[EM] Russ´s Better-Than-Expectaton derivation (was "no subject")

[MIKE OSSIPOFF]

I´d said:

>So the voter using that strategy votes for a candidate if that candidate is 
>so good that s/he would rather have that candidate in office than hold the 
>election.

Russ replied:

You never answered my question about what it would mean to not "hold the
election."

I reply:

That´s correct. I didn´t reply because the answer is so obvious.

Russ continues:

Does that mean the incumbent stays in office, or does it mean
that the government ends and anarchy begins?

I reply:

What did I say? :-)

Let me walk you through this, Russ:

>So the voter using that strategy votes for a candidate if that candidate is 
>so good that s/he would rather have that candidate in office than hold the 
>election.

I didn´t say that the voter would rather have the incumbant in office, or no 
one at all in office. What did I say? I said if the voter would rather have 
that candidate in office than hold the election. Not the incumbant, not no 
one in office, but that candidate in office.

Nor did I say that the voter has the power to put that candidate into office 
instead of holding the election. I merely said if that voter would rather 
have that candidate in office than hold the election.

>One can come up with situations in which that isn´t optimal. But it 
>maximizes one´s utility expectation if certain approximations or 
>assumptions are made. One usual assumption is that there are so many voters 
>that one´s own ballot won´t change the probabilities significantly. By one 
>approach, it´s also necessary to assume that the voters are so numerous 
>that ties & near-ties will have only 2 members, and that Weber´s Pij = 
>Wi*Wj, the product of the win-probabilities of i & j.

That's called dropping second-order terms, the product of two small
quantities.

I reply:

The assumption that Weber´s Pij  =  the product of Wi*Wj is called dropping 
second-order terms, the product of two small quantities? :-)

And, about the assumption that any ties or near-ties will have only 2 
members, that isn´t really called dropping second-order terms, the product 
of two small quantities, because we aren´t calculating the probability of a 
tie. We´re merely noting that the ties and near-ties with three members are 
less likely, and ignoring them. If calculating those less likely 
probabilities involves a product of two small numbers, then we´d drop the 
terms consisting of those products _if we were calculating the probability 
of a tie or near-tie_. But we aren´t.

>But, instead of the last 2 assumptions named in the previous paragraph, it 
>would also be enough to assume that when your vote for a candidate 
>increases his win-probability, it decreases everyone else´s win-probability 
>by a uniform factor.
>
>That´s the approach that Russ used, except that he didn´t state that 
>assumption.

You reply:

Yes I did. I said that the other winning probability ratios should
remain unchanged.

I reply:

You didn´t say why. You didn´t say they needed to remain unchanged because 
that´s an assumption on which your derivation depends. You said, and I 
quote: "...so each individual probability needs to be normalized by dividing 
by 1 + delta Pj to keep the sum of all probabilities at unity without 
changing the probability ratios." You stated the goal of not changing the 
probaility ratios, but you didn´t say anything to indicate that the 
assumptoin that all the non-j win-probabilities are reduced by the same 
factor is the assumption that makes your derivation possible.

Mike Ossipoff

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