No price envy in the multi-unit object allocation problem with non-quasi-linear preferences

We consider the problem of allocating multiple units of an indivisible object among a set of agents and collecting payments. Each agent can receive multiple units of the object, and has a (possibly) non-quasi-linear preference on the set of (consumption) bundles. We assume that preferences exhibit both nonincreasing marginal valuations and nonnegative income effects. We propose a new property of fairness: no price envy. It extends the standard no envy test (Foley, 1967) over bundles to prices (per-unit payments), and requires no agent envy other agents' prices to his own in the sense that if he has a chance to receive some units at other agents' prices, then he gets better off than his own bundle. First, we show that a rule satisfies no price envy and no subsidy for losers if and only if it is an inverse uniform-price rule. Then, we identify the unique maximal domain for no price envy, strategy-proofness, and no subsidy for losers: the domain with partly constant marginal valuations. We further establish that on the domain with partly constant marginal valuations, a rule satisfies no price envy, strategy-proofness, and no subsidy for losers if and only if it is a minimum inverse uniform-price rule. Our maximal domain result implies that no rule satisfies no price envy, strategy-proofness, and no subsidy for losers when agents have preferences with nonincreasing marginal valuations. Given this negative observation, we look for a minimally manipulable rule among the class of rules satisfying both no price envy and no subsidy for losers in the case of preferences with nonincreasing marginal valuations. We show that a rule is minimally manipulable among the class of rules satisfying no price envy and no subsidy for losers if and only if it is a minimum inverse uniform-price rule. Our results provide a rationale for the use of the popular minimum uniform-price rule in terms of fairness and non-manipulability.