Voters.Army – My New Attempt to make Election Reform Sexy
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You might recall that I created an election reform website, VotersTakeCharges in 2024. It did not fare so well. In the spring of 2025, I tried Voters.News, not much better.
Recently, I combined election reform, including campaign finance reform (the Montana Plan to eliminate corporate election spending), with economic reforms at Voters.Army. This is more popular.I am advocating for BTR-Score for single-winner elections, but I need some help with a few questions:
Who invented BTR-Score, and how can I contact this person?
Would BTR-Score be a good way to choose four candidates in a blanket primary?If so, would the BTR tournament need to be run four times (removing the winner each time)?
A proportional method might be fairer, but I place a high value on simplicity, and I see a need for a diverse set of winners to advance to the general election. I am trying to provide an alternative to Top 4 primaries. A Top 4 ballot initiative lost in Colorado in November 2024.
Thank you, Greg Wasleski
Why I advocate for BTR-Score
One big reason: in August, the Denver City Council came one vote short of sending Ranked Choice Voting to the voters for City Council elections (6 votes for, 7 against). I think that BTR-Score is at least one vote better than RCV.
My Criteria for Single Winner Voting Methods:
Fairness
Reliability
No Favorite Betrayal
Compliance with state constitutions that require election winners to have the highest, greatest, largest number of votes, or a plurality of the votes.
High Voter Expression
Simplicity, Easy to Explain
Sizzle, Easy to SellBTR-Score is easy to explain, and the voters’ ability to rate all candidates gives it plenty of sizzle. After a quick BTR-Score explanation, you can add history and prestige with a sentence or two about Ramon Llull and the Marquis de Condorcet.
Of all the voting system criteria, Condorcet speaks most directly to fairness and reliability. A candidate who beats all other candidates one on one should win. If a “beat all” candidate loses, people will bitch, and rightfully so. In elections with no “beats all” winner, BTR-Score will elect a deserving winner, the winner of the runoff tournament, in a fair and transparent manner.
Yes, as a hybrid model, BTR-Score will fail more than a few important criteria; however, I believe it will be fair and reliable. High voter expression, simplicity, sizzle, and no favorite betrayal are critical to the adoption of a new voting method in the United States.
Concerning Utility and Majority:
Assuming a three-way cycle, the top seed of the BTR tournament will have a 50% chance of winning. The other two candidates have a 25% chance of winning as they must face each other to have a shot at the top seed. This gives the highest utility candidate an advantage, but each contest in the tournament is a two-candidate plurality vote (a majority vote if you ignore the ballots that rate the two candidates equally).Ranked Choice Voting eliminates ties on the ballots, but it is not a true majority system because some of the ballots may be spent before the winner is determined, and the capricious way that some second-choice votes are counted and some are not.
I do not consider a plurality election with a top two runoff to be a true majority election because there are two separate sets of voters voting on separate days. Also, I do not believe in Santa Claus, although he did finish sixth in the first round of the infamous 2022 Alaskan Congressional special election (top four primary with an RCV final). BTW The winner, Mary Peltola, is running for the US Senate. It could be an extremely important race.
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@gregw BTR/score (or BTR/approval) is an excellent system, although it is not stable ex ante under majority preference—only a maximal lottery is. Maximal lotteries also satisfy participation and Condorcet (they can do that because they are inherently non-deterministic in the case of Condorcet cycles—those properties are incompatible for deterministic methods).
BTR was invented by Nicolaus Tideman. He is still around, I don’t know how accessible he is but he is certainly involved in voting theory. Sorting/tie-breaking by score or approval in BTR is an obvious extension.
For primaries, a specified multi-winner method is needed, which could be a “natural” extension of a single-winner method. Peeling winners of BTR off recursively is one option, although it would be more stable to prioritize candidates in the Smith/bipartisan set (the naive recursion can lead to results that violate multi-winner Smith compliance). A PR/multi-winner method would probably be theoretically preferable, maybe some others more versed in multi-winner methods can comment on options for simplicity.
Something on my mind, for any Condorcet method, even a maximal lottery, being intrinsically stable under majority preference after a winner is chosen is simply impossible with Condorcet cycles. My thinking lately is, this implies that stability requires a supplementary mechanism that compensates dissatisfied majorities in the event of Condorcet cycles, specifically to the extent that majority grievances are sufficiently reduced. However, I don’t know what that mechanism ought to be or how it ought to be enforced, and serious consideration of that enters the interface between technical voting theory, real politics, and law. I mused about that here: https://www.votingtheory.org/forum/topic/591/maximal-lotteries/8
Just food for thought. I’m glad your reform efforts are picking up steam!
P.S.: While I do like Condorcet methods, my opinion is that realistic and highly impactful reforms would be easiest to implement by pushing for approval voting. We discussed that point here as well and it seems to have broad agreement, but obviously that’s just my personal interpretation and some disagree for their own reasons: https://www.votingtheory.org/forum/topic/495/approval-voting-as-a-workable-compromise/20?_=1768708850097
Ultimate approval would still require a multi-winner primary system.
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Thank you for the help!
