Boris Johnson’s appointment of a cabinet dominated by hardline Brexiters suggests he will adopt an uncompromising approach to Brexit, one that will be hard to deliver given the current composition of the House of Commons. But what if the goal is to change the Commons arithmetic via an early election? Could a “delivering Brexit” campaign win him a parliamentary majority and, if so, where would he need to win and what would the risks be?

The balance of power in the polls is far more fragmented now than when Theresa May called an early election in June 2017. Both the Conservatives and Labour have fallen sharply and now find themselves in a four-way competition with the Brexit party, which rose to 20% support within months of its foundation, and the Liberal Democrats, whose anti-Brexit stance has become more popular as opinion has polarised. The Greens, another strongly anti-Brexit party, have also gained substantial ground.

The adoption of an uncompromising approach to Brexit suggests Johnson’s strategy is to consolidate support from Leave voters by squeezing the newcomer Brexit party as much as possible, while hoping that the Remain vote remains divided.

The first problem is that an uncompromising hard Brexit strategy puts quite a lot of Conservative seats at risk

The British electoral system rewards only the party that comes first, so Johnson can gain a majority even on a much lower overall share of the vote than May, provided his opponents fall back more and geography breaks his way. It is easy to see how a Conservative consolidation of the Leave vote could help deliver strongly Leave-backing Labour marginals, particularly if competition from the Brexit party eats into the local Labour vote. But a divided Remain vote could also offer openings in strongly Remain areas. If disaffected Labour Remainers defect to the Liberal Democrats in large numbers, the Conservatives could gain these seats even while going backwards substantially themselves.

The logic of such a strategy is fairly clear – under first past the post, a consolidated Leave vote will overcome a fragmented Remain vote often enough to deliver the Conservatives victory in many seats and thus a majority overall. A similar dynamic – consolidating the independence vote while the unionist vote split in multiple directions – enabled the SNP to turn 50% of the vote into 95% of the seats in Scotland in 2015. Yet such a strategy also comes with very substantial risks.

The first problem for Johnson is that an uncompromising hard Brexit strategy puts quite a lot of Conservative seats at risk. “No deal”, and Johnson himself, is unpopular in Scotland, so many of the Scottish Conservative MPs elected in 2017 would be at risk and the Liberal Democrats could pick up a number of Remain-leaning Conservative seats in southern England. Every seat lost on these strongly Remain flanks would increase the number of gains the strong Leave strategy will need to deliver. This in turn is a problem, because there are far fewer traditional marginal seats than there used to be, so Johnson might need victories in seats where the Conservatives start well behind.

We also do not know how the public would react to a “unite the Brexiters” strategy. Johnson and his advisers would hope that the Brexit party vote will melt away when people are offered a Conservative campaign that fits their preferences. This isn’t certain – the Brexit party, like Ukip before it, draws support from disaffected and low-trust voters. Many might not trust Johnson and might prefer to stick to Nigel Farage instead. Geography could exacerbate this problem; if, for example, Brexit party voters in safe Tory seats find Johnson more attractive than Brexit party voters in Labour-held marginals, he might end up with extra votes that don’t translate into extra seats.

The prospect of a no-deal Brexit might also transform the electoral dynamics on the Remain side, leading voters to back whichever party looks best placed locally to thwart an outcome many of them see as disastrous. This would end the current fragmentation on the Remain side and could open the door both to a Labour recovery in key Labour v Conservative marginals, and to a renewed Liberal Democrat challenge in Conservative v Liberal Democrat marginals.

If both of these dynamics kick in, Johnson’s gamble could backfire dramatically, consolidating Remain support while Leave support remains fragmented, and removing the Conservatives from office altogether. With polling more fragmented, and voters more volatile, than ever before, it is impossible to know how an election campaign would play out. A dramatic gamble may be needed to break the Brexit deadlock, but such gambles can easily backfire, as we saw in 2017. An autumn 2019 election would be an even bigger leap of faith.

• Robert Ford is professor of political science at the University of Manchester