Why do three buses always come along at once? This game explains

Typical. Bloody typical. Two London buses serving the same route, bunched up in the Clapton area. Image: Felix O, via Flickr.

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.

 
 
 
 

How the pandemic is magnifying structural problems in America's housing market

Justin Sullivan/Getty Images

Long before Covid-19, the United States suffered from a housing crisis. Across the country, working class and low-income Americans struggled to pay rent, while the possibility of home ownership receded into fantasy. In hot markets, affordability became a struggle for even the middle class: In California, 41 percent of the population spends over a third of their income on housing costs. 

The coronavirus pandemic will only make these trends worse as millions are unable to work and the economy dives into a recession. Building could slow down in the medium term, as construction loans (risky bets in the best of times) become harder to come by. Unsubsidised affordable housing is often owned by small landlords, who are more likely to struggle during recessions, prompting flips to home ownership or sales to rental empires. 

New York Times reporter Conor Dougherty documented America’s longstanding housing crisis – and California’s efforts to battle it – in his book Golden Gates, which debuted just before the pandemic hit. “My sense is that right now coronavirus is magnifying a lot of things that were already happening,” Dougherty says.  


While Covid-19 adds new pressures, he says that many of the same issues we were facing still loom over the issue, from developers crowding the higher end of the market, to escalating construction costs, to stagnating wages and vulnerable service-sector jobs that leave ordinary Americans struggling to keep a roof over their heads. “That’s my larger message,” Dougherty says. “I think the structural problems continue to be a much bigger deal than the cyclical problem in housing.”

CityMetric spoke with Dougherty about how his thinking has changed since Covid-19, Donald Trump’s pro-suburban rhetoric, and the apparent exodus from San Francisco. 

I’ve really been struck by how strong the housing market seems to be despite the epic economic crisis we are facing. Costs seem to be higher everywhere. I've heard realtors talk about bidding wars like they haven't seen before in Philly, where I live. But perhaps that's just pent up demand from the big shutdowns?

What you have is an economy that has bifurcated. You have fewer middle-income jobs, more lower-income service jobs, and more higher-end jobs in software and finance. That's how our economy looks and that's a problem that is going to take the rest of our lives to solve. In the meantime, we have this housing market where one group of people have so much more money to spend than this other group. Cities reflect that. 

What's important about this bifurcation isn't just that you have gross inequality, but that these people have to live next to each other. You cannot be someone's Uber driver and telecommute. You cannot clean someone's house remotely. These lower-end service workers have to occupy the same general housing market as the super-high-end workers. 

All the pandemic has done is thrown that even more out of whack by creating a situation where one group of people is buying and expanding homes or lowering their home cost by refinancing, while another group are at income zero while trying to live in the same housing market with no demand for their services. When you see home prices booming and an eviction tsunami coming in the same newspaper, that tells you the same thing the book was trying to show you.

Does America writ large have the same housing shortage crisis as California and the Bay Area more specifically? There are other super hot markets, like New York City, Boston, or Seattle. But in Philly, or in Kansas City, is there really a lack of supply? 

There are three kinds of cities in America. There are the really out of control, fast-growing, rich cities: the Bay Area, Seattle, New York. There are declining Detroits and Clevelands, usually manufacturing-centric cities. Then there are sprawling Sun Belt cities. This book is by and large concerned with the prosperous cities. It could be Minneapolis, it could be Nashville. But the housing crisis in places like Cleveland is much more tied to poverty, as you pointed out. 

Those kinds of cities do have a different dynamic, although they still do have the same access to opportunity issues. For instance, there are parts of Detroit that are quite expensive, but they're quite expensive because that's where a lot of the investment has gone. That's where anybody with a lot of money wants to live. Then you have Sun Belt cities like Dallas and Houston, which are starting to become a lot more expensive as well. Nothing like the Bay Area, but the same forces are starting to take root there. 

I think that the Bay Area is important because throughout history, when some giant American industry has popped up, people have gone to Detroit or Houston. Now tech, for better or for worse, has become the industrial powerhouse of our time. But unlike Detroit in its time, it's very hard for people to get close to and enjoy that prosperity. There's a certain kind of city that is the future of America, it has a more intellectual economy, it's where new productive industries are growing. I think it's an outrage that all of them have these housing crises and it's considered some insane luxury to live there. 

A recent Zillow study seemed to show there hasn't been a flood of home sales in the pandemic that would signify a big urban exodus from most cities, with the glaring exception of San Francisco. Do you think that could substantially alleviate some of the cost pressure in the city proper?

On the one hand, I think this is about the general economy. If unemployment remains over 12% in San Francisco, yes, rent is going to be a lot cheaper. But is that really the reality we're all looking for? If restaurants and bars that were key to the city's cultural life remain shut, but rent is cheaper, is that what everyone wants? I bet you when this is all over, we're going to find out the tech people left at a much lower rate than others. Yes, they can all work from home, but what do you think has a bigger impact on a city: a couple of companies telling people they can work from home or the total immolation of entire industries basically overnight?

I don't want to make predictions right now, because we're in the middle of this pandemic. But if the city of San Francisco sees rents go down, well, the rent was already the most expensive in the nation. It falls 15%, 20%? How much better has that really gotten? Also, those people are going to go somewhere and unless they all move quite far away, you're still seeing these other markets picking up a lot of that slack. And those places are already overburdened. Oakland's homeless problem is considerably worse than San Francisco's. If you drive through Oakland, you will see things you did not think possible in the United States of America. 

