[EM] Approval-Condorcet hybrid encouraging truncation

[Steve Eppley]

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