{i,m} (t)+M_{m,a} \left( t \right) } } \right) }} \right) \times WA_{m,a} \ge 0 \end{aligned}$$

(16)

$$\begin{aligned}&B_m \left( t\right) =\sum _{a=1}^8 {N_{m,a} (t)\times WA_{m,a} \ge 0} \end{aligned}$$

(17)

$$\begin{aligned}&R_c\left( t\right) ={\textit{SSB}}_c (t)\times e^{\propto _c -\beta _c \times {\textit{SSB}}_c (t)+\gamma \times \varepsilon } \end{aligned}$$

(18)

$$\begin{aligned}&R_m\left( t\right) =\alpha _m \times {\textit{SSB}}_m (t)\times e^{-\beta _m \times {\textit{SSB}}_m (t)} \end{aligned}$$

(19)

The optimisation constraints define the population dynamics (Eq. 15), catches affecting the stock (Eq. 16), non-negativity of the biomass (Eq. 17) and recruitment (Eqs. 18 or 19). The first three equations apply to all three species, Eq. (18) refers to cod, and Eq. (19) applies to herring and sprat. By assumption, only cod in age groups 3–8 are harvested, and the harvesting of herring and sprat considers age groups 2–8.

Under non-cooperation, countries compete against each other and rationally maximise their own payoff. Each country must take into account the choices of others because they affect the availability of fish that can be harvested. Each country i optimises its individual net present value according to the function:

$$\begin{aligned} \textit{NPV}_i=\max \limits _{F_{i,m} \left( t\right) }\sum _{t=1}^{50} {\frac{\sum _m Q_{i,m} (t)-C_{i,c} (t)-C_{i,h+s} (t)}{\left( {1+r_i } \right) ^{t-1}}} \end{aligned}$$

(20)

which applies to all countries. The optimisation is subject to Eqs. (15)–(19).

The third game structure is a partial cooperation in which two of the countries cooperate and the third country is a singleton, or free rider. The cooperative coalition maximises its joint payoff, and the cooperating countries have a game against the singleton, which adopts its best reply strategy, i.e., it maximises its individual payoff. For a coalition structure in which players 1 and 2 cooperate and player 3 is a free rider, the objective functions are:

$$\begin{aligned} \textit{NPV}_{12}= & {} {\mathop {\mathop {\max }\limits _{F_{1,m}\left( t\right) }}\limits _{F_{2,m}\left( t\right) }} \sum _{t=1}^{50} {\sum _{i=1}^2 {\frac{\sum _m Q_{i,m}(t)-C_{i,c} (t)-C_{i,h+s} \left( t \right) }{\left( {1+r_i } \right) ^{t-1}}} } \end{aligned}$$

(21)

$$\begin{aligned} \textit{NPV}_3= & {} \max \limits _{F_{3,m} (t)} \sum _{t=1}^{50} {\frac{\sum _m Q_{3,m} (t)-C_{3,c} (t)-C_{3,h+s} (t)}{\left( {1+r_3 } \right) ^{t-1}}} \end{aligned}$$

(22)

Again, optimisation is subject to Eqs. (15)–(19). The maximisation problems for other partial coalition structures are formed similarly. In the case of full non-cooperation and partial cooperation, the players have a competitive game against the outsiders of their own coalition and end up in a Nash equilibrium. In the Nash equilibrium, it is profitable for neither the members nor the non-members to change their fishing strategies when the behaviour of others remains unchanged. The fishing strategy of each coalition is the strategy that maximises its payoff given the behaviour of the others.

4 Results of the Partition Function Game

4.1 Partition Function Approach

A partition function takes into account the whole coalition structure, not just the coalition in question. Thus, the positive externalities of the game will be internally considered. The simultaneous-move ‘open-membership’ game introduced by Yi and Shin (2000) is used in this paper as the rule of the coalition formation game. The game consist of two stages: In the first stage, players simultaneously choose whether they will join the coalition or act as singletons.⁷ If they decide to join the coalition, they can do so freely without the permission of its members, which is referred to as an ‘open-membership game’. In the second stage, the coalition and/or singleton(s) play against each other by choosing the optimal fishing mortalities that maximise their own payoffs, taking into account the behaviour of the other player(s). The optimisation is conducted as a dynamic open-loop game with MATLAB’s fmincon toolbox and solved by using backward induction.

