Being curious about how PR might work, I've taken a lot of real world data and applied PR to it in many instances.
As I have said, "real world data" means people are voting in different circumstances and we don't really get a picture of what they would do if they had a real PR option instead. But it indicates a few things anyway.
A major lesson is that FPTP strongly tends to exaggerate the support the leading parties get. Even granting that people only vote for them (not at all true generally, but very true in the USA) the leading party tends to get extra seats that turn small majorities into large ones and manufacture fake majorities out of whole cloth.
US elections, if people did not change their voting patterns as they surely would, make for subtle but still significant differences.
Take the most recent, 2016 US Congress. Currently, the outcome of the 2016 election is that the Republicans hold 241 seats, 23 more than the 218 needed for a majority out of 435, and the Democrats hold every single other remaining seat with 194. The Republicans thus hold 55.54 percent of the House. But in fact, of all votes cast they won only 49.1 percent! The Republicans hold almost 25 percent more seats than the Democrats do, but in the popular vote, their margin of victory over the Democrats' 48 percent is just over 2 percent--thus their advantage is multiplied by over a factor of ten!
Among PR methods, I feel the Hamilton method of greatest remainders is the most appropriate for a legislature or any deliberative body of any scale or scope. It is most inclusive, resulting in the most voters getting the result of having at least one representative in the body, leaving the least with no direct representation. Other methods tend to transfer seats from some rag tag small parties to the leading one or two. I think the importance of including the outliers ought to outweigh a very small increment in direct power of the leading parties, and I think the often expressed fear this will lead to chaos is quite mythical.
Using Hamilton's method, if the nation voted exactly as it did in 2016 we would have a Congress that looks like this:
Republican 214
Democratic 209
Libertarian 6
Green 2
Constitution party, Legal Marijuana Now Party, Reform and Conservative parties, 1 each
Four independents:
David Walker, Oregon district 3
Alan LaPolice, Kansas district 1
Frederick O. Maycock, Massachusetts district 1
Preston Picus, California district 12
In terms of power, the Republicans fall 4 seats short of a majority, and would need to gain some allies. Fortunately for them, it was a right wing vote all over, and allying with the Libertarians alone could do the trick. Alternatively, if the Libertarians demand too high a price, I would think between the likelihood at least half those Independents are pretty conservative and the certainty that 3 of the 4 one-seat parties are, they would have many options to close the gap to their liking. Democrats would need to pick up 9 allies, and the numbers just aren't there without massive compromise.
If one looks at the 2016 vote more closely, it often happened--not a high percentage, but in a dozen cases or so--that a candidate would run unopposed in their district. It makes sense in the context of FPTP but this is unlikely to happen in PR, even a PR system (such as one I have thought up and recommend) that uses districts as part of its basic machinery.
Britain is perhaps a more interesting case study. In the USA, third party or independent victories in Congress races are historically very rare, at least when a given party system has evolved and stabilized. When they win at all, they tend to be very few in number, literal handfulls. Not so in Britain; there too each seat is won FPTP, but nevertheless significant numbers of candidates contest and a fair number of third party seats are routinely won. But their numbers tend to be way out of whack, with party delegations often having only a passing semblance to the support they actually got.
Out of 650 seats in the Westminister Parliament, one is exceptional in that the office of Speaker of the House has a special procedural role, and is customarily exempted from partisan challenges in their district--turnout for their elections is thus remarkably low! Setting that seat aside there are 649 in play, and a party needs 325 seats to rule as a majority in it its own right.
The election of May 7, 2015 returned the following results in terms of seats in Parliament:
Conservative 330
Labour 232
UK Independence (behind Brexit) 1
Liberal Democrat 8
Scottish Nationalist 56
Green 1
DUP (N. Ireland) 8
Plaid Cymru 3
Sinn Fein 4
UUP (N.I) 2
SDLP 3
One independent
Thus, eleven parties and an independent emerge. If you wonder why the parties are listed in this strange order and not in terms of seats, the table I took them from
here is in order of number of total popular votes cast, not seat outcomes. That is, the Liberal Democrats got more votes, by a lot, than the Scottish Nationalists did, and the Greens with one seat got more than the conservative Democratic Union Party of Ulster which got 8 seats.
A proportional outcome using the Hamilton method would look like this:
Conservative 239
Labour 197
UK Independence (behind Brexit) 82
Liberal Democrat 51
Scottish Nationalist 30
Green 24
DUP (N. Ireland) 3
Plaid Cymru 3
Sinn Fein 2
UUP (N.I) 2
SDLP 2
Alliance party 1
In truth, the Conservatives only won less than 37 percent of the popular vote in Britain in 2015, yet somehow wound up dominating Parliament single handed with a margin of 5 members!
In truth, based on the proportional outcomes, the UKIP movement would have become a kingmaker, being absolutely necessary for forming a majority with the Tories, and even so they would be scrambling to scrape up another crucial vote even if the conservative DUP had joined them as well. But Labour would have an even harder struggle to rule.
But this was not good enough for them, and they called a snap election for June 8th of last year, hoping to pick up more seats. Ironically, they lost seats and wound up needing to scrounge for some third party to ally with them to form a Government. They lost 13 seats, Labour gained 30.
Conservative 317
Labour 232
Liberal Democrat 12
Scottish Nationalist 35
UKIP 0
Green 1
DUP (N. Ireland) 10
Sinn Fein 7
Plaid Cymru 4
Independent 1
SDLP 0
UUP (N.I) 0
But the proportional results would look like this:
Conservative 275
Labour 260
Liberal Democrat 48
Scottish Nationalist 20
UKIP 12
Green 11
DUP (N. Ireland) 6
Sinn Fein 5
Plaid Cymru 4
Independent 1
SDLP 2
UUP (N.I) 2
Alliance party 1
Irony on irony--the Tories did in fact gain some voting support--but lost seats by FPTP anyway. Analyzing this, we see that UKIP is just a shadow of its former self, and now it is the moderate Liberal Democrats who are the kingmakers. Whichever of the top two parties that party aligns with can rule in coalition, picking up a few more votes from fourth parties. Labour is in a better position to do this actually.
These examples then give some sense of the different outcomes, and the drastic divergence between actual popular support and the illusion of it refracted through even highly competitive FPTP voting.
And should dispel the illusion that coalition government is a curse of PR FPTP is exempt from. OTL the Tories had to form a coalition despite the FPTP system that had previously handed them government quite out of line with their actual popular support--though counting UKIP's strength before the Brexit vote, clearly Britain was in a conservative mood overall, just as the USA was in 2016.