This may be obvious to some, and maybe someone has even already posted about it, but, in case it hasn�t been posted yet, I�ll mention it now:
That Better-Than-Expectation maximizes a voter�s utility expectation if voting for j reduces the win probabilities of all the non-j candidates by the same factor, can be demonstrated in a briefer way:
Here, Pj represents the probability that j will win, instead of representing the probability that j will win if we don�t vote for j.
If voting for j reduces the win probabilities of the non-j candidates by the same factor, then the expectation if j doesn�win ("Enonj) is unchanged, because the lottery among the non-j is unchanged. All that�s changed are Pj and, as a result, 1 - Pj.
E, the overall expectation is Enonj(1 - Pj) + UjPj.
So E must be between Enonj and Uj. So iff Uj > Enonj, then Uj > E.
Uj > E is the necessary and sufficient condition for Uj > Enonj.
If Uj > Enonj, then, since voting for j increases Pj and decreases 1 - Pj, then, voting for j must increase that voter�s overall expectation, E.
This could also be worded more briefly, if a litle less completely, by just saying that if candidate j is better than E, and therefore is better than what can be expected if j doesn�t win then, obviously it�s better to vote for j and make j more likely to win.
Mike Ossipoff
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