Incentive Equilibrium in Bioresource Sharing Problem: Model

1. Incentive equilibrium in bioresource management problem Anna N. Rettieva annaret@krc.karelia.ru Science advisor: Prof. Vladimir V. Mazalov Institute of Applied Mathematical Research Karelian Research Centre Russian Academy of Sciences

2. Incentive equilibrium in bioresource sharing problem: model The dynamic of the fishery is described by the equation: (1) The population evolves in accordance with Verhulst model of the form: The players’ net revenues over a fixed time period [0,T] are (2) Theorem 1. The cooperative equilibrium of the problem (1)-(2) is (3) with satisfying the differential equation Karelian Research Center of the RAS, Institute of Applied Mathematical Research

3. Incentive equilibrium in bioresource sharing problem The strategy of the player i is a casual mapping if Definition. A strategy pair is called the incentive equilibrium at Proposition. The incentive equilibrium of the problem (1)-(2) is where are defined in (3) when and conjugate variables satisfy the next system of equation Karelian Research Center of the RAS, Institute of Applied Mathematical Research

4. Incentive equilibrium in bioresource sharing problem Let’s consider the second player’s deviation We find the center’s strategy in the following form Then the first player’s punishment strategy is It yields Theorem 2. The incentive equilibrium of the problem (1)-(2) is where Karelian Research Center of the RAS, Institute of Applied Mathematical Research

5. Incentive equilibrium in bioresource sharing problem : Numerical results Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Karelian Research Center of the RAS, Institute of Applied Mathematical Research

6. Incentive equilibrium in bioresource sharing problem : References T. Basar, G.J. Olsder. Dynamic noncooperative game theory. Academic Press, New York, 1982. C.W. Clark, Bioeconomic modelling and fisheries management, Wiley, New York, 1985. H.Ehtamo, R.P.Hamalainen, A cooperative incentive equilibriumfor a resource management problem, J. of Econ. Dynam. andContr.,17 (1993), 659-678. R.P.Hamalainen, V.Kaitala, A.Haurie, Bargaining on whales: A differential game model with Pareto optimal equilibria, Oper. Res. Lett. 3, 1 (1984), 5-11. V.V. Mazalov, A.N. Rettieva, A fishery game model with age distributed population: reserved territory approach, Game Theory and Applications 9 (2003), 56-72. V.V.Mazalov, A.N.Rettieva, Dynamic game methods for reserved terrirory Optimization, Survey in Appl. and Indust. Mathem. 12, 3 (2005), 610-625. V.V.Mazalov, A.N.Rettieva, A fishery game model withmigration: reserved territory approach, Game Theory andAppl., 10 (2005), 97-108. V.V.Mazalov, A.N.Rettieva, Nash equilibrium in bioresourcemanagement problem, Mathem. model., 18 (5) (2006),73-90. D.K. Osborn. Cartel problems, American Economic Review 66 (1976), 835-844. L.S.Pontryagin, V.G.Boltyanskii, R.V.Gamrelidze, I.F.Mischenko, Mathematical theory of optimal processes, Moscow: Nauka, 1976. Karelian Research Center of the RAS, Institute of Applied Mathematical Research