Bounding the approach to oligarchy in a variant of the yard-sale model (Q6620726)

The article provides for a relevant reference to researchers in econophysics, mathematical economics and economics researchers in general.\N\NThe authors introduce a modification to the classical yard-sale model of stochastic transactions between agents. In the original model, as reviewed by the authors, given a population of N agents, at each round of the transaction game, two agents are selected at random without replacement and their respective wealth values are updated in a way that the lowest wealth value of the pair multiplied by a factor is randomly transferred from one agent to the other, with the agent in the interaction pair that will receive the added wealth and the agent that will transfer the wealth being selected at random with 1/2 probability. The multiplication factor used in the transaction model is the square root of the transaction intensity.\N\NThe authors change this model to include an adaptive component, namely, if the lowest wealth in the agent pair is below 1, the lowest wealth value multiplied by the square root of the transaction intensity is selected, like in the original model, however, if it is higher than or equal to 1 the square root of the lowest value is used instead. In this way, the wealth transfer becomes a piecewise nonlinear function of the lowest wealth in the agent pair. In this case, if the lowest wealth is below 1 there is a linearly increasing transfer of wealth, however, if the lowest wealth is equal to or above 1, the transfer of wealth grows with the square root factor, which means that, for higher values of wealth, the transfer of wealth grows more slowly than the linear wealth transfer.\N\NIntuitively, from an economics standpoint, this means that between two agents with higher values of wealth the wealth transfer is lower than the original linear model. As the authors show, the expected value of the transaction is zero, while the variance depends upon the piecewise function.\N\NIn this case, introducing a time interval for each round, the variance is equal to the value of the transaction intensity multiplied by the time interval in turn multiplied by the square of the piecewise nonlinear wealth kernel introduced by the authors. Given the above-described piecewise function, this last factor is equal to the square of the lowest wealth value of the agent pair, in the case where the lowest wealth value is below 1, but it is equal to the lowest wealth value, in the case where the lowest wealth value is equal to 1 or higher, this is because of the square root in the second branch of the piecewise wealth kernel.\N\NThe authors then derive a Fokker-Planck equation for the normalized agent distribution and prove that the Gini coefficient is monotone increasing under the dynamics given by the Fokker-Planck equation (Theorem 4.1), they also prove that the rate of change in the Gini coefficient can be bounded (Theorem 4.3) and that the bound can be carried over into a bound on that coefficient at a future time.\N\NFrom an econophysics standpoint, besides the proof of the bound and condition for it, the authors present a relation of similarity to entropy-entropy production for nonlinear Fokker-Planck equations in the form of a Gini-Gini production inequality for the modified yard-sale model, which the authors show that it holds for a broader class of models. In this generalization, the transaction kernels need not assume independence of the wealthier agent, also showing compatibility with models in which microtransactions are assumed over a continuous rather than discrete set of outcomes.\N\NThere are two major conditions for the wealth transfer kernel under which the generalization holds for the bounded rate of change in the Gini coefficient, the kernel and its square value must be at most equal to the lowest value of the two wealth values being compared in the transaction process (Corollary 4.4).\N\NThe authors also expand on the model to incorporate taxation, performing an asymptotic analysis in comparison with the original nonmodified yard-sale model. Taxation involves a redistribution scheme that operates towards mean wealth reversion in the agent population.\N\NThe authors review two taxation schemes, one in which taxation pools the total wealth and redistributes it equally, the other in which taxation operates locally between agents towards their pairwise mean wealth. Both schemes lead to the same continuum equation of motion. As shown by the authors, their modified yard-sale model with taxation agrees with the original yard-sale model with taxation which means that their modified model can match actual wealth distributions, that is, it has an empirical matching.\N\NThe article thus provides for a mathematical and methodological background and results that can be employed both by those researching on wealth distribution and wealth inequality dynamics in economic wealth transaction agent-based models and those researching on the applications of statistical mechanics and agent-based modeling in economics in general. The proof that the authors' results hold for a wide range of transaction kernels (Corollary 4.4) is evidence of robustness of these results and make them relevant to both economics in general and econophysics in particular.