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We investigate the repeated averaging model for money exchanges: two agentspicked uniformly at random share half of their wealth to each other. It isintuitively convincing that a Dirac distribution of wealth (centered at theinitial average wealth) will be the long time equilibrium for this dynamics. Inother words, the Gini index should converge to zero. To better understand thisdynamics, we investigate its limit as the number of agents goes to infinity byproving the so-called propagation of chaos, which links the stochasticagent-based dynamics to a (limiting) nonlinear partial differential equation(PDE). This deterministic description has a flavor of the classical Boltzmannequation arising from statistical mechanics of dilute gases. We prove itsconvergence toward its Dirac equilibrium distribution by showing that theassociated Gini index of the wealth distribution converges to zero with anexplicit rate.

Explicit decay rate for the Gini index in the repeated averaging model