195 research outputs found
Primary Facets Of Order Polytopes
Mixture models on order relations play a central role in recent
investigations of transitivity in binary choice data. In such a model, the
vectors of choice probabilities are the convex combinations of the
characteristic vectors of all order relations of a chosen type. The five
prominent types of order relations are linear orders, weak orders, semiorders,
interval orders and partial orders. For each of them, the problem of finding a
complete, workable characterization of the vectors of probabilities is
crucial---but it is reputably inaccessible. Under a geometric reformulation,
the problem asks for a linear description of a convex polytope whose vertices
are known. As for any convex polytope, a shortest linear description comprises
one linear inequality per facet. Getting all of the facet-defining inequalities
of any of the five order polytopes seems presently out of reach. Here we search
for the facet-defining inequalities which we call primary because their
coefficients take only the values -1, 0 or 1. We provide a classification of
all primary, facet-defining inequalities of three of the five order polytopes.
Moreover, we elaborate on the intricacy of the primary facet-defining
inequalities of the linear order and the weak order polytopes
Choice by Lexicographic Semiorders
We propose an extension of Tversky's lexicographic semiorder to a model of boundedly rational choice. We explore the connection with sequential rationalisability of choice, and we provide axiomatic characterisations of both models in terms of observable choice data.lexicographic semiorders, bounded rationality, revealed preference, choice
Choice by lexicographic semiorders
In Tversky's (1969) model of a lexicographic semiorder, preference is generated by the sequential application of numerical criteria, by declaring an alternative x better than an alternative y if the first criterion that distinguishes between x and y ranks x higher than y by an amount exceeding a fixed threshold. We generalize this idea to a fully-fledged model of boundedly rational choice. We explore the connection with sequential rationalizability of choice (Apesteguia and Ballester 2009, Manzini and Mariotti 2007), and we provide axiomatic characterizations of both models in terms of observable choice data.Lexicographic semiorders, bounded rationality, revealed preference, choice
Balance constants for Coxeter groups
The - Conjecture, originally formulated in 1968, is one of the
best-known open problems in the theory of posets, stating that the balance
constant (a quantity determined by the linear extensions) of any non-total
order is at least . By reinterpreting balance constants of posets in terms
of convex subsets of the symmetric group, we extend the study of balance
constants to convex subsets of any Coxeter group. Remarkably, we conjecture
that the lower bound of still applies in any finite Weyl group, with new
and interesting equality cases appearing.
We generalize several of the main results towards the - Conjecture
to this new setting: we prove our conjecture when is a weak order interval
below a fully commutative element in any acyclic Coxeter group (an
generalization of the case of width-two posets), we give a uniform lower bound
for balance constants in all finite Weyl groups using a new generalization of
order polytopes to this context, and we introduce generalized semiorders for
which we resolve the conjecture.
We hope this new perspective may shed light on the proper level of generality
in which to consider the - Conjecture, and therefore on which methods
are likely to be successful in resolving it.Comment: 27 page
Four Variations on Graded Posets
We explore the enumeration of some natural classes of graded posets,
including all graded posets, (2+2)- and (3+1)-avoiding graded posets,
(2+2)-avoiding graded posets, and (3+1)-avoiding graded posets. We obtain
enumerative and structural theorems for all of them. Along the way, we discuss
a situation when we can switch between enumeration of labeled and unlabeled
objects with ease, generalize a result of Postnikov and Stanley from the theory
of hyperplane arrangements, answer a question posed by Stanley, and see an old
result of Klarner in a new light.Comment: 28 page
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