Re: Questions about Majority-Beat vs Plurality-Beat Condorcet

On 2026-04-05 15:29, Toby Pereira via Election-Methods wrote:
> My understanding was that Maximal Lotteries (a non-deterministic 
> Condorcet method) did pass participation. 
> https://en.wikipedia.org/wiki/Maximal_lotteries 
> <https://en.wikipedia.org/wiki/Maximal_lotteries>

Here's my attempt to check Moulin's proof as recounted by Schulze: 
http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html

using https://voting.ml/ to calculate maximal lotteries.

https://voting.ml/?profile=3ADBC-3ADCB-4BCAD-5DBCA
gives the following distribution:
	p(A): 7/15 (0.42)
	p(B): 1/3  (0.33)
	p(C): 0
	p(D): 1/5  (0.2)

B is elected with positive probability, so adding 6 BDAC voters:

https://voting.ml/?profile=3ADBC-3ADCB-4BCAD-5DBCA-6BDAC

makes D win with certainty due to Condorcet.

This violates the participation criterion as stated at 
https://electowiki.org/wiki/Participation_criterion.

As I understand it, there's a utilitarian weakening of participation 
that maximal lotteries passes, but I don't know precisely how it is 
defined. https://pub.dss.in.tum.de/brandt-research/fishburn_slides.pdf 
gives it as follows

"No agent can obtain more expected utility (for all vNM representations) 
by abstaining from an election".

But if "for all vNM representations" means "for every representation", 
this seems wrong, because the 6 BDAC voters could have relative (vNM) 
utilities:
	B: 0.97 + 6 epsilon
	D: 0.01 - epsilon
	A: 0.01 - 2 epsilon
	C: 0.01 - 3 epsilon
In this case, their utility if they don't vote is roughly 1/3; after 
they vote, it is 0.01 - epsilon.

On the other hand, if it means "there exists at least one utility 
assignment consistent with the new voters' rank so that they are not 
harmed", then the criterion is very weak.

BTW, since the maximal lotteries Wikipedia page was in large part 
written by our all-time ML fan CLC, who has elsewhere demonstrated a 
need to win his arguments by any means necessary, I would be suspicious 
of what it says. At least I would be unsurprised if it lacks nuance.

-km
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