So much for the argument that first-past-the-post voting delivers stable majorities. The results are all in (except for Thirsk and Malton, where voting was delayed due to the death of a candidate), but Britain is none the wiser as to what its next government will look like.

The sort of horse trading that is now going on is familiar to countries with more democratic electoral systems, but is still unusual in the UK (and Australia), so it’s worth going through the process systematically. There are three key decision points, which should yield one of four possible outcomes. Let’s take them in turn.

A. Conservative-Liberal Democrat coalition

Only one possible combination (other than a Tory-Labour hook-up, which no one is considering) yields a stable majority: an agreement between the Conservatives and Liberal Democrats, who have 363 seats between them. It makes sense that negotiations have begun there. Facing turbulent times, Britain needs stable and economically responsible government, so other things being equal this would probably be the best outcome.

Left to themselves, David Cameron and Nick Clegg could probably work out a deal; both temperamentally and philosophically they seem not far apart. But their followers will be much harder to reconcile, and in particular the Tories are unlikely to concede much ground on electoral reform. Having just won 9% of the seats with 23% of the vote, Clegg has to treat reform as a top priority; he may never get a better chance.

If the Tory-Lib Dem talks fail to reach agreement, we come to the second decision point: can the Tories’ opponents put together a majority?

B. Grand centre-left coalition

Labour and the Lib Dems between them won 52% of the vote, but only 315 seats: not a majority. Adding in their Northern Irish allies (the SDLP and Alliance) brings them to 319. The nominal figure for a majority is 326, but because the five Sinn Fein MPs don’t take their seats and the Speaker only votes to break a tie, a government could survive with 322.

So a grand centre-left coalition would need to attract support or abstentions from some combination of the Greens (one seat), Plaid Cymru (3), the Scottish Nationalists (6) and the Unionist independent. Since none of those are likely to deal with the Conservatives, the arithmetic is tempting. From the Lib Dems’ point of view, it would only need to hold together for long enough to pass electoral reform.

In a country such as Australia, with tight party discipline, it wouldn’t be too hard to make this sort of project work. But in Britain, where MPs routinely vote against the party line, its prospects look pretty dim. It would also start with huge political disadvantages: the pro-Tory media would paint it as a “coalition of the losers”, and any subsequent election would almost certainly produce big gains to the Conservatives.

If Gordon Brown fails to put together a majority (or near-majority) coalition, he will resign and advise the Queen to send for Cameron. That brings up the third decision point: can Cameron survive his first test in the House of Commons?

C. Conservative minority government

Minority government is the Tories’ preferred option. It worked for Harold Wilson in 1974: get yourself on the government benches, make a few announcements, rebuild your finances a bit, then hold a fresh election at the earliest opportunity. With no change to the electoral system, they can reasonably expect that to yield a majority.

The stumbling block is the need to survive an initial vote in the Commons — presumably on the Queen’s speech outlining the government’s program, which is scheduled for May 25. If the centre-left parties can’t agree on a coalition themselves, they may feel obliged to give Cameron a pass. But they may not, and reaching an agreement to oppose someone is usually easier than resolving on a common program.

D. Chaos

If a Tory government falls on its first test, the country enters uncharted territory. Cameron would probably recommend a fresh election, but the Queen would be under no obligation to agree and would probably give Brown — or a new Labour leader — another chance to form a minority government.

A few divisions in the Commons might give the parties a better idea of where they stand, and enable some combination that had previously been rejected to be given another try. But the numbers are so close that, unless Conservatives and Liberal Democrats can work together, Britain is likely to be heading back to the polls sooner rather than later.

Peter Fray

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