[EM] A Proportionally Fair Consensus Lottery for which Sincere Range Ballots are Optimal

[Warren Smith]

On 11/21/09, Jobst Heitzig <heitzig-j at web.de> wrote:
> Even simpler is this:
>
> Method "Top-3 approval sincere runoff" (T3ASR)
> ==============================================
>
> 1. Each voter separately supplies
>    a "nomination" approval ballot and a "runoff" range ballot.
>
> 2. From all "nomination" ballots, determine
>    the options A,B,C with the top-3 approval scores a>b>c.
>
> 3. Let p be the proportion of nomination ballots
>    which approve of C but not of B.
>
> 4. If on at least half of all "runoff" ballots we have a rating
>    r(A) > p*r(C) + (1-p)*r(B), then option A wins.
>
> 5. Otherwise draw a "nomination" ballot.
>    If it approves of C but not of B, C wins, otherwise B wins.
>
> Most of the time, this will elect one of the top-2 approval options, and
> only rarely the 3rd placed.

--you could also use two range ballots (a "sincere" one and "insincere" one)
and change "approve of C but not B" to "scores C above B" in the rules.

> One can then also compute and publish some kind of "index of sincerity"
> by comparing the submitted approval and range ballots.

--now by comparing the two ("sincere" and "insincere") range ballots.

PROPERTIES:
I. Either the top or second-top or third-top range-voting winner,
based on the insincere ballots, will always win.
II. Strategic voters (as well as honest ones) will always provide
all-sincere scores (positive-slope linear function of true utilities)
on their sincere ballots.
III. The voters will elect the insincere range-voting winner EXCEPT
if a majority of them sincerely prefer a certain lottery to that
candidate, in which case the lottery winner wins.
III. So sincere voters will usually elect the sincere range-voting winner EXCEPT
when a majority of them sincerely prefer a certain lottery to that
candidate, the lottery winner wins.
IV. I therefore expect similar behavior and similar Bayesian-Regret
performance, to "range plus later top-2-runoff" voting.   I.e. good
performance.
V. The method is majoritarian, since any majority can rule by bullet voting
on both ballots.

If desired, one can bias the method so that it becomes arbitrarily
close to being equivalent to ordinary insincere range voting as
follows.  In rule 4, also make A win if
a biased coin toss (bias level q) is "heads."   Then make the
heads-probability q=0.999.

Also, one can bias it to make it arbitrarily close to equivalent to
ordinary insincere "range+later top-2-runoff" voting as follows.  In
the lottery that elects B or C with probabilities p and 1-p, instead
make p just be a random variable chosen
from [0,1] which has positive density everywhere in (0,1) but expectation value
0.999 [regardless of the ballots].

It should be clear that using such "biasing" techniques we can cause
this kind of voting method to be "arbitrarily close" to essentially
any voting method we want
(i.e. elects the same winner 99.9% of the time, where 99.9% is any
constant and can be made arbitrarily near 1)  but still inspire
sincere range voting on the sincere range ballots.


-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html