[EM] Saari reply

[Tarr, Adam]

Dave wrote:

> I missed any believable proof of this equivalence.
>  
> MANY voters can understand wanting to be able to say:  I 
> prefer Nader; I  can tolerate Gore as a second choice; my 
> dislike for Bush  requires that I 
> mark him as less desirable than Gore.
> 
> Condorcet lets me say this, and expect to be heard when the 
> votes are counted.
> 
> IRV lets me say this, but I cannot depend on their noticing.
> 
> Approval does not even let me say it.

The argument for Approval and Cardinal Rankings being equivalent breaks down
to this: the best play is to give every candidate either the maximum or the
minimum ranking.  There are lots of ways to argue this, but here's a
heuristic argument followed by an anecdotal example:

- Inevitably, the election comes down to two candidates.  You want your
preference between those two candidates to have a maximum impact.  This
reguires that one of those candidates gets the maximum rank, and one gets
the minimum.

- If you know who those two candidates will be (i.e. perfect or good
information case) then your ranking of the other candidates is more or less
irrelevant to strategy.  But presumably you will rank candidates you like
more than your favorite front runner at least as high, and candidates you
like less than your least favorite frontrunner at least as low.  You might
float the remaining cadidates between them freely, but if you have any
inkling that one of the frontrunners might not end up being one of the top
two, then it pays to put those candidates at the maximum or minimum ranking,
depending on whether you prefer that candidate to the one that is sure to be
a top two candidate.

- If you do not know who the top two candidates will be (i.e. zero or poor
information case) then you still have the highest expected impact on the
race by putting half the candidates at the highest ranking and half at the
lowest ranking.  Your highest expected utility comes from placing the cutoff
at the point of average expected utility, rather than at half the
candidates.

My anecdotal example is very simple, but also fairly realistic.  Say the
preferences (utilities) of the public go something like this:

36%  Al 10, Ralph 3, George 0
48%  George 10, Al 0, Ralph 0
16%  Ralph 10, Al 5, George 0

If everyone votes in Cardinal rankings just like that, the results will be

George 4.8
Al 4.4
Ralph 2.68

And George wins the election.  If, on the other hand, the 10% that backs
Ralph gives Al the full 10, the results go to

Al 5.2
George 4.8
Ralph 2.68

Which demonstrates my point - why give the frontrunner you prefer less than
your full support?  It's irrelevant at best and costs you your best result
at worst.

-Adam

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