Iterated Threshold Reapproval Voting

Iterated Threshold Reapproval Voting (ITRV) is a single-winner voting method I invented in an effort to improve upon STAR Voting with an eye towards strategic straightforwardness. It is complicated enough that I would not recommend it for governmental elections, but I do find it elegant and conceptually interesting.

Algorithm

Each voter writes a score for each candidate; scores can be any real number (including negative numbers), and unscored candidates receive a 0. There are as many rounds of tabulation as there are candidates, minus 1, and in each round a candidate is eliminated. In each round:

1. For each ballot, the average score of the remaining candidates is calculated. This is used to determine a virtual approval ballot in which every candidate with an above-average score receives a vote.
2. The candidate with the fewest votes on these virtual approval ballots is eliminated.

The last candidate remaining is declared the winner. For example, suppose a voter assigns the following scores:

• A: 10
• B: 3
• C: -1
• D: -100

In the first round, this voter votes for every candidate except D. Let’s suppose D is eliminated. Then, in the second round, the voter will only vote for A. If A is eliminated, the voter will vote for B in the final round.

Iteraterated Threshold Reapproval Voting (ITRV) has the obvious variant Smith//ITRV that uses the same ballots and in which the ITRV algorithm is used to elect a candidate from the Smith set.

Motivation

A nice feature of STAR voting is that all ballots which express a preference between the finalists count equally in the runoff, regardless of how strategically savvy the voters are. However, a basic awareness of which candidates are viable is quite important in the scoring. Consider a four-candidate election in which the vast majority of voters consider candidate D to be the worst by far. Here, it’s much better to vote A: 5, B: 4, C: 0, D: 0 than A: 5, B: 4, C: 3, D: 0; the former ballot counts two and a half times as much for A over C during the scoring round, and added opposition to the non-viable D is superfluous. However, there’s a simple way of making the two ballots equally impactful: eliminate the non-viable candidate D, and then rescale the ballots so that both the highest and lowest scores are used. This approach yields the cardinal version of Baldwin’s method.

There’s another way in which strategic voters yield more influence in STAR that Bladwin’s method doesn’t address: strategic exaggeration within the set of viable candidates. In a three-candidate STAR election, you’re better off not giving any candidate a 2 or a 3; it’s best to exaggerate those scores to 1 or 4. This suggests another way to increase the influence of non-strategic voters: have the tabulation algorithm take care of this strategic exaggeration for the voter. The most natural amount of strategic exaggeration to employ is the maximal amount, such that only the highest and lowest scores get used in a given round of tabulation, and this is just Approval Voting. The only remaining question is where to set the approval threshold. Sticking with the philosophy of having the algorithm take care of strategy for the voter and using the optimal strategy when all candidates are believed to be equally viable yields the ITRV algorithm.

Once we’re reducing everything to approvals, there’s no need to limit voters to a particular scale. We can adopt the approach allowed by IRNR and let voters simply write down their utilities. This results in a voting method that is maximally expressive while making it effective to vote honestly.

Pass/fail criteria

I typically do not consider an analysis of pass/fail criteria to be a fruitful approach to evaluating voting methods, so I made no attempt to have ITRV pass any given criterion. As a result, it does not pass very many. ITRV fails Condorcet, Favorite Betrayal, Later No Harm, Monotonicity, the Participation Criterion, and Independence of Clones (it could be modified to pass the latter by changing how the approval threshold is determined). It passes the Equality Criterion and a weaker version of Later No Harm: It is possible to give a nonzero amount of support for any number of candidates in a way that adding a second choice cannot hurt your first choice, adding a third choice cannot hurt your first two choices, etc. In an N-candidate ITRV election, you can give your kth choice a score of N^(N-k) to have your ballot function like an IRV ballot and only provide support for one candidate at a time. (For example, in a 4-candidate election you could give scores of 256, 64, 16, and 1. A similar approach would allow voters to vote Coomb’s-style.)

Strategic voting in ITRV

While ITRV eliminates the most straightforward forms of strategic voting by having the algorithm do them for you, it is sometimes possible to gain an advantage with sufficiently good knowledge of how everyone else is voting. In the three-candidate setting, strategic voting is almost exactly the same as in STAR — the only real difference is that it’s never optimal to score two candidates the exactly same (excluding when you’re literally indifferent between two of the candidates) — not just almost never. In general, strategic voting usually requires knowledge that goes beyond simply knowing who is and is not viable; knowing that Bob is a front-runner does nothing to inform strategic voting by itself, but knowing that Bob is favored to win in the final round if he faces anyone other than Andy (who would beat him, but could easily be knocked out before the final round) enables several potential strategies.

