Braess’ Paradox is a famous result in game theory which states that in a network where users selfishly seek to lower their travel times, the Nash equilibrium flows may increase after a new arc/road is added.

Braess’ Paradox can be demonstrated physically with springs, as seen in this nifty YouTube video:

When searching for information about Braess’ Paradox, I found a delightful post on Anna Nagurney’s blog.

In the basketball world, according to Bill Simmons of ESPN, there is the Ewing Theory. According to Simmons: The theory was created in the mid-’90s by Dave Cirilli, a friend of his who was convinced that Patrick Ewing’s teams (both at Georgetown and with New York) inexplicably played better when Ewing was either injured or missing extended stretches because of foul trouble. Simmons has a primer, Ewings Theory 101, which lists examples in basketball history where the removal of a top player (paradoxically) results in a better outcome for the basketball team.

Brian Skinner, a physicist at the University of Minnesota, wrote an article, “The Price of Anarchy in Basketball” in which he developed an analogy, through a model, between certain basketball plays and the Braess paradox, in order to further explore the Ewing Theory.

I’ve been enjoying the NBA basketball playoffs since my favorite team (the Chicago Bulls) won their first round series and beat the Miami Heat in the first game in the second round. The Bulls are without Derrick Rose, the best player on the Bulls. I hope there is not a “Derrick Rose Effect” that would cause the Bulls to be worse when he returns.

Are you aware of other interesting applications of Braess’ paradox?

I recently had the pleasure of reading the book Traffic by Tom Vanderbilt (@tomvanderbilt) as my summer beach read based on the recommendation of a reader or tweep. It is an excellent book. The book is about all aspects of traffic, including operational, psychological, and human factors. Research by several OR professors were included in this book, including research by Anna Nagurney (UMass), Richard Larson (MIT), and John Kobza (Texas Tech). I rarely read books about operations research, so this book was a real treat.

I won’t summarize the entire book, but I will discuss a couple of small parts.

The book starts with a argument for who one should be a late merger at merge points on the expressway, a habit that I picked up after testing a few approaches. The argument here is a mix of traffic theory and psychology.  The traffic arguments is that the total throughput at a merge points depends on how people merge. In the US, we tend to merge way before we have to and assume that anyone who violates this rule is a jerk. Germans like to coordinate their merges to increase efficiency. Efficiency is the German Way. It turns out that it is most efficient to maintain two lanes of traffic up until the merge point, where drivers in the two lanes take turns. This is exactly what happens in the merging lanes in I-95 south of DC when driving toward Richmond, where it works pretty well. The total throughput across all lines of traffic is best if drivers obey this rule, and experiments validate this approach.  One of the US states changed their merge signs during the summer construction season to urge drivers to merge late and take turns. They noticed a large improvement with throughput. Other experiments had limited success, since drivers ignored the new instructions, merged too early, and were aggressive with drivers who obeyed the new rule.  So why do drivers like to merge early? Merging, according to psychologists, is the single most stressful aspect of routine driving.  When we attempt to merge, we can’t help but wonder, “What if I cannot get into the lane?”

I enjoyed the part of the book about networks and congestion. One of the main themes was that networks make driving non-intuitive. We have all heard that adding capacity does not reduce congestion. Tom does a wonderful job of making this intuitive. One reason is that more congestion encourages people to take unneeded trips, live farther from work, etc. More capacity = more miles driven = more congestion. Another reason is due to Braess’ paradox, a game theory model for congestion. Anna Nagurney’s work was cited here, and she has blogged about this before, so I will only briefly elaborate. Here, adding a new superhighway that promises shorter travel times to all if it is not congested, then everyone will want to use it (the Nash equilibrium). The congested road will make everyone’s driving time worse.

Another network example dealt with the monorails at Disney that operate on a simple network. For safety, the trains have to slow down if they get close to other trains. If a train catches up to another train in the network, it has to wait for the first train to make its stops. Disney found that service would speed up if they reduced the number of trains in their system by 1.

