[EM] There's nothing wrong with Average Rating.

[Ken Johnson]

election-methods-electorama.com-request at electorama.com wrote:

Message: 5
Date: Thu, 26 Feb 2004 01:47:40 -0500
To: election-methods-electorama.com at electorama.com
From: Adam Tarr <atarr at purdue.edu>
Subject: Re: [EM] Re: Election-methods digest, Vol 1 #517 - 3 msgs

Ken Johnson wrote:

>>...
>>
>>On the other hand, it may be that the optimum-rating function looks more 
>>and more like a discontinuity as the number of voters increases, so I 
>>could be wrong - maybe CR does reduce effectively to Approval for large 
>>voting populations.
>  
>

My intuition is that that would be the case.  Basically, it comes down to 
this.  After some, possibly messy, calculations, you will derive some 
expected utility for voting for candidate B.  If that expected utility is 
positive, then you should maximize your utility by giving B 100.  If that 
expected utility is negative, you should maximize your utility by giving B 
0.  Only when that expected utility is zero (i.e. your feelings toward B 
are exactly equal to your expectation of the election outcome) does a vote 
between zero and 100 make sense - of course you can put B anywhere in that 
case.

Your mention of "large voting populations" is crucial, since it allows me 
to assume that the marginal utility for an additional ranking point given 
to candidate B remains constant through the entire 0->100 range.  On a 
small committee, this may not be the case, so it's possible (I'm not sure) 
that intermediate rankings make sense in that case.
...

-Adam



I did some "messy calculations" and convinced myself that the optimum CR 
strategy is indeed identical to Approval, at least under two special 
circumstances: (1) large voter population, and (2) if I have no 
knowledge of how other people might vote. This doesn't entirely address 
the issue I raised earlier, which considered situations where I have a 
good idea of which candidates are more or less likely to get elected, 
but I can kind of see how the more general analysis would turn out. 
Basically, what it boils down to is that when the number of voters is 
large, each candidate's probability of success is approximately a linear 
function of my ballot rating for B. Hence, the statistical "expectation 
value" for the winner's rating (based on my sincere rating standard) is 
also linear, and if you look for a maximum of a linear function, 
naturally you find yourself marching off towards plus or minus infinity. 
So the optimum is at one of the limits of the valid rating range. The 
point at which the slope of the expectation value function goes from 
positive to negative is where my sincere B rating is midway between my A 
and C ratings, equivalent to Approval.

If I have prior knowledge of the candidates' chances of winning, then 
the expectation value function will still be linear in my ballot rating 
for B, but the optimal threshold dividing my "approved" vs "unapproved" 
rating levels will probably not be midway between my A and C ratings. 
The strategy gets more complicated, but I would still give B an extreme 
rating. However, if there are few voters (or if I am part of a large 
voting block), the expectation value function is actually nonlinear, 
possibly admitting a maximum between the rating limits.

Ken Johnson

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