Essays in Applied and Computational Game Theory

University of Minnesota Ph.D. dissertation.June 2019. Major: Economics. Advisor: Jan Werner. 1 computer file (PDF); viii, 123 pages.This dissertation considers computational and applied aspects of cooperative and non-cooperative game theory. The first chapter discusses a novel applied game theory approach within the field of vulnerability disclosure policy. I introduce a three-player game between software vendors, software users, and a hacker in which software vendors attempt to protect software users by releasing updates, i.e. disclosing a vulnerability, and the hacker is attempting to exploit vulnerabilities in the software package to attack the software users. The software users must determine whether the protection offered by the update outweighs the cost of installing the update. Following the model set up, I describe why low-type software users, software users that do not get much value out of the software and are thus not very damaged by an attack, prefer Non-Disclosure, and Disclosure can only be an optimal policy in cases when the cost to the hacker of searching for a zero-day vulnerability is small. Many economic problems are inherently non-linear, so in the second chapter we introduce the MGBA, the Modular Groebner Basis Approach, which is a solution technique from Algebraic Geometry that can be used to ``triangularize'' polynomial systems. The MGBA is a computational tool that overcomes the typical computational problems of intermediate coefficient swell and solving for lucky primes that can limit the ability to compute Groebner bases. The Groebner basis is an all-solution computational technique that can be applied to many fields in economics. This chapter focuses on applying the MGBA to Bertrand games with multiple equilibria and a manifold approach to solving dynamic programming problems. Advances in computational power and techniques have greatly benefited both economic theory, in allowing economists to solve more realistic models, and data analysis, such as machine learning. However, the field of cooperative game theory has fallen behind. Therefore, in the final chapter, I introduce the compression value, a computationally efficient approximation technique for the non-transferable utility (NTU) Shapley value. This algorithm gives a reasonable approximation of the NTU Shapley value if the initial guess of Pareto weights is near the actual solution

Investigation of Game-Theoretic Mechanisms for the Valuation of Energy Resources

Electricity systems are facing the pressure to change in response to the effects of new technology, particularly the proliferation of renewable technologies (such as solar PV systems and wind generation) leading to the retirement of traditional generation technologies that provide stabilising inertia. These changes create an imperative to consider potential future market structures to facilitate the participation of distributed energy resources (DERs; such as EVs and batteries) in grid operation. However, this gives rise to general questions surrounding the ethics of market structures and how they could be fairly applied in future electricity systems. Particularly the most basic question "how should electricity be valued and traded" is fundamentally a moral question without any easy answer. We give a survey of philosophical attitudes around such a question, before presenting a series of ways that these intuitions have been cast into mathematics, including: the Vickrey-Clarke-Groves mechanism, Locational Marginal Pricing, the Shapley Value, and Nash bargaining solution concepts. We compared these different methods, and attempted a new synthesis that brought together the best features of each of them; called the 'Generalised Neyman and Kohlberg Value' or the GNK-value for short. The GNK value was developed as a novel bargaining solution concept for many player non-cooperative transferable utility generalised games, and thus it was intrinsically flexible in its application to various aspects of powersystems. We demonstrated the features of the GNK-value against the other mathematical solutions in the context of trading the immediate consumption/generation of power on small sized networks under linear-DC approximation, before extending the computation to larger networks. The GNK value proved to be difficult to compute for large networks but was shown to be approximable for larger networks with a series of sampling techniques and a proxy method. The GNK value was ethically compared to other mechanisms with the unfortunate discovery that it allowed for participants to be left worse-off for participating, violating the ethical notion of 'euvoluntary exchange' and 'individual rationality'; but was offered as an interesting innovation in the space of transferable utility generalised games notwithstanding. For sampling the GNK value, there was a range of new and different techniques developed for stratified random sampling which iteratively minimise newly derived concentration inequalities on the error of the sampling. These techniques were developed to assist in the computation of the GNK value to larger networks, and they were evaluated in the context of sampling synthetic data, and in computation of the Shapley Value of cooperative game theory. These new sampling techniques were demonstrated to be comparable to the more orthodox Neyman sampling method despite not having access to stratum variances