# Economic Segregation Under the Action of Trading Uncertainties

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. A Brief Excursion on Kinetic Modeling of Opinion Formation

## 3. Kinetic Modeling of Uncertain Trading Activity

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Statistical Study of the Marginal Wealth Distribution

#### 4.1. Literature Approaches

#### 4.2. Bayesian Approach

## 5. Numerical Results Based on Markov Chain Monte Carlo

#### 5.1. Estimation of the Marginal Function

- The support was shifted by 0.3 (which in the case of the function (27) is the value of the parameter $\delta $).
- The support was multiplied by a factor equal to 10 to highlight the shape of the wealth distribution.

#### 5.2. Estimation of the Posterior Function

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bassetti, F.; Toscani, G. Explicit equilibria in a kinetic model of gambling. Phys. Rev. E
**2010**, 81, 066115. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bassetti, F.; Toscani, G. Explicit equilibria in bilinear kinetic models for socio-economic interactions. ESAIM Proc. Surv.
**2014**, 47, 1–16. [Google Scholar] [CrossRef] [Green Version] - Bisi, M.; Spiga, G.; Toscani, G. Kinetic models of conservative economies with wealth redistribution. Commun. Math. Sci.
**2009**, 7, 901–916. [Google Scholar] [CrossRef] [Green Version] - Cáceres, M.J.; Toscani, G. Kinetic approach to long time behavior of linearized fast diffusion equations. J. Stat. Phys.
**2007**, 128, 883–925. [Google Scholar] [CrossRef] - Chatterjee, A.; Chakrabarti, B.K.; Manna, S.S. Pareto law in a kinetic model of market with random saving propensity. Physica A
**2004**, 335, 155–163. [Google Scholar] [CrossRef] [Green Version] - Chatterjee, A.; Chakrabarti, B.K.; Stinchcombe, R.B. Master equation for a kinetic model of trading market and its analytic solution. Phys. Rev. E
**2005**, 72, 026126. [Google Scholar] [CrossRef] [Green Version] - Chakraborti, A. Distributions of money in models of market economy. Int. J. Mod. Phys. C
**2002**, 13, 1315–1321. [Google Scholar] [CrossRef] [Green Version] - Chakraborti, A.; Chakrabarti, B.K. Statistical mechanics of money: How saving propensity affects its distribution. Eur. Phys. J. B
**2000**, 17, 167–170. [Google Scholar] [CrossRef] [Green Version] - Cordier, S.; Pareschi, L.; Piatecki, C. Mesoscopic modelling of financial markets. J. Stat. Phys.
**2009**, 134, 161–184. [Google Scholar] [CrossRef] [Green Version] - Dragulescu, A.; Yakovenko, V.M. Statistical mechanics of money. Eur. Phys. J. B
**2000**, 17, 723–729. [Google Scholar] [CrossRef] [Green Version] - Düring, B.; Matthes, D.; Toscani, G. Kinetic equations modelling wealth redistribution: A comparison of approaches. Phys. Rev. E
**2008**, 78, 056103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Düring, B.; Matthes, D.; Toscani, G. A Boltzmann-type approach to the formation of wealth distribution curves. Riv. Mat. Univ. Parma
**2009**, 8, 199–261. [Google Scholar] [CrossRef] [Green Version] - Gupta, A.K. Models of wealth distributions: A perspective. In Econophysics and Sociophysics: Trends and Perspectives; Chakrabarti, A., Chakraborti, B.K., Chatterjee, A., Eds.; Wiley VHC: Weinheim, Germany, 2006; pp. 161–190. [Google Scholar]
- Ispolatov, S.; Krapivsky, P.L.; Redner, S. Wealth distributions in asset exchange models. Eur. Phys. J. B
**1998**, 2, 267–276. [Google Scholar] [CrossRef] [Green Version] - Maldarella, D.; Pareschi, L. Kinetic models for socio–economic dynamics of speculative markets. Physica A
**2012**, 391, 715–730. [Google Scholar] [CrossRef] [Green Version] - Matthes, D.; Toscani, G. On steady distributions of kinetic models of conservative economies. J. Stat. Phys.
**2008**, 130, 1087–1117. [Google Scholar] [CrossRef] - Slanina, F. Inelastically scattering particles and wealth distribution in an open economy. Phys. Rev. E
**2004**, 69, 046102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Toscani, G.; Brugna, C.; Demichelis, S. Kinetic models for the trading of goods. J. Stat. Phys.
**2013**, 151, 549–566. [Google Scholar] [CrossRef] [Green Version] - Pareto, V. Cours d’Économie Politique; Librairie Droz: Geneva, Switzerland, 1964. [Google Scholar]
- Bassetti, F.; Toscani, G. Mean field dynamics of collisional processes with duplication, loss and copy. Math. Mod. Meth. Appl. Sci.
