Linear quadratic game of exploitation of common renewable resources with inherent constraints

Rajani Singh, Agnieszka Wiszniewska-Matyszkiel

DOI: http://dx.doi.org/10.12775/TMNA.2017.057

Abstract


In this paper, we analyse a linear quadratic multistage game of extraction of a common renewable resource -- a fishery -- by many players with inherent state dependent constraints for exploitation and an infinite time horizon. To the best of our knowledge, such games have never been studied. We analyse the social optimum and Nash equilibrium for the feedback information structure and compare the results obtained in both cases. For the Nash equilibria, we obtain a value function that is contrary to intuitions from standard linear quadratic games. In our game, we face a situation in which the social optimum results in sustainability, while the Nash equilibrium leads to the depletion of the fishery in a finite time for realistic levels of the initial biomass of fish. Therefore, we also study an introduction of a tax in order to enforce socially optimal behaviour of the players. Besides, this game constitutes a counterexample to simplifications of techniques often used in computation of Nash equilibria and/or optimal control problems.

Keywords


Common renewable resources; Nash equilibrium; social optimality; linear quadratic dynamic games with constraints; Bellman equation; Pigouvian taxation

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References


H. Abou-Kandil, Closed-form solution for discrete-time linear-quadratic Stackelberg games, J. Optim. Theory Appl. 65 (1990), 139–147.

T. Başar, A. Haurie and G. Zaccour, Nonzero-sum differential games, Handbook of Dynamic Game Theory, Birkhäuser, Basel, 2016, DOI: 10.1007/978-3-319-27335-8 5-1.

T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, second edition, Academic Press, London, 1995.

R. Bellman, Dynamic Programming, Princeton University Press, Princeton, 1957.

D. Blackwell, Discounted dynamic programming, Ann. Math. Statistics 36 (1965), 226–235.

C. Carraro and J.A. Filar (eds.), Control and Game-Theoretic Models of the Environment, Ann. Internat. Soc. Dynam. Games, vol. 2, Birkhäuser, Boston, 1995.

B. Chen and P.A. Zadrozny, An anticipative feedback solution for the infinite horizon, linear-quadratic, dynamic, Stackelberg game, J. Econom. Dynam. Control 26 (2002), 1397–1416.

A. De Zeeuw and F. Van Der Ploeg, Difference games and policy evaluation: a conceptual framework, Oxford Economic Papers 43 (1991), 612–636.

E.J. Dockner, S. Jørgensen, N.V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.

J.C. Engwerda, On the open-loop Nash equilibrium in LQ-games, J. Econom. Dynam. Control 22 (1998), 729–762.

J.C. Engwerda, LQ Dynamic Optimization and Differential Games, John Wiley and Sons, Chichester, 2005.

C. Fershtman and M.I. Kamien, Dynamic duopolistic competition with sticky prices, Econometrica 55 (1987), 1151–1164.

R.D. Fischer and L.J. Mirman, A strategic dynamic interaction. Fish wars, J. Econom. Dynam. Control 16 (1992), 267–287.

O. Górniewicz, A. Wiszniewska-Matyszkiel, Verification and refinement of a two species Fish Wars model, Fisheries Research (2017), DOI: 10.1016/j.fishres.2017.10.021.

R.P. Hämäläinen, Nash and Stackelberg solutions to general linear-quadratic two player difference games. I. Open-loop and feedback strategies, Kybernetika 14 (1978), 38–56.

G. Hardin, The tragedy of the commons, Science 162 (1968), 1243–1248.

A. Haurie, J.B. Krawczyk and G. Zaccour, Games and Dynamic Games, World Scientific, Hackensack, 2012.

R. Isaacs, Differential Games, Wiley, New York, 1965.

G. Jank and H. Abou-Kandil, Existence and uniqueness of open-loop Nash equilibria in linear-quadratic discrete time games, IEEE Trans. Automat. Control 14 (2003), 267–271.

S. Jørgensen and G. Zaccour, Developments in differential game theory and numerical methods: economic and management applications, Comput. Manag. Sci. 4 (2007), 159–181.

F. Kydland, Noncooperative and dominant player solutions in discrete dynamic games, Internat. Econom. Rev. 16 (1975), 321–335.

D. Levhari and L.J. Mirman, The great fish war: an example using a dynamic Cournot–Nash solution, Bell Journal of Economics 11 (1980), 322–334.

N.V. Long, Dynamic games in the economics of natural resources: a survey, Dyn. Games Appl. 1 (2011), 115–148.

N.V. Long, Applications of dynamic games to global and transboundary environmental issues: a review of literature, Strategic Behaviour and the Environment 2 (2012), 1–59.

P.V. Reddy and G. Zaccour, Open-loop Nash equilibria in a class of linear-quadratic difference games with constraints, IEEE Trans. Automat. Control 60 (2015), 2559–2564.

N.L. Stokey, R.E. Lucas Jr. and E.C. Prescott, Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge, 1989.

A. Wiszniewska-Matyszkiel, A Dynamic game with continuum of players and its counterpart with finitely many players, Ann. Internat. Soc. of Dynam. Games (A.S. Nowak and K. Szajowski, eds.), Birkhäuser, 7 (2005), 455–469.

A. Wiszniewska-Matyszkiel, Common resources, optimality and taxes in dynamic games with increasing number of players, J. Math. Anal. Appl. 337 (2008), 840–841.

A. Wiszniewska-Matyszkiel, On the terminal condition for the Bellman equation for dynamic optimization with an infinite horizon, Appl. Math. Lett. 24 (2011), 943–949, DOI: 10.1016/j.aml.2011.01.003.

A. Wiszniewska-Matyszkiel, Open and closed loop Nash equilibria in games with a continuum of players, J. Optim. Theory Appl. 160 (2014), 280–301, DOI: 10.1007/s10957013-0317-5.


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