Q: Why do buses come in threes? A: Mathematical inevitability

That's the problem with the internet in summer, isn't it? Nothing but repeats. 

Anyway, here's something we originally published in June 2015. But it's dead good, honest.

The old joke that you wait ages for one bus, then three come along at once, is bordering on cliché. But it's also, as it turns out, true – not just because of bad planning, but also because of maths.

The phenomenon is so common, in fact, that it has a choice of names. Bus bunching, clumping, convoying, platooning – all relate to the depressing reality that, over any length of time, buses serving a single route are likely to end up tootling along directly behind each other.

The reasons why this should be can be difficult to get your head around – so Lewis Lehe, a postgrad working on a PhD in transport engineering at the University of California, Berkeley, has built a game of sorts to demonstrate. You can play it here. But if, like us, you are both lazy and impatient, here's how it works, with a few helpful screenshots to illustrate. 

The game features two buses, serving a circular route with four stops. At the start of the game, the two buses are evenly spaced, at opposing ends of the loop. Passenger flows at each stop are identical to those opposite: when one bus has to pause for a set period at one stop, the other is pausing for the same length of time across the map.

The result is that the two buses are always carrying the same number of passengers. Left to their own devices, they will remain equally spaced forever, in perfect equilibrium.

And we know this, because at this point we got distracted, had a conversation on Twitter, went to make a coffee, and then suddenly remembered we were in the middle of doing something about buses, flicked back to the game in a panic, and discovered that it looked like this:

Our game was still ticking along nicely, the two buses now holding seven passengers apiece.

Eternal equilibrium is really boring, though, so let's mess things up a bit. What happens if we delay Bus One, just for a single second?

Err, well, this as it turns out. This is a couple of moments after the unexpected delay.

Two things to notice here. One is that the gap between the two buses has closed, slightly: that momentary delay allowed Bus Two to catch up, slightly. The other thing is that Bus One now has more passengers.

Those two things are actually features of a single phenomenon. In those few moments in which Bus One was delayed, Bus Two started to catch up – not by much, but for enough for it to have an effect. The consequence is that it's now reaching stops which last saw a bus relatively recently – and where there are fewer passengers for it to pick up.

But here's the insidious thing: that process is self-perpetuating.

Think this through for a moment. If Bus Two is serving stops that have relatively fewer passengers waiting, it can get moving again faster. Bus One, by contrast, is serving stops that haven't seen a bus in a while (it was delayed, after all), and so it's picking up more people. That means it needs to stop for longer, both to let its larger contingent of people off, and to collect the next lot.

And so, Bus Two gets closer...

...and then catches up altogether.

 

Look at the distribution of passengers now. Bus One is now arriving at stops with populations bigger than some small Chinese cities. Bus Two, just behind, arrives moment later to find them deserted. Without intervention, this situation will persist, essentially, forever.


In real life, of course, there would be an intervention (Bus Two would almost certainly overtake, as soon as there’s room). And real life is messier than this model in other ways, too. Bus routes are served by more than two buses; passenger flows aren't so evenly spread.

But it’s this inevitable messiness that provides that slight delay that kicks the whole process off. A slight disparity in passenger numbers, a traffic jam, bad luck with the lights; whatever the cause, one bus will be delayed, and those behind it will start to catch up.

Until suddenly you get three buses showing up at once and everyone makes tutting noises.

There are ways of mitigating this feedback loop. One is setting maximum or minimum stopping times in advance, to regulate the service. Another is building in waiting time at one end of a route, so that a bus doesn't immediately turn around and set off again (in effect, doing the second half of a circular route), thus bringing a measure of predictability to the time it takes to do an entire circuit.

These things can help – but the ubiquity of bus bunching suggests that they can’t elimate the problem altogether.

So the next time someone in your hearing asks why it is that buses all come along at once, you can reply, "Mathematical inevitability". Then you can look smug about it, as they, presumably, decide that they’d rather walk.

 
 
 
 

Meet the YIMBY campaigners hoping to ease the housing crisis

Some houses, being built. Image: Getty.

The nimby is a wearily familiar political breed. Though individuals may support new housing and infrastructure projects in theory, they oppose them in practice (“not in my backyard”). For fear of consequences such as a fall in property values, locals reliably revolt against proposed developments – and politicians retreat. The net result is that cities and countries are denied the housing they need. For the past decade, the UK has fallen far short of the 250,000 new homes required annually to meet demand.

But the nimby has now met its dialectical opposite: the yimby. In contrast to their opponents, yimbys not merely tolerate but welcome development (“yes in my backyard”). The earliest known usage of yimby was in a 1988 New York Times article (“Coping in the Age of Nimby”) and the first organisation was founded in 2007 (Yimby Stockholm). Sister groups have since been established in Toronto, San Francisco, Sao Paulo, Sydney, Helsinki and, most recently, London.

John Myers, a 44-year-old former barrister and financial analyst, co-founded London Yimby with four others last year. They were inspired by the capital’s dysfunctional property market (London is the most expensive major global city for buying or renting) and the success of groups elsewhere.

“We saw what was happening in the States,” Myers said when we spoke. “The San Francisco group has just had three new laws passed in California to get more housing built. There are now more than 30 US cities with yimby groups… There really is a feeling in the air that something has to be done.” Myers lives in a small mortgaged house in Camden, north London, but most of the group’s volunteers are private or social housing tenants and range from “the very young to retired grandparents”.

“The big problem with the housing crisis,” Myers told me, “the dirty little secret that politicians don’t like to talk about is that, actually, people quite like house prices to go up.”


In 2013, shortly after launching the Help to Buy scheme, the former chancellor George Osborne told the cabinet: “Hopefully we will get a little housing boom and everyone will be happy as property values go up” (the average London house now costs £484,362). Though the exorbitant price of housing (such that there are now more outright owners than mortgagors) has become an electoral problem for the Tories, homeowners remain an obstacle to development.

In a recent report for the Adam Smith Institute (“Yes In My Back Yard”), Myers made three proposals to win over this bloc: allowing individual streets to grant themselves planning permission to extend or replace buildings; permitting local parishes to develop “ugly or low amenity” sections of the green belt; and devolving planning powers to city-region mayors.

“There are ways to get support from local people for high-quality developments but we have a system right now that doesn’t try and get that support,” Myers said. “It just imposes measures from the top down.”

In some US cities, yimbys have antagonised anti-gentrification campaigners by supporting luxury developments. There is a tension between the aim of greater supply and that of greater affordability. Myers argued that it was crucial to have “clear rules on what percentage [of affordable housing] is required up front, so it gets priced into the land and taken out of the landowner’s pocket”.

The replacement of stamp duty with a land value tax, he added, would leave both “the buyer and the seller better off: the buyer doesn’t have to scrape a deposit together and the seller doesn’t have the price reduced by the amount of stamp duty”.

That some Conservatives are now prepared to consider previously heretical measures such as building on the green belt and borrowing £50bn for housing investment may herald a new era. The yimby bulldozer is beginning to dislodge the nimbys from their privileged perch. 

This article previously appeared in our sister title, the New Statesman.

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