[EM] A PR quality gauge for lotteries

[Forest Simmons]

Just a note to those who think it strange that a multi-winner PR lottery
method might have relevance in the single winner case.

A candidate is a bundle of attributes.  The voter gives various weights to
those respective attributes.  The weighted average is a kind of
expectation.  If the voters rate the candidates according to those
expectations, then the lottery method applies.  Nobody has to know whether
or not the voters actually calculated the expectations or just went with
their gut feelings.

On Wed, Dec 7, 2016 at 1:05 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

> Dear Friends,
>
> Given a collection beta of cardinal ratings ballots, where the ratings are
> on a scale from zero to one, and a lottery L on the candidates, here's a
> way to gauge the Proportional Representation quality Q of the lottery L
> relative to the collection of ballots beta:
>
> For x between zero and one, let G(x) be the fraction of ballots for which
> the lottery expectation is greater than x.
>
> The quality gauge Q is given by
>
> the integral over the interval 0<x<1 of the integrand G(x)^(ln(1-x)/ln(x)).
>
> Note that Q is always in the interval [0, 1], and attains the value of one
> only if G(x) is identically one of the interval 0<x<1.
>
> If the lottery is uniform on a sub-set of candidates, then Q gauges the PR
> Quality of that slate of candidates.
>
> In particular if that subset is a singleton, then Q is a gauge of the
> quality of the single candidate as a representative of the electorate.
>
> If all of the ballots are voted approval style, and the lottery is
> restricted to a single candidate C, then the function G(x) will be
> identically equal to the fraction of the ballots that approve C.  Therefore
> the candidate with the greatest approval will get the greatest value of Q.
> In other words, the approval winner will win in a single winner approval
> style election.
>
> It does not follow (and is emphatically not the case in general) that a
> slate of more than one candidate wins solely on the basis of most total
> approval.
>
> Is it true that a candidate rated at 100% on fifty percent of the ballots
> gets the same Q as a candidate rated at fifty percent on 100 percent of the
> ballots?
>
> No.  The latter gets a Q value of 50%, while the former gets the smaller
> value of Q equal to the integral over 0<x<1 of the integrand
>
> (1/2)^(ln(1-x)/ln(x))
>
> which is less than 48 percent.
>
> In general, if the lottery expectation is constant over all ballots, then
> that constant value will be the value of Q.
>
> Now try it on some of your own favorite lotteries.
>
> In another email I'll explain where the exponent of the integrand comes
> from.
>
> My Best,
>
> Forest
>
>
>
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