Some advances in mathematical models for preference relations

Preference relations, and their modeling, have played a crucial role in both social sciences and applied mathematics. A special category of preference relations is represented by cardinal preference relations, which are nothing other than relations which can also take into account the degree of relation. Preference relations play a pivotal role in most of multi criteria decision making methods and in the operational research. This thesis aims at showing some recent advances in their methodology. Actually, there are a number of open issues in this field and the contributions presented in this thesis can be grouped accordingly. The first issue regards the estimation of a weight vector given a preference relation. A new and efficient algorithm for estimating the priority vector of a reciprocal relation, i.e. a special type of preference relation, is going to be presented. The same section contains the proof that twenty methods already proposed in literature lead to unsatisfactory results as they employ a conflicting constraint in their optimization model. The second area of interest concerns consistency evaluation and it is possibly the kernel of the thesis. This thesis contains the proofs that some indices are equivalent and that therefore, some seemingly different formulae, end up leading to the very same result. Moreover, some numerical simulations are presented. The section ends with some consideration of a new method for fairly evaluating consistency. The third matter regards incomplete relations and how to estimate missing comparisons. This section reports a numerical study of the methods already proposed in literature and analyzes their behavior in different situations. The fourth, and last, topic, proposes a way to deal with group decision making by means of connecting preference relations with social network analysis.

A flexible consensus scheme for multicriteria group decision making under linguistic assessments

The present paper proposes a flexible consensus scheme for group decision making, which allows one to obtain a consistent collective opinion, from information provided by each expert in terms of multigranular fuzzy estimates. It is based on a linguistic hierarchical model with multigranular sets of linguistic terms, and the choice of the most suitable set is a prerogative of each expert. From the human viewpoint, using such model is advantageous, since it permits each expert to utilize linguistic terms that reflect more adequately the level of uncertainty intrinsic to his evaluation. From the operational viewpoint, the advantage of using such model lies in the fact that it allows one to express the linguistic information in a unique domain, without losses of information, during the discussion process. The proposed consensus scheme supposes that the moderator can interfere in the discussion process in different ways. The intervention can be a request to any expert to update his opinion or can be the adjustment of the weight of each expert`s opinion. An optimal adjustment can be achieved through the execution of an optimization procedure that searches for the weights that maximize a corresponding soft consensus index. In order to demonstrate the usefulness of the presented consensus scheme, a technique for multicriteria analysis, based on fuzzy preference relation modeling, is utilized for solving a hypothetical enterprise strategy planning problem, generated with the use of the Balanced Scorecard methodology. (C) 2009 Elsevier Inc. All rights reserved.

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces. (C) 2002 Elsevier Science B.V. All rights reserved.

Lexicographic decomposition of chains and the concept of a planar chain

In an earlier paper [Journal of Mathematical Economics, 37 (2002) 17-38], we proved that if a preference relation on a commodity space is non-representable by a real-valued function then that chain is necessarily a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain. In this paper, we study the class of planar chains, the simplest example of which is the Debreu chain (R-2, <(l)). (C) 2002 Elsevier Science B.V. All rights reserved.

Some things couples always wanted to know about stable matchings (but were afraid to ask)

It is well-known that couples that look jointly for jobs in the same centralized labor market may cause instabilities. We demonstrate that for a natural preference domain for couples, namely the domain of responsive preferences, the existence of stable matchings can easily be established. However, a small deviation from responsiveness in one couple's preference relation that models the wish of a couple to be closer together may already cause instability. This demonstrates that the nonexistence of stable matchings in couples markets is not a singular theoretical irregularity. Our nonexistence result persists even when a weaker stability notion is used that excludes myopic blocking. Moreover, we show that even if preferences are responsive there are problems that do not arise for singles markets. Even though for couples markets with responsive preferences the set of stable matchings is nonempty, the lattice structure that this set has for singles markets does not carry over. Furthermore we demonstrate that the new algorithm adopted by the National Resident Matching Program to fill positions for physicians in the United States may cycle, while in fact a stable matchings does exist, and be prone to strategic manipulation if the members of a couple pretend to be single.

