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

[Rob Lanphier]

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.
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