The game follows the structure and definitions considering the partition and valuation function applied by Eyckmans and Finus (2009). The set of players is denoted as $N=\left\{ {1,\ldots ,n} \right\} =\left\{ {1,2,3} \right\}$, the coalition S is a subset of N, individual members of the coalition are denoted as $i\in \hbox {S}$, and singletons are denoted as $j\in N\backslash \hbox {S}$. The partition function assigns the aggregate payoff to the coalition and the singleton. In this case, it corresponds to the objective functions in Eqs. (21) and (22). Thus, the partition function is written as follows:

$$\begin{aligned} \textit{NPV}:S\mapsto \textit{NPV}\left( S \right) =\left( {\textit{NPV}_S \left( S \right) ,\textit{NPV}_j \left( S \right) } \right) \in {\mathbb {R}}^{1+\left( {n-s} \right) } \end{aligned}$$

(23)

The valuation function $v\left( S \right)$ is a function that assigns individual payoffs to the coalition members. For the purposes of this paper, the valuation function is defined in terms of the objective function as presented in general terms in Eq. (20):

$$\begin{aligned} \left\{ {{\begin{array}{ll} \sum _{i\in S}v_i \left( S \right) =\textit{NPV}_s \left( S \right) &{} \\ v_j \left( S \right) =\textit{NPV}_j \left( S\right) &{}\quad \forall j\in N\backslash S \\ \end{array}}}\right. \end{aligned}$$

(24)

The valuation function ensures group rationality because it ensures that the entire worth of the coalition is allocated among the members.

4.2 Base Case Results

Under the base case results, the three countries optimise the fishing mortalities of each species under the low salinity conditions. $\textit{NPV}_j \left( S \right)$ shows the individual payoff (valuation function), $\textit{NPV}_{s/j} \left( S \right)$ assigns the payoff to each coalition S or singleton j (partition function), and $\textit{NPV}\left( S\right)$ provides a value for the whole coalition structure, including singletons. The coalition $\left\{ {Den,Pol,Swe}\right\}$ is the grand coalition, the structure $\left\{ {Den}\right\} ,\left\{ {Pol}\right\} ,\left\{ {Swe}\right\}$ refers to non-cooperation, and the rest of the structures illustrate different partial cooperation structures. NPV represents the aggregate payoff in the 50-year simulation period. As expected, the results show that the game exhibits positive externalities, as the formation of a coalition also increases the payoff of a non-merging player (Table 1).

Table 1 Valuation functions, partition functions and stabilities with optimal fishing mortalities and low salinity (base case)$^{\mathrm{a}}$

The common terminology of internal and external stability (IS and ES) (see, for example, Eyckmans and Finus 2009, Kulmala et al. 2013) is applied here. The stability of a coalition S, for all its valuations, is defined as:

$$\begin{aligned} v_i \left( S \right)\ge & {} v_i \left( {S\backslash \left\{ i \right\} } \right) \quad \forall i\in S \end{aligned}$$

(25)

$$\begin{aligned} v_j \left( S \right)\ge & {} v_j \left( {S\cup \left\{ j \right\} } \right) \quad \forall j\in N\backslash S \end{aligned}$$

(26)

Thus, coalition S is stable when no one inside the coalition has an incentive to leave it (IS) and no one outside the coalition has an incentive to join it (ES). If a coalition is simultaneously IS and ES, then it is also the Nash equilibrium of the game, as it is the mutually best response for all players, with no incentives to deviate. Unstable cooperation may obtain stability if the benefits are shared differently among the players using a specific sharing rule for the side payments, and unstable coalitions may thus be potentially internally stable (PIS) or potentially externally stable (PES). In our case, a coalition S is potentially stable for its partition function NPV if:

$$\begin{aligned} \pi _S\left( S\right)\ge & {} \sum _{i\in S} \textit{NPV}_i \left( {S\backslash \left\{ i \right\} } \right) \quad \forall i\in S \end{aligned}$$

(27)

$$\begin{aligned} \pi _S \left( {S\cup \left\{ j \right\} } \right)< & {} \sum _{i\in S\cup \left\{ j \right\} } \textit{NPV}_i \left( S\right) \qquad \forall j\in N\backslash S \end{aligned}$$

