Majoritarianism, Utilitarianism, and the Paradox of Democracy

I’m sure all of you are aware of this, but I just wanted to examine the concept of majoritarianism versus utilitarianism myself to make sure that I grasp the situation. The concepts of "majoritarianism’’ and "utilitarianism’’ can be explained well using an anecdote, which I am borrowing from @psephomancy in his questionnaire linked here:

Suppose that three friends plan to chip in equal amounts of money to buy and share a pizza. They are just able to afford a single topping pizza if they all chip in. Two of the friends love pepperoni, and like mushrooms (but not as much as they love pepperoni). The third friend loves mushrooms and hates pepperoni. The question is, what topping should they get on their pizza?

According to the questionnaires, around 90% of people seem to agree that the friends should get the mushroom topping. Some ethical explanations are that the friends should get a topping that everybody is at least fine with if possible, that it is better for two of the three friends to have slightly suboptimal outcomes than for one friend to be extremely disappointed, and that “the total amount of happiness will be greater.” Ordering mushrooms instead of pepperoni would be called the "utilitarian’’ solution to this problem. Getting mushrooms has a "balancing’’ effect on the distribution of happiness in the group, while still pleasing everybody reasonably well. The outcome is sociable, and everybody in the group feels appropriately represented. There are additional positive consequences to this—good faith is established across the coalition through compromise and sharing, making it generally more stable and effective. Even though two of the friends have suboptimal outcomes, they may themselves benefit from utilitarian arrangements in future situations—effectively, the risks of negative outcomes to the parties in general contexts are distributed “more fairly.”

In contrast, the "majoritarian’’ solution would be if the friends collectively decided to order the pepperoni. The basic ethical explanation is that more people prefer pepperoni over mushrooms than vice versa. (Or in this specific instance, maybe the third friend can just pick off the pepperoni—he will still feel sad that he gets no mushrooms, though). In this case, where the parties involved are friends, the majoritarian solution would hardly be socially acceptable. However, parties involved in collective decision-making are far from always friendly. In this example, the two friends who prefer pepperoni might be referred to as the "block majority,’’ and given that they were selfishly only concerned with getting their favorite pizza, they would benefit from the majoritarian solution at the expense of the minority. People are generally less keen to support a utilitarian outcome when the parties involved are not “friends” per se, when decisions are to be made only a small number of times rather than periodically over a long period of time, or when they themselves are a member of the block majority. There is a double standard individuals hold for themselves versus others—subject X will want everybody else to be utilitarian when it serves his own interests, but he will probably support majoritarianism when it’s convenient. In terms of evaluating voting systems, it’s a prisoner’s dilemma.

As opposed to utilitarianism, majoritarianism tends to cause internal pressures to build up from within coalitions, making them weaker and less effective. Minorities will feel inadequately represented, and if the block majority remains stable over time—which is quite likely if they are the ones controlling the decision-making procedure—the minority will continue to build up resentment for the block majority. Such an arrangement is referred to as the "tyranny of the majority.’’ Short of violent oppression of minorities by the majority, coalitions ruled by tyrannical majorities can only be stabilized by external pressures that compensate against the internal pressures caused by the instability of the system. Otherwise, they will either dissolve through abandonment or rebellion, or else devolve, however steadily, into full-blown totalitarianism. Even in cases where there is no clear block majority, it is still possible for smaller factions to conglomerate according to shared interests in order to secure power over competing factions. In a majoritarian system, this behavior is encouraged. In other words, the public response to majoritarianism is to form a small number of large political parties, which implies a tendency toward political oligopoly, duopoly, or fascism.

It seems clear that utilitarianism is a socially superior ethical framework to majoritarianism. However, it is generally difficult to design a system that will remain utilitarian for long. The so-called “paradox of democracy" occurs when superficially utilitarian systems lead to unambiguously majoritarian outcomes, primarily due to strategic alignment of the block majority. For example, superficially “utilitarian” voting systems usually enable voters to be more expressive with their ballots by allowing them to indicate "degrees’’ of preference in some manner. However, it is exactly this freedom of expression that can allow the block majority to strategize and out-compete the more diffuse voter pool. On the other hand, less expressive systems tend to give poor results unless they are majoritarian to begin with. Thus there is the cynical argument for majoritarian systems that they simply eliminate the middle man—voters don’t need to burden themselves with worrying about strategies that tend to end them up with the same results anyway. This of course is slightly mendacious, since it can be very difficult for block majorities to align strategically in some utilitarian systems. In this case, though, voters are burdened with playing a strategic mind game in order to vote, which can make expressing their ballots more frustrating and difficult, and can also make the "meaning’’ of their ballots more arbitrary. Even then there is no guarantee that the block majority cannot eventually find a way to secure political victory.

