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

[Kristofer Munsterhjelm]

On 2025-03-05 06:45, Rob Lanphier 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.

That's pretty straightforward, isn't it? Suppose there are two 
candidates, A and B. The possible Approval ballots are:
		1. (nobody)
		2. A
		3. B
		4. A, B

If we suppose that voters want to have some effect on the count by 
voting, there's no reason to cast ballot number one or four. So one can 
either vote for A or for B, which is the same as in FPTP.


As for ties, it's hard to tell. Let's say we have two major candidates 
and a bunch of smaller ones. Then if the smaller ones can't win under 
Approval, the tie rate would be the same: in FPTP, the voters vote for 
their preferred frontrunner to keep the other side from winning; and in 
Approval they vote for their preferred frontrunner plus everybody they 
prefer to that frontrunner.

If there are more than two viable candidates, then FPTP will get 
chaotic. More candidates could tie at the top in Approval than in FPTP, 
depending on how strategic the voters are.

On the FPTP side, you could argue that, with strategic voters, some 
butterfly flapping its wings would cause two of the candidates to be 
seen as "viable", then the self-fulfilling prophecy would reduce to the 
two-party scenario above. (But it doesn't have to; e.g. Perot in 1992.)

On the other hand, for Approval, if there's repeated polling and a 
Condorcet winner, the strategic equilibrium *could* make that CW the 
undisputed Approval winner, which would reduce the tie rate. Whether 
that happens depends on whether Approval indeed does home in on the CW. 
If it doesn't, then the fact that there are numerous viable candidates 
would imply that there are more pairs of candidates who are in the range 
to produce a tie, and then having more candidates would *increase* the 
tie rate.

There are also longer term effects. You might get multiparty rule - 
this, I think, depends on how serious the Burr dilemma would be in 
practice - with the candidates clustering closer around the center. If 
it's harder to tell the center candidates apart, that would also 
increase the chance of a tie.

But then some of the candidates might decide to stand out from the crowd 
by making a particular issue "their thing". That would draw the voters 
into more clearly separated clusters which could, in turn, reduce tie rates.

-km