(28)

which means that a coalition S is PIS if the value of the coalition is greater than or equal to the sum of the free-rider payoffs, and PES if it is impossible for any country to buy cooperation. When a coalition simultaneously satisfies PIS and PES, it is possible to have a stable coalition by reallocating the cooperative benefits between the countries, which can be performed without even specifying a transfer rule. Moreover, no one has enough resources to buy cooperation (Fuentes-Albero and Rubio 2010). It has been shown (e.g., in Weikard 2009) that when a PIS coalition cannot be expanded to another PIS coalition, the coalition is externally stable.⁸

It is apparent that the non-cooperative coalition structure is internally stable, as no player can deviate further. However, external stability is not satisfied because all countries would be better off by joining any partial cooperation. Furthermore, all partial coalition structures are internally stable, as no one is better off by deviating, but the only coalition that simultaneously satisfies IS and ES is between Denmark and Sweden, but it cannot be enlarged to the grand coalition due to Poland’s free-rider incentive. Thus, this structure is the only feasible solution without side payments and is the Nash equilibrium of the game. Partial coalitions including Poland satisfy internal stability, but because Denmark and Sweden have an incentive to join those coalitions and form the grand coalition, they are not externally stable. The grand coalition, which assigns the highest aggregate payoff and would therefore be the best solution, can be stable when side payments are allowed, as it satisfies PIS and satisfies PES by definition.

When Denmark is a member of a coalition, it always harvests the majority of the total cod catches inside the coalition (Table 2). The co-members with Denmark harvest less and gain lower payoffs, but it is still beneficial for them to cooperate, as a deviation would lead to a decrease in their payoffs. Except in the grand coalition, the fishery is most heavily concentrated on the valuable cod stock, and its fishing mortality is the highest. The fishing pressure for herring exceeds that of other species under full cooperation. As the level of cooperation increases, the fishing mortalities of the species become more homogenous, and the intensity of cod harvesting simultaneously decreases. Therefore, the cod biomass is at its lowest when the countries are engaging in non-cooperation and is highest under the grand coalition. According to the simulation, the historical maximum value of the cod biomass cannot be reached, even under full cooperation, which may be due to the assumption of a constant salinity condition, which is unfavourable for cod recruitment.

Table 2 Fishing mortalities, biomasses and cod catches (mean values from 2039–2053) with optimal fishing mortalities and low salinity (base case)

The simulated cod catches in the last column of Table 2 (average catches from 2039–2053) substantially exceed the current quotas in most cases. This result is due to the fairly low level of the quotas, which do not therefore reflect the long-term perspective very well. Since 1990, the total quotas have ranged between 45,000 and 220,000 t; the 2013 quota was 88,700 t. In addition, it is important to keep in mind that the results presented here are averages, as the model does not converge to a steady state in the simulation period. However, if the simulation period is extended to 150 years, it is possible to find a steady state solution, which is, unfortunately, not a very applicable solution for this type of empirical problem due to changes, e.g., in prices, technologies and fish stocks over such a long time period.

4.2.1 Comparison with Status Quo Fisheries

It is essential to compare the optimal base case results for the fishery with the current fishing mortalities, i.e., the status quo (SQ) fishery. The current fishing mortalities for Denmark, Poland and Sweden, respectively, are 0.24, 0.22 and 0.13 for cod, 0.01, 0.05 and 0.07 for herring, and 0.06, 0.18 and 0.11 for sprat. These fishing mortalities are derived by assuming that only these three countries act in the Baltic Sea fishery.⁹ By running the model using these constant SQ fishing mortalities, the total NPV in the simulation period is ${\EUR }3{,}018$ million (Table 3). This result reveals that the Baltic Sea fishery is not currently economically optimal, as the aggregate payoff is much lower than the optimal situation would yield (${\EUR }5,152$ million in the optimal cooperative situation) but is higher than the situation under non-cooperation (${\EUR }2,244$ million). Therefore, the current fishery management may be functioning reasonably well, although improvements are possible. All partial coalition structures yield higher profits than the SQ fishery. For the fishery to be more profitable in the long run, cod and sprat should be harvested at a lower intensity, but herring could be harvested more aggressively.

