74,864 research outputs found
Minimal Stable Sets in Tournaments
We propose a systematic methodology for defining tournament solutions as
extensions of maximality. The central concepts of this methodology are maximal
qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy
of tournament solutions, which encompasses the top cycle, the uncovered set,
the Banks set, the minimal covering set, the tournament equilibrium set, the
Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new
tournament solution, the minimal extending set, which is conjectured to refine
both the minimal covering set and the Banks set.Comment: 29 pages, 4 figures, changed conten
A 2.75-Approximation Algorithm for the Unconstrained Traveling Tournament Problem
A 2.75-approximation algorithm is proposed for the unconstrained traveling
tournament problem, which is a variant of the traveling tournament problem. For
the unconstrained traveling tournament problem, this is the first proposal of
an approximation algorithm with a constant approximation ratio. In addition,
the proposed algorithm yields a solution that meets both the no-repeater and
mirrored constraints. Computational experiments show that the algorithm
generates solutions of good quality.Comment: 12 pages, 1 figur
Who Can Win a Single-Elimination Tournament?
A single-elimination (SE) tournament is a popular way to select a winner in
both sports competitions and in elections. A natural and well-studied question
is the tournament fixing problem (TFP): given the set of all pairwise match
outcomes, can a tournament organizer rig an SE tournament by adjusting the
initial seeding so that their favorite player wins? We prove new sufficient
conditions on the pairwise match outcome information and the favorite player,
under which there is guaranteed to be a seeding where the player wins the
tournament. Our results greatly generalize previous results. We also
investigate the relationship between the set of players that can win an SE
tournament under some seeding (so called SE winners) and other traditional
tournament solutions. In addition, we generalize and strengthen prior work on
probabilistic models for generating tournaments. For instance, we show that
\emph{every} player in an player tournament generated by the Condorcet
Random Model will be an SE winner even when the noise is as small as possible,
; prior work only had such results for . We also establish new results for significantly more
general generative models.Comment: A preliminary version appeared in Proceedings of the 30th AAAI
Conference on Artificial Intelligence (AAAI), 201
Consensus theories: an oriented survey
This article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly – but not exclusively – concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de Mathématique Sociale (CAMS).Consensus theories ; Arrowian results ; aggregation rules ; metric consensus rules ; median ; tournament solutions ; restricted domains ; lower valuations ; median semilattice ; complexity
Recognizing Members of the Tournament Equilibrium Set is NP-hard
A recurring theme in the mathematical social sciences is how to select the
"most desirable" elements given a binary dominance relation on a set of
alternatives. Schwartz's tournament equilibrium set (TEQ) ranks among the most
intriguing, but also among the most enigmatic, tournament solutions that have
been proposed so far in this context. Due to its unwieldy recursive definition,
little is known about TEQ. In particular, its monotonicity remains an open
problem up to date. Yet, if TEQ were to satisfy monotonicity, it would be a
very attractive tournament solution concept refining both the Banks set and
Dutta's minimal covering set. We show that the problem of deciding whether a
given alternative is contained in TEQ is NP-hard.Comment: 9 pages, 3 figure
Robust Bounds on Choosing from Large Tournaments
Tournament solutions provide methods for selecting the "best" alternatives
from a tournament and have found applications in a wide range of areas.
Previous work has shown that several well-known tournament solutions almost
never rule out any alternative in large random tournaments. Nevertheless, all
analytical results thus far have assumed a rigid probabilistic model, in which
either a tournament is chosen uniformly at random, or there is a linear order
of alternatives and the orientation of all edges in the tournament is chosen
with the same probabilities according to the linear order. In this work, we
consider a significantly more general model where the orientation of different
edges can be chosen with different probabilities. We show that a number of
common tournament solutions, including the top cycle and the uncovered set, are
still unlikely to rule out any alternative under this model. This corresponds
to natural graph-theoretic conditions such as irreducibility of the tournament.
In addition, we provide tight asymptotic bounds on the boundary of the
probability range for which the tournament solutions select all alternatives
with high probability.Comment: Appears in the 14th Conference on Web and Internet Economics (WINE),
201
Ordinally consistent tournament solutions
A set ranking method assigns to each tournament on a given set an ordering of the subsets of that set. Such a method is consistent if (i) the items in the set are ranked in the same order as the sets of items they beat and (ii) the ordering of the items fully determines the ordering of the sets of items. We describe two consistent set ranking methods
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