Convergence Analysis of Heterogeneous Decision-making Populations Under the Coordinating Best-response and Imitation Update Rules

Abstract

This thesis emphasis is on coordination games. In a coordination game, selecting the same strategy or decision as the opponent is mutually beneficial for both parties. We studied the problem of equilibrium convergence in such games in both discrete and continuous (time) cases.

In the first Chapter, we provide a brief introduction to the field of game theory. We discuss different categories of agents based on their levels of rationality and decision-making strategies, along with a variety of games. Additionally, we address important issues and challenges within this field.

The second Chapter of this work is dedicated to a heterogeneous mixed population of imitators and best-responders. In this model, agents’ update rules are assumed to be discrete functions of time. Imitators refer to agents who simply replicate the strategy of another agent with the highest payoff, while best-responders pick the strategies that maximise their individual outcomes. Suggesting the concept of ’sections’--a consecutive sequence of agents with similar strategies– helped us in establishing convergence to an equilibrium state. This convergence was demonstrated under any arbitrary asynchronous activation sequence within a linear network. The proof was then extended to networks with ring, starike, and sparse-tree structures. However, the question of equilibrium convergence for other network structures remains an open challenge.

In the third Chapter, we examined a large well-mixed population of imitators within a coordination context. Our analysis is grounded in the assumption that imitation here is driven by dissatisfaction. Equivalently, agents with lower payoffs are more dissatisfied and have more tendency to change and imitate higher earners within the population. The analysis reveals the presence of three fixed points, of which two are stable and one is a saddle point. The stable manifold of the unstable fixed point is also calculated. Additionally, It is demonstrated that starting from any initial state, the population eventually converges towards one of these introduced fixed points.