Posts made by cfrank

This probably already exists but I like the idea. It’s a rank order system based on Tideman’s bottom-2 runoff that proceeds recursively.

Informal description: Eliminates Condorcet losers in steps, each time from the largest possible pool of candidates (which pool of a given size to examine must be addressed).

0. Let N be the number of candidates.

1. From the N candidates with the fewest first-place rankings, determine if a Condorcet loser exists.

2. If no Condorcet loser exists, reduce N by 1 and repeat from step 1.

3. If a Condorcet loser exists, eliminate them from the election, let N be the number of remaining candidates, and repeat from step 1 until a single candidate remains.

Just like Tideman's Bottom-2 Runoff, this system is Condorcet compliant and also satisfies the Condorcet loser criterion. I believe this is due to Smith compliance. Variants can use alternative criteria (as opposed to the number of first-place votes) to determine which subset of candidates to examine. For example, a more general/robust method might only consider Condorcet losers among non-Bucklin winners.

I’m actually not certain whether this system turns out to be equivalent to the bottom-2 runoff. I believe it is generally different, but I haven’t constructed a distinguishing example yet. (1/19/2023)

The only reason for considering such a system is to enhance the robustness of the bottom-2 runoff against Condorcet-cycle induced non-monotonicity.

@rob yes, I do think it's easy to veer into political ideology territory, which is just a small step away, and it should be avoided or approached very cautiously. I do think sometimes an evaluation of the effects that a voting system might have on the quality of representation or government processes is valid, but should be done without asking loaded or aggressive questions, and probably there are other general policies in that regard that we should discuss in this forum. I may be guilty of this kind of rhetoric myself, I have no catalog of past transgressions to exhibit but I won't pretend or claim otherwise. It isn't easy to avoid making ideologically oriented judgments even without realizing. In any case moving forward I personally intend to avoid it, and I think we all should probably do the same and let mistakes be recognized. There are plenty of issues that are less controversial to examine here.

@Jack-Waugh, in my opinion (just having read through this now, and not to scold you) it does seem to me like your implicit claim would warrant more evidence. Anyway, I'll throw my cents in about what may be answers to your original questions. Even if the people of Germany hypothetically were not in charge of their government, it seems like a non-sequitur to conclude therefore that PR and voting rights would be either politically significant or a waste of time. Let's put the topic of Germany entirely aside: PR is used in 40/43 of the European nations.

@mosbrooker Just for future reference it might help to tag us! You can type the "at" symbol and then look for the name of whoever you would like to direct your comments to.

Just to address your comments, I do think there are issues with the way representation is handled in the U.S., but there is also a very thoughtful literature documenting the rationale behind the structure of the federal government. The founders of the U.S. lived in a very different socioeconomic context and, as with anybody, were not capable of predicting all of the challenges that would face the nation in terms of its democratic legitimacy, but they also did consider a lot of important concepts and addressed them in a way that I think would be very difficult to consider anything but extremely successful. After all, the United States has become a globally dominant superpower with a robust economy and an overall incredible standard of living, notwithstanding significant wealth inequality and all the problems that come along with that, among other things (ahem, public education, health care, cough). Surely they did something structurally right. And in fact, they did predict most of the core issues that the nation is struggling with presently.

Don't let me just rant, I want to talk about your senator comment. The United States is a federation composed of individual states with distinct laws. The Senate is meant as a body that represents each state on equal footing, independent of population or economic status. The populace of each state is supposed to obtain representation in the House. You may be right that states should have more than 2 senators each, and we probably also need more representatives in the House, since our population is significantly larger than it ever has been! But that is a different way to address the problem. If we essentially do away with the Senate, we won’t really be a federation anymore. As to why that might be a bad idea, you could look to the Federalist Papers. I have my own individual reasons for very much despising the very concept of a large nation, and also large corporations, for that matter. I think each is the suboptimal result of a prisoner’s dilemma.

