We study the problem of designing voting rules that take as input the ordinal preferences of n agents over a set of m alternatives and output a single alternative, aiming to optimize the overall happiness of the agents. The input to the voting rule is each agent’s ranking of the alternatives from most to least preferred, yet the agents have more refined (cardinal) preferences that capture the intensity with which they prefer one alternative over another. To quantify the extent to which voting rules can optimize over the cardinal preferences given access only to the ordinal ones, prior work has used the distortion measure, i.e., the worst-case approximation ratio between a voting rule’s performance and the best performance achievable given the cardinal preferences.
The work on the distortion of voting rules has been largely divided into two “worlds”: utilitarian distortion and metric distortion. In the former, the cardinal preferences of the agents correspond to general utilities and the goal is to maximize a normalized social welfare. In the latter, the agents’ cardinal preferences correspond to costs given by distances in an underlying metric space and the goal is to minimize the (unnormalized) social cost. Several deterministic and randomized voting rules have been proposed and evaluated for each of these worlds separately, gradually improving the achievable distortion bounds, but none of the known voting rules perform well in both worlds simultaneously.
In this work, we prove that one can in fact achieve the “best of both worlds” by designing new voting rules, both deterministic and randomized, that simultaneously achieve near-optimal distortion guarantees in both distortion worlds. We also prove that this positive result does not generalize to the case where the voting rule is provided with the rankings of only the top-t alternatives of each agent, for t < m, and study the extent to which such best-of-both-worlds guarantees can be achieved.
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A Truthful Cardinal Mechanism for One-Sided Matching
We revisit the well-studied problem of designing mechanisms for one-sided matching markets, where a set of n agents needs to be matched to a set of n heterogeneous items. Each agent i has a value vij for each item j, and these values are private information that the agents may misreport if doing so leads to a preferred outcome. Ensuring that the agents
have no incentive to misreport requires a careful design of the matching mechanism, and mechanisms proposed in the literature mitigate this issue by eliciting only the ordinal
preferences of the agents, i.e., their ranking of the items from most to least preferred. However, the efficiency guarantees of these mechanisms are based only on weak measures
that are oblivious to the underlying values. In this paper we achieve stronger performance guarantees by introducing a mechanism that truthfully elicits the full cardinal preferences of the agents, i.e., all of the vij values. We evaluate the performance of this mechanism using the much more demanding Nash bargaining solution as a benchmark, and we
prove that our mechanism significantly outperforms all ordinal mechanisms (even non-truthful ones). To prove our approximation bounds, we also study the population monotonicity of the Nash bargaining solution in the context of matching markets, providing both upper and lower bounds which are of independent interest.
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- Award ID(s):
- 1909538
- NSF-PAR ID:
- 10155725
- Date Published:
- Journal Name:
- ACM-SIAM Symposium on Discrete Algorithms
- Volume:
- 31
- Page Range / eLocation ID:
- 2096-2113
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611975994.129
