9. January 2017

Utility theory and pedestrian behaviour in London

A not-so-serious attempt at explaining pedestrian behaviour in London using utility theory. Ever wondered why certain pedestrians jump so readily in front of passing lorries? Ever wonder why some are keen to collide with approaching cyclists? Sorry, I don’t have the answer, either. But I am still curious. And in a mood to rant. Just a little bit.

For those who live in London, it is a common sight. Go to a random junction and observe what pedestrians do. There will be parents pushing prams over a busy junction in Nappy Valley a.k.a. Clapham Junction, no matter whether there is traffic or not (the toddlers will find out first, I feel sorry for them). There will be confident suits with stylish earphones at King’s Cross, waiting on the street rather than on the pavement (the crucial step ahead of everyone else). And so on.

For some pedestrians, ignorance has no bounds. By stepping on the road instead of waiting on the pavement, they put themselves into the path of cyclists who are bound to the space between curb and where the cars go. A common sight where Northcote Road meets Battersea Rise: cyclists are approaching the junction at high speed, coming down a slope, and need to negotiate space with cars. And then suddenly, some pedestrians decide to wait more comfortably on the road. Until they find out that there is traffic in form of cyclists. Surprise, surprise.

Then there is the category of self-confident (or over-confident, if you ask me), uber-cool urban cowboys (cowgirls) who display total ignorance of other traffic participants (be it cyclists, cars, buses, or lorries). For them, stepping onto the road while waiting is compulsory, and crossing must happen as early as possible at all costs. After all, in London, running over pedestrians is neither customary nor is it permitted by law. Hence, these cowboys might think — assuming they are aware of their behaviour — that drivers will stop for that reason. Fortunately, for them and for me and other bystanders, the drivers indeed stop or evade such cowboys! For a cowboy, every centimetre headstart counts. And once there is the opportunity to wait on the traffic island in the middle, no matter whether there is any chance to cross the remaining lanes or space remaining on that traffic island, go! Any progress is progress. Waiting is display of weakness.

An example of such a cowboy was the trolley-pulling guy in the smart suit listening to music with a bored I-am-the-King-of-the-Bongo face, who intended to cross Euston Road: first, he steps onto the road (several lorries pass by, the drivers swerving a fraction to the left to avoid running him over), then spots his chance to cross (mistake, the lorry was actually just waiting for the junction to clear), but loses his courage (or comes to his senses, matter of perspective; the lorry driver brakes generously) and returns back to his starting block position (on the road, of course).

This does not make sense to me. So I got a bit curious. What’s the benefit of such behaviour? Will utility theory explain it? We know that there are many cases where utility theory is insufficient (see Kahneman’ and Tversky’s work on prospect theory). But I have already stated, this is not a scientific article. I’m just using utility theory as a model for thinking.

Let’s assume that time is money, and that I as a pedestrian can make £1 more by crossing the street now rather than when there is no traffic or the lights are green because you will save a minute that way. Let’s furthermore assume that I will not sell my life for any sum in the world (very realistic; that’s the same as saying my life is worth an infinite amount of pounds), so that if the lorry driver doesn’t brake for whatever reason, my loss is infinite. So here is my utility table as a pedestrian, and I have the choice to wait or cross.

wait cross
break 0 +1
go 0 -∞

After I have made my decision, the driver of the approaching lorry will either break or go. Let’s say that in only 1 out of 100,000 dangerous situations where I decided to cross, unable to see me, the driver will accidentally not break.

wait cross
break0% 99.999%
go 100% 0.001%

What is the expected value of either of my decisions?

E(wait) = 0% * 0 + 100% * 0 = 0 [£]
E(cross) = 99.999% * 1 + 0.001% * -∞ = -∞ [£]

Clearly, and not unexpectedly at all (common sense?), in this particular model, crossing makes no sense. Certainly, using infinity here is a choice that can be discussed. But again, I don’t think you need utility theory here to make the point. There are two interesting events here (Black Swan events: very improbable, but also very impactful) when crossing, and the unfortunate outcome of the unlikely one should in principle dominate our decision-making.

wait cross
break> X
go Y

Apparently, as “gathered evidence” shows, this is not the case. Not all pedestrians in London follow common sense. Is this new insight? No. Do I have to write about that? No. Did I warn you that I was about to rant? Yes.