Opps – Correction

I made a mistake in an earlier post on this thread:

I stated that in a BTR-Score election with a 3-way cycle, the highest seed in the cycle would have a 50% chance of winning, and the two lower seeds would each have a 25% chance of winning.

This is incorrect. The highest seed in the cycle would always win.

Consider this BTR-Score tournament with a 3-way cycle:

Candidate A is the highest seed in the cycle.
Candidate B is the 2nd highest seed in the cycle.
Candidate C is the 3rd highest seed in the cycle.

In every such election, B and C will face off in the tournament; the winner will then face A in the deciding contest.

In a 3-way cycle, each of the candidates wins one match and loses one match to the other two candidates. The winner of the B vs C contest uses its only win to prevail; therefore, that winner has no chance of defeating A.

2 Possibilities:
A > B > C > A - B beats C then B loses to A
C > B > A > C - C beats B then C loses to A

A BTR-Score election with a Condorcet winner rewards majority.

A BTR-Score without a Condorcet winner rewards utility.

It would be possible, but extremely weird, for a BTR-Score election to have a 3-way cycle that does not include the top seed, the candidate with the highest total score.

@toby-pereira yes and that’s interesting in itself. I thought it would be about compensating disaffected majorities since that’s what the Chatbot said lol

@toby-pereira your interpretation is correct, and yes with an even number of voters and non-strict rankings, ties can occur and that can induce non-uniqueness. For example there may be two separate, disconnected Condorcet cycles of different sizes in the Smith set for instance.

With ties in other methods, the resolution of ties is typically standard because the set over which the ties occur are discrete—uniformly sample one from among the tied candidates. But for maximal lotteries, when non-uniqueness holds there is a continuum of admissible lotteries as you indicated. The analog of a uniform distribution in this case would be using Jeffrey’s prior, which is why I think that’s the “right” way to go.

But yes it ultimately doesn’t really matter how the maximal lottery used is chosen since they are all maximal, but that’s also kind of the issue, because a choice has to be made.

Hi @cfrank - as @Toby-Pereira mentioned, there is electowiki for consolidating ideas from over here. At this point, I'm already relying on some pretty substantial help from LLMs for the tools I use to put together the electorama news. The latest draft edition is here:

https://electowiki.org/wiki/ElectoramaNews/2026-January

The prose in the ElectoramaNews is pretty much all mine, but the code I've "written" to provide the summaries of each of the forums (like my summary of this forum) was mostly generated from a vibecoded script I run periodically. My hope is that I'll be able to provide a more-and-more sophisticated summary of the various forums every month, and maybe I can have my tools call the LLMs to provide the summaries of the discussions. Right now, the "summary" in many sections is basically just a link to every forum post or email or whatever.

Hopefully, by regularly publishing ElectoramaNews, we'll be able to drive some traffic over here to the Voting Theory forum.