[EM] the "meaning" of a vote (or lack thereof)

[Kristofer Munsterhjelm]

(Resubmitting to the list as Michael Allan suggested :-)

Michael Allan wrote:
> Warren Smith wrote:
>> --no. A single ballot can change the outcome of an election. This 
>> is true in any election method which is capable of having at least 
>> two outcomes.
>> Proof: simply change ballots one by one until the outcome changes. 
>> At the moment it changes, that single ballot changed an election 
>> outcome. QED.
> 
> Your proof is flawed, of course. It assumes the election method would
>  allow one to "change ballots one by one until the outcome changes". 
> Such gross manipulations are not permitted by the rules of any 
> election method. The rules grant to the voter a single vote, and that
>  is all.

I haven't been following the list all that closely of late because I
have been busy with other things, therefore I'm replying here instead of
to list. If you think my mail would be of use on list even given the
time delay, just say so and I'll put it up there, too :-)

Anyway, to prove that a single ballot can change the outcome of an
election, it is sufficient to find a single example where this is the
case. Such a case is easy to construct for most methods. For instance,
this will do in every method that passes Majority:

50: A > B
50: B > A

which is a tie. Now add a single A>B vote, or alter an B>A vote to
become A>B, and A wins. Hence a single vote altered the outcome of the
election.

This can also be used to validate Warren's proof. Say that we have one
set of ballots X_a, where A is the unique winner, and another set of
ballots X_b, where A is not the unique winner. Then by permuting X_a
into X_b one vote at a time, there will be a set of "adjacent" ballot
sets (differing only by a single vote). Call these X_a', and X_a'',
where X_a' has A as the unique winner, but X_a'' does not. Then if there
was an election, and the submitted ballots just happened to form X_a',
then the alteration or addition of a single ballot could turn X_a' into
X_a'', and then that would prove that a single ballot could alter the
outcome.

You might say that these voting situations are very rare indeed, so that
a single vote *most of the time* does not affect the outcome. However,
"most of the time" is not the same thing as "always".