**2015**, 25, 1887–1925. [Google Scholar] [CrossRef] - Carrillo, J.A.; Fornasier, M.; Rosado, J.; Toscani, G. Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal.
**2010**, 42, 218–236. [Google Scholar] [CrossRef] [Green Version] - Ha, S.-Y.; Jeong, E.; Kang, J.-H.; Kang, K. Emergence of multi-cluster configurations from attractive and repulsive interactions. Math. Models Methods Appl. Sci.
**2012**, 22, 1250013. [Google Scholar] [CrossRef] - Kashdan, E.; Pareschi, L. Mean field mutation dynamics and the continuous Luria-Delbrück distribution. Math. Biosci.
**2012**, 240, 223–230. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Toscani, G. A kinetic description of mutation processes in bacteria. Kinet. Relat. Models
**2013**, 6, 1043–1055. [Google Scholar] [CrossRef] - Castellano, C.; Fortunato, S.; Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys.
**2009**, 81, 591–646. [Google Scholar] [CrossRef] [Green Version] - Chakrabarti, B.K.; Chakraborti, A.; Chakravarty, S.R.; Chatterjee, A. Econophysics of Income and Wealth Distributions; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Naldi, G.; Pareschi, L.; Toscani, G. Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Pareschi, L.; Toscani, G. Interacting Multiagent Systems. Kinetic Equations & Monte Carlo Methods; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Sen, P.; Chakrabarti, B.K. Sociophysics: An Introduction; Oxford University Press: Oxford, UK, 2014. [Google Scholar]
- Herty, M.; Tosin, A.; Visconti, G.; Zanella, M. Reconstruction of traffic speed distributions from kinetic models with uncertainties. In Mathematical Descriptions of Traffic Flow: Micro, Macro and Kinetic Models; Tosin, A., Puppo, G., Eds.; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Piccoli, B.; Tosin, A.; Zanella, M. Model-based assessment of the impact of driver-assist vehicles using kinetic theory. arXiv
**2019**, arXiv:1911.04911. [Google Scholar] - Tosin, A.; Zanella, M. Uncertainty damping in kinetic traffic models by driver-assist controls. arXiv
**2019**, arXiv:1904.00257. [Google Scholar] - Tosin, A.; Zanella, M. Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles. Multiscale Model Simul.
**2019**, 17, 716–749. [Google Scholar] - Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G. Non-Maxwellian kinetic equations modeling the evolution of wealth distribution. Math. Models Methods Appl. Sci.
**2020**, 30, 685–725. [Google Scholar] [CrossRef] - Bouchaud, J.F.; Mézard, M. Wealth condensation in a simple model of economy. Physica A
**2000**, 282, 536–545. [Google Scholar] [CrossRef] [Green Version] - Cordier, S.; Pareschi, L.; Toscani, G. On a kinetic model for a simple market economy. J. Stat. Phys.
**2005**, 120, 253–277. [Google Scholar] [CrossRef] [Green Version] - Bisi, M. Some kinetic models for a market economy. Boll. Unione Mat. Ital.
**2017**, 10, 143–158. [Google Scholar] [CrossRef] - Düring, B.; Pareschi, L.; Toscani, G. Kinetic models for optimal control of wealth inequalities. Eur. Phys. J. B
**2018**, 91, 265. [Google Scholar] [CrossRef] [Green Version] - Düring, B.; Toscani, G. Hydrodynamics from kinetic models of conservative economies. Physica A
**2007**, 384, 493–506. [Google Scholar] [CrossRef] [Green Version] - Düring, B.; Toscani, G. International and domestic trading and wealth distribution. Commun. Math. Sci.
