Nobody won. That is the basic definition of a hung parliament. The newly elected members might not see it that way. The leaders of the three main parties might couch the results of last week's election in historical and statistical terms that make them feel better. But the fact remains: nobody won.

The Conservatives have the most plausible claim to some kind of victory. They took the highest national share of the vote and gained 97 seats. But Mr Cameron was battling to restore majority Conservative rule. He campaigned vigorously against a hung parliament, all but demanding unchecked power. He was rebuffed: 10.7 million people voted for Tory government; more than 15 million people did not.

But the non-Tory vote was divided, largely between Labour and Liberal Democrats. Despite many local skirmishes, there is a strain of cousinly feeling in both parties that sees the Tories as a common enemy. From that impulse now springs the idea that Labour and the Lib Dems could join forces to prevent Mr Cameron from taking power.

The parliamentary arithmetic just about makes it possible. But the ambition relies on the belief that voters who endorsed the Liberal Democrats would prefer Labour to continue in government if the alternative is a switch to the Tories. That might be true for some, the majority even. But there is no way of discerning what psychological motive impels people to vote. Meanwhile, millions of people deserted Labour. The party lost 91 MPs. Despite a disappointing haul of seats, Nick Clegg won more votes than any leader in his party's recent history. This ballot was not an endorsement of the status quo.

Who, then, should govern? The circumstances are unique, not in terms of the electoral outcome – Parliament has been hung before – but in terms of the cultural background. Britain has just held the most presidential campaign in its history and emerged with the result that most demands recognition of Parliament's primacy in our constitution. The televised leaders' debates put almost exclusive emphasis on the question of who should be prime minister. If the election is construed as a referendum on that point, David Cameron won.

The Tories also have the strongest parliamentary mandate of any single party. For that reason, Nick Clegg did the honourable thing by quickly restating his belief that Mr Cameron should have the first opportunity to try to form a government.

But the fact remains that victory, under the electoral system we have, means securing a Commons majority. Constitutionally, no other metric matters. If the Conservatives believe that share of vote and lead over the nearest rival should have some moral weight in deciding a winner, they have already conceded a vital point about the need for electoral reform: the proportion of overall support in the country as a whole matters.

At the start of the campaign, this newspaper argued it should be the last fought under the first past the post system. The result clearly vindicates that view.

Labour won 8.6m votes and 258 seats; the Lib Dems got 6.8m votes and 57 seats. It takes nearly four times as many people to deliver a seat for the Lib Dems as it does for Labour or the Tories. A profound injustice is committed when the system that is supposed to amplify the voice of the electorate distorts it instead.

Mr Clegg is right to insist on a substantial commitment to reform as a condition of participation in government with either of the two bigger parties. Mr Brown has shrewdly acknowledged that imperative and offered a referendum on proportional representation – a system that would fairly allocate seats in Parliament according to votes cast.

There has long been support for PR in the Labour ranks, but it would be naive to assume that Mr Brown's readiness to embrace the idea is driven by anything other than a cynical attempt to woo Mr Clegg away from Mr Cameron and thereby stay on as prime minister.

None the less, the commitment cannot be unmade. This is a significant milestone on the road to fairer politics.

The Tories by contrast are confused about electoral reform. It cannot have escaped their notice that they have suffered as a result of the system they are determined to keep. It is Labour whose results are most inflated by systemic bias. The Tories insist that first past the post delivers clear results, when it has just failed to do exactly that. Conservatives have always grumbled that coalition politics means shadowy deals between parties cobbled together in dingy corridors. The opposite is now proven. Mr Clegg has made explicit his priorities and Mr Cameron has voiced with decent candour the areas where he imagines collaboration is possible.

The process is already more transparent than the one that usually turns back-of-envelope doodles into policy.

The Tories might generally disapprove of coalitions, but they have also said they will seek to govern in the national interest and the nation has instructed that power be shared.

What Mr Cameron means when he talks about the national interest is that Britain's economic situation is too critical to allow for any delay in forming a government. In particular, the Conservatives are fixated on deficit reduction as a moral imperative to avert catastrophe. The spectacle of Greece, now erupting in civil unrest as its parliament inflicts brutal austerity measures, is cited in evidence.

That is a perverse reading of the situation. The Greek riots were triggered by over-aggressive budget cuts inflicted by a government with insufficient moral authority. If anything, the lesson is that painful measures can only be implemented by a government that is trusted by the communities that will be affected.

That is a problem for the Tories. They have only one seat in Scotland. They failed to break through in the north of England and made fewer inroads in Wales than they hoped. London was also sceptical of Mr Cameron.

Even if they could survive in the Commons as a minority administration, the Tories have not advanced enough from their bases in the southern shires to claim a national mandate for their divisive economic agenda. They need a coalition for moral authority, not just parliamentary votes.

The power to confer that authority rests with Nick Clegg. He should withhold it. The Tories have no greater claim to economic competence than any other party, nor any greater credibility on the deficit. They have simply shouted louder about it. Their manifesto was full of unfunded pledges; their priorities, as indicated by capricious tax breaks for the rich, suggest a flimsy grasp of what counts as fairness in austerity.

Meanwhile, there are vast gaps between Lib Dems and Tories on nearly every other aspect of policy. The Lib Dems seek sensible co-operation with the rest of Europe, for example, while the Tories prefer obstruction. Mr Cameron's party is allied in the European parliament with fringe nationalists, described by Mr Clegg crudely, but not inaccurately, as "nutters". The Observer today publishes an extraordinarily obdurate memo prepared for William Hague's first encounter with his European counterparts that indicates how their policies would guarantee confrontation. That is not in the national interest.

But surely the greatest obstacle is electoral reform? Many Tories feel existential hostility to the kind of radically realigned politics that might emerge in a new, more representative democracy. Mr Cameron's entire political career and his bearing during the campaign express an underlying confidence that, by virtue of historical entitlement, the Tories get regular turns at the levers of power. Electoral reform might end that guarantee. By contrast, electoral reform is Labour policy. So is a more judicious and fair deficit reduction strategy much more in tune with Lib Dem ideas.

There is much ideological overlap between the two parties and no constitutional obstacle to a coalition between them. The Conservatives would howl that they have been deprived their victor's spoils. But they won only the right to try to form a government, which Nick Clegg has given them. If Mr Clegg then finds it easier to deal with Labour he is free to do so. A Lib-Lab coalition could claim to have marshalled an overall majority of votes casts. That, together with their combined parliamentary weight, would grant a genuine mandate to govern.

The problem is Mr Brown. Whatever convention says about his right to continue as prime minister, the campaign has flayed his authority. The Lib Dems could not plausibly enter a coalition with Labour if the administration that emerged were seen as a rickety continuation of the one that has just been punished by voters. Any ensuing referendum on electoral reform might be construed as a plea to rubber-stamp Gordon Brown's prolonged hold on power – and be rejected.

If a Lib-Lab pact is to have any chance of survival, Mr Brown must signal unequivocally that he seeks to continue only for as long as it takes to get a new constitutional order in place. He must give clear advance notice of his resignation, stating that he will continue as prime minister in a caretaker capacity only. He must accept a timetable, no longer than two years, for a referendum on electoral reform and a new general election, in which Labour will be led by someone else.

Combined, the Liberal Democrats and Labour have the affinity on policy, the electoral mandate and the unique historic opportunity to usher in a new era of fairer, better governance for Britain. Mr Brown must offer Mr Clegg partnership in an administration of real national renewal and make the vital concession needed to secure it – a guarantee of his own departure.

Mr Clegg should accept those terms. That is how the national interest is best served after the election that nobody won.

• More election comment from Cif at the polls