@RobBrown has expressed the analogy between the Condorcet criterion and using the median to determine the result of an election for points over a real numbered line segment (as opposed to using, say, the arithmetic mean). I think this is a good analogy, in that both methods suppress tactical voting and hence encourage “honesty.” However, these methods are also both majoritarian (we can discuss examples if anybody wants to). I have been trying to think about a related spectrum along which voting systems can be polarized according to the expressivity of ballots in the context of the voting algorithm that operates on them, which I am informally calling “elasticity”—on one end are the “rigid” systems, for example ranked choice methods; and on the other end are the “loose” systems, for example, most range methods. To illustrate this concept more, I would consider Cardinal Baldwin to be weakly loose, score to be strongly loose, and Condorcet methods to be rigid. Obviously these are relative terms, and I am comparing these systems to what I imagine would be ideal. What I’ve noticed is that systems that are too rigid tend to produce poor results unless they are majoritarian, and systems that are too loose come across superficially as utilitarian, but will eventually (though perhaps only after an unreasonable amount of time) devolve into majoritarianism once the block majority aligns in strategy, whereas in the meantime strategic voting is something of a free-for-all.

It seems to me that one of the main things we want to do by studying voting systems is to produce a system that is more than “superficially” utilitarian—for example, a system that is at once not so rigid so that majoritarianism is preferable, and at the same time not too loose so that strategy is prevalent or that it devolves into majoritarianism anyway. In other words we want to resolve the paradox of democracy. In my opinion, true utilitarianism in voting systems exists at the balance between rigidity and looseness, rather than as one end of a spectrum.

I suggested a method for evaluating voting systems that might help us sort through systems that are too rigid or too loose according to how well they conform to a standard for utilitarianism outlined in a particular kind of social experiment here:

Hopefully we will be able to participate in it once the results for the mock election are in.

As usual, anything else you would like to add is welcome!

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I am being careful here also to address what I am calling “superficially utilitarian” systems. Even systems that appear utilitarian can lead to majoritarianism due to strategic alignment of the block majority. That is the “paradox of democracy.”

SV = 100% utilitarian
Condorcet = 100% majoritarian
however, there are many middle grounds.

I use this example, to distinguish the middle ground:
50: A[5] B[0] C[3]
30: A[3] B[5] C[0]
30: A[0] B[5] C[3]
Win A: the method is more utilitarian.
Win B: the method is more majoritarian.

Paradox: “superficially utilitarian systems” can lead to majoritarianism due to strategic alignment of the block majority.
To verify it’s necessary that:

  • there is a block of voters that is larger than 50%.
  • such a block of voters must know that it is indeed that large.
  • such a block of voters must coordinate to use a certain strategy.

These conditions are very difficult to satisfy, but it is also true that the problem exists anyway (in other forms) and the only solution is to try to minimize the strategies in utilitarian system.

I take this opportunity to resume an interrupted discussion with @cfrank in which we are looking for a utilitarian system, not subject to strategies, simple and without too arbitrary concepts (such as normalizations). To meet these demands I have recently developed FAIR-V, try to take a look at it.

In general, I think you need a simulator that tests various types of strategies on a voting system to understand exactly how much subject to strategies is, and how many times the utilitarian candidate wins.

There will always be a block majority larger than 50% because smaller factions can always conglomerate according to shared interests, which happens often in British Parliament. It’s easy for such factions to learn that they are large enough collectively to sway elections in their favor, and also easy for them to coordinate as long as their common incentives are strong enough, i.e. when it matters.

I think a good voting system should eliminate the strategic benefits of forming subcoalitions while still providing a utilitarian outcome.

I will check out your voting system. What do you think of the social experiment in this context?

There will always be a block majority larger than 50% because smaller factions can always conglomerate according to shared interests, which happens often in British Parliament.

I am talking about elections, in which millions of unknown people vote. The parliamentary context is different, and the problem you described can actually happen more easily in that context.

What do you think of the social experiment in this context?

I prefer to deal with the parliamentary context separately from the electoral one.

