We consider a social choice setting in which agents and alternatives are represented by points in a metric space, and the cost of an agent for an alternative is the distance between the corresponding points in the space. The goal is to choose a single alternative to (approximately) minimize the social cost (cost of all agents) or the maximum cost of any agent, when only limited information about the preferences of the agents is given. Previous work has shown that the best possible distortion one can hope to achieve is 3 when access to the ordinal preferences of the agents is given, even when the distances between alternatives in the metric space are known. We improve upon this bound of 3 by designing deterministic mechanisms that exploit a bit of cardinal information. We show that it is possible to achieve distortion 1+sqrt(2) by using the ordinal preferences of the agents, the distances between alternatives, and a threshold approval set per agent that contains all alternatives for whom her cost is within an appropriately chosen factor of her cost for her most-preferred alternative. We show that this bound is the best possible for any deterministic mechanism in general metric spaces, and also provide improved bounds for the fundamental case of a line metric.
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Best of Both Distortion Worlds
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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- Award ID(s):
- 2047907
- PAR ID:
- 10491121
- Publisher / Repository:
- 24th ACM Conference on Economics and Computation
- Date Published:
- Journal Name:
- 24th ACM Conference on Economics and Computation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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