David Pérez-Román

Measuring Consensus in Weak Orders

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.

Consensus-based clustering under hesitant qualitative assessments

In this paper, we consider that agents judge the feasible alternatives through linguistic terms - when they are confident in their opinions - or linguistic expressions formed by several consecutive linguistic terms - when they hesitate. In this context, we propose an agglomerative hierarchical clustering process where the clusters of agents are generated by using a distance-based consensus measure.

Measuring consensus in a preference-approval context

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.

SOME CONSENSUS MEASURES AND THEIR APPLICATIONS IN GROUP DECISION MAKING

In this paper we consider that a group of decision makers rank a set of alternatives by means of weak orders for making a collective decision. Since decision makers could have very different opinions and it should be important to reach a consensuated decision, we have introduced indices of contribution to consensus for each decision maker for prioritizing them in order of their contributions to consensus. These indices are defined by means of a consensus measure which assigns a number between 0 and 1 to each subset of decision makers. For putting in practice this idea, we have introduced a class of consensus measures based on distances on weak orders and we have analyzed some of their properties. We have illustrated the weighted decision procedure with an example.

Ordinal proximity measures in the context of unbalanced qualitative scales and some applications to consensus and clustering

In this paper, we introduce ordinal proximity measures in the setting of unbalanced qualitative scales by comparing the proximities between linguistic terms without numbers, in a purely ordinal approach. With this new tool, we propose how to measure the consensus in a set of agents when they assess a set of alternatives through an unbalanced qualitative scale. We also introduce an agglomerative hierarchical clustering procedure based on these consensus measures.

Consensus measures generated by weighted Kemeny distances on weak orders

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.

Consensus-based hierarchical agglomerative clustering in the context of weak orders

In this paper, we introduce a new hierarchical agglomerative clustering process in the setting of weak orders. This process is based on consensus measures induced by weighted Kemeny distances that associate a number between 0 and 1 to each subset of weak orders. Then, clusters are sequentially formed according to the consensus among the corresponding weak orders. The process is illustrated with the data obtained in an experiment carried out during the 2006 Public Choice Society election.

Consensus Reaching Processes under Hesitant Linguistic Assessments

In this paper, we introduce a flexible consensus reaching process when agents evaluate the alternatives through linguistic expressions formed by a linguistic term, when they are confident on their opinions, or by several consecutive linguistic terms, when they hesitate. Taking into account an appropriate metric on the set of linguistic expressions and an aggregation function, a degree of consensus is obtained for each alternative. An overall degree of consensus is obtained by combining the degrees of consensus on the alternatives by means of an aggregation function. If that overall degree of consensus reaches a previously fixed threshold, then a voting system is applied. Otherwise, a moderator initiates a consensus reaching process by inviting some agents to modify their assessments in order to increase the consensus.

A consensus reaching process in the context of non-uniform ordered qualitative scales

In this paper, we consider that a group of agents judge a set of alternatives by means of an ordered qualitative scale. The scale is not assumed to be uniform, i.e., the psychological distance between adjacent linguistic terms is not necessarily always the same. In this setting, we propose how to measure the consensus in each subset of at least two agents over each subset of alternatives. We introduce a consensus reaching process where some agents may be invited to change their assessments over some alternatives in order to increase the consensus. All the steps are managed in a purely ordinal way through ordinal proximity measures.

Aggregating opinions in non-uniform ordered qualitative scales

Abstract This paper introduces a new voting system in the setting of ordered qualitative scales. The process is conducted in a purely ordinal way by considering an ordinal proximity measure that assigns an ordinal degree of proximity to each pair of linguistic terms of the qualitative scale. Once the agents assess the alternatives through the qualitative scale, the alternatives are ranked according to the medians of the ordinal degrees of proximity between the obtained individual assessments and the highest linguistic term of the scale. Since some alternatives may share the same median, an appropriate tie-breaking procedure is introduced. Some properties of the proposed voting system have been provided.