Majority Rule will induce non empty choice set if individual preferences are single peaked

Is this statement true? I have some trouble in understanding the meaning of 'single peakedness' in context of this statement. Does choice set here imply that we need to find the, say, favorite alternative amongst the given ones? I understannd that Majority Voting is riddled with intransitivity. What does 'non empty' choice set imply ?

What I understand is that for single peaked alternatives, individual has a particular alternative of his choice and the alternatives that are away from the peak would be preferred less. My concepts seem to be quite shaky. Please bear with me.


Suppose that A={a,b,c,....,z} is a finite set of social alternatives, and let P={>1,>2,....,>N} be a profile of strict preference orders on $A$ (where the set {1,2,...,N} indexes the voters). We say that the profile P is single-peaked if there is some way to order the alternatives in A (e.g. in alphabetical order) such that, for each of the preference orders >n in P, there is some "ideal point" (say, h) such that

a<n b <n c <n ... <n g <n h >n i >n j >n .... >n y >n z.

Note that the condition does not impose any particular preferences between, say, g and i, or indeed between any element of {a,b,...,g} and any element of {i,j,..,z}. Note also that different voters could have different ideal points. (So voter n has ideal point h in the above exampe, but maybe voter m has ideal point d....) Note also that the question here is not whether a single preference ordering >n is "single-peaked" ---the question is that entire the profile P is single-peaked, meaning that we can find a single ordering of A such that all the preferences in P are single-peaked with respect to this ordering. (In the above example, I used alphabetical order, but this was just for simplicity. The question is only whether some order exists.)

Why is this condition useful? Suppose you try to construct a social preference order >0 by determining the comparison between each pair of alternatives through a simple majority vote (as recommended by Condorcet). So for example, for alternatives b and g, we will stipulate that b >0 g if and only if b >n g for a majority of the voters {1,2,...,N}. (For simplicity, suppose N is odd, so that majority vote never results in a tie.) We know that in general, this method does not result in a transitive preference relation. (This is Condorcet's Paradox.) However, if the profile P is single-peaked, then this method of "pairwise majority vote" always yields a transitive social preference order. (This was discovered by Duncan Black.) Furthermore, the "maximal" element of the resulting social preference order is the median of the ideal points of the voters. (This is the famous Median voter theorem.)

In summary, if the profile P is single-peaked, then not only does majority rule induce a non-empty choice set (as you said), but this choice set is rationalized by a complete and transitive social preference order (obtained by pairwise majority votes), and furthermore we can say exactly what the choice set is (it is the median of the ideal points of the voters).


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