Continuous representability of semiorders

In the framework of the analysis of orderings whose associated indifference relation is not transitive, we study the structure of a semiorder, paying attention to the problem of its continuous representability through a real-valued function and a positive threshold. For the case of connected topological spaces, we obtain a full characterization of the continuous representability of semiorders without extremal elements.     

Semiorders and thresholds of utility discrimination: Solving the Scott–Suppes representability problem

We find an internal characterization of the so-called Scott–Suppes representability of a semiorder, defined as the existence of a numerical representation for a semiorder through a real-valued utility function and a positive constant threshold of discrimination.     

Continuous representability of interval orders and biorders

We furnish a characterization of the representability of an interval order through a pair of continuous real-valued functions which in addition represent two total preorders associated to the given interval order. Our techniques lean on the key concept of a biorder. We introduce the concept of a natural topology for an interval order, and through such concept we extend the classical biorder approach to the continuous case.     

Representing approximate ordering and equivalence relations

The central question considered is: given appropriate precisations of the ideas of an empirical system's approximately satisfying laws of measurement with error at most ? (for some ? ≥ 0), and of a real-valued function over its domain providing an approximate representation of its basic operations and relations with error at most δ, can it be shown that satisfaction of the laws with ‘sufficiently small’ error insures numerical representability with arbitrarily small error? Positive answers are given in the cases of ordinal and nominal measurement, together with some indications of the sizes of the errors involved. Problems of extending the theory to more complex types of measurement are discussed, some open problems and conjectures are formulated, and a relation between the ‘approximate representation’ and ‘stochastic choice model’ approaches to measurement with fallible data is established.     

Difference measurement spaces

An absolute-difference measurement space is a pair (X, e) where the real-valued function e on X² satisfies conditions which are shown in the paper to be necessary and sufficient for its representability by the absolute distance on the real line. A positive-difference measurement space is a pair (X, l), where the real-valued function l on X² satisfies conditions necessary and sufficient for its representability by positive distances on the real line. The conditions imposed on e and l make these functions extensive measurements of proximity and dominance, the two basic predicates of social enquiry. Another way of treating these conditions is to translate them to the formal language of multivalued logic. The translation is easy and the sentences obtained have plausible intuitive meanings such as reflexivity, symmetry, and transitivity. The two sets of conditions thus become formal theories of proximity and dominance. Our difference measurement spaces are relational structures for the multi-valued logic and models of the two formal theories. Thus proximity and dominance are considered dichotomous in principle and the multiple truth-values represent degrees of error. We suggest adopting multivalued logic as a framework within which the problem of measurement error can be treated together with the formal axiomatization of social and phychological theories.     

A bijection between a set of lexicographic semiorders and pairs of non-crossing Dyck paths

It is a known result that the set of distinct semiorders on n elements, up to permutation, is in bijective correspondence with the set of all Dyck paths of length 2n. I generalize this result by defining a bijection between a set of lexicographic semiorders, termed simple lexicographic semiorders, and the set of all pairs of non-crossing Dyck paths of length 2n. Simple lexicographic semiorders have been used by behavioral scientists to model intransitivity of preference (e.g., Tversky, 1969). In addition to the enumeration of this set of lexicographic semiorders, I discuss applications of this bijection to decision theory and probabilistic choice.     

A selection of maximal elements under non-transitive indifferences

In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of “undominated maximals” (cf., Peris &; Subiza, 2002). Provided that an agent’s binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alternatives is finite. In the case of semiorders, proceeding in this way is the same as using Luce’s selected maximals.We present a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain types of continuous semiorders is very intuitive and accommodates the well-known “sugar example” by Luce.     

The Scott-Suppes theorem on semiorders

Luce introduced semiorders as a natural generalization of weak orders. The basic representation theorem for semiorders was established in 1958 by Scott and Suppes. In this note this theorem is given a simple contructive proof.     

Lexicographic aggregation of semiorders

It is well known that the lexicographic aggregation of complete preorders gives a complete preorder. Unfortunately, this property is no longer true when the lexicographic rule is applied to semiorders. In this paper we study some ways of defining a lexicographic aggregation of semiorders and the properties of the resulting structures.     

Biased extensive measurement: The general case

We develop a theory of biased extensive measurement which allows us to prove the existence of a ratio-scale without transitivity of indifference and with a property of homothetic invariance weaker than independence. These representations, which cover the cases of interval orders and of semiorders, reveal a unique biasing function smaller or equal to 1 that distorts extensive measurement and explains departures from its standard axioms. We interpret this biasing function as characterizing the qualitative influence of the underlying measurement process and we show that it induces a proportional indifference threshold.     

Real Interval Representations

This paper presents several necessary and sufficient conditions for real interval representability of biorders, interval orders, and semiorders. Let A and X be nonempty sets. We consider two types of interval representations for P⊂A×X. The first concerns the existence of two mappings, F: A→J_↓ and F: X→J_↑, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)∩F (x)= ∅, where J_↓ and J_↑ respectively denote the set of all real intervals that are unbounded below and the set of all real intervals that are unbounded above. The second yields two mappings, F: A→J_↓ and G: X→J_↓, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)⊂G(x). Specializations of those representations include the cases of A=X for interval orders and semiorders.     

On jnd representations of semiorders

This paper determines automorphisms and endomorphisms associated with constant jnd (just noticeable difference) semiorder representations. Building on the constant jnd case, a novel representation theorem is given for countable semiorders which provides insight into the role of more complex jnd functions. The significance of the representation problem for countable structures is clarified by discussion of relations among observations, representations, and theories. The stress is on motivation, and on general methods applicable to many measurement models.     

