In social choice and voting theory, the profile of voters' rankings obviously determines all the head-to-head outcomes among n candidates. A theorem by McGarvey states that, given any head-to-head results you want, there is a profile of voters that will produce it. Very cool.

My question is... If you impose the condition that all voters unanimously prefer candidate A to candidate B, how does that limit the possible head-to-head results? Is anyone aware of any work done on this question?

I've come across an nxn matrix whose entry in row i and col j is min(i,j). For example, the 4x4 version would be
1 & 1 & 1 & 1\\
1 & 2 & 2 & 2\\
1 & 2 & 3 & 3\\
1 & 2 & 3 & 4
Does anybody recognize this class of matrices? Does it have a name? I'd like to research what is known about it.

that there exist irrationals \(a\) and \(b\) such that \(a^{b}\) is rational:
\(\sqrt{2}\) is irrational. Let \(z=\sqrt{2}^{\sqrt{2}}\). If \(z\) is rational, the claim is proved. If not, then \(z^{\sqrt{2}}=2\) proves the claim.


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