[EM] Questions about Majority-Beat vs Plurality-Beat Condorcet

[Kevin Venzke]

Thanks Joshua and Kristofer for trying to solve it.

Having now seen Moulin's paper, I see that for some reason both his definition of an
election method and his definition of Participation seem to be deterministic. I'm
not sure why. The proof doesn't seem to require it (at least as Markus and Douglas
Woodall interpret it). Moulin also never mentions determinism directly; he asserts
plainly "We show that every Condorcet consistent method ... must generate the
paradox among four or more candidates." and uses italics to emphasize, "This
criticism applies to *all* [italics] voting rules consistent with Condorcet's
principle" and "we show that *all* Condorcet consistent voting rules are subject to
the No Show Paradox ..."

Moulin didn't invent the paradox; maybe someone earlier defined things the same way,
or maybe they didn't.

A definition from Woodall that works more as I expect is this:

Participation: If further ballots are added that are all solidly committed
to the same set X of candidates, then the probability that the elected
candidate is in X should not be reduced

In other words if you add a ballot, the probability that the winner comes from the
top N ranks should not decrease, no matter what N is.

One could define participation criteria involving the underlying utilities. Although
I'd rather not have rank ballot criteria that can't be applied without knowing
utilities. Makes it hard to prove anything.

Kevin
votingmethods.net



Le dimanche 5 avril 2026 à 22:40:43 UTC−5, Joshua Boehme via Election-Methods <election-methods at lists.electorama.com> a écrit :
> Here's what I get:
> 
> #1: 7/15 A, 5/15 B, 3/15 D
> #2: D
> #3: B
> #4: A
> #5: 5/13 A, 7/13 B, 1/13 C
> #6: C
> #7: B
> 
> The following are cases where adding votes with candidate X at the top of
> the ballot causes X to go from a positive probability of winning to a zero
> probability:
> 
> #1 -> #2
> #1 -> #4
> #5 -> #6
> #5 -> #7
> 
> The general pattern here is that a lottery over the additional voters'
> first, second, and third choices switches to a definitive win by their
> second choice. Without knowing the underlying utilities of the voters,
> whether or not they prefer that resulting outcome is impossible to
> determine. [1]
> 
> [1] Once you go down that rabbit hole, it gets harder to stand by Condorcet
> / Smith as a strict requirement. Indeed, it's easy to construct examples of
> elections -- not necessarily in these particular cases -- where *every
> voter* prefers, say, a random ballot lottery to a Condorcet winner