Stochastic Choice and Consideration Sets

We model a boundedly rational agent who suffers from limited attention. The agent considers each feasible alternative with a given (unobservable) probability, the attention parameter, and then chooses the alternative that maximises a preference relation within the set of considered alternatives. We show that this random choice rule is the only one for which the impact of removing an alternative on the choice probability of any other alternative is asymmetric and menu independent. Both the preference relation and the attention parameters are identi fied uniquely by stochastic choice data.

Choice, Deferral and Consistency

In this paper we study decision making in situations where the individual’s preferences are not assumed to be complete. First, we identify conditions that are necessary and sufficient for choice behavior in general domains to be consistent with maximization of a possibly incomplete preference relation. In this model of maximally dominant choice, the agent defers/avoids choosing at those and only those menus where a most preferred option does not exist. This allows for simple explanations of conflict-induced deferral and choice overload. It also suggests a criterion for distinguishing between indifference and incomparability based on observable data. A simple extension of this model also incorporates decision costs and provides a theoretical framework that is compatible with the experimental design that we propose to elicit possibly incomplete preferences in the lab. The design builds on the introduction of monetary costs that induce choice of a most preferred feasible option if one exists and deferral otherwise. Based on this design we found evidence suggesting that a quarter of the subjects in our study had incomplete preferences, and that these made significantly more consistent choices than a group of subjects who were forced to choose. The latter effect, however, is mitigated once data on indifferences are accounted for.

Biased representation of homothetic preferences on homogeneous sets

In the homogeneous case of one-dimensional objects, we show that any preference relation that is positive and homothetic can be represented by a quantitative utility function and unique bias. This bias may favor or disfavor the preference for an object. In the first case, preferences are complete but not transitive and an object may be preferred even when its utility is lower. In the second case, preferences are asymmetric and transitive but not negatively transitive and it may not be sufficient for an object to have a greater utility for be preferred. In this manner, the bias reflects the extent to which preferences depart from the maximization of a utility function.

Comparison and evaluation of states of health

Starting from a finite or countable set of states of health, and assumingthe existence of an objective transitive preference relation on that set,we propose a way of performing interpersonal comparisons of states ofhealth. In so doing, we first consider the population divided into types,and consider that two individuals of a different type have a comparablestate of health whenever they sit at the same centile of their respectivetype. A way of comparing and evaluating states of health for differentgroups is then proposed and rationalized. This can be viewed as both analternative and an extension of the traditional QALY approach.

Core Retionalizability in Two-Agent Exchange Economies.

We provide a characterization of selection correspondences in two-person exchange economies that can be core rationalized in the sense that there exists a preference profile with some standard properties that generates the observed choices as the set of core elements of the economy for any given initial endowment vector. The approach followed in this paper deviates from the standard rational choice model in that a rationalization in terms of a profile of individual orderings rather than in terms of a single individual or social preference relation is analyzed.

Top trading with fixed tie-breaking in markets with indivisible goods

We study markets with indivisible goods where monetary compensations are not possible. Each individual is endowed with an object and a preference relation over all objects. When preferences are strict, Gale's top trading cycle algorithm finds the unique core allocation. When preferences are not necessarily strict, we use an exogenous profile of tie-breakers to resolve any ties in individuals' preferences and apply Gale's top trading cycle algorithm for the resulting profile of strict preferences. We provide a foundation of these simple extensions of Gale's top trading cycle algorithm from strict preferences to weak preferences. We show that Gale's top trading cycle algorithm with fixed tie-breaking is characterized by individual rationality, strategy-proofness, weak efficiency, non-bossiness, and consistency. Our result supports the common practice in applications to break ties in weak preferences using some fixed exogenous criteria and then to use a 'good and simple' rule for the resulting strict preferences. This reinforces the market-based approach even in the presence of indifferences because always competitive allocations are chosen.

