AK84 actually addressed that question way back in post 6. 
The percentage of popular vote is irrelevant. It’s the number of seats in the House that determines who forms the government.
If a party wins a majority of the seats they form the government, even if they don’t have a majority of the popular vote. With first-past-the-post, that’s actually fairly common: the party that wins a strong plurality in the popular vote will normally have a majority of seats.
With respect to your question about what happens when there’s a minority, I think the negotiation process is already addressed in posts 6, 9, 12, 130, and 147.
Is there something that isn’t clear there?
Nitpick: it’s “Chief Justice of the United States.”
Check out C.J. Sansom’s Dominion for a very interesting, and chilling, what-if take on that meeting and its aftermath.
A phrase which would make any American lawyer’s head spin.
Hmm. I’ve read a lot about Churchill over the years, and never came across that story. Cite, please?
I haven’t got the exact page, but I read it in William Manchester’s The Last Lion.
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Several factors apply:
Who was in government before? Did they gain or lose more or fewer seats than expected in the election? This can inform whether a party has been repudiated or rewarded by the electorate.
What policy overlaps are there between the parties? Parties that have more in common than not are more likely to be natural bedfellows. Parties that share policy areas with multiple larger parties can act as kingmakers and decided which policy areas they want to prioritise.
What are the main issues facing the country? What messages chimed most with the public? This is linked to the two former points, but parties themselves learn about the electorate from the election.
What are the leading personalities like? If they’re personally amicable, that can aid the likelihood of a deal. Otherwise, it can hamper even if the policy areas overlap. Unless there’s an agreement for a coalition on the condition that a leader quits.
Then there’s the mundane mathematics of parliamentary seat share. Sometimes, all the parties that tick all or most of the boxes above only just make a majority, or may fall below it. Sometimes, a minority government, with the others offering confidence and supply, might be more practical.
So that’s a brief outline of some of the things that may factor in to government formation. The Head of State will normally keep out of the picture - it’s a job for politicians to hammer out their own problems. But some countries have time limits on how long negotiations can go on for before another election must happen. Others permit the outgoing administration to hobble along but denied of its mandate to make policy innovations until a full government is agreed.
You might get a scenario where no majority is possible and no government combination can survive parliamentary censure. Then it’s back to the electorate to resolve it. The electorate may give a clear answer - giving more votes this time round to a group of parties likely to coalesce - or may not give a clear answer and politicians will have to deal with it. The electorate may well punish a party that is being deliberately obtuse in negotiations or one that sold its soul too easily for power.
In other countries where coalitions are routine many parties have natural bedfellows. I believe the FDP in Germany would nearly always coalesce with the CDU when it was able to form a government. I think that’s changing, though.
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That was the case up until the 1960s, when the FDP veered more towards social rather than economic liberalism (in the European, rather than American sense), and allied with the SDP under Brandt and Schmidt. Then they veered back again, though their support levels have dipped on occasion below the point where they’d be much use to any coalition. After the last, very inconclusive, general election, there was much talk of a so-called “Jamaica” coalition (i.e., black for the CDU + gold for the FDP + green for the Greens) before eventually the CDU/SDP grand coalition was renewed, somewhat uneasily on both sides.
Thanks. I don’t remember that story.
Oh, it’s in the diaries of Maisky, the Soviet Ambassador, too.
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Yeah, I like doing that. 
If there isn’t a clear majority to one party, or to an existing coalition, the HOS would be expected to give the parties a couple of days to work out a clear majority.
Between the call of the election and the formation of the new government, the machinery of government would be in care-taker mode.
In Aus (also UK and Canada?) , if there still isn’t a clear majority, the previous government would be invited to form a new government (continue government). They would either fail or succeed in the House.
As I mentioned above, the new government /could/ start making radical changes without waiting for parliament: this might be very contentious in a situation where they were forming a minority government without a clear mandate, but aside from being politically and constitutionally contentious, they’d be fully occupied with negotiating support from the minor parties.
Compared to say Belgium or Holland the situation in the UK is super easy to understand.
(Except for the whole Brexit thing)
In 2010 Belgium needed 541 days to form a coalition.
The current government in Holland took 225 days of negotiations.
Everything still goes on, government workers get paid, etc. Just no new mayor legislation is introduced. A cabinet waiting for a new government is called “demissionair”. The representatives in parliament are sworn in right after the election results are in. (Prime) ministers stay on until a new government is formed.
In Holland the King had (before 2012, in Belgium this is still current) a minor role in this process as he had to appoint an “informateur”; usually a former politician representing the largest party. He/She tries to find common ground with other parties to make a “regeerakkoord”: a kind of mission statement for a new government where most goals are formulated and mayor new policies are formulated (this is the part that takes so long) Representatives of parties in a coalition have no legal obligation to honour this accord, but it is expected. When the “informateur” finds enough common ground a “formateur” (usually the PM to be) is appointed, the formateur presents their cabinet to the King.
When it is not feasible to create a majority government, it is also possible to have a minority government. This is not very practical as they have to find a mayority for every decision in parliament and opposition parties can send ministers home with a vote of no confidence without even one vote from the “ruling” parties. The third option is a “zakenkabinet” : a non-polical government. This option is always talked about when formation takes longer than 10 day (i.e. always). The last time that really happened was in 1864.
