The Possibility of Social Choice for 3 Alternatives

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Arrow allows for the existence of tied alternatives via his social choice function, C(S), which selects the most preferred alternative or set of alternatives from an ordering, but not for the existence of tied social orderings. However, as Arrow himself acknowledges, the Axiom of Completeness demands that "either xPy or yPx or both." Similarly, for three alternatives completeness would demand the consideration of orderings of the form xPyPz, yPzPx or both as well as combinations of all other possible social orderings. When tie social orderings are allowed as part of the range of a SWF, it can be shown that a rational SWF which is compliant with a strengthened version of Arrow's Axioms and Criteria is possible for the case of three alternatives.
These results can be extended to the general case of an arbitrary number of alternatives. We have demonstrated elsewhere an algorithm which provides solutions in the general case and shown that the solutions meet a strengthened version of Arrow's Axioms and Criteria. We have also proven that the general algorithm provides solutions for any number of alternatives and voter/consumers, and, therefore, that social choice is possible.

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