**2008**, 6, 1043–1058. [Google Scholar] [CrossRef] [Green Version] - Torregrossa, M.; Toscani, G. Wealth distribution in presence of debts. A Fokker–Planck description. Commun. Math. Sci.
**2018**, 16, 537–560. [Google Scholar] [CrossRef] [Green Version] - Bobylëv, A.V. The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules. Dokl. Akad. Nauk SSSR
**1975**, 225, 1041–1044. [Google Scholar] - Cercignani, C. The Boltzmann Equation and Its Applications; Springer Series in Applied Mathematical Sciences; Springer: New York, NY, USA, 1988; Volume 67. [Google Scholar]
- Dolfin, M.; Leonida, L.; Outada, N. Modeling human behavior in economics and social science. Phys. Life Rev.
**2017**, 22–23, 1–21. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G. Fokker–Planck equations in the modelling of socio-economic phenomena. Math. Models Methods Appl. Sci.
**2017**, 27, 115–158. [Google Scholar] [CrossRef] [Green Version] - Torregrossa, M.; Toscani, G. On a Fokker–Planck equation for wealth distribution. Kinet. Relat. Models
**2018**, 11, 337–355. [Google Scholar] [CrossRef] [Green Version] - Toscani, G. Entropy dissipation and the rate of convergence to equilibrium for the Fokker–Planck equation. Quart. Appl. Math.
**1999**, LVII, 521–541. [Google Scholar] [CrossRef] [Green Version] - Toscani, G.; Villani, C. Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys.
**1999**, 203, 667–706. [Google Scholar] [CrossRef] - Villani, C. A Review of Mathematical Topics in Collisional Kinetic Theory. In Handbook of Mathematical Fluid Dynamics; Friedlander, S., Serre, D., Eds.; Elsevier: New York, NY, USA, 2002; Volume 1. [Google Scholar]
- Albi, G.; Pareschi, L.; Zanella, M. Uncertainty quantification in control problems for flocking models. Math. Probl. Eng.
**2015**, 15, 850124. [Google Scholar] [CrossRef] [Green Version] - Bellomo, N.; Colasuonno, F.; Knopoff, D.; Soler, J. From a systems theory of sociology to modeling the onset and evolution of criminality. Netw. Heterog. Media
**2015**, 10, 421–441. [Google Scholar] [CrossRef] - Bellomo, N.; Herrero, M.A.; Tosin, A. On the dynamics of social conflicts looking for the Black Swan. Kinet. Relat. Models
**2013**, 6, 459–479. [Google Scholar] [CrossRef] [Green Version] - Bellomo, N.; Soler, J. On the mathematical theory of the dynamics of swarms viewed as complex systems. Math. Models Methods Appl. Sci.
**2012**, 22, 1140006. [Google Scholar] [CrossRef] - Gualandi, S.; Toscani, G. Pareto tails in socio-economic phenomena: A kinetic description. Economics
**2018**, 12, 1–17. [Google Scholar] [CrossRef] [Green Version] - Gualandi, S.; Toscani, G. Human behavior and lognormal distribution. A kinetic description. Math. Models Methods Appl. Sci.
**2019**, 29, 717–753. [Google Scholar] [CrossRef] [Green Version] - Chatterjee, A.; Chakrabarti, B.K.; Manna, S.S. Money in gas-like markets: Gibbs and Pareto laws. Phys. Scr.
**2003**, 106, 36. [Google Scholar] [CrossRef] [Green Version] - Lim, G.; Min, S. Analysis of solidarity effect for entropy, Pareto, and Gini indices on two-class society using kinetic wealth exchange model. Entropy
**2020**, 22, 386. [Google Scholar] [CrossRef] [Green Version] - Patriarca, M.; Chakraborti, A.; Kaski, K.; Germano, G. Kinetic theory models for the distribution of wealth: Power law from overlap of exponentials. In Econophysics of Wealth Distributions; Chatterjee, A., Yarlagadda, S., Chakrabarti, B.K., Eds.; Springer: Berlin/Heidelberg, Germany, 2005; pp. 93–110. [Google Scholar]
- Patriarca, M.; Chakraborti, A.; Germano, G.O. Influence of saving propensity on the power-law tail of the wealth distribution. Physica A
