An LP-based inconsistency monitoring of pairwise comparison matrices

A distance-based inconsistency indicator, defined by the third author for the consistency-driven pairwise comparisons method, is extended to the incomplete case. The corresponding optimization problem is transformed into an equivalent linear programming problem. The results can be applied in the process of filling in the matrix as the decision maker gets automatic feedback. As soon as a serious error occurs among the matrix elements, even due to a misprint, a significant increase in the inconsistency index is reported. The high inconsistency may be alarmed not only at the end of the process of filling in the matrix but also during the completion process. Numerical examples are also provided.

An inconsistency control system based on incomplete pairwise comparison matrices

Incomplete pairwise comparison matrix was introduced by Harker in 1987 for the case in which the decision maker does not fill in the whole matrix completely due to, e.g., time limitations. However, incomplete matrices occur in a natural way even if the decision maker provides a completely filled in matrix in the end. In each step of the total n(n–1)/2, an incomplete pairwise comparison is given, except for the last one where the matrix turns into complete. Recent results on incomplete matrices make it possible to estimate inconsistency indices CR and CM by the computation of tight lower bounds in each step of the filling in process. Additional information on ordinal inconsistency is also provided. Results can be applied in any decision support system based on pairwise comparison matrices. The decision maker gets an immediate feedback in case of mistypes, possibly causing a high level of inconsistency.

The Case for Reforming Euro Area Entry Criteria

The global economic and financial crisis has raised further concerns about the euro-entry criteria, in addition to other factors, such as the effective tightening of the criteria due to the enlargement of the EU from 12 to 27 members, the highly unfavourable property of business cycle dependence, the internal inconsistency of the criteria due to the structural price level convergence of Central and Eastern European countries, and the continuous violation of the criteria by euro-area members. The interest rate criterion became a highly volatile measure. Many US metropolitan areas would fail to qualify to be members of the US monetary union by applying the currently used inflation criterion to the US. It is time to reform the criteria and to strengthen their economic rationale within the legal framework of the EU treaty. A good solution would be to relate all criteria to the average of the euro area and simultaneously to extend the compliance period from the currently considered one year to a longer period.

Pairwise comparison matrices: an empirical research

Our research focused on testing various characteristics of pairwise comparison (PC) matrices in controlled experiments. About 270 students have been involved in the test exercises and the final pool contained 450 matrices. Our team conducted experiments with matrices of different size obtained from different types of MADM problems. The matrix elements have been generated by different questioning orders, too. The cases have been divided into 18 subgroups according to the key factors to be analyzed. The testing environment made it possible to analyze the dynamics of inconsistency as the number of elements increased in a given case. Various types of inconsistency indices have been applied. The consequent behavior of the decision maker has also been analyzed in case of incomplete matrices using indicators to measure the deviation from the final ranking of alternatives and from the final score vector.

Consistency of the decision-maker in pair-wise comparisons

Most authors assume that the natural behaviour of the decision-maker is being inconsistent. This paper investigates the main sources of inconsistency and analyses methods for reducing or eliminating inconsistency. Decision support systems can contain interactive modules for that purpose. In a system with consistency control, there are three stages. First, consistency should be checked: a consistency measure is needed. Secondly, approval or rejection has to be decided: a threshold value of inconsistency measure is needed. Finally, if inconsistency is ‘high’, corrections have to be made: an inconsistency reducing method is needed. This paper reviews the difficulties in all stages. An entirely different approach is to elaborate a decision support system in order to force the decision-maker to give consistent values in each step of answering pair-wise comparison questions. An interactive questioning procedure resulting in consistent (sub) matrices has been demonstrated.