More than just arguably, election season is mostly toxic due to the "Choose One" voting system, which fuels the competitive and toxic 2-party system that all but disenfranchises moderate voters. This is exactly the problem most of us here want to address by, hopefully, somehow changing the voting system to something that is effective at producing satisfactory representation, and does not suffer from the flaws that induce significant conflicts of interest in voters. European citizens are intrigued to watch U.S. election season like it is a reality TV show. I think they have their own institutional problems as well, but the contrast is at the very least interesting.

Continuous voting has pros and cons, and it's also obviously sensitive to what is considered a "vote." Generally continuous voting would be more sensitive to changes in public opinion, which makes it intrinsically unstable (I'm repeating myself). Maybe you want the status quo to change, but in theory you might want to change it to something. If the system is not stable, whatever you change it to isn't liable to stick around for very long, either. Because it will be unpredictable, it's not easy to preclude it from devolving into something even worse than what it was initially! (Ex: Bolshevik revolution?)

In the end, I don't think we want a revolution. At least as far as what I would like to see, that would be stable and incremental changes that are constructive rather than destructive. I think tearing down the current infrastructure (1) is totally infeasible and (2) would be extremely wasteful anyway. There are many things about our government that work very well, but corruption and institutional issues have eroded its accountability to the public and general welfare.

Benjamin Franklin purportedly said, when asked what the founders had created, “a republic, if you can keep it.” I think the Achilles’ heel of the Constitution was and is the interstate commerce clause, I’m also not alone in this, nor am I by any means an expert. But I also would contest, at the risk of verging on political ideology territory, that movements toward “pure” democracy have been, in my opinion, detrimental, at least in California. I would say that the proposition system has done very little but empower special interest groups to dupe the masses with fine print, and to strip representatives of responsibility without doing the same to their status and power. Anyway, when the masses vote for public goods but also vote not to pay for them, the state is forced to borrow money, and inevitably gets tangled up in all sorts of nonsense, namely internal corruption and crippling debt. Direct democracy does not come structurally equipped with checks and balances. This apology will terminate my rant: sorry.

@mosbrooker all of your examples have been of small groups of people. I already conceded that a system like what you describe is a reasonable solution to delegating executive power and authority in a small tribe, it’s efficient enough and it has historical examples—chiefdoms.

Chiefdoms do not function without oppression when they grow large. The Aztec and Mongolian Empires are sufficient examples. If they aren’t upheld by authoritarian violence, they fragment and crumble, or in any case they generally can’t compete with alternative forms of government without internal violence or oppression. That’s why you don’t see anything like what you describe on a large scale unless you ignore gross oppression. As egalitarian chiefdoms grow larger, they become increasingly unstable.

Also I think the vast majority of people would agree that when fairness of an outcome among alternatives is impossible, an unbiased randomized selection of the alternatives is procedurally fair, and in fact is among the most procedurally fair methods that exist. That’s why coin tosses are used to break two-way ties, neither party can reasonably grumble and say “that’s not fair!” Both had an equal chance, one had to win and the other had to lose. If party #2 wants to flip the table, I don’t think most people would be thrilled to play with them again.

There are ways other than randomness. You could have some contest between the front runners, maybe what, they have a foot race? Wrestling? A game of chess? Fight to the death? It becomes almost just as arbitrary.

I also do not understand how you can consider appointments fair but random selection unfair. Power by appointment is nepotism, bribery and favoritism waiting to happen.

@mosbrooker I think you are mistaken, as democracy is absolutely not simple at all. Your ideal is good and it would be nice if people were empathetic and universally cooperative, and didn’t bandy together in exclusive groups to exert power and control over others, but unfortunately that isn’t true. People do bandy together in groups to exert power and control over others. If we could stop that from happening, we would hardly need any government at all, and people would just organize their societies spontaneously without any conflict. That doesn’t seem possible in a world where life is short and resources are finite.

