Duality in an asset exchange model for wealth distribution

Abstract

Asset exchange models are agent-based economic models with binary transactions. Previous investigations have augmented these models with mechanisms for wealth redistribution, quantified by a parameter χ, and for trading bias favoring wealthier agents, quantified by a parameter ζ. By deriving and analyzing a Fokker–Planck equation for a particular asset exchange model thus augmented, it has been shown that it exhibits a second-order phase transition at ζ∕χ=1, between regimes with and without partial wealth condensation. In the “subcritical” regime with ζ∕χ<1, all of the wealth is classically distributed; in the “supercritical” regime with ζ∕χ>1, a fraction 1−χ∕ζ of the wealth is condensed. Intuitively, one may associate the supercritical, wealth-condensed regime as reflecting the presence of “oligarchy,” by which we mean that an infinitesimal fraction of the total agents hold a finite fraction of the total wealth in the continuum limit.In this paper, we further elucidate the phase behavior of this model – and hence of the generalized solutions of the Fokker–Planck equation that describes it – by demonstrating the existence of a remarkable symmetry between its supercritical and subcritical regimes in the steady-state. Noting that the replacement {ζ→χ,χ→ζ}, which clearly has the effect of inverting the order parameter ζ∕χ, provides a one-to-one correspondence between the subcritical and supercritical states, we demonstrate that the wealth distribution of the subcritical state is identical to that of the corresponding supercritical state when the oligarchy is removed from the latter. We demonstrate this result analytically, both from the microscopic agent-level model and from its macroscopic Fokker–Planck description, as well as numerically. We argue that this symmetry is a kind of duality, analogous to the famous Kramers–Wannier duality between the subcritical and supercritical states of the Ising model, and to the Maldacena duality that underlies AdS/CFT theory.