Judgment Aggregation. Gabriella Pigozzi

Чтение книги онлайн.

СКАЧАТЬ and z first and then to compare the winning alternative with y, we would get a different result, namely y would be the winning alternative.

      Condorcet’s second main contribution employs probability theory to deal with the ‘jury problem’. Voters are seen like jurors who vote for the ‘correct’ alternative (or the ‘best’ candidate). The idea that groups make better decisions than individuals dates back to Rousseau [Rou62], according to whom, in voting, individuals express their opinions about the ‘best’ policy, rather than personal interests. Condorcet approached Rousseau’s position in probabilistic terms and aimed at an aggregation procedure that would maximize the probability that a group of people take the right decision. This led Condorcet to formulate the result now known as the Condorcet Jury Theorem, which provides an epistemic justification to majority rule [GOF83]. The theorem states that, when all jurors are independent and have a probability of being right on the matter at issue, which is higher than random, then majority voting is a good truth-tracking method. In other words, under certain conditions, groups make better decisions than individuals, and the probability of the group taking the right decision approaches 1 as the group size increases.

      So, interestingly, Condorcet showed at the same time the possibility of majority cycles, a negative result around which much of the literature on social choice theory built up, and a positive result like the Condorcet Jury Theorem, which gives an epistemic justification to majority voting.⁴

       Dodgson

      From the overview so far, the reader may have gotten the impression that the early developments of social choice theory were exclusively due to French scholars. But this is not the case. Indeed many English mathematicians have also studied the subject: Eduard John Nanson, Francis Galton and, more importantly, the Rev. Charles Lutwidge Dodgson (better known as ‘Lewis Carroll’, author of Alice’s Adventures in Wonderland), to whom we now turn.

      Black gives a careful analysis of Dodgson’s life and of the circumstances that raised the interests of a Mathematics lecturer at Christ Church college in Oxford for the theory of elections. In particular, Black discovered three of Dodgson’s previously unpublished pamphlets and, thanks to his extensive research, could conclude that Dodgson ignored the works of both Borda and Condorcet.

      Dodgson referred to well-known methods of voting (like plurality and Borda’s method) and highlighted their deficiencies. For him the main interest of the theory of elections resided in the existence of majority cycles. He suggested a modification of Borda’s method to the effect of introducing a ‘no election’ alternative among the existing ones [Dod73], the idea being that in case of cycles, the outcome should be ‘no election’. He then claimed that if there is no Condorcet winner, his modified method of marks should be used [Dod74].

      Later Dodgson proposed a method based on pairwise comparison that may seem to contradict the ‘no election’ principle he introduced earlier. However, as Arrow suggests [Arr63], this approach could be used when we do not wish to accept ‘no election’ as a possible outcome. The new method (now known as Dodgson rule) selects the Condorcet winner (whenever there is one) and otherwise finds the candidate that is ‘closest’ to being a Condorcet winner [Dod76]. The idea is to find the (not necessarily unique) alternative that can be made a Condorcet winner by a minimum number of preference switchings in the original voters’ preferences. A switch is a preference reversal between two adjacent positions. In order to illustrate the method, let us consider one of the examples made by Dodgson himself.

      Consider the preference profile given in Figure 1.3. Each row represents a group of voters with the same preferences, given in a left to right order. The number in the first column indicates the size of each group. In this example, there are eleven voters and four alternatives (a, b, c, and d). As the reader can easily check, the majority is cyclical (adcba) and none of the alternatives is a Condorcet winner. However, if the voter holding preference dcba switches alternatives c and b (marked by an asterisk) in her preference ranking, b becomes a Condorcet winner. Alternative c also can be made a Condorcet winner by one switch, so b and c are the only Dodgson winners (a and d each need four switches to be preferred to every other alternative by some strict majority).

1.1.2 MODERN SOCIAL CHOICE THEORY

      We have mentioned how Robbins’s claim [Rob38] that interpersonal utilities could not be compared undermined what constituted the predominant utilitarian approach to welfare economics until the Thirties: this amounted to say that there is social improvement when everyone’s utility goes up (or, at least, no one’s utility goes down when someone’s utility goes up) [Sen95].

      It thus appeared that social welfare must be based on just the n-tuple of ordinal interpersonally non-comparable, individual utilities. […] This “informational crisis” is important to bear in mind in understanding the form that the origin of modern social choice theory took. In fact, with the binary relation of preference replacing the utility function as the primitive of consumer theory, it made sense to characterise the exercise as one of deriving a social preference ordering R from the n-tuple of individual orderings {Ri} of social states. [Sen86, p. 1074]

      Figure 1.3: An example of Dodgson’s rule.

      The need for functions of social welfare defined over all the alternative social states was made explicit by Abram Bergson [Ber38, Ber66] and Paul Samuelson [Sam47]. Economists turned to the mathematical approach to elections explored by Condorcet, Borda, and Dodsgon only when—following the informational restriction decreed by Robbins—they searched for methods to aggregate binary relations of preference into a social preference ordering. Thus, social choice theory stemmed from two distinct problems—how to select the winning candidate in an election, and how to define social welfare—and the relations between these problems became clear only in the 1950s.

      Young economist and future Nobel prize winner, Kenneth Arrow defined a social welfare function as a function that maps any n-tuple of individual preference orders to a collective preference order. His axiomatic method outlined the requirements that any desirable social welfare function should satisfy.⁵ In 1950 he proved what still is the major result of social choice, the “General Possibility Theorem,” now better known as Arrow’s impossibly result⁶ [Arr50, Arr63]. The theorem shows that there exists no social welfare function that satisfies only just a small number of desirable conditions.

      Let us informally present these conditions: the first is that a social welfare function must have a universal domain, that is, it has to accept as input any combination of individual preference orders. Another commonly accepted requirement is the Pareto condition, which states that, whenever all members of a society rank alternative x above alternative y, then the society must also prefer x to y. The independence of irrelevant alternatives condition states that the social preference over any two alternatives x and y must depend only on the individual preferences over those alternatives x and y (and not on other—irrelevant—alternatives).⁷ Finally, non-dictatorship requires that there exists no individual in the society such that, for any domain of the social welfare function, the collective preference is the same as that individual’s preference (i.e., the dictator). Arrow’s celebrated result shows that no social welfare function can jointly satisfy these conditions.⁸

1.2 A NEW TYPE OF AGGREGATION

1.2.1 FROM THE DOCTRINAL СКАЧАТЬ