Social Choice_SRF - Social Ranking Functions Matt Van Essen...

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Social Ranking Functions Matt Van Essen University of Alabama Van Essen (U of A) SRF 1 / 21
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Social Ranking Functions Previously we have been interested in choosing a winner (or winners) from a collection of individual preference lists. In this lecture we introduce a very close extension of that idea ° choosing a social ranking. In many applications, it is desirable to know how alternatives compare to one another in a social ranking. Example College Football Rankings Example Sorority Rush Van Essen (U of A) SRF 2 / 21
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Social Ranking Functions In our simple model, we have assume that individual preferences are consistent °i.e., transitive. Speci±cally, for each individual i and any a , b , c 2 A , if a ° i b and b ° i c , then we must have a ° i c . This assumption, along with completeness and asymmetry, yields individual rankings over the alternatives. We explore the idea of creating a social ranking of the alternatives. A function that takes the rankings of individuals as its input and returns a single ranking as its output will be called a social ranking function. Van Essen (U of A) SRF 3 / 21
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Condorcet²s Paradox Given the way we derived individual preferences, a natural attempt at creating a social ranking is to de±ne a social preference relation ° S , where a ° S b is read ³ a is socially preferred to b We can then use a voting procedure to de±ne ° S . We might de±ne ° S , for example, as society prefers a to b if a beats b in a head-to-head competition using majority rule. Van Essen (U of A) SRF 4 / 21
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Condorcet²s Paradox This procedure turns out to be doomed! In fact, one of the surprising realizations of early work in social choice is that, while individual preferences may be consistent, social preferences de±ned by majority rule need not be!
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