A fair successor is one who has “consent to rule by the people.” Defining what it means to have the consent of the people in a way that can be efficiently verified is not easy, so we use the proxy of a vote. I don’t see how your system avoids an oligarchy. What happens if leader A states “if I am not approved, I appoint leader B,” and B states “if I am not approved, I appoint leader A”? This situation is impossible to reconcile with the details you’ve described, and according to it the pair A and B are in power perpetually, which is as close to a dictatorship as you can get without being one.

The only reasonable response to this, unless the dictators are very highly benevolent, is an insurrection by the people to either disregard or overthrow the system. But then there is no need for a leader to begin with, this is just direct democratic rule—which, I agree, is efficient and probably usually good for very small societies.

At the risk of continuing this rant, somebody might try to point to Ancient Greek society as a candidate or example for this kind of democratic system. But the truth is that democracy meant something very different to Ancient Greek people than what it means to us today, and their society was actually highly oligarchical. Something that is historically overlooked quite often is how very new the democratic republic is as a form of government. It was not that long ago that most men simply could not vote. I would recommend reading some of the Federalist and anti-Federalist papers, as well as John Stuart Mill’s “On Liberty.”

@mosbrooker allowing appointments of the successor doesn’t make any sense to me. Don’t take this the wrong way, I think the fantasy doesn’t take into account very real problems that exist with democracy. There are no checks and balances, there’s no constitution, there’s no protection for minorities, and there is no avenue to establish representation, or to create, enforce or validate laws. To me it seems mostly like a vacuous space for something more substantial to fill. What stops the society from devolving into an authoritarian mafia-run police state, or from being destroyed by warlike neighbors who want to take their crops, enslave their men and rape their women?

@mosbrooker the main problem here is how to establish the incentives for representatives to make decisions that are in the interest of the general public. For me, the goal of (real) democracy is to stabilize a society away from the extremes of anarchy and authoritarianism. We may need a better way, but we don’t need the “best” way (especially considering that there is no such thing).

Informal methods like this make sense for small groups where interpersonal social ties and influences are significant, but for large and heterogenous populations those interpersonal social ties become heavily diluted.

Dictators don’t reign primarily by playing whack-a-mole, although they do this as well. The most significant source of dictatorial power is having the seat at the very top of a pyramid, just above an oppressive (and often internally dysfunctional and corrupt) block oligarchy.

A different issue has to do with stability and competency. The whims of the masses are fickle and can be easily swayed by media bias and propaganda, and enabling immediate changing of seats can destroy the coherency of a policy-making apparatus. People are already skeptical about the prospect of term limits being imposed in Congress for that reason. New blood isn’t always a good idea when it means a lack of experience, and direct majoritarian democracy is a bad idea because it enables oppression of minorities without protection or recourse.

What happens in your hypothetical society when competing factions campaign to take control of leadership? In my mind, without a constitution and a formal electoral power structure in place, it devolves into a state of populist demagoguery (and it might do that anyway.) I think just historically that kind of society is only appropriate and stable for a small homogeneous tribe, which is why that’s the only context in which it can be found.

To tie things to current events, the reason Putin hates “The West” (read: liberal democracy) is that it is the single biggest direct threat to his authoritarian regime. All of the culture war nonsense is almost fully pretense or self-deception, and with a psycho-narc vampire like Putin it could go 50/50 either way, at any given time, take your pick.

@toby-pereira @mosbrooker and indeed (as you know) this ping-ponging is exactly the game Putin played with Medvedev and the Russian "presidency."

@mosbrooker this kind of system is only feasibly democratic when the population is very small and homogeneous.

This is a concept I had in mind which may already have been described, although not all of the logistics are necessarily hashed out and there may be issues with it. The idea is described below, but first I want to make a connection to “cake-cutting.” The standard cake-cutting problem is when two greedy agents are going to try to share a cake fairly without an external arbiter. An elegant solution is a simple procedure where one agent is allowed to cut the cake into two pieces, and the other agent is allowed to choose which piece to take for themselves. The first agent will have incentive to cut the cake as evenly as discernible, since the second agent will try to take whichever piece is larger. In the end, neither agent should have any misgivings about their piece of cake.