A tudásreprezentáció és a tudásmegosztás információtechnológiai megoldásai a személyazonosság-menedzsment területén (Knowledge Representation and Sharing for Identity Management Domain)

A személyazonosság-menedzsment napjaink fontos kutatási területe mind elméleti, mind gyakorlati szempontból. A szakterületen megvalósuló együttműködés, elektronikus tudásáramoltatás és csere hosszú távon csak úgy képzelhető el, hogy az azonos módon történő értelmezést automatikus eszközök támogatják. A szerző cikkében azokat a kutatási tevékenységeket foglalja össze, amelyeket - felhasználva a tudásmenedzsment, a mesterséges intelligencia és az információtechnológia eszközeit - a személyazonosság-menedzsment terület fogalmi leképezésére, leírására használt fel. Kutatási célja olyan közös fogalmi bázis kialakítása volt személyazonosság-menedzsment területre, amely lehetővé teszi az őt körülvevő multidimenzionális környezet kezelését. A kutatás kapcsolódik a GUIDE kutatási projekthez is, amelynek a szerző résztvevője. ______________ Identity management is an important research field from theoretical and practical aspects as well. The task itself is not new, identification and authentication was necessary always in public administration and business life. Information Society offers new services for citizens, which dramatically change the way of administration and results additional risks and opportunities. The goal of the demonstrated research was to formulate a common basis for the identity management domain in order to support the management of the surrounding multidimensional environment. There is a need for capturing, mapping, processing knowledge concerning identity management in order to support reusability, interoperability; to help common sharing and understanding the domain and to avoid inconsistency. The paper summarizes research activities for the identification, conceptualisation and representation of domain knowledge related to identity management, using the results of knowledge management, artificial intelligence and information technology. I utilized the experiences of Guide project, in which I participate. The paper demonstrates, that domain ontologies could offer a proper solution for identity management domain conceptualisation.

A simplified implementation of the least squares solution for pairwise comparisons matrices

This is a follow up to "Solution of the least squares method problem of pairwise comparisons matrix" by Bozóki published by this journal in 2008. Familiarity with this paper is essential and assumed. For lower inconsistency and decreased accuracy, our proposed solutions run in seconds instead of days. As such, they may be useful for researchers willing to use the least squares method (LSM) instead of the geometric means (GM) method.

On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper.

On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices

The aim of the paper is to obtain some theoretical and numerical properties of Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices (PRM). In the case of 3 × 3 PRM, a differentiable one-to-one correspondence is given between Saaty’s inconsistency ratio and Koczkodaj’s inconsistency index based on the elements of PRM. In order to make a comparison of Saaty’s and Koczkodaj’s inconsistencies for 4 × 4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n × n PRM is formulated, the elements aij (i < j) of which were randomly chosen from the ratio scale ... ... with equal probability 1/(2M − 1) and a ji is defined as 1/a ij . By statistical analysis, the empirical distributions of the maximal eigenvalues of the PRM depending on the dimension number are obtained. As the dimension number increases, the shape of distributions gets similar to that of the normal ones. Finally, the inconsistency of asymmetry is dealt with, showing a different type of inconsistency.

On pairwise comparison matrices that can be made consistent by the modification of a few elements

Pairwise comparison matrices are often used in Multi-attribute Decision Making forweighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.

Munkavásárlás vagy dolgozóbérlés? Elmélkedés a munkaviszony közgazdasági értelmezéséről. = Purchase of work or renting of workers?

Munkaszolgáltatások adásvételeként értelmezendő-e a munkaviszony vagy dolgozók bérléseként? A közgazdaság-tudomány álláspontja következetlen: jellemzően dolgozóbérlésnek deklarálja, ugyanakkor ideálisan munkaszolgáltatások adásvételeként gondolja el. Az írás érvelése szerint e következetlenség vélhető oka az abból fakadó szaktudományi dilemma, hogy a tömeges dolgozóbérlés színtereként értelmezett munkaerőpiac versenyzői egyensúlya elvi lehetetlenség. _______ Should the employment relationship be interpreted as a form of purchase of labour services or as one of renting workers? Economics typically portrays it as the latter, but idealizes it as the former. This apparent inconsistency can presumably be attributed to the dilemma arising from the theoretical incompatibility of the core economic concept of competitive equilibrium and the reality of mass renting of people in the labour market.

