An Econophysics Model of Financial Bubbles

Usually financial crises go along with bubbles in asset prices, such as the housing bubble in the US in 2007. This paper attempts to build a mathematical model of financial bubbles from an econophysics, and thus a new perspective. I find that agents identify bubbles only with a time delay. Furthermore, I demonstrate that the detection of bubbles is different on either the individual or collective point of view. Second, I utilize the findings for a new definition of asset bubbles in finance. Finally, I extend the model to the study of asset price dynamics with news. In conclusion, the model provides unique insights into the properties and developments of financial bubbles.

The <i>q</i>-Exponential Social Discounting Functions of Gain and Loss

Social discounting has been attracting attention in behavioral psychology, econophysics, and neuroeconomics. Several mathematical models have been proposed for social discounting;exponential discounting, hyperbolic discounting, a q-exponential discounting model based on Tsallis’ statistics. In order to experimentally examine the mathematical characteristics of the q-exponential social discounting models for gain and loss in humans, we estimated the parameters of the q-exponential social discounting models by assessing the points of subjective equality (indifference points) at seven social distances. We observed that gain was more steeply social-discounted than loss. Usefulness of the q-exponential social discounting model in social physics, econophysics, and cultural neuroeconomics are discussed.

The <i>q</i>-Exponential Probability Discounting of Gain and Loss

Probability discounting is defined as the devaluation of outcomes as the probability of receiving or paying those decreases. A q-exponential probability discounting model based on Tsallis’ statistics has been proposed in econophysics (Takahashi, 2007, Physica A). We examined (a) fitness of the models to behavioral data of probability discounting of both gain and loss;and (b) relationships between parameters in the q-exponential probability discounting model across gain and loss. Our results demonstrated that, for both gain and loss, the q-exponential model better fits the behavioral data than exponential and hyperbolic functions, and there is the sign effect in q-exponential probability discounting. Relationships between Kahneman-Tversky’s prospect theory in behavioral economics and the q-exponential probability discounting are high-lightened.

The explosion in cryptocurrencies:a black hole analogy

Using an analogy between finance and astrophysics,this study aims to investigate whether there exists a mechanism that can describe the explosive increase in the number of traded cryptocurrencies and the cryptocurrency market in general.In physics,the Schwarzschild radius indicates that black holes are constantly expanding because of their mass increase.Enriching this analogy,we consider the cryptocurrency market as a self-gravitational body whose mass is denoted by(1)the number of traded cryptocurrencies and(2)in terms of increasing market capitalization for a given number of traded cryptocurrencies.By analyzing weekly snapshot data of all traded cryptocurrencies from January 4,2009,to June 14,2020,we find evidence that the above-mentioned mechanism exists.The results clearly indicate the self-gravitational property of the cryptocurrency market,which is direct evidence toward the hypothesis that the changes in the traded cryptocurrencies are a positive function of the previous period’s number of traded cryptocurrencies.

An empirical behavioral order‑driven model with price limit rules

We propose an empirical behavioral order-driven(EBOD)model with price limit rules,which consists of an order placement process and an order cancellation process.All the ingredients of the model are determined based on the empirical microscopic regularities in the order flows of stocks traded on the Shenzhen Stock Exchange.The model can reproduce the main stylized facts in real markets.Computational experiments unveil that asymmetric setting of price limits will cause the stock price to diverge exponentially when the up price limit is higher than the down price limit and to vanish vice versa.We also find that asymmetric price limits have little influence on the correlation structure of the return series and the volatility series,but cause remarkable changes in the average returns and the tail exponents of returns.Our EBOD model provides a suitable computational experiment platform for academics,market participants,and policy makers.

Mathematical Neurolaw of Crime and Punishment: The <i>q</i>-Exponential Punishment Function

Whether people tend to punish criminals in a socially-optimal manner (i.e., hyperbolic punishment) or not is unknown. By adopting mathematical models of probabilistic punishment behavior (i.e., exponential, hyperbolic, and q-exponential probability discounting model based on Tsallis thermodynamics and neuroeconomics, Takahashi, 2007, Physica A;Takahashi et al., 2012, Applied Mathematics), we examined 1) fitness of the models to behavioral data of uncertain punishment, and 2) deviation from the socially optimal hyperbolic punishment function. Our results demonstrated that, the q-exponential punishment function best fits the behavioral data, and people overweigh the severity of punishment at small punishing probabilities and underweigh the severity of punishment at large punishing probabilities. In other words, people tend to punish crimes too severely and mildly with high and low arrest rate (e.g., homicide vs. excess of speed limit), respectively. Implications for neuroeconomics and neurolaw of crime and punishment (Takahashi, 2012, NeuroEndocrinology Letters) are discussed.

