[EM] expectation with sincere voting, table.

[MIKE OSSIPOFF]

EM list--

A certain way of voting will change one's utility expectation
in a calculatable way, compared to what that expectation would
be if one didn't vote. And if one didn't vote in a 0-info
election, his expectation would be the average of the candidates'
utilities. So we can find out what a voter's expected utility
would be, with various methods, if he votes sincerely.

I'm posting here some results for Approval, Plurality, IRV,
& Borda.

The table entries should be multiplied by the probability that
a certain 2 candidates are so close for 1st place that we
can affect the outcome when we vote. That gives the amount by
which voting sincerely would improve the voter's utility expectation
over what it would be if he didn't vote.

Call A's utility 1, and call C's utility 0. B's utility is the
first table entry, first column. Each column after that is the
table entry for a method. The methods are named above their
columns. The B values, in the 1st column, aren't in numerical
order, but are ordered to show a pattern.

B  Approval  Plurality  IRV  Borda
.4  1.6       1.6        4/3   3
.6  1.6       1.4        4/3   3
.25 1.75      1.75       4/3   3
.75 1.75      1.25       4/3   3
.1  1.9       1.9        4/3   3
.9  1.9       1.1        4/3   3

***

Voting sincerely in Approval means voting for all the above-mean
candidates. Just as voting for one's favorite in Plurality
with 0-info is the utility-expectation maximizing strategy,
but is also sincere, that's also true of Approval's above-mean
strategy.

A sincere voter always does better with Approval than with
IRV. He usually does better with Plurality than with IRV.
He does better with Plurality than with IRV as long as B is
less than 2/3. In other words, he does better with Plurality than
with IRV twice as often as not.

The formula for the table entry with Plurality is 2-B. With
Approval it's 2-min(B, 1-B).

Let's find out how Condorcet & Margins score in that regard.
And get some tables with more candidates.

Because that's a computer job, and because there are a number
of computer programmers on this list, someone else will probably
post those answers before I do.

***

Borda does especially well, but, with its uniquely abominable
reputation with majority rule, defensive strategy, &
corruption-encouragement, with its unique ability to fail to
elect a 1st choice majority candidtate, and with its ability
to make co-operation/defection strategy dilemmas where even
Plurality wouldn't--Borda's good score here isn't enough to save
it.

***

Mike Ossipoff




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