the death of majoritarianism

clay shentrup
9 min readFeb 11, 2024
a group of shadowy suited figures preside over the death of majority rule

escaping the subjectivity trap

when it comes to the design of democracy, there’s a lot of contentious debate. but how can we debate subjective opinions? how can there be any objective truth about what constitutes the “best” voting method if this is fundamentally a philosophical question of what one values?

the answer lies in internal consistency. if any two ostensibly subjective statements contradict each other, then they cannot both be true. this is the basis for the logical technique known as reductio ad absurdum, or “reducing to an absurdity”.

majoritarianism is a contradiction

herein we shall see that the majority criterion itself leads to such an absurdity, rendering it objectively false. that is, the mere notion that an electorate “prefers” x over y if a majority of its members prefer x over y is proven to be false.

irrelevant alternatives

imagine the following preferences.

35% x y z
33% z x y
32% y z x

who does the electorate prefer here?

suppose you claim it’s x. okay then, let’s look at this alternative scenario in which y has been caught committing a terrible crime, and thus plummeted to last place among all voters, but without any change whatsoever to their opinions on x or z.

35% x z y
33% z x y
32% z x y

we can simplify even further by combining the second and third rows:

35% x z y
65% z x y

logically, x must still be the preferred candidate here, because we have the exact same voters with the exact same preferences as in the first example — the only difference is that y has gotten less popular. and yet a huge 65% majority of voters have z as their first choice.

this is a mathematical proof that an electorate can prefer a candidate different from the favorite of a majority of its members. and it has been known in the field of social choice theory for centuries. yet this proof is both widely unknown among electoral reform activists, and they often have a difficult time grasping and accepting it when it’s presented to them.

concurrent contradiction

now you might object that it’s unreasonable to demand a consistent outcome between these two hypothetical election scenarios, given the huge change in preferences for y . we would counter that this shouldn’t matter, because the group’s preference between x and z should only be a function of the individual voters` preferences between x and z (which are consistent in both scenarios), and not a function of their opinion of an “irrelevant alternative”, y. i mean think about it: if we’re trying to assess whether apples or oranges are more popular among a class of students, why should we need to ask their opinion of an irrelevant option such as broccoli? this is called independence of irrelevant alternatives in the field of social choice theory, where broccoli is the irrelevant alternative.

still not convinced?

but perhaps you don’t find that argument convincing. very well then. we can turn to a more concrete concurrent scenario. imagine we have these three simultaneous majority opinions.

  1. a landslide 67:33 majority prefers the democrat candidate for mayor.
  2. a landslide 65:35 majority prefers the democrat candidate for city attorney.
  3. an even bigger landslide 68:32 majority opposes having both the democrat for mayor and the democrat for city attorney. (e.g. because they believe this will lead to self-dealing corruption, compared to having a city attorney from a different party, thus less beholden to internal party politics.)


just in case you thought the aforementioned scenario was mathematically impossible, i now present to you a trivial proof that it’s quite possible indeed.

# voters | their preferences
35% dm ra > rm ra > dm da > rm da
33% rm da > rm ra > dm da > dm ra
32% dm da > dm ra > rm da > rm ra

for instance, the first row says that 35% of voters have democrat mayor and republican attorney as their first choice.

no escape

now that we’ve demonstrated that this concurrent contradiction scenario is indeed possible, who should win? while a huge majority want a democrat for mayor, and a huge majority want a democrat for city attorney, an even larger majority would prefer either the democrat for mayor and republican for city attorney, or the republican for mayor and democrat for city attorney.

the simple reality is that no matter what outcome you pick, a super-majority of voters will fail to get their way in one of these three binary choices.

cardinal voting

in voting theory, cardinal voting methods are those which use scores (e.g. 0–5 stars), such as star voting or approval voting (binary approve/disapprove). those which use rankings (1–2–3) are called ordinal voting methods.

people sometimes complain that cardinal voting methods are somehow unfair because they can allow a majority-favored candidate to lose. but as we see, in so doing, they’re simply demonstrating a lack of familiarity with the deep science of voting theory. logic dictates that the majority isn’t always right, no matter how unsettling that might seem.

a right, not an obligation

moreover, it’s crucial to see that score voting and related voting methods can never deny a majority the right to get their way if they want to. all they have to do is give their preferred candidate a maximum score, and everyone else a minimum score. then they can force majority rule.

here’s an example. imagine a voter who slightly favors the democrat to the green, but is so worried about getting the republican, that they also give the green a high score, or approve green as well as the democrat under approval voting. now imagine this switches the winner from democrat to green, even tho a majority of voters ultimately favor the democrat. while they might be upset in hindsight, and wish they hadn’t supported the green, they should not demand to switch to a voting method that would “fix” that problem in the future by rescinding their right to support additional candidates, such as the green in this scenario. because it may very well have gone the other way: doing so might have indeed helped to elect the green instead of the republican, thus protecting them from getting their worst option.