Single-basined choice

Single-basined preferences generalize single-dipped preferences by allowing for multiple worst elements. These preferences have played an important role in areas such as voting, strategy-proofness and matching problems. We examine the notion of single-basinedness in a choice-theoretic setting. In conjunction with independence of irrelevant alternatives, single-basined choice implies a structure that conforms to the motivation underlying our definition. We also establish the consequenes of requiring single-basined choice correspondences to be upper semicontinuous, and of the revealed preference relation to be Suzumura consistent. Journal of Economic Literature.

Desempenho dos serviços de atenção primária do município de Natal/RN para o diagnóstico e controle da tuberculose

Tuberculosis (TB) is one of the most important health problems being faced worldwide. In Brazil, the responsibility for the actions of to diagnosis and control of this disease was transferred to the municipalities within the Primary Health Care (PHC), aiming at improvement in epidemiological indicators, requiring reorientation of the practice of family health teams and requiring methodologies to analyze the extent to which components of the PHC are being achieved. Thus, this study aims to analyze the performance of primary care services in the city of Natal-RN for the diagnosis and control of TB, from the perspective of health professionals (doctors and nurses). The study is descriptive, cross-sectional and quantitative. Data collection was conducted from March to July 2011 and involved 121 health professionals working in 52 health units (family health unit, basic health unit and mixed units). The instrument is structured based on the Primary Care Assessment Tool (PCAT), validated and adapted to assess attention to TB in Brazil, and includes questions regarding the Structure and Process components of health services. For quantitative analysis, it was constructed indicators, whose response patterns are followed according to the Likert scale between one and five, which meant the degree of preference relation (or agreement) of the claims. Values between 1 and 3 were considered unsatisfactory for the indicator, between 3 and less than 4, regular, and between 4 and 5, satisfactory. With regard to inputs and equipment, the units had satisfactory condition for form (  = 4.26), consultation (  = 4.02) and basic basket (  = 4.24); regular condition to pot (  = 3.56) and unsatisfactory conditions for transportation tickets (  = 1.50) and sputum smear microscopy (  = 2.42) and X-rays (  = 1.07). In relation to actions, there was satisfactory development for those focused on the individual patient. Actions aimed at the collective level, as the search for respiratory symptoms (RS), monitoring of contacts and guidelines for the community ranged from regular to unsatisfactory (  = 3.16 -  = 1.34). With regard to training, 94,2% received training to identify RS. As regards the time for diagnosis, the median time elapsed between the identification of RS and the beginning of treatment it was 22 days. In relation to the difficulties faced by professionals in the diagnosis of TB, 56,2% reported that they are related only to health services, especially for the failure in the rearguard laboratory and in the specialized services reference, the lack of human and material resources and low performing an active search. The professionals perceive the performance of diagnosis and control of TB, permeated with limitations and barriers to organizational and operational character of various sizes, emerging the need for effective coordination of various sectors and key stakeholders of TB care, to adoption of a new intersectoral strategies that aim to increase the responsiveness of the PHC, providing the best performance in service delivery to the user, family and community, and ensuring effective action and resolving the needs of this population group.

As ações de controle da tuberculose na atenção primária à saúde: a visão do doente

