How much inequality in income is fair? A microeconomic game theoretic perspective

The increasing inequality in income and wealth in recent years, and the associated excessive pay packages of CEOs in the US 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 questions of what kind of income distribution we ought to see, at least under ideal conditions, in a free market environment, and whether this distribution is fair. We propose a novel microeconomic game theoretic framework that addresses these questions and proves that the lognormal distribution is the fairest inequality of pay in an organization comprising of homogeneous agents, under ideal free market conditions at equilibrium. We also show that for a population of two different classes of agents, the equilibrium 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. We compare our predictions with empirical data on global income inequality trends provided by Piketty and others. Our analysis suggests that the Scandinavian countries, and to a lesser extent Switzerland, Netherlands and Australia, have managed to get close to the ideal distribution for the bottom ∼99% of the population, while the US and UK remain less fair at the other extreme. Other European countries such as France and Germany, and Japan and Canada, are in the middle. Our theory also shows the deep and direct connection between potential game theory and statistical mechanics through entropy, which we identify as 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.