  We investigate the repeated averaging model for money exchanges: two agents
picked uniformly at random share half of their wealth to each other. It is
intuitively convincing that a Dirac distribution of wealth (centered at the
initial average wealth) will be the long time equilibrium for this dynamics. In
other words, the Gini index should converge to zero. To better understand this
dynamics, we investigate its limit as the number of agents goes to infinity by
proving the so-called propagation of chaos, which links the stochastic
agent-based dynamics to a (limiting) nonlinear partial differential equation
(PDE). This deterministic description has a flavor of the classical Boltzmann
equation arising from statistical mechanics of dilute gases. We prove its
convergence toward its Dirac equilibrium distribution by showing that the
associated Gini index of the wealth distribution converges to zero with an
explicit rate.
