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Game Theory, Statistical Mechanics and Income Inequality

The widening inequality in income distribution in recent years, and the associated excessive pay packages of CEOs in the U.S. and elsewhere, is of growing concern among policy makers as well as the common person. However, there seems to be no satisfactory answer, in conventional economic theories and models, to the fundamental question of what kind of pay distribution we ought to see, at least under ideal conditions, in a free market environment and whether this distribution is fair. We propose a game theoretic framework that addresses these questions and show that the lognormal distribution is the fairest inequality of pay in an organization comprising of homogenous agents, achieved at equilibrium, under ideal free market conditions. We also show that for a population of two different classes of agents, the final distribution is a combination of two different lognormal distributions where one of them, corresponding to the top 3-5% of the population, can be misidentified as a Pareto distribution. Our theory also shows the deep and direct connection between potential game theory and statistical mechanics through entropy, which is a measure of fairness in a distribution. This leads us to propose the fair market hypothesis, that the self-organizing dynamics of the ideal free market, i.e., Adam Smith's "invisible hand", not only promotes efficiency but also maximizes fairness under the given constraints.