We study the random multi-unit assignment problem in which the number of goods to be distributed depends on players' preferences, obtaining several results that also apply to its corresponding version with fixed number of goods, commonly referred to as the course allocation problem. Although efficiency, envy-freeness, and group strategy-proofness can be achieved in the domain of dichotomous preferences, two standard results disappear: the egalitarian solution cannot be supported by competitive prices, so the competitive solution can no longer be computed with the Eisenberg-Gale program maximizing the Nash product, and the competitive allocation with equal incomes is no longer unique. The egalitarian solution is more appealing than the competitive one in this setup because it is Lorenz dominant, unique, and impossible to manipulate by groups. Moreover, it can be adapted to satisfy a new fairness axiom that arises naturally in this context.
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