Additive and multiplicative properties of scoring methods for preference aggregation

The paper reviews some additive and multiplicative properties of ranking procedures used for generalized tournaments with missing values and multiple comparisons. The methods analysed are the score, generalised row sum and least squares as well as fair bets and its variants. It is argued that generalised row sum should be applied not with a fixed parameter, but a variable one proportional to the number of known comparisons. It is shown that a natural additive property has strong links to independence of irrelevant matches, an axiom judged unfavourable when players have different opponents.

On the additivity of preference aggregation methods

The paper reviews some axioms of additivity concerning ranking methods used for generalized tournaments with possible missing values and multiple comparisons. It is shown that one of the most natural properties, called consistency, has strong links to independence of irrelevant comparisons, an axiom judged unfavourable when players have different opponents. Therefore some directions of weakening consistency are suggested, and several ranking methods, the score, generalized row sum and least squares as well as fair bets and its two variants (one of them entirely new) are analysed whether they satisfy the properties discussed. It turns out that least squares and generalized row sum with an appropriate parameter choice preserve the relative ranking of two objects if the ranking problems added have the same comparison structure.

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.