Abstract
Game theory offers a rigorous mathematical framework for analyzing how rational players make optimal decisions in competitive environments. Its application spans numerous fields, including economics, political science, psychology, computer science, and biology. At the core of game theory, the Nash equilibrium provides a fundamental basis for understanding strategic interactions in non-cooperative games. However, the presence of multiple Nash equilibria in a single game often results in outcomes that may seem counterintuitive or undesirable. This challenge has prompted the development of various equilibrium refinements designed to select more reasonable outcomes.One notable refinement is the concept of proper equilibrium. Building on trembling hand perfect equilibrium, the notion of proper equilibrium further refines the concept by imposing stricter conditions on trembles, ensuring that less costly errors are considerably more likely than those associated with greater cost, while perfect equilibrium only eliminates implausible equilibria by introducing small perturbations that ensure every pure strategy is assigned a positive probability. Proper equilibrium holds particular importance in games presented in their reduced normal form, as proper equilibria not only generate sequential strategies but also yield quasi-perfect outcomes when full information about previous moves is available. However, there are only a few papers in the literature that examine the computation of a proper equilibrium, and even fewer studies have demonstrated the effectiveness of such computational methods. In this thesis, we introduce extensions of tracing procedures to select a proper equilibrium with smooth paths with arbitrarily chosen starting points.
Harsanyi introduced the linear tracing procedure as a method for selecting a Nash equilibrium in 1975. This method constructs a continuous path of mixed strategy profiles, linking the best response defined by the prior belief to a Nash equilibrium. The linear tracing procedure represents a fundamental methodology in game theory, providing both theoretical foundations for equilibrium selection and practical computational approaches for identifying Nash equilibria in strategic interactions. While researchers have successfully adapted this technique to various refinement concepts, including perfect equilibria, its application to proper equilibrium computation remains an important unsolved challenge. This methodological gap is particularly significant given that proper equilibria offer the most compelling refinement for eliminating implausible outcomes in games with multiple Nash equilibria.
Given the foundational role of Harsanyi's linear and logarithmic tracing procedures in equilibrium selection theory, this paper introduces extensions of these procedures to select proper equilibria in normal-form games. We propose extensions of the linear tracing procedure by constructing a perturbed game with continuously differentiable polynomial functions, where each player optimizes a linear combination of the payoff function based on the strategies given by others and the payoff function based on a prior belief about the strategies. By applying the optimality condition, fixed-point argument, and variable transformation to the perturbed game, we establish the existence of a smooth path starting from a unique point and approaching a proper equilibrium. To enhance procedural efficiency, we develop extensions of the logarithmic tracing procedure tailored to approximate the extension of the linear tracing procedure. Additionally, we incorporate a continuously differentiable exponential function into the perturbed games and utilize error-bound analysis to establish smooth paths to proper equilibria.
Numerical experiments demonstrate the efficiency and stability of our methods.
| Date of Award | 2 Sept 2025 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Chuangyin DANG (Supervisor) |
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