I think it’s becoming easier and easier for people to find others with common interests and band together according to those interests, but honestly I’m not sure how the internet will continue to change the way people interact. I mean take all of us for example, I’ve never met anybody here in person but I know that we share an interest in voting systems and are participating in a group centered around that topic. Once the Democratic and Republican parties are out of power, I have no idea what will fill the void. My guess is that over time, a majoritarian system or a superficially utilitarian system will continue to pressure the public to form large political parties. It might take a while but once the mud settles, I don’t know. Millions of unknown people vote now but are still influenced by the two major parties. We don’t know the full scope or cause of the Duverger effect.

I wasn’t totally clear about context but I meant in terms of evaluating the utilitarian character of generic voting systems.

I wasn’t totally clear about context but I meant in terms of evaluating the utilitarian character of a general voting system.

Ok, but then as you say it’s a paradox.

  1. a utilitarian system must allow the voter to assign ratings with different weight to candidates.
  2. but by satisfying 1, a majority group can strategically give maximum points to one candidate, and minimum to all the others.

I think this is an inevitable problem, so one can only try to reduce it.

In FAIR-V I try to solve the problem by making the voter follow this logic:
“I support at most all candidates, excluding the worst, until this is eliminated; when the worst is eliminated, I support all the others except the second worst; and so on”.
In short: voter fight to eliminate the worst, rather than to elect the best.
Again though, voters might group together and say “1 candidate is the best of all, while all the others are worst”, but doing so would be risky because if their only supported candidate were to lose, their vote would become null in deciding who should win among the remaining candidates; for this I say that the group should be more than 50% of the voters to have an interest in using a similar strategy.

I think the experiment I outlined might address that paradox in some way. What about the following:

Everybody rates the candidates. Then the voters are shown the distributions of ratings for each candidate incognito, with enough random noise so that nobody can tell which candidate is which. Based on that information, there is a second plurality vote for the incognito winner, who then is the winner of the election.

For example, say there were votes as follows:

5: A[5] B[1] C[0]
10: A[5] B[4] C[0]
15: A[0] B[3] C[5]

Then the distributions would be

X: 0[15] 5[15]
Y: 0[15] 5[15]
Z: 1[5] 3[15] 4[10]

Then some random noise is added:

X: 0[14] 1[1] 2[3] 3[1] 4[0] 5[15]
Y: 0[15] 1[1] 2[0] 3[2] 4[1] 5[15]
Z: 0[1] 1[7] 2[1] 3[14] 4[10] 5[2]

Which candidate do you think would win this election?

So you make the method nondeterministic, and then the second choice of the voters is also random because they don’t know in the least who X,Y,Z are.

All this randomness I think would ultimately favor the majority candidate (and not the utilitarian one).

If my honest vote is this:
A[5] B[4] C[0]
but I vote strategically like this:
A[5] B[0] C[0]
there is no way to understand that I like B more than C and a little less than A (if A is the majority winner).

The real solution must be found for those contexts in which there are groups of voters who cannot themselves be the majority.
In such contexts it is necessary to ensure that voters are encouraged to indicate their first choices.
Eg if a group votes honestly like this:
A[5] B[3] C[0] but he learns that the majority candidates are probably B and C, then the voting system must:

  1. encourage the voter to indicate A as the first choice, possibly above B.
  2. make sure that B is always favored at most over C.

A 1st and B favored at most means that A and B will both have the maximum points and C the minimum, but if C were to be eliminated then A should have max points while B at 0 points, and this is exactly what FAIR-V does, pushing the voter not to mask his real first choices (even knowing the frontrunners).

In the example I provided with

X: 0[14] 1[1] 2[3] 3[1] 4[0] 5[15]
Y: 0[15] 1[1] 2[0] 3[2] 4[1] 5[15]
Z: 0[1] 1[7] 2[1] 3[14] 4[10] 5[2]

you don’t think voters would be more keen to elect Z over X or Y? Choosing X or Y is risky, it’s basically a 50/50 chance whether you love or hate them. Choosing Z is safe, you have a very good chance of liking Z reasonably well.

Z represents the utilitarian winner, which is B. But let’s look at your older example:

55: A[5] B[4] C[0]
45: A[0] B[5] C[0]

Now the distributions are like this:

A: 0[45] 1[0] 2[0] 3[0] 4[0] 5[55]
B: 0[0] 1[0] 2[0] 3[0] 4[55] 5[45]
C: 0[100]

and incognito with random noise, they would appear as something like

X: 0[47] 1[2] 2[1] 3[2] 4[3] 5[56]
Y: 0[1] 1[2] 2[5] 3[2] 4[53] 5[48]
Z: 0[104] 1[5] 2[2] 3[5] 4[1] 5[1]

Choosing Z is stupid, choosing X is risky. The only reasonable choice is Y, which represents B, the utilitarian winner.

