Random Multi-Unit Assignment with Endogenous Quotas

We study the random multi-unit assignment problem in which the number of goods to be distributed depends on players' preferences. In this setup, the egalitarian solution is more appealing than the competitive equilibrium with equal incomes because it is Lorenz dominant, unique in utilities, and impossible to manipulate by groups when agents have dichotomous preferences. Moreover, it can be adapted to satisfy a new fairness axiom that arises naturally in this context. Both solutions are disjoint. Two standard results disappear. The competitive solution can no longer be computed with the Eisenberg-Gale program maximizing the Nash product, and the competitive equilibrium with equal incomes is no longer unique in its corresponding utility profile.