do we have more info on this topic?

• Zero-Knowledge Proof (ZKP)
• using ZKP in voting systems

Here is an example:
https://github.com/microsoft/electionguard

==
https://medium.com/edge-elections/what-is-a-zero-knowledge-proof-aebe33cb47af
Verifying a secret’s truthfulness without revealing the secret itself
Have you ever tried to demonstrate to another person that you know a secret without revealing the secret itself? It might seem difficult to convince them that you have specific knowledge without telling them what that knowledge is. However, this is the exact scenario for many digital and online processes in which the system must verify that certain information is correct or true without disclosing the potentially sensitive content (such as personal data). This is precisely where zero-knowledge proofs (ZKPs) come in.

A ZKP is a cryptographic protocol between a prover, who we can call Peggy, and a verifier, Victor, in which the former tries to convince the latter that some statement is true without revealing any information other than the truthfulness of the statement. The theory behind ZKPs is explained rather well in the article How to explain zero-knowledge protocols to your children.

The story tells of a magic cave whose entryway forks into two dark winding passages: one to the left and the other to the right. Apparently, deeper into the cave, the two paths migrate back toward one another and almost connect to form a continuous, circuitous path, but remain separated by a wall. Therefore, if you decide to run into one passage, you must get out of the cave by turning around and taking the same passage back.

Nevertheless, the cave holds a secret: whispering the magic words “open sesame,” the wall between the passages opens and they become connected. Peggy tells Victor that she knows the magic words that open the wall, but he wants her to demonstrate that this is indeed true. In order to do so, Victor proposes a challenge: he will wait at the entry of the cave and will ask Peggy to go into the cave using the left-hand passage and to come out using the right-hand passage. This is only possible if Peggy knows the magic words, so if she successfully completes the challenge, Victor can be convinced that Peggy knows the magic words despite neither of them telling the other what the magic words are.

In this way, a zero-knowledge proof should satisfy the following properties:

Completeness: If the statement is true, the prover can succeed in convincing the verifier that it is true.
Soundness: If the statement is false, a prover cannot succeed in convincing the verifier that it is true.
Zero-knowledge: If the statement is true, a verify does not learn anything more about the statement other than that it is true.

How are zero-knowledge proofs used in online voting systems?
Zero-knowledge proofs are a key element when talking about verifiability or vote correctness in online voting systems (to learn more about security requirements for online voting, check out this article).

Vote correctness
Let’s think of a referendum in which the voter is presented with a list of questions, and they have to answer either yes or no to each one of them. In order to encrypt these answers, the voting device (a mobile phone, a laptop, etc.) understands the answer yes as a 1 and the answer no as a 0. In this kind of system, it is important to demonstrate that the value encrypted is indeed a 1 or a 0, in order to prevent a malicious voting device from voting more than once for a question, e.g.: encrypting a choice with the number 3, giving 3 yes votes to a single question. In order to do that, the system generates a zero-knowledge proof to prove that the value encrypted is either a 1 or a 0 without giving any clue about the exact choice that was encrypted.

Universal verifiability
In order to provide universal verifiability, an online voting system should generate evidence which enables anybody to check that the election results have been calculated from votes cast by eligible voters, and that these votes have not been manipulated during the counting process. This evidence is usually provided in zero-knowledge proofs.

Independently of the anonymization mechanism used during the counting process (such as mixing or homomorphic tallying), votes must be decrypted. Decryption is a sensitive operation that needs a private key to be executed. This key must be protected in order to preserve each voter’s privacy. If we want to verify that the decryption process has been done correctly and that no votes have been modified, added, or deleted during the process, zero-knowledge proofs are generated in order to demonstrate that the decrypted votes are those that were encrypted, without leaking any information about the election private key and the contents of each vote.

Zero-knowledge proofs are just one of the many security protocols that can be implemented in an election with online voting, and they specifically help to offer verifiability and a certain level of transparency while still protecting voter privacy and overall election integrity. If you’re interested in taking a deep dive into ZKPs and other security mechanisms and processes used for online voting, take a look at our resource center.