A new epistemic model

Meier (2012) gave a "mathematical logic foundation" of the purely measurable universal type space (Heifetz and Samet, 1998). The mathematical logic foundation, however, discloses an inconsistency in the type space literature: a finitary language is used for the belief hierarchies and an infinitary language is used for the beliefs. In this paper we propose an epistemic model to fix the inconsistency above. We show that in this new model the universal knowledgebelief space exists, is complete and encompasses all belief hierarchies. Moreover, by examples we demonstrate that in this model the players can agree to disagree Aumann (1976)'s result does not hold, and Aumann and Brandenburger (1995)'s conditions are not sufficient for Nash equilibrium. However, we show that if we substitute selfevidence (Osborne and Rubinstein, 1994) for common knowledge, then we get at that both Aumann (1976)'s and Aumann and Brandenburger (1995)'s results hold.

A Kullback-Leibler relatív entrópia függvény alkalmazása páros összehasonlítás mátrix egy prioritásvektora meghatározására (Applying the Kullback-Leibler relative entropy function for determining priorities for the pairwise comparison matrix)

A dolgozatban a döntéselméletben fontos szerepet játszó páros összehasonlítás mátrix prioritásvektorának meghatározására új megközelítést alkalmazunk. Az A páros összehasonlítás mátrix és a prioritásvektor által definiált B konzisztens mátrix közötti eltérést a Kullback-Leibler relatív entrópia-függvény segítségével mérjük. Ezen eltérés minimalizálása teljesen kitöltött mátrix esetében konvex programozási feladathoz vezet, nem teljesen kitöltött mátrix esetében pedig egy fixpont problémához. Az eltérésfüggvényt minimalizáló prioritásvektor egyben azzal a tulajdonsággal is rendelkezik, hogy az A mátrix elemeinek összege és a B mátrix elemeinek összege közötti különbség éppen az eltérésfüggvény minimumának az n-szerese, ahol n a feladat mérete. Így az eltérésfüggvény minimumának értéke két szempontból is lehet alkalmas az A mátrix inkonzisztenciájának a mérésére. _____ In this paper we apply a new approach for determining a priority vector for the pairwise comparison matrix which plays an important role in Decision Theory. The divergence between the pairwise comparison matrix A and the consistent matrix B defined by the priority vector is measured with the help of the Kullback-Leibler relative entropy function. The minimization of this divergence leads to a convex program in case of a complete matrix, leads to a fixed-point problem in case of an incomplete matrix. The priority vector minimizing the divergence also has the property that the difference of the sums of elements of the matrix A and the matrix B is n times the minimum of the divergence function where n is the dimension of the problem. Thus we developed two reasons for considering the value of the minimum of the divergence as a measure of inconsistency of the matrix A.

The Soft Budget Constraint. An introductory study to volume IV of the Life’s Work series

The author’s ideas on the soft budget constraint (SBC) were first expressed in 1976. Much progress has been made in understanding the problem over the ensuing four decades. The study takes issue with those who confine the concept to the process of bailing out loss-making socialist firms. It shows how the syndrome can appear in various organizations and forms in many spheres of the economy and points to the various means available for financial rescue. Single bailouts do not as such generate the SBC syndrome. It develops where the SBC becomes built into expectations. Special heed is paid to features generated by the syndrome in rescuer and rescuee organizations. The study reports on the spread of the syndrome in various periods of the socialist and the capitalist system, in various sectors. The author expresses his views on normative questions and on therapies against the harmful effects. He deals first with actual practice, then places the theory of the SBC in the sphere of ideas and models, showing how it relates to other theoretical trends, including institutional and behavioural economics and theories of moral hazard and inconsistency in time. He shows how far the intellectual apparatus of the SBC has spread in theoretical literature and where it has reached in the process of “canonization” by the economics profession. Finally, he reviews the main research tasks ahead.