# Fishermen’s Profits Maximization: Case of Generalized Nash Equilibrium of a Non-symmetrical Game

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DOI: 10.1007/s10441-014-9223-y

- Cite this article as:
- El Foutayeni, Y. & Khaladi, M. Acta Biotheor (2014) 62: 325. doi:10.1007/s10441-014-9223-y

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## Abstract

In the present paper, we consider a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. Each fisherman tries to find the fishing effort which maximizes his profit at biological equilibrium without any consultation with others, but all of them have to respect two constraints: (1) the sustainable management of the resources ; and (2) the preservation of the biodiversity. With all these considerations, our problem leads to a generalized Nash equilibrium problem. The objective is to show that even when a fisherman \(i\) provides a fishing effort equal to twice the fishing effort of a fisherman \(j\), then the profit of fisherman \(i\) is not necessarily double that of fisherman \(j\).

### Keywords

Population dynamics Bio-economic equilibrium model Generalized nash equilibrium Sustainable management of the resources Preservation of the biodiversity Fisherman’s profit maximization## 1 Introduction

The management of renewable biological resources implies both the objectives of the conservation of a population and the maximization of the benefits coming from its exploitation. To achieve these objectives, there are several studies of bio-economic models using various mathematical tools. In Purohit et al. (2007) deal with the problem of optimal harvesting of a fishery consisting of two competing fish species, each of them obeys the Gompertz law of growth. A dynamic reaction model is developed from a capital-theoretic view point taking taxation as a control measure to regulate exploitation of the fishery. The existence, as well as the stability of the possible steady states is examined. Bionomic equilibrium of the system is determined and the optimal harvest policy is studied with the help of Pontryagin’s Maximum Principle. In Merinoa et al. (2007) use the GAMEFISTO simulation model to present a tool to improve the small scale fisheries bio-economic simulation techniques. The main novelty of the model is the implementation of game theoretic techniques for forecasting the fishing effort trends and consequently, the fish population levels and the economic outcome, including landings, profit and net profits. The model assigns individual fishing strategies to individual vessels according to their technical characteristics. The fishermen within a fishing fleet exploiting a single stock are assumed to be the decision agents, who share not only a fish population (modeled through stock externality) but also a market, through an offer-demand function (market externality). In previous work Bulte (2003), extended the traditional G-S model of open access by defining a non-concave harvesting function. The main of the paper was to demonstrate the possible existence of multiple equilibria and perverse comparative statics and show that small changes in the underlying economic parameters may trigger large jumps in species abundance. More recently, in the paper El Foutayeni et al. (2012), we defined a bio-economic equilibrium model for several fishermen who catch three species; these species compete with each other for space and/or food. The natural growth of each species is modeled using a logistic law. We find the fishing effort that maximizes the profit of each fisherman. The existence of the steady states and its stability are studied using eigenvalue analysis. In El Foutayeni and Khaladi (2012) we presented a bio-economic equilibrium model for several fish populations taking into consideration the fact that the prices of fish populations vary according to the quantity harvested. We introduced a mathematical model; studied the existence and stability of the equilibrium point and calculating the fishing effort that maximizes the profit of the fisherman exploiting all fish populations.

The goal of the present paper is to consider a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. Specifically, we consider a bio-economic equilibrium model of a fish population exploited by several fishermen represented by their fishing efforts \((E_{i})_{i=1,\ldots ,n}\), where \(n\) is the number of fishermen. The objective is to find the fishing effort \(E_{i}^{*}\) maximizing each fisherman’s profit \(\pi _{i}^{*}(E^{*})\), at biological equilibrium, without any consultation between the fishermen, but all of them have to respect the sustainable management of the resources. Each fisherman strives to maximize its profit choosing a fishing effort strategy. With these considerations, our problem leads to a generalized Nash equilibrium problem. The fishing effort \(E^{*}=\left( E_{1}^{*},\ldots ,E_{n}^{*}\right) \) depends on: (a) the catchability coefficients \(q_{i}\); (b) the costs of fishing \(c_{i}\); and (c) the price of fish population \(p\). It allows us to discuss the trends of individual fishing effort in terms of competitiveness. In our previous works El Foutayeni et al. (2012), El Foutayeni and Khaladi (2012), it has been shown that if the fishermen provides the same fishing effort \(E_{i}=E_{j}\), then all fishermen earn the same profits \(\pi _{i}=\pi _{j}\). In this work we assume that the fishermen provides distinct fishing efforts \(E_{i}\ne E_{j}\). The main objective of this paper is to show that even when a fisherman \(i\) provides a fishing effort equal to twice the fishing effort of a fisherman \(j \), while not necessarily the first (fisherman \(i\)) earns double the second (fisherman \(j\)).

The paper is organized as follows. In the next section, we present a bio-economic equilibrium model describing the dynamics of a fish population exploited by several fishermen seeking to maximize their profits. We give the steady states of biological model and present bio-economic equilibrium model of fishery. We show that finding a solution of bio-economic equilibrium model is equivalent to solving a generalized nash equilibrium problem (GNEP). In Sect. 3 we show that the problem GNEP has a unique solution and we compute this solution. In Sect. 4 we give numerical simulations of the mathematical model and discussion of the results. Finally, we give a conclusion in Sect. 5.

