Iīd 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:
Thatīs correct. I didnīt 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 didnīt 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 isnīt optimal. But it maximizes oneīs utility expectation if certain approximations or assumptions are made. One usual assumption is that there are so many voters that oneīs own ballot wonīt change the probabilities significantly. By one approach, itīs also necessary to assume that the voters are so numerous that ties & near-ties will have only 2 members, and that Weberīs 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 Weberīs 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 isnīt really called dropping second-order terms, the product of two small quantities, because we arenīt calculating the probability of a tie. Weīre 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 weīd drop the terms consisting of those products _if we were calculating the probability of a tie or near-tie_. But we arenīt.
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 elseīs win-probability by a uniform factor.
Thatīs the approach that Russ used, except that he didnīt state that assumption.
You reply:
Yes I did. I said that the other winning probability ratios should remain unchanged.
I reply:
You didnīt say why. You didnīt say they needed to remain unchanged because thatīs 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 didnīt 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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