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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 thisassignment problem satisfying weak budget balance, individual rationality, andminimizing the budget imbalance. This calls for designing an appropriate rebatefunction. Our main result is an impossibility theorem which rules out linearrebate functions with non-zero efficiency in heterogeneous object assignment.Motivated by this theorem, we explore two approaches to get around thisimpossibility. In the first approach, we show that linear rebate functions withnon-zero are possible when the valuations for the objects are correlated. Inthe second approach, we show that rebate functions with non-zero efficiency arepossible if linearity is relaxed.

On Optimal Linear Redistribution of VCG Payments in Assignment of Heterogeneous Objects