Maximal Lotteries and BTR-Score
In the case of a BTR-Score election 3-way cycle, the probabilities of winning for each of the three candidates are set in stone. The survivor of the first N - 2 rounds (N = number of candidates) of the bottom 2 runoff tournament has a 1 in 4 chance of winning. The candidate with the second highest total score (2nd seed) also has a 1 in 4 chance. The candidate with the highest score (top seed) has a 2 in 4 chance of winning.A Maximal Lotteries vote with a 3-way cycle should produce is more nuanced set of probabilities. Maximal Lotteries also has impressive criteria compliance. Significant improvements over BTR-Score. Unfortunately, I do not understand Maximal Lotteries enough to sell it to the public. Can you point me in the right direction?
Approval Voting
Approval voting is a great method; it asks one critical question: do you approve of this candidate are not?I believe that BTR-Score is an easier sell because it enables more voter expression. It has more sizzle. In an Approval race you can only indicate your favorite candidate by approving that candidate only; too much like plurality. More expressive voting methods like BTR-Score provide more information about voter preferences to the public, the press, and politicians.
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@gregw with the caveat that I strongly encourage anyone to correct me: a maximal lottery is the unique kind of single-winner method that satisfies the following properties:
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Condorcet-consistency;
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Reinforcement: if two electorates independently select the same outcome, combining them does not change that outcome; and
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Participation.
That’s the main selling point. If we want reinforcement, Condorcet-consistency, and participation all together in a single-winner method, then maximal lotteries are forced. Without randomization, Condorcet-consistency and participation are already incompatible. But if we allow randomization, then once we require Condorcet-consistency, reinforcement, and participation, not only is it possible, but we actually have no other choice but to use a maximal lottery.
Why that’s true is because of how maximal lotteries are defined—they are exactly the undominated mixed strategies of the majority margin game. That’s the technical/mathematical machinery behind the result, which may not itself be easy to sell per se. But the result is pretty compelling—trust aside, acceptance or rejection becomes primarily a question of whether we need to satisfy those three fairly intuitive properties. If you demand all three, you're forced to reject determinism and to accept maximal lotteries.
Maximal lotteries can’t solve the fact that Condorcet cycles exist, but they do guarantee the strongest possible form of stability compatible with majority rule: no alternative decision rule can be majority-preferred on procedure. Stable preference by majority on outcome is not nominally possible when Condorcet cycles exist. If you want ex-post stability of majority preference when cycles exist, you need supplementary structure that actually changes the decision problem (compensation, bargaining, agenda constraints, etc.).
As a bonus, they also satisfy independence of clones. In fact, if you require independence of clones instead of participation in the list above, the same uniqueness result holds. (With a slight caveat—you need to consider all maximal lotteries over candidates, so you could choose one at random). Importantly as well, in the generic case, the set of maximal lotteries from the majority margin matrix Mij is continuous in its entries. There are abrupt boundaries that can be crossed, but those boundaries have measure zero in the space of all majority margin matrices (they are almost guaranteed not to occur in any real election with many voters).
Lastly, they satisfy the Smith criterion. Even the Landau criterion. Actually, they induce a slightly stronger criterion called the “bipartisan” criterion—the “bipartisan set” is exactly the set of candidates that can attain nonzero probability under a maximal lottery, and it is a (sometimes strict) subset of the Landau set, which itself is a (sometimes strict) subset of the Smith set.
I stress single-winner because designing principled multi-winner extensions of maximal lotteries under comparable axioms remains an active research problem.
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The benefits of Maximal Lotteries are impressive. What would be the basic procedure to use Maximal Lotteries in a public election?
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@gregw I think the most practical way to proceed would be to just compute the full majority margin matrix. The final Mij would need to be the legal object outputted from the counting process.
If there’s a Condorcet winner, you’re done. Otherwise, the randomization process would need to follow a “commit” then “reveal,” where a secret seed is generated and committed to before the resulting lottery is known from multiple independently verifiable sources. The selection can be audited by comparing the seed’s output with the inverse CDF of the published maximal lottery distribution (and proof that it is actually a maximal lottery).
To determine the maximal lotteries from the Smith/Landau restricted majority margin matrix, the maximal lotteries are the mixed strategy Nash equilibria of the game with payoff matrix equal to the majority margin matrix. Since this is a zero-sum game, it follows that the Nash equilibrium is the minimax solution by von Neumann's minimax theorem. This is a feasibility problem that is polynomial-time computable via linear programming as a convex optimization problem. I’ll probably put a small script together that works soon, there are almost certainly existing ones online.
One caveat mentioned is that maximal lotteries aren’t always unique. One could compute the Jeffrey’s prior over the maximal lottery set and sample one accordingly, equivalently that yields a “canonical” choice of maximal lottery.
But basically, it goes (1) pre-commit to an auditable distributed seed with information asymmetry, (2) compute Mij, (3) reveal the winner with an auditable certificate of validity (either by showing they are the Condorcet winner, or that they were genuinely produced by the maximal lottery procedure via the distributed seed in (1)).
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Thank you for the plan!
Maximal Lotteries has unprecedented criterion compliance but we will need some human progress before we get the public to buy in.
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@gregw definitely. I think the main persuasive route is in framing the argument. I tried to present the most persuasive argument I could think of for the principle. There might be more persuasive arguments, and there might be persuasive arguments to the contrary that should be considered.
Still, if the principle is accepted, then the technical details required to actually implement it should be acceptable as long as they’re presented well—that gets difficult, because it really is a technical problem whose solution is not immediately obvious to people outside relevant fields (math, computer science, cryptography, economics, etc.).
