Ragnar Arnason, Gylfi Magnusson, Sveinn Agnarsson*

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Ragnar Arnason, Gylfi Magnusson, Sveinn Agnarsson*

The Norwegian Spring Spawning Herring Fishery:

A Stylised Game Model

A paper submitted for publication in the Marine Resource Economics**

*Department of Economics, University of Iceland IS-101 Reykjavik.

Telephone: 354-525-4539

E-mail:

**This paper is a substantially revised version of a paper originally given at the conference on The Management of Straddling and Highly Migratory Fish Stocks and the UN Agreement, Bergen, May 19. – 21.1999. We would like to thank the participants at that conference, in particular, professors Bjorndal and Munro and two anonymous referees for useful comments.

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Abstract

This paper presents an empirically-based game-theoretic model of the exploitation of the Norwegian Spring Spawning Herring stock, also known as the Atlanto-Scandian herring stock. The model involves five exploiters; Norway, Iceland, the Faroe Islands, the EU and Russia and an explicit, stochastic migratory behaviour of the stock. Under these conditions Markov Perfect (Nash) equilibrium game strategies are calculated and compared to the jointly optimal exploitation pattern. Not surprisingly, it turns out that the solution to the competitive game is hugely inefficient leading very quickly to the virtual exhaustion of the resource. The scope for co-operative agreements involving the calculation of Shapley values is investigated. It turns out that although the grand coalition of all players maximizes overall benefits such a coalition can hardly be stable over time unless side payments are possible.

Keywords: Fisheries economics, migratory fish stocks, fisheries game theory, multi-nation fisheries games, high seas fishing, natural resource extraction games

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0.Introduction

The Norwegian spring-spawning (Atlanto-Scandian) herring stock is potentially one of the largest and biologically most productive fish stocks in the world. During the early 1950s its total biomass ranged between 15 and 20 million metric tonnes and its spawning stock averaged 10 million metric tonnes (Patterson 1998, Bjorndal et al. 1998). Although annual catches during the 1950s were in excess of 1 million metric tonnes, average fishing mortality was usually less than 0,1.

In the 1960s, new harvesting technology, involving sonar and the powerblock, led to greatly increased exploitation of the stock. Several European fishing nations participated in the fishery with Norway, Iceland and the USSR being the most prominent. In the late 1960s, the stock suffered a collapse apparently due to a combination of overfishing and deteriorating environmental conditions. In spite of a moratorium on fishing from the spawning stock imposed in 1969, the stock continued declining reaching a nadir of 71.000 metric tonnes and a spawning stock of 2.000 metric tonnes in 1972 (Patterson, 1998). Since then, the stock has recovered and the current spawning stock is now close to its previous size of 10 million metric tonnes.

The Atlanto-Scandian herring is highly migratory. The adult stock spawns off western Norway in February to April (see map in Figure 1). After spawning the adult stock embarks on feeding migrations westward and northward following the zooplankton blooms across the North Atlantic. The feeding period normally ends in September at which time the stock commences migrations to its wintering area. There the adult stock stays until January each year when it migrates to the spawning grounds off western Norway.

Although the above describes the essential features of the Atlanto-Scandian herring’s migratory pattern, the exact migratory routes and distances have been somewhat variable. Although not fully understood, it appears that this migratory variability depends primarily on two factors: (i) spawning stock size and (ii) environmental conditions especially the availability of feed and ocean thermoclines. A stylised migratory pattern based on the migratory behaviour for a sizeable spawning stock is illustrated in Figure 1.

It is primarily during the feeding migrations from May to September each year[1] that the Atlanto-Scandian herring becomes subject to international fishing pressure. On leaving the Norwegian EEZ, the herring enters international waters (the herring loophole, see Figure 1). It then enters one or more of the EEZs of the Faroe Islands, Jan Mayen (Norway) and Iceland. During this period, the herring tends to form dense schools that are particularly suitable for purse-seine fishing. In the herring loophole, access to the stock is basically open to all. This is followed by sequential but somewhat stochastic exclusive national access by the three countries with adjacent EEZs, Iceland, the Faroe Islands and Norway.

This obviously defines a fairly intricate game-theoretic situation. First of all, the game is dynamic or evolutionary, in the sense that the opportunities (or moves) available to each player depend on the size of the stock and, consequently, his moves and those of the other players’ in previous time periods. Secondly, over the course of the year, the set of moves available to each player depends on the location of the stock. Thus, if the stock is located within a country’s EEZ, the other players do not have access to the stock and are reduced to the role of observers. Thirdly, any co-operative agreement the players may manage to arrange is potentially threatened by (i) the entry of new players wanting to take advantage of a growing stock and (ii) altered migratory behaviour of the herring which will change the respective national threat-points and may render the existing co-operative sharing untenable.