So this is my attempt to apply that kind of procedure to political parties and representatives. Forgive my lack of education regarding how political parties work:

• There should be a government body that registers political parties and demands the compliance of all political parties to its procedures in order for them to acquire seats for representation;
• (Eyebrow raising, but you might see why...) Every voter must register as a member of exactly one political party in order to cast a ballot (?);
• Each political party A is initially reserved a number of seats in proportion to the number of voters with membership in A; the fraction of seats reserved for A is P(A). however
• For each pair of political parties A and B (where possibly B=A), a fraction of seats totaling P(A~B):=P(A)P(B) will be reserved for candidates nominated by A, and elected by B; these seats will be called ambassador seats from A to B when B is different from A, and otherwise will be called the main platform seats for A;
• Let there be a support quota Q(A~B) for the number of votes needed to elect ambassadors from A to B, and call P(A~B) the ambassador quota of party A for B. If E(A~B) is the fraction of filled A-to-B ambassador seats (as a fraction of all seats), I.e. nominees from A who are actually elected by members of B, then A will only be allowed to elect P(A~A)*min{min{E(A~B)/P(A~B), E(B~A)/P(B~A)}: B not equal to A} of its own nominees. That is, the proportion of reserved main-platform seats that A will be allowed to fill is the least fraction of reserved ambassador seats it fills in relation to every other party, including both the ambassadors from A to other parties, and the ambassadors from other parties to A.

This procedure forces parties to also nominate candidates that compromise between different party platforms in order to obtain seats for any main-platform representatives. If a party fails to meet its quota for interparty compromises, it will lose representation. On the flip side, this set up will also establish high incentives for other parties to compromise with them in order to secure their own main-platform representation. In total, this system would give parties high incentives to compromise with each other and find candidates in the middle ground, which will serve as intermediaries between their main platforms.

Basically, here the outlines indicate seats open to be filled by candidates who are nominated by the corresponding party, and the fill color indicates seats open for election by the corresponding party:

Seats with outlines and fills of non-matching color are ambassador seats, and seats with matching outline and color are main platform seats. In terms of party A, by failing to nominate sufficiently-many candidates who would meet the support quota Q(A~B) to become elected as ambassadors from A to B, or by failing to elect enough ambassadors from B to A, party A restricts its own main platform representation and that of B simultaneously. By symmetry the reciprocal relationship holds from B to A. Therefore all parties are entangled in a dilemma: to secure main-platform representation, parties must nominate a proportional number of candidates who are acceptable enough to other parties to be elected as ambassadors.

To see that all needed seats are filled in the case of a stalemate, where parties refuse to nominate acceptable candidates to other parties and/or refuse to elect ambassadors, the election can be redone with the proportions being recalculated according to the party seats that were actually filled.

The support quotas collectively serve as a non-compensatory threshold to indicate sufficient levels of inter-party compromise. Ordinary PR is identical to PR with ambassador quotas but with all support quotas set to zero, whereby there is no incentive to nominate compromise candidates.

The purpose of this kind of procedure is twofold: firstly, it should significantly enhance the cognitive diversity of representatives, and secondly, it should significantly strengthen more moderate platforms (namely those of the ambassadors) that can serve as intermediaries for compromises between the main platforms of parties. Every party A has a natural “smooth route” from its main platform to the main platform of every other party: The main platform of A should naturally be in communication with ambassadors from A to B, who should naturally communicate with ambassadors from B to A, who should naturally communicate with the main platform of B.

Also, this procedure gives small parties significant bargaining power in securing representation. Large parties will have much more representation to lose than the small parties that are able to secure seats if the small parties refuse to elect any ambassadors, so rationally speaking, large parties should naturally concede to nominating sufficiently many potential ambassadors whose platforms are closer to the main platforms of those small parties. The same rationale holds for the potential ambassadors nominated by small parties, who also should tend to have platforms closer to the main platform of the small party.

