# 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

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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