**2006**, 369, 723–736. [Google Scholar] [CrossRef] [Green Version] - Ghosh, A.; Chatterjee, A.; Inoue, J.I.; Chakrabarti, B.K. Inequality measures in kinetic exchange models of wealth distributions. Physica A
**2016**, 451, 465–474. [Google Scholar] [CrossRef] [Green Version] - Ferrero, J.C. The monomodal, polymodal, equilibrium and nonequilibrium distribution of money. In Econophysics of Wealth Distributions; Chatterjee, A., Yarlagadda, S., Chakrabarti, B.K., Eds.; Springer: Cernusco, Italy, 2005; pp. 159–167. [Google Scholar]
- Ferrero, J.C. The individual income distribution in Argentina in the period 2000–2009. A unique source of non stationary data. arXiv
**2010**, arXiv:1006.2057. [Google Scholar] - Bellomo, N.; Bingham, R.; Chaplain, M.A.; Dosi, G.; Forni, G.; Knopoff, D.A.; Lowengrub, J.; Twarock, R.; Virgillito, M.E. A Multi-Scale Model of Virus Pandemic: Heterogeneous Interactive Entities in a Globally Connected World. Math. Mod. Meth. Appl. Sci.
**2020**. [Google Scholar] [CrossRef] - Dimarco, G.; Pareschi, L.; Toscani, G.; Zanella, M. Wealth distribution under the spread of infectious diseases. Phys. Rev. E
**2020**, 102, 022303. [Google Scholar] [CrossRef] - Toscani, G. Kinetic models of opinion formation. Commun. Math. Sci.
**2006**, 4, 481–496. [Google Scholar] [CrossRef] [Green Version] - Pareschi, L.; Toscani, G.; Tosin, A.; Zanella, M. Hydrodynamics models of preference formation in multi-agent societies. J. Nonlin. Sci.
**2019**, 29, 2761–2796. [Google Scholar] [CrossRef] [Green Version] - Toscani, G.; Tosin, A.; Zanella, M. Opinion modeling on social media and marketing aspects. Phys. Rev. E
**2018**, 98, 022315. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Albi, G.; Pareschi, L.; Zanella, M. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinet. Relat. Models
**2017**, 10, 1–32. [Google Scholar] [CrossRef] [Green Version] - Ben-Naim, E. Opinion dynamics: Rise and fall of political parties. Europhys. Lett.
**2005**, 69, 671–677. [Google Scholar] [CrossRef] [Green Version] - Ben-Naim, E.; Krapivsky, P.L.; Redner, S. Bifurcation and patterns in compromise processes. Physica D
**2003**, 183, 190–204. [Google Scholar] [CrossRef] [Green Version] - Ben-Naim, E.; Krapivsky, P.L.; Vazquez, F.; Redner, S. Unity and discord in opinion dynamics. Physica A
**2003**, 330, 99–106. [Google Scholar] [CrossRef] - Weisbuch, G.; Deffuant, G.; Amblard, F. Persuasion dynamics. Physica A
**2005**, 353, 555–575. [Google Scholar] [CrossRef] [Green Version] - Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G. Wright–Fisher–type equations for opinion formation, large time behavior and weighted Logarithmic–Sobolev inequalities. Ann. IHP Anal. Non Linéaire
**2019**, 36, 2065–2082. [Google Scholar] [CrossRef] [Green Version] - Villani, C. Contribution à L’étude Mathématique des Équations de Boltzmann et de Landau en théorie Cinétique des gaz et des Plasmas. Ph.D. Thesis, Univ. Paris-Dauphine, Paris, France, 1998. [Google Scholar]
- Villani, C. On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal.
**1998**, 143, 273–307. [Google Scholar] [CrossRef] - Tosin, A.; Zanella, M. Boltzmann-type models with uncertain binary interactions. Commun. Math. Sci.
**2018**, 16, 962–984. [Google Scholar] [CrossRef] [Green Version] - Kac, M. Probability and Related Topics in the Physical Sciences; New York Interscience: London, UK, 1959. [Google Scholar]
- Carrillo, J.A.; Pareschi, L.; Zanella, M. Particle based gPC methods for mean-field models of swarming with uncertainty. Commun. Comput. Phys.
**2019**, 25, 508–531. [Google Scholar] [CrossRef] - Carrillo, J.A.; Pareschi, L.; Zanella, M. Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties. Viet. J. Math.
**2019**, 47, 931–954. [Google Scholar] [CrossRef] [Green Version] - Bernardo, J.M.; Smith, A.F.M. Bayesian Theory; John Wiley and Sons: Chichester, UK, 2009. [Google Scholar]
- Hastings, W.K. Monte Carlo sampling methods using Markov chains and their applications. Biometrik
**1970**, 57, 97–109. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Beta probability distribution functions plotted for different values of parameters $\alpha $ and $\beta $.(

