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, strategyproofness, 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 uniformprice rule. Our results provide a rationale for the use of the popular minimum uniform-price rule in terms of fairness and non-manipulability.