A numerical experiment on how to optimally wait for traffic lights.

You are commuting on foot in Manhattan. You wish to go from point A to point B. The only thing standing in your way is the numerous traffic signals. How should you strategize? To start with, do you go south or east first?

Since there’s no obvious advantage or disadvantage going either way, you decide to follow the green light. All goes well until you’ve reached point C or point D, where finally, there is a clear winner.

I am not sure about you, but between point C and point D, I would very much prefer the former.

Why? Because traffic signals aren’t evenly distributed. Usually, main streets have a larger share of green light, while the minor ones get more red. Each road has a certain level of “importance” associated with it, as demonstratively indicated by the width of the lane in the plot above.

So you say to yourself: it seems a really good idea to end up in point D. To pull it off, does it make sense to wait for a few seconds in one or more of the preceding crossroads? More generally, is there an optimal decision?

To find out, let’s take a virtual walk. Note that since you have to walk the full Manhattan distance in any case, the time spent on actual walking is irrelevant for comparison purposes.

Strategy 1: Let the Light Guide Your Way

You are a brave soul. You don’t want to be bound by any rules. So you keep it simple. You follow wherever the green light tells you to go, unless you’ve already reached the boundary, in which case you wait for the signals to turn green (if you have to).

Strategy 2: Conditional Wait

You are walking eastbound along one of the major streets. More likely than not, you run into a green light. But you also notice that the green light that allows you to walk straight ahead is dying out, and soon enough, you will be able to make a turn and go southbound. You decide to seize the opportunity, because waiting a few seconds for something this rare seems a good deal.

To fully specify the conditional wait strategy, we introduce two new parameters:

• the “rarity” of a green light, defined as the ratio of the current wait time to maximum wait time. The lower it is, the more likely you are inclined to wait for it when it’s about to turn green.
• the relative “importance” of a road, defined as the ratio of the absolute “importance” of the two crossing roads. The lower it is, the less likely you will be able to cross it with ease - one more reason you might want to wait if the green light isn’t too far away.

Strategy 3: Ride Along Main Street

Main Street, defined as the street of the highest “importance” (either horizontally or vertically), has a nice property to it. If you walk along Main Street, chances are that you will face green traffic signals more often than it would have been otherwise on any other street - you might even encounter several green lights in a row if you get lucky.

Putting it All Together

Numerical simulation suggests that you simply cannot beat strategy 1. Strategy 2 gradually gets worse as we increase the two artificial thresholds that enable it to behave differently from strategy 1. Strategy 3, while by no means sound like a crazy idea, performs the worst (*).

So next time you are in doubt, remember to let the (green) light guide your way.

(*) It’s worthy noting that the introduction of correlation between green lights along Main Street, as is usually the case in real life, might make a significant difference.