  In the manuscript, we are interested in using kinetic theory to better
understand the time evolution of wealth distribution and their large scale
behavior such as the evolution of inequality (e.g. Gini index). We investigate
three type of dynamics denoted unbiased, poor-biased and rich-biased dynamics.
At the particle level, one agent is picked randomly based on its wealth and one
of its dollar is redistributed among the population. Proving the so-called
propagation of chaos, we identify the limit of each dynamics as the number of
individual approaches infinity using both coupling techniques [48] and
martingale-based approach [36]. Equipped with the limit equation, we identify
and prove the convergence to specific equilibrium for both the unbiased and
poor-biased dynamics. In the rich-biased dynamics however, we observe a more
complex behavior where a dispersive wave emerges. Although the dispersive wave
is vanishing in time, its also accumulates all the wealth leading to a Gini
approaching 1 (its maximum value). We characterize numerically the behavior of
dispersive wave but further analytic investigation is needed to derive such
dispersive wave directly from the dynamics.
