When there are just $2$ candidates, the majority function possesses all of the mathematical properties that seem desirable in a voting rule (e.g., May’s Theorem and Theorem 32). Unfortunately, as soon as there are $3$ (or more) candidates the problem of social choice becomes much more difficult. For example, suppose we have candidates $a$, [...]

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Ryan O'Donnell: Good catch, thank you Xi.Ryan O'Donnell: Thank you! Sorry for the delay in replying.Ryan O'Donnell: Hi Ming. Here S stands for a fixed (non-random) subset of [...Xi Wu: typo: "our definition of $\mathbf{Inf}_i[f]$ from Chapter 2....Chengyu: Ex 2.c It should be "Suppose ... is an LTF with $\textbf{E}...Ming: I confuse the notation S in Fact 1.7. I wonder that the sym...Ryan O'Donnell: Yes, thanks! Sorry for the delay in replying.