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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