79 research outputs found
A quantitative Gibbard-Satterthwaite theorem without neutrality
Recently, quantitative versions of the Gibbard-Satterthwaite theorem were
proven for alternatives by Friedgut, Kalai, Keller and Nisan and for
neutral functions on alternatives by Isaksson, Kindler and Mossel.
We prove a quantitative version of the Gibbard-Satterthwaite theorem for
general social choice functions for any number of alternatives. In
particular we show that for a social choice function on
alternatives and voters, which is -far from the family of
nonmanipulable functions, a uniformly chosen voter profile is manipulable with
probability at least inverse polynomial in , , and .
Removing the neutrality assumption of previous theorems is important for
multiple reasons. For one, it is known that there is a conflict between
anonymity and neutrality, and since most common voting rules are anonymous,
they cannot always be neutral. Second, virtual elections are used in many
applications in artificial intelligence, where there are often restrictions on
the outcome of the election, and so neutrality is not a natural assumption in
these situations.
Ours is a unified proof which in particular covers all previous cases
established before. The proof crucially uses reverse hypercontractivity in
addition to several ideas from the two previous proofs. Much of the work is
devoted to understanding functions of a single voter, and in particular we also
prove a quantitative Gibbard-Satterthwaite theorem for one voter.Comment: 46 pages; v2 has minor structural changes and adds open problem
The Geometry of Manipulation — A Quantitative Proof of the Gibbard Satterthwaite Theorem
We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10−4∈2 n −3 q −30, where ∈ is the minimal statistical distance between f and the family of dictator functions.
Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely.
Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies
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
A Topological Proof of The Gibbard-Satterthwaite Theorem
We give a new proof of the Gibbard-Satterthwaite Theorem. We construct two
topological spaces: one for the space of preference profiles and another for
the space of outcomes. We show that social choice functions induce continuous
mappings between the two spaces. By studying the properties of this mapping, we
prove the theorem
A Generalized Probabilistic Gibbard-Satterthwaite Theorem
Friedgut, Kalai, and Nisan have proved that social choice functions can be successfully manipulated by random preference reordering with non- negligible probability. However, their results require two restrictions: the social choice function must be neutral, and the election must have at most 3 alternatives. In this thesis we focus on removing the latter restriction and generalizing the results to elections with any number of candidates. We also provide a survey of related work analyzing and comparing results from a number of authors
Voting with Coarse Beliefs
The classic Gibbard-Satterthwaite theorem says that every strategy-proof
voting rule with at least three possible candidates must be dictatorial.
Similar impossibility results hold even if we consider a weaker notion of
strategy-proofness where voters believe that the other voters' preferences are
i.i.d.~(independent and identically distributed). In this paper, we take a
bounded-rationality approach to this problem and consider a setting where
voters have "coarse" beliefs (a notion that has gained popularity in the
behavioral economics literature). In particular, we construct good voting rules
that satisfy a notion of strategy-proofness with respect to coarse
i.i.d.~beliefs, thus circumventing the above impossibility results
A Smooth Transition from Powerlessness to Absolute Power
We study the phase transition of the coalitional manipulation problem for
generalized scoring rules. Previously it has been shown that, under some
conditions on the distribution of votes, if the number of manipulators is
, where is the number of voters, then the probability that a
random profile is manipulable by the coalition goes to zero as the number of
voters goes to infinity, whereas if the number of manipulators is
, then the probability that a random profile is manipulable
goes to one. Here we consider the critical window, where a coalition has size
, and we show that as goes from zero to infinity, the limiting
probability that a random profile is manipulable goes from zero to one in a
smooth fashion, i.e., there is a smooth phase transition between the two
regimes. This result analytically validates recent empirical results, and
suggests that deciding the coalitional manipulation problem may be of limited
computational hardness in practice.Comment: 22 pages; v2 contains minor changes and corrections; v3 contains
minor changes after comments of reviewer
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