# [EM] expectation error corrected

MIKE OSSIPOFF nkklrp at hotmail.com
Sun Apr 2 13:04:04 PDT 2000

```EM list--

I've found the mistaken assumption that made Borda look
improbably good in the comparison of utility expectations for
sincere voters.

When I first posted the table, I said that, to get the improvement
in utility expectation for a sincere voter, the table entry
should be multiplied by the probability that some arbitrarily-
chosen pair of candidates are the 2 frontrunners, and are so
close that 2 same-voting voters could reverse their comparison
with eachother, changing the winner from one to the other.

There were other assumptions that I didn't state because it
didn't occur to me that I was making them: Each voter votes
between each pair, and votes a vote-difference of one between
each pair.

I said to multiply the table entry by that probability because
I considered that to be the probability, for each candidate-pair,
that, in any of those voting systems, 2 same-voting voters could
change the winner between those 2. That isn't true, because
the 2 assumptions in the previous paragraph aren't true.

But that assumption is still ok as a standard assumption, as
long as the table entry for each method is adjusted in accordance
with the way that assumption differs from the truth with that
method.

Making those adjustments, Approval is better than Borda all the
time. Approval is better than IRV all the time. Plurality is
better than IRV slightly more than half the time.

***

How do the different methods winner-changing probabilities differ
from the standard one that I defined? In Borda, it isn't true
that the voters are only voting a vote difference of 1 between
each pair of candidates. Each voter votes, among the 3
candidate pairs, vote differences of 1, 1, & 2. So each voter's
average vote-difference is 4/3. So there are 4/3 times as many
votes voted among each pair in Borda than I'd assumed. That makes
it 3/4 as likely that the pivotal voters (the 2 voters whose
expectation change we're estimating, and whose probability of
changing the outcome we're interested in) will be able to change
the outcome. So Borda's 3 should be multiplied by 3/4 to get
9/4 = 2.25

The other methods have to be adjusted similarly. In Approval and
Plurality, each voter is only voting between 2 candidate pairs,
not all 3. So the assumption that everyone votes between each
candidate pair isn't true. There are only 2/3 as many votes between
each pair as I'd assumed, and so Approval's & Plurality's
table entries should be multiplied by 3/2.

With IRV, part of its determination needs to be adjusted in that
way.

An immediately subsequent posting will describe how I got
the results that I got, and will include a new table.

Mike Ossipoff

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