[EM] Probability of ties in approval voting vs FPTP?

[Etjon Basha]

Hi Rob,

I suppose you're interested in the chance of tied-at-the-top scenarios,
instead of any ties. If the later, I'm sure there will be many candidates
with exactly 4 friends who will get exactly 5 votes.

In the former case, I can't really see why increasing the number of
candidates would decrease the chance of ties if we also increase the
voter's ability to vote for many candidates. It's a wash.

In general, I think of Approval as a more efficient search algorithm that
looks for exactly the same ideal winner as FPTP.

Given the higher efficiency and the zero cost of fielding candidates close
to the supposed median, I would expect ties to be somewhat more likely.

I love jousting with the reasoning models, though I stick to the free ones.
My experience with Gemini and DeepSeek is that they are competent ar
critiquing a theory of mine, but still can't see major flaws I myself find
out about later on, though they will acknowledge these if brought to their
attention.

My own 2 cents.

Regards,

On Wed, 5 Mar 2025, 4:46 pm Rob Lanphier, <roblan at gmail.com> wrote:

> Hi folks,
>
> One of the debates that has broken out on the Center for Election
> Science's Discord server is a debate about the likelihood of ties in
> approval voting elections vs the likelihood of ties in FPTP elections.
>
> I've been playing around with ChatGPT, and learned a lot while going back
> and forth with it.  In short, it would seem approval reduces the risk of
> ties when there are more candidates, with a significant caveat (which I
> note below).  Since approval has fewer problems with vote splitting, it's
> likely to have more candidates.  Thus the folks that believe that ties are
> less likely in approval have a point that I'll have to concede.
>
> However, some of the models get skewed in a two-candidate election because
> naive models consider votes for "", "A", "B", and "AB" to be different,
> even though "" and "AB" are effectively identical votes (effectively
> abstentions).  After I twisted ChatGPT's arm, it conceded that
> two-candidate elections are identical under approval and FPTP, and provided
> me a proof.  I haven't stepped through the proof yet, but I'm inclined to
> believe it.  There was a lot of truthiness to it, at first glance.
>
> The caveat noted above: when I pressed ChatGPT to update its model to
> allow for multiple sequential elections (where voters and candidates adjust
> their strategy based on previous elections), then approval elections become
> MUCH more likely to produce ties.  My speculation is that it is because the
> candidates adopt consensus positions (i.e. they move toward the center of
> the distribution).  Since approval doesn't punish clones, it seems the
> long-term equilibrium settles around candidates clustering in the middle of
> the N-dimensional spectrum, regardless of the value of N, and regardless of
> the number of candidates.  Approval's relative lack of vote splitting also
> makes it very clone friendly.
>
> In my ChatGPT discussion, we agreed that the simulations provided also
> provide strong evidence of Duverger's Law applying to FPTP, but not
> approval.  With FPTP, candidates benefit by clustering around two points
> rather than one point, but with approval, the best strategy for candidates
> is to find a single point in the center.
>
> Many of you are more stubborn than ChatGPT, and more likely to push back.
> I'm curious where all y'all stand on this topic.  Thoughts?  Is ChatGPT
> hallucinating again?  Are ties more likely or less likely under approval
> voting when compared to FPTP voting, or is it about the same?
>
> Rob
> p.s. Email me privately if you want an invite to the Center for Election
> Science's Discord server.  They used to have an open invite URL at
> https://electionscience.org/discord , but that wasn't working the last
> time I checked.
> p.p.s. Those who want to join a Discord server and talk about this, but
> not joint the CES server, I'd encourage you to join the Electorama server:
> https://electorama.com/discord .  This server isn't as active, but it's
> got a lot of smart people on it.
> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>
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