Hyperplane arrangements in preference modeling

A standard representation of a family B of partial orders on a given finite set X is as a set of vertices of a cube. The metric and order structures on B inherited from the cube are often used in applications. In this paper, following Stanley [(1996). Hyperplane arrangements, interval orders, and trees. Proceedings of the National Academy of Sciences of the United States of America, 93, 2620-2625], we represent relations in B by regions and cells of a hyperplane arrangement arising from numerical representations of the partial orders in B. To illustrate this approach, we establish wellgradedness of some families of generalized semiorders. Although the families of linear and weak orders are not well graded, our approach allows the recasting of such concepts as well graded families of sets.     

On A Semantic Interpretation Of Kant's Concept Of Number

What is central to the progression of a sequence is the idea of succession, which is fundamentally a temporal notion. In Kant's ontology numbers are not objects but rules (schemata) for representing the magnitude of a quantum. The magnitude of a discrete quantum 11...11 is determined by a counting procedure, an operation which can be understood as a mapping from the ordinals to the cardinals. All empirical models for numbers isomorphic to 11...11 must conform to the transcendental determination of time-order. Kant's transcendental model for number entails a procedural semantics in which the semantic value of the number-concept is defined in terms of temporal procedures. A number is constructible if and only if it can be schematized in a procedural form. This representability condition explains how an arbitrarily large number is representable and why Kant thinks that arithmetical statements are synthetic and not analytic.     

Interval representations for interval orders and semiorders

This paper discusses two types of real interval representations for interval orders and semiorders ? on a set X of arbitrary cardinality. In each type, each x in X is mapped into a real interval F(x). The first model is: x ? y iff a < b for all a in F(x) and all b in F(y). The second is: x ? y iff sup F(x) < infF(y). Necessary and sufficient countability conditions are presented for the second model for interval orders and for semiorders; simpler sets of these conditions are shown to be sufficient for the first model. Some special properties for the representations are noted, including two monotonicity properties for the semiorder representation.     

Some representation problems for semiorders

Suppose we have a number representation of a semiorder 〈A, P〉 such that aPb iff f(a)+δ(a) < f(b), for all a, b ∈ A, where δ is a nonnegative function describing the variable jnd. Such an f (here called a closed representation) may not preserve the simple order relation $\text{R}{}^1$ generated by 〈A, P〉, i.e., $\text{aR}{}^1\text{b}$ but f(a) > f(b) for some f, δ and a, b ∈ A. We show that this “paradox” can be eliminated for closed and closed interval representations. For interval representations it appears to be impossible. That is why we introduce a new type of representation (an R-representation) which is of the most general form for number representations that preserve the linear structure of the represented semiorders. The necessary and sufficient condition for an R-representation is given. We also give some independent results on the semiorder structure. Theorems are proved for semiorders of arbitrary cardinality. The Axiom of Choice is used in the proofs.     

Preference Structures and Co-comparability Graphs

Order structures such as linear orders, weak orders, semiorders and interval orders are often considered as models of a decision maker's preferences. In this paper we introduce and study new order structures characterized by their symmetric part belonging to certain classes of co-comparability graphs. We outline possible interpretations and suggest special representations of these structures and we point out their potential use for approximating relations obtained through a multicriteria aggregation procedure. We provide various characterizations of the new structures (as well as of older ones) in terms of minimal forbidden configurations and by algebraic conditions.     

A logistic approach to knowledge structures

Models for quantitative (or numerical) testing like e.g. educational testing have a relatively long tradition in psychology, while the qualitative (or nonnumerical) approach to psychometrics is more recent. The approach presented in this paper can be regarded as an attempt to integrate, to some extent, the numerical and nonnumerical fields. In numerical testing a subject is characterized by some real-valued parameter representing her level or ability. In the nonnumerical approach the knowledge state of an individual is represented by the subset of problems that the individual is capable of solving. We propose a model in which the relationship between the ability levels and the knowledge states is worked out on a probabilistic basis. The central idea is that the ability parameters and the knowledge states are not independent. A logistic model is derived which specifies the probabilities of the knowledge states conditional on the ability levels. We show that the Rasch model arises as a special case of the proposed model.     

Parametrized preference structures and some geometrical interpretation

Order structures such as linear orders, semiorders and interval orders are often used to model preferences in decision-making problems. In this paper we introduce a family of preference structures where the mutual indifference threshold belongs to a specific family parametrized by extended reals α. This family includes interval orders (α=1), tangent circle orders (α=0) and a new preference structure called ‘diamond order’ (α=−∞). All these preference relations present an asymmetric part which is shown to be always quasi-transitive and to be transitive for α > 1. Diamond orders present ‘forbidden configurations’ which can occur in the case of tangent circle orders. © 1997 John Wiley & Sons, Ltd.     

Computing the real solutions of Fleishman's equations for simulating non-normal data

Fleishman's power method is frequently used to simulate non-normal data with a desired skewness and kurtosis. Fleishman's method requires solving a system of nonlinear equations to find the third-order polynomial weights that transform a standard normal variable into a non-normal variable with desired moments. Most users of the power method seem unaware that Fleishman's equations have multiple solutions for typical combinations of skewness and kurtosis. Furthermore, researchers lack a simple method for exploring the multiple solutions of Fleishman's equations, so most applications only consider a single solution. In this paper, we propose novel methods for finding all real-valued solutions of Fleishman's equations. Additionally, we characterize the solutions in terms of differences in higher order moments. Our theoretical analysis of the power method reveals that there typically exists two solutions of Fleishman's equations that have noteworthy differences in higher order moments. Using simulated examples, we demonstrate that these differences can have remarkable effects on the shape of the non-normal distribution, as well as the sampling distributions of statistics calculated from the data. Some considerations for choosing a solution are discussed, and some recommendations for improved reporting standards are provided.