# Uncertainty and Complex Dynamics in Econophysics

Abstract

During the recent history of humanity, on the one hand economic systems of advanced civilizations have achieved a great capacity in supplying their citizens with unimaginable wellness, but on the other hand their devastating power of destraction has made us doubt about the fragile security that we think they grant us. The wounds left after the last financial crisis of 2008 are not healed yet, but they serve us as a reminder that unlike what many people believe or want to believe, this economic beast is not ready to remain immobile, trapped in an equilibrium state for the rest of its life. However, these tragic events sometimes produce positive outcomes, they force us to return to the designing table and reconsider our models, paradigms, theories and even our knowledge. I was in charge of a small familiar business when the crisis hit Spain. I felt on my own flesh the huge social and economic recession that was generated by this catastrophic event. My curiosity and anxiety for knowing, drove me to question everything I knew or thought, borrowing ideas from a variety of scientific fields that in principle do not have anything to do with economics. This thesis is the result of this long search for the truth. This thesis has been developed during my years, researching in the Nonlinear Dynamics, Chaos Theory and Complex Systems research group of the Universidad Rey Juan Carlos. In this thesis we develop and study nonlinear models in economics capable of producing so complex dynamics, that our own understanding of predictability is challenged. Next, I will briefly describe the structure of this work. Chapter 1. Introduction In this chapter we will build the pillars of this work. We will describe how standard economics theory has been challenged by the emerging science of complexity. This new scientific paradigm has changed the way economists look and understand the entire economic system and it also opened the door for physical scientists to apply their own tools and methods to study economic phenomena, giving way to the novel econophysics approach. In this work, we have used it broadly, for this reason, concepts like complex systems, interdisciplinarity, chaos, basins of attraction, fractals, agent based models (ABM) and emergence are briefly introduced. These ideas will recurrently come up along the whole thesis. Chapter 2. The supply based on demand method In this chapter we introduce the supply based on demand dynamical model (SBOD). This model is one of the props of this thesis and we invoke it in Chapter 3. For this reason, we describe very carefully the SBOD model. We explain the reasoning behind it, and the motivations that led us to develop this model in the first place. Afterwards we describe the structure of the model and how it works. Finally, we study in detail the complex market dynamics produced by this model and the important observation that the last bifurcation means market collapse. Chapter 3. Preventing the crash with partial control The partial control method was developed by our research group with the collaboration of Prof. James Yorke from the University of Maryland. This novel control method has been successfully applied to a variety of systems in science and engineenering achieving remarkable results. Here we apply for the first time the partial control method in an economic context, applying it on the supply based on demand model introduced in Chapter 2. We demonstrate that the firm is capable of controlling the market from the bottom up, applying much smaller control than the market noise. Chapter 4. When repetition is the best strategy This chapter is devoted to the study of the system proposed by the Complexity Challenge Team from the Santa Fe Institute at Spring 2018. This intriguing problem is an extension of the original El Farol Bar problem in which more complexities are introduced. There are three pools and each one of them has different rewarding schemes that depend on the attendance to the pool and in some stochastic functions. Each agent must locate itself at each time step in one of the pools with the goal of maximizing its total balance, which is the reward paid by the pool minus some cost. To study this problem, we have developed an agent based model that integrates 13 types of strategies. Chapter 5. Making predictions on fractal basins There are situations where our idea of deterministic predictability is challenged by the mere existence of complex structures on the phase space of some dynamical systems. When these basins of attraction present fractal structure, the predictability of any event in these fractal regions depends on the accuracy in which we measure the initial conditions, and this is a technical problem. In this chapter, we study the relationship between the global predictability and the local measure of uncertainty in a dynamical system that presents the Wada property. We demonstrate that the probability of ending up in each basin of attraction converges to a fixed probability when we increase the accuracy of the measurement or the phase space resolution. This means that globally, the probabilities of each final state of the system are fixed, although locally we will never be able of predicting the orbits. Chapter 6. Results and Discussion This chapter is devoted to the description of the main results of this thesis. We also propose some possible research lines for a future work. Chapter 7. Conclusions The main conclusions of this thesis are summarized in this chapter.

Description

Tesis Doctoral leída en la Universidad Rey Juan Carlos de Madrid en 2019. Directores de la Tesis: Miguel ´Ángel Fernández Sanjuán y Juan Sabuco

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