Assuming that these best practices are followed:
https://www.fairvote.org/best_practices_for_releasing_rcv_election_results

• Release a preliminary round-by-round tally on Election Night
• Continue to release preliminary tallies as more votes are counted
• Conduct vote total checks with each release of preliminary results
• Publish the full ballot record so that anyone can verify the result
• Make use of existing tools for visualizing RCV results
• Clearly communicate expectations, timelines, and results

To what extend are these points still valid / invalid:
● Less accountability due to required centralized tabulation. RCV presents serious barriers to local oversight of elections, compromising checks and balances. Local elections officials are unable to generate the ballot sub-totals required for local audits. In contrast, all other voting methods considered here allow local tabulation (e.g, precinct-level tallies), which ensures that there can be checks and balances with a multitude of witnesses at each local tally and ease of audits at the local level.
● Lack of precinct summability results in compromised election security. Ballot centralization presents serious issues for election security.
Complexity in the many round tabulation process means that any errors or tampering which may occur are much harder to detect, and that less people overseeing the election would be able to identify an issue even if they were looking in the right place.

Assuming that each precinct publishes periodically preliminary RCV totals (publishing 'raw data' maybe a challenge) - questions:

• Q1: Theoretically we should be able to relatively easilly audit the election - right?
• Q2: Are the 'RCV best practices published by FairVote (releasing RCV election results 'good enough'?
• Q3: What else is missing in the 'RCV audit best practices'
• Q4: To what extend the challenges related to 'RCV transparency / security challenge / overseeing the election' remain in place even after publishing the best practices?
• Q5: I do not understand this objection "Local elections officials are unable to generate the ballot sub-totals required for local audits".

As a devils advocate - security challenges are present with every voting method (including STAR Voting)
finding errors, potential tampering, etc.

I agree that a central tabulation requirement for RCV is a huge bummer - but security issue / auditing / checking totals - is not that different than any other voting method - right?

Maybe I do not understand the challenge though:
RCV example - precinct 1
3: B>A
1: A>B
RCV example - precinct 2
2: B>A
8: A>B

RCV example - totals from all precincts:
5: B>A
9: A>B

How is it different (from Audit perspective / checking totals) from STAR Voting:
STAR Voting example Precinct 1
3: B=5, A=4
1: A=5, B=4

STAR Voting example Precinct 2
2: B=5, A=4
8: A=4, B=4

STAR Voting example - totals from all Precincts:
5: B=5, A=4
9: A=5, B=4

AI Claude - it helped me to create this response:

The distinction you're drawing is real, and I think it's underrated: formal Smith compliance is a property of the ballot→winner map, so it can't say anything about sincere preferences the method never sees. Worth separating out explicitly.

Two things I'd push on before the IRV conclusion, though.

First, the baseline. Ranked Pairs and Stable Voting elect the sincere CW 100% of the time on sincere ballots; IRV doesn't. So a shared ~70% isn't a tie — for the Condorcet methods the entire shortfall requires someone to have lied, while for IRV a chunk of it is the method missing with everyone honest. Which raises the diagnostic question: what's your profile generator? If IRV is reaching 70% under adaptive strategy, the model is probably impartial culture. That's where IRV always looks best (in my own runs it beats STAR there at 3 candidates, 96.7% vs 89.7% — happy to share) and it's also the model that manufactures cycles at rates no real electorate shows. Under 1-D spatial at 7 candidates, IRV elects the sincere CW 47% of the time with fully sincere ballots. Model choice swings IRV ~50 points in my measurements — wider than the entire 65–75% band you're describing as convergence.

Second, the strategy space. Burial is the one attack IRV is immune to by construction (later-no-harm — it never reads your lower ranks while your top choice is alive), so testing burial and finding IRV robust is close to definitional. IRV's real exposure is compromising and center squeeze. Wolk/Quinn/Ogren's PVSI measures per-strategy incentive and finds IRV's favorite-betrayal incentive positive (~3%) while Smith/Minimax is disincentivized across every strategy tested — the opposite of convergence. Running adaptive compromising symmetrically would be the strong version of your test.

