2,047 research outputs found
Improved Metric Distortion for Deterministic Social Choice Rules
In this paper, we study the metric distortion of deterministic social choice
rules that choose a winning candidate from a set of candidates based on voter
preferences. Voters and candidates are located in an underlying metric space. A
voter has cost equal to her distance to the winning candidate. Ordinal social
choice rules only have access to the ordinal preferences of the voters that are
assumed to be consistent with the metric distances. Our goal is to design an
ordinal social choice rule with minimum distortion, which is the worst-case
ratio, over all consistent metrics, between the social cost of the rule and
that of the optimal omniscient rule with knowledge of the underlying metric
space.
The distortion of the best deterministic social choice rule was known to be
between and . It had been conjectured that any rule that only looks at
the weighted tournament graph on the candidates cannot have distortion better
than . In our paper, we disprove it by presenting a weighted tournament rule
with distortion of . We design this rule by generalizing the classic
notion of uncovered sets, and further show that this class of rules cannot have
distortion better than . We then propose a new voting rule, via an
alternative generalization of uncovered sets. We show that if a candidate
satisfying the criterion of this voting rule exists, then choosing such a
candidate yields a distortion bound of , matching the lower bound. We
present a combinatorial conjecture that implies distortion of , and verify
it for small numbers of candidates and voters by computer experiments. Using
our framework, we also show that selecting any candidate guarantees distortion
of at most when the weighted tournament graph is cyclically symmetric.Comment: EC 201
Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation
We consider the following problem: There is a set of items (e.g., movies) and
a group of agents (e.g., passengers on a plane); each agent has some intrinsic
utility for each of the items. Our goal is to pick a set of items that
maximize the total derived utility of all the agents (i.e., in our example we
are to pick movies that we put on the plane's entertainment system).
However, the actual utility that an agent derives from a given item is only a
fraction of its intrinsic one, and this fraction depends on how the agent ranks
the item among the chosen, available, ones. We provide a formal specification
of the model and provide concrete examples and settings where it is applicable.
We show that the problem is hard in general, but we show a number of
tractability results for its natural special cases
Communication, Distortion, and Randomness in Metric Voting
In distortion-based analysis of social choice rules over metric spaces, one
assumes that all voters and candidates are jointly embedded in a common metric
space. Voters rank candidates by non-decreasing distance. The mechanism,
receiving only this ordinal (comparison) information, should select a candidate
approximately minimizing the sum of distances from all voters. It is known that
while the Copeland rule and related rules guarantee distortion at most 5, many
other standard voting rules, such as Plurality, Veto, or -approval, have
distortion growing unboundedly in the number of candidates.
Plurality, Veto, or -approval with small require less communication
from the voters than all deterministic social choice rules known to achieve
constant distortion. This motivates our study of the tradeoff between the
distortion and the amount of communication in deterministic social choice
rules.
We show that any one-round deterministic voting mechanism in which each voter
communicates only the candidates she ranks in a given set of positions must
have distortion at least ; we give a mechanism achieving an
upper bound of , which matches the lower bound up to a constant. For
more general communication-bounded voting mechanisms, in which each voter
communicates bits of information about her ranking, we show a slightly
weaker lower bound of on the distortion.
For randomized mechanisms, it is known that Random Dictatorship achieves
expected distortion strictly smaller than 3, almost matching a lower bound of
for any randomized mechanism that only receives each voter's
top choice. We close this gap, by giving a simple randomized social choice rule
which only uses each voter's first choice, and achieves expected distortion
.Comment: An abbreviated version appear in Proceedings of AAAI 202
Algorithm Design for Ordinal Settings
Social choice theory is concerned with aggregating the preferences of agents into a single outcome. While it is natural to assume that agents have cardinal utilities, in many contexts, we can only assume access to the agents’ ordinal preferences, or rankings over the outcomes. As ordinal preferences are not as expressive as cardinal utilities, a loss of efficiency is unavoidable. Procaccia and Rosenschein (2006) introduced the notion of distortion to quantify this worst-case efficiency loss for a given social choice function.
We primarily study distortion in the context of election, or equivalently clustering problems, where we are given a set of agents and candidates in a metric space; each agent has a preference ranking over the set of candidates and we wish to elect a committee of k candidates that minimizes the total social cost incurred by the agents.
In the single-winner setting (when k = 1), we give a novel LP-duality based analysis framework that makes it easier to analyze the distortion of existing social choice functions, and extends readily to randomized social choice functions. Using this framework, we show that it is possible to give simpler proofs of known results. We also show how to efficiently compute an optimal randomized social choice function for any given instance. We utilize the latter result to obtain an instance for which any randomized social choice function has distortion at least 2.063164. This disproves the long-standing conjecture that there exists a randomized social choice function that has a worst-case distortion of at most 2.
When k is at least 2, it is not possible to compute an O(1)-distortion committee using purely ordinal information. We develop two O(1)-distortion mechanisms for this problem: one having a polylog(n) (per agent) query complexity, where n is the number of agents; and the other having O(k) query complexity (i.e., no dependence on n). We also study a much more general setting called minimum-norm k-clustering recently proposed in the clustering literature, where the objective is some monotone, symmetric norm of the the agents' costs, and we wish to find a committee of k candidates to minimize this objective. When the norm is the sum of the p largest costs, which is called the p-centrum problem in the clustering literature, we give low-distortion mechanisms by adapting our mechanisms for k-median. En route, we give a simple adaptive-sampling algorithm for this problem. Finally, we show how to leverage this adaptive-sampling idea to also obtain a constant-factor bicriteria approximation algorithm for minimum-norm k-clustering (in its full generality)
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