I was trying to work out the smallest majority required to pass a Constitutional amendment. I expected the answer to be 2/3rd or 3/4ths, but came up with 11%:

NOTE: in the below, I am referring to state legislators, not US Senators or US Representatives.

  • It requires "the Legislatures of three fourths of the several States" to pass (ratify) an amendment. I realize no amendment has actually been passed this way, but it is theoretically possible.

  • There are 50 states. Three fourths of 50 is 37.5, so we round up to 38 states.

  • The 38 least-populated states make up 42% of the entire US population. Like in the Senate (and unlike in the House), size does not matter. Wyoming counts the same as California.

  • Most (all?) of these 38 states are broken up into legislative districts. You need to win only 51% of these districts to pass the amendment. Legislative approval is by simple majority, and the governor does not participate.

  • In each of these 51% of all districts, you need only 51% of the vote to elect a legislator.

  • The 51% least populated districts in a state make up at most 51% of the population (probably less, since legislative districts have approximately the same number of people, but not exactly).

  • Therefore, you need at most 51% of the people in 51% of the districts for a state to pass an amendment. That's about 26% of the people in a state.

  • Therefore, you need 26% of the people in 38 states (making up 42% of the population) to pass an amendment. That works out to about 11%

What, if anything is wrong with this calculation? Several notes/disclaimers:

  • I'm using "51%" above to mean "more than 50%". In districts with a million people, even one vote over 50% works.

  • I realize this can't be done realistically, so this is primarily a theoretical question.

  • 11% is close to an absolute extreme, but what if 30% of the population supported an amendment, and at least some of those people were willing to move or declare residency in other areas, and the entire effort was coordinated through the Internet. Would this have a real chance of working? I strongly suspect the 70% opposed would amend Article V itself, but it would be an interesting project.

  • Not all states have 100+ member legislative districts (although most do), so "51%" in this case means "more than half".

EDIT: A joke answer to this question: you need roughly 2000 people, provided they are all state legislators or something similar. Obviously, that's not the answer I'm looking for.

EDIT: A "realistic" way this could happen (to answer @NL7 concerns):

  • Suppose 11% of the US electorate is zealously determine to pass an amendment, and are willing to put a reasonable amount of time, effort, and money into doing so.

  • The 11% rearrange themselves (moving and/or declaring residency) to form the majorities I describe above.

  • One of the 11% then runs for various state legislator offices. This is one of the zealots running for office: he doesn't have a party affiliation and his greatest ambition is to pass this amendment, not to achieve higher office.

  • By the argument above, the amendment would pass.

In other words, the zealous 11% chooses the candidate. They don't just hope that the existing candidate follows the majority vote.

Keep in mind Constitutional amendments can significantly change political power: the 11%ers might end up giving themselves a lot more political power than they would have than just running for higher office.

NB: The word "gerrymandering" does not appear in this question.

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    You are correct in saying only about 50% of eligible voters actually vote. I'm assuming that if someone tries this, voter participation might increase dramatically. As you note, residency requirements would be a fairly easy challenge. Article V of the Constitution allows state legislatures to pass Constitutional amendments with no interference from the Senate or House or President. The United States is a republic, not a democracy. – barrycarter Mar 20 '14 at 16:50
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    Yes, you are correct - and that it what the framers intended. They were more interested in widespread geographical support, moreso than actual majoritarianism. May not think it is fair, but it works... – Affable Geek Mar 20 '14 at 17:22
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    The only flaw I see in this argument is a technicality. You need to win only 51% of these districts to pass the amendment. You don't have to win any districts to pass an amendment, unless state law requires them to be put to a popular vote. This should be You need only 51% of these legislators... In addition, some states are bicameral, so you would need the people in the state to elect a majority of state senators in support of it, which may require a different amount of people than for a majority of representatives. – Bobson Mar 20 '14 at 19:53
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    I think what is the smallest percent of the US population that would be able to enact a constitutional amendment then post this as the answer to that... that would be a question better suited to our QA style – SoylentGray Mar 20 '14 at 19:57
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    FWIW: This isn't entirely theoretical. It is quite likely that a majority of Americans were against the 18th Amendment when it passed. – T.E.D. Jan 26 '17 at 17:16
up vote 1 down vote accepted

Your plan revolves around controlling the state legislatures. Article V gives Congress the ability to specify that an amendment must be passed via convention instead of via state legislatures, and if this sort of thing is openly being done, they would probably specify that.