Psychophysical Neuroeconomics of Decision Making: Nonlinear Time Perception Commonly Explains Anomalies in Temporal and Probability Discounting

Anomalies in decision over time (e.g., “hyperbolic time discounting”) and under risk (e.g., Allais paradox and hyperbolic probability discounting) have been attracting attention in behavioral and neuroeconomics. We have proposed that psychophysical time commonly explains anomalies in both decisions (Takahashi, 2011, Physica A;Takahashi et al., 2012, J Behav Econ & Finance). By adopting the q-exponential time and probability discounting models, our psychophysical and behavioral economic experiment confirmed that nonlinear distortion of psychophysical time is a common cause of the anomalies in decision both over time and under risk (i.e., intertemporal choice and decision under risk). Implications for psychophysical neuroeconomics and econophysics are discussed.

Econophysical Applications for Economic Progress:Monopoly and Competition Cases

This study attempts to investigate the relationship between monopoly and competition and the philosophy of progress,using the methods,models,and terms from physics.The term“progress”is a newly adopted term and is defined as the value of an increase in the production rate per unit of time.It is shown that to achieve progress,it is necessary to increase the production on a non-linear basis over time.Therefore,it is enough to have many firms that interact with each other under the influence of“market forces”.It is important to have a high level of university education,a legal environment for competition and indestructible antitrust laws.Even without strong science,you can make progress by acquiring technology and creating conditions for competition in the market.A factor that strongly influences progress,of course,is technology.For the development of technology,there is a great need for science.Science is a very powerful factor that affects the non-linear change in the economic development.It is shown that during the transition period from an absolute monopoly to an imperfect monopoly the system becomes more complex,and its output characteristics become non-linear as a function of time.It was found that the relationship between monopoly and competition is very simple,and there is no contradiction between them.Initially,the market is born as a monopoly,and then with the creation of similar firms,competition,as the natural market process,begins between them.

Income Inequality Generates Chaos

The present paper is an attempt to bridge the gulf between economics and econophysics. That is, constructing a chaos-based theoretical model, we show the behavior of the goal-driven agents exhibits the behavior of the purpose-free agents. Main conclusion is: economy becomes chaos if 1) capital gain of the middle class people is large enough for them to consume eight times as much as their income gain and 2) market for the middle class people is large enough and number of the middle class people is 16 times as large as the amount of products made by one producer.

An Experimental Comparison of Quantum Decision Theoretical Models of Intertemporal Choice for Gain and Loss

In mathematical physics and psychology, “quantum decision theory” has been proposed to explain anomalies in human decision-making. One of such quantum models has been proposed to explain time inconsistency in human decision over time. In this study, we conducted a behavioral experiment to examine which quantum decision models best account for human intertemporal choice. We observed that a q-exponential model developed in Tsallis’ thermodynamics (based on Takahashi’s (2005) nonlinear time perception theory) best fit human behavioral data for both gain and loss, among other quantum decision models.

Economic importance and structural robustness of the international pesticide trade networks

Pesticides are agricultural inputs that can significantly reduce yield losses,regulate plant growth,effectively liberate agricultural productivity,and improve food security.The availability of pesticides in economies worldwide is ensured by redistribution through international trade,with different economies playing different roles in this process.In this study,we measured and ranked the importance of economies using nine node metrics.We found that the clustering coefficient was negatively correlated with the other eight node metrics,whereas the other eight node metrics were positively correlated with each other and could be grouped into three communities(betweenness;in-degree,PageRank,authority,and in-closeness;out-degree,hub,and out-closeness).We further investigated the structural robustness of international pesticide trade networks proxied by large component size under three types of shocks to economies(node removal in descending,random,and ascending orders).The results showed that,except for the clustering coefficient,international pesticide trade networks are relatively robust under shocks to economies in ascending order but fragile under shocks to economies in descending order.By contrast,removing nodes with a clustering coefficient in ascending and descending order yielded similar robustness curves.Moreover,the structural robustness related to large component size evolved over time and exhibited an inverse U-shaped pattern.

Eigen microstates and their evolutions in complex systems

Emergence refers to the existence or formation of collective behaviors in complex systems.Here,we develop a theoretical framework based on the eigen microstate theory to analyze the emerging phenomena and dynamic evolution of complex system.In this framework,the statistical ensemble composed of M microstates of a complex system with N agents is defined by the normalized N×M matrix A,whose columns represent microstates and order of row is consist with the time.The ensemble matrix A can be decomposed as■,where r=min(N,M),eigenvalueσIbehaves as the probability amplitude of the eigen microstate U_I so that■and U_I evolves following V_I.In a disorder complex system,there is no dominant eigenvalue and eigen microstate.When a probability amplitudeσIbecomes finite in the thermodynamic limit,there is a condensation of the eigen microstate UIin analogy to the Bose–Einstein condensation of Bose gases.This indicates the emergence of U_I and a phase transition in complex system.Our framework has been applied successfully to equilibrium threedimensional Ising model,climate system and stock markets.We anticipate that our eigen microstate method can be used to study non-equilibrium complex systems with unknown orderparameters,such as phase transitions of collective motion and tipping points in climate systems and ecosystems.