Randomizing in [EM] not infallible, but too attractive

Hugh R. Tobin htobin at earthlink.net
Wed Oct 7 00:00:25 PDT 1998

(Having composed this message, I now see that Blake has already made the
points and provided another example.)

Markus Schulze wrote:

> The problem with Condorcet[EM] is the fact, that the random-fill
> strategy _always_ works. The random-fill strategy cannot
> back-fire. That means: A voter is _never_ punished for using
> this strategy, but _sometimes_ rewarded for using this strategy.
> I would never have criticized Condorcet[EM], if the random-fill
> strategy had worked only on average.
> [Random-fill strategy means: Even if you don't have different
> opinions about your least favourite candidates, give different
> ranking to your least favourite candidates.
> The random-fill strategy always works, because the worst defeats
> of your least favourite candidates _sometimes_ increase but
> _never_ decrease, if you rank these candidates differently.]
> Markus Schulze

I agree that the random-fill issue makes Condorcet [EM] unacceptable. 
But to be precise, the strategy could backfire, because random votes
will not balance exactly so as to be equivalent to half-votes.  Suppose
that there are 2 voters with preference profile A>B=C, and that if each
votes that way, then there is a circular tie B>A>C>B in which A is the
[EM] winner, even though A's loss to B is by a wider margin than B's one
vote loss to C. (This may happen due to votes with the profile A=B>C,
even if everyone follows random-fill at the bottom of ballots.)  Then it
is possible that if each such A voter randomizes between B and C, and
each happens to pick A>B>C, B will be made Condorcet winner. 

2 A>B=C
4 A>B>C
9 B>A>C
10 A>C>B
10 B>C>A
7 A=B>C
10 C>B>A
11 C>A>B

B 29 - A 27; A 32 - C 31; C 31 - B 30.  A wins in EM.  But change the
top two voters to A>B>C, and B wins.  I hope I have the numbers right.

We should criticize [EM] even though randomizing works only on average. 
It need not be infallible to distort the balloting and put a premium on
knowledge of insincere strategies.

-- Hugh Tobin

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