One methodological question that I think drives your whole result: what are the strategic blocs maximizing? If the objective is "displace the CW," an adaptive search will always find burials. If it's the bloc's own expected utility, they'll often decline, because burial into a margin-based Condorcet rule frequently backfires and elects the buried candidate. Without backfire priced into the decision rule, 72% is an upper bound on damage rather than a prediction of behavior.

Related: "Smith-compliant" may be too coarse to be your independent variable. On the Alaska 2022 numbers, the same burial attack succeeds or fails depending purely on the completion rule — margin-based rules shrug it off, a Hare/runoff completion falls for it. Finding that everything in the bucket lands at ~70% might be telling you the bucket is wrong.

Also worth noting your own numbers don't quite support the conclusion: best variants 72%, IRV 70%, Ranked Pairs/Stable Voting high-60s. That's a wash within noise, not IRV outperforming — and it's a wash before correcting for IRV's sincere-ballot penalty.

None of which is a defense of Smith compliance as a strategy guarantee — Gibbard–Satterthwaite already forbids strategy-proofness for everyone, so that was never on the table. But I'd land somewhere different from you: the useful question isn't the hit rate, it's the price — how big a coordinated bloc, how good the polling, how bad the backfire, and does it leave fingerprints. On that measure burial is a heist and center squeeze is a Tuesday.

Would genuinely like to see the code and the generator. If you publish it I'll run it against my harness.

I am member of member of STAR Voting Local Chapter - https://www.facebook.com/groups/209427010120253

I am interested in:

• Voting software simulation, voting file formats
• How to make small, single winner elections simple and fun - web voting and voting simulations
• Voting apps on iPhone / Android

I like Approval Voting for its simplicity and STAR Voting for its Expressiveness and Accuracy.

Phone +1-919-602-5424
email: masiarek@gmail.com

What to do with floats in STAR Voting?

Single winner
See example 'test_original_example' at https://github.com/Equal-Vote/starpy/blob/main/STAR_Test.py

Example:

• A[1.0] B[3] C[4.9]

Option 1 - Use 'as-is' - assume more accurate data is a good thing?

• if allow - how many digits?

Option 2 - round it to nearest integer?

I assume that a warning message should be issued "Scores provided as 'floats' - expected scores as whole numbers {0, 1, 2, 3, 4, 5}.

• no warning in this case - correct?
  A[1.0] B[3.00] C[4.0000]

How about STAR Multi-winner?
Any special considerations related to Multi-winner?

Again AI Claude assisted

@cfrank This is a really generous reply, and it corrects me on two things I got wrong, so let me start there.

Your objective was already utility-maximizing for each faction, not CW-displacement — so the question I made the most noise about was one you'd already answered. And your generator wasn't impartial culture, it was stress profiles. Both of my sharpest points were aimed at problems you didn't have. Apologies for assuming.

I spent some time with your plots, and I think they make your own case more precisely than the summary does — including in one place where they cut against your original conclusion.

The crossover only happens at the PSRO endpoint. Reading off your panels (so ±1 point): at 0% strategic the Smith methods are at 100% CW rate and 0.992 normalized utility against IRV's ~59% and 0.944. At 25/50/75/90% they're at ~90/81/74/70% and 0.987→0.955, against IRV flat at ~61–64% and ~0.937–0.944. So on both the hit rate and the utility measure, Smith methods dominate IRV at every level of strategic participation you tested short of the endpoint — including 90%, where nine voters in ten are strategic. That's a much stronger statement than "the sincere advantage can remain large," and it's your data, not mine.

