With fewer potential paths to No 10 than Ed Miliband, the Tory leader needs a lot more to change in the next 20 days than his Labour opponent does

Labour and the Conservatives are still virtually tied in the polls. The Guardian’s latest average of polls has David Cameron’s party on 33.7% and Labour on 33.6%. Once these figures are translated into seats, the Guardian’s projection puts the Tories on 269 seats and Labour on 272.

It is self-evident that the race to become the largest party is extremely close – but when it comes to who will have the numbers needed to form a government, the odds are stacked against Cameron. As things stand, Ed Miliband holds the better cards.

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The number to keep in mind here is 326 – any government needs the backing of a majority of MPs in parliament.

The “anti-Tory” bloc of Labour, the SNP, Plaid Cymru, the Green party and the SDLP) – those parties that would vote a Conservative government down – adds up to 333. Piling up all possible sources of support (Lib Dems, Ukip and the DUP) brings Cameron’s tally to 311.

The difference between the two blocs may seem small, but it could be decisive. However, the factor to consider here is the different paths that the Miliband bloc and the Cameron bloc have to surpass in order to reach the threshold of 326 seats.

For Cameron there aren’t many ways of closing that 20-seat deficit other than by halving the party’s current projected losses to Labour. As the graphic below shows, on current polls the Conservatives are set to lose more than 40 seats to Labour. That needs to drop to no higher than 20-25.

The alternative is for the Conservatives to win more seats from other parties. The problem is that the party the Tories are best placed to gain from otherwise are the Lib Dems – and Nick Clegg’s party is already included in the Cameron bloc above. So, while in 21 of the 29 seats that the Lib Dems are projected to retain the Tories are in second place, it makes no difference. Any Conservative gain in seats such as Cheltenham and North Cornwall would make no net difference to the Cameron bloc’s total count.

The same of course is also true of the Labour-SNP bloc of seats: even if Nicola Sturgeon’s party were to win in 30-40 constituencies, and not in the 50 or more implied in the polls, most of those seats would be staying with Labour. That means that the overall balance between the two blocs is unchanged – ie with Miliband’s bloc ahead. In this case, though, any softness in the SNP vote would also boost Labour’s chances of becoming the largest party.

Cameron is unlikely to be able to count on the Lib Dems substantially eating into the gap between the two blocs either, by winning seats from either Labour or the SNP. If we look at the 28 seats the Lib Dems are projected to lose, eight are to Labour and 10 to the SNP:

To make a difference to the overall scores between the blocs, the Lib Dems would need to claw back most of these. On current polling, this is highly unlikely.

What this all computes to is that Cameron needs a lot to go his way in the next three weeks. First, he needs lift off in the polls and to open up a substantial lead over Labour. Tory strategists were hoping to see their party ahead by this stage of the campaign on the back of a strong economic message and the so called incumbency factor.

But there is no evidence to support this. Labour in 2010 was polling at 29% with ICM in March, at 31% in the first poll in April. At the election, Gordon Brown’s party scored 30%. In effect, no change.

Labour in 2005 polled 37% in the first poll in April. In the ensuing general election, Tony Blair’s party achieved 36%. In 2001, Labour slipped back from a high base – polling in 47% in May. At the time of the election in June, Blair’s party scored 42%.

Does this apply to the Tories too? Go back to 1997 and the party was polling 32% in the last poll of March, and 34% in the first poll of April. Come May and the general election and John Major achieved 31%. You get the point. Based on the limited data available, there is no discernible incumbency effect.

Another Conservative trump card was supposed to be Miliband’s dire personal ratings. Make no mistake: with three weeks to go, Cameron remains more popular (or less unpopular to be precise) than Miliband and is perceived as the more competent between the two leaders.

Crucially, however, the gap between the two has narrowed since the leaders’ debates. The Tories’ belief that the British public’s doubts over Miliband’s suitability for the premiership may still end up being prophetic, yet right now any such concerns aren’t putting a dent in the polling deadlock.

The Conservatives also need the Greens to make a comeback. The Green party’s support comes overwhelmingly from people who have switched from Labour and the Lib Dems. In tight contests where every vote counts, even a relatively benign 3-5% for the Greens could keep a seat in the hands of the Tories.

However, the Green trend is heading in the opposite direction to where Cameron needs it to go. In February Natalie Bennett’s party was polling on 7% nationally. It is now on 5.5%.

The Tories also need Ukip’s support to drop substantially. Farage’s party has fallen two points since early February to 13%. Ukip voters have come from both the Tories and Labour, which means Cameron needs the drop in the party’s support to accelerate, and to fall in the right constituencies.

These are all substantial changes. In truth, significant shifts in public opinion over a short period of time (ie an election campaign) do not normally happen. The exception – that proves the rule – was “Cleggmania” in 2010, although even then the bounce in Lib Dem support following the first leaders’ debate was only partly sustained at the ballot box.

The arithmetic is of course one thing, the politics another. A scenario where the Conservatives are the largest party by 20 or more seats, but Miliband becomes PM, would undoubtedly lead the Tory party (under whichever leader) to drum up questions over the “legitimacy” of a Labour-led government.

But whatever the rhetoric of the future, the simple fact is that Cameron needs a lot more to change over the next 20 days than Miliband does. This matters greatly because you need the numbers in the Commons to win a confidence vote and hence maintain the authority to govern.