[EM] Approval-Condorcet hybrid encouraging truncation

[Forest Simmons]

I must be missing something.  Could you give an example in which the
approval winner is not the winner of the method?

Forest

On Mon, 10 Mar 2003, Steve Eppley wrote:

> On 10 Mar 2003 at 11:36, Kevin Venzke wrote:
> > My recent "MinMax" message concluded with a
> > half-hearted attempt at a system combining Approval
> > and Condorcet.  I have a much better proposal now,
> > although I'm not entirely certain of its merits.
> -snip-
>
> I have another way of combining Approval and Condorcet,
> actually a family of voting methods, plus a criterion they
> satisfy that is stronger than Mike Ossipoff's "Strong
> Defensive Strategy Criterion" (a variation of which I call
> the Minimal Defense criterion).
>
>    1. Each voter is allowed to (non-strictly) order the
>    candidates from top to bottom, and optionally
>    may insert a "dividing line" anywhere in her ordering
>    (that partitions the candidates into two subsets,
>    those over the line and those under).
>
> Given a touchscreen voting interface, it would be
> straightforward to implement #1, since the dividing line
> could be dragged and dropped into the desired position just
> like any other candidate.  Given paper ballots that would
> be optically scanned, the following format would suffice:
>
>                 <--BETTER                  WORSE-->
>    Bradley          (X)   ( )   ( )   ( )   ( )
>    Nader            (X)   ( )   ( )   ( )   ( )
>    Gore             ( )   (X)   ( )   ( )   ( )
>    Bush             ( )   ( )   (X)   ( )   ( )
>    Buchanan         ( )   ( )   ( )   ( )   ( )
>    McCain           ( )   (X)   ( )   ( )   ( )
>    Dole             ( )   ( )   (X)   ( )   ( )
>    Keyes            ( )   ( )   (X)   ( )   ( )
>       DIVIDING LINE:   ( )   (X)   ( )   ( )
>
> Each voting method in the family constructs a social
> ordering consistent with the following:
>
>    2. For all pairs of candidates, say x & y, y is socially
>    ordered over x if the number of votes that rank "y over
>    x" exceeds the number of votes that rank "x over y"
>    and the number of votes that rank "y over the dividing
>    line over x" exceeds the number of votes that
>    rank "x over the dividing line over y."
>
> It's not actually necessary to construct such a social
> ordering, as long as the following condition is met:
>
>    For all candidates x, x must not be elected if there
>    exists a candidate y such that the number of votes
>    that rank "y over x" exceeds the number of votes
>    that rank "x over y" and the number of votes that
>    rank "y over the dividing line over x" exceeds the
>    number of votes that rank "x over the dividing line
>    over y."
>
> Assuming only one winner is to be elected, it's always
> possible to satisfy condition 2 (or the revised wording)
> since the subset of pairings that meet condition 2 is
> acyclic. (I have a proof of acyclicity, but it's tedious so
> I won't post it without a request.)
>
> The strong criterion satisfied by these methods is:
>
>    Sincere Defense: For all subsets X of the candidates,
>    all subsets C of voters and all candidates y,
>    if C includes more than half of the voters and
>    every member of C prefers y over every candidate in X,
>    then there must exist a way that the members of C
>    can vote that ensures all candidates in X will lose
>    and does not require any member of C to misrepresent
>    any preferences.
>
> Sincere Defense is stronger than Minimal Defense, which is
> stronger than Strong Defensive Strategy Criterion, since
> Minimal Defense and SDSC allow a majority coalition to
> misrepresent some preferences (by downranking candidate(s)
> to ensure their defeat).
>
> For an example of a preference order method that can be
> tweaked to satisfy Sincere Defense, MAM and other
> variations of Ranked Pairs can be tweaked to allow each
> voter to insert the dividing line in her ranking as in #1
> above, and to give utmost precedence to every pairwise
> majority that meets the condition in #2 above.
>
> I'm concerned that some voters wouldn't use the dividing
> line strategically as intended, and instead treat it as
> some sort of "sincere approval" dividing line.  In that
> case, the dividing line may not have much force because
> condition 2 wouldn't be met by as many pairwise majorities.
> For instance, some Nader voters might rank Gore and Bush
> below the line even though ranking Gore over the line would
> be more effective (by creating a majority voting "Gore over
> the line over Bush" that would ensure the defeat of Bush,
> who is the Nader voters' "greater evil").  So even though
> many voting methods could be tweaked to be in the family
> that satisfies Sincere Defense, only the best methods
> should be considered.  In particular, by tweaking a method
> that satisfies Minimal Defense, the "minimal defensive
> strategy" of downranking X can be simultaneously employed
> as a second line of defense.
>
> -- Steve Eppley
>
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