[EM] Briefer demonstration for Better-Than-Expectation

[MIKE OSSIPOFF]

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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