[EM] Consistency, Truncation, etc. (was CR ballots, etc.)

Jobst Heitzig heitzig at mbox.math.uni-hannover.de
Wed Sep 26 02:19:18 PDT 2001

I would like to defend my opinion that voters should have the option 
to express "undecidedness" about certain pairs of alternatives, or, in
other words, to "abstain" from some but not all pairwise decisions.

First of all, I think one should distinguish at least to situations:
1) Few alternatives (less than 10, say), no possibility for write-ins
2) Many alternatives and possibly write-ins

1) In the first situation, voters know (the names of) the alternatives. A 
voter might be well-informed to make up a complete ranking. She also might
not care at all. Or she might have strong feelings concerning some
pairings but at the same time feel unsufficiently informed about some
other alternatives. In this last case, she either might want to assure
that these unknown alternatives only get a chance when they have a strong
support, or she might want to delegate the evaluation of these
alternatives to the other voters - this being perhaps mostly a question of
how much she trusts the other voters to make a good or correct decision.

Let me give an example: 
Two extreme candidates A,B, and an "intermediate" one C. 
14 voters have A>C>B
35 voters have A>B and don't know what to make of C
14 voters have B>C>A
35 voters have B>A and don't know what to make of C
 2 voters have C>A=B.
This might occur when, before the election, the media focused on A and B,
so that many voters are uninformed about the moderate candidate C.

Now what happens when the 70 voters are deemed to rank C last when they
don't want to rank him equal to their supported candidate?
Under plurality and approval, A and B would both get 49 votes, C only 2.
Under IRV, there would be the same tie. C would also be both the Borda
loser and the Condorcet loser.

On the contrary, suppose the 70 undecided voters would wish to delegate
the decision about C to the other voters and would be allowed to express 
this wish by not ranking C. In other words, the expressed preferences
would be given by the following quasi-orders:

   A                 B
   |      A          |      B          C
   C  ,   |   C  ,   C  ,   |   C  ,   |
   |      B          |      A         A=B
   B                 A

 (14x)    (35x)    (14x)    (35x)    (2x)

Let us analyse this in the following way. The matrix of strict preferences
is this:

   >   A   B   C             
   A      49  14
   B  49      14
   C  16  16

49 support the argument that A is better than B, but this is countered by
an equally strong argument that B is better than A, so there is no
"social" decision between A and B. But there is one between C and both A
and B: More voters prefer C to A or B, resp., than vice versa. Hence C is
an acceptable compromise! (I have formalized such reasoning in my paper.)

|| What I mean to show by this is that how the situation for the
|| candidates looks like when we analyse the ballots, highly depends on
|| whether we let the voters abstain from certain pairwise decisions or
|| force them to rank all alternatives last about which they feel
|| unsufficiently informed! 

This is not to say that truncated ballots should be interpreted as
abstention (although I suggested just that in my last posting...). But
they should neither be interpreted as ranking all others last. They should
just not be used without an option for the voter to make clear what he

Now for 2): Of course we cannot require the voters to answer a question
about each pair of alternatives when there are too many and perhaps even
some the voters does not know about when he votes (that is, write-ins).
However, even when truncated rankings are used on the ballot, one can
still ask the voter if he wishes to rank all yet unmentioned alternatives
(including all write-ins) last or if he wishes to abstain from these
pairwise decisions. 

In a more sophisticated method, a voter might be given the possibility to
express her preferences like this:

       / \
      /   \       C
     B1   B2     /
      \   /     /
       \ /     /
     (rest)   /
        |    /
        |   /
        |  /
(In this, A would be the voter's preferred party's candidate, B1 and B2
would be the candidates of the second major party, and C could be the
voter's husband)

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