In Vermont, for example, the delegates to such a convention would be elected at-large instead of from districts that could be gamed. The way things stand, the amendment would be defeated. This state law could be changed, but you would need more than the bare legislative majority you propose - you'd need enough votes to override the inevitable veto (the governor isn't on your side, since you don't have half the state's population.) Now you need 2/3 of the legislature instead of a simple majority. So, instead of 26% of the state (51% of 51%), you need 34% (51% of 2/3), and if you need to do this in all states it changes your final 11% number to about 14%.

On the other hand, you probably won't need to do this in all states; for example New Mexico law says that the legislators are the delegates, so you wouldn't need to change that law.

Not all states have 100+ member legislative districts (although most do), so "51%" in this case means "more than half"

Actually, no state has 100+ in its less populous house. Minnesota has the largest state senate at 67 members, and Alaska is smallest at 20. This may push your final number up very slightly, maybe a tenth of a percentage point or so.

The 11% rearrange themselves (moving and/or declaring residency) to form the majorities I describe above.

If you move in significant numbers, however, you run into a problem. Let's say there are 50,000 people in a legislative district, and of those, 20,000 people are in favor already. If you move in 5,001 people to try to get over 50%, that's not enough, because by adding population you've increased the number needed for a majority. You're going to have to move in a minimum of 10,001 people. But now, the district contains 60,001 people instead of 50,000, and still only gets to elect one legislator. You've diluted the power of your voters by about 20%. This could easily add a few percentage points. (And if your plan is to wait for redistricting to even things out... remember that the opposition controls redistricting since you don't have the state yet.)

  • This is the best answer I've seen, thanks! – barrycarter May 5 at 16:39
  • I just read Wisconsin's convention rules. It's crazy. The legislature appoints legislators as delegates (the governor gets to appoint 1 delegate, but it must be a legislator.) And then the legislature appoints a committee, also made up exclusively of legislators, to tell the delegates what to do. If a rule is proposed in the convention (not necessarily the amendment itself, just a convention rule), and the committee tells the delegates to vote against it, the delegates must vote against it. If it passes anyway, the delegates go home. – D M May 5 at 18:29
  • Someone should try this in real life and see how close they get. Remove current President from office may drive enough people to do it. – barrycarter May 6 at 16:29
  • @barrycarter Not only is that impractical for the obvious reason that moving is burdensome and expensive (and the initial distribution seems unfavorable - many liberals live in big cities), it's doubly impractical to use it to remove a President, because many state senate terms are 4 years long. By the time enough people are elected, he'll be facing an election anyway. Plus it's just a bad idea to pass an amendment to get rid of one guy - that's what impeachment is for. The minimum number needed for that might be interesting - but again you run into issues with Senate term length. – D M May 7 at 5:47

There are two primary problems with your calculation, the first of which is overestimating the minimum support required and the second of which is dramatically underestimating the plausible lower limits.

First, you are assuming that state legislators will vote yes if a bare majority of their districts support the amendment. This elides the fact that only a plurality is necessary in all states (which are all FPTP). But it also simply assumes that legislators will vote based on the opinions of their constituents. This is not true, given that huge numbers of eligible voters don't have strong opinions on issues and that even people with reasonably well formed views may not flip their vote (even in competitive primaries) based on those views. There is also no rule that the legislators voting for an amendment must survive reelection for their votes to be valid.

The true minimum minority that can amend the constitution is literally the number of legislators needed - or 2/3 of Congress plus >50% of the legislators in 3/4 of the states (counting only the states with the least numerous legislatures). This number is under 3,500 and possibly under 3,000. Which would imply that a minority of roughly 0.001% is needed to pass an amendment.

