The USA shifts to adopt a form of proportional representation in the period 1890-1950, and it becomes the national norm in all elections, leading perhaps to a quasi-proportional Presidential election system as well.
A specific opportunity exists, I suspect. From late in the 19th century to well into the 20th, into the 1940s in fact, momentum existed in the form of cities adopting a PR system for city councils and the like. The analysis of the rise, and more to the point, near total fall, of this movement tends by consensus to assume it was inherently a sideshow, but I think it could have gone otherwise quite plausibly. I suspect that part of the problem that caused the movement to stagnate and start falling, really put paid to by the Cold War sentiment and very disingenuous campaigns in favor of returning to first past the post as the patriotic option, related to the form of PR adopted. The Single Transferable Vote system was the OTL model.
I suggest a different model might have sustained the momentum, jumped to electing state legislatures, and thus put pressure on changing how the House and ultimately even the US Senate was elected. If it could go that far then I think the process would eventually at some opportunity go to completion with an Amendment or two mandating the alternate system for the House and even an adapted quasi-proportional system preserving the equality of all states in the Senate as well, and thus perhaps lay the groundwork for elimination of the current state EV system for electing the President in favor of a more directly democratic and national system.
Specifically, I for one find STV to be a bit strange and confusing in the manner it might achieve a truly proportional representative body. Perhaps I could get to like it, but analyses suggest that PR did not achieve the great increases in voter turnout and general engagement PR advocates predicted by its nature, showing statistically that turnout was largely unaffected. (What did happen under PR was a great improvement of demographic representation, with African Americans and other minorities appearing on governing bodies they had hitherto been excluded from). I suggest though that maybe voters were indeed put off by STV's complexities and obscure workings, but that there was a countervailing tendency for them to feel more engaged indeed, and the two factors cancelled in general outcome.
What then if there were an alternative approach to PR that seemed simpler and more transparent in its workings? Might we not then see more momentum and enthusiasm for switching over and taking it farther? Furthermore, I gather that if STV is going to enable a proportional outcome, it is by means of creating large districts which elect many officers, so a city council of say 50 would be elected in 5 districts each electing ten councilors instead of 50 districts each electing one. This may or may not work well for a city (I dislike any system calling itself proportional that subdivides the electorate like that, a truly proportional system should include the whole electorate in arriving at proportionality) but it is a bit awkward to adapt to say a state legislature, and breaks down completely trying to apply it to Congress.
In my proposed alternate system, the philosophy is that one creates just half as many districts as there are offices. (I actually then say and add one more office, to make the total odd, but that is not strictly necessary). Then people vote as in familiar FPTP for a single candidate in their district, but in addition to selecting a single FPTP plurality winner in each district, their votes are then tallied by party as well. Each candidate they could vote for is also a candidate for a party (and I would make due and flexible provision for independent candidates to form meaningful coalitions with others so as to capture votes across the city or state or nation as well) and out of the larger total number of seats to win, the parties are assigned these proportionally. I strongly urge using Hamilton's system as the most inclusive. Then, to make up the roster of all seats a party has won, the strongest vote winners in the districts where their party did not win the plurality race are taken in order of total number of votes each got, so that the parties are each represented in the body by the candidates who polled the strongest for that party. Thus they will tend to come from locations that are particularly strong in supporting that party.
I happen to have done a case study based on Michigan in 2012 where I can discuss how the outcome would look down to the names and parties of the legislators from each district. If this AH idea sparks any interest, or someone wants to discuss it by PM or in Chat, I would be quite pleased to. It would help illustrate reasons why I think the outcome would be pleasing to large minorities and yet "feel" more like normal American FPTP in process and some aspects of outcome, such as an attachment of specific office holders to specific districts. But I forego it for now to stick to the ATL proposal.
Being district related, transitioning say a state legislature is a matter of either doubling the size of the body (thus doubling the number of representatives per citizen) or consolidating two former districts into one, and then proceeding much as before.
Being proportional, the evils done by misapportionment of districts (a flagrant problem in the late 19th and first half of the 20th century before SCOTUS finally cracked down on it) and by gerrymandering are largely wiped out; one way or another groups of voters who are numerous enough to merit a few representatives will be able to be heard. Therefore the process of attempting to draw right sized and fair districts is much relaxed; arbitrary methods that are simple can be used without doing harm.
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For purposes of ATL development then I propose that in the late 19th century, major reform societies and leagues that campaigned for proportional city government adopted a system like this instead of STV, and cities that OTL went over to STV used this system instead. And that instead of a push in terms of turnout and engagement, the cities that adopt it experience a significant rise in voter participation. Furthermore, overall the form of government of these cities is remarkably improved, and reformers in other cities who either never went PR at all OTL or did so later do it sooner and more, so it becomes increasingly normal across many regions for cities and towns to be run in this way.
Gradually the pressure is on, especially in the early 1910s, for some states to adapt their legislatures to use the same form, some doubling their legislature size, others consolidating their districts, and after transition the majority of these states have good experiences with the outcomes; citizens feel they have better options and are better connected to state government.
Thus they start to look beyond, in large states with high numbers of Congress members apportioned to them anyway, to consider electing their Congress delegations in the same way, so that half are elected in districts twice the size of before and the other half plus one when the number is odd make the partisan proportion of the total delegation match as much as the size of the delegation allows.
By the Great Depression a majority of states have gone proportional, a majority of the House is elected proportionally state by state (this is already being recognized as imperfect yet a big improvement on before) and there is a movement to mandate a form of the system for all states, integrating the whole national popular vote, which requires an Amendment of course.
FDR or some other butterflied champion of the majority who enjoys massive support either wants this reform themselves or is forced by the expediencies of building a *New Deal alliance to champion it, and shepherds it through the adoption process.
By this time a reform of the Senate, perhaps demonstrated on a local scale in some states, is also in the works and this too passes.
WWII elections are carried out under the new national system, and postwar there are a number of new third parties that win small but consistent shares, the postwar political order is thus transformed. The Cold War involves an organized movement to try to push back and restore FPTP, and has some success in some places, but cannot overturn the amended form, the courts come to recognize it as more properly democratic and republican than going back to FPTP and frown on that more and more, and by the 1960s the transition is irreversible. Conservatives often lament it but talking about trying to reverse it is like talking about abolishing Social Security or eliminating the income tax or minimum wage, extreme radical talk; the vast majority are adapted to it and take it as normal and inevitable.