The general election is being fought on multiple fronts. One battleground is seats where Conservatives are vulnerable to swings to the Liberal Democrats. Making this problem more difficult for the Conservatives is the newly announced ‘Remain Alliance’.

There is disagreement over the number of seats that could be taken from the Conservatives by a Liberal Democrat-Green-Plaid Cymru alliance: UK in a Changing Europe senior fellow John Curtice put their potential seat gains at six or so.

The Remain Alliance itself has announced a pact in 60 seats overall.

The Remain Alliance could make the difference in 11 Conservative seats

The first question is where this Remain Alliance could be a critical factor in eating into Boris Johnson’s hopes of a majority.

Best for Britain’s MRP constituency poll—based on a 9-point lead for the Conservatives over Labour, with the Lib Dems on 18% and the Greens on 6%—is a good place to start. There has been some debate about its use, but it is currently the best granular data we have for seat-by-seat analysis in England and Wales.

The poll predicts that, without a Remain Alliance, the Liberal Democrats would end up with 19 seats (including those in bold in the table below). Given the party ended the last parliament with 20 MPs, largely owing to defections, this would hardly represent a surge.

Yet if the entire predicted vote share for the Liberal Democrats, the Green Party and Plaid Cymru moved en masse to the Liberal Democrats in the constituencies below, they would pick up a further 11 seats from the Conservatives (all those not in bold), giving them 30 overall.

Voter behaviour is difficult to predict

The obvious caveat to this, of course, is that Green and Plaid voters won’t all flock to the Liberal Democrats, especially because their parties are unlikely to gain any seats themselves from the alliance but merely help defend the ones they already hold.

Yet this could also work the other way. There is the possibility that the pact could be, to use political science jargon, an ‘additive coalition’—that is, more than the sum of its parts, motivating voters to turn out who would otherwise abstain.

A further number of seats may not have enough Lib Dem, Green and Plaid voters to win but may do so if they could also attract Labour voters. In traditional Conservative seats, the Remain Alliance may be an attractive proposition to Labour voters who want to cast a decisive ballot.

There are a further eight seats where the Lib Dems finished in second in 2017 in which a Remain Alliance could be within touching distance of the Conservatives: the MRP poll suggests (in order of likelihood) Wells, Chelmsford, Eastleigh, Taunton Deane, South East Cambridgeshire, North Cornwall, Wokingham and Berwick-Upon-Tweed are within 5%—a further 2.5% swing—for the Liberal Democrats.

Trends in Scotland and Northern Ireland could support a Remain coalition

Factor in a likely strong performance for the Scottish National Party (SNP) – where, as Professor Nicola McEwen has pointed out, polls indicate the Scottish Conservatives are 8% down on their 2017 vote share.

Add in the reasonable chance of one or two pro-Remain MPs being returned in Northern Ireland – where losses for the DUP to ‘pro Remain’ parties are possible in Belfast South and Belfast East – and a ‘rainbow alliance’ in the House of Commons begins to take shape.

But Labour is key to the Remain Alliance achieving its goal

The key problem for the Remain Alliance is the elephant in the room: the Labour Party. To achieve what it wants, i.e. remain in the EU, these parties would need enough Labour MPs to retain their seats to form a collective majority after the election.

However, in a swathe of seats, the polling suggests the Liberal Democrats—polling double the votes they received in 2017—are eating into Labour’s vote share.

A seat like Ipswich is a case in point: it is a two-horse race which Labour won back in 2017 on a wafer-thin majority. The Lib Dems standing here, despite having little chance of winning, could easily open the door to the Conservatives.

The Lib Dems could also reduce Labour gains by standing in marginals like Hastings and Rye—again a two-horse race between Labour and the Conservatives.

Taking Conservative-Lib Dem seats in isolation, the table above suggests an alliance would reduce the Conservatives to 301 seats and give a theoretical Labour-SNP-Lib Dem-Plaid-Green coalition a collective 331 seats.

Yet the MRP model suggests the Conservatives will get over 350 seats: on these numbers, the Labour Party would lose more than double the number of constituencies that a Remain Alliance could hope to capture.

As a result, the big strategic problem facing the Remain Alliance is whether they can (and want to) encourage their supporters, implicitly or explicitly, to put their cross in the Labour box, where their candidate has a little chance.

This would include seats, such as Burnley and Colchester, that in the past decade have had Liberal Democrat MPs, but where now Labour are in a better position.

Convincing their supporters to vote Labour in particular seats without a reciprocal agreement could be even more difficult than forming the Remain Alliance itself. Yet if these parties really want a referendum on Brexit, it will be Labour that ultimately holds the key to it.

By Matt Bevington and Dr Alan Wager, researchers at the UK in a Changing Europe.