[EM] Deriving Better-Than-Expectation from Weber´s strategic values

[MIKE OSSIPOFF]

First, Weber´s utilitly expectation-maximization method:


Because Russ used "candidate j" to represent the candidate for whom we´re 
considering voting, I used that too in my derivation that I just posted. But 
here I´ll let i be the candidate for whom we´re considering voting.

By "vote-expectation" I mean the expected benefit of voting for candidate i.

As with the other derivation, there´s the assumption that there are so many 
voters that your ballot won´t change the probabilities significantly.

There´s also the assumption that there are so many voters that any tie or 
near-tie will be between only 2 candidates.

And there´s the assumption that Weber´s Pij (which I´ll define soon) = 
Wi*Wj, the product of the win-probabilities of i & j.

At least as I define it here, Pij is the probability that you can make or 
break a tie between i and j, in i´s favor, by voting for i and not for j.

Let´s  assume that Pji = Pij.

Looking at it with respect to j only, what´s the benefit of voting for i?

If you make or break an ij tie, in i´s favor, that will benefit you by 
(1/2)(Ui - Uj).

So, with respect to j,  the expected benefit of voting for i and not for j 
is:

(1/2)Pij(Ui - Uj).

The 1/2 is present in all such terms, and so we can drop it.

But what if you´ve voted for j. What then is the benefit, with respect to j, 
of voting for i?

Well, if you don´t vote for i, then you´re voting for j and not for i, and 
the formula above applies, negatively. Pij(Uj-Ui), since Pji = Pij. Pij(Uj - 
Ui) = -Pij(Ui-Uj).

If you also vote for i, then you´re ceasing to vote for j and no for i, so 
you´re eliminating that
-Pij(Ui - Uj). So the expected benefit, with respect to j,  of voting for i, 
when you´ve voted for j, is Pij(Ui-Uj). The same as if you hadn´t voted for 
j. So the matter of whether or not you vote for j doesn´t affect the benefit 
of voting for i, with respect to j.

To repeat, then, with respect to j, the expected benefit of voting for i is 
Pij(Ui - Uj).

So, to find the overall expected benefit of voting for i, we sum that 
expression over all j.

Sum, j<>i, Pij(Ui - Uj).

That´s Weber´s formula. I use Merrill´s name for that quantity: Strategic 
value. That sum is the strategic value of i.

If that sum is positive, then you benefit from voting for i. If it´s 
negative, then you lose by voting for i.

So, in Approval, vote for i if i´s strategic value is positive.

Incidentally, in Plurality, vote for the candidate with the greatest 
strategic value.

Now let´s assume that Pij = kWi*Wj, the product of the winning-probabilities 
of i & j. As I said, this derivation depends on that assumption. That´s the 
key to this derivation, just as the assumption of a uniform factor of 
win-probability reduction was key to the derivation in my previous posting.

That´s a reasonable assumption. The more likely a candidate is to win, the 
better a contender s/he must be. And the better contender s/he is, the more 
likely s/he is to be in a tie or near-tie.

So let´s replace the Pij by kWi*Wj. But let´s leave out the k, since it´s 
present in all terms of that type.

So we have:

Sum, j<>i, (Wi*Wj(Ui - Uj) > 0   That´s the condition for voting for i

Multiply it out:

Sum, j<>i(WiWjUi - WiWjUj) > 0

WiUi = Ei, i´s expectation-contribution. WjUj = Ej.

So Sum, j<>i, (WjEi - WiEj) > 0

Writing it as 2 sums:

Sum, j<>i, WjEi - Sum, j<>i, WiEj > 0

In the summation, of course all the non-i candidates take their turn as j. 
So j changes during the summation, but i doesn´t. For the summation, i just 
refers to one candidate, because it´s a summation over j. So the things 
involving only i are constants for the purpose of the summation.

Taking the constants out of the summations:

Ei*Sum, j<>i, Wj - Wi*Sum,j<>i, Ej > 0

Well, for j<>i, the sum of the Wj is 1 - Wi, since the sum of all the win 
probabilities is 1.

And the sum of the Ej = E - Ei for the same reason. E is the sum of the 
expectation contributions of all the candidates.

Substituting those in the previous inequality:

Ei(1-Wi) - Wi(E - Ei) > 0

Substituting WiUi for Ei:

WiUi(1 - Ui) - Wi(E - WiUi) > 0

Solve that for Ui. You get:

Ui > E

Mike Ossipoff

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