We study social choice mechanisms in an implicit utilitarian framework with a metric constraint, where the goal is to minimize Distortion, the worst case social cost of an ordinal mechanism relative to underlying cardinal utilities. We consider two additional desiderata: Constant sample complexity and Squared Distortion. Constant sample complexity means that the mechanism (potentially randomized) only uses a constant number of ordinal queries regardless of the number of voters and alternatives. Squared Distortion is a measure of variance of the Distortion of a randomized mechanism.Our primary contribution is the first social choice mechanism with constant sample complexity and constant Squared Distortion (which also implies constant Distortion). We call the mechanism Random Referee, because it uses a random agent to compare two alternatives that are the favorites of two other random agents. We prove that the use of a comparison query is necessary: no mechanism that only elicits the top-k preferred alternatives of voters (for constant k) can have Squared Distortion that is sublinear in the number of alternatives. We also prove that unlike any top-k only mechanism, the Distortion of Random Referee meaningfully improves on benign metric spaces, using the Euclidean plane as a canonical example. Finally, among top-1 onlymore »
Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties
We study social choice rules under the utilitarian distortion framework, with an additional metric assumption on the agents' costs over the alternatives. In this approach, these costs are given by an underlying metric on the set of all agents plus alternatives. Social choice rules have access to only the ordinal preferences of agents but not the latent cardinal costs that induce them. Distortion is then defined as the ratio between the social cost (typically the sum of agent costs) of the alternative chosen by the mechanism at hand, and that of the optimal alternative chosen by an omniscient algorithm. The worst-case distortion of a social choice rule is, therefore, a measure of how close it always gets to the optimal alternative without any knowledge of the underlying costs. Under this model, it has been conjectured that Ranked Pairs, the well-known weighted-tournament rule, achieves a distortion of at most 3 (Anshelevich et al. 2015). We disprove this conjecture by constructing a sequence of instances which shows that the worst-case distortion of Ranked Pairs is at least 5. Our lower bound on the worst-case distortion of Ranked Pairs matches a previously known upper bound for the Copeland rule, proving that in the more »
- Award ID(s):
- 1637397
- Publication Date:
- NSF-PAR ID:
- 10026403
- Journal Name:
- EC '17 Proceedings of the 2017 ACM Conference on Economics and Computation
- Page Range or eLocation-ID:
- 287 to 304
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Publisher's Version of Record
- https://doi.org/10.1145/3033274.3085138
