[EM] Verification of a voting outcome for VoteFair.

[Kristofer Munsterhjelm]

On 04/03/2012 07:22 AM, Richard Fobes wrote:

> Conclusion 11: VoteFair ranking is consistent with the Condorcet-Kemeny
> method because the Condorcet-Kemeny method does not specify an overall
> ranking, nor does it specify who should win this election. (It only
> specifies that choice A is the lowest-ranked choice, and this is
> consistent with the VoteFair ranking result.)

Is that true? Speaking of Kemeny as an optimization method, the problem 
is specified as:

Determine the transitive ordering of all candidates,
so that the sum of, for each pairwise preference X consistent with the 
ordering, the number of voters agreeing with preference X, is maximized.

(At least in absence of ties)

That seems to make it pretty likely that Kemeny specifies a full 
ordering. It has to, in order to calculate the score that is to be 
optimized.

But anyway, I'll try to find an example where:

- VoteFair elects A,
- VoteFair has no ties in its social ordering,
- Kemeny finds another candidate X as the winner,
and
- There is no Kemeny-optimal ordering that puts A first.

Would that suffice to show that VoteFair isn't Kemeny?

> Conclusion 12: VoteFair ranking calculates a fair result within the
> limitations of the preference information available, and does so within
> the context of the goal of maximizing the Condorcet-Kemeny sequence score.

It doesn't actually maximize that sequence score, however; it falls one 
short. It does provide the same winner as one of the sequences that do, 
I see that.

> Thank you Kristofer for this interesting case!
>
> This is an excellent example of the unclear (muddled) preferences that I
> have referred to in other messages. It clarifies what I've said before,
> which is that if there were a 50-candidate election that had this kind
> of circular ambiguity throughout the candidates (which is what can make
> it harder to quickly find the highest sequence score), and if VoteFair
> ranking failed to find the sequence with the highest score (assuming
> only one such sequence), then a runoff election between the fully
> calculated Condorcet-Kemeny winner and the VoteFair ranking winner would
> be difficult to predict.

You could say the same of other Condorcet methods. E.g. you could say 
"If Ranked Pairs fails to find the winner according to Kemeny, the 
outcome of a runoff election between the fully calculated Kemeny winner 
and Ranked Pairs would be difficult to predict". Still doesn't save 
Ranked Pairs from not being Kemeny, though! :-)