Traffic is a fascinating book about many OR-related topics. I highly recommend it if you are looking for a book to read. The blog about the book is here. I may write another post or two about this book. Stay tuned.

HT to @iamreddave for recommending the book to me.

Yesterday, Peter K. requested that I put the US driving safety statistics in an international perspective. Here are four figures on this topic.

The driving fatality rates in the EU by region. These rates include fatalities caused by driving, and therefore, pedestrians killed by cars are included. “The total road death toll was cut by 48 % between 1991 and 2008 and has fallen by 31 % since the year 2000.” Eastern Europe looks kinda dangerous here, and Italy doesn’t live up to its dangerous driving reputation.

Here are the traffic fatalities per 100K inhabitants. Only about half of the people who died were in cars, although I suspect that less developed countries contributed to more pedestrian deaths.

Here are the traffic fatalities per 100M miles driven. The last figure didn’t normalize the fatalities according to how much people drive in different countries. This figure, on the other hand, better reflects how good the drivers are in different countries. However, countries with high speed limits (think of the Autobahn) may fare a little worse, since accidents at high speeds tend more be more deadly. Again, half of the victims were not in cars. Therefore, you might not want to be a pedestrian in UAE.

This is my favorite figure. Here I tested whether people who drove more got more practice, and therefore, led to safer driving conditions in their countries, a possible upside to driving a lot [link to data from the EU countries]. The x-axis shows the number of cars per 1000 inhabitants, a proxy for how much people drove (This isn’t perfect, but I couldn’t find the actual number of miles driven). The y-axis has per capita car fatality rates (per 100K). There is certainly a negative relationship here, with more cars associated with fewer fatalities. However, this only explains ~19% of the variation.

My post yesterday on why teenagers don’t have drivers licenses continues to generate discussion (thank you readers!). I’m going to continue the discussion here.

Despite the poor economy and expensive gas, Americans still love their cars. This love goes way, way back to the invention of cars (see my other blog post on super-highways). Here are some other interesting figures about Americans and their cars.

There are more cars than licenses drivers in the US. This is not a recent trend: there have been more cars than drivers since 1971. This gap is getting smaller, but it’s still wide, historically speaking. My husband pointed out that commercial and government vehicles may be included here. Another report shows that cars outnumbered drivers c. 2000.

This figure shows the miles of road (red). More recently, the total number of lane-miles (green) are shown, which takes multi-lane roads into account. This is a bit misleading, since adding a second lane does not double the road capacity. Compare the total number of road-miles to the total number of miles driven (blue). This suggests that there may be a road shortage. However, building more roads only encourages people to drive even more. Still, Americans love to drive more than ever.

The average annual number of miles driven per driver has steadily increased over time. Only recently people have been cutting back on their mileage, probably due to a poor economy and expensive gas. However, we have only cut back to our 1998 driving level.

Look at the purple line here. This is the average mpg relative to 1987 cars (the 1.0 level), and low is good. This shows that the average mpg has been more or less constant since 1980. This is largely because of the growing popularity of SUVs and minivans. We’ll have to see if the trend of tiny cars and Cash For Clunkers improves these values in the future.

Car, light truck (read: SUVs) and motorcycle fatalities per mile driven. This should convince you not to get around by motorcycle! You cannot see the car fatalities rates well. The fatality rate on rural rates is more than twice as large as the fatality rate on urban roads. This is probably due to urban congestion (congestion = slow driving = safer accidents) and the danger surrounding 2-lane rural highways (2-lane highways = head on collisions at high speeds). Both rural and urban fatality rates have come way down since 1980, probably due to air bags and other safety features (and despite of road rage). The urban fatality rate has dropped faster than the rural fatality rate. Again I would suspect urban congestion here. This is the upside to traffic. Takeaway: Driving is safer than ever.