In recent years, a number of fishing nations have participated in the Atlanto-Scandian herring fishery. The most important of these are Norway (about 60% of the total harvest), Iceland (about 15%), Russia (about 11%), EU nations[2] (about 8%) and the Faroe Islands (about 5%). A few years ago, these nations agreed on setting and sharing an overall quota in this fishery. The agreed quota shares are roughly in conformance with recent historical catch shares. This agreement, however, is not intended to be permanent, in particular the quota shares are periodically renegotiated. Given the high likelihood of altered migratory behaviour of the stock and the possibility of new entrants, it is unclear how stable this agreement can be.

Our intention in this paper is to study the fisheries game situation in which the exploiters of Atlanto-Scandian herring fishery find themselves. Our approach is to devise a simple model of the situation based on the measurable realities of the fishery. Since the model is quite simple and its key relationships imperfectly estimated we prefer to refer to this model as a stylised portrayal rather than an empirical model of the fishery. Subsequently, on the basis of this stylised model, we seek equilibrium strategies for each of the players under a variety of competitive and co-operative situations and study the implications for the fishery. Although designed for the Atlanto-Scandian herring fishery, our modelling framework is in fact quite general and can with little modifications be used to study multi-player, migratory fisheries games in general.

The structure of the paper is as follows. In section 1 we provide an overview of our game-theoretical framework for studying multi-player, migratory fishery games and describe the numerical solution methods we employ. In section 2, we outline the empirical content of our model. In section 3, we present our results from simulating the Atlanto-Scandian herring fisheries game involving the current five exploiters (e.g. Norway, Iceland, the Faroe Islands, the EU and Russia). Finally, in section 4 we briefly discuss the main results of the paper.

1Theory

Considerable research has been conducted into the strategic aspects of the exploitation of fish stocks (Clark 1976, Levhari and Mirman 1980, Hannesson 1993). Kaitala (1986) provides a survey of the use of game theory to analyze the exploitation of fish stocks prior to 1986. This paper studies the special situation of strategic interaction where the fish stocks are strongly migratory.

We regard the situation as a game between various fishing agents, each of whom is trying to maximize the present value of their net returns. We describe the evolution of the game in terms of Markov perfect equilibria and utilize recently developed methods for analyzing such equilibria, example of which can be found in Ericson and Pakes (1995), Pakes and McGuire (1994), Pakes (1994) and Rust (1994 and 1996). According to these methods, the agents select decision rules that prescribe their reaction to changes in the state variables, in this case the size of the fish stock and its location. Furthermore, each decision rule gives a best response to the decision rule of all the other agents. Agents' controls are usually either fishing effort or the amount of biomass caught. The setup is general enough to allow for more state variables such as several species and cohorts and more than one control per agent. However, computational limitations may prevent the implementation of these extensions. The Markov perfect equilibrium assumption means that agents cannot commit themselves for extended periods.[3] When coalitions are introduced it will be assumed that coalitions do not co-operate with each other or with single players. Coalition agreements are assumed to be binding.[4]

Our particular setup focuses on the importance of the migratory behaviour of fish stocks and in particular whether a fish stock, at a point of time, is located within the EEZ of a particular country or in high seas. Authors that have introduced EEZs or other ways of ensuring the excludability of potential exploiters include Fischer and Mirman (1994), Kennedy (1987), Kennedy and Pasternak (1991), Krawczyk and Tolwinsky (1993) and Naito and Polasky (1997).

In addition to finding the (competitive) Markov perfect equilibrium, we also calculate the jointly optimal solution. No attempt is made to model how the jointly optimal solution could be implemented, except by calculating Shapley values (Shapley, 1953). Several authors, including Kaitala and Pohjola (1988), have looked at the possibility of side payments to support a solution that is a Pareto improvement on the competitive outcome.

1.1The Basic Model

We are concerned with modelling the harvesting from a migratory fish stock by more than one exploiter (nation).[5] Compared to the usual bioeconomic fisheries models, this implies two additional features; (i) variable catchability depending on the location of the stock at each point of time and (ii) strategic behaviour by each of the exploiters of the stock.

The following equations represent the essential structure of our model:

Biomass growth:

(1),

where x represents the size (biomass) of the fish stock, y is the catch, t denotes time and the index i refers to the different exploiters. The function G(.), of course, represents the natural growth of the biomass.[6]

Harvesting costs:

The generic form of the harvesting cost function employed is:

(2),

where dt represents the distance from the base of the exploiter to the centre of the fish stock at time t. More specific assumptions on the effect of the three variables, catches, stock size and distance on cost will be introduced later.

