629 research outputs found
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
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
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
Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results
A preference profile is single-peaked on a tree if the candidate set can be
equipped with a tree structure so that the preferences of each voter are
decreasing from their top candidate along all paths in the tree. This notion
was introduced by Demange (1982), and subsequently Trick (1989) described an
efficient algorithm for deciding if a given profile is single-peaked on a tree.
We study the complexity of multiwinner elections under several variants of the
Chamberlin-Courant rule for preferences single-peaked on trees. We show that
the egalitarian version of this problem admits a polynomial-time algorithm. For
the utilitarian version, we prove that winner determination remains NP-hard,
even for the Borda scoring function; however, a winning committee can be found
in polynomial time if either the number of leaves or the number of internal
vertices of the underlying tree is bounded by a constant. To benefit from these
positive results, we need a procedure that can determine whether a given
profile is single-peaked on a tree that has additional desirable properties
(such as, e.g., a small number of leaves). To address this challenge, we
develop a structural approach that enables us to compactly represent all trees
with respect to which a given profile is single-peaked. We show how to use this
representation to efficiently find the best tree for a given profile for use
with our winner determination algorithms: Given a profile, we can efficiently
find a tree with the minimum number of leaves, or a tree with the minimum
number of internal vertices among trees on which the profile is single-peaked.
We also consider several other optimization criteria for trees: for some we
obtain polynomial-time algorithms, while for others we show NP-hardness
results.Comment: 44 pages, extends works published at AAAI 2016 and IJCAI 201
Mass differentiated reading skills instruction in high school
Thesis (M.Ed.)--Boston University
N.B.: Page 3 Misnumbered
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
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