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Analytic treatment of a trading market model

We mathematically analyze a simple market model where trading at each point in time involves only two agents with the sum of their money being conserved and with neither parties resulting with negative money after the interaction process. The exchange involves random re-distribution among the two players of a fixed fraction of their total money. We obtain a simple integral nonlinear equation for the money distribution. We find that the zero savings and finite savings cases belong to different universality classes. While the zero savings case can be solved analytically, the finite savings solution is obtained by numerically solving the integral equation. We find remarkable agreement with results obtained by other researchers using sophisticated numerical techniques.