4,616 research outputs found
Answer Set Programming and Combinatorial Voting
We show how Logic Programming with Ordered
Disjunction (LPOD), the extension of answer
set programming for handling preferences, may
be used for representing and solving collective
decision making problems. We present the
notion of combinatorial vote problem in the
context of LPOD and define various types of
vote rules, used as decision criteria for
determining optimal candidate for a group of voters.
15 min presentatio
Complexity of coalition structure generation
We revisit the coalition structure generation problem in which the goal is to
partition the players into exhaustive and disjoint coalitions so as to maximize
the social welfare. One of our key results is a general polynomial-time
algorithm to solve the problem for all coalitional games provided that player
types are known and the number of player types is bounded by a constant. As a
corollary, we obtain a polynomial-time algorithm to compute an optimal
partition for weighted voting games with a constant number of weight values and
for coalitional skill games with a constant number of skills. We also consider
well-studied and well-motivated coalitional games defined compactly on
combinatorial domains. For these games, we characterize the complexity of
computing an optimal coalition structure by presenting polynomial-time
algorithms, approximation algorithms, or NP-hardness and inapproximability
lower bounds.Comment: 17 page
Recognising Multidimensional Euclidean Preferences
Euclidean preferences are a widely studied preference model, in which
decision makers and alternatives are embedded in d-dimensional Euclidean space.
Decision makers prefer those alternatives closer to them. This model, also
known as multidimensional unfolding, has applications in economics,
psychometrics, marketing, and many other fields. We study the problem of
deciding whether a given preference profile is d-Euclidean. For the
one-dimensional case, polynomial-time algorithms are known. We show that, in
contrast, for every other fixed dimension d > 1, the recognition problem is
equivalent to the existential theory of the reals (ETR), and so in particular
NP-hard. We further show that some Euclidean preference profiles require
exponentially many bits in order to specify any Euclidean embedding, and prove
that the domain of d-Euclidean preferences does not admit a finite forbidden
minor characterisation for any d > 1. We also study dichotomous preferencesand
the behaviour of other metrics, and survey a variety of related work.Comment: 17 page
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Mathematical Programming formulations for the efficient solution of the -sum approval voting problem
In this paper we address the problem of electing a committee among a set of
candidates and on the basis of the preferences of a set of voters. We
consider the approval voting method in which each voter can approve as many
candidates as she/he likes by expressing a preference profile (boolean
-vector). In order to elect a committee, a voting rule must be established
to `transform' the voters' profiles into a winning committee. The problem
is widely studied in voting theory; for a variety of voting rules the problem
was shown to be computationally difficult and approximation algorithms and
heuristic techniques were proposed in the literature. In this paper we follow
an Ordered Weighted Averaging approach and study the -sum approval voting
(optimization) problem in the general case . For this problem we
provide different mathematical programming formulations that allow us to solve
it in an exact solution framework. We provide computational results showing
that our approach is efficient for medium-size test problems ( up to 200,
up to 60) since in all tested cases it was able to find the exact optimal
solution in very short computational times
Optimal Opinion Control: The Campaign Problem
Opinion dynamics is nowadays a very common field of research. In this article
we formulate and then study a novel, namely strategic perspective on such
dynamics: There are the usual normal agents that update their opinions, for
instance according the well-known bounded confidence mechanism. But,
additionally, there is at least one strategic agent. That agent uses opinions
as freely selectable strategies to get control on the dynamics: The strategic
agent of our benchmark problem tries, during a campaign of a certain length, to
influence the ongoing dynamics among normal agents with strategically placed
opinions (one per period) in such a way, that, by the end of the campaign, as
much as possible normals end up with opinions in a certain interval of the
opinion space. Structurally, such a problem is an optimal control problem. That
type of problem is ubiquitous. Resorting to advanced and partly non-standard
methods for computing optimal controls, we solve some instances of the campaign
problem. But even for a very small number of normal agents, just one strategic
agent, and a ten-period campaign length, the problem turns out to be extremely
difficult. Explicitly we discuss moral and political concerns that immediately
arise, if someone starts to analyze the possibilities of an optimal opinion
control.Comment: 47 pages, 12 figures, and 11 table
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