On the equilibrium structure of some problems in game theory and economics

Moreno de Guerra Beato, José M. (2026) On the equilibrium structure of some problems in game theory and economics. PhD thesis, University of Glasgow.

Abstract

The thesis studies equilibrium in a variety of settings relevant to economics and game theory.

Chapter 1 presents structure theorems for optimisation problems. These results are in the spirit of the celebrated theorem by Mertens and Kohlberg concerning the existence of a homeomorphism between the space of games and the graph of the Nash correspondence. In particular, chapter 1A establishes an analogue of the theorem by Mertens and Kohlberg for the case of simultaneous linear programmes. Meanwhile, chapter 1B presents a generalisation to monotone variational inequalities. Specifically, the chapter proves that given a closed translation invariant subset of the space of variational monotone inequalities, there exists a homeomorphism between the set and the restriction of the graph of the correspondence that associates to each variational inequality in the given set its set of solutions. Chapter 1B also introduces a weakening of monotonicity, ω-monotonicity, and an associated structure theorem which is used to derive a generalisation of Kohlberg-Mertens due to Predtetchinski for games with continuous actions and concave differentiable pay-offs.

Chapter 2 establishes that given an algebraic curve on the square [0, 1]2 which does not intersect the top or bottom of the square, i.e., does not intersect {0, 1} × [0, 1], there exists a binary game whose Nash equilibrium set contains a component that projects injectively onto the original curve. Moreover, individual coordinates of the inverse of this projection are affine functions. If, in addition to the previous condition, the curve intersects each fibre {t} × (0, 1), for every t ∈ [0, 1], at least once. Then the whole Nash equilibrium set projects injectively onto the curve instead. Several novel examples of these results are also provided. Notably, we show that there exists a binary game whose set of equilibria is equivalent to a set consisting of a horizontal line segment and a parabolic segment meeting at a zero-degree angle. This equivalence solves a conjecture proposed by Levy.

Chapter 3 modifies an equilibrium notion due to Azevedo and Gottlieb for competitive markets with adverse selection. Following Azevedo and Gottlieb, an equilibrium is characterised by firms making no profits from traded contracts; consumers choosing their most preferred option and robustness to certain perturbations. Unlike Azevedo and Gottlieb, where agents are required to behave optimally in the perturbation, we only require that agents behave ε-optimally. Equilibrium is shown to exist under the assumptions that agents’ utility is uniformly Lipschitz in price, and uniformly Lipschitz in contract characteristics. Equilibrium is shown to have the same properties as the original definition by Azevedo and Gottlieb; notably, the equilibrium price vector is a Lipschitz function of contract characteristics. A significant advantage however is that the set of equilibria are shown to satisfy natural continuity properties in the set of contract characteristics. This property is shown to not hold for Azevedo-Gottlieb equilibria. More generally, the equilibrium notion is simultaneously restrictive enough to satisfy the regularity properties Azevedo-Gottlieb equilibrium does, while being significantly easier to verify the robustness to perturbation stipulated in their definitions.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: H Social Sciences > HB Economic Theory; Q Science > QA Mathematics
• Colleges/Schools: College of Social Sciences > Adam Smith Business School
• Supervisor's Name: Levy, Dr. John and Bogomolnaia, Professor Anna
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-86219
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 09 Sep 2026 14:43
• Last Modified: 09 Sep 2026 14:45
• Thesis DOI: 10.5525/gla.thesis.86219
• URI: https://theses.gla.ac.uk/id/eprint/86219