Ordinal Approach to Characterizing Efficient Allocations, An

The invisible hand theorem relates nothing about the attributes of the optimal allocation vector. In this paper, we identify a convex cone of functions such that order on vectors of exogenous heterogeneity parameters induces component-wise order on allocation vectors for firms in an efficient market. By use of functional analysis, we then replace the vectors of heterogeneities with asymmetries in function attributes such that the induced component-wise order on efficient allocations still pertains. We do so through integration over a kernel in which the requisite asymmetries are embedded. Likelihood ratio order on the measures of integration is both necessary and sufficient to ensure component-wise order on efficient factor allocations across firms. Upon specializing to supermodular functions, familiar stochastic dominance orders on normalized measures of integration provide necessary and sufficient conditions for this component-wise order on efficient allocation. The analysis engaged in throughout the paper is ordinal in the sense that all conclusions drawn are robust to monotone transformations of the arguments in production.arrangement monotone; functional analysis; market structure; ordinal analysis; simplex; symmetry

The role of Walsh structure and ordinal linkage in the optimisation of pseudo-Boolean functions under monotonicity invariance.

Optimisation heuristics rely on implicit or explicit assumptions about the structure of the black-box fitness function they optimise. A review of the literature shows that understanding of structure and linkage is helpful to the design and analysis of heuristics. The aim of this thesis is to investigate the role that problem structure plays in heuristic optimisation. Many heuristics use ordinal operators; which are those that are invariant under monotonic transformations of the fitness function. In this thesis we develop a classification of pseudo-Boolean functions based on rank-invariance. This approach classifies functions which are monotonic transformations of one another as equivalent, and so partitions an infinite set of functions into a finite set of classes. Reasoning about heuristics composed of ordinal operators is, by construction, invariant over these classes. We perform a complete analysis of 2-bit and 3-bit pseudo-Boolean functions. We use Walsh analysis to define concepts of necessary, unnecessary, and conditionally necessary interactions, and of Walsh families. This helps to make precise some existing ideas in the literature such as benign interactions. Many algorithms are invariant under the classes we define, which allows us to examine the difficulty of pseudo-Boolean functions in terms of function classes. We analyse a range of ordinal selection operators for an EDA. Using a concept of directed ordinal linkage, we define precedence networks and precedence profiles to represent key algorithmic steps and their interdependency in terms of problem structure. The precedence profiles provide a measure of problem difficulty. This corresponds to problem difficulty and algorithmic steps for optimisation. This work develops insight into the relationship between function structure and problem difficulty for optimisation, which may be used to direct the development of novel algorithms. Concepts of structure are also used to construct easy and hard problems for a hill-climber

VIPSCAL: A combined vector ideal point model for preference data

In this paper, we propose a new model that combines the vector model and theideal point model of unfolding. An algorithm is developed, called VIPSCAL, thatminimizes the combined loss both for ordinal and interval transformations. As such,mixed representations including both vectors and ideal points can be obtained butthe algorithm also allows for the unmixed cases, giving either a complete idealpointanalysis or a complete vector analysis. On the basis of previous research,the mixed representations were expected to be nondegenerate. However, degeneratesolutions still occurred as the common belief that distant ideal points can be represented by vectors does not hold true. The occurrence of these distant ideal points was solved by adding certain length and orthogonality restrictions on the configuration. The restrictions can be used both for the mixed and unmixed cases in several ways such that a number of different models can be fitted by VIPSCAL.unfolding;ideal point model;vector model

Economic Explanation, Ordinality and the Adequacy of Analytic Specification

This paper examines the implicit links between models containing ordinal variables and their underlying unquantified counterparts that are necessary to make the former viable theoretical constructions. It is argued that when the underlying unquantified structure is unknown, the permissible transformations of scale applicable to the ordinal variables have to be restricted beyond that which is permitted by dint of the ordinality itself. The possibility of an underlying structure being known but unspecified is also considered. In the case of the efficiency wage model, the only usable transformations of the ordinal effort scale are those which are multiples of each other.Ordinal variable, unquantified variable, effort, efficiency wage theory

Nash bargaining in ordinal environments

We analyze the implications of Nash’s (1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (1977), we introduce a weaker independence of irrelevant alternatives axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley-Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker independence of irrelevant alternatives axiom. We also analyze the implications of other independence axioms

Homogeneity analysis with k sets of variables: An alternating least squares method with optimal scaling features

Homogeneity analysis, or multiple correspondence analysis, is usually applied to k separate variables. In this paper, it is applied to sets of variables by using sums within sets. The resulting technique is referred to as OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: (1) nominal; (2) ordinal; and (3) numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and non-linear multivariate analysis techniques are shown to be special cases of OVERALS. Disadvantages of the OVERALS method include the possibility of local minima in some complicated special cases, a lack of information on the stability of results, and its inability to handle incomplete data matrices. Means of dealing with some of these problems are suggested (i.e., an alternating least squares algorithm to solve the minimization problem). An application of the method to data from an epidemiological survey is provided

Multiple Approaches to Absenteeism Analysis

Absenteeism research has often been criticized for using inappropriate analysis. Characteristics of absence data, notably that it is usually truncated and skewed, violate assumptions of OLS regression; however, OLS and correlation analysis remain the dominant models of absenteeism research. This piece compares eight models that may be appropriate for analyzing absence data. Specifically, this piece discusses and uses OLS regression, OLS regression with a transformed dependent variable, the Tobit model, Poisson regression, Overdispersed Poisson regression, the Negative Binomial model, Ordinal Logistic regression, and the Ordinal Probit model. A simulation methodology is employed to determine the extent to which each model is likely to produce false positives. Simulations vary with respect to the shape of the dependent variable\u27s distribution, sample size, and the shape of the independent variables\u27 distributions. Actual data,based on a sample of 195 manufacturing employees, is used to illustrate how these models might be used to analyze a real data set. Results from the simulation suggest that, despite methodological expectations, OLS regression does not produce significantly more false positives than expected at various alpha levels. However, the Tobit and Poisson models are often shown to yield too many false positives. A number of other models yield less than the expected number of false positives, thus suggesting that they may serve well as conservative hypothesis tests

DISTANCE MEASURES IN AGGREGATING PREFERENCE DATA

The aim of this paper is to present aggregation methods of individual preferences scores by means of distance measures. Three groups of distance measures are discussed: measures  which use preference distributions for all pairs of objects (e.g. Kemeny’s measure, Bogart’s measure), distance measures based on ranking data (e.g. Spearman distance, Podani distance) and distance measures using permissible transformations to ordinal scale (GDM2 distance). Adequate distance formulas are presented and the aggregation of individual preference by using separate distance measures was carried out with the use of the R program