Your "outside sincere Smith rate" panel is the cleanest thing in the set. IRV sits flat at roughly 36–40% at every level, including 0%. It elects outside the sincere Smith set about 40% of the time with nobody lying at all. That flat line is the whole reason the convergence appears: strategy has very little left to take from IRV, because IRV has already spent it. The Smith methods start at 0% and are driven up to 25–30% by strategy; IRV starts at 40% and stays. Those are different failures wearing the same number.

Two smaller things. First, I think there may be an off-by-one between your prose and your plot: you quote Ranked Pairs at 97/90/81/74 for 25/50/75/90%, but the plot reads roughly 90/81.5/73.5/70 at those positions — your sequence looks shifted one notch. Doesn't change the shape, but at 90% strategic it's ~70%, not 74%.

Second, and I think this matters for the conclusion: at the PSRO endpoint your best method is IRV + fresh runoff, not IRV. It leads on all three panels — ~69% CW rate, 0.964 utility, and the lowest outside-sincere-Smith rate of any method at that endpoint (~31%, while plain IRV is at ~39.5% and Ranked Pairs is at ~56%). So even taking the adaptive regime entirely at face value, the finding isn't "IRV is robust to sophisticated strategy." It's "a second round on fresh ballots is robust to sophisticated strategy" — a claim about two-round structure rather than about Smith compliance, and one that would apply just as well to score-plus-runoff designs. I'd be curious whether that holds up as you vary the runoff's ballot.

On your mechanism — the CW being pushed out of the Smith set before completion runs. That's a sharp claim and I hadn't separated it out, so I measured it. Single coordinated bloc, burial, rational (it only submits if it beats voting honestly), every challenger tried, swept over 3/5/7/9 candidates at your 71 voters. Ejection is the minority regime everywhere: 19–36% of successful burials remove the CW from the reported Smith set, and the other two-thirds to four-fifths leave the CW sitting inside it, where the completion rule still decides. Interestingly the ejected share barely moves with field size (23%→26% in 1-D from 3 to 9 candidates) even as raw displacement climbs from 20% to 88%. A wider field makes burial much easier without much changing where it lands.

The obvious limit: that's one bloc doing a plain bury-to-last, not several factions best-responding over rank and score offsets. Ejection is clearly something a stronger search buys. Which I think locates our remaining disagreement precisely and answerably: how much of the ejection rate is purchased by adaptive multi-faction optimization, over and above single-bloc burial? If PSRO is ejecting at 60–70% where a naive bloc ejects at 25%, that's a real and quotable finding about what adaptivity does, and it would be worth a paper on its own. Code's here if it's useful: https://masiarek.github.io/star-voting-library/07_Concepts/topics/compliance_vs_strategic_preservation.html

On the AI point — I think this is the most interesting thing in your post and I don't want to wave it away, because the direction is right. Three things give me pause about how far it goes:

Computation isn't the binding constraint. In my runs a successful burial needed 34–40% of the electorate to rank someone they genuinely like dead last. AI can tell you that's the optimal play; it can't get 40% of voters to cast it on trust. That cost is social, and it's the one that doesn't fall with compute.

The information required is about opponents, not preferences. PSRO converges because it iterates against a live opponent mixture, observing what the other factions actually do. A real electorate votes once, without seeing anyone's final policy. Better tools don't close that gap — they arguably widen the variance, since everyone is now optimizing against a guess about everyone else, and a burial aimed at the wrong equilibrium is exactly the one that backfires.

Cheaper attacks make cheaper detection, and that cuts toward pairwise methods. A successful burial's signature is a cycle appearing in a race whose pre-election pairwise polling showed a clean head-to-head winner. The thing detection needs is a published pairwise matrix — which is precisely what Condorcet methods emit as a byproduct and what IRV doesn't. If the threat model is "strategic optimization gets cheap," I'd want the method that publishes the most auditable structure, not the least.

None of which touches your core point, which I think is right and which I've now written up: formal compliance is a property of the cast ballots, "the sincere winner wins" is a property of the electorate plus its behaviour, and treating the first as a guarantee of the second is sloppy. That's a correction advocates on my side of this should absorb rather than argue with. I just don't think it demotes Smith compliance — your own 0–90% panels are about as strong an argument for it as I've seen.

