# The Possibility of Social Choice for 3 Alternatives

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 The Possibility of Social Choice for 3 Alternatives by John Clifton Lawrence General Algorithm POB 230351 Encinitas, CA 92023 Phone/fax: 760-633-3778 Web site: http://www.genalg.com E-mail: jlawrence@genalg.com  1998 by John Clifton Lawrence September 15, 1998 Revised: 2006 PO Box 126155 San Diego, CA 92112 Email: j.c.lawrence@cox.net Web Site: http://www.socialchoiceandbeyond.com Blog: http://willblogforfood.typepad.com Abstract In “Social Choice and Individual Values” Arrow (1951) discusses the possibility of ties for binary orderings. In particular, either xRy, yRx or both xRy and yRx are possible solutions when m (the number of alternatives) = 2 where R is the social “preference or indifference” operator. This is demanded by the axiom of completeness. Let us write “both xRy and yRx” as {xRy, yRx}. In the preceding sentence the word “and” is the English connective as distinguished from the logical and which we write “AND. ” If half the voter/consumers have xRiy and half have yRix (where Ri is the individual “preference or indifference” operator), it would be natural to assume (as one possibility) that the social ordering is {xRy, yRx} which we define as a tie. By extension, for three alternatives, if half the voter/consumers have xPiyPiz and half have yPixPiz, it would be natural to assume (as one possibility) that the social ordering is the tie {xPyPz, yPxPz}. This reduces correctly to the binary solutions {xPy, yPx}, xPz and yPz when the appropriate alternative is removed both at the individual and the social levels. Arrow only considers ties among alternatives via his social choice function, C(S), and not ties among orderings. Since he demands orderings as the solutions for a Social Welfare Function (SWF), it would be more natural to consider ties among orderings which are also demanded by the axiom of completeness. Considerations of ties among orderings leads to the possibility of legitimate SWFs which are presented for m = 3 and which comply with the axioms of connectivity and transitivity and a strengthened version of Arrow's 5 criteria. Download 108.5 Kb.Share with your friends:

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