Finally, this system creates significant incentives for voters to learn about the platforms of candidates from other parties who stand to reserve seats for representatives.

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

I'm just saying, dividing people into discrete groups and then trying to deal with it based on group isn't always helpful.

This is true when the groups are imposed by force or external factors, but when people naturally form collectives due to the possibility of aligning their intentions for mutual benefit (and given that said collectives are not going to war with each other...) group behavior can be very, very helpful indeed.

I do understand your issue with the avenues for extremists to impede collaboration. In my conception, as long as there is sufficient diversity in a representative body, the clout behind extreme platforms will not be enough to stop broader decisions from being made. This of course is assuming that the diversity present in the actual population of "electable" candidates for that body has a strong center that can bridge the gaps between more extreme positions. A Condorcet/median seeking method does seem like it would build up a very good bridge, but there doesn't seem to be much of anything on either side of it.

I do wonder about the structure of interparty politics. Do parties have ambassadors for other parties? It seems like something like that would help with a lot of the difficulties.

I mentioned this concept in another thread, but for example, what if there were a coalition of parties, where elected representatives from each party would come in two generic kinds, one being elected internally by members of the party, and others being elected externally by members outside of the party, with some kind of quota of the form "To admit X internally elected representatives, each party requires to admit Y externally elected representatives"? If voters could freely choose party alignment before elections, seats could be apportioned according to party membership, and maybe quotas could be determined by some function of the number of seats apportioned to a party and the number of seats apportioned to every other party.

Maybe for each pair (A,B) of parties, party A requires at least Y representatives elected from party B in order to have X representatives elected from party A. Can you imagine what that kind of thing would do to the democratic and republican parties? They would both be forced to nominate middle ground candidates in order to be allowed to elect their own more extreme representatives. The ratios and seat appointments could be chosen to guarantee proportional representation for complying parties. Under a system like this, parties could only gain representation by accruing membership and by compromising with other parties.

@toby-pereira I totally agree with you here. And a point to supplement or maybe just reiterate your stance on “benevolent dictators,” there’s a definite survivorship bias there. The business dictators who fail don’t show up in the news, while the ones who succeed are often lauded as geniuses without due regard for the role of pure luck or factors totally unrelated to their supposed acumen. The same goes for stock market investors—many who come out on top end up believing they have a special intuition to “outsmart” the market, but when compared with randomly generated portfolios, their inclinations usually fare no better, and often do worse (due to their relative lack of diversity, they are not as robust against market volatility).

So basically, when uncertainty is involved, failure doesn’t always indicate a poor strategy, not any more than success indicates a good one. This reduces my confidence that business dictatorships actually tend to promote the success of a company.

@rob I do think diversity can turn down the heat, since it will naturally introduce centrists and moderates. At the same time, individuals who are centrist will probably tend to have their own class of characteristic biases that can be checked by those further from the center. For example, centrists may consider certain issues unimportant, dismiss certain ways of thinking, be indecisive or even generally apathetic. I think it's better for the diverse perspectives to be heard and shot down by an equally diverse cohort than not to be heard at all. Rather than rely on a Condorcet method to identify the center, the moderate center could be allowed to emerge naturally.

I think novel ideas come from team work, and that cognitive diversity is the single most valuable resource available to a team. For example, teams have been shown repeatedly to outperform experts when it comes to solving hard and novel problems. Engagement with diversity tension and disagreement is what refines the directions of projects. It is true that teams tend to have fairly well-defined goals, so that there is naturally a degree of “like-mindedness” in a team. I think the goals of representative legislators should be to compromise to create laws that are congruent with the values and interests of their constituents.

Restricting to moderates might filter out the ideas and voices from less central ideological regions, which I think is bad, even if I disagree with what those voices might be saying. In fact, I think it's bad exactly because I might disagree with what those voices might be saying.

