[EM] Forced strictly-dishonest strategy is common in Schulze-beatpaths voting

[Michael Poole]

Warren Smith writes:

> I will sketch a proof that, in Schulze beatpaths voting in "random"
> N-candidate V-voter elections (V-->infinity, N fixed):
>   with probability > a positive constant C (where C goes to 1 as N-->infinity):
>      at least a constant fraction K of the voters (where K goes to 3/4
> as N-->infinity)
>      will regard it as strategically forced that they order D>E for at
> least one
>      candidate-pair {D,E} for which they honestly prefer E>D.  By
> forced I mean,
>      they'll feel if they don't do this, they'll have lower expected utility.

Can you follow up with a proof sketch of the fraction of debaters who
have no ethical compunctions about inventing a scenario and then
arguing that they can prove how a supermajority of a population will
*feel* about that invented scenario?

Beyond the obvious inefficiency, one reason that current electoral
systems are loath to repeat elections (even in substantially similar
form, such as runoffs) is that repetition permits a variety of
strategic considerations between iterations, including focused
violence or intimidation.  Outlining the extent to which this is true
in "random" elections with arbitrarily large numbers of both
candidates and voters is not particularly informative.

As the number of candidates increases without bound, the "distance"
from any given point to the candidates, or between candidates, tends
to decrease -- much in the same way that distance between points
converges to unity in high-dimensional space.  Given that Condorcet
methods are susceptible to order reversal, that is exactly the kind of
scenario where you would expect it to be more likely to have an
effect, but the per-voter benefit averaged over "random" elections
goes down as the number of candidates goes up.

So accepting, arguendo, that 75% of voters might -- a posteriori --
gain expected utility from strategic order reversal, to conclude how
they would feel about that requires an argument that they care more
about the vanishingly small gain in utility than they do about honesty
in voting.

Michael Poole