My brain melts when reading this article: Are Condorcet and Minimax Voting Systems the Best? by Richard B. Darlington:
https://arxiv.org/pdf/1807.01366.pdf

What is 'minmax-TD' and how does it compare to STAR Voting?
Are the conclusions correct in this article?

thank you
adam
https://arxiv.org/pdf/1807.01366 - current version: [v10] Mon, 9 May 2022 13:44:20 UTC (506 KB)

Do we have a list of GitHub links with Voting methods, test cases and expected results?

This is what I found so far:

• https://github.com/endolith/elsim/tree/master/elsim/methods

@keith-edmonds https://www.votingtheory.org/forum/topic/241/zero-knowledge-encryption-using-in-voting-methods?_=1664064515062

My main objection - loss of expressive ballot.

Plurality voting is among the least expressive voting methods there is available.

2CV method seems to force voter to express opinion on two candidates only (lost expressiveness of the voting ballot).

You may very well have opinions about some or all of the candidates on a ballot, and yet you only get a say about two candidates only.

Other voting methods allow you to express yourself in different kinds of ways — for example:

• approval voting allows voters to choose any number of candidates;
• RCV lets you rank candidates by preference;
• score or range voting asks voters to give each candidate a score, with the winner being determined by the candidate with the highest total or average score.
• STAR Voting - multiple candidates - both score and preference !

Again AI Claude assisted

@cfrank This is a really generous reply, and it corrects me on two things I got wrong, so let me start there.

Your objective was already utility-maximizing for each faction, not CW-displacement — so the question I made the most noise about was one you'd already answered. And your generator wasn't impartial culture, it was stress profiles. Both of my sharpest points were aimed at problems you didn't have. Apologies for assuming.

I spent some time with your plots, and I think they make your own case more precisely than the summary does — including in one place where they cut against your original conclusion.

The crossover only happens at the PSRO endpoint. Reading off your panels (so ±1 point): at 0% strategic the Smith methods are at 100% CW rate and 0.992 normalized utility against IRV's ~59% and 0.944. At 25/50/75/90% they're at ~90/81/74/70% and 0.987→0.955, against IRV flat at ~61–64% and ~0.937–0.944. So on both the hit rate and the utility measure, Smith methods dominate IRV at every level of strategic participation you tested short of the endpoint — including 90%, where nine voters in ten are strategic. That's a much stronger statement than "the sincere advantage can remain large," and it's your data, not mine.

Your "outside sincere Smith rate" panel is the cleanest thing in the set. IRV sits flat at roughly 36–40% at every level, including 0%. It elects outside the sincere Smith set about 40% of the time with nobody lying at all. That flat line is the whole reason the convergence appears: strategy has very little left to take from IRV, because IRV has already spent it. The Smith methods start at 0% and are driven up to 25–30% by strategy; IRV starts at 40% and stays. Those are different failures wearing the same number.

Two smaller things. First, I think there may be an off-by-one between your prose and your plot: you quote Ranked Pairs at 97/90/81/74 for 25/50/75/90%, but the plot reads roughly 90/81.5/73.5/70 at those positions — your sequence looks shifted one notch. Doesn't change the shape, but at 90% strategic it's ~70%, not 74%.

Second, and I think this matters for the conclusion: at the PSRO endpoint your best method is IRV + fresh runoff, not IRV. It leads on all three panels — ~69% CW rate, 0.964 utility, and the lowest outside-sincere-Smith rate of any method at that endpoint (~31%, while plain IRV is at ~39.5% and Ranked Pairs is at ~56%). So even taking the adaptive regime entirely at face value, the finding isn't "IRV is robust to sophisticated strategy." It's "a second round on fresh ballots is robust to sophisticated strategy" — a claim about two-round structure rather than about Smith compliance, and one that would apply just as well to score-plus-runoff designs. I'd be curious whether that holds up as you vary the runoff's ballot.