Migrations and the location of fish stock:

Several modelling assumptions are possible, but to keep the presentation reasonably simple let us initially assume that the fish migrate in a deterministic fashion, so that location in each period is a function of the location in the previous period. A more general stochastic type of migrations is discussed in section 1.3 below.

Let lt = (lxt,lyt) represent the location of the stock at time t, where x denotes the x-coordinate and y denotes the y-coordinate of the location. Then a simple deterministic presentation of migrations is given by the differential equation:

(3).

Location of exploiters:

It seems plausible to assume that the exploiters operate from a number of fixed ports or locations . Note that in principle each country may have fleets operating out of different ports so that the number of these locations may exceed the number of national exploiters. The exploitation pattern may and presumably will shift over time as the fleets embarking from each exploitation point vary between zero and a positive number over time.

Distance from exploiter to centre of fish stock:

Ignoring the curvature of the globe (which is reasonable for relatively short distances) we represent the distance between the ports of exploiter i and the location of the stocks by the expression:

.

Prices:

We provisionally assume that all prices including the price of landed fish, p, and the discount factor, ,[7] are constant. This assumption is easy to relax.

Profits each period:

.

Net present value of future profits:

.

1.2Solution method

In order to facilitate the appreciation of the method we employ to obtain explicit numerical solutions to the migratory fisheries game it is useful to consider first relatively simple game situations. In section 1.3 below we extend the model to include stochastic migrations and the restrictions imposed by exclusive economic zones.

Case 1: One exploiter

First we will consider the situation of one exploiter referred to as exploiter i. In this situation, presumably, the exploitation of the stock will be optimal (given the location of this exploiter).

The problem for one exploiter is easily solved using dynamic programming. In particular, note that the net present value of future profits can be split into two parts, the profits this year and the present value of all future profits, as follows:

.

It is important to notice that this system has two state variables; the size of the fish stock and its location. Profits will be a function of these two variables. To maximize the net present value of profits, exploiters will have to find the optimal catch, given the size of the fish stock and its location. Mathematically:

.

This is a straightforward contraction mapping that can be solved numerically with the help of a computer. The form of is known, given the above equations for cost, distance and the price of fish. The forms of the X and l functions are also known. The only unknown is thus . This can be found by iterative techniques. We start with a guess for on the right hand side and use that to compute the on the left hand side. The guess for the left hand side thus found is then used as a guess for the right hand side and a new guess for the left hand side found. This is repeated until the 's on the left and right hand side are deemed sufficiently similar. A Fortran program has been written that performs these calculations.[8]

Having found we have implicitly derived the decision rule for the exploiter:

where

.

To maximize profits, the harvesting activity should be concentrated on the period when the stock is closest to the home port of the exploiter. This rule is in general modified by capacity constraints (in this paper no capacity constraints are assumed) and the rate of discount.

Case 2: Two or more exploiters that cooperate

This is a straightforward extension of case 1. The only change is that the relevant profit function is now the sum of the two exploiters´ individual profit functions and there are two locations and harvests to maximize over. Consequently, essentially the same method as in the single exploiter case can be used to solve this problem. Having gone through that exercise we find the individual and total exploitation rule each period as:

all i.

.

Case 3: Two or more exploiters that compete

The simplest assumption is that each exploiter takes the decision rule (i(xt, lt), i.e. catch as a function of stock size and the location of the stock) of his opponents as given and chooses his decision rule without taking into account that his choice of decision rule may affect the choice of a decision rule by the other exploiter.[9] In effect, this means that exploiter 1 behaves as if the growth function for the stock is:

(4).

Exploiter 1 then finds his optimal decision rule, 1, given this "growth" function. We have found an equilibrium if each i’s is the best response to all the other decision rules (j’s). This is referred to as a Nash equilibrium of the competitive game (Nash 1951).

To calculate this, we need a somewhat more complicated process than in cases 1 and 2. For the two exploiter game, we start with any decision rule for exploiter 2. One possibility might be the decision rule for exploiter 2 if he were the sole exploiter. Given this, we solve the problem for exploiter 1 in the same way as in case 1 but using the new "growth function", i.e. equation (4), given above. This yields his decision rule, i.e. 1(xt, lt). Then we use this decision rule to find the optimal decision rule for exploiter 2 and so on. This process is repeated until it converges, i.e. the changes in the two decision rules between iterations are deemed sufficiently small. This then represents the Nash equilibrium of the game.

With more than two exploiters, n, say, we start with any set of n-1 decision rules. On this basis we find the decision rule for the n-th exploiter, then the decision rule for exploiter number n-1, given the initial guess for the first n-2 exploiters and the one calculated for exploiter n. This is repeated until we have found a decision rule for all exploiters. Then we start with the n-th exploiter again and repeat the process until it converges in the sense that the changes in each exploiter's decision rule between iterations is arbitrarily small.