# Majority Rule and Single Peakedness

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