## More on the reversal problem

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• The reverse of a vote is the vote that would logically be cast by the same voter if the goal were to elect the worst, rather than the best, candidate from that
Message 1 of 50 , Mar 24, 2013
The reverse of a vote is the vote that would logically be cast by the
same voter if the goal were to elect the worst, rather than the best,
candidate from that voter's point of view. We would expect a proper
voting system to reverse the order of the resulting tabulation of
results if all votes are reversed. E.g., if, for a given set of
ballots, the order in which the candidates finish is A, B, C, D, then,
if all votes are reversed, the finishing order should also be
reversed: to D, C, B, A.

Consider how various voting schemes handle reversals.

1. Score voting. If the range is from 0 to k, then, if a vote assigns
x to a candidate, the assignment changes to k-x on the reversed
ballot. It is obvious that reversing all ballots will properly invert
the order of the election results. For c candidates, if candidate
finishes in ith place, then for the reverse ballots, that candidate
will finish in position c-i+1.

2. Approval voting. This is the score voting case where k=1, so it too
handles reversal properly. (In the reverse vote, every approved
candidate is not approved , and every non-approved candidate is
approved.)

3. Borda can be considered as a special case of score, where the score
assigned by the voter to a candidate is limited to a rank. (In one
version, a candidate assigned rank r is given the score c-r). So Borda
also handles reversal properly.

4. IRV. Fails big time, in that it is possible that reversal
(reversing the ordering of all votes) won't change the winner. In the
example below, both first and last place candidates remain the same

5 B>C>D>A
4 C>D>A>B
3 A>B>D>C
2 D>A>B>C

Trying out a bunch of IRV examples of 3 or 4-candidate races, I have
the impression that the likelihood that the results of reversing all
ballots is very often (maybe more than 50% of the cases) not a correct
reversal of the results. E.g., look at the IRV election below.

6 A>B>C>D
5 B>C>A>D
3 C>D>B>A
1 D>C>A>B

The IRV results are: BACD. For reversed votes, the finishing order is
DABC (as opposed to the actual reversed order, DCAB). Perhaps some
simulations might be run to get a handle on the probabilities.

6. Plurality. We can't talk even about the reversal behavior for
plurality voting because there is no valid reverse of a plurality
vote. E.g., in a 4-candidate race, the reverse of a vote for A would
logically be the set {B, C, D}. This would be a valid approval vote,
but would be an over-vote in a plurality election.

This reflects a major weakness of plurality voting; voters can't vote
maximally to defeat one of the candidates. This can be done partially
in a ranked voting system, by listing the hated candidate last. But
this is not as strong as what can be done in score, where the hated
candidate can be given a 0, and all other candidates given the top
score. Or under approval, where all candidates can be approved, except
for the hated one.

Steve
............

Stephen H. Unger
Professor Emeritus
Computer Science and Electrical Engineering
Columbia University
............
• ... That would certainly be true, does not contradict what I wrote, and the concept of repeated voting with lowered approval cutoff was extensively discussed
Message 50 of 50 , Apr 6, 2013
At 08:12 AM 4/5/2013, Bruce Gilson wrote:
>On Fri, Apr 5, 2013 at 9:38 AM, Abd ul-Rahman Lomax
><abd@...>wrote:
>
> > Bucklin appeared, in many analyses, to not pass reversal because of
> > mindless ranking without regard for approval.
> >
> >
>In fact, when Steven Brams discusses Bucklin in his book, *Mathematics and
>Democracy*, mostly in conjunction with his nearly identical system called
>"Fallback Voting," he makes it clear that only approved candidates are
>ranked by the voter.
>
>So at least one person outside this group has noted this.

That would certainly be true, does not contradict what I wrote, and
the concept of repeated voting with lowered approval cutoff was
extensively discussed by students of approval voting.

However, reviewing Mathematics and Democracy, I'm struck by what
seems to be entirely missing from it: any consideration of repeated
ballot, seeking an explicit majority, whether absolutely (Robert's
Rules) or as a goal (i.e., runoff systems that terminate, usually on
the second ballt, with a plurality result if a majority is not found
-- or that *force* a majority by disallowing any other votes that for two).

It's a glaring omission. It's not just Brams. The characteristics of
runoff voting were widely ignored in voting systems discussions,
including Bayesian Regret calculations that assumed the same voting
electorate with the same preferences. I can think of no example of
someone proposing runoff voting using an advanced voting system,
until I started doing it, and that blind spot seems to have existed
in the past.

Strangely, the actual behavior of runoff voting and how it differed
from IRV was either ignored, or was *explained* by some deficiency in
turnout, or manipulability by monied interests, ignoring the actual
situations. Yet Robert's Rules of Order is totally aware of the
problem of *not* holding a real runoff, using IRV. Their concern is
not abstract, they are parliamentarians. They also criticize IRV for
center squeeze, and their own preferential voting process *still
seeks a true majority or the election must be repeated.*

Almost by accident, in Arizona (it wasn't an accident, we have a man
on the ground there, who did yeoman work), we may be getting
Approval/Runoff, with a mandatory top two runoff. Technically, of
course, this is Top Two RAnge, which outperforms Range in simulations
with realistic voting patterns. We can probably do better, but this
is very good to start! (I've suggested that Bucklin for the primary,
at least, is an obvious move.)

(Pure Range with *accurate and commensurable* utility expression is,
by definition, optimal for summed social utility. But there are few
ways to get that expression. Interestingly, an inconvenient runoff
election does push results toward absolute utility, because if the
choices are not greatly different in utility, the voter has less
motivation to vote, hence expressed votes will tend to have higher
preference strength behind them.)

Reading over Brams I see that PAV is essentially what I've proposed,
but I've suggested a true hybrid ballot that allows equal ranking.
He's suggested, essentially, a ranked ballot with an explicit
approval cutoff. I've simply suggested using a Range ballot with
midrange as explicit approval cutoff. That allows ranking within the
approved and disapproved sets, yet the "sincere" ballot is easy to
vote, except for the *intrinsic difficulty*, deciding how to
categorize the candidates, which gets tougher when there are three or
more. (And there are always, in fact, many, if write-ins are
allowed.) Runoff systems make the decision in the primary a bit
easier, approval becomes a binary decision: would I prefer the
election of this candidate to a runoff being held?)

Brams also mentions voting after a series of pre-election polls. If
we assume that the polls are accurate, this is essentially repeated
ballot, but based on a sample instead of the full electorate.
However, that assumption can be a problem!

If, instead of a poll, it's a real election, in a runoff system
primary, we can be reasonably confident that a majority result is
real! Multiple majorities are a problem, but the hybrid ballots I've
suggested can find ways of making that choice, if it's considered
excessively inconvenient to refer it to the voters in a runoff.

(Polls should not simply collect favorite information, that will
distort them. Range polls are much better, that's been obvious for
some time. A Range 2 poll -- +/-, default 0 -- showed what
vote-for-one polls could not show, true relative position of
candidates, and strength of support and opposition.)
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