## 2 The Bio-Economic Equilibrium Model

The description of the bio-economic equilibrium model is divided into three parts: The first one is the mathematical model and hypotheses; the second one is the steady states of the system; and the third one is the bio-economic equilibrium model of fishery.

### 2.1 The Mathematical Model and Hypotheses

It is interesting to note that according to the literature, the fishing effort depends on several variables, namely for example: number of hours spent fishing; search time; number of hours since the last fishing; number of days spent fishing; number of operations; number of sorties flown; ship; technology; fishing gear; crew; etc. However, in this paper, the fishing effort is treated as a unidimensional variable which includes a combination of all these factors.

### 2.2 The Steady States of the System

### 2.3 The Bio-Economic Equilibrium Model of Fishery

The description of the bio-economic equilibrium model of fishery is divided into four parts: The total revenue, the total cost, the profit and the bio-economic equilibrium model optimization.

#### 2.3.1 Total Revenue

#### 2.3.2 Total Cost

#### 2.3.3 Profit

#### 2.3.4 Bio-Economic Equilibrium Model Optimization

- (a)
the first one is the preservation of biodiversity of fish population, expressed mathematically as \(\frac{B}{K}=1-\frac{1}{r}\sum \nolimits _{j=1}^{n}q_{j}E_{j}>0\);

- (b)and the second one is that the optimal fishing effort \(E^{*}=(E_{1}^{*},\ldots ,E_{n}^{*})\) must satisfy the following conditionswith the conditions that \((E_{i})_{i=1,\ldots ,n}\geqslant 0\).$$\begin{aligned} \pi _{i}(E_{1}^{*},\ldots ,E_{i}^{*},\ldots ,E_{n}^{*})\geqslant \pi _{i}(E_{1}^{*},\ldots ,E_{i},\ldots ,E_{n}^{*})\quad {\text{ for\,all }}\quad i=1,\ldots ,n; \end{aligned}$$

With all these considerations, our bio-economic equilibrium model can be translated into the following mathematical problem

## 3 Computing the Generalized Nash Equilibrium

We recall that the GNEP is an extension of the nash equilibrium problem (*NEP*), in which each fisherman’s strategy (fishing effort) set is dependent on the rival fishermen strategies. Mathematically, \((E_{1}^{*},\ldots ,E_{n}^{*})\) is called generalized Nash equilibrium point, if and only if, \(E_{i}^{*}\) is a solution of the problem \((P_{i})\) for \({(}E_{j}^{*})_{j=1,\ldots ,n;\;j\ne i}{\ }\)are given.

\(A=(a_{ij})_{1\leqslant i,j\leqslant n}\) where \(a_{ii}=2\) and \(a_{ij}=1\) for all \(i\ne j\);

\(b=(b_{i})_{1\leqslant i\leqslant n}\) where \(b_{i}=r(\frac{c_{i}}{pKq_{i}}-1) \).

Here, \(\mathbf {-1}\) denotes the vector \((-1,\ldots ,-1)\in \mathrm {IR}^{n}\) and \(\mathbf {0}\) denotes the vector \((0,\ldots ,0)^{T}\in \mathrm {IR}^{n}\).

We note that the \((LCP)\) problem is called a Linear Complementarity Problem \(Lcp(M,q)\) (for more details see Cottle and Dantzig (2006), Cottle et al. (1992), El Foutayeni and Khaladi (2010), El Foutayeni and Khaladi (2012), and Lemke (1965)).

For solving the linear complementarity problem \(Lcp(M,q)\) we can demonstrate that the matrix \(M\) is P-matrix (Recall that a matrix \(M\) is called P-matrix if all of its principal minors are positive) and we will use the following result: a linear complementarity problem \(Lcp(M,q)\) has a unique solution for every \(q\) if and only if \(M\) is a P-matrix (For demonstration we can see Murty (1971)).

## 4 Numerical Simulations of the Mathematical Model and Discussion of the Results

We consider four different fishermen (players), represented by their fishing effort \(E_{i}, i\in \{1,\ldots ,4\}\). The fishermen are assumed to fish the same fish population, they differ, however, in terms of their (a) \(q_{i}\) the catchability coefficient of each fisherman; (b) \(c_{i}\) the harvesting costs per fishing effort employed by each fisherman; (c) and \(E_{i}^{*}\) the fishing effort strategy.

