From the binomial reshuffling model to Poisson distribution of money

Fei Cao; Nicholas F. Marshall; Fei Cao; Nicholas F. Marshall

doi:10.3934/nhm.2024002

Networks and Heterogeneous Media

2024, Volume 19, Issue 1: 24-43. doi: 10.3934/nhm.2024002

Previous Article Next Article

Research article

From the binomial reshuffling model to Poisson distribution of money

Fei Cao ^{1
,
,},
Nicholas F. Marshall ²

1.
Department of Mathematics and Statistics, University of Massachusetts Amherst, 710 N. Pleasant St, MA 01003, USA
2.
Department of Mathematics, Oregon State University, 368 Kidder Hall, Corvallis, OR 97331, USA

Received: 06 July 2023 Revised: 14 December 2023 Accepted: 20 December 2023 Published: 08 January 2024

We present a novel reshuffling exchange model and investigate its long time behavior. In this model, two individuals are picked randomly, and their wealth $ X_i $ and $ X_j $ are redistributed by flipping a sequence of fair coins leading to a binomial distribution denoted $ B\circ (X_i+X_j) $. This dynamics can be considered as a natural variant of the so-called uniform reshuffling model in econophysics. May refer to Cao, Jabin and Motsch (2023), Dragulescu and Yakovenko (2000). As the number of individuals goes to infinity, we derive its mean-field limit, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. Our aim of this work is then to prove (using a coupling argument) that the distribution of wealth converges to the Poisson distribution in the $ 2 $-Wasserstein metric. Numerical simulations illustrate the main result and suggest that the polynomial convergence decay might be further improved.
- Econophysics,
- Markov chain,
- Agent-based model,
- mean-field limit,
- coupling,
- wasserstein metric
Citation: Fei Cao, Nicholas F. Marshall. From the binomial reshuffling model to Poisson distribution of money[J]. Networks and Heterogeneous Media, 2024, 19(1): 24-43. doi: 10.3934/nhm.2024002

Related Papers:

Abstract

We present a novel reshuffling exchange model and investigate its long time behavior. In this model, two individuals are picked randomly, and their wealth $ X_i $ and $ X_j $ are redistributed by flipping a sequence of fair coins leading to a binomial distribution denoted $ B\circ (X_i+X_j) $. This dynamics can be considered as a natural variant of the so-called uniform reshuffling model in econophysics. May refer to Cao, Jabin and Motsch (2023), Dragulescu and Yakovenko (2000). As the number of individuals goes to infinity, we derive its mean-field limit, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. Our aim of this work is then to prove (using a coupling argument) that the distribution of wealth converges to the Poisson distribution in the $ 2 $-Wasserstein metric. Numerical simulations illustrate the main result and suggest that the polynomial convergence decay might be further improved.

