{"id":22348,"date":"2020-04-20T15:02:34","date_gmt":"2020-04-20T19:02:34","guid":{"rendered":"http:\/\/chcollins.com\/100Billion\/?p=22348"},"modified":"2022-08-01T07:42:48","modified_gmt":"2022-08-01T11:42:48","slug":"why-so-many-aggressive-drivers","status":"publish","type":"post","link":"https:\/\/chcollins.com\/100Billion\/2020\/04\/why-so-many-aggressive-drivers\/","title":{"rendered":"Why So Many Aggressive Drivers?"},"content":{"rendered":"

Asked and Answered 8.0<\/em><\/strong><\/h4>\n

You’ve been there.\u00a0 You’re driving on the expressway at a reasonable pace, a bit faster than some drivers, slower than others.\u00a0 You move into the left lane to pass a slightly slower car.\u00a0 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.\u00a0 You think, where did this guy come from?\u00a0 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!<\/p>\n

At this point, you make one of two choices, depending on the kind of driver you are and your mood that day.\u00a0 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.<\/p>\n

Doesn’t it seem like aggressive drivers are everywhere?\u00a0 It makes you wonder what kind of life experiences create such angry, impatient, bullying people.\u00a0 And can there really be that many of them out there?<\/p>\n

\"\"<\/a>To explore this situation, we’re going to do a little thought experiment here at the stay-at-home office of Asked and<\/em> Answered<\/em>.\u00a0 Here’s the setup.\u00a0 You and 999 other drivers are going to take a 60-mile trip on the same stretch of interstate highway.\u00a0 The posted speed limit is 60 mph, but most of you drive some other speed.\u00a0 In our scenario (see diagram<\/a>), 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.\u00a0 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.<\/p>\n

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?\u00a0 How many vehicles will you encounter and of what type?<\/p>\n

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 <\/em>in traffic engineering texts<\/a>.\u00a0 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.\u00a0 This concept is the key to the ignition, if you will.<\/p>\n

\"\"<\/a>The chart at right<\/a> is a simplified example of a time-space diagram.\u00a0 Travel time is on the horizontal axis and total distance traveled in that time is on the vertical axis.\u00a0 Each of the colored lines (refer to color scheme above) represents a trip made by one of the drivers in our scenario.\u00a0 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.\u00a0 The silver car started first, followed by green, blue and finally red.<\/p>\n

We will assume each driver maintained constant speed — otherwise these would be curves instead of lines.\u00a0 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.\u00a0 Whenever two trip-lines cross, it means one car passes another.\u00a0 Here, we see that Red encountered and then passed Blue around Mile 25.\u00a0 Red then caught up with Silver at Mile 60, the end of the trip for both drivers.\u00a0 And on this trip, Red and Green never saw each other.<\/p>\n

\"\"<\/a>One more example before we turn to the original question.\u00a0 The chart at right<\/a> shows the trip-lines for 40 cars on a 60-mile trip.\u00a0 Four of every five drivers (silver) drive 60 mph; the fifth (pink) drives 90 mph.\u00a0 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.<\/p>\n

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.\u00a0 Based on his encounters with other cars during his trip, a Silver driver could easily conclude that most drivers are Pinks.\u00a0 Conversely, a Pink driver might think that over 90% of drivers are Silvers.<\/p>\n

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.\u00a0 The formula is:<\/p>\n

E<\/em>IJ<\/sub> =\u00a0 f<\/em>J<\/sub> C D<\/em> S\u0394<\/sub><\/em> \/ (S<\/em>I<\/sub> S<\/em>J<\/sub>)<\/p>\n

where<\/div>\n
E<\/em>IJ<\/sub> = the expected number of encounters Driver I will have with drivers of type J
\nf<\/em>J<\/sub> \u00a0 = the fraction of drivers of type J in the general population
\nC\u00a0\u00a0 = <\/em>the average number of cars per hour (of all types) passing a given point
\nD\u00a0\u00a0 = <\/em>the distance of the trip in miles
\nS<\/em>I\u00a0\u00a0\u00a0 <\/sub>= the (constant) speed in miles\/hour of a driver of type I
\nS<\/em>J\u00a0\u00a0 <\/sub>= the (constant) speed in miles\/hour of a driver of type J
\nS\u0394\u00a0 <\/sub><\/em> = the absolute difference in the speeds of Drivers I and J<\/div>\n

\u00a0<\/p>\n

Now let’s return to the question posed at the start.\u00a0 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.\u00a0 In the figure below, the actual driver population is shown in the middle, along with each driver’s perception<\/em> of the population based on the sample of cars — including her own — that she encountered during the trip.<\/p>\n

\"\"<\/a>A few things to note here.\u00a0 First, based on her encounters on the road, every driver thinks that her own driver-type is decisively in the minority.\u00a0 This is because she will neither pass or be passed by anyone driving at the same speed.\u00a0 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.<\/p>\n

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).\u00a0 In our scenario, both Silver and Blue drivers believe that more than 25% of drivers are Reds, when the actual figure is 10%.\u00a0 Similarly, both Green and Red drivers believe that Silvers comprise 45% or more of the driving population.\u00a0 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.\u00a0 Remember this the next time you complain about all the crazies on the road.<\/p>\n

I would be remiss here if I didn’t mention something about the real-world distribution of driving speeds.\u00a0 The most recent data I could find is from a 2015 federal survey<\/a> of traffic speeds on various classes of roads, compared to the speed limits on those roads.\u00a0 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).<\/p>\n

So the hypothetical driver pool that I presented in my original question is not much of a departure from reality.\u00a0 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.\u00a0 However, I think my general observations still hold:\u00a0 relative speed skews a driver’s perspective of the driver pool.\u00a0 The greater the speed difference, the more prevalent that type of driver appears to be to you.<\/p>\n

Thanks for reading.\u00a0 I trust all your questions have now been answered.\u00a0 Except of course, the most important one:\u00a0 what makes aggressive drivers be that way?<\/p>\n","protected":false},"excerpt":{"rendered":"

Asked and Answered 8.0 You’ve been there.\u00a0 You’re driving on the expressway at a reasonable pace, a bit faster than some drivers, slower than others.\u00a0 You move into the left lane to pass a slightly slower car.\u00a0 Just about the … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[58],"tags":[],"class_list":["post-22348","post","type-post","status-publish","format-standard","hentry","category-asked-and-answered"],"_links":{"self":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts\/22348","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/comments?post=22348"}],"version-history":[{"count":76,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts\/22348\/revisions"}],"predecessor-version":[{"id":22442,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts\/22348\/revisions\/22442"}],"wp:attachment":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/media?parent=22348"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/categories?post=22348"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/tags?post=22348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}