Tuberculosis is a disease of great impact on the world context today. In Brazil, the disease management was directed to the Primary Health Care, due to the determination of the Ministry of Health to decentralize health actions for primary care. Thus, since the actions of diagnosis, treatment and control of the disease should happen in this context, however, there are still many barriers that may hinder the realization of these determinations. This study aims to analyze the development of tuberculosis control activities conducted in the services of primary health care from the patient's vision. This is a descriptive, cross-sectional and quantitative study. The population consists of 517 tuberculosis patients treated in units of Primary Health Care in the city of Natal-RN; the sample consists of 93 TB patients. The collect instrument is structured, based in The Primary Care Assessment Tool (PCAT), validated in Brazil and adapted to assess attention to TB in Brazil, with modifications. This instrument was divided into blocks: the first one describes the socio-demographic information of patients with TB and the second one describes the health services working in control, diagnosis and treatment of TB, and includes issues related to the dimensions of primary care: access, bond, services, coordination of care, guidance to the community and family focus. For quantitative analysis, were built indicators for each item of the instrument. The response patterns are followed according to the Likert scale, which was assigned a value between one and five meant that the degree of preference relation (or agreement) of the statements. Values between 1 and 3 were considered unsatisfactory for the indicator, between 3 and less than 4, regular, and between 4 and 5, satisfactory. The results indicate that 62.37% of patients are male, 27.96% aged 41 to 50 years old, and 34.41% unemployed, with low education and low family income. It was found that the reference hospital services are the front door to the patient (59.14%), and are also the local diagnosis of the disease (72.04%). On access, the conditions satisfactory found are: the number of times the patients need to pick up the health care issue, the marking and the facility to get a consultancy in the HS, assistance provided without harm to the individual's attendance labor and facilities related to the proximity between the residence and services; were considered unsatisfactory conditions related to travel to the HS, and on hours and days of operation of services. As for the cast of services were satisfactory and regular actions related to the request for examination to become viable in the first HS, the availability of pot to perform smear and medicines for the treatment, as well as consultations control and receiving information about the disease and the treatment performed; it is considered unsatisfactory the performance of the home care for patients with TB by the HS that acts as a front door, for implementation of the Directly Observed Treatment (DOT), home visits during treatment, the provision of transportation allowance to the patient and the existence of groups for TB patients. Regarding the coordination of care, resulted in regular the action of referring the patient to other HS to obtain examinations, and as unsatisfactory referral to obtain medications. The relationship bond between patient and health team were considered satisfactory in the majority or regular. As for the family and community focus, is satisfactory only the indicator relating to questions from professionals to the patient about the existence of respiratory symptoms in the family. It is considered that there is need for greater commitment from government entities to the incentives required to TB control, as well as the availability of necessary inputs and training of human resources working in the PHC in the ongoing quest to strengthen primary care, as a place of broader host needs to contact the user with the actions and health professionals. It is recommended the adoption of management mechanisms possible to expand the capacity of the health PHC, promoting the service delivery to the user and ensuring attention to population health.

Predação em capítulos de picão-preto e sua relação com o tamanho e o grau de agrupamento das plantas

Com o objetivo de verificar possíveis correlações entre níveis de predação de capítulos de B. pilosa e o tamanho das plantas, bem como com o seu grau de agrupamento, o presente trabalho foi desenvolvido em áreas ruderais nos arredores da cidade de Botucatu, SP. em cada coleta, foram obtidos 15 indivíduos em fase reprodutiva, sendo dez deles provenientes de agrupamentos e cinco isolados, no período de março a setembro de 1993, totalizando seis coletas. Cada planta foi caracterizada quanto a parâmetros biométricos por meio de mensurações, contagens e determinação da biomassa das diferentes estruturas, avaliando-se também a ocorrência de predação nos capítulos. Nas duas condições de agrupamento, o tamanho das plantas foi altamente variável havendo, porém, maior freqüência nas menores classes de tamanho. de modo geral, não houve diferença significativa entre plantas agrupadas e isoladas no que se refere aos parâmetros biométricos analisados. Plantas maiores produziram maior número de capítulos e o nível de predação correlacionou-se positivamente com o tamanho das plantas, independentemente do grau de agrupamento das populações. A distribuição agrupada não condicionou, portanto, maiores níveis de predação, uma vez que plantas maiores dos dois grupos foram preferencialmente atacadas. Isto corrobora a Hipótese do Vigor proposta para explicar relações de preferência entre plantas e seus herbívoros.