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

TL;DR

This paper investigates the design of rebate functions in VCG mechanisms for assigning heterogeneous objects, proving an impossibility for linear rebates with efficiency and proposing alternative approaches involving correlation and non-linearity.

Contribution

It establishes an impossibility theorem for linear rebate functions with efficiency in heterogeneous object assignment and explores conditions under which such rebates are feasible.

Findings

Linear rebate functions with non-zero efficiency are impossible in the general case.

Correlated valuations enable linear rebate functions with efficiency.

Relaxing linearity allows for efficient rebate functions.

Abstract

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 non-zero are possible when the valuations for the objects are correlated. In the second approach, we show that rebate functions with non-zero efficiency are possible if linearity is relaxed.