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.