Advanced game theory applications: equilibrium enumeration for 2 × 2 × 2 games and Empirical Game-Theoretic Analysis of a pricing game

This thesis comprises two main projects. In the first project, we analyse the Nash equilibria of three-player games in their simplest form, known as 2 × 2 × 2 games. We explore the best-response surfaces for each player in detail and develop an algorithm that computes the complete set of Nash equilibrium components, improving upon previous algorithms that only guarantee finding a single equilibrium in these games. In addition to the algorithm, we implemented a software tool that visually represents these games in 3D using best-response surfaces and calculates and displays all Nash equilibria. Another significant contribution of this project is our theoretical proof establishing an upper bound of nine on the number of Nash equilibria in non-generic 2 × 2 × 2 games. Moreover, we extend the definition of degeneracy from the two-player setting to these games and provide directions for extending the upper bound result to the broader class of non-degenerate games. The second project presents an Empirical Game-Theoretic Analysis (EGTA) of a classical multi-round duopoly pricing game with an infinite strategy space. We employ the Policy-Space Response Oracles (PSRO) framework to iteratively construct a finite approximation of this infinite-strategy game, referred to as the “meta-game.” Within this framework, single-agent reinforcement learning (RL) is utilised to compute best-response strategies against Nash equilibria of the evolving meta-game. Starting with custom implementations of basic RL algorithms, we encountered limitations that motivated the adoption of advanced learning methods better suited to capture the pricing game’s complexity. We discuss the strengths and weaknesses of each method applied, alongside techniques developed to enhance their performance. Through comprehensive experiments varying RL algorithms and model specifications, we analyse the resulting meta-games, the estimated equilibria, and the emergent behaviours of RL-trained pricing strategies. We further explore conditions leading to the emergence of collusive behaviour among the trained strategies. To support future research, we release our framework publicly, facilitating similar analyses by adapting it to other duopoly pricing games. Our framework integrates seamlessly with the advanced RL algorithms provided by the Stable-Baselines3 library and automates the tracking and analysis of learning dynamics, meta-game approximations, and emerging equilibria.