Aggregation Methods for Fuzzy Judgments

By Syed, A.; Beg, I. et al. | Fuzzy Economic Review, May 2016 | Go to article overview

Aggregation Methods for Fuzzy Judgments

Syed, A., Beg, I., Khalid, A., Fuzzy Economic Review

Keywords: judgment aggregation, fuzzy, collective decision

JEL Code: C02, C60, D71

1.INTRODUCTION AND LITERATURE REVIEW

Aggregation theories have been of considerable importance for the social scientists since the 18th century, with significant research work done in many disciplines. The theory shed light on how judgments are collected into a set of consistent group judgment and how democratic majority is unable to give one. This problem arises as one of the most fundamental barrier of collective decision making in various situations, ranging from the most basic collective decision making parliaments to small one case based committees and from economic expert panel to court judges' committee. Judgment aggregation is also explained as a tool that looks at each of the interdisciplinary problems that they have in common.

The latest attention to judgment aggregation was triggered by the so called "Doctrinal Paradox" or "Discursive Dilemma" which was explained by Kornhauser (1992). This problem was primarily faced when the situation was multifaceted. In literature it was first discussed by Kornhauser and Sager (1986), where they explicated a three member court. This court had to arrive at a judgment in a breach of contract case; comprising of three propositions which had to be true in order for the defendant to be held liable. Further explaining this case, a propositions means a statement on which decision is to be made. Here propositions are denoted by "a", "b" and 'c'. Where 'a' states that the defendant was not liable to do an act, and 'b' states that the defendant did that act. The conclusion premise, c, states that whether a person is liable to pay compensation or not. Judges are then required to state whether those premises are true or false called judgment, together the decision on all the premises written individual or collective is called as judgment set. Important to know is that 'c' is true if both a and b are true, which means that if a person was liable not to do an act and still did it hence he is liable to pay compensation. Together (a λ b) hold a person guilty of breach of contract. The following example portrays the Doctrinal paradox through Table 1.

Analyzing Table 1, it can be assessed that first judge holds only first premise to be true. Second judge find both the premises to be true, however the third apprehends first to be false but second to be true. Judgment set of majority voting holds two premises to be true but the collection of third turns out to be false, resulting in logically inconsistent judgment set {TTF}, inconsistency here signifies the lack of rationality. Likewise Table 1 majority outcome means that the person was liable not to do an act and did that act but is not guilty of the breach of contract. Pettit (2003) explains this paradox as 'discursive dilemma'. But the question was how general this doctrinal paradox, is it limited to some specific case of majority outcomes or is it not possible for any majority to reach a consistent outcome. In response to this List and Pettit (2002) presented a model of judgment aggregation, which was inspired by Arrowian Social Choice Theory, explained in Arrow (1963) and illustrated a generic form proving the impossibility theorem and inability of the majority outcome to reach a consistent judgment set. After List and Pettit (2004) discussed model of judgment aggregation, much debate started assessing if judgment aggregation is familiar with the preference aggregation. Preference aggregation is endorsed with problem of cyclical preferences, judgment aggregation similarly bear the consequences of inconsistent majority output. The concept of "discursive dilemma" resembles Condorcet paradox. Dietrich and List (2007) explained the analogue of Arrow's theorem in theory of judgment aggregation.Dietrich (2016) later spoke about agenda sensitivity issues and proved new impossibility theorem based on the previous one.

Arrow (1963) started working in the field of social welfare functions and analyzed how many of them satisfy specific axioms. …

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