[EM] Probability of Condorcet Winners (was: Re: Approval Voting fish (3), selected replies)

[Bart Ingles]

Norman Petry wrote:

> 1) The frequency of Condorcet winners declines gradually as the number of
> candidates increases.  Extrapolating from my data, I expect that an election
> having a "0.999999999999999999999999999999999999999999999" probability of no
> Condorcet winner would require tens of thousands of candidates.
> 
> 2) Increasing the "dimensionality" of the data reduces the likelihood of
> voting cycles. The one-dimensional case is the most difficult, always.
> 
> 3) As truncation increases ("involvement" approaches 0), voting cycles
> become slightly more probable.
> 
> 4) There will be no Condorcet winner about 5% of the time, under most
> reasonable assumptions.  These few cases will require a suitable tiebreaker,
> and as Mike has pointed out, we have several good ones to choose from.
> 
> Of course, like any simulation, mine is based on certain assumptions, and
> the results can be questioned on those grounds.  The basic assumption I have
> made is that a sincere, but somewhat "lazy" electorate is voting.  That is,
> voters will always rank according to sincere preferences, but each voter
> will truncate his/her ballot at some randomly-determined point.  For these
> results to be similar to real life, the specific method chosen should be as
> strategy-proof as possible, so that the expressed preferences are close to
> the sincere preferences.


Adding a 'random society' component to the model would increase the
probability of cycles.  It looks like a purely random model reaches 50%
at around 10 or 11 candidates (it would still take many more to reach
the figure shown above, though).

Of course there is the question of whether a real-life setting that
appears to be partially random is really just one with many unidentified
dimensions.  The more purely dimensional the model, the worse FPP and
STV are at picking Condorcet winners.  This is especially true if
candidates are competing heavily to position themselves as center
candidates, since FPP and STV have a tendency to squeeze out centrists.

Bart