The point is that the HoS will ask whoever got the most seats to form the government, typically. He may hold off if there is a group that claims that together, they will promise to vote together and support someone different. If labour has 45 seats, Conservatives 40, and Chess 15 (seats out of the total 100, let’s say) but Chess announces (and presumably tells HoS) “we will support the conservatives” then common sense suggests the Labour leader step aside and say “I cannot guarantee I will have the confidence of the house” and suggest the HoS ask the Conservatives. In more complicated parliaments (like Israel) the back and forth negotiating can last weeks or months, and can shift at any time.
I suppose Labour could say “screw it, make me PM and I’ll give it a try” and see if the Chess guys would rather vote with them than bring down the government. That could just delay the inevitable, though and make the Labour leader look even worse for the next election. However, convention (at least in Canada) seems to be if the government falls too soon after the last election (so, somewhere before between 6 months to a year, depending on the politics going on) the HoS will let the next party or coalition give it a go instead of calling an election. And it could be the next person called on would be a coalition or cooperative group of 3 smaller parties (although it would be a weird parliament that had a government without the two largest parties participating).
So answer to OP - THERE ARE NOT FIXED RULES. Just tradition.
Another point - sometimes the HoS will appoint a caretaker that is apolitical, a respected unaffiliated elder statesman or bureaucrat or whomever to run the government until the next election can be called. IIRC Italy did this many years ago, and the guy ran the government so well for quite a few months that the joke was “why do we need politicians?”
Informateur! That’s the word I meant. Not rapporteur. Good grief. Getting my European politics mixed up with my monarchical politics!
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At Westminster, it would depend on whether Labour was the outgoing government or opposition. The outgoing government stays in place until the PM in place formally hands in a resignation at the Palace. The opposition isn’t in a position to force the issue, and if they tried, there’d be frantic backstairs phone calls to say “Keep the Palace out of this until you can demonstate by one means or another that you actually have a guarantee of the confidence of the third party”.
The outgoing government might try to hang on and test their position in Parliament without having already come to some reasonably secure agreement with a third party (or more), but it is, as you say, a high-risk strategy.
There’s no route by which an opposition can force itself in without such a clear agreement, or waiting for a confidence vote in Parliament to bring down the outgoing government,.at which point the negotiating merry-go-round sets off again.
IIRC, this is the exact situation in British Columbia. In the last election, in 2017:
However, NDP and Green struck a deal. The Liberals got as far as opening the legislature, lost on a confidence vote, and the NDP became the government (was asked by the Lieutenant Governor to form a government).
Green, Chess, what’s the difference?
And you do it so well!
Yeesh. Good thing those countries don’t have nuclear weapons. In US military contingency planning there’s great emphasis placed on always knowing who, and where, the President is. Could be an interesting novel in which, say, a caretaker British PM is called upon to make major military decisions, or even respond to a nuclear attack, while the party leaders are wrangling over who will succeed him or her.
In that circumstance the PM is still the PM even when his government is on shaky territory in parliament and the military will obey the commands of the government as if they were from the Queen herself.
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I’ve never heard of an instance of a caretaker PM operating in a fit of pique. I guess they are out there.
Most are poster examples of co-operative government as significant decisions are made in consultation with prospective incoming government.
Since the introduction of MMP in NZ elections, the top MPs in each party have been on the party lists in positions such that if they lose their electorate seat, they can return as a list MP. If they retain their seat, then that place from the list can be filled by someone who didn’t get an electorate.
It hasn’t happened yet, and may never do as the party leader tends to stand in a safe seat.
If none of the above makes sense, then please check the Wikipedia article for NZ’s electoral system.
NZ, 2017 September 23rd. The results of the general election had National with 56 seats, Labour with 46 and New Zealand First with 9, Greens 8 and ACT 1
As there were 120 seats in Parliament, National did not have enough to form a majority, and in fact had been in a coalition before the election with 60 seats in a 121 seat Parliament
Both National and Labour negotiated with NZ First to get their support. The leader of NZ First has long had an antipathy to National even though their main policies are quite similar, so it was almost a certainty he would not enter coalition with National and in the end he didn’t. It took nearly 3 weeks for this result though. He’s well known for playing his cards close and not answering reporters’ questions with anything resembling a straight answer, so kept the suspense going as long as he thought would benefit him.
So now Labour/NZ First have 55 seats in a minority coalition government with the Greens supporting them in confidence and supply.
The leader of Labour is the PM, and the leader of NZ First is deputy PM. The Greens got a couple of ministers out of the agreement too.
National supporters still moan that as National has the most seats, they should be ruling, denying the numbers on the ground. If their party can’t get the confidence of Parliament, then they can’t govern.
As a footnote regarding the number of seats in Parliament. The MMP system can produce what is called overhang, when a political party wins more electorate seats than their list vote percentage would give them. There are a fixed number of electorates, currently 60. So they get say 10 electorates, but their list support will only get them 9 list seats. As the seats are proportioned according to the list vote, that electorate seat is more than they would get from the normal allocation of seats. That extra seat becomes an overhang, and Parliament now has 121 seats, 60 electorate seats and 61 list seats, so the other parties receive the appropriate proportion according to the list vote.