# Isn't it trivial that preferences are single-peaked over two alternatives?

I'm studying Down's Median Voter Theorem which states that if voters have single-peaked preferences over two alternatives arranged along a one-dimensional political spectrum, a majority rule voting system selects the outcome most preferred by the median voter.

As I understand it, single-peaked requires that each agent has an ideal choice, and for each agent, outcomes that are further from his ideal choice are preferred less. But isn't this trivially satisfied when two alternatives are considered (provided agents are not indifferent): If I prefer Clinton to Trump, then my preferences are single peaked over all possible choices.

I understand that singele-peakedness is far from trivial in $N>2$ alternatives, but $N=2$ is required for MVT. From wikipedia:

The third assumption (3) is that voters are only choosing between two options. This is important because when there are more than two choices for voters, the median voter may not have voted for the most popular option.

• Since you don't provide a link to the Wikipedia source, I cannot say anything about that source. But the median voter is certainly not about voting over two policy outcomes alone. – Michael Greinecker Apr 7 '18 at 12:59
• It's literally the wikipedia page on MVT. – pafnuti Apr 7 '18 at 13:28
• That Wikipedia page has no formal statement of the MVT, and it is fairly pointless to argue about such a "theorem." – Michael Greinecker Apr 7 '18 at 14:09
• I'm not arguing about the ""theorem"", I'm seeking clarification about single-peakedness when the number of alternatives is 2. – pafnuti Apr 7 '18 at 14:13
• Every preference profile over two alternatives is single-peaked if there are no indifferences. – Michael Greinecker Apr 7 '18 at 14:42