Rankings, representing preferences over a set of candidates, are widely used in many information systems, e.g., group decision making. It is of great importance to evaluate the consensus of the obtained rankings from multiple agents. There is often no ground truth available for a ranking task. An overall measure of the consensus degree enables us to have a clear cognition about the ranking data. Moreover, it could provide a quantitative indicator for consensus comparison between groups and further improvement of a ranking system. In this paper, a novel consensus quantifying approach, without the need for any correlation or distance functions, is proposed based on a concept of q-support patterns of rankings. The q-support patterns represent the commonality embedded in a set of rankings. A method for detecting outliers in a set of rankings is naturally derived from the proposed consensus quantifying approach. Experimental studies are conducted to demonstrate the effectiveness of the proposed approach.
We introduce a general framework for measuring the degree of diversity in the preferences held by the members of a group. We formalise and investigate three specific approaches within that framework: diversity as the range of distinct views held, diversity as aggregate distance between individual views, and diversity as distance of the group's views to a single compromise view. While similarly attractive from an intuitive point of view, the three approaches display significant differences when analysed using both the axiomatic method and empirical studies. This paper surveys approaches to preference diversity measurement. Applying preference diversity axiomatics, a generalization of the Alcalde-Unzu and Vorsatz (2016) criterion, is developed. It is shown that all previously used indices violate this criterion. Two new indices (geometric mean based and leximaxbased)are developed that satisfy a new criterion. Leximax-based orders act as a polarization index and are compared with ’s (2015) polarization index. The paper concludes by formulating a new open question: the preference profile reconstruction conjecture. We consider measuring the degree of homogeneity for preference-approval profiles which include the approval information for the alternatives as well as the rankings of them. A distance-based approach is followed to measure the disagreement for any given two preference-approvals. Under the condition that a proper metric is used, we propose a measure of consensus which is robust to some extensions of the ordinal framework. This paper also shows that there exists a limit for increasing the homogeneity level in a group of individuals by simply replicating their preference-approvals. In this paper, we axiomatically study how to measure the similarity of preferences in a group of individuals. For simplicity, we refer to this as the cohesiveness. First, we provide axioms that characterize a family of linear and additive measures whose intersection is a partial ordinal criterion similar to first order stochastic dominance. The introduction of some additional properties isolates a one-parameter subfamily. This parameter evaluates the effect on the cohesiveness if one individual changes his ranking on a single pair of objects, as a function of how many of the remaining individuals in the group rank the first object over the second and vice versa. Finally, we characterize the focal measures of this subfamily separately showing that they coincide with measures constructed using two, at first sight, totally different approaches suggested in the literature. Motivation. Human beings do not live in isolation and they have to take many decisions collectively. Examples include the election of firm representatives, the decision of where to build a new school, and the task of how to share natural resources. Equally, the satisfaction of a single individual usually depends on the performance of the group; just think of the problem of shirking in team production. To obtain a good collective performance and, as a consequence, a high individual satisfaction, it is important that collective decisions are taken with consensus. This is because in many instances, it is not beneficial for the society as a whole if the decision is imposed by a subset of the individuals—even if this subset includes more than half of the collective— as there may be other alternatives that are more accepted by the rest of the members and that increase the overall satisfaction. In this paper we analyze the consensus in groups of decision makers that rank alternatives by means of weak orders. We have introduced the class of weighted Kemeny distances on weak orders for taking into account where the disagreements occur, and we have analyzed the properties of the associated consensus measures.  In this chapter we focus our attention in how to measure consensus in groups of agents when they show their preferences over a fixed set of alternatives or candidates by means of weak orders (complete preorders). We have introduced a new class of consensus measures on weak orders based on distances, and we have analyzed some of their properties paying special attention to seven well-known distances.