On your mechanism — the CW being pushed out of the Smith set before completion runs. That's a sharp claim and I hadn't separated it out, so I measured it. Single coordinated bloc, burial, rational (it only submits if it beats voting honestly), every challenger tried, swept over 3/5/7/9 candidates at your 71 voters. Ejection is the minority regime everywhere: 19–36% of successful burials remove the CW from the reported Smith set, and the other two-thirds to four-fifths leave the CW sitting inside it, where the completion rule still decides. Interestingly the ejected share barely moves with field size (23%→26% in 1-D from 3 to 9 candidates) even as raw displacement climbs from 20% to 88%. A wider field makes burial much easier without much changing where it lands.

The obvious limit: that's one bloc doing a plain bury-to-last, not several factions best-responding over rank and score offsets. Ejection is clearly something a stronger search buys. Which I think locates our remaining disagreement precisely and answerably: how much of the ejection rate is purchased by adaptive multi-faction optimization, over and above single-bloc burial? If PSRO is ejecting at 60–70% where a naive bloc ejects at 25%, that's a real and quotable finding about what adaptivity does, and it would be worth a paper on its own. Code's here if it's useful: https://masiarek.github.io/star-voting-library/07_Concepts/topics/compliance_vs_strategic_preservation.html

On the AI point — I think this is the most interesting thing in your post and I don't want to wave it away, because the direction is right. Three things give me pause about how far it goes:

Computation isn't the binding constraint. In my runs a successful burial needed 34–40% of the electorate to rank someone they genuinely like dead last. AI can tell you that's the optimal play; it can't get 40% of voters to cast it on trust. That cost is social, and it's the one that doesn't fall with compute.

The information required is about opponents, not preferences. PSRO converges because it iterates against a live opponent mixture, observing what the other factions actually do. A real electorate votes once, without seeing anyone's final policy. Better tools don't close that gap — they arguably widen the variance, since everyone is now optimizing against a guess about everyone else, and a burial aimed at the wrong equilibrium is exactly the one that backfires.

Cheaper attacks make cheaper detection, and that cuts toward pairwise methods. A successful burial's signature is a cycle appearing in a race whose pre-election pairwise polling showed a clean head-to-head winner. The thing detection needs is a published pairwise matrix — which is precisely what Condorcet methods emit as a byproduct and what IRV doesn't. If the threat model is "strategic optimization gets cheap," I'd want the method that publishes the most auditable structure, not the least.

None of which touches your core point, which I think is right and which I've now written up: formal compliance is a property of the cast ballots, "the sincere winner wins" is a property of the electorate plus its behaviour, and treating the first as a guarantee of the second is sloppy. That's a correction advocates on my side of this should absorb rather than argue with. I just don't think it demotes Smith compliance — your own 0–90% panels are about as strong an argument for it as I've seen.

AI Claude - it helped me to create this response:

The distinction you're drawing is real, and I think it's underrated: formal Smith compliance is a property of the ballot→winner map, so it can't say anything about sincere preferences the method never sees. Worth separating out explicitly.

Two things I'd push on before the IRV conclusion, though.

First, the baseline. Ranked Pairs and Stable Voting elect the sincere CW 100% of the time on sincere ballots; IRV doesn't. So a shared ~70% isn't a tie — for the Condorcet methods the entire shortfall requires someone to have lied, while for IRV a chunk of it is the method missing with everyone honest. Which raises the diagnostic question: what's your profile generator? If IRV is reaching 70% under adaptive strategy, the model is probably impartial culture. That's where IRV always looks best (in my own runs it beats STAR there at 3 candidates, 96.7% vs 89.7% — happy to share) and it's also the model that manufactures cycles at rates no real electorate shows. Under 1-D spatial at 7 candidates, IRV elects the sincere CW 47% of the time with fully sincere ballots. Model choice swings IRV ~50 points in my measurements — wider than the entire 65–75% band you're describing as convergence.

