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.


To fix the housing crisis, we need to decide what success would look like

Building houses in Ilford, 1947. Image: Getty.

Recent years have seen growing public and political recognition that there is a crisis in housing. This has led to a widening debate on the causes and potential solutions.

However, within this debate there has been little in the way of a consensus view of what constitutes the current housing crisis – or what a “crisis-free” housing system might look like. There seems little clear idea of any measurable goal. The nearest we have as a target to aim at has been a series of aspirational numbers for new-build homes, with limited clarity on what to expect if we were to hit those numbers.

Clarity about what success would look like is essential. Without a framework for what we need and want from housing, our ability to understand and fix it appropriately will be compromised. A lack of clarity also increases the risk of unintended consequences from misguided policy interventions.

The current housing debate is, to quote UCL’s Michael Edwards, “bedevilled by rival simplifications”. There are several, quite often competing explanations for why we have a housing crisis. For many it is our failure to build homes at the same rate as projected household formation. This failure might be assigned to the planning system, the greenbelt, housebuilder business models, the land market, or NIMBYs.

For others, the crisis is a result of falling interest rates, rising credit supply, low income growth, wealth and income inequality, tax incentives, or simply our fixation on house price growth. For some, there is no shortage of homes, rather a poor distribution. And for others there isn’t really a housing crisis.

Despite the apparent contradictions in this mix of positions, each of the arguments that support these various views may hold significant elements of truth. Housing is a complex and interconnected system within the economy and society. There is no simple single housing market: there are multiple markets defined by location, property type, tenure, and price. Therefore, there is no simple single housing crisis. Instead we have multiple overlapping issues affecting different parts of the country in different ways and to varying degrees.

There may be factors that influence all housing markets across the UK, indeed across much of the globe. There will be others that impact more locally and within specific housing sectors.

So, for instance, there is growing acceptance by many experts that the cost and availability of credit has been one of the biggest, if not the biggest, drivers of increases in national house prices over the last twenty years.

But it is not the only factor. The growth in buy-to-let has contributed to the financialisation of housing, with homes increasingly thought of as an investment rather than a place for people to live. A lack of supply is predominantly an issue for London and its surrounds, but there are localised shortages elsewhere, particularly of specific types or tenure of homes.

Planning (including a lack of) and the land market limit the responsiveness of supply to rising demand. Housing is unevenly distributed, mostly across generations but also spatially and within generations. Some areas don’t need a net increase in housing but desperately need existing poor-quality homes improved or replaced. In many areas the biggest issue is low (or negative) income growth and employment insecurity.

All these issues and others play a part in defining “the housing crisis”. Having a framework for what we need and want from housing, combined with an understanding of the complexities and interactions that run through the housing market, is essential to resolving the problems they create.

The problem with ‘households’

A misunderstanding of the complexities of housing can be found in one of the most frequently stated explanations for the crisis: a lack of new supply compared with household projections.

Unfortunately, this argument is flawed. Household projections are not a measure of housing demand. The effective demand for new housing is determined by the number of people or companies willing and financially able to buy property. Meanwhile new supply only accounts for around 12 per cent of total transactions and probably less of available homes for sale.

Importantly, even if some analysis may suggest there is no shortage of supply, that does not mean there is no need for new supply. Household projections are a statistical construct based on the past, not a direct measure of future housing demand. But they are still important if used appropriately within a framework for what we need and want from housing.

If we are more explicit about the role of household projections in measuring housing need and the assumptions they contain, then the ‘supply versus household projections’ argument might be recast as a debate on changing household sizes and the consumption of housing (both in terms of space and multiple properties).

This then implies that we should be clearer about the minimum acceptable amount of housing people need, while also accounting for what they want. Should younger people still expect to form households at the same rate and size as their parents? The assumptions and projections around future household sizes should be moved from the background, where they are typically only discussed by planners and researchers, to the centre of the debate.

They should be just one part of a framework for success that explicitly states what we need and want from housing – not just in terms of size but also cost, tenure, quality, security, and location – and better defines the minimum we are prepared to accept. That will provide a clearer understanding of where housing is failing to meet those requirements and help set objectives for how to fix it. These could then be applied appropriately across different markets.

“Rather than trying to return to the relatively short-lived 20th century ideal of mass home-ownership, perhaps we should be focussing our efforts on making renting cheaper”

If measurement against the framework shows that households are not able to form at an appropriate rate and size relative to what they need, then we probably need to increase supply while possibly encouraging older households to move out of larger homes. If rents are too expensive then we may need to reform the rental sectors and increase supply. If housing quality is poor, then we need to work harder at improving and replacing existing stock. If many areas are struggling due to low (or negative) income growth and employment insecurity, then we probably need to look beyond just housing. It might even question whether we need to rebalance the economy and infrastructure investment away from London and its commuter zone.

Having a framework for success may even highlight which issues we can fix and which we can’t. For example, it looks likely that we are stuck with a low interest rate and hence high house price to income market. Under those conditions, prospective first-time buyers will continue to struggle to raise a deposit and access home-ownership irrespective of how much new supply can be realistically delivered.

Rather than trying to return to the relatively short-lived 20th century ideal of mass home-ownership, perhaps we should be focussing our efforts on making renting cheaper, higher quality, and more secure as a long-term home. Increasing new supply would be an important tool in achieving that outcome.

When we have a framework for what success could look like, our ability to understand and fix housing appropriately will be dramatically improved. It would be an important step towards making housing available, affordable, and appropriate for everyone that needs it. It would also be more useful than simply setting a nice round number national target for new homes.

Neal Hudson is an independent housing analyst, who tweets as @resi_analyst. This article originally appeared on his blog.