Almost Quasi-Linear Utilities in Disguise: An Extension of Roberts' Theorem

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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, which is defined by a threshold, 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. We show that the original characterization of VCG mechanism (as well as the recent work of Ma et al. [EC 2018] on parallel utility domains) is an immediate corollary of our generalized characterization. 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 than a new non-quasi-linear domain extension.
 
This work also highlights a common misconception in the community attributing the VCG result to the usage of transferable utility. Our result shows that these results extend naturally to the non-transferable utility domain. That is, 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 also provide simple characterizations of the types which are represented and positive-represented by quasi-linear utility functions. We show characterizations both in utility function terms and in preference terms, by that supplying a full comparison of the different classes.

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.
 
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