Second, the strategy space. Burial is the one attack IRV is immune to by construction (later-no-harm — it never reads your lower ranks while your top choice is alive), so testing burial and finding IRV robust is close to definitional. IRV's real exposure is compromising and center squeeze. Wolk/Quinn/Ogren's PVSI measures per-strategy incentive and finds IRV's favorite-betrayal incentive positive (~3%) while Smith/Minimax is disincentivized across every strategy tested — the opposite of convergence. Running adaptive compromising symmetrically would be the strong version of your test.

One methodological question that I think drives your whole result: what are the strategic blocs maximizing? If the objective is "displace the CW," an adaptive search will always find burials. If it's the bloc's own expected utility, they'll often decline, because burial into a margin-based Condorcet rule frequently backfires and elects the buried candidate. Without backfire priced into the decision rule, 72% is an upper bound on damage rather than a prediction of behavior.

Related: "Smith-compliant" may be too coarse to be your independent variable. On the Alaska 2022 numbers, the same burial attack succeeds or fails depending purely on the completion rule — margin-based rules shrug it off, a Hare/runoff completion falls for it. Finding that everything in the bucket lands at ~70% might be telling you the bucket is wrong.

Also worth noting your own numbers don't quite support the conclusion: best variants 72%, IRV 70%, Ranked Pairs/Stable Voting high-60s. That's a wash within noise, not IRV outperforming — and it's a wash before correcting for IRV's sincere-ballot penalty.

None of which is a defense of Smith compliance as a strategy guarantee — Gibbard–Satterthwaite already forbids strategy-proofness for everyone, so that was never on the table. But I'd land somewhere different from you: the useful question isn't the hit rate, it's the price — how big a coordinated bloc, how good the polling, how bad the backfire, and does it leave fingerprints. On that measure burial is a heist and center squeeze is a Tuesday.

Would genuinely like to see the code and the generator. If you publish it I'll run it against my harness.

@cfrank Interesting construction, but I think stage 2 is close to null in the common case,
and it's worth checking before the simulation numbers carry any weight.

Take the ordinary situation: A = {a} (one score winner), B = {c} (a Condorcet
winner exists), a ≠ c. Both sets are published, so every voter in (2.1) knows
branch A elects a and branch B elects c. But c is by definition the candidate a
majority prefers to a — that's what being a Condorcet winner means. So sincere,
consistent voters hand branch B a majority every time, and SSVV elects the
Condorcet winner. The ex ante veil you're relying on only exists to the extent
that |A| or |B| exceeds 1.

That leaves the cycle case, where the veil is genuine — and there (2.2) is an
approval ballot over the Smith set, which is Smith//Approval. Same winner, no
second election.

Which makes me want to ask about the simulations: under sincere voting SSVV
should score identically to Smith//Approval. Was that in the comparison set, and
how were stage-2 voters modeled? If they're sincere and the electorate is stable
between rounds, I don't see where a difference could come from.

One other thing that might be worth modeling: stage 1 is now a qualification
contest with two doors, and burial has a payoff it doesn't have in a one-shot
Condorcet method — manufacturing a cycle moves you from "loses deterministically"
to "is in B and can win the fresh approval round."

None of which kills the idea, if the goal is legitimacy rather than optimality.
"Publish both answers and let the electorate ratify one" is a genuinely different
thing from "compute a better winner," and it's a reasonable thing to want. I'd
just pitch it on those grounds — the simulation framing invites the reduction
above.

@toby-pereira is it OK to mention your names at: https://github.com/masiarek/star-voting-library/blob/master/00_start_here/topics/condorcet/ordered_majority_rule_irv.md

Bummer - there are no perfect voting methods

Ballots:
       A,B,C
       1,4,3
       2,3,1
       5,0,4

Winner
  STAR      = A
  RCV-IRV   = B   (differs from STAR)
  Approval  = B   (differs from STAR)
  Condorcet = B   (differs from STAR)

Is this a good commentary:
https://docs.google.com/document/d/1DVhn9sbPpbXJCjN1fgFZzyKSfFbt97ksXWlh9K1wlOU/edit?tab=t.0

On a different note - it is probably impossible to come up with a contrived example where each method has a different winner - right?