In terms of being more substantial and less controversial, I can only speak generally to the point that such agreements will be more informed and therefore more robust. I will try to think of an example but I am mostly operating on my own thoughts.

In terms of consensualism, I'm coming to believe that single-winner systems can't really accomplish that. I also think that allowing diversified representation would increase voter satisfaction and faith in government, since voters would be able to at least point to some group of people in office who more than just roughly embody their values in government, and know that their voices are being heard and addressed.

One can look at one of the core issues with our current system in Congress in two ways: one might think that the issue is a lack of moderation, or perhaps a lack of cognitive diversity. Both are true, I think, but I also think addressing cognitive diversity will naturally address the lack of moderation, and not necessarily the other way around.

@rob that is true that I am more concerned with positive representation in Congress, I think cognitive diversity is a safe way to “turn down heat” in government while simultaneously offering routes for important or novel issues to be addressed. In my opinion, agreements between individuals of dissimilar ideologies are often more substantial and less controversial than those between like-minded individuals. I think those are the kinds of agreements that should be influence legislation. When it comes to things that need to get done quickly, nonpartisan executive authority is needed and I think that’s what that branch of government is for.

@rob I think certainly X achieves better representation in the second case. His interests are almost exactly embodied by one of the three candidates. So are the interests of the other voters near the two other candidates.

Ideally we would have a representative body whose ideological distribution is an informative compression of the electorate’s own distribution. Both centrist and more polar representatives would be present, not just one or the other. Middle ground representatives facilitate compromise while those further from the center contribute significantly to cognitive diversity and introduce issues to be compromised about.

This is just a half-formed spitball of an idea, but I think in the context of parties, it would be cool or at least interesting to have some kind of blocked and cross-over voting structure, where parties can win seats via at least two kinds of candidates, with one kind being elected solely from inside the party, and a second kind being elected solely from outside the party. This could basically set up a system of inter-party ambassadors and possibly encourage inter-communication and more compromise. In any case it would produce more representation of any ideological overlaps between parties.

@rob I think something we agree on, although we might express the concept differently, is that what it means to be properly or adequately represented in government or in general, whether absolutely or to any given degree, is not fully clear. To me the meaning of representation appears to hinge on the meaning of consent, which is heavily social. A functional definition or acceptable proxy of that social concept to apply to voting is basically a pillar of the discussion here.

Correct me if I mistake your view when I suggest that you are more or less alleging that agreement on a “reasonable” procedure is in itself an acceptable proxy for consent to representation, and you have illustrated your conception of what constitutes a reasonable procedure with analogies and well-defined criteria such as Condorcet compliance—Because such a procedure is reasonable, then if any such procedure is agreed upon, problems relating to adequacy of representation are effectively solved. Is that accurate?

If so, then I think this is one instance where we disagree. To me, adequacy of representation is an outcome issue, and explicitly not a procedural one. In my opinion it is highly dependent on context whether a single winner method is appropriate to achieve representation of any degree, even if it is for example a Condorcet method. A simple poll on the quality of representation voters receive from government officials on something like a “very poor,” “poor,” “fair,” “good,” “excellent” scale could formalize those feelings in any case. I would be very surprised if a PR system did not outperform a single-winner system in such a poll almost always.

Single-winner systems are still necessary for the delegation of distinct executive roles, but I don’t think they can be very effective in establishing good representation. And the more I think about it, the more merit I feel in @Andy-Dienes’ comment about single-winner systems and Duverger’s law.

@rob I think you could consider two groups of people such as those who have reasonable confidence that they are "adequately represented" in government, versus those who have reasonable confidence in the opposite direction. Then maybe there’s also some middle zone of suspicious, uninformed and misinformed people, and you could lump them into the mix. But I think we want to make the first of those groups as large as feasible and the second of those groups as small as feasible.

