Games, fair and square: constructive approaches to equilibrium stability, fairness and computation

Anand Kumar, Dhivya (2026) Games, fair and square: constructive approaches to equilibrium stability, fairness and computation. PhD thesis, University of Glasgow.

Abstract

Game theory has evolved from analysing simple strategic interactions to addressing the computational and normative challenges of complex multi-agent systems. This thesis advances three interconnected dimensions of modern game theoretic analysis: the stability of equilibria under perturbations, the algorithmic discovery of equilibrium structures in large scale games, and the design of fair mechanisms for resource allocation. Together, these contributions bridge theoretical foundations with computational practice, offering constructive solutions to fundamental problems in
strategic decision making.
The first contribution revisits hyperstability, a refinement concept for equilibria robust against structural perturbations. Building on Hauk and Hurkens (2002), we propose a systematic alternative to adding complex mixed strategies, which they do to eliminate, following perturbations, certain equilibrium components; this alternative utilises the addition of pure strategies, along with binary, enforcing players. By introducing pure strategy duplicates paired with auxiliary disciplinary players using a construction inspired by Levy games (Levy, 2016), we enforce we enforce an imitation of mixed strategies through strategic interactions. This framework avoids manual mixture calculations, generalises to N-player games, and provides a transparent method for identifying hyperstable equilibria.
The second contribution addresses equilibrium discovery in Polymatrix Games. We establish that determining the existence of a pure strategy Nash equilibrium in these games is NP-complete. To navigate this complexity, we show that a specific parametrisation of Lemke (1965)’s algorithm implements the Linear Tracing Procedure (van den Elzen and Talman, 1999), enabling principled computation from arbitrary priors. To overcome the bias of forward tracing toward dominant basins, we introduce the Tracing Backwards algorithm. By using discovered equilibria as "seeds" for breadth first exploration, we uncover a novel property termed fertility: the capacity of an equilibrium to lead to new discoveries. Our analysis reveals that while pure equilibria
are easy to find, they exhibit low fertility compared to completely mixed equilibria, which act as critical hubs in the strategic landscape.
The third contribution evaluates fair division problems involving indivisible goods and monetary transfers. We conduct an empirical comparison of three mechanisms: Bundled Auction (BA), Bid & Sell (B&S), and Divide & Choose (D&C). Our simulations demonstrate that mechanism responsiveness, i.e., the ability to adapt to heterogeneous agent preferences, is the primary driver of social welfare. While D&C performs well in additive environments with independent valuations, B&S emerges as the superior mechanism in complex settings with strong complementarities or substitution effects. Its continuous pricing structure effectively captures "unbundling surplus" that discrete mechanisms fail to exploit, proving that the computational overhead of sophisticated mechanisms yields substantial welfare gains, validating their practical value in real world allocation problems.
Collectively, this thesis offers a constructive journey through equilibrium analysis: from establishing when equilibria are strategically robust, to developing algorithms that systematically map equilibrium manifolds in large games, to designing mechanisms that efficiently and fairly allocate scarce resources. These contributions address theoretical, computational, and applied challenges in modern game theory, providing tools and insights for analysing strategic interactions in increasingly complex multi agent systems.

• 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: Levy, Dr. John, Moulin, Professor Hervé and Von Stengel, Professor Bernhard
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-86063
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 24 Jun 2026 10:23
• Last Modified: 24 Jun 2026 10:34
• Thesis DOI: 10.5525/gla.thesis.86063
• URI: https://theses.gla.ac.uk/id/eprint/86063