Table 3 Status quo fishery with current fishing mortalities and low salinity

4.2.2 Cost Effectiveness of the Countries

An obvious question is why Denmark should optimally account for the majority of the harvests. This result can be explained by comparing the harvesting efficiencies of the countries, i.e., the price–cost ratios, for each species with and without the effect of a discount rate (Table 4). In the former case, the price–cost ratios are divided by the country-specific discount rates, resulting in indicators for price–cost ratios that are adjusted with different time preferences. The ratios of herring and sprat are not comparable with those for cod, as the units of the cost parameters differ. Denmark is the most efficient as well as the most patient country because it harvests all of the species at the highest price–cost ratio, whereas Poland has the lowest efficiency for herring and sprat and Sweden for cod. However, before the time adjustment, Denmark and Sweden have the same price–cost ratios for herring and sprat, but when the effect of time preferences is included, Denmark becomes by far the most profitable country because it has the lowest discount rate i.e., the lowest rate of time preference, thus resulting in higher profits in the long run. It is notable that even if the efficiencies of the countries vary, it is never only the most efficient country that harvests because of the non-linear cost function of cod harvesting, which also affects the fishing of other species.

Table 4 Price–cost ratios without/with the effect of the discount rate

4.2.3 Comparison Under a Single-Species Context

By running the same simulation under a single-species context for cod, herring and sprat separately, i.e., without any interactions between the species, the stability conditions are weakened. Now, none of the game structures simultaneously satisfies IS and ES, and furthermore, none of the cooperative coalition structures is even IS when compensation payments are not present (Table 5).¹⁰ This result contrasts with that from the multispecies game, in which all partial coalition structures satisfy internal stability. The reason for this result is that neither Poland nor Sweden benefits if they are cooperating with Denmark, which is the most effective country in harvesting cod; thus, Poland and Sweden will deviate from cooperation. In addition, the partial coalition between Poland and Sweden breaks down because Poland gains too little from cooperation. Therefore, it is impossible to have a stable coalition structure without side payments, but when compensation is allowed, all partial coalitions in cod game can be stabilised. The same holds for the grand coalition in herring and sprat games. Importantly, a single-species cod game is not PIS under the grand coalition, but when the prey species are added to the model, thus resulting in a multispecies game, even full cooperation can be stabilised by allowing side payments. Therefore, a multispecies approach improves the scope of cooperative agreements.

Table 5 Stabilities under single-species context and low salinity

4.3 High Salinity Conditions

Under higher salinity conditions, i.e., when the climate change impact is lower, cod recruitment is improved, and the stability conditions have changed from the previous findings. The partial coalitions including Denmark are no longer internally stable, as Poland and Sweden now have the incentive to deviate (Table 6). Therefore, the stability conditions are weakened because none of the coalition structures simultaneously satisfy IS and ES. However, it is still possible to stabilise the grand coalition through side payments. Under better salinity conditions, the harvesting concentrates even more strongly on cod due to the positive environmental impact, which further weakens the prey stocks (Table 7). In all cases, the cod biomass exceeds its historical maximum value. This result is due to a quite optimistic scenario in which high salinity conditions are assumed to remain over the whole simulation period. Nevertheless, this scenario illustrates that improved recruitment conditions for cod could help the stock reach its good historical state and increase profits.

Table 6 Valuation functions, partition functions and stabilities with optimal fishing mortalities and high salinity

Table 7 Fishing mortalities, biomasses and cod catches (mean values from 2039–2053) with optimal fishing mortalities and high salinity

Table 8 shows how much higher or lower (%) each country’s payoffs are inside a coalition compared to free riding. In most cases, the cooperative payoffs appear to be higher under low salinity conditions, as countries benefit more from cooperation when the cod biomass is moderate. Therefore, the incentive to cooperate is higher when salinity is at the current low level; thus, ongoing climate change may not hinder a binding agreement. In our case, climate change only affects cod harvesting by increasing costs, not the prices of fish. Thus, it increases the cost–benefit ratio c / p, which has been shown to improve the chances for cooperation (Finus et al. 2011). Additionally, Finus et al. (2011) showed that whenever the cost-benefit ratio is higher, the need for cooperation is lower, as are the economic rents. This result is contradictory to our case, in which the need for cooperation increases under low salinity simultaneously with higher chances to establish cooperation. The result conflicts with the common concept of the ‘paradox of cooperation’, according to which cooperation is more likely when the gains from it are lower, and thus the need for cooperation increases (Barrett 1994; Finus 2000). Our finding may be explained by the multispecies nature, which affects countries’ behaviour.

