Majority Rule and Vote Cycling with Non-Single Peaked Preferences

Is majority rule the best way to represent voters preferences for a given set of policies?

Let’s look at a particular voting scenario to illustrate this. Consider a set of policies A,B,C following a sequence from  (A) less to more extreme (C) - maybe tax rates or some level of spending- with (B) being the intermediate policy. Suppose voters have the following preferences:


VOTER X: A >B >C  'single peaked '

VOTER Y: C>A>B

VOTER Z: B>C >A  'single peaked'

Both voters X and Z have single peaked preferences. As we move away from their optimal choice the strength of their preferences or utility decreases. However, voter Y does not exhibit single peaked preferences. They prefer the extreme policy C most, but their next preferred policy is in the direction of the other extreme A. They prefer the intermediate policy B least.

If the voters were voting on these issues, voter X would prefer law A over law B and law B over law C. In shorthand – A >B >C. To summarize all of the choices of the voters we see that 2/3 of the voters have preference A >B, 2/3 of the voters have preference B > C, but when voting A vs. C, 2/3 have preference C >A.

See if you follow the application of this. If we vote on policies in a pairwise fashion and have two elections and the first is made between policy B and C, then B will win (2/3 of the voters have preference B > C). If this is followed by a second election A vs. B (Because C was eliminated in the first election) then A will be the law that ultimately passes by majority rule.


Now if the order is changed, in which the first election is between A and B, A will win (because 2/3 of the voters rank A > B). Then in the second election when A goes against C, C will be the law that passes by majority rule (again because 2/3 of the voters have preference C >A).

So when voting on these policies, the process becomes arbitrary. The outcome depends on the order of the vote, so a cycling of choices ensues. According to public choice economist Gordon Tullock, any outcome can be obtained in majority voting by at least one voting method. This indicates that majorities can be irrational and dangerous unless preferences are all single peaked. 

Reference:

Lemieux, Pierre, The Public Choice Revolution. Regulation, Vol. 27, No. 3, pp. 22-29, Fall 2004. Available at SSRN: https://ssrn.com/abstract=604046