Conditions For Majority Rule

The first condition for majority rule is to drop the unrestricted domain condition U, since it is possible to construct preference profiles for which majority rule violates the ordering conditions.

If the citizens agree to arrange themselves on a spectrum from liberal to moderate to conservative the standard voting paradoxes cannot occur. (However, the median or "swing" voter may tend to become a dictator.) Otherwise, the final outcome can be determined by Parliamentary maneuvers regarding voting agendas.

Majority rule is the only SWF satisfying the following 3 conditions:

A Anonymity: The SWF produces the same social ordering for two profiles P1 and P2 if P2 can be devired from P1 by exchanging the preference orders among the citizens.

Violated by "chairman votes only to break a tie"

A stronger form of D (nondictatorship)

N Neutrality: If each citizen's pairwise ordering for x & y in a profile P1 is the same as that citizen's pairwise ordering of w & z in another profile P2, then the SWF gives the same social ordering for x & y in P1 that it does for w & z in P2.

Violated by most forms of 2/3 majority rules

A stronger form of CS (Citizen's Sovereignty

PR Positive Responsiveness: If the SWF rates x at least as good as y for profile P1, and the only difference in the ratings of x and y between P1 and P2 is that some citizens have moved x up in their orderings, then the SWF prefers x to y in P2.

A stronger form of PA (Positive Association)