Social Choice with Non Quasi-linear Utilities

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

Almost Quasi-linear Utilities in Disguise: Positive-representation An Extension of Roberts' Theorem

This work deals with the implementation of social choice rules using dominant strategies for unrestricted preferences. The seminal Gibbard-Satterthwaite theorem shows that only few unappealing social choice rules can be implemented unless we assume some restrictions on the preferences or allow monetary transfers. When monetary transfers are allowed and quasi-linear utilities w.r.t. money are assumed, Vickrey-Clarke-Groves (VCG) mechanisms were shown to implement any affine-maximizer, and by the work of Roberts, only affine-maximizers can be implemented whenever the type sets of the agents are rich enough. In this work, we generalize these results and define a new class of preferences: Preferences which are positive-represented by a quasi-linear utility. That is, agents whose preference on a subspace of the outcomes can be modeled using a quasi-linear utility. We show that the characterization of VCG mechanisms as the incentive-compatible mechanisms extends naturally to this domain. Our result follows from a simple reduction to the characterization of VCG mechanisms. Hence, we see our result more as a fuller more correct version of the VCG characterization. This work also highlights a common misconception in the community attributing the VCG result to the usage of transferable utility. Our result shows that the incentive-compatibility of the VCG mechanisms does not rely on money being a common denominator, but rather on the ability of the designer to fine the agents on a continuous (maybe agent-specific) scale. We think these two insights, considering the utility as a representation and not as the preference itself (which is common in the economic community) and considering utilities which represent the preference only for the relevant domain, would turn out to fruitful in other domains as well.Comment: An extended abstract of this work is forthcoming in: The 15th Conference on Web and Internet Economics (WINE 2019

Pareto efficient combinatorial auctions: dichotomous preferences without quasilinearity

We consider a combinatorial auction model where preferences of agents over bundles of objects and payments need not be quasilinear. However, we restrict the preferences of agents to be dichotomous. An agent with dichotomous preference partitions the set of bundles of objects as acceptable} and unacceptable, and at the same payment level, she is indifferent between bundles in each class but strictly prefers acceptable to unacceptable bundles. We show that there is no Pareto efficient, dominant strategy incentive compatible (DSIC), individually rational (IR) mechanism satisfying no subsidy if the domain of preferences includes all dichotomous preferences. However, a generalization of the VCG mechanism is Pareto efficient, DSIC, IR and satisfies no subsidy if the domain of preferences contains only positive income effect dichotomous preferences. We show the tightness of this result: adding any non-dichotomous preference (satisfying some natural properties) to the domain of quasilinear dichotomous preferences brings back the impossibility result