**b**) marginal distribution functions computed for each of the Beta distributions.

**Figure 2.**We report the sensitivity analysis on the marginal distribution in (

**b**) performed for different parameters of the Beta distribution in (

**a**) obtained for the length of the support fixed to five.

**Figure 3.**Sensitivity analysis performed on different values of the length of the support using a Beta density function with $\alpha =0.1,\beta =0.1$ as the opinion distribution.

**Figure 5.**Posterior estimation through Metropolis–Hastings for different prior distributions: (

**a**) Beta distribution with parameters $\alpha =0.1$ and $\beta =0.1$, (

**b**) Beta distribution with parameters $\alpha =3$ and $\beta =3$, (

**c**) Beta distribution with parameters $\alpha =2$ and $\beta =8$, (

**d**) Beta distribution with parameters $\alpha =8$ and $\beta =2$.

**Table 1.**Gini index for each inverse Gamma distribution derived from the corresponding Beta distribution with parameters shown in the legend.

Prior Parameters | Gini Index |
---|---|

$\alpha =0.1$, $\beta =0.1$ | 0.55 |

$\alpha =3$, $\beta =3$ | 0.61 |

$\alpha =2$, $\beta =8$ | 0.56 |

$\alpha =8$, $\beta =2$ | 0.67 |

**Table 2.**Estimation of the posterior parameters of the Beta distribution fixing the prior distribution.

Prior Parameters | Posterior Parameters |
---|---|

$\alpha =0.1$, $\beta =0.1$ | $\widehat{\alpha}=0.41$, $\widehat{\beta}=0.22$ |

$\alpha =3$, $\beta =3$ | $\widehat{\alpha}=3.41$, $\widehat{\beta}=3.17$ |

$\alpha =2$, $\beta =8$ | $\widehat{\alpha}=2.62$, $\widehat{\beta}=8.33$ |

$\alpha =8$, $\beta =2$ | $\widehat{\alpha}=8.28$, $\widehat{\beta}=2.23$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ballante, E.; Bardelli, C.; Zanella, M.; Figini, S.; Toscani, G.
Economic Segregation Under the Action of Trading Uncertainties. *Symmetry* **2020**, *12*, 1390.
https://doi.org/10.3390/sym12091390

**AMA Style**

Ballante E, Bardelli C, Zanella M, Figini S, Toscani G.
Economic Segregation Under the Action of Trading Uncertainties. *Symmetry*. 2020; 12(9):1390.
https://doi.org/10.3390/sym12091390

**Chicago/Turabian Style**

Ballante, Elena, Chiara Bardelli, Mattia Zanella, Silvia Figini, and Giuseppe Toscani.
2020. "Economic Segregation Under the Action of Trading Uncertainties" *Symmetry* 12, no. 9: 1390.
https://doi.org/10.3390/sym12091390