Table 8 Benefits from cooperation (%)

4.4 Sensitivity Analysis

The robustness of the model is tested through sensitivity analysis, where the key parameters are changed to estimate their effects on the results. There is significant uncertainty, especially over the cost parameters, and the assumption of constant harvesting costs over time is particularly hampered by continuously increasing oil prices. Therefore, a sensitivity analysis is conducted in which the country-specific cost parameters of cod harvesting increase with the predicted increase in oil prices. The basis of the prediction is that the oil price will increase by 2.5 % annually until 2020 and 1.5 % annually from 2021–2030 (Oxford Economics 2010). In this paper, it is assumed that the increase will continue at the same 1.5 % pace onwards from 2030, and based on data provided by the European Commission (2013), fuel costs are assumed to account for 18 % of total fishing costs. Therefore, the cost parameters of cod increase according to these data. Additionally, the analysis tests how the model behaves with discount rates starting from the country-specific initial value and declining smoothly to zero instead of using fixed discount rates. This approach is suggested to be applicable when future economic growth is uncertain (Gollier et al. 2008). Furthermore, the results are compared with those from more symmetric countries, where the countries only differ in their cod harvesting costs. The analysis reveals that one of the most uncertain parameters, a cost parameter, seems to not be overly sensitive regarding the stability of the grand coalition, which remains attainable even with the increasing costs (Table 9). In contrast, the stability conditions are to some extent sensitive to the assumptions when considering the discount rates and the symmetry of the countries, as the grand coalition no longer satisfies PIS under these scenarios and is therefore an unattainable solution. Additionally, the assumption about decreasing discount rates violates IS conditions under partial coalition structures, which can now be stabilised only through transfers. When the countries are more symmetric, and the only difference is in their cod harvesting costs, all partial coalitions are Nash equilibria with and without a payments scheme.

Table 9 Sensitivity analysis

5 Discussion and Conclusions

In this study, a dynamic multispecies partition function game was presented with three asymmetric players optimising the rents from fishery in the Baltic Sea by setting their fishing strategies. The stability of coalition formation among the players was explored. A partial coalition between Denmark and Sweden was found to be internally and externally stable, even without side payments, and was therefore the Nash equilibrium of the game. If side payments are allowed, the grand coalition could also be a stable solution. This result is not uncommon in the literature; e.g., Pintassilgo et al. (2010) showed that with more than two asymmetric players, fully or partially cooperative coalition structures can be stable. In contrast, Kulmala et al. (2013) argued that four asymmetric players can form a stable partial coalition in the Baltic salmon fishery when an Almost Ideal Sharing Scheme (AISS) is used, but the grand coalition remains unachievable even under a transfer scheme. Thus, there is a divergence in the results, as stability depends on the number of players, the level of asymmetry and efficiency, and the time preferences of the countries, i.e., discount rates. Importantly, when the model is run for each of three species separately, i.e., in a single-species context, and the scope of cooperation is compared with the results attained from the multispecies simulations, the stability conditions are weakened. Therefore, when these stocks are managed jointly in a multispecies context, the scope of cooperation is improved, as then the benefits from harvesting other species also play a role.

The current Baltic Sea fishery management system yields higher payoffs than the non-cooperative simulation, but it is still not economically optimal, as the optimised cooperative payoffs exceed the payoffs under the current management scheme. This result is due to inefficient management following MSY recommendations, according to which the fishing mortalities are not set at the economic optimum. For the fishery to be more profitable, Denmark should account for a larger share of the cod catches, as it is the most efficient country in harvesting, and it should pay compensation to Poland and Sweden, which would give up part of their fisheries.

