Stochastic Maps, Wealth Distribution in Random Asset Exchange Models and the Marginal Utility of Relative Wealth

We look at how asset exchange models can be mapped to random iterated function systems (IFS) giving new insights into the dynamics of wealth accumulation in such models. In particular, we focus on the "yard-sale" (winner gets a random fraction of the poorer players wealth) and the "theft-and-fraud" (winner gets a random fraction of the loser's wealth) asset exchange models. Several special cases including 2-player and 3-player versions of these `games' allow us to connect the results with observed features in real economies, e.g., lock-in (positive feedback), etc. We then implement the realistic notion that a richer agent is less likely to be aggressive when bargaining over a small amount with a poorer player. When this simple feature is added to the yard-sale model, in addition to the accumulation of the total wealth by a single agent ("condensation"), we can see exponential and power-law distributions of wealth. Simulation results suggest that the power-law distribution occurs at the cross-over of the system from exponential phase to the condensate phase.