17,618 research outputs found
Simple Dynamics for Plurality Consensus
We study a \emph{Plurality-Consensus} process in which each of anonymous
agents of a communication network initially supports an opinion (a color chosen
from a finite set ). Then, in every (synchronous) round, each agent can
revise his color according to the opinions currently held by a random sample of
his neighbors. It is assumed that the initial color configuration exhibits a
sufficiently large \emph{bias} towards a fixed plurality color, that is,
the number of nodes supporting the plurality color exceeds the number of nodes
supporting any other color by additional nodes. The goal is having the
process to converge to the \emph{stable} configuration in which all nodes
support the initial plurality. We consider a basic model in which the network
is a clique and the update rule (called here the \emph{3-majority dynamics}) of
the process is the following: each agent looks at the colors of three random
neighbors and then applies the majority rule (breaking ties uniformly).
We prove that the process converges in time with high probability, provided that .
We then prove that our upper bound above is tight as long as . This fact implies an exponential time-gap between the
plurality-consensus process and the \emph{median} process studied by Doerr et
al. in [ACM SPAA'11].
A natural question is whether looking at more (than three) random neighbors
can significantly speed up the process. We provide a negative answer to this
question: In particular, we show that samples of polylogarithmic size can speed
up the process by a polylogarithmic factor only.Comment: Preprint of journal versio
The Pareto Frontier for Random Mechanisms
We study the trade-offs between strategyproofness and other desiderata, such
as efficiency or fairness, that often arise in the design of random ordinal
mechanisms. We use approximate strategyproofness to define manipulability, a
measure to quantify the incentive properties of non-strategyproof mechanisms,
and we introduce the deficit, a measure to quantify the performance of
mechanisms with respect to another desideratum. When this desideratum is
incompatible with strategyproofness, mechanisms that trade off manipulability
and deficit optimally form the Pareto frontier. Our main contribution is a
structural characterization of this Pareto frontier, and we present algorithms
that exploit this structure to compute it. To illustrate its shape, we apply
our results for two different desiderata, namely Plurality and Veto scoring, in
settings with 3 alternatives and up to 18 agents.Comment: Working Pape
Computing Majority with Triple Queries
Consider a bin containing balls colored with two colors. In a -query,
balls are selected by a questioner and the oracle's reply is related
(depending on the computation model being considered) to the distribution of
colors of the balls in this -tuple; however, the oracle never reveals the
colors of the individual balls. Following a number of queries the questioner is
said to determine the majority color if it can output a ball of the majority
color if it exists, and can prove that there is no majority if it does not
exist. We investigate two computation models (depending on the type of replies
being allowed). We give algorithms to compute the minimum number of 3-queries
which are needed so that the questioner can determine the majority color and
provide tight and almost tight upper and lower bounds on the number of queries
needed in each case.Comment: 22 pages, 1 figure, conference version to appear in proceedings of
the 17th Annual International Computing and Combinatorics Conference (COCOON
2011
Fast plurality consensus in regular expanders
Pull voting is a classic method to reach consensus among vertices with
differing opinions in a distributed network: each vertex at each step takes on
the opinion of a random neighbour. This method, however, suffers from two
drawbacks. Even if there are only two opposing opinions, the time taken for a
single opinion to emerge can be slow and the final opinion is not necessarily
the initially held majority.
We refer to a protocol where 2 neighbours are contacted at each step as a
2-sample voting protocol. In the two-sample protocol a vertex updates its
opinion only if both sampled opinions are the same. Not much was known about
the performance of two-sample voting on general expanders in the case of three
or more opinions. In this paper we show that the following performance can be
achieved on a -regular expander using two-sample voting. We suppose there
are opinions, and that the initial size of the largest and second
largest opinions is respectively.
We prove that, if ,
where is the absolute second eigenvalue of matrix and
is a suitable constant, then the largest opinion wins in steps with high probability.
For almost all -regular graphs, we have for some
constant . This means that as increases we can separate an opinion
whose majority is , whereas majority is required for
constant.
This work generalizes the results of Becchetti et. al (SPAA 2014) for the
complete graph
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