What happens in a hung Parliament
We are 9 days out from the election, people look at the opinion polls wanting to know who is going to win, who is going to form the next government. The simple answer is that at present they can't tell us - we look set for a hung Parliament and who will form the government will depend to a certain extent upon negotiations between the parties, rather than the levels of support the parties receive.
Ironically the present electoral maths look set to give is an excellent illustration of the arguments used by both supporters of PR and its opponents. For PR's supporters we look likely to get a hugely unproportional result - Labour could possibly end up with the most seats with the fewest votes, the Lib Dems in second place, but with under 100. For PR's opponents, who argue that PR leads to governments being decided in secret discussions behind closed doors, we are heading into an election where measuring public opinion cannot tell us who will triumph - for that will depend upon the negotiations after the election.
If I can't give you any polling evidence on what the result of a hung Parliament will be, I can at least offer guidance on what will happen! The way a hung Parliament plays out is guided by some key constitutional principles:
1) The prime minister remains the Prime Minister until he resigns. Even if he has lost his majority or is no longer the largest party, the PM remains PM until he resigns. It is his right, if he wishes, to wait until Parliament reassembles and to try and get approval for a Queen's speech, even if he does not lead the largest party.
2) The Queen's government must continue. When the Prime Minister resigns the Queen immediately invites someone else to replace him, in the knowledge that they will accept. The Palace will not allow there to be a period without government.
3) The Queen will not involve itself in anything that could be construed as being partisan, and does not personally involve herself in negotiations - though the Palace will closely follow the progress of negotiations.
4) Should the Prime Minister resign, the Queen will invite the person most capable of commanding a majority in the Commons (or at least, getting a Queen's Speech and budget past the House). That will normally be the leader of the largest party, but it doesn't have to be.
5) Should a Prime Minister loose a vote of confidence, or something regarded as a vote of confidence like the vote on the Queens Speech, they must resign or request a dissolution. A dissolution remains the personal power of the monarch, and she may refuse if the Parliament has only just been elected and there is a chance of an alternative government.
Putting all that into practice, this means that in a hung Parliament Gordon Brown will remain Prime Minister during negotiations. What that does not mean is Brown automatically getting first dibs at negotiations or arranging a coalition. Negotiations between the parties do not have a formal structure and are up to the parties themselves, if Nick Clegg wishes to play Labour and the Conservatives off against each other at the same time, or refuse to negotiate with Brown, or go straight to dealing with Cameron - he can.
If a coalition or pact commanding a majority in the House emerges, then one way or the other it will become the government, regardless of Brown being the sitting PM. If it is not a Labour led coalition then in theory it could come down to them waiting for Parliament to reassemble and forcing Brown out in a confidence vote, but in practice Brown would accept the inevitable and resign with dignity once it became clear that his position was not tenable.
The instance where Gordon Brown's position as incumbent does make a difference is if there is no agreement to a coalition or a pact. As the sitting Prime Minister, Gordon Brown would then be the leader to go before Parliament and essentially dare them to vote down the Queen's speech, leaving the other parties to consider whether it is in their strategic interests to vote the government out or bide their time and suffer it to continue for the time being.
If a party does end up without a majority, daring the Commons to vote them out, the threat they hold over the other parties is the that of a dissolution and a second election. The Queen does have the right to refuse such a dissolution under certain circumstances (basically if Parliament is still young and there is an alternative government that may be able to command a majority). Essentially, if Brown went before the Commons, lost a vote of confidence, and asked for a dissolution it would be refused, and David Cameron offered the chance to try and form a government instead. It's less clear whether Cameron would be granted a dissolution if he in turn was defeated - in 1974 the opinion of the Palace was that they would have been very hard pressed to refuse Wilson had he requested one. I expect in practice Cameron would be granted one unless an alternate government with an agreed majority was obvious.
The final thing to consider are the rules of the political parties themselves, or two specific rules in particular. Firstly the Labour party - Nick Clegg has implied that one requirement for him to agree a deal with Labour would be a change of leader. In the Labour party's rules, if they are in government and the leader becomes permanently unavailable, then the cabinet and NEC can pick one of the cabinet as leader until a full leadership contest can be arranged - in other words, if Brown resigned as Labour leader during coalition negotiations he could in theory be swiftly and easily replaced within the party rules.
The second issue is the Liberal Democrat party's rules. Formally Cameron and Brown have a free hand in negotiations, Clegg does not. The Southport Resolution in the Lib Dem rules requires him to get the support of 75% of the Parliamentary Liberal Democrat party, and 75% of the party's Federal executive (and failing that the support of two-thirds of the wider party) in order to enter into any agreement that "could affect the party’s independence of political action" - taken as meaning a coalition agreement. While all the leaders would in practice need to take their parties with them, only Clegg would have such a formal process to deal with somehow.
That's the background - beyond that, all is speculation.
UPDATE: Thanks to Mark Pack for correcting me on the mysteries of the Lib Dem rule book. If Clegg did not get the 75% support from his Parliamentary party and executive, he'd then need two-thirds support of a special conference, and then failing that, of the wider party. On the other point that has been raised, outgoing Prime Ministers have in the past offered the monarch advice on who they should invite to succeed them, however, this is informal advice ("advice with a small a" in the terms the Palace would use) that the Queen may ignore, not the formal Advice from a minister to the monarch that the Queen is compelled to follow.