We need three small, illustrative elections to demonstrate each ‘problematic’ box separately (avoid ‘Less Problematic’ Exhausted Ballots).

Assumptions:

• ranking limit 3 (or 5 if easier to re-create)
• number of candidates 5 (or less if easier)
• all voters correctly rank all the ballots (no spoilers, no bullet voting)
• all voters rank all ranks within Ranking Limit (3 or 5)

Context:

• article https://docs.google.com/document/d/1ASC5BS10rCfAYZWGeCyS7dKdKc4p5wwI6DHs4F7ScGc/edit

• presentation https://docs.google.com/presentation/d/1ipof9WSSy0GenVKWfKLu_jlYwUIT0TU0WVSAap0VZ5Q/edit#slide=id.g27f6bb33467_0_1

Comments, feedback - very much appreciated!

Book Chaotic Elections - Saari, page 34:

Strange Plurality voting notation:
A > C > B supported by a 5 : 4 : 2 tally.

Have you seen it somewhere?

Additionally, it is strange to see 'ranks' for plurality voting results/winners (instead of 'chose-one').

8:Ava>Cedric>Deegan>Bianca>Eli
6:Ava=Bianca=Cedric>Eli>Deegan
6:Eli>Ava>Bianca=Cedric=Deegan
6:Deegan>Bianca=Cedric>Eli>Ava
4:Bianca>Ava>Eli>Deegan>Cedric
3:Eli>Deegan>Bianca=Cedric>Ava
2:Deegan=Eli>Bianca=Cedric>Ava

1. preference matrix at https://electowiki.org/wiki/Ranked_Robin:
   

2. preference matrix at - https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html
   

Youtube Video – [02:50..]

Ranks
G,B,P,R,O
18:1,5,4,2,3
12:5,1,4,3,2
10:5,2,1,4,3
9: 5,4,2,1,3
4: 5,2,4,3,1
2: 5,4,2,3,1

Ranks converted into Scores
G,B,P,R,O
18:5,1,2,4,3
12:1,5,2,3,4
10:1,4,5,2,3
9: 1,2,4,5,3
4: 1,4,2,3,5
2: 1,2,4,3,5

Voting Method and Winner
Plurality - Green
Two-Round Runoff- Blue
RCV IRV - Purple
Borda - Red
Range - Red
Approval - Red
Condorcet - Orange
STAR Voting - Orange

Here are my calculations: https://docs.google.com/document/d/1icQZ1efJV4XX7fD0_OTjNnW7uhfFV4lxbz5afTmqcyg/edit

ordered majority rule - Condorcet, IIA, monotonicity-not desirable properties ?!

A Majority Rule Philosophy for Instant Runoff Voting
https://arxiv.org/abs/2308.08430

Conclusions
IRV is the only voting method to satisfy ordered majority rule, which ensures the election of a candidate from the majority coalition while preventing opposition voters from influencing the choice of candidate from the faction they oppose. Ordered majority rule determines the relative social order between any two candidates by counting only ballots from those voters who do not prefer another major candidate, while ignoring all minor candidates.
Ordered majority rule is incompatible with the Condorcet criterion and independence of irrelevant alternatives because Condorcet and IIA demand that the relative social ranking between two candidates should depend on the relative social ranking of these candidates on all ballots. In addition, it is incompatible with monotonicity because monotonicity demands that if supporters of minor candidates have influence over lower ranked candidates, then supporters of major candidates should have equal or greater influence, while IRV does not grant influence over the social ranking of two candidates by voters who prefer a different major candidate. For situations where allowing supporters of a major candidate to have influence over the relative social ranking between other major candidates is deemed inappropriate, compliance with Condorcet, independence of irrelevant alternatives, and monotonicity are not desirable properties of a voting system.