There are, realistically, only two likely outcomes in the forthcoming general election – either a Conservative majority or a hung parliament. If the first, the path is clear: Brexit by the end of January and five more years of Conservative government.

But what if there is a hung parliament? Boris Johnson will have lost his majority. He might well resign immediately, since his chances of securing support from other parties, possibly even the DUP, are small. And even if he were inclined to somersault by offering the Liberal Democrats a second referendum, his MPs would probably veto so blatant a reversal of the policy on which they were elected.

If Johnson resigns, the Queen would normally be expected to call on the leader of the largest opposition party – almost certainly Labour. The only way in which a Jeremy Corbyn government could be avoided in these circumstances would be if a majority of members of parliament were to indicate on paper that they would support an alternative candidate for the premiership. That is unlikely, but not perhaps impossible, in a House of Commons that could include a record number of MPs without allegiance to either Labour or the Conservatives.

Yet Boris Johnson would not be constitutionally required to resign until defeated on a no-confidence vote in the Commons; and until he resigns, there is no vacancy.

Were Johnson to be defeated, MPs would have to discover an alternative PM who can secure the confidence of the Commons

It would then be for the politicians rather than the Queen to sort out the situation, as they did in the previous hung parliaments of 1923-4, 1929, February 1974, 2010 and 2017. The Queen would not play an active part. After the inconclusive election of 2010, the Queen’s non-intervention was symbolised by her remaining at Windsor for the five days of coalition negotiations. That, of course, was before enactment of the Fixed-Term Parliaments Act, which emphasises that the Queen’s role is symbolic. The politicians decide. The Queen endorses.

Were Johnson to be defeated in a no-confidence vote, MPs would have 14 days to discover an alternative prime minister who can secure the confidence of the Commons. That alternative would almost certainly be Corbyn.

Might the Liberal Democrats play a key role in a hung parliament? At first sight, it would seem they would be in a pivotal position – as they were in 2010, when they decided to join a coalition with David Cameron rather than allowing Gordon Brown to continue in office.

But the 2010 precedent is deceptive. More often in their history, the Liberals (predecessors of the Lib Dems), found themselves unable to exert leverage. In 1923-4, 1929 and 1974, they failed to support the continuation of Conservative governments that had lost their majorities. The consequence on each occasion was a minority Labour government in which the Liberal vote was squeezed. In the October 1924 election the party lost 118 of its 158 seats, while in October 1974 it lost nearly a million of the six million votes it had won in the February election earlier that year.

David Steel, the former Lib Dem leader, once said that asking his party to choose between the two major parties would be like asking Cinderella to choose between the two ugly sisters; and in this election campaign Lib Dem leader Jo Swinson has insisted she would not put either Johnson or Corbyn into power. But, in a hung parliament, would the Lib Dems really vote against a Corbyn government? They would be cross-pressured. Supporting a Labour leadership would certainly be unpalatable for them: defectors from Labour such as Luciana Berger and Chuka Umunna left the party because they believed Corbyn was unfit to govern. But if the Lib Dems were to reject Corbyn in a confidence vote, there would be a second election which they would be ill-equipped to fight, since they would be seen as spoilers.

Corbyn, moreover, is offering a second referendum on EU membership. That is an incentive the Lib Dems would find difficult to resist. Therefore they would probably support a Corbyn government on a case-by-case basis until a referendum bill had been enacted. That would take some time. Labour would first renegotiate, then pass a referendum bill. The 2015 bill providing for the 2016 referendum took seven months to pass through parliament. The Electoral Commission then tests the question, after which the campaign would begin. So Corbyn would be safe for at least a year.

As long ago as 1917, a Liberal presciently remarked “There are only two sovereign words in politics – these words are ‘yes’ and ‘no’. Our leaders are apt to avoid saying ‘yes’ and ‘no’, and they are apt to try to reconcile the two.” That remark encapsulates the Lib Dems’ dilemma, which is that of a third party seeking to work within an adversarial political structure, buttressed by the first-past-the-post electoral system.

The party’s dilemma can only be resolved by replacing the adversarial structure with a consensual one sustained by proportional representation. This has long been the Lib Dems’ goal, and there’s a chance things might at last be turning in their favour. As the politics professor Matthew Goodwin has pointed out, Britain’s departure from the EU is fragmenting the British party system and, paradoxically, making it more like those on the continent. A multi-party system fits ill with an electoral system designed for just two parties.

But until the Lib Dems’ dilemma is resolved by constitutional and electoral reform, they will have little bargaining power; and a hung parliament in which they hold the balance of power will almost certainly result in a minority government led by Jeremy Corbyn.

• Vernon Bogdanor is professor of government, King’s College London. His books include, Beyond Brexit: Towards a British Constitution