5,801 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
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
An Analysis Framework for Metric Voting based on LP Duality
Distortion-based analysis has established itself as a fruitful framework for
comparing voting mechanisms. m voters and n candidates are jointly embedded in
an (unknown) metric space, and the voters submit rankings of candidates by
non-decreasing distance from themselves. Based on the submitted rankings, the
social choice rule chooses a winning candidate; the quality of the winner is
the sum of the (unknown) distances to the voters. The rule's choice will in
general be suboptimal, and the worst-case ratio between the cost of its chosen
candidate and the optimal candidate is called the rule's distortion. It was
shown in prior work that every deterministic rule has distortion at least 3,
while the Copeland rule and related rules guarantee worst-case distortion at
most 5; a very recent result gave a rule with distortion .
We provide a framework based on LP-duality and flow interpretations of the
dual which provides a simpler and more unified way for proving upper bounds on
the distortion of social choice rules. We illustrate the utility of this
approach with three examples. First, we give a fairly simple proof of a strong
generalization of the upper bound of 5 on the distortion of Copeland, to social
choice rules with short paths from the winning candidate to the optimal
candidate in generalized weak preference graphs. A special case of this result
recovers the recent guarantee. Second, using this generalized
bound, we show that the Ranked Pairs and Schulze rules have distortion
. Finally, our framework naturally suggests a combinatorial
rule that is a strong candidate for achieving distortion 3, which had also been
proposed in recent work. We prove that the distortion bound of 3 would follow
from any of three combinatorial conjectures we formulate.Comment: 23 pages An abbreviated version appears in Proceedings of AAAI 202
Beyond the worst case: Distortion in impartial culture electorate
{\em Distortion} is a well-established notion for quantifying the loss of
social welfare that may occur in voting. As voting rules take as input only
ordinal information, they are essentially forced to neglect the exact values
the agents have for the alternatives. Thus, in worst-case electorates, voting
rules may return low social welfare alternatives and have high distortion.
Accompanying voting rules with a small number of cardinal queries per agent may
reduce distortion considerably.
To explore distortion beyond worst-case conditions, we introduce a simple
stochastic model, according to which the values the agents have for the
alternatives are drawn independently from a common probability distribution.
This gives rise to so-called {\em impartial culture electorates}. We refine the
definition of distortion so that it is suitable for this stochastic setting and
show that, rather surprisingly, all voting rules have high distortion {\em on
average}. On the positive side, for the fundamental case where the agents have
random {\em binary} values for the alternatives, we present a mechanism that
achieves approximately optimal average distortion by making a {\em single}
cardinal query per agent. This enables us to obtain slightly suboptimal average
distortion bounds for general distributions using a simple randomized mechanism
that makes one query per agent. We complement these results by presenting new
tradeoffs between the distortion and the number of queries per agent in the
traditional worst-case setting.Comment: 27 pages, 2 figure
Sequential Deliberation for Social Choice
In large scale collective decision making, social choice is a normative study
of how one ought to design a protocol for reaching consensus. However, in
instances where the underlying decision space is too large or complex for
ordinal voting, standard voting methods of social choice may be impractical.
How then can we design a mechanism - preferably decentralized, simple,
scalable, and not requiring any special knowledge of the decision space - to
reach consensus? We propose sequential deliberation as a natural solution to
this problem. In this iterative method, successive pairs of agents bargain over
the decision space using the previous decision as a disagreement alternative.
We describe the general method and analyze the quality of its outcome when the
space of preferences define a median graph. We show that sequential
deliberation finds a 1.208- approximation to the optimal social cost on such
graphs, coming very close to this value with only a small constant number of
agents sampled from the population. We also show lower bounds on simpler
classes of mechanisms to justify our design choices. We further show that
sequential deliberation is ex-post Pareto efficient and has truthful reporting
as an equilibrium of the induced extensive form game. We finally show that for
general metric spaces, the second moment of of the distribution of social cost
of the outcomes produced by sequential deliberation is also bounded
Distortion in Social Choice Problems: The First 15 Years and beyond
The notion of distortion in social choice problems has been defined to measure the loss in efficiency-typically measured by the utilitarian social welfare, the sum of utilities of the participating agents-due to having access only to limited information about the preferences of the agents. We survey the most significant results of the literature on distortion from the past 15 years, and highlight important open problems and the most promising avenues of ongoing and future work
- β¦