TY - CHAP

T1 - Measure-valued solutions to a harvesting game with several players

AU - Bressan, Alberto

AU - Shen, Wen

N1 - Funding Information:
Acknowledgements The work of the first author was partially supported by NSF, with grant DMS-0807420. The work of the second author was partially supported by NSF grant DMS-0908047. This paper was written as part of the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–2009.

PY - 2011

Y1 - 2011

N2 - We consider Nash equilibrium solutions to a harvesting game in one-space dimension. At the equilibrium configuration, the population density is described by a second-order O.D.E. accounting for diffusion, reproduction, and harvesting. The optimization problem corresponds to a cost functional having sublinear growth, and the solutions in general can be found only within a space of measures. In this chapter, we derive necessary conditions for optimality, and provide an example where the optimal harvesting rate is indeed measure valued. We then consider the case of many players, each with the same payoff. As the number of players approaches infinity, we show that the population density approaches a well-defined limit, characterized as the solution of a variational inequality. In the last section, we consider the problem of optimally designing a marine park, where no harvesting is allowed, so that the total catch is maximized.

AB - We consider Nash equilibrium solutions to a harvesting game in one-space dimension. At the equilibrium configuration, the population density is described by a second-order O.D.E. accounting for diffusion, reproduction, and harvesting. The optimization problem corresponds to a cost functional having sublinear growth, and the solutions in general can be found only within a space of measures. In this chapter, we derive necessary conditions for optimality, and provide an example where the optimal harvesting rate is indeed measure valued. We then consider the case of many players, each with the same payoff. As the number of players approaches infinity, we show that the population density approaches a well-defined limit, characterized as the solution of a variational inequality. In the last section, we consider the problem of optimally designing a marine park, where no harvesting is allowed, so that the total catch is maximized.

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U2 - 10.1007/978-0-8176-8089-3_20

DO - 10.1007/978-0-8176-8089-3_20

M3 - Chapter

AN - SCOPUS:84879619926

T3 - Annals of the International Society of Dynamic Games

SP - 399

EP - 423

BT - Annals of the International Society of Dynamic Games

PB - Birkhauser

ER -