Majoritarian decisions. Penalizing the disagreement (original) (raw)
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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.
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We characterize two lexicographic-type preference extension rules from a set X to the set ? of all orders on this set. Elements of X are interpreted as basic economic policy decisions, whereas elements of ? are conceived as political programs among which a collectivity has to choose through majority voting. The main axiom is called tournament-consistency, and states that whenever majority pairwise comparisons based on initial preferences on X define an order on X, then this order is also chosen by a majority among all other orders in ?. Tournament-consistency thus allows to predict the outcome of majority voting upon orders from the knowledge of majority preferences on their components.
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In this paper we introduce a multi-stage decision making procedure where decision makers' opinions are weighted by their contribution to the agreement after they sort alternatives into a fixed finite scale given by linguistic categories, each one having an associated numerical score. We add scores obtained for each alternative using an aggregation operator. Based on distances among vectors of individual and collective scores, we assign an index to decision makers showing their contributions to the agreement. Opinions of negative contributors are excluded and the process is reinitiated until all decision makers contribute positively to the agreement. To obtain the final collective weak order on the set of alternatives, we weigh the scores that decision makers assign to alternatives by indices corresponding to their contribution to the agreement.
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Consensus is understood as a unanimous agreement by all experts in a group. The goal of consensus is not the selection of several options but to develop one decision that suits the interests of the entire group under consideration. In this paper, it is assumed that collective preferences are developed with the help of commonly used ordered weighted averaging operators but resultant relations do not exhibit any property of a consensus type. That is, consensus is not reached at the first attempt of ranking alternatives. Under such circumstances, the measure of distance to consensus can be successfully used to determine how far a group collectively is from consensus. The aim of this paper is to compare, where possible, the distance to consensus of collective relations compiled with the help of most commonly used ordered weighted averaging operators. The innovative part is that this measure helps in defining an upper and lower bound of distance to consensus of the resultant collective relations. With such an analysis at hand, experts can choose a suitable ordered weighting averaging operator to formulate a collective relation that exhibits a lower distance to consensus.
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