Presented with this situation I believe the voters will tend to elect a utilitarian winner in order to mitigate the risks of a negative outcome for themselves. Choosing uniformly at random would be a terrible idea, since you can use the information about how the candidates are rated to make a more informed choice. You should choose a candidate with a distribution that balances your aversion to risk against your potential for high gains.

The only randomness with any real effect is the incognito labeling. The random noise added will hardly affect the shape of the distribution, its only function is to prevent voters from discovering who is who by comparing their own ballot with the distribution.

In this voting system you assume that the initial votes are honest, so it is normal for the winner to be B.
Once understood how the system works, the group of 55 voters, instead of voting like this:
55: A[5] B[4] C[0]
would vote like this:
55: A[5] B[0] C[0]
and your subsequent random noise doesn’t change the fact that A would seem better than B.

Also, if it were true (and in my opinion it’s true) that:

Choosing Z is stupid, choosing X is risky. The only reasonable choice is Y, which represents B, the utilitarian winner.

then all the incognito procedure is useless because in reality you already know that Y will be chosen and therefore you might as well have it chosen by the voting system instead of having people vote twice.

The voters now have incentives to vote honestly. If the voters do not indicate their honest preferences, then they are being extremely risky. In the example you gave, the voters would be faced with this:

X: 0[45] 1[0] 2[0] 3[0] 4[0] 5[55]
Y: 0[55] 1[0] 2[0] 3[0] 4[0] 5[45]
Z: 0[100] 1[0] 2[0] 3[0] 4[0] 5[0]

Noise might lead to this:

X: 0[47] 1[1] 2[3] 3[2] 4[4] 5[54]
Y: 0[55] 1[3] 2[1] 3[2] 4[2] 5[44]
Z: 0[98] 1[2] 2[4] 3[1] 4[3] 5[1]

So now voters are faced with a conundrum. Sure, it is “more likely” that they preferred the incognito candidate Y over X, but the choice is still very risky. There is still a nearly 50% chance that their preference is actually reversed. Unless they know that they are in a clear majority, by voting like that, they are making the situation far more risky for themselves. Essentially they have left themselves no option but to face risks that would otherwise be intolerable.

For example, would you choose to engage in a one-time game where you have an 80% chance of winning $125, and a 20% chance of losing $500? Or even a one-time game where you have a 55% chance of winning $100 and a 45% chance of losing $100?

In these instances, calculating expected values doesn’t help you make an informed decision. Expected values are only useful for games that are repeated “many times.” For these one-time games, you have to face the risks involved and decide whether you will tolerate them. If the risks are intolerable to you, you would choose not to play.

There are other problems with this system though that I’m still trying to think about. For example, groups could try to indicate scores that can make a candidate easily identifiable, for example, agreeing to vote their favorite candidate as a 3, and then choosing the candidate with the most 3s, or something along those lines.

I understand your underlying logic better, but in the real world I still think it doesn’t work to avoid strategies.

This context indicated by you:
X: 0 [47] 1 [1] 2 [3] 3 [2] 4 [4] 5 [54]
Y: 0 [55] 1 [3] 2 [1] 3 [2] 4 [2] 5 [44]
Z: 0 [98] 1 [2] 2 [4] 3 [1] 4 [3] 5 [1]
it would be risky only if people didn’t logically vote for X (which is the candidate with the most points), but if people are not logical, then this system can always fail.

Basically, you put in some noise to make A’s defeat possible even though he actually has the majority.
This is a problem because in case the following votes are honest (not strategic as assumed before):
55: A [5] B [0] C [0]
45: A [0] B [5] C [0]
then there should be no risk that B win rather than A.

A voting system must also be deterministic because one must be sure that there are no “external” alterations; a certain set of ballots must give only one result.

I definitely prefer the case “55% win $100 and 45% lose $100”. In an election context, losing $500 can be irrecoverable.
This risk you are talking about is actually already there even without the need for this system, specifically the risk consists in believing that the group of 55 voters actually has the majority (55%), when this may not be true.

Moreover, always considering the example, if 55 voters use this strategy: A[5] B[0] C[0] does not risk anything since the winner would be X or Y (they win if it’s X, they don’t lose if it’s Y).

But that’s a Catch-22, the initial votes won’t necessarily be honest otherwise. As you point out, even with the system they might not be.

