Redistribution Mechanisms for the Assignment of Heterogeneous Objects

There are p heterogeneous objects to be assigned to n competing agents (n > p) each with unit demand. It is required to design a Groves mechanism for this assignment problem satisfying weak budget balance, individual rationality, and minimizing the budget imbalance. This calls for designing an appropriate rebate function. Our main result is an impossibility theorem which rules out linear rebate functions with non-zero efficiency in heterogeneous object assignment. Motivated by this theorem, we explore two approaches to get around this impossibility. In the first approach, we show that linear rebate functions with nonzero efficiency are possible when the valuations for the objects have some relationship. In the second approach, we show that rebate functions with non-zero efficiency are possible if linearity is relaxed.