Speaking of markets beyond San Francisco, you have a chapter about how difficult it is to build housing in the municipalities around big cities – many of which were just founded to hive off their tax revenues from low-income people.

That’s why you see Oregon, California, or the Democratic presidential candidates talking about shaking this up and devising ways to kick [zoning] up to a higher level of government. We've always done this whenever we've had a problem that seems beyond local governance. Like voting rights: you kick it to a higher body when the local body can't or won't solve it. 

But for better or for worse, this suburban thing is part of us now. We cannot just undo that. This notion of federalism and local control, those are important American concepts that can be fiddled with at the edges, but they cannot be wholesale changed. 

The first time I ever met Sonja Trauss [a leader of the Bay Area YIMBY group], she told me she wasn't super concerned about passing new laws but that the larger issue was to change the cultural perception of NIMBYism. We were living in a world where if you went to a city council meeting and complained about a multifamily development near your single-family house, you were not accosted for trying to pump up your property values or hoard land in a prosperous city. You were seen as a defender of the neighbourhood, a civically-minded person.

What is significant about YIMBYism is that the cultural tide is changing. There is this whole group of younger people who have absorbed a new cultural value, which is that more dense housing, more different kinds of people, more affordable housing, more housing options, is good. It feels like the tide is turning culturally and the movement is emblematic of that. I think that value shift will turn out to have been much more lasting than anything Scott Wiener ever does. Because the truth is, there are still going to be a bunch of local battles. Who shows up and how those places change from within probably will turn out to be more important. 

As you said, we've been seeing a lot of Democratic candidates with proposals around reforming zoning. How does Joe Biden's plan compare to the scope of the ambition in the field? 

There are two big ideas that you could pull from all the plans. First, some kind of renter's tax credit. It is obscene that we live in a country where homeowners are allowed to deduct their mortgage interest, but renters aren't. It is obscene that we live in a world where homeowners get 30-year fixed mortgages that guarantee their house payment pretty much for life and renters don't. If we think that it's a good idea to protect people from sudden shocks in their housing costs, that is as good of an idea for renters as it is for homeowners. 

I tell people that in this country, homeowners are living in the socialist hellscape of government intervention and price controls. Renters are living in the capitalist dream of variable pricing and market forces. Homeowners think they're living in this free market, but actually they're in the most regulated market – there are literally price controls propping up their market mortgages. 

Then there is Section 8 housing. Right now homeowners get access to the mortgage interest deduction. That programme is available to as many people as can use it, yet only about a quarter of the people eligible for Section 8 can get it. I think rectifying that is hugely important and a lot of the plans talked about that. 

The second big idea is using the power of the purse to incentivise people to more robustly develop their regions. You should have higher density housing in fancy school districts, near job centres, near transit. We're going to use the power of the purse to incentivise you, within the bounds of your own local rules, to do this right. Of course, that’s what Donald Trump is running against when he talks about Affirmatively Furthering Fair Housing (AFFH). 

When I was a local reporter in Philly, the city went through with that AFFH regulation despite Trump and HUD Secretary Ben Carson not being interested in enforcing it anymore. The city produced a fat report that maybe a few people read, but I don't think it changed policy. It's this phantom that Trump is running against, an ideal version of the policy that did not exist. It's also a phantom no one's heard of until Trump started tweeting about it. 

It’s been bizarre to watch. But Trump does seem to recognise that suburban politics don’t neatly fit into a red or blue construct. People who live in Texas and claim to want a free market system will turn around and erect local regulation to make sure nobody can build apartments near them. People in the Bay Area who claim to be looking for a more diverse place will use different logic, anti-developer logic, to keep apartments being built near them. 

People like that regardless of how they feel about things nationally. The bluntness with which Trump is doing it is discordant with the electorate and quixotic because people don't know what he's talking about. But the basic things he recognises – can I make voters feel like their neighbourhoods are threatened – he's onto something there. As with many things Trump, his tactics are so off-putting that people may ultimately reject them even if under the surface they agree.

You hear people on the left say the scary thing about Trump is that one day a good demagogue could come along. They're going to actually tax private equity people and they're actually going to build infrastructure. They're going to actually do a lot of popular stuff, but under a racist, nationalist banner. I think the suburban thing is a perfect example of that. There's a lot of voters even in the Bay Area who [would support that policy] in different clothing.

The world has changed completely since Golden Gates debuted just a few months ago. Has your thinking about housing issues changed as a result of the seismic disruptions we are living through?

The virus has done little more than lay itself on top of all of the problems I outline in the book. Whether we have an eviction tsunami or not, a quarter of renters were already spending more than half their income on rent. There's a chapter about overcrowded housing and how lower-income tenants are competing with each other by doubling, tripling, and quadrupling up for the scant number of affordable apartments. We now know that overcrowded housing is significantly more of a risk [for Covid-19] than, say, dense housing. If you live in a single-family home with 15 people in it, that's a lot more dangerous than 40 apartments in a four-story building.

Housing is just a proxy for inequality, it's a way of us building assets for one group at the exclusion of another. It is an expression of the general fraying of American society. I don't feel like that larger message has been affected at all, it's only been enhanced by the pandemic. With the caveat that this can all change, it just doesn't seem to me like there's some uber housing lesson we can learn from this – other than having a bunch of people crowded together is a really bad idea. 

Jake Blumgart is a staff writer at CityMetric.