Second, as a practical matter constitutional amendments are very controversial and tend to attract lots of opposition. State legislators all want to become representatives, senators, governors and presidents. So they do not just narrowly care about opinion in their districts (let alone a bare majority of local voters), but about opinion in their broader region (especially if they want to run for a House seat that covers that area), and the state (if they want to run for Senate or Governor), opinion in nearby metropolises (where they will need to fundraise) and opinion in big cities like NYC, Chicago, LA, SF (again, where a presidential candidate, Senator, or high-profile Congressman will want to fundraise). Moreover, they need to care about the broader opinion in their political party, because that coalition includes the pundits, commentators, endorsers, and fundraisers who will help catapult some politicians to greater heights.

So although there is little formal need to rely on any sort of support, and only a narrow number of votes are needed to elect the minimum number of people voting for an amendment, as a practical matter there really needs to be a strong groundswell for an amendment. This will mean a supermajority among politically engaged voters and either apathy or support from partially engaged voters. Legislators aren't realistically going to shoot themselves in the foot by foisting an unpopular amendment on people they will soon rely on for money and support and votes.

  • I was asking as a purely theoretical lower bound. I made several other assumptions I didn't place in the question (eg, suppose this issue polarizes the nation to the point where it supersedes any other reason people vote). I still maintain that my calculation, as a theoretical calculation, is correct, and that it shows that a small number of voters (not necessarily 11%) can make a big difference. – barrycarter Mar 30 '14 at 2:00
  • I edited my original question to give a better idea of what I meant: a zealous 11% willing to move and elect their own candidates. – barrycarter Mar 30 '14 at 2:11
  • A much smaller and more committed group, if willing to conceal their full preferences before election and to be voted out afterward, could do it. – NL7 Mar 30 '14 at 3:55
  • True, but I'm saying 11% can do it openly and honestly (even though that flies in the face of political tradition) – barrycarter Mar 30 '14 at 13:08
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    This wrinkle also occurs to me: to the extent that state Senate districts and state House districts (or the upper/lower equivalents everywhere but Nebraska) are less than conveniently overlapping, you might find that the required votes to get 51% in the lower house districts will push the upper house districts beyond 51%. This requires an incredibly detailed look at district boundaries and population patterns. – NL7 Mar 31 '14 at 16:33

In theory, yes.

However, constitutional amendments are not like local city council elections. The citizens in general pay attention to them. They will get lots of publicity, especially if the amendment (which is the final word in law, even the Supreme Court cannot override it) is provocative to some. By their very nature, most constitutional amendments make fundamental changes to the government, not something voters are likely to ignore.

The last constitutional amendment to fail ratification by the states was the DC voting rights amendment. Authored in 1978, in the midst of the Carter administration, it failed to meet the seven year ratification deadline. It would have given DC two senators and a congressman. There were valid reasons to oppose ratification, chiefly giving two very influential senators to a very small segment of the population.

In theory, a constitutional amendment could pass with a small minority, if voter turnout and concern was very low, just as in theory, you could win 500 million dollars in the next lottery. The chances of either of those scenarios happening are just about zero.

I believe you are obsessing over the ideological illusions that states are governed by their citizens, or a majority thereof. And since your calculation illustrates a "numeric" or "technical" example of how this is not the case, you're experiencing ideological consternation.

Well, fear not. For centuries and millenia, states have been ruled by minorities, with the populace having influence - stronger or weaker - over the state. Indeed, like @NL7 suggests, it is technically possible for just 3,500 people to pass an amendment; the question is how much influence is necessary to make those people pass that amendment, or to put it in the terms of your question - how much of the population. But this question too I would reverse on its head, and claim that it's not 50%, or 11%, but the extent of social power necessary to influence enough people to vote - legislators, or the public.

As an example, consider this segment from "Yes Prime Minister", which discusses what could be a referendum, if Britain had referenda. With the right circumstances of social discourse, education, media coverage, and so on - powerful or interested minorities make people believe or support one thing or the other, and they vote accordingly. Now, I don't want to be too cynical - if you wanted people to support a constitutional amendment which relinquishes sovereignty to Canada, you'd need a huge amount of social influence over a prolonged period of time; but some of the existing initiatives for constitutional amendments are challenging mostly because of active minority groups applying pressure in both directions rather than staunch popular opinion.

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