You’re right that people are not logical. That’s another problem lol. But the noise is only there to prevent voters from identifying candidates.

I understand the appeal with deterministic systems, but I’m not totally concerned with it personally. Call it impractical, but I actually feel it is the opposite. I would rather have a nondeterministic system produce reliably fair results than a deterministic one that doesn’t.

If there is no choice but to face the risk, then voters still would choose to vote for the majority candidate. I agree that there is a range where error is likely, if noise were to push one candidate over another, although usually that wouldn’t happen.

And that is true. At least rather than a 100% chance of A winning, they can only achieve 50%. Still not perfect, but I think it’s an improvement.
That’s also only in this case. In other cases, the voters may be reducing the chance that a secondarily preferred candidate wins over a less preferred candidate by voting dishonestly. So overall they have dramatically reduced incentives to strategize. If standard utilitarian voting systems tend to avoid strategy, then I think this system will only do better in that regard.

But still I’m just trying to throw ideas out there. These are hard problems to solve and I don’t expect to find a perfect solution or even a good one without a lot of trial and error and thought provoking conversations, so thank you for engaging with the idea even if you think it’s impractical.

You’re right that people are not logical. That’s another problem lol.

ok, one more reason to avoid using a system whose result also depends on this illogicality.

But the noise is only there to prevent voters from identifying candidates.

So I don’t think it’s necessary because voters already don’t know exactly how big is the group that supporting a certain candidate (they only know roughly).

I would rather have a nondeterministic system produce reliably fair results than a deterministic one that doesn’t.

Which I don’t think is what you suggested here.
Apart from that, I am extremely convinced that people opposed to the winner would always have something to complain about the nondeterminism part of the system.

If standard utilitarian voting systems tend to avoid strategy, then I think this system will only do better in that regard.

On the other hand, I think that this system adds nothing that is actually useful.
Without the noise, which is not really necessary, there are 2 possibilities:

  • among the ingoniti candidates, the voters logically vote the one with the most points, so the fact that there is such a vote doesn’t change anything (you could immediately win the candidate with the greater sum and you have the SV).
  • among the unknown candidates, some voters do not logically vote and this I don’t think can lead to better results or a reduction of strategies.

In my opinion the solution is to create a system that pushes people to indicate their second choices, while knowing the frontrunners; and to push people I think you need something well defined (deterministic).

I’m not sure why the logical voters would behave as you suggest. Accounting for risk is a logical thing to do. The system I am suggesting depends on voters behaving logically, not illogically. I’m not sure why the noise is unnecessary, it prevents voters from identifying candidates, which is an essential apparatus of the system. But you may be right for large electorates.

Logical voters will not only vote for the candidates with the most points, that’s a prescription without empirical justification. Logical voters will assess the distributions of each candidate, weigh the risks involved with selecting each candidate, and if possible choose one with low risk and high or moderate reward, which by its very nature will tend to be a utilitarian candidate. Investors who behave as you suggest on the other hand will go bankrupt, because the tremendous risks they are taking will inevitably catch up with them unless they quit while they’re ahead or change their behavior.

I’m not sure why the noise is unnecessary, it prevents voters from identifying candidates, which is an essential apparatus of the system. But you may be right for large electorates.

Yes, I am talking about elections with many voters; in reality even with only 100 voters, after having combined the points of the votes, it is not so trivial to recognize one’s favorite candidate. I would prefer to avoid noise to simplify the analysis of the voting system, assuming that candidates are in any case unrecognizable even without noise.

Logical voters will assess the distributions of each candidate, weigh the risks involved with selecting each candidate, and if possible choose one with low risk and high reward, which by its very nature will tend to be utilitarian.

Ok, you basically assume that some voters for example would use the sum, while others the median, etc in short, everyone uses their own method but I don’t understand how this can lead to improvements or more honest votes.
If you don’t know how voters estimate risk, then the outcome seems largely random to me (which is not a good thing); to this are added the illogical voters.

I assume voters would weigh risks against rewards in essentially the same way that I would, perhaps with different personal thresholds for risk tolerance. It’s hard to formalize mathematically but people do it all the time. You might weigh the risk of getting caught by a police officer against the reward of not having to wait for a red light when nobody seems to be around, etc. Businesses and investors operate by weighing risks against rewards all the time, in the end the free market determines who does so effectively. I think people can generally agree about which distributions are risky and which distributions would be favorable with high probability and low risk. It doesn’t necessarily come down to people choosing their favorite methods, I mean they can if they want, but I don’t think it’s random at all, it’s a rational process.