Fairness and efficiency in a random assignment: Three impossibility results

This paper considers the problem of allocating N indivisible objects among N agents according to their preferences when transfers are not allowed, and studies the tradeoff between fairness and efficiency in the class of strategy-proof mechanisms. The main finding is that for strategy-proof mechanisms the following efficiency and fairness criteria are mutually incompatible: (1) Ex-post efficiency and envy-freeness, (2) ordinal efficiency and weak envy-freeness and (3) ordinal efficiency and equal division lower bound. Result (1) is the first impossibility result for this setting that uses ex-post efficiency; results (2) and (3) are more relevant for practical implementation than similar results in the literature. In addition, for N = 3 the paper strengthens the characterization result by Bogomolnaia and Moulin (2001): the random serial dictatorship mechanism is the unique strategy-proof, ex-post efficient mechanism that eliminates strict envy between agents with the same preferences.