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 mechanism truthfully implements the value-maximizing choices, assuming utilities of agents are quasi-linear in money. We study social choice with payments where utilities are non-quasi-linear.
The main result of this paper is a tight characterization of the maximal non-quasi-linear utility domain, which we call the largest parallel domain, for which where there exist non-dictatorial mechanisms that are strategy-proof, onto, deterministic, individually rational and that do not not make positive transfers to the agents. In particular, mechanisms satisfying the above conditions must be dictatorial when the type domain is quasi-linear together with any single non-parallel type. We also provide utility domains that differ very slightly from quasi-linearity for which individual rationality and no payment to the agents can be relaxed and the dictatorship result remains.