so there you have it. complaining that cardinal voting methods can allow a majority to voluntarily give up their right to force majority rule would be like going a whole year without getting into a car wreck, and then feeling bitter that you paid for auto insurance. sure, in hindsight that might seem like a waste of money. but you couldn’t see the future, and you made a probabilistic decision that made sense given the uncertainty you had at that time. getting down to brass tacks, the question is, do you now decide that insurance is a waste of money going forward? probably not.

maximizing happiness

another thought-provoking way to think about it is to imagine a series of two elections. in the first election, you’re in the majority, but you have a very mild preference between the two candidates. in the second election, you’re in the minority, but you hugely prefer your candidate to the other. if you could somehow trade your majority win in the first election in exchange for getting the minority candidate in the second election, you’d be happier in total.

you could think of it kind of like horse trading. imagine you were to go back in time and find a voter in the opposite predicament: they were the vocal minority in the first election, and the tepid majority in the second election. you pledge to vote for their candidate in the first election, in exchange for them voting for your candidate in the second election. you’d both give up majority rule in the less important election where you were in the majority, in exchange for receiving the majority in the critical election in which you were the minority. you’d both be happier in net.

a voting method like star voting is effectively allowing voters something akin to this horse trading. a cardinal voting method focuses on maximizing voter happiness rather than insisting on majority rule at all costs. thus even if you’re not moved by the logical proof above, you have a compelling interest in supporting a score-based voting method for your own personal satisfaction.

but doesn’t instant runoff voting fix this?

first, note that instant runoff voting is commonly called “ranked choice voting”. but this is a misnomer, since there are a multitude of ranked voting methods — and most experts actually consider instant runoff to be one of the worst of them. so we’ll stick with the proper, or at least more specific, name.

now at this stage, a lot of proponents of i.r.v. (instant runoff voting) will counter that their voting method lets them have their cake and eat it too, because their “insurance policy” of supporting a second candidate only counts if their favorite has already been eliminated. in their mind, that makes it totally safe to support additional candidates, while still ensuring a “majority winner”.

but that is also false. here’s a simple example to demonstrate this.

# voters | their preferences
35% left center
33% right center
32% center

we’ve bundled all the center voters together here, leaving their second choice unspecified. in this simplified scenario, center is preferred by a landslide majority to both left and right. yet center is eliminated due to having just slightly fewer first-place votes. never mind the debate about whether this is “right” outcome. the point is, it wasn’t the insurance policy voters thought. the winner will come down mostly to who earns the most second choice voters from center supporters.

if left wins, then right voters will wish they had insincerely voted for center, because then center wouldn’t have been eliminated, and would have won — so they’d have gotten their second choice instead of their third.

if right wins, then left voters will wish they had insincerely voted for center, because then center wouldn’t have been eliminated, and would have won — so they’d have gotten their second choice instead of their third.

this is similar to how many people preferred candidates like elizabeth warren in the 2020 democratic primary, but voted for biden because polling showed him performing much better head-to-head against trump. it’s technically called the compromise strategy in social choice theory. the opposite, where trump supporters vote for warren to try to pit trump against the weaker candidate, is called pushover strategy.

a more stark example

if somehow you’ve made it this far and still maintain the notion that ranked voting methods like instant runoff voting uphold “majority rule”, then this last section is for you.

here is a 4-candidate instant runoff voting election (candidates named a,b,c,d):

# voters | their preferences
35% A > C > D > B
17% B > C > D > A
32% C > D > B > A
16% D > B > C > A

instant runoff voting selects candidate b as the winner, beating a in the final round, 65% to 35%.

but wait!

a huge 67% majority of voters would rather have candidate c than candidate b. and candidate c received nearly twice as many first-place votes as candidate b, 32% to 17%. and an even larger 83% super-majority of voters would rather have candidate d than b (and d got just a little fewer first-place votes than b). so the claim that irv “elects majority winners” is seriously misleading. also…

  • a is a spoiler (if he would drop out of the race, c would win instead of b).
  • the first row of voters have an incentive to betray candidate a by pretending candidate c is their actual favorite — then they get their second choice instead of their last.
  • the third row of voters have an incentive to betray candidate c by pretending candidate d is their favorite — then they get their second choice instead of their third.
  • the first row of voters made a big mistake by voting honestly. suppose of the 35 first-row voters, 20 had simply refused to vote. that move (not voting) would actually have been better for them than voting honestly because it would have caused c to win (whom they prefer over b). their honest “b is worst” votes actually caused b to win! also, c is the condorcet “beats-all” winner, but doesn’t make it to the final round: 65% majority says c>a; 67% majority says c>b; 84% majority says c>d. and a is the condorcet “lose-to-all” loser, but makes it to the final round (65% majorities say others>a).


the only logically consistent view of voting methods is that the goal is to maximize expected voter satisfaction. the insistence on majoritarianism may feel deeply intuitive and connected to instinctive notions of fairness, but we can see upon closer examination that there’s a much more nuanced reality. accepting that reality is to our own benefit as voters.



clay shentrup

advocate of score voting and approval voting. software engineer.