Three essays in microeconomic theory

Song, Xihao (2025) Three essays in microeconomic theory. PhD thesis, University of Glasgow.

Abstract

This thesis explores foundational problems in microeconomic theory and advances understanding of complex decision-making processes in different environments with different parties through mathematical modelling.

Chapter 2 proves a new representation theorem for continuous‐time preference over consumption or portfolio streams that take values in the probability simplex, allowing for both fully continuous and piecewise‐continuous paths. Unlike existing results, our construction only relies on standard axioms. This framework is applicable to problems like portfolio allocation under a fixed budget or consumption under liquidity shocks—where payoffs may jump at discrete dates. We also show that one agent is more impatient than another if and only if their discount factor is larger.

Chapter 3 explores the asymptotic behaviour of strictly dominated strategies in random games of two players, in which the elements are i.i.d. selected from a probability distribution. As the game size increases, if the number of strategies for the two players is similar, in a sense we make precise, the probability of having strictly dominated strategies approaches 0. This is particularly evident in large nearly square games where the row size is proportionate to the column size within a specific range. Consequently, the probability of a game being dominance solvable diminishes to zero. Furthermore, we show that this result is very nearly tight: small deviations lead to a non-zero chance of strictly dominated strategies, while larger deviations make their existence nearly certain. Our findings first emphasize the significance of the parameters in the underlying probability distribution.

Chapter 3 also examines the asymptotic behaviour of the fixed proportion q of dominated strategies as M,N →∞, where M and N are the row and column sizes. Specifically, for the row player, we show that the probability of existence of q-portion-dominated strategy approaches 0 as N ≥ M/(ln(M))α for some α > 0 and it approaches 1 when M grow much faster than (N/(1−δ −q))N for some δ > 0.

Finally, Chapter 3 proposes a simple algorithm for detecting strictly dominated strategies in finite games, a topic of interest in economics and computer science since Yu and Zeleny 1975. This algorithm improves the conventional approach by utilizing minimum and maximum comparisons to reduce the expected time complexity.

Chapter 4 introduces a social planner into the random allocation framework who does not receive any object but may influence the allocation toward his preference. We present two guiding principles to clarify the conditions under which the social planner’s opinion cannot be dismissed and when an agent’s opinion must be respected. Conformity-Priority Efficiency (CPF) asserts that for any given object, we should reward the agent who conforms to social expectation more than others, which strengthens the notion of ordinal efficiency. Indistinguishability Fairness (IF) requires that for any object, if we can’t distinguish agents due to social expectation and support of lottery, then they should be treated equally. Then, we construct a simple Flow algorithm to characterize them precisely.

Chapter 4 also examine the classic random allocation problem and, adhering to the principle of interim favouring rank, propose an alternative variant to the probabilistic setting to eliminate an unfair scenario where agents who rank objects higher receive more favourite objects with positive probability. We introduce the property of interim favouring support, which is satisfied by the fractional adaptive Boston rule. Additionally, we propose a new fairness criterion, termed equal support equal claim, to characterize the fractional adaptive Boston rule.

Finally, Chapter 4 introduces a new efficiency notion, interim efficiency, which is stronger than ex-post Pareto efficiency but weaker than ordinal efficiency. We construct the algorithm that is easy to apply in the lab, the Random Flow mechanism, to achieve interim efficiency. Numerical analysis shows that random flow results in less envy across preference profiles than the random priority mechanism.

Together, these contributions demonstrate the versatility of game theory in addressing complex decision-making, resource allocation, and strategic refinement across diverse contexts.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: H Social Sciences > HB Economic Theory
• Colleges/Schools: College of Social Sciences > Adam Smith Business School
• Supervisor's Name: Hayashi, Professor Takashi and Levy, Dr. John
• Date of Award: 2025
• Depositing User: Theses Team
• Unique ID: glathesis:2025-85464
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 07 Oct 2025 09:11
• Last Modified: 07 Oct 2025 09:15
• Thesis DOI: 10.5525/gla.thesis.85464
• URI: https://theses.gla.ac.uk/id/eprint/85464