Distribution of money for the uniform reshuffling and immediate exchange models

Friday, September 13, 2019 - 2:00pm


Nicolas Lanchier
Associate Professor
School of Mathematics and Statistical Sciences
Arizona State University


The uniform reshuffling and immediate exchange models consist of systems of economical agents located on a graph. These agents possess coins and randomly engage in pairwise monetary transactions with their neighbors. In the first model, the combined amount of coins of the two interacting neighbors is redistributed randomly and uniformly between the neighbors, while in the second model, the two interacting neighbors choose simultaneously a random number of their coins to give to the other neighbor. Physicists used numerical simulations to conjecture that, at least for large well-mixing systems (large complete graphs), the distribution of money converges to the exponential distribution for the uniform reshuffling model and to the gamma distribution for the immediate exchange model. In this talk, we give rigorous proofs of these two conjectures but also extend these results to all connected graphs, not just the complete graph. Although the models are mathematically challenging due to the inclusion of space and stochasticity, they are also simple in their formulation and do not include any parameter. In particular, as minimal models, they can easily be applied to other contexts. For instance, thinking of each vertex as a patch and each coin as an individual results in population dynamics models of metapopulations.