Calculation of optima and equilibria in dynamic resource extraction problems

Exploitation or extraction of common-property renewable resources is one of the biggest challenges in society. It encompasses a wide range of various problems among other things, the phenomenon known as the tragedy of the commons. Most importantly, the extraction and consumption of common natural renewable resources have a strong impact on the quality of life and well-being of both, the current and future generations. From the mathematical point of view, the only tool to deal with the whole spectrum of phenomena arising in such types of problems, in which there are at least two independent decision makers in a common resource extraction problem, are dynamic games, since both dynamic optimization methods and static games encompass only fractions of aspects of those problems. In the dissertation, we propose several models of dynamic games and dynamic optimisation problems, modelling the exploitation of common renewable resources by taking into account various aspects of the problem: • Many players in commons. Increasing number of players regarded as decomposition of the decision making structures. To be more specific, if we consider the same mass of individuals, decomposed into units of decreasing size: from consumers, through North and South, actual countries, regions etc. and finally actual decision makers. • Relation between the Nash equilibria and the social optima and ways of solving the tragedy of the commons by Pigovian taxation or a tax-subsidy system. • Taking into account information: feedback form, closed loop, delayed information. • Self-enforcing environmental agreements with a delay in observation of defection. • Completing and correcting previous results in this research field or finding counterexamples to common beliefs and methodological simplifications. In dynamic games, the strategy of a player is a function which defines his/her behaviour at each time instant in the time interval considered in the game. Therefore, calculation of both, the social optima and the Nash equilibria requires solving the dynamic optimisation problems. However, finding a Nash equilibrium in dynamic games requires solving a set of dynamic optimisation problems, coupled by finding a fixed point of the resulting best response correspondence in some functional space of the profiles of strategies. Due to this coupling, the problem becomes much more complicated than the analogous dynamic optimisation problems. There are quite a few results in nonzero-sum dynamic games, and if the constraints appear (which is natural in real life problems, especially resource extraction problems), then the results are very rare. Therefore, unexpected behaviour of the solution may appear (irregularity, discontinuity, the nonexistence of equilibria of a certain type, existence of many equilibria, lack of convergence). So, we try to fill in the gaps in the simplifications of dynamic games. The dissertation also contains counterexamples to some methods and hypotheses that are regarded as correct and used to solve dynamic games. With the strong motivation behind the chosen problems, in Chapter 1, we introduce the game and some preliminary knowledge of game theory, brief literature review and the mathematical optimisation tools that are used to solve the game models in the dissertation. In Chapter 2, we present a discrete time, infinite horizon, a linear-quadratic dynamic game model with many players and with linear state-dependent constraints on decisions of players. In this model, players can be regarded as countries or firms. There are either finitely many players or a continuum of players. The model has an obvious application in a common fishery extraction problem where the players sell their catch at a common market. We solve the social optimum problem for n-players and for the continuum of players. When it comes to the Nash equilibrium problem, we are only able to solve it for the continuum of players case. For n-players case, we are not able to calculate it for n ≥ 2, only negative results can be proven: that the Nash equilibrium strategies and the value functions are not of assumed regularity with respect to the state variable and showing that presence of even a very simple and obvious constraints on strategies may result in a very complicated form of the value functions and the Nash equilibria. While looking for Nash equilibria, the social optima, we have also found a very simple counterexample to the correctness of a procedure often used in dynamic game theory literature. We also calculate the enforcement of a social optimum profile by various type of Pigouvian tax or a tax-subsidy system, both for n-players and for the continuum of players. Non-existence of a symmetric feedback Nash equilibrium of assumed regularity in the linear-quadratic problem considered in Chapter 2 seems to be inherited from the finite time horizon truncations of the game, so in Chapter 3, we solve a feedback Nash equilibrium problem in a very simple 2-stage, 2-player linear-quadratic dynamic game being a truncation of the model which was studied in Chapter 2 with the infinite time horizon. As a result, we found that the presence of simple linear state-dependent constraints results in the nonexistence of a continuous symmetric feedback Nash equilibria, whereas the existence of the continuum of discontinuous symmetric feedback Nash equilibria. Our result is counter-intuitive to the common belief in the continuity of Nash equilibria for linear-quadratic dynamic games with concave payoffs. While previous two Chapters deal with the specific value of the discount factor β, given by the so called golden rule, in Chapter 4, we solve the social optimum problem from Chapter 2 for more general class of linear-quadratic dynamic games with only one player, called social planner and for more general β instead of the golden rule β. So, we consider a discrete time linear-quadratic dynamic optimisation problem with linear state-dependent constraints. We solve the problem in the infinite time horizon and its finite horizon truncations. Although it seems simple in its linear-quadratic form, calculation of the optimal control is nontrivial. In Chapter 5, we study a general class of dynamic optimization problems. We derive general rules stating what kind of errors in calculation or computation of the value function does not lead to errors in calculation or computation of optimal control. This general result concerns not only errors resulting from using the numerical methods but also errors resulting from some preliminary assumptions related to constraints on the value functions. The results are illustrated by a motivating example of discrete time Fish Wars model, proposed by Levhari and Mirman, with singularities in payoffs. In Chapter 6, we study a continuous time version of the Fish Wars model with the infinite time horizon, linear state equation and state-dependent linear constraints on controls. We calculate the social optimum and a Nash equilibrium which always leads to the depletion of the resource even if the social optimum results in its sustainability. We propose two ways of solving the problems of enforcing social optimality: either by a tax-subsidy system or by an environmental agreement even if we assume that it takes time to detect any defection of a player. We also propose a general algorithm for finding the financial incentives enforcing the socially optimal profile in a large class of differential games.