# Ivars Peterson's MathTrek

March 18, 2002

## Highway Relativity

There may be more to zipping along in the fast lane than meets the casual eye.

Freeway drivers often sense that cars in an adjacent lane are moving faster than those in their own lane. That's certainly true when the average speed of the cars in the next lane is significantly higher than that of cars in a driver's lane. However, the same impression can arise even when the average speed in both lanes is identical.

In 1999, Donald A. Redelmeier of the University of Toronto and Robert J. Tibshirani of Stanford University suggested that such an effect arises from faulty intuition rooted in a perceptual illusion.

To address the question of whether drivers can accurately judge if they are in a lane slower than the next lane on a congested roadway, the researchers focused on an individual driver's perceptions while operating a vehicle in heavy traffic.

In their computer simulations, the researchers created two lanes of traffic with identical characteristics, except that vehicles had different initial spacings in the two lanes. They assumed that a vehicle would accelerate if traveling slower than its target speed and when no other vehicle was within a certain minimum distance. They then tracked each vehicle each second to determine where it was and what it was doing--accelerating, decelerating, or staying at a constant speed.

In general, cars tended to spread out when going quickly and to pack together when moving slowly.

The researchers found that the 1-second time intervals during which a given vehicle was observed to be overtaken occurred more often than the 1-second time intervals during which that vehicle was observed to overtake another vehicle--even though the total number of vehicles that passed it was balanced by the number that were overtaken by it. In effect, drivers spent much more time watching other cars whiz by, creating the illusion that the next lane was moving faster.

"We suggest that this illusion occurs because more time is generally spent being overtaken. . .by other vehicles than is spent in overtaking them," Redelmeier and Tibshirani concluded in a Nature report. "Knowing that this effect is illusory might encourage drivers to resist small temptations to change lanes."

Other perceptual factors may contribute to the illusion. Because drivers look forward rather than backward, vehicles that are overtaken move quickly out of sight (and out of mind), whereas vehicles that move ahead remain in view for a longer time.

The researchers added, "Even if attention was not focused in particular directions and was evenly spaced in time, human psychology may make being overtaken (losing) seem more salient than the corresponding gains."

In responding to the findings of Redelmeier and Tibshirani, Nick Bostrom of Yale University argued that the impulse to change lanes has more to do with the fact that cars in one lane are typically faster than those in the other.

Again using the observation that cars traveling at higher speeds are usually separated by longer distances than are slower cars, Bostrom noted that this leads to a higher density of cars in lanes moving at slower average speeds. In other words, over a given stretch of road, there are likely to be more cars in the slower lane than in the faster lane.

"This in turn means that, when driving on a two-lane road system, more of the average driver's time is spent in the slower lane," he wrote in the November 2001 issue of the online math magazine Plus.

"If you are driving on the motorway and think of your present observation as a random sample from all the observations made by all the drivers, then chances are that your observation will be made from the viewpoint that most drivers have, which is the viewpoint of the slow-moving lane (where more cars are likely to be)," Bostrom contended. Ergo, the cars in the other lane are moving faster!

Basically, Bostrom's argument is that it is unrealistic to assume that vehicles typically move at the same average speed in two adjacent lanes.

Additional insights have come from recent work by Bryan Dawson and Troy Riggs of Union University in Jackson, Tenn. They took up the question of what a driver moving at a certain speed would perceive to be the average speed of the other vehicles on the same highway.

Dawson and Riggs based their traffic model on the assumption that drivers gauge the average speed from the cars around them rather than all the cars on the highway. So, a driver going much faster than average would overtake many slower cars and only a few faster cars. Conversely, a slow driver would overtake just a few cars and be passed by many cars moving at a faster pace.

In their mathematical analysis, Dawson and Riggs assumed that the speeds of vehicles on a highway are normally distributed--with a mean of 68 miles per hour and with 68 percent of the vehicle speeds between 66 and 70 miles per hour (a standard deviation of 4 miles per hour).

The mathematicians calculated that, under the given conditions, a driver traveling at 65 miles per hour would "perceive" the average speed of the traffic to be about 70 miles per hour, rather than 68 miles per hour. "Hence, the observed average speed is higher than the actual average speed of traffic," Dawson remarks.

In general, "if you are traveling at a speed near the actual average speed, then as you increase your speed, it seems as if the average speed of the other vehicles decreases," he adds. "Conversely, as you decrease your speed, it seems as if the average speed of the other vehicles increases." Yet the actual average speed of the traffic remains unchanged. Different speed distributions give similar results.

"An implicit assumption in our work is that the driver is able to accurately assess the speed of each of the cars that he or she passes or that pass him or her." Dawson says. "As a result, the effect we observe is purely a mathematical one." Such an effect could very well enhance the adjacent-lane perceptual illusion postulated by Redelmeier and Tibshirani.

The difference between a driver's speed and the perceived average speed of other vehicles seems to be sensitive to traffic density. In heavy traffic, changes in perceived difference of speed may be noticed relatively quickly. "If you are driving at approximately the same speed that you perceive to be the average," Dawson says, "then the accelerator and brake of your vehicle appear [about] twice as sensitive as normal."

That may be one reason why driving in heavy traffic can be so nerve-wracking. "Our perceptions don't match reality," Dawson says. "Things seem more 'touchy.'"

References:

Bostrom, N. 2001. Cars in the next lane really do go faster. Plus (November). Available at http://www.pass.maths.org.uk/issue17/features/traffic/index.html.

______. 2000. Why cars in the next lane do go faster. Available at http://www.anthropic-principle.com/preprints/lane/cars.html.

Clevenson, L., M. Schilling, A. Watkins, and W. Watkins. 2001. The average speed on the highway. College Mathematics Journal 32(May):169-171.

Dawson, B., and T.D. Riggs. 2002. Highway relativity. Abstracts of Papers Presented to the American Mathematical Society 23(No. 1):154.

Klarreich, E. 2002. Driven to delusions. New Scientist 173(Jan. 19):11.

Redelmeier, D.A., and R.J. Tibshirani. 2000. Are those other drivers really going faster? Chance 13(No. 3):8-14. Available at http://www.public.iastate.edu/~chance99/133.redelmeier.pdf.

______. 1999. Why cars in the next lane seem to go faster. Nature 401(Sept. 2):35.

Comments are welcome. Please send messages to Ivars Peterson at ip@sciserv.org.

A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles.