Sylenko I. V. Markov Perfect Equilibrium in a resource extraction game with power utilities of the agents. –– Qualification scientific work in the form of manuscript.
Thesis for doctor of philosophy degree in speciality 113 –– Applied math- ematics. –– NaUKMA, Kyiv, 2023.
The thesis is devoted to researching a class of nonzero-sum stochastic games, which is referred to in the scientific game theory literature as a game of resource extraction or a game of capital accumulation. From a theoretical standpoint a noteworthy characteristic of the game lies in uncountability of both its state space and the players’ action spaces. Being motivated by prac- tical applications, the game concept models an economic problem of strategic interaction with the following interpretation. Several agents commonly pos- sess a certain renewable asset (a capital or a resource), which generates advan- tage from its utilization. Throughout a sequence of discrete time moments all players simultaneously decide on the resource quantity for the current per- sonal consumption, according to which an instantaneous payoff is received by means of evaluating a constant individual utility function. The quantity of the resource available for consumption in each moment of time is deter- mined accordingly to a known stochastic law of motion, which depends on the previous resource amount and the extraction values of the players. The total reward of every participant equals to a discounted sum of all received instantaneous payoffs, the expected value of which each of them aims to maximize when choosing a personal strategy at the beginning of the game. The main purpose behind the research is to enrich the available results of the (Stationary) Markov Perfect Equilibrium existence in the game, which is considered a solution concept in this type of noncooperative games. The equilibrium existence in a general framework of resource extraction games is an open problem, yet due to having a scientific and practical interest it is being investigated extensively.
The resource extraction game model considered in the thesis belongs to a previously unexamined class, which is characterized by an arbitrary number of participants, an unbounded state space, unbounded utility functions and a transition probability in form of a stochastic kernel dependent on a joint investment of the players. The model is constructed with a pair of special assumptions concerning the players’ utility functions and the stochastic law of motion between states, both of which are commonly applied in economic modelling. First, it is assumed that the players’ preferences are isoelastic (or CRRA according to the economic terminology) in the shape of strictly concave power functions. Second, every subsequent quantity of the available resource (i. e. a state of the game) is a linear transformation of the amount left after the players’ consumption (i. e. their joint investment ) influenced by a multiplicative random shock, which is identically distributed at each stage of the game. Such stochastic process is exploited in economic modelling and financial literature under the term geometric random walk.
The proposed model is studied in both symmetric and nonsymmetric game settings. For the symmetric game model a unique Markov Perfect Equilib- rium is constructed using the method of backward induction in the case when the horizon of the game is finite. By showing that the equilibrium policy functions and the corresponding value functions of the players are monotoni- cally convergent as the time horizon goes to infinity, a symmetric Stationary Markov Perfect Equilibrium existence in non-randomized strategies is estab- lished for the infinite horizon game model. It is then proven that the obtained equilibrium is not Pareto optimal, and a formula for calculating the socially optimal strategy profile is revealed. It is also confirmed that the players tend to overconsume when being in the equilibrium comparing to their socially op- timal behaviour. According to this result, the symmetric game model admits the presence of “the tragedy of the commons”. In the non-symmetric infi- nite horizon game model a Stationary Markov Perfect Equilibrium existence in pure strategies is concluded via a different approach, and is additionally deduced to be unique among the strategy profiles, consisting of linear policy functions. The employed proving method, while being distinct from the one used in the case of a symmetric game model, similarly provides an algorithm for constructing the equilibrium strategy profile, which is especially useful for applying the results in practice.
Keywords: nonzero-sum stochastic games, resource extraction, capital ac- cumulation, stationary Nash equilibrium, Markov perfect equilibrium, power utility, geometric random walk, discrete time, discounted payoff, cooperative game theory.