Model biological parameters

Biological parameters | Value | Unit |
---|---|---|

\(r\) | 0.850000 | Per year |

\(q_{1}\) | 0.004000 | Kg per unit of effort |

\(q_{2}\) | 0.002828 | Kg per unit of effort |

\(q_{3}\) | 0.003464 | Kg per unit of effort |

\(q_{4}\) | 0.002000 | Kg per unit of effort |

\(K\) | 1,000.000 | Kg |

\(B_{0}\) | 100.0000 | Kg |

Model economic parameters

Economic parameters | Value | Unit |
---|---|---|

\(p\) | 1.000000 | Euros per Kg |

\(c_{1}\) | 0.500000 | Euros per unit of effort |

\(c_{2}\) | 0.186760 | Euros per unit of effort |

\(c_{3}\) | 0.071996 | Euros per unit of effort |

\(c_{4}\) | 0.250000 | Euros per unit of effort |

The fishing effort, harvest, revenue, cost and profit

Fisherman No. 1 | ||
---|---|---|

\(E_{1}^{*}\) | 30.252051 | Unit of effort |

\(H_{1}^{*}\) | 32.353068 | Kg |

\(R_{1}^{*}\) | 32.353068 | Euros |

\(C_{1}^{*}\) | 15.126026 | Euros |

\(\pi _{1}^{*}\) | 17.227042 | Euros |

Fisherman No. 2 | ||
---|---|---|

\(E_{2}^{*}\) | 60.504659 | Unit of effort |

\(H_{2}^{*}\) | 45.754568 | Kg |

\(R_{2}^{*}\) | 45.754568 | Euros |

\(C_{2}^{*}\) | 11.299850 | Euros |

\(\pi _{2}^{*}\) | 34.454718 | Euros |

Fisherman No. 3 | ||
---|---|---|

\(E_{3}^{*}\) | 60.504105 | Unit of effort |

\(H_{3}^{*}\) | 56.037159 | Kg |

\(R_{3}^{*}\) | 56.037159 | Euros |

\(C_{3}^{*}\) | 04.356029 | Euros |

\(\pi _{3}^{*}\) | 51.681129 | Euros |

Fisherman No. 4 | ||
---|---|---|

\(E_{4}^{*}\) | 60.504103 | Unit of effort |

\(H_{4}^{*}\) | 32.353068 | Kg |

\(R_{4}^{*}\) | 32.353068 | Euros |

\(C_{4}^{*}\) | 15.126026 | Euros |

\(\pi _{4}^{*}\) | 17.227042 | Euros |

Figure 4 shows also that if the fishermen provide a fishing efforts \(E_{2}=E_{3}=E_{4}=2E_{1}=60.504659\) (unit of effort), then (1) the first fisherman spends \(C_{1}=15.126026\) Euros, its income equal to \(R_{1}=32.353068\) Euros, its harvest is equal to \(H_{1}=32.353068\,\hbox {kg}\) (because we have taken \(p=1\) Euros per kg) and its profit is equal to \(\pi _{1}=17.227042\) Euros; (2) the second fisherman spends \(C_{2}= 0.747000C_{1}\,\bar{(}=11.299850\) Euros), its income equal to \(R_{2}=1.414200R_{1}\) (=45.754568 Euros), its harvest is equal to \(H_{2}=1.414200H_{1}\) (= 45.754568 kg) and its profit is equal to \(\pi _{2}=2\pi _{1}\) (= 34,454718 Euros); (3) the third fisherman spends \(C_{3}=0.288000C_{1}\,\bar{(}=04.356029\) Euros), its income equal to \(R_{3}=1.732100R_{1}\) (= 56.037159 Euros), its harvest is equal to \(H_{3}=1.732100H_{1}\) (= 56.037159 kg) and its profit is equal to \(\pi _{3}=3\pi _{1}\) (= 51.681129 Euros); (4) and the fourth fisherman spends \(C_{4}=C_{1}\,(=15.126026\) Euros), its income equal to \(R_{4}=R_{1}\) (= 32.353068 Euros), its harvest is equal to \(H_{4}=H_{1}\) (= 32.353068 kg) and its profit is equal to \(\pi _{4}=\pi _{1}\) (= 17.227042 Euros). So we can conclude that even when the fisherman \(i\) (\(i=2,3,4\)) provides the fishing effort equal to twice the fishing effort of the fisherman \(1\), then it will not necessarily to conclude that the fisherman \(i\) earns double the fisherman \(1\).

## 5 Conclusion

In this paper, we have considered a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. At biological equilibrium, we have calculated the fishing effort which maximizes the profit for each fisherman. To do so, we took into consideration the fact that all the fishermen have respected the sustainable management of the resources and the preservation of the biodiversity. This gave us a generalized Nash equilibrium problem. In section Numerical simulations, we have shown that even when the fisherman \(i\) provides the fishing effort equal to twice the fishing effort of the fisherman \(j\), then it will not necessarily to conclude that the first (fisherman \(i\)) earns double the second (fisherman \(j\)). Specifically we have shown that (scenario 1) if fisherman \(2\) provides fishing effort \(E_{2}=2E_{1}\), then he earns double profit \(\pi _{2}=2\pi _{1} \); (scenario 2) if fisherman \(3\) provides fishing effort \(E_{3}=2E_{1}\), then he earns the triple profit \(\pi _{3}=3\pi _{1}\); (scenario 3) and if fisherman \(i\) provides fishing effort \(E_{4}=2E_{1}\), then he earns the same profit \(\pi _{4}=\pi _{1}\).