1. Incentive equilibrium in bioresource management problem
Anna N. Rettieva
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
Definition. A strategy pair
is called the incentive equilibrium at if
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
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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 equilibrium for a resource management problem, J. of Econ. Dynam. and Contr., 17 (1993),
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),
V.V. Mazalov, A.N. Rettieva, Dynamic game methods for reserved terrirory
Optimization, Survey in Appl. and Indust. Mathem. 12, 3 (2005),
V.V. Mazalov, A.N. Rettieva, A fishery game model with migration: reserved territory approach, Game Theory and Appl., 10 (2005),
V.V. Mazalov, A.N. Rettieva, Nash equilibrium in bioresource management problem, Mathem. model., 18 (5) (2006),
D.K. Osborn. Cartel problems, American Economic Review 66 (1976),
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