Social Choice with Non Quasi-linear Utilities

TL;DR

This paper characterizes the largest domain of non-quasi-linear utilities where VCG-like mechanisms can exist, extending Roberts' theorem and showing that slight deviations from quasi-linearity lead to dictatorship results.

Contribution

It provides a tight characterization of the maximal non-quasi-linear utility domain, called the largest parallel domain, and extends Roberts' theorem to these domains.

Findings

Maximal non-quasi-linear utility domain identified as the largest parallel domain.

Extended Roberts' theorem to parallel domains.

Any reasonable mechanism becomes a dictatorship with slight deviations from quasi-linearity.

Abstract

Without monetary payments, the Gibbard-Satterthwaite theorem proves that under mild requirements all truthful social choice mechanisms must be dictatorships. When payments are allowed, the Vickrey-Clarke-Groves (VCG) mechanism implements the value-maximizing choice, and has many other good properties: it is strategy-proof, onto, deterministic, individually rational, and does not make positive transfers to the agents. By Roberts' theorem, with three or more alternatives, the weighted VCG mechanisms are essentially unique for domains with quasi-linear utilities. The goal of this paper is to characterize domains of non-quasi-linear utilities where "reasonable" mechanisms (with VCG-like properties) exist. Our main result is a tight characterization of the maximal non quasi-linear utility domain, which we call the largest parallel domain. We extend Roberts' theorem to parallel domains, and use the generalized theorem to prove two impossibility results. First, any reasonable mechanism must be dictatorial when the utility domain is quasi-linear together with any single non-parallel type. Second, for richer utility domains that still differ very slightly from quasi-linearity, every strategy-proof, onto and deterministic mechanism must be a dictatorship.