[EM] Truncation

[Adam Tarr]

Bart Ingles wrote:

>Adam Tarr wrote:
> > Specifically, there is the remarkable fact that a voter in a winning
> > votes-based Condorcet voting system can NEVER be hurt by fully expressing
> > their preferences.  There are cases where fully voting your preferences can
> > fail to help you, but it can never actually hurt you.
>
>Never is a strong word.

True, and I lack a rigorous proof, but every empirical example I have seen 
supports this.  Your example is no different...

>  How about the following:
>
>(projected vote percentages shown; assumed accurate to within +/- 5
>percentage points)
>
>45%   A  B  C
>  5%   B  A  C
>  5%   B  C  A
>45%   C  B  A

There's no sense in talking about uncertainty and ties; it only confuses 
the issue.  Let's just assume one or the other camp has more votes, and see 
if in that light, either side has an incentive to truncate.  If neither 
side has such an incentive, then neither side has that incentive in the 
toss-up case as well (since they know they either have more votes or less 
votes than the other guy).  Without loss of generality, I'll give A the 
edge, which gives us:

46 ABC
5 BAC
5 BCA
44 CBA

Pairwise votes are:

B 56 > C 44
B 54 > A 46
A 51 > C 49

In this case, B is the Condorcet winner.

If both sides truncate we get

46 A
5 BAC
5 BCA
44 C

Pairwise votes are:

A 51 > C 49
A 46 > B 10
C 44 > B 10

Now A wins the election in either winning votes or margins (don't stop the 
presses yet).

Now, if only the A camp truncates:

46 A
5 BAC
5 BCA
44 CBA

Pairwise votes are:

B 54 > A 46
A 51 > C 49
C 44 > B 10

B wins the election in winning votes, and C wins the election in margins.

Finally, if only the C camp truncates:

46 ABC
5 BAC
5 BCA
44 C

Pairwise votes are:

B 56 > C 44
A 51 > C 49
A 46 > B 10

A wins the election in winning votes or in margins.

OK, let's look at the decision matrix.  Here is the pairwise matrix of 
decisions for each camp, and the candidate elected, for each method:

(I apologize in advance if the tables look lousy.  Try cutting and pasting 
into a text editor with uniform character spacing if it looks bad.  I used 
the "terminal" font type if that helps.)

The top row is the tactics of the ABC faction, the left column is the 
tactics of the CBA faction.  T = truncate, NT = do not truncate.

Margins methods:

    | T | NT |
---|---|----|
T  | A | A  |
---|---|----|
NT | C | B  |
-------------

Winning Votes methods:

    | T | NT |
---|---|----|
T  | A | A  |
---|---|----|
NT | B | B  |
-------------

OK, so what can we conclude from this?  If the CBA voters truncate, they 
always get A elected in either system.  This is a "strictly dominated 
strategy" to use the game theory name.  There's no way the B voters should 
truncate, regardless.

In winning votes methods, truncation for the ABC voters makes no difference 
(i.e. does not hurt them, even if it fails to help them).  In margins 
methods, truncating can prove costly for an ABC voter.  This is neither 
here nor there for the purposes of my analysis; I can show you a 
counter-example where truncation can help in margins cases.

The point is, nowhere here do we get any suggestion that a voter in a 
winning votes method can be helped by truncation.  My initial contention 
still stands.

>  But then I don't see truncation as necessarily a bad thing. If 
> truncation can defeat a "hated middle" candidate, it addresses my main 
> misgiving about the Condorcet methods.

Much in the same way that we can't differentiate between the indifferent 
voter and the lazy voter, we cannot distinguish between the "respected (if 
unglamorous) compromise middle" and the "hated (yet still) compromise 
middle".  Smart CBA voters in an approval election will still approve B, to 
defeat A, anyway.  What method would actually prevent B from winning when 
the voters act in a logical manner?  Even plurality and IRV encourage CBA 
voters to dump C for B if they have perfect information.

-Adam
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