Joe Biden has never been coy about his desire to be president, even as he has wavered about whether he will actually make a third run for the Oval Office. But as the former vice-president nears a decision, he is apparently homing in on the worst possible understanding of the 2020 primary race in order to decipher a plausible path for him to become the Democratic nominee.

It is not entirely his fault because the way he is thinking about it is extremely common. It is, however, utterly mistaken.

New reporting from Edward-Isaac Dovere of The Atlantic informs us of Biden’s assessment of the race:

“Biden and his aides think Bernie Sanders, who is a year older, might help neutralise the issue of Biden’s age. Sanders is considered likely to join a progressive field that includes Elizabeth Warren, Kamala Harris, Kirsten Gillibrand, Cory Booker, and possibly Sherrod Brown sometime soon. A Sanders candidacy would likely encourage Biden to run because he doesn’t agree with the Vermont senator’s policies and thinks they’re losing politics.

“In a primary, Biden and his advisers believe, all those left-tilting candidates would divide support and leave him with a sizeable number of more moderate voters.”

A third person who has been in touch with Biden’s top aides said: “We’ve got a little bit of our own lane. We just need to go own it. He’s well aware that this isn’t going to be easy, he’s going to have to fight for it, but I don’t think he’s viewing this thing through the lens of matching up against any one candidate.”

Let’s talk about that idea of “lanes”. The idea that a primary contest is determined by who can win among a series of non-overlapping groups of voters has been around for a long time, but the “lanes” metaphor particularly came into its own early in the 2016 Republican contest. Who would be the candidate of evangelical Christians, of tea partiers, of establishment Republicans? At the time it seemed to make sense to think of them as distinct groups who would be attracted to different candidates for different reasons. If someone could occupy a particular lane, they could hold the voters situated there and then seek to grab another lane as well.

Then Donald Trump came along, and it turned out that there were no “lanes”. There was Trump — and everyone else.

The real problem, though, is the idea Biden and his aides have that if he can hold the moderates, the more liberal candidates will divide the liberals and then he will sneak through. This implies not just that this is a contest determined by ideology but also that you can win a presidential primary with the support a plurality of voters — which isn’t really the case.

OK, it’s mathematically possible that you could. But it will not ever happen because the plurality strategy is built for a single primary such as the ones for every office except president. If you are running for Senate as the only moderate, you could indeed get the nomination if a bunch of liberals split the liberal vote. The problem is that presidential primaries play out over a period of about four months, from early February to early June.

As that process goes along, voters will coalesce around a smaller and smaller number of candidates, meaning many of the ones Biden is counting on to split the vote will drop out or fade into irrelevance. In fact, some of them are likely to drop out before the counting even starts, and others will drop out right after one or two failures. Biden should know that: when he ran for the 1988 Democratic nomination, he withdrew from the race before voting started in the wake of a plagiarism scandal, and when he ran again 20 years later, he came in fifth in the Iowa caucuses with less than 1 per cent of the delegates, then dropped out of the race the next day.

As the race winnows down, voters in states that haven’t voted will continually reassess their choices, and a voter supporting one liberal candidate will likely move to another liberal candidate if her first choice drops out. Even though the race will be long, the winnowing could be quick. We may start with 15 candidates, but don’t be surprised if soon after Iowa, New Hampshire and South Carolina, we’re down to four or five.

Biden’s strategy is also premised on the idea that moderate Democratic voters will make their judgments on ideology and pick him over more liberal candidates on that criterion alone. The biggest problem with that strategy is that, with the sole possible exception of Sanders, the candidates who have staked out more liberal positions are not out to the left of the Democratic electorate. Things such as Medicare-for-all and liberal immigration policies are now accepted by large majorities of Democrats.

Even if it were possible to find and hold a significant number of moderate Democrats, to do so Biden would have to spend his time arguing against liberalism when the entire party has become more liberal. That’s going to be a tough argument to win with.

Let’s be clear about something: Biden was an underappreciated vice-president who deserved more credit than he got for his performance in that position. In a job where it is easy to just fade into the woodwork, Biden was energetic and effective, using his long government experience and personal relationships to help to advocate for and implement Barack Obama’s policies; none more so than the 2009 stimulus, which could have been a disaster but instead was a tremendous success, thanks in no small part to Biden’s efforts.

But every politician has their strengths and weaknesses, and Biden showed with two failed attempts that running for president is not something he is all that good at. Before he has even decided to make a third attempt, he has already making clear that he doesn’t have much of an idea how to go about it.

• Paul Waldman is an opinion writer for the Plum Line blog