# Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems

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## Abstract

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**PACS**87.23.Ge; 05.90.+m; 89.90.+n

## 1. Introduction

## 2. Recalling Some Results

#### 2.1. Linear constraint

#### 2.2. Quadratic constraint

## 3. Multi-Agent Systems and Equiprobability: General Derivation of the Asymptotic Distribution

## 4. Gamma Distributions, Economic Gas Models and Geometry: A Speculation

**Figure 1.**Normalization constant ${c}_{b}$ versus b, calculated from Equation (28). The asymptotic behavior is: ${lim}_{b\to 0}{c}_{b}=\infty $, and ${lim}_{b\to \infty}{c}_{b}=1$. This last asymptote is represented by the dotted line. The minimum of ${c}_{b}$ is reached for $b=3.1605$, taking the value ${c}_{b}=0.7762$.

**ECONOMIC MODEL A:**The first one is the saving propensity model introduced by Chakraborti and Chakrabarti [4]. In this model a set of N economic agents, having each agent i (with $i=1,2,\cdots ,N$) an amount of money, ${u}_{i}$, exchanges it under random binary $(i,j)$ interactions, $({u}_{i},{u}_{j})\to ({u}_{i}^{\prime},{u}_{j}^{\prime})$, by the following the exchange rule:

**ECONOMIC MODEL B:**The second one is a model introduced in [6]. In this model a set of N economic agents, having each agent i (with $i=1,2,\cdots ,N$) an amount of money, ${u}_{i}$, exchanges it under random binary $(i,j)$ interactions, $({u}_{i},{u}_{j})\to ({u}_{i}^{\prime},{u}_{j}^{\prime})$, by the following the exchange rule:

## 5. Other Geometrical Questions

**Figure 2.**The factor ${g}_{b}(N)$ versus b for $N=10,40,100$, calculated from Equation (56). Observe that ${g}_{b}(N)=0$ for $b=0$, and ${lim}_{b\to \infty}{g}_{b}(N)=1$.

**Figure 3.**The factor ${g}_{b}(N)$ versus N for $b=10,40,100$, calculated from Equation (56). Observe that ${g}_{b}(N)=1$ for $N=1$, and ${lim}_{N\to \infty}{g}_{b}(N)=0$.

## 6. Conclusions

## Acknowledgements

## References

- Mantegna, R.; Stanley, H.E. An Introduction to Econophysics: Correlations and Complexity in Finance; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Yakovenko, V.M. Econophysics, Statistical Mechanics Approach to. In Encyclopedia of Complexity and System Science; Meyers, R.A., Ed.; Springer: Berlin, Germany, 2009; pp. 2800–2826. [Google Scholar]
- Dragulescu, A.; Yakovenko, V.M. Statistical mechanics of money. Eur. Phys. J.
**2000**, B17, 723–729. [Google Scholar] [CrossRef] - 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] - Patriarca, M.; Chakraborti, A.; Kaski, K. Statistical model with a standard Gamma distribution. Phys. Rev. E
**2004**, 70, 016104–016105. [Google Scholar] [CrossRef] - Patriarca, M.; Heinsalu, E.; Chakraborti, A. The ABCD’s of statistical many-agent economy models. 2006; arXiv:physics/0611245. [Google Scholar]
- Angle, J. The inequality process as a wealth maximizing process. Physica A
**2006**, 367, 388–414, (references therein). [Google Scholar] [CrossRef] - Gonzalez-Estevez, J.; Cosenza, M.G.; Lopez-Ruiz, R.; Sanchez, J.R. Pareto and Boltzmann-Gibbs behaviors in a deterministic multi-agent system. Physica A
**2008**, 387, 4637–4642. [Google Scholar] [CrossRef] - Pellicer-Lostao, C.; Lopez-Ruiz, R. Economic models with chaotic money exchange. In Proceedings of the ICCS 2009, Baton Rouge, LA, USA, 2009; Part I. pp. 43–52.
- Lopez-Ruiz, R.; Sañudo, J.; Calbet, X. Geometrical derivation of the Boltzmann factor. Am. J. Phys.
**2008**, 76, 780–781. [Google Scholar] [CrossRef] - Lopez-Ruiz, R.; Calbet, X. Derivation of the Maxwellian distribution from the microcanonical ensemble. Am. J. Phys.
**2007**, 75, 752–753. [Google Scholar] [CrossRef] - Harremoës, P. Projections maximizing Tsallis entropy. AIP Conf. Proc.
**2007**, 965, 90–95. [Google Scholar]

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**MDPI and ACS Style**

López-Ruiz, R.; Sañudo, J.; Calbet, X.
Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems. *Entropy* **2009**, *11*, 959-971.
https://doi.org/10.3390/e11040959

**AMA Style**

López-Ruiz R, Sañudo J, Calbet X.
Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems. *Entropy*. 2009; 11(4):959-971.
https://doi.org/10.3390/e11040959

**Chicago/Turabian Style**

López-Ruiz, Ricardo, Jaime Sañudo, and Xavier Calbet.
2009. "Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems" *Entropy* 11, no. 4: 959-971.
https://doi.org/10.3390/e11040959