[EM] How I got my expectation table

[MIKE OSSIPOFF]

EM list--

As I was saying before, the utility expectation gain that
you get by voting between a particular pair of candidates can
be assumed equal to Pij(Vi-Vj)(Ui-Uj). That assumption is used
by academic authors on the subject.

Though, for a particular method, Pij isn't really the same
as the probability that 2 same-voting voters could change the
win from j to i, if they each vote a vote-difference of 1
between i & j, and if everyone votes a vote-difference of 1
between each candidate pair, it's still convenient to use that
standard factor, and to consider each term to be multiplied by
that unwritten factor. Then, when making the determination
for a particular method, each term of the determination has
to be multiplied by an adjustment factor, as I described in
my previous posting.

Of course only with Borda is Vi-Vj anything other than 1.

***

Plurality:

You're voting between A & B, and between A & C. (Va-Vb) is 1,
as it is in every method other than Borda, and Pij is something
that I'm not including, because, at first, I'm just considering
that standard factor that I mentioned to be in front of each
term. So that just leaves (Ua-Ub). Ua-Uc is 1. Ua-Ub is 1-B.
So, adding those, the whole amount, for the two pair votes, is
1+1-B = 2-B. For the reason that I described in the previous
posting, that has to be multiplied by 3/2. So it's 3/2(2-B).

When B varies from 1 to 0, then, Plurality's table entries vary
from 3/2 to 3.

***

Approval:

If B is greater than .5, then it's above average, and you
vote A over C and B over C. Adding up the utility differences,
it's (Ua-Uc) + (Ub-Uc). That's 1 + B.

[Sometimes I use A & B to mean Ua & Ub]

If B is less than .5, then it's below average, and you
vote A over B and A over C. Adding up the utility differences,
it's (Ua-Ub) + (Ua-Uc). That's 1 + (1-B) = 2-B.

If B = .5, then both of those 2 expressions equal 1.5

If B is 0, then the 2nd expression applies, and it equals 2.
If B is 1, then the 1st expression applies, and it equals 2.

So, before adjustment, Approval varies from 1.5 to 2. But
multiplying by 3/2, it varies from 2.25 to 3.

***

I'll send this now, and send separate postings for Borda
& IRV, immediately after this one.

***

Mike Ossipoff

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