Obviously what constitutes adequate representation to a person is highly individual and depends on values and special interests, but it has a pretty unequivocal normative meaning once hypocrisy is removed. I think the second group is pretty close to what I would consider “the minority,” even though it has nothing to do with whatever fraction of the electorate it constitutes. PR even with parties seems to pretty decently address adequately representing a much larger fraction of the current minority than the present situation could hope to accomplish.

Also, there just has to be some kind of non-compensatory means of narrowing down the pool of candidates. I think political parties and primaries are reasonable and I'm not sure how else it would be done. The problem to me is a small number of large parties with a serious lack of accountability to public interests, not necessarily the very existence of parties. It would be ideal to be able to have a smooth spectrum of choices and the ability to elect based on the candidates alone, but I don’t think that can be accomplished without untenable sacrifices in efficiency.

When it comes down to it, potential candidates need to gain recognition, support and traction with a significant base, otherwise they have no chance of even getting on the map, since there is limited bandwidth. Granular parties don’t seem problematic to me, I think particularly binary ones though just in theory don’t have enough resolution to represent public interests, even ignoring externalities like polarization.

@rob only an act of Congress, only the same requirement for a declaration of war! Lol.

Hopefully local legislatures steadily incorporate PR, I think certain state governments are just as farcically representative as the whole nation.

@andy-dienes I see, interesting. I’ll have to learn more about multi-winner systems. Are there any you like in particular?

@bternarytau I returned to thinking about this. I think that if the voting system is homogenous—meaning that scaling the ballot set leaves the winner invariant—and generic, meaning that almost every distribution of ballot sets has a neighborhood of ballot sets of nearby distributions that give the same winner—then it seems very reasonable that the participation criterion would imply monotonicity. And these two criteria seem satisfied by pretty much any reasonable voting system.

I don’t consider this a formal proof but a sketch of plausibility, and this kind of reasoning might somehow lead to a formal proof that for a generic homogeneous voting system, participation almost always implies monotonicity.

If a homogeneous, generic system is not monotonic, then there are two ballot sets B1 and B2 of the same size such that the winner of B1 is x, and such that every ballot in B2 has an identical corresponding copy in B1 except for exactly K ballots for some K, which are all the same as their corresponding ballots in B1 except for having a higher rating of x; however, despite this, the winner of B2 is y≠x.

Now scale up the ballot sets (making the same number of copies of each ballot) by a factor of (1+E) for some positive E to make B1’ and B2’ with the same properties by homogeneity, except now there are K(1+E) ballots in B2’ whose corresponding copies in B1’ rate x lower. Let this set of K(1+E) ballots be called X, and its complement is (B2’-X).

Also, suppose actually that these K(1+E) ballots rate x at maximum. This seems plausible to be able to find.

Note that if the winner of (B2’-X) is x, then we are done, since this is a no-show paradox—the K(1+E) voters can make their favorite x win by abstaining from voting. Therefore we may assume that the winner of the election from the ballots in (B2’-X) is different from x, say z.

Now let a small subset H of X be changed so that the distribution of ballots in the union Hu(B2’-X) of H and (B2’-X) approaches the same distribution of ballots as are in B1.

As H is increased in size, the distribution of ballots in Hu(B2’-X) must be able to approach that of B1 (with respect to whatever metric you consider), and since the system is generic, at some point, the winner of the election in Hu(B2’-X) must switch from z to x, possibly through some intermediaries, but at some point it will almost certainly stabilize at x.

As this subset is grown, the winner of the election based on all of the ballots in B2’ also changes from y to x, also possibly through some intermediaries, but at some point it also almost certainly will stabilize at x.

Now, it seems virtually impossible that these stabilizations will occur simultaneously. If the stabilization of Hu(B2’-X) occurs first, then we are done. If the stabilization occurs second, then right at the boundary of stabilization, if there are any voters outside of X who prefer the intermediary winner to x, they should also abstain from voting, which will push the distribution back the other way.

There are obviously some extra assumptions made here, but I think this is at least convincing that almost all reasonable voting systems will not satisfy participation if they are not also monotonic.