The role of cooperation in spatially explicit economical systems

This paper is concerned with a model in econophysics, the subfield of statistical physics that applies concepts from traditional physics to economics. In our model, economical agents are represented by the vertices of a connected graph and are characterized by the number of coins they possess. Agents independently spend one coin at rate one for their basic need, earn one coin at a rate chosen independently from a fixed distribution $\phi$ and exchange money at rate $\mu$ with one of their nearest neighbors, with the richest neighbor giving one coin to the other neighbor. If an agent needs to spend one coin when her fortune is at zero, she dies, i.e., the corresponding vertex is removed from the graph. Our first results focus on the two extreme cases of lack of cooperation $\mu = 0$ and perfect cooperation $\mu = \infty$ for finite connected graphs. These results suggest that, when overall the agents earn more than they spend, cooperation is beneficial for the survival of the population, whereas when overall the agents earn less than they spend, cooperation becomes detrimental. The infinite one-dimensional system is also studied. In this case, when the agents earn less than they spend in average, the density of agents that die eventually is bounded from below by a positive constant that does not depend on the initial number of coins per agent or the level of cooperation.