Asked and Answered 8.0

You’ve been there.  You’re driving on the expressway at a reasonable pace, a bit faster than some drivers, slower than others.  You move into the left lane to pass a slightly slower car.  Just about the time you draw even, you glance up at your rear-view mirror and see a set of fierce headlights bearing down and closing in on you by the second.  You think, where did this guy come from?  Barely ten seconds later he’s riding your tail, making sure you know in no uncertain terms that you’re in his way, so get your ass moving already!

At this point, you make one of two choices, depending on the kind of driver you are and your mood that day.  You either finish your pass without changing speed, no matter how much the guy behind you tries to intimidate you, or, you decide this person is bad news and you speed up to get out of his way and let him go roaring by — which he will.

Doesn’t it seem like aggressive drivers are everywhere?  It makes you wonder what kind of life experiences create such angry, impatient, bullying people.  And can there really be that many of them out there?

To explore this situation, we’re going to do a little thought experiment here at the stay-at-home office of Asked and Answered.  Here’s the setup.  You and 999 other drivers are going to take a 60-mile trip on the same stretch of interstate highway.  The posted speed limit is 60 mph, but most of you drive some other speed.  In our scenario (see diagram), 30% of drivers drive at 60 mph; 30% drive at 65 mph; 30% drive at 70 mph; and the remaining 10%, the most aggressive ones, drive at 80 mph.  We will assume that the four types of drivers enter the on-ramp in random order and at a steady rate of 4 cars per minute.

Now, decide which type of driver fits you best (denoted by silver, blue, green or red) and then answer the question: what will your driving experience be like?  How many vehicles will you encounter and of what type?

To answer this, I originally thought that I would have to write a computer simulation of the problem and keep track of hundreds of cars as they navigated 60 miles of interstate. But then I stumbled upon time-space diagrams in traffic engineering texts.  Simply put, such diagrams capture how a vehicle (or any number of them) covers a stretch of road based on the vehicle’s speed profile.  This concept is the key to the ignition, if you will.

The chart at right is a simplified example of a time-space diagram.  Travel time is on the horizontal axis and total distance traveled in that time is on the vertical axis.  Each of the colored lines (refer to color scheme above) represents a trip made by one of the drivers in our scenario.  In this example, one driver of each speed-type drove the 60-mile trip, with the drivers starting out 5 minutes apart from each other.  The silver car started first, followed by green, blue and finally red.

We will assume each driver maintained constant speed — otherwise these would be curves instead of lines.  Now, consider the trip-line of the red car, the 80-mph driver (Red) who started at the 15-minute mark and finished his trip at the 60-minute mark.  Whenever two trip-lines cross, it means one car passes another.  Here, we see that Red encountered and then passed Blue around Mile 25.  Red then caught up with Silver at Mile 60, the end of the trip for both drivers.  And on this trip, Red and Green never saw each other.

One more example before we turn to the original question.  The chart at right shows the trip-lines for 40 cars on a 60-mile trip.  Four of every five drivers (silver) drive 60 mph; the fifth (pink) drives 90 mph.  From the line crossings, we see that the typical pink car passes 10 silver cars, while a silver car encounters 2 or 3 pink ones.

What we learn from this example is that a driver’s perception of the other types of drivers on the road is distorted by the relative speeds of the drivers.  Based on his encounters with other cars during his trip, a Silver driver could easily conclude that most drivers are Pinks.  Conversely, a Pink driver might think that over 90% of drivers are Silvers.

This brings us to the formula I derived for the expected number of times a given driver will encounter drivers of other types during a constant-speed trip.  The formula is:

E_IJ =  f_J C D S_Δ / (S_I S_J)

Now let’s return to the question posed at the start.  We defined 4 types of drivers based on their speeds: 30% (Silver) drive 60 mph, the next 30% (Blue) do 65 mph, 30% (Green) do 70 mph, and 10% (Red) do 80 mph.  In the figure below, the actual driver population is shown in the middle, along with each driver’s perception of the population based on the sample of cars — including her own — that she encountered during the trip.

A few things to note here.  First, based on her encounters on the road, every driver thinks that her own driver-type is decisively in the minority.  This is because she will neither pass or be passed by anyone driving at the same speed.  While she may follow another car of similar speed for many miles, her own vehicle and the one she sees right in front of her would be the extent of her experience with like drivers.

A second observation is that every type of driver over-estimates the proportion of both speedsters and slowpokes (unless you happen to be one of those types).  In our scenario, both Silver and Blue drivers believe that more than 25% of drivers are Reds, when the actual figure is 10%.  Similarly, both Green and Red drivers believe that Silvers comprise 45% or more of the driving population.  As is evident from the formula, the greater the speed difference, the more encounters one is likely to have with a given type of driver.  Remember this the next time you complain about all the crazies on the road.

I would be remiss here if I didn’t mention something about the real-world distribution of driving speeds.  The most recent data I could find is from a 2015 federal survey of traffic speeds on various classes of roads, compared to the speed limits on those roads.  The data for limited-access highways suggest that 30% of drivers do not speed (Silver), 25% exceed the speed limit by up to 5 mph (Blue), 25% exceed it by no more than 10 mph (Green), and 20% drive more than 10 mph over the limit (Red).

So the hypothetical driver pool that I presented in my original question is not much of a departure from reality.  The main difference is that, in real life, there is a continuum of driving speeds, and most drivers do not maintain a fixed speed for a whole trip.  However, I think my general observations still hold:  relative speed skews a driver’s perspective of the driver pool.  The greater the speed difference, the more prevalent that type of driver appears to be to you.

Thanks for reading.  I trust all your questions have now been answered.  Except of course, the most important one:  what makes aggressive drivers be that way?