[EM] Alright, next try. Range voting fix, version 2.
Kevin Venzke
stepjak at yahoo.fr
Sat Dec 10 19:51:23 PST 2005
Hi,
--- Abd ul-Rahman Lomax <abd at lomaxdesign.com> a écrit :
> It has been alleged that Range is not good if voters exaggerate.
> However, suppose that these voters are exaggerating. They can't
> exaggerate any more in the minimum and maximum ratings: normalization
> would make no difference in this election. The only question is how
> to rate the middle candidate. And it seems intuitively obvious to me
> that the optimum rating is the expected utility of that candidate's
> election, compared to the min and max candidates. How, exactly, do
> you "exaggerate" the middle? If you exaggerate minimum, you could
> cause the middle candidate to lose to your least favorite. And if you
> exaggerate maximum, you are failing to indicate your preference for
> your favorite.
The fact that you can make a strategic error when deciding how to rate
middle candidates doesn't mean that there is no optimum strategy.
A sincere rating makes sense when you expect the other voters' knowledge
to complement your own. But usually in a public election we assume
that voters know what they want, and are trying to out-maneuver the
other voters rather than reach a compromise with them.
Looking at this particular scenario:
>Voter 1:
>A: 100
>B: 40
>C: 0
>
>Voter 2:
>A: 45
>B: 0
>C: 100
>
>Voter 3:
>A: 0
>B: 100
>C: 60
Let's determine a strategy for voter 3. Let's guess that there's a 1/3
chance that a voter (each of the other two) will give their middle
choice the top rating, a 1/3 chance of the sincere rating, and a 1/3
chance of a 0 rating.
Here are the outcomes we can get prior to adding in the third vote:
("1t" means "first voter rates middle choice top," etc.)
1t 2t: A 200, B 100, C 100
1t 2s: A 145, B 100, C 100
1t 2b: A 100, B 100, C 100
1s 2t: A 200, B 40, C 100
1s 2s: A 145, B 40, C 100
1s 2b: A 100, B 40, C 100
1b 2t: A 200, B 0, C 100
1b 2s: A 145, B 0, C 100
1b 2b: A 100, B 0, C 100
if we add 0, 100, 100, we get these winners:
ABC, BC, BC, AC, C, C, AC, C, C
if we add 0, 100, 60:
AB, B, B, A, C, C, A, C, C
if we add 0, 100, 0:
AB, B, B, A, A, B, A, A, ABC
We know that A is worth 0, B 100, C 60 to us.
ABC tie is worth 53.33. AB is 50. AC is 30. BC is 80.
expectation for rating C top: 513.33/9 = 57.036
expectation for rating C sincere: 490/9 = 54.444
expectation for rating C bottom: 403.33/9 = 44.814
So that given these assumptions about how voter 1 and 2 will vote, it
is better for the third voter to rate C at 100 than sincerely at 60.
It's possible I made a math error somewhere, but you can see where
the gain would be: In rating C 60 instead of 100, the solo wins that B
obtains (2/9ths probability) don't offset the penalties from C not
sharing in ties with A (1/3rd probability).
Kevin Venzke
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