Single-Peakedness and Consensus
Let us reconsider the so-called "Paradox of Voting" presented in Chapter 2. Three alternatives A, B, and C are ordered by three individuals as follows: for number 1, ABC; for number 2, BCA; for number 3, CAB. By majority vote A is preferred to B, and also B is preferred to C. Transitivity for social choice would require that A be preferred to C. But in fact by majority vote C is preferred to A.
This paradox was advanced as an example of the difficulty which Arrow's Possibility Theorem generalizes. It is conceivable that if we discover the source of intransitivity in this situation we may in turn generalize for the Possibility Theorem itself.
It has been suggested that the difficulty stems from an inadequate analysis of the voting situation presented, and a consequently unreasonable identification of social choice. To elaborate:
We are not 'forced' to say that A is preferred to C [when A is preferred to B and B preferred to C ] unless we first postulate a 'community' preference that is different from the preferences of the people in it. We are no more 'forced' to say that the 'community' prefers A to C than we are 'forced' to say it prefers C to A, or B to C. Two people prefer A to B; two (a different pair) prefer B to C; and two (still a third pair) prefer C to A; conversely two people would vote against A as the most desirable alternative, another pair would vote against B as the most desirable alternative, and a third pair would vote against C as the most desirable alternative. In this situation one can only say