Consider a world in which most inequality results from random chance. Society hands large sums of money to some people and little to others for no particular reason. Economic analysis suggests that the relation between output and inequality follows an inverse-U-shaped curve. At very low levels of inequality, increases in inequality generate greater social output. At the maximal level of inequality, where all of the output goes to a single person, output is higher than zero but low. As inequality increases from zero incentives, output increases. The higher inequality will induce a few "superstars" with a chance to reach the top of the income hierarchy to work harder and produce more, but it will demotivate everyone else. The natural question to ask next is how income inequality in the United States compares to the output-maximizing level and to the socially desirable level. The relevant theory is the theory of prizes in tournaments when competitors vary in their abilities.