[Paper Review] Mathematical analysis of Soros's theory of reflexivity

C.P. Kwong|ArXiv.org|Jan 28, 2009

Philosophy and History of Science15 references7 citations

TL;DR

This paper provides a rigorous mathematical analysis of George Soros's theory of reflexivity using discrete dynamical systems, modeling the cognitive and manipulative functions as interdependent feedback loops. It demonstrates that fixed points in the system explain key behaviors such as boom-bust cycles, validating reflexivity as a dynamic, non-equilibrium process rather than a static equilibrium model.

ABSTRACT

The mathematical model proposed by George Soros for his theory of reflexivity is analyzed under the framework of discrete dynamical systems. We show the importance of the notion of fixed points for explaining the behavior of a reflexive system governed by its cognitive and manipulative functions. The interrelationship between these two functions induces fixed points with different characteristics, which in turn generate various system behaviors including the so-called "boom then bust" phenomenon in Soros's theory.

Motivation & Objective

• To provide a formal mathematical framework for Soros's theory of reflexivity, which has been largely ignored in academic economics.
• To analyze the behavior of the reflexive system governed by the cognitive and manipulative functions using discrete dynamical systems theory.
• To clarify the role of fixed points in determining system stability, convergence, or divergence in reflexive processes.
• To resolve ambiguity in Soros's original formulation by rigorously analyzing the recursive feedback structure of his model.
• To demonstrate that the 'boom then bust' phenomenon arises naturally from the dynamics of the system's fixed points.

Proposed method

• Modeling Soros's theory using the pair of equations: y = f(x) (cognitive function) and x = φ(y) (manipulative function), forming a feedback loop.
• Transforming the system into a discrete dynamical system by iterating the functions over time steps to generate sequences {x_i} and {y_i}.
• Defining the state of the system as (x_i, y_i) and analyzing its forward orbit on the Euclidean plane.
• Applying standard results from discrete dynamical systems theory, particularly focusing on fixed points and their stability.
• Using graphical and analytical techniques to study the behavior of the system near fixed points and under different functional forms.
• Analyzing the interplay between the cognitive and manipulative functions to explain emergent phenomena like market bubbles and crashes.

Experimental results

Research questions

• RQ1 How can Soros's theory of reflexivity be formalized as a discrete dynamical system?
• RQ2 What is the role of fixed points in determining the long-term behavior of a reflexive system?
• RQ3 Under what conditions does the system converge to equilibrium, and when does it diverge into boom-bust cycles?
• RQ4 How do the characteristics of the cognitive and manipulative functions influence system stability?
• RQ5 Can the 'boom then bust' phenomenon in financial markets be mathematically derived from the reflexive feedback mechanism?

Key findings

• Fixed points in the system correspond to states where the cognitive and manipulative functions agree on a consistent price, representing potential equilibria.
• The existence of fixed points is not precluded by reflexivity; rather, they are central to understanding system behavior, contradicting Soros’s initial skepticism.
• Stable fixed points lead to convergence, where market participants' perceptions and actions align over time.
• Unstable fixed points or lack of convergence can lead to divergent behavior, explaining the 'boom then bust' cycle in financial markets.
• The system's behavior is highly sensitive to the functional forms of f and φ, with nonlinearities amplifying instability and feedback loops.
• The analysis confirms that the reflexive system is inherently non-equilibrium, as the feedback loop prevents the system from settling into a single, stable equilibrium.