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More on the reversal problem

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  • Stephen Unger
    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

      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
      when all votes are reversed.

      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.


      Stephen H. Unger
      Professor Emeritus
      Computer Science and Electrical Engineering
      Columbia University
    • Abd ul-Rahman Lomax
      ... 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
        > > 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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