[EM] Saari reply

[Blake Cretney]

On Thu, 2002-06-20 at 13:45, Tarr, Adam wrote:
> Dave wrote:
> 
> > I missed any believable proof of this equivalence.
> >  

I'm going to give an outline of a proof, where I make various
simplifying assumptions, but I think the main issue will be resolved,
and it will be clear how to deal with more complicated situations.

First, I am going to restrict myself to 3 candidates.  Only two will be
front-runners, but I have no idea which (in fact all cases will be
assumed equally likely).

Now, let's label the three candidates A,B,C.  Without loss of
generality, A is my favourite, C is my least favourite, and B is in
between.  I'm going to use a scale for utility where A's utility is 1
and C's is 0.  B's is utility is assigned to the variable u.  Also, I'm
going to assume that u>.5.  The corresponding argument for u<.5 should
be obvious from my proof.  The ballot will also allow the voter to
assign any rating between 0 and 1 inclusive.

So, obviously I should rate A at 1 and C at 0, and the only question is
how I should rate B.  It would seem I should rate B at u, so I'm going
to try to prove that this is false.  I'm going to argue that the
expected utility from rating B at 1 is higher.  You might have other
ideas for how B should be rated, but for now I am only going to argue
against the B=u rating.

If the front-runner situation is A vs. C, it doesn't matter how I rate
B, so I will ignore this possibility.  There are two remaining
possibilities:  A vs. B and B vs. C.  The vote of B=1 is obviously more
likely to cause B to win over C, but sacrifices the possibility of
causing A to win over B.

A plausible assumption (which I could defend in more detail) is that for
a large electorate, the probability of a difference in rating causing a
difference in outcome is proportionate to the size in the difference of
rating.  I am using p as the chance of a full rating difference causing
a change.  Of course, p is unknown, but between 0 and 1.  But it
follows, for example, that a ratings difference of u, has a p*u chance
of having an effect, based on the assumption that the chance of a change
in outcome is proportional to the ratings difference.

B=u:
 p*(1-u)*(1-u)[for A vs. B] 
     + p*u*u  [for B vs. C]
 -------------
      2

B=1:
 p*0*(1-u)  [for A vs. B]
 +p*1*u     [for B vs. C]
 ------
  2

So, we get averages of
B=u:
  (p*(1-u)*(1-u)+p*u*u)/2
  p*(1-2u+u^2+u^2)/2
  p*(1-2u+2u^2)/2

B=1:    p*u/2

So, which is greater?  Well remember that u<1, so u-1<0.  Also, recall
that u>.5

u>.5
.5<u
1<2u
u-1 > 2*u*(u-1) [u-1<0]
u-1 > 2u^2-2u
u> 1-2u+2u^2
p*u/2 > p*(1-2u+2u^2)/2 [p>0]

So, the expected utility benefit of the B=1 strategy is higher than the
B=u strategy.

---
Blake Cretney


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