References

[1]	F. Bassetti, G. Toscani, Mean field dynamics of interaction processes with duplication, loss and copy, Math. Models Methods Appl. Sci., 25 (2015), 1887–1925. https://doi.org/10.1142/S0218202515500487 doi: 10.1142/S0218202515500487
[2]	F. Cao, S. Reed, A biased dollar exchange model involving bank and debt with discontinuous equilibrium, arXiv: 2311.07851, [Preprint], (2023) [cited 2024 Jan 08]. Available from: https://doi.org/10.48550/arXiv.2311.07851
[3]	F. Cao, S. Motsch, Derivation of wealth distributions from biased exchange of money, Kinet. Relat. Models, 16 (2023), 764–794. https://doi.org/10.3934/krm.2023007 doi: 10.3934/krm.2023007
[4]	F. Cao, PE. Jabin, S. Motsch, Entropy dissipation and propagation of chaos for the uniform reshuffling model, Math. Models Methods Appl. Sci., 33 (2023), 829–875. https://doi.org/10.1142/S0218202523500185 doi: 10.1142/S0218202523500185
[5]	F. Cao, Explicit decay rate for the Gini index in the repeated averaging model, Math. Methods Appl. Sci., 46 (2023), 3583–3596. https://doi.org/10.1002/mma.8711 doi: 10.1002/mma.8711
[6]	F. Cao, PE. Jabin, From interacting agents to Boltzmann-Gibbs distribution of money, arXiv: 2208.05629, [Preprint], (2022) [cited 2024 Jan 08]. Available from: https://doi.org/10.48550/arXiv.2208.05629
[7]	F. Cao, S. Motsch, Uncovering a two-phase dynamics from a dollar exchange model with bank and debt, SIAM J. Appl. Math., 83 (2023), 1872–1891. https://doi.org/10.1137/22M1518621 doi: 10.1137/22M1518621
[8]	E. A. Carlen, E. Gabetta, G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas, Comm. Math. Phys., 199 (1999), 521–546. https://doi.org/10.1007/s002200050511 doi: 10.1007/s002200050511
[9]	J. A. Carrillo, G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75–198. https://ddd.uab.cat/record/44032
[10]	L P. Chaintron, A. Diez, Propagation of chaos: A review of models, methods and applications. I. Models and methods, Kinet. Relat. Models, 15 (2022), 895–1015. https://doi.org/10.3934/krm.2022017 doi: 10.3934/krm.2022017
[11]	A. Chakraborti, B. K. Chakrabarti, Statistical mechanics of money: how saving propensity affects its distribution, Eur. Phys. J. B, 17 (2000), 167–170. https://doi.org/10.1007/s100510070173 doi: 10.1007/s100510070173
[12]	A. Chatterjee, B. K. Chakrabarti, S. S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A, 335 (2004), 155–163. https://doi.org/10.1016/j.physa.2003.11.014 doi: 10.1016/j.physa.2003.11.014
[13]	S. Chatterjee, P. Diaconis, A. Sly, L. Zhang, A phase transition for repeated averages, Ann. Probab., 50 (2022), 1–17. https://doi.org/10.1214/21-AOP1526 doi: 10.1214/21-AOP1526
[14]	R. Cortez, J. Fontbona, Quantitative propagation of chaos for generalized Kac particle systems, Ann. Appl. Probab., 26 (2016), 892–916. https://doi.org/10.1214/15-AAP1107 doi: 10.1214/15-AAP1107
[15]	R. Cortez, Particle system approach to wealth redistribution, arXiv: 1809.05372, [Preprint], (2018) [cited 2024 Jan 08]. Available from: https://doi.org/10.48550/arXiv.1809.05372
[16]	R. Cortez, F. Cao, Uniform propagation of chaos for a dollar exchange econophysics model, arXiv: 2212.08289, [Preprint], (2022) [cited 2024 Jan 08]. Available from: https://doi.org/10.48550/arXiv.2212.08289
[17]	T. M. Cover, J.A. Thomas, Elements of information theory, John Wiley & Sons, 1999. https://doi.org/10.1002/0471200611
[18]	G. Da Prato, An introduction to infinite-dimensional analysis, Springer Science & Business Media, 2006. https://doi.org/10.1007/3-540-29021-4
[19]	A. Dragulescu, V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. J. B, 17 (2000), 723–729. https://doi.org/10.1007/s100510070114 doi: 10.1007/s100510070114
[20]	G. Gabetta, G. Toscani, B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Stat. Phys., 81 (1995), 901–934. https://doi.org/10.1007/BF02179298 doi: 10.1007/BF02179298
[21]	B. T. Graham, Rate of relaxation for a mean-field zero-range process, Ann. Appl. Probab., 19 (2009), 497–520. 10.1214/08-AAP549 doi: 10.1214/08-AAP549
[22]	E. Heinsalu, P. Marco, Kinetic models of immediate exchange, Eur. Phys. J. B, 87 (2014), 1–10. https://doi.org/10.1140/epjb/e2014-50270-6 doi: 10.1140/epjb/e2014-50270-6
[23]	P E. Jabin, B. Niethammer, On the rate of convergence to equilibrium in the Becker–Döring equations, J Differ Equ, 191 (2003), 518–543. https://doi.org/10.1016/S0022-0396(03)00021-4 doi: 10.1016/S0022-0396(03)00021-4
[24]	N. Lanchier, Rigorous proof of the Boltzmann–Gibbs distribution of money on connected graphs, J. Stat. Phys., 167 (2017), 160–172. https://doi.org/10.1007/s10955-017-1744-8 doi: 10.1007/s10955-017-1744-8
[25]	N. Lanchier, S. Reed, Rigorous results for the distribution of money on connected graphs, J. Stat. Phys., 171 (2018), 727–743. https://doi.org/10.1007/s10955-018-2024-y doi: 10.1007/s10955-018-2024-y
[26]	N. Lanchier, S. Reed, Rigorous results for the distribution of money on connected graphs (models with debts), J. Stat. Phys., 176 (2019), 1115–1137. https://doi.org/10.1007/s10955-019-02334-z doi: 10.1007/s10955-019-02334-z
[27]	T. M. Liggett, Interacting particle systems, New York: Springer, 1985.
[28]	D. Matthes, G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087–1117. https://doi.org/10.1007/s10955-007-9462-2 doi: 10.1007/s10955-007-9462-2
[29]	M. Merle, J. Salez, Cutoff for the mean-field zero-range process, Ann. Probab., 47 (2019), 3170–3201. https://doi.org/10.1214/19-AOP1336 doi: 10.1214/19-AOP1336
[30]	G. Naldi, L. Pareschi, G. Toscani, Mathematical modeling of collective behavior in socio-economic and life sciences, Berlin: Springer Science & Business Media, 2010. https://doi.org/10.1007/978-0-8176-4946-3
[31]	G. Peyré, M. Cuturi, Computational optimal transport: With applications to data science, Found. Trends Mach. Learn., 11 (2019), 355–607. http://dx.doi.org/10.1561/2200000073 doi: 10.1561/2200000073
[32]	R. Pymar, N. Rivera, Mixing of the symmetric beta-binomial splitting process on arbitrary graphs, arXiv: 2307.02406, [Preprint], (2023) [cited 2024 Jan 08]. Available from: https://doi.org/10.48550/arXiv.2307.02406
[33]	A. Santos, Showing convergence of an infinite ODE system, MathOverflow 2022. Available from: https://mathoverflow.net/q/432268.
[34]	A S. Sznitman, Topics in propagation of chaos, in Ecole d'été de probabilités de Saint-Flour XIX—1989, Berlin: Springer, (1991), 165–251. https://doi.org/10.1007/BFb0085166
[35]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Berlin: Springer Science & Business Media, 2012. https://doi.org/10.1007/978-1-4612-0645-3

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)