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

[Rob Lanphier]

Hi Etjon,

You're correct that I'm only interested in tied-at-the-top scenarios (i.e.
ties that matter in the case of approval).  I was almost willing to drop
the subject yesterday (after a debate on the CES Discord that had dated
back to February), but on a lark, decided to run my intuition by ChatGPT.
After ChatGPT initially said I was wrong, I became more interested.

My hunch is that increasing the number of clone candidates increases the
odds that there will be two popular clones that tie at the top.  Most of my
back and forth with ChatGPT seems to be the two of us trading hunches with
one another, rather than coming up with anything definitive.  Here's the
proof that it provided that two-candidate elections are identical in FPTP
and approval:

>
> *Two-Candidate Elections Have Identical Tie Probabilities in FPTP and
> Approval Voting*
> *Theorem:*For a two-candidate election, Approval Voting and
> First-Past-The-Post (FPTP) produce the same probability of a tie, assuming
> that each voter votes sincerely and independently.
>
> *Proof:**Define the Election Setup:*
>
>    - Two candidates, *A* and *B*.
>
>
>    - *n* voters, each with some *utility function* that ranks A and B.
>
>
>    - Each voter independently decides how to vote based on this
>    preference.
>
> *FPTP Voting Mechanism:*
>
>    - Each voter selects exactly one candidate, choosing the one with the
>    higher utility.
>
>
>    - The probability of a tie is determined by the likelihood that *exactly
>    half* of the voters choose A and the other half choose B.
>
> *Approval Voting Mechanism:*
>
>    - Each voter approves a candidate if their utility exceeds a personal
>    approval threshold.
>
>
>    - In a *two-candidate race*, every voter has *only three possible
>    voting behaviors*:
>
>
>    - Approve only A.
>
>
>    - Approve only B.
>
>
>    - Approve both A and B (which is equivalent to abstaining in terms of
>    affecting the outcome).
>
>
>    - The only votes that *change the election result* are those where a
>    voter picks only A or only B.
>
>
>    - If the proportion of voters approving only A vs. only B is *exactly
>    50-50*, a tie occurs.
>
> *Equivalence of Tie Probability:*
>
>    - The critical observation is that *"approving both" and "approving
>    neither" do not change the relative count between A and B*.
>
>
>    - Thus, the set of voters *who actually cast a deciding vote* follows
>    the same 50-50 condition as in FPTP.
>
>
>    - Since the probability of a tie in FPTP depends only on the exact
>    split of votes between A and B, and this is the same deciding factor in
>    Approval Voting (ignoring those who approve both), *the probability of
>    a tie must be identical in both systems*.
>
>
> *Conclusion:*For *two-candidate elections*, Approval Voting does not
> reduce or increase the probability of ties compared to FPTP. The
> probability is determined by the randomness in how voters split their
> preferences between the two candidates, and the Approval Voting mechanism
> does not introduce any additional bias that would change this.


Once ChatGPT proved to itself that two-candidate elections were equivalent,
it seemed more willing to entertain the possibility that 3-or-more
candidate elections might be equivalent-ish, though as I tried to think
through how to nail Jello to the wall, and get a proof of my conjecture
that 3-or-more candidates is also equivalent, I realized that they may not
be.  I'm now just vaguely unsure, even after spending a lot of time
exploring the topic.

ChatGPT has generally provided better answers for me, but I know that all
of the LLMs are rapidly improving (and likely in fits and starts at
different rates).  I've gotten lazy and rely on ChatGPT more and more for
my LLMing (the same way I still frequently still rely on
Google...inertia).  I'd be really interested in hearing what you learn from
going back and forth with Gemini and/or DeepSeek on this topic.  Many LLMs
seem to love constructing short Python scripts to simulate something that
backs up their conjecture, and then using that as "proof" of their point.
It's worth pressing them for mathematical points if/when possible.  When I
tried asking Gemini about this topic, it immediately shut the conversation
down, stating "*I can't help with that right now. I'm trained to be as
accurate as possible but I can make mista**kes sometimes. While I work on
perfecting how I can discuss elections and politics, you can try Google
Search.*"  I've found Gemini needs to be tricked into discussing the
election methods.

Rob
p.s. check out my reddit post pointing back to this conversation:
https://www.reddit.com/r/EndFPTP/comments/1j3wor0/em_probability_of_ties_in_approval_voting_vs_fptp/

On Wed, Mar 5, 2025 at 12:30 AM Etjon Basha <etjonbasha at gmail.com> wrote:

> 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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