Compensation or side payments, which are essential in our simulations to achieve the most profitable outcome (grand coalition), are observed in real world fisheries. The compensation can be a monetary transaction, or it can be non-monetary in the form of fishery rights, where one country assigns the rights to some other species to another country. The latter type of compensation has been used, for instance, in bilateral agreements between Norway and Russia (Armstrong 1994). Another successful example of side payments providing a stable management system was the fur seal agreement in the Northeast Pacific from 1911–1984, where the low cost harvester, in this case the United States, transferred furs to Canada and Japan to commit them to stopping harvesting (Barrett 2003). The conclusion from our model is in line with these conclusions, namely that the most efficient country must compensate other countries to reach the most profitable solution. Side payments can be difficult to implement, as countries may not want to give up an actual fishery, which might cause unemployment in the fishery sector, but non-monetary compensation in the form of a change in fishery rights could be easier to adopt.

When the salinity level is higher (i.e., the climate change impact is lower), the stability conditions are weakened compared with the previous findings. None of the coalition structures simultaneously satisfy IS and ES, but using side payments, it is still possible to stabilise the grand coalition. Higher salinity conditions reduce the incentive to cooperate, as countries benefit more from cooperation when the cod biomass is lower. Thus, ongoing climate change may surprisingly have a positive effect on the likelihood of a binding agreement, as with a lower cod biomass, countries have a greater incentive to cooperate. This result contrasts with that of Brandt and Kronbak (2010), who concluded that when climate change has a negative impact on the resource rent, the set of possible stable, cooperative agreements declines, and cooperation is therefore less likely. Nevertheless, our result is interesting and provides insights into how deteriorated cod recruitment conditions could affect the agreements. However, it raises the need for further research using different species and ecosystems to determine which factors actually influence the scope of cooperation. It must also be kept in mind that the assumption about better cod recruitment conditions for the whole simulation period may be too optimistic and is very unlikely to occur. This aspect could be improved by assuming salt-water pulses, which dynamically affect the salinity level in the sea, instead of using a constant salinity parameter.

According to the sensitivity analysis, the stability conditions do not seem to be overly sensitive when the cod fishing costs increased over the simulation period together with the future oil price, and the grand coalition still remained attainable. Nevertheless, it is important to highlight that the cost parameters for cod harvesting used here are uncertain, and there is a need for better approximations. In addition, the cost parameters of the herring and sprat fisheries were assumed to be symmetric among the countries, which is most likely not the case. In future research, it would be important to include uncertainty in the model and to allow stochastic changes. When decreasing discount rates or more symmetric countries are assumed in the sensitivity analysis, the grand coalition no longer satisfies PIS. The result considering symmetry is somewhat in line with that of Pintassilgo et al. (2010), who demonstrated that with symmetric players, it is impossible to form a grand coalition that includes more than two players. However, when the players have asymmetric fishing costs, full cooperation is achievable, even with three or more players. According to Pintassilgo et al. (2010), such cooperation is not only a result of the externality becoming internalised in the grand coalition, but the gains from a cost-effective allocation of fishing efforts would now also be maximised. Thus, larger cost asymmetries indicate larger gains from cooperation (Pintassilgo et al. 2010). Here, it is shown that this not only applies to the asymmetry of fishing costs but also to other types of asymmetry, in our case discount rates and prices, which also affect cost efficiencies.

In addition to the suggestions for future research mentioned above, there are other issues to consider. As mentioned previously, Baltic cod has recently suffered from malnutrition due to its improved stock size and a lack of prey biomass, as currently the geographic overlap of cod and its prey species is relatively small (ICES 2013). Therefore, a study that takes into account spatial considerations would be important for future research. Additionally, an analysis in a setting of a dynamic membership game where the membership decision is revised periodically is needed. Our result that contradicts the ‘paradox of cooperation’ calls for further research to determine whether this collision is valid in other multispecies studies.

Generally, this paper contributes to a relatively new stream of literature on multispecies (or ecosystem) exploitation and coalition formation in a dynamic setting. By its application, the paper demonstrates the importance of including the interlinkages between species, which requires a dynamic model that is not based on equilibrium use. However, the area is still underexplored, and among other topics, multiple coalition structures require further attention.

Appendix

See the Tables 10, 11, 12 and 13.

Table 10 Biological and economic parameter values

Table 11 Initial values of cod population dynamics

Table 12 Initial values of herring population dynamics

Table 13 Initial values of sprat population dynamics