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Limiting distributions of generalized money exchange models on hypergraphs

Part of: Special processes Mathematical economics Limit theorems

Published online by Cambridge University Press:  11 December 2025

Hironobu Sakagawa
Hironobu Sakagawa*
Affiliation:
Keio University
*
*Postal address: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan. Email: sakagawa@math.keio.ac.jp
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Abstract

The money exchange model is a type of agent-based model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model, the uniform reshuffling model, and the uniform saving model, all of which are types of money exchange model, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that, under appropriate scaling, the asymptotic wealth distribution converges to an exponential distribution for the uniform reshuffling model, and to either an exponential distribution or a gamma distribution depending on the tail behavior of the number of coins given/saved in the immediate exchange model and the random saving model, which generalizes the uniform saving model. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.

Keywords

Econophysicsinteracting particle systemstationary distributionequivalence of ensembleslocal limit theorem

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60F99: None of the above, but in this section 91B80: Applications of statistical and quantum mechanics to economics (econophysics)

Information

Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 26
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Bingham, N., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Chakrabarti, B. K, Chakraborti, A., Chakravarty, S. R. and Chatterjee, A. (2013). Econophysics of Income and Wealth Distributions. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Chakraborti, A. and Chakrabarti, B. K. (2000). Statistical mechanics of money: how saving propensity affects its distribution. Eur. Phys. J. B 17, 167–170.CrossRefGoogle Scholar
Cirillo, P., Redig, F. and Ruszel, W. (2014). Duality and stationary distributions of wealth distribution models. J. Phys. A 47, 085203.CrossRefGoogle Scholar
Davis, B. and McDonald, D. (1995). An elementary proof of the local central limit theorem. J. Theoret. Probab. 8, 693–701.CrossRefGoogle Scholar
Dragulescu, A. A. and Yakovenko, V. M. (2000). Statistical mechanics of money. Eur. Phys. J. B 17, 723–729.CrossRefGoogle Scholar
Durrett, R. (2019). Probability: Theory and Examples, 5th ed. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Friedli, S. and Velenik, Y. (2018). Statistical Mechanics of Lattice Systems. A Concrete Mathematical Introduction. Cambridge University Press, Cambridge.Google Scholar
van Ginkel, B., Redig, F. and Sau, F. (2016). Duality and stationary distributions of the “immediate exchange model” and its generalizations. J. Stat. Phys. 163, 92–112.CrossRefGoogle Scholar
Greenberg, M. and Gao, H. O. (2024). Twenty-five years of random asset exchange modeling. Eur. Phys. J. B 97, 69.CrossRefGoogle Scholar
Heinsalu, E. and Patriarca, M. (2014). Kinetic models of immediate exchange. Eur. Phys. J. B 87, 170–179.CrossRefGoogle Scholar
Katriel, G. (2015). The immediate exchange model: an analytical investigation. Eur. Phys. J. B 88, 19–24.CrossRefGoogle Scholar
Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren Math. Wiss., Vol. 320. Springer-Verlag, Berlin.Google Scholar
Lanchier, N. (2017). Rigorous proof of the Boltzmann-Gibbs distribution of money on connected graphs. J. Stat. Phys. 167, 160–172.CrossRefGoogle Scholar
Lanchier, N. (2024). Stochastic Interacting Systems in Life and Social Sciences. De Gruyter, Berlin.CrossRefGoogle Scholar
Lanchier, N. and Reed, S. (2018). Rigorous results for the distribution of money on connected graphs. J. Stat. Phys. 171, 727–743.CrossRefGoogle Scholar
Lanchier, N. and Reed, S. (2019). Rigorous results for the distribution of money on connected graphs (models with debts). J. Stat. Phys. 176, 1115–1137.CrossRefGoogle Scholar
Lanchier, N. and Reed, S. (2024). Distribution of money on connected graphs with multiple banks. Math. Model. Nat. Phenom. 19, Paper No.10.CrossRefGoogle Scholar
Landim, C., Sethuraman, S. and Varadhan, S. R. S. (1996). Spectral gap for zero-range dynamics. Ann. Probab. 24, 1871–1902.CrossRefGoogle Scholar
Mukhin, A. B. (1991). Local limit theorems for lattice random variables. Theory Probab. Appl. 36, 698–713.CrossRefGoogle Scholar
Patriarca, M., Chakraborti, A. and Kaski, K. (2004). Statistical model with standard $\Gamma$ distribution. Phys. Rev. E 70, 016104.CrossRefGoogle ScholarPubMed
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.Google Scholar
Pymar, R. and Rivera, N. Mixing of the symmetric beta-binomial splitting process on arbitrary graphs. preprint arXiv:2307.02406.Google Scholar
Quattropani, M. and Sau, F. (2023). Mixing of the averaging process and its discrete dual on finite-dimensional geometries. Ann. Appl. Probab. 33, 1136–1171.CrossRefGoogle Scholar
Redig, F. and Sau, F. (2017). Generalized immediate exchange models and their symmetries. Stochastic Process. Appl. 127, 3251–3267.CrossRefGoogle Scholar
Yakovenko, V. M. and Barkley Rosser, J. J. (2009). Colloquium: statistical mechanics of money, wealth, and income. Rev. Mod. Phys. 81, 1703–1725.CrossRefGoogle Scholar