There is an important similarity between strategic voting in ITRV and strategic voting in IRV: In each case, strategic voting requires identifying a candidate who is liable to get eliminated in one round but is still capable of influencing the outcome of the election in a later round (either by winning the election or, less commonly, by knocking out a more viable candidate). There is also an important difference: ITRV is repeated Approval Voting, IRV is repeated Plurality, and a single round of Approval is far better at identifying the best candidates than a single round of Plurality. Therefore, situations where strategy matters should be less common under ITRV than under IRV.

A couple potential dishonest strategies:

• Pushover: Cast a ballot such that you vote for only your favorite candidate and a chosen pushover candidate in every round (so long as your favorite and a more viable candidate are both in the race). An example ballot would be Favorite: 100, Pushover: 99, Everyone else: somewhere from 0 to 10. This is akin to voting Favorite: 5, Pushover: 4, Everyone else: 0 or 1 in STAR. As with STAR, this is a usually terrible strategy that can backfire horribly. First, the pushover might crowd your favorite out of the final round, such that, if you and likeminded voters had cast something closer to honest IRV-style ballots instead of using the pushover strategy, your favorite would have reached the final round and won. Second, in a potential final round between the pushover and a candidate other than your favorite who you prefer to the pushover, the pushover gets your vote — so, especially if many people are using this strategy, the “pushover” might win. The pushover strategy only really makes sense if you expect your favorite to get the most support in the penultimate round while still being heavily disadvantaged in the final round against a candidate other than the chosen pushover.
• Compromise: If you think your favorite has an excellent chance of making it to the final round but a worse chance of winning the final round than a compromise candidate (should they make it there), you could score that compromise candidate very highly to help them reach the final round instead of your favorite. As mentioned above, this scenario should be less common in ITRV than in IRV.

Simulations

In terms of Voter Satisfaction Efficiency, ITRV is comparable to many Condorcet methods and slightly outperforms STAR. I have not done simulations with ITRV and strategic voting since it’s difficult to come up with a strategy that the entire electorate could use that would be expected to outperform honesty. ITRV has better VSE than Smith//ITRV, which I did not expect prior to running the simulation.

These VSE data are from 30,000 simulations with 7 candidates and 51 voters. None of the strategies utilized polling data.

• Plurality 0.6737732
• Plurality + Top 2 0.8449519
• Approval (bad strategy) 0.8935806
• IRV 0.9019897
• Approval (better strategy) 0.9412792
• Approval + Top 2 0.9443711
• Score 0.9692215
• Borda Count 0.9703541
• Schulze 0.9766078
• STAR 0.9779458
• Smith//IRV 0.9793172
• Raynaud 0.9802274
• Ranked Robin 0.9809177
• Smith//ITRV 0.9809934
• ITRV 0.9814414
• Ranked Pairs 0.9816964
• Smith//Minimax 0.9818341
• Smith//Score 0.9825240

Overall opinion

The expressiveness of ITRV is unsurpassed, and it does an excellent job of giving maximally honest voters about as strong of a voice as strategic voters. It is also a top-tier method in terms of electing good winners. However, it also has major weaknesses.

First, the tabulation is rather involved. ITRV doesn’t come close to being precinct summable, and hand tabulation is utterly impractical. These issues are hypothetically manageable but still pose a substantial hurdle.

Second, I suspect a great many voters won’t actually believe that they can write down their honest utilities and things will work out well for them; some would probably give their favorite a score of one billion or something just in case that might help, and would end up casting a strategically suboptimal ballot as a result of this ill-considered strategizing. Moreover, it could be difficult to convey how to vote to voters who are unfamiliar with the concept of a utility function. Even the significance of negative numbers could be problematic since we want our elections to be accessible to people with a very poor math background.

Third, the ideal method of filling out a ballot (writing in scores with a pen that don’t even need to be integers) introduces a lot of practical difficulties. For example, it is extremely bad if a stray mark looks like a decimal point or a minus sign. It is possible to only allow integer scores without reducing the potential expressiveness, but switching to a conventional scoring ballot (where voters fill in bubbles) would impair this voting method significantly and make strategic voting far more of a factor.

Ultimately, I am highly skeptical that ITRV would be suitable for most governmental elections. It could still be useful in cases where the extra complexity is acceptable and as a relatively neutral choice for voting on voting methods. It could also be useful conceptually. Optimistically, it might inspire a new voting method that is simple enough for governmental use. Less optimistically, it could be conceptually useful in an analogous way to Borda Count, as a concrete illustration of how a voting method can have major strengths at the cost of crippling flaws.