The 2015 election is certainly an interesting one for this consideration because the different between the proportion of the votes each party won compared to the proportion of the seats they won, has probably reached its most extreme.
There are a couple of well-known factors in UK elections which were apparent in 2015. First is that the 'conversion rate', i.e. how many seats a party gets compared to how many votes they received, has always been highest for the Conservative Party. The Labour Party and to an even greater extent, the Liberal Democrats, often have tens of thousands of 'wasted' votes that win them no additional seats. In this election, such vagaries can be seen at the extreme. The SNP (Scottish National Party) gained 1.45 million votes which won them a total of 56 seats. In contrast, UKIP (UK Independence Party) won 3.861 million, more than double what the SNP got, but only won 1 seat. This explains UKIP's sudden support for the introduction of some kind proportional representation. This is despite the defeat of the proposal of adopting the Alternative Vote (AV) method in the referendum of 2011.
The other known factor is the 'shy Conservative'. The factor that, in opinion polls, people who are liable to vote Conservative, are much less likely to say they will than supporters of other parties. This explains the fact that the Conservatives received 2% more of the vote than polls had predicted.
Before I launch into the analysis, there are some caveats. These calculations are made on the basis of direct proportion, so assuming constituencies with equal populations. The most simple approach is used rather than ones such as AV which alter the proportionality to some extent.
In addition, I assume that, unlike, for example, in Germany, there is no bar on how few votes a party can get in order to win a seat. In (West) Germany only parties that achieve above 5% of the vote have been allowed to sit in the Bundestag. This has meant the absence of numerous small parties at national level and effectively a two-party system since 1949.
The first figure is the number of seats the parties would have gained in a simple proportional representation system. The figures in brackets are the actual number of seats that the party gained is shown in square brackets.
Note, one seat - that held by the Speaker, was not really contested. The current Speaker is a Conservative but he does not vote with his party.
2015: 650 seats [Conservative Government]
- Conservatives: (36.9%); 240 seats 
- Labour (30.4%); 198 seats 
- UKIP (12.6%); 82 seats 
- Liberal Democrats (7.8%); 51 seats 
- SNP (4.7%); 31 seats 
- Green (3.8%); 25 seats 
- Plaid Cymru (0.6%); 4 seats 
- DUP (0.6%); 6 seats 
- Sinn Fein (0.6%); 5 seats 
- UUP (0.4%); 4 seats 
- SDLP (0.3%); 2 seats 
- Alliance (0.2%); 1 seat 
As always, I caution that, of course, with a different system in place, other parties may have come forward. As a result there might have been a different range of parties on offer that there were in our world. If we look back to the 2010 election, if then there had been proportional representation then, UKIP would have had 20 seats in Parliament. Whereas with our system UKIP only gained 2 seats and then only in 2014. Similarly, back in 2010, with proportional representation, the Greens would have got 7 seats and now would have 25. Thus, representation in the House of Commons would reflect the trends that have been so commented on in recent weeks, but, because of the system in use, have not come to fruition.
Under proportional representation, Labour would have won 198 seats in 2015, compared to 189 in 2010. This would still reflect Labour's difficulty in increasing their support, notably in England. In 2015, voting for Labour did rise a little over that of 2015 but this was heavily counter-balanced by Labour's losses in Scotland. In our 2015 election they declined in number of seats, a net loss of 26 seats. The 48 seats they lost compared to their representation in 2010, were not only located in Scotland but in England as well. With proportional representation system, despite a modest rise, the failure to break through the 200-seat mark might have been enough to see off Ed Miliband even more than the poor performance did in our world.
Proportional representation would not have altered the severe decline for the Liberal Democrats; they would now have a third of the seats they had back in 2010. However, they would have retained 51 seats rather than just 8. In this alternative, it would have still be likely that a Conservative-Liberal Democrat coalition would have been formed back in 2010, though then with the Liberal Democrats three times stronger than in our world. Persisting with this scenario into 2015, they may have gone back into coalition, but, unlike in 2010, alone, they could not have secured either the Conservatives or Labour a majority.
Even with proportional representation, the often-envisaged Labour-SNP-Green coalition would be no more feasible in forming government in 2015 than in our world. Indeed, the SNP would only have 31 seats rather than 56, so would be weaker by far than in our current parliament. Thus, with proportional representation in place what seems would have been the most likely government would have been a Conservative-UKIP-DUP coalition to give David Cameron a small majority. Given that they had lost two-thirds of their seats as a result of the election (though a better result than losing over four-fifths in our world), the Liberal Democrats, are likely to have been reluctant to have participated in this more right-wing coalition being formed in 2015.
The first-past-the-post system has given the SNP 25 out of 56 of the seats they won. It has denied UKIP 82 seats and the Greens 24 seats. As Douglas Carswell, the lone UKIP MP now, noted, this has meant, combined, 5 million voters for these parties have been denied something even approaching the level of representation they might expect. In large part those seats have gone to the Conservatives and SNP.
The analysis of how different proportional representation would have impacted has shown that the scenarios discussed in recent weeks were not massively misplaced. This is because, as I have seen frequently mentioned, polls measure the anticipated proportion of the vote. As yet, the only good way of predicting the share of seats is the exit poll which, this time was only out by 15 seats, and that neglected the persistent 'shy Conservative' difficulty. Given that one of those Conservative seats was won by a majority of just 27 votes, it was pretty decent estimation, though, of course, insufficiently close to silence those who at every election howl for the banning of 'inaccurate' polls. In this case, they worked very much in the Conservatives' favour by making their warning about the advance of the SNP sound convincing and getting out their core voters especially in the very tight marginals that Labour had to take in order to approach winning.
This time round, the BBC has done similar analysis: http://www.bbc.co.uk/news/election-2015-32601281
drawing on work done by the Electoral Reform Society using the D'Hondt Method of proportional representation approach. Their figures are not radically different to mine Conservatives: 256; UKIP: 83; SNP: 31. However, their work corrects my views based on my previous analysis regarding 'conversion' rates. This shows that only this year, 2015, have the Conservatives needed fewer votes than Labour to win a seat. Contrary to my sense of what was happening in 1997, they needed about 30,000 more votes to get a seat compared to Labour. Now Labour needs about 6,000 votes per seat more than the Conservatives. This does suggest that the party on track to win gains the better conversion rate. I wonder what the impact of boundary changes will make on this conversion rate. For UKIP the rate is running at over 3.5 million votes per seat, 100 times greater than for the Conservatives.