33 research outputs found
General solutions for choice sets: The Generalized Optimal-Choice Axiom set
In this paper we characterize the existence of best choices of arbitrary binary relations over non finite sets of alternatives, according to the Generalized Optimal-Choice Axiom condition introduced by Schwartz. We focus not just in the best choices of a single set X, but rather in the best choices of all the members of a family K of subsets of X. Finally we generalize earlier known results concerning the existence (or the characterization) of maximal elements of binary relations on compact subsets of a given space of alternatives.Generalized Optimal-Choice Axiom; maximal elements; acyclicity; consistency; ≻-upper compactness
A characterization of the existence of generalized stable sets
The generalized stable sets solution introduced by van Deemen (1991) as a generalization of the von Neumann and Morgenstern stable sets solution for abstract systems. If such a solution concept exists, then it is equivalent to the admissible set appeared in game theory literature by Kalai and Schmeidler (1977). The purpose of this note is to provide a characterization for the existence of the generalized stable sets solution
Szpilrajn-type theorems in economics
The Szpilrajn "constructive type" theorem on
extending binary relations,
or its generalizations by Dushnik and Miller [10],
is one of the best known theorems in
social sciences and mathematical economics.
Arrow [1], Fishburn [11],
Suzumura [22], Donaldson and Weymark [8] and
others
utilize Szpilrajn's
Theorem and the Well-ordering principle to obtain more general "existence type" theorems
on
extending binary relations. Nevertheless, we are generally interested not only
in the existence of linear extensions of a binary relation R, but in something more:
the conditions of the preference sets and the properties which satisfies
to be "inherited" when one passes to any member of some
\textquotedblleft interesting\textquotedblright
family of linear extensions of R.
Moreover,
in extending a preference relation , the problem will often be how to incorporate some additional preference data with a minimum
of disruption of the existing structure or how to extend the relation so that some desirable new condition is fulfilled. The key to addressing these kinds of problems is
the szpilrajn constructive method.
In this
paper, we give two general
"constructive type" theorems on
extending binary relations, a
Szpilrajn type and a Dushnik-Miller
type theorem, which generalize and give a "constructive type" version of all the well known extension
theorems in the literature
A Solution to the Completion Problem for Quasi-Pseudometric Spaces
The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given T0 quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness
A note on the Gao et al. (2019) uniform mixture model in the case of regression
© 2019, The Author(s). We extend the uniform mixture model of Gao et al. (Ann Oper Res, 2019. https://doi.org/10.1007/s10479-019-03236-9) to the case of linear regression. Gao et al. (Ann Oper Res, 2019. https://doi.org/10.1007/s10479-019-03236-9) proposed that to characterize the probability distributions of multimodal and irregular data observed in engineering, a uniform mixture model can be used. This model is a weighted combination of multiple uniform distribution components. This case is of empirical interest since, in many instances, the distribution of the error term in a linear regression model cannot be assumed unimodal. Bayesian methods of inference organized around Markov chain Monte Carlo are proposed. In a Monte Carlo experiment, significant efficiency gains are found in comparison to least squares justifying the use of the uniform mixture model
Characterization of the Generalized Top-Choice Assumption (Smith) set
In this paper, I give a characterization of the Generalized Top-Choice Assumption set of a binary relation in terms of choice
from minimal negative consistent superrelations. This result provides a characterization of Schwart's set
in tournaments