[EM] IRV expectation
MIKE OSSIPOFF
nkklrp at hotmail.com
Sun Apr 2 14:15:16 PDT 2000
EM list:
With IRV, you have 2 chances of changing the outcome: In the
1st elimination, and in the pair comparison of the 2 candidates
left after that elimination.
Again, I'm leaving out Pij, because I'm considering each term
to have the unwritten standard factor in front of it.
Vi-Vj is always 1. So we just have to write Ui-Uj
In the 1st elimination, though, we don't change from one outcome
to another, but rather from one lottery to another.
For instance, if we make B be eliminated instead of A, then
we get the AC lottery instead of the BC lottery.
So let me write the sum of the utility differences that we
could achieve:
The utility of the AB lottery is (A+B)/2.
Let me write the utility difference if we make B get
eliminated instead of A:
(A+C) - (B+C)
Of course C cancels out, and we just have A-B.
I'm going to multiply the whole sum by 1/2 once, rather than
each term.
So we get, for our utility expectation change in the 1st
elimination:
1/2( (A-B) + (A-C) + (B-C) ).
Collecting terms:
1/2(2A-2C) = A-C
Since A (which I sometimes use for Ua) is 1 & C is 0, that
expression equals 1.
***
But say we don't change the outcome in the 1st elimination.
There's still one more chance. The 2nd elimination will
be just one contest between one pair, and so we just have
one chance to influence a pair-comparison, rather than 3.
But it could be any of the 3. So the utility expectation change
in the 2nd elimination is:
1/3( (A-B) + (A-C) + (B-C) ).
Collecting terms:
1/3(2A-2C). Since A = 1 & C = 0, that comes to 2/3.
***
It remains to adjust the 1 & the 2/3:
In the 1st elimination, it's like Plurality. Each voter is
only voting beteen 2 pairs, not 3. Also, even though we're
interested in frontrunners for elimination, it's still as
if we were interested in frontrunners. So we multiply
the 1 by 3/2, and so the utility expectation change for
the 1st elimination is 3/2.
In the 2nd elimination, it's different in 2 ways. For one thing,
each voter is voting between all the pairs, and so we don't
need to make the 3/2 adjustment. For another thing, we don't
need for the 2 candidates to be frontrunners in some field of
candidates. They need only be within 1 vote of eachother.
The standard assumption was that the 2 candidates need to be
frontrunners and within one vote of eachother. With 3 candidates,
there are 2 ways 2 candidates could be within one vote of eachother:
They could be the top 2 or the bottom 2. One of those lets us
change the outcome and one doesn't, so changing the outcome is
1/2 as probable as it would be if we didn't need for them to
be frontrunners among 3. With just 2 candidates, in the 2nd
elimination, we don't have that problem, and all we need is
for the 2 candidates to be within 1 vote of eachother, and so
we double the value for the 2nd elimination, to 4/3.
Adding the adjusted values for the 2 eliminations:
3/2 + 4/3 = 9/6 + 8/6 = 17/6 = 2 & 5/6 = 2.8333...
***
That isn't the 2.25 that I said in the earlier message. I
must have made an error before. Maybe I divided the 2nd
elimination result by 2, mistakeningly copying the 1/2 that
the 1st elimination result had to be multiplied by.
Anyway, from this, the table entry for IRV is 2.8333...
That makes IRV usually better than Approval by this standard,
unless there's still a mistake in this IRV determination.
Neeless to say, that doesn't make me start advocating IRV :-)
There are too many other differences that make IRV look really
bad. FBC, SARC, WDSC, Corruption-encouragement, non-median
candidates winning at Myerson-Weber equilibrium, Social utility,
to name some differences.
Social utility has a more concrete meaning than this
utility expectation change for a sincere voter. It tells us
how we can expect to do in some future election after the
new method is adopted. It tells us our utility expectation, at
this time, for that future election.
But I hope someone can tell me that there's an error in
this posting.
Mike Ossipoff
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