2,136 research outputs found
Finding Preference Profiles of Condorcet Dimension via SAT
Condorcet winning sets are a set-valued generalization of the well-known
concept of a Condorcet winner. As supersets of Condorcet winning sets are
always Condorcet winning sets themselves, an interesting property of preference
profiles is the size of the smallest Condorcet winning set they admit. This
smallest size is called the Condorcet dimension of a preference profile. Since
little is known about profiles that have a certain Condorcet dimension, we show
in this paper how the problem of finding a preference profile that has a given
Condorcet dimension can be encoded as a satisfiability problem and solved by a
SAT solver. Initial results include a minimal example of a preference profile
of Condorcet dimension 3, improving previously known examples both in terms of
the number of agents as well as alternatives. Due to the high complexity of
such problems it remains open whether a preference profile of Condorcet
dimension 4 exists.Comment: Corrected typos, updated references, and added conclusio
Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers
We propose a Condorcet consistent voting method that we call Split Cycle.
Split Cycle belongs to the small family of known voting methods that
significantly narrow the choice of winners in the presence of majority cycles
while also satisfying independence of clones. In this family, only Split Cycle
satisfies a new criterion we call immunity to spoilers, which concerns adding
candidates to elections, as well as the known criteria of positive involvement
and negative involvement, which concern adding voters to elections. Thus, in
contrast to other clone-independent methods, Split Cycle mitigates both
"spoiler effects" and "strong no show paradoxes."Comment: 71 pages, 15 figures. Added a new explanation of Split Cycle in
Section 1, updated the caption to Figure 2, the discussion in Section 3.3,
and Remark 4.11, and strengthened Proposition 6.20 to Theorem 6.20 to cover
single-voter resolvability in addition to asymptotic resolvability. Thanks to
Nicolaus Tideman for helpful discussio
Condorcet Domains, Median Graphs and the Single Crossing Property
Condorcet domains are sets of linear orders with the property that, whenever
the preferences of all voters belong to this set, the majority relation has no
cycles. We observe that, without loss of generality, such domain can be assumed
to be closed in the sense that it contains the majority relation of every
profile with an odd number of individuals whose preferences belong to this
domain.
We show that every closed Condorcet domain is naturally endowed with the
structure of a median graph and that, conversely, every median graph is
associated with a closed Condorcet domain (which may not be a unique one). The
subclass of those Condorcet domains that correspond to linear graphs (chains)
are exactly the preference domains with the classical single crossing property.
As a corollary, we obtain that the domains with the so-called `representative
voter property' (with the exception of a 4-cycle) are the single crossing
domains.
Maximality of a Condorcet domain imposes additional restrictions on the
underlying median graph. We prove that among all trees only the chains can
induce maximal Condorcet domains, and we characterize the single crossing
domains that in fact do correspond to maximal Condorcet domains.
Finally, using Nehring's and Puppe's (2007) characterization of monotone
Arrowian aggregation, our analysis yields a rich class of strategy-proof social
choice functions on any closed Condorcet domain
Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers
We introduce a new Condorcet consistent voting method, called Split Cycle. Split Cycle belongs to the small family of known voting methods satisfying independence of clones and the Pareto principle. Unlike other methods in this family, Split Cycle satisfies a new criterion we call immunity to spoilers, which concerns adding candidates to elections, as well as the known criteria of positive involvement and negative involvement, which concern adding voters to elections. Thus, relative to other clone-independent Paretian methods, Split Cycle mitigates “spoiler effects” and “strong no show paradoxes.
Approximately Strategyproof Tournament Rules: On Large Manipulating Sets and Cover-Consistence
We consider the manipulability of tournament rules, in which n teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all binom{n}{2} matches. Prior work defines a tournament rule to be k-SNM-? if no set of ? k teams can fix the ? binom{k}{2} matches among them to increase their probability of winning by >? and asks: for each k, what is the minimum ?(k) such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) k-SNM-?(k) tournament rule exists?
A simple example witnesses that ?(k) ? (k-1)/(2k-1) for all k, and [Jon Schneider et al., 2017] conjectures that this is tight (and prove it is tight for k=2). Our first result refutes this conjecture: there exists a sufficiently large k such that no Condorcet-consistent tournament rule is k-SNM-1/2. Our second result leverages similar machinery to design a new tournament rule which is k-SNM-2/3 for all k (and this is the first tournament rule which is k-SNM-(<1) for all k).
Our final result extends prior work, which proves that single-elimination bracket with random seeding is 2-SNM-1/3 [Jon Schneider et al., 2017], in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is 2-SNM-1/3 and cover-consistent (the winner is an uncovered team with probability 1)
Guilbaud's Theorem : An early contribution to judgment aggregation
In a paper published in 1952, the French mathematician Georges-Théodule Guilbaud has generalized Arrow's impossibility result to the "logical problem of aggregation", thus anticipating the literature on abstract aggregation theory and judgment aggregation. We reconstruct the proof of Guilbaud's theorem, which is also of technical interest, because it can be seen as the first use of ultrafilters in social choice theory.Arrow's theorem, aggregation rule, judgment aggregation, logical connexions, simple game, ultrafilter.
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