OPINION AGGREGATION IN FUZZY FRAMEWORK AND INCOMPLETE PREFERENCES

Abstract

Judgment Aggregation and Preference Aggregation are emerging research areas in many disciplines. Both the theories interrogate the consistency of the collective outcome produced by rational experts. The main idea is that there does not exist any method of aggregation which guarantees consistent collective choices and satisfies certain minimal conditions. This finding, which is now famous to be known as the Discursive Dilemma, is a generalization of classic Condorcet’s paradox, discovered by the Marquis de Condorcet in the 18th century. The early surveillance that enhanced the current development of the field goes back to some work in jurisprudence sparked by Lewis Kornhauser in 1992. The prob- lem was reconstructed and developed further by Christian List, Pettit and Brennan as a more general problem of majority inconsistency. List and Petit proved the first social choice theoretics impossibility results similar to those of Arrow and Sen’s impossibility theorem. This work was reinforced and extended in 2006 by several authors, beginning with Pauly and Van Hees and Dietrich. Some strong results on the theory of strategy-proof social choice were put forward by Nehring and Puppe in 2002 which later on helped produce important corollaries for the theory of judgment aggregation. The work produced by Pigozzi in 2007 and Konieczny and Pino Perez in 2002 suggests that the theory of judgment aggregation and belief merging share similar objective. Moreover, judgment aggregation can be interconnected with probability aggregation as recommended by McConway in 1981 and Mongin 1995. Its link- age with abstract aggregation was highlighted by Wilson in 1975 and Rubinstein and Fishburn in 1986. But modern axiomatic social choice theory was founded by ivArrow. List and Pettit’s work in 2004 and Dietrich and List’s 2007 paper can be con- sulted for the understanding of relationship that exists between Arrovian preference aggregation and judgment aggregation. In the parallel framework of Preference aggregation, experts are encouraged to provide complete and consistent preference relations. On the other hand, demanding a complete preference relation is an idealistic assumption which may not be prob- able in actuality. Incomplete preferences provided by experts were once discarded which lead to biased collective relations which did not represent choices of experts. To complete such a relation, it is imperative to consider consistency of the resultant completed relation. Literature proposes several methods for completing incomplete fuzzy preference relations and emphasises on their importance in decision making. Zai-Wu et al study a goal programming approach to complete intuitionistic fuzzy preference relations. Alonso et al give an estimation procedure for two tuple fuzzy linguistic preference relations. Two methods for estimating missing pairwise prefer- ence values given by Fedrizzi and Giove and Herrera et al are compared by Chiclana. Chiclana deduced that Fedrizzi’s method to estimate missing values based on res- olution of optimization is a special case of Herrera’s estimation method based on known preference values. Herrera proposed a method to estimate missing values in an incomplete fuzzy preference relation when (n − 1) preference values are provided by the expert. A more general condition which includes the case where a complete row or column is given. Estimated preference values that surpassed the unit interval were taken care of with a transformation function. However, consistency of the resultant relation is not assured. Moreover, this can void the originality of preference values provided by experts. Following the trend, this thesis focuses on solutions to inconsistent, indecisive and paradoxical outcomes in judgment aggregation and emphasises on methods to resolve incomplete preference relations provided by experts such that the resultant relations are also consistent. This thesis is based on four research papers and it builds on the following questions: • Can the problem of belief aggregation be molded into a framework where complete, consistent and non-paradoxical outcomes are attainable. v• How can Incomplete preference and multiplicative preference relations be com- pleted into complete and additive consistent or Saaty’s consistent relations. While ranking consistent relations, can we categorize some ranking methods that are equally efficient or better for these relations. • Provided additive reciprocal relations, how far are the collective relations from consensus. To interrogate the above mentioned problems, we have divided this dissertation into seven chapters. Chapter 1 is essentially an introduction aimed at recalling some basic definitions and facts where we fix notations and introduce terminologies to be used in the sequel. Chapter 2 is concerned with background and literature review of the work in judg- ment and preference aggregation. The impossibility theorem is listed along with examples of Majority rule and Dictatorship rule to assert how aggregation rules fail to satisfy collective rationality along with other minimal conditions. In the third chapter, we introduce belief aggregation in fuzzy framework. We pro- pose a distance based approach and study how this structure helps in producing collectively rational outcomes without compromising on systematicity or anonymity. With the help of the illustrated method, the resultant outcomes are consistent and the solutions are free of ties. Chapter 4 introduces an upper bound condition which ensures complete and con- sistent collective preference and multiplicative preference relations. The chapter proposes that if preference values provided by experts are ”expressible” then the in- complete relation can be completed using consistency properties. The upper bound ensures that the resultant relation is complete with expressible values such that no value transgresses the unit interval and that the completed relation is consistent. In chapter 5, our focus is on the relations completed in chapter 4. We term such relations as RCI preference and multiplicative preference relations and discuss per- formance of some ranking methods on complete RCI relations. It is highlighted in the chapter that complete RCI preference relations are additive transitive and com- plete RCI multiplicative preference relations satisfy Saaty’s consistency. For the purpose of comparing ranking methods on these relations, Column wise addition vimethod is introduced and compared with the performance of Fuzzy borda rule and Shimura’s method of ranking. For complete RCI multiplicative preference relations, Fuzzy borda rule for multiplicative preferences is defined and the mentioned proce- dure is recurred. A ranking method is confirmed to be better than the others if it produces lesser number of ties among alternatives. Chapter 6 deals with additive reciprocal preference relations which are more general than additive consistent relations. Several preference relations are compiled using ordered weighted averaging operators to formulate different collective relations. In the absence of complete consensus, the metric of distance to consensus is employed to measure how far are the collective relations from consensus. Chapter 7 concludes the dissertation and gives insight to some possible future work.