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Braess’s paradox (wikipedia.org)
134 points by unhammer on Sept 25, 2018 | hide | past | favorite | 37 comments

Simple explanation: when you win five minutes by switching from road A to road B, other drivers on road B might lose more than five minutes in total from the extra congestion you cause. So the total time spent by all drivers will increase, and so will average time (if number of drivers is constant). So removing some connections from the road network, to prevent some drivers from optimizing their route, can reduce average driving time.

Furthermore, when a driver chooses to take the new route, this does not create any opportunity for the drivers disadvantaged by this choice to do better for themselves (they also have the option of switching to the new route themselves, but that opportunity was created by the creation of the new route, not by the responses to it by other drivers.)

This paradox may have some relevance to the effect of navigation apps, especially those that attempt to route around congestion, as they effectively create routes in the sense that the vast majority of drivers would not have considered using them without the app.

I wonder if current navigation apps are smart enough to only propose a new route to a portion of the total number of drivers in that location.

Ultimately, the only way to have optimal flow is to delegate all itinerary decisions to a central authority...

Have you noticed how good the time estimates have gotten? Clearly they're streering other traffic out of or into the way to keep it even.

I started noticing this very abruptly - that Google no longer redirects me from congested roadways to low-traffic parallel roads. Now, as soon as I take the exit to a less congested road, my estimated drive time might drop by 5-10 minutes, so clearly Google knows that it isn't sending me down the optimal route. I don't think this used to be the case.

Interesting - I noticed recently, for the first time, that when I modified a 'fastest' proposed route using local knowledge, I found the predicted time dropped, and that has happened several time since.

This may be the result of better predictive modeling, and particularly in recognizing that dumping a lot of traffic into a low-capacity route is pointless. It is probably also a political response to the fact that residential areas don't like having a lot of traffic routed through them, and are pressing for the power to regulate it.

Better predictions could also explain toast0's observations even when global optimization is not being attempted.

At the very least, the estimated time for given routes would update dynamically, as a proposed alternative route became less efficient.

Thank you for that explanation. In a way it reminds me of the cost of context switching. Sure, it seems quicker to do a task right now, but the overall cost will go up because of what you left behind.

From reading the article, they do not fix the numbers of cars, so it is likely that the number of cars has a greater effect on flow.

Improvement in roads always lead to more cars, beyond what was expected. Entire suburbs get created when roads improve!

From the article:

"the removal of main roads does not cause deterioration of traffic .../... some motorized travels are not transferred on public transport and simply disappear ("evaporate")."

In transport tycoon (openTTD) when train station started to become congested adding more tracks usually didn't solve the problem, best way was having two uninterrupted tracks almost into station then spread out into 15-wide station (or whatever max was allowed) with short buffers before and after station.

TTD is great for modeling systems like that; what I like(d) to do was just create a loop track, two tracks wide (one for each direction), with stop lights every x tiles, making sure to have the same length trains at all times.

This is fascinating. The book Traffic [1] talks a great deal about how adding road capacity (e.g. more lanes) can result in zero change in congestion because more people choose to take more trips.

But this paradox appears to hold traffic constant, and uses game theory to show how more connections (not lanes) can result in worse congestion too.

It really is amazing how something as simple-seeming as roads and traffic, where it feels like simple common sense ought to apply, winds up being so deeply and fundamentally counter-intuitive.

[1] https://www.amazon.com/Traffic-Drive-What-Says-About/dp/0307...

> adding road capacity (e.g. more lanes) can result in zero change in congestion because more people choose to take more trips

This is not the paradox nor even an argument against adding road capacity though.

Even if congestion or travel times stay the same, now more people are able to travel to places they want to go. Traffic might not have improved, but some measure of quality of life did.

Depends on the impact of sitting in traffic on quality of life.

You can still forgo the trip though. It's up to each individual participant to decide whether or not they want to make the trip. The ability for more people to take the trip at the same opportunity cost. Adding a lane or a road is little different than increasing the frequency of trains on a subway.

Peak demand will always saturate whatever capacity you have but the amount of stuff you move at peak capacity is greater so you spend less time fully saturated

> Peak demand will always saturate whatever capacity you have but the amount of stuff you move at peak capacity is greater so you spend less time fully saturated

That doesn't follow.

I believe this paradox applies to any congested highway with a robust network of side streets around it. If everyone could agree to take side streets for half of their trip, the side streets could easily absorb the extra traffic without slowing down, and the highway would be able to flow freely, shortening everyone's total commute time. But no single person can improve their commute by switching to the side streets alone.

What? It's usually the first switcher who benefits the most from using the side streets, because they see zero congestion, and, if anything, it's more of a problem (for the cut-through neighborhoods) of too many people switching.

If the side streets are currently faster than the congested highway that's true. I'm talking about situations where it isn't, where the fixed cost of stoplights and speed limits ensures it they will never actually be faster than the congested highway.

Another relevant discussion, albeit from a more game theoretic view: https://news.ycombinator.com/item?id=17741641.

Beyond two submissions sharing the same subject-title, do you think it'd be possible to find all "relevant" discussions -- say, back-links or forward-links -- as well and post them?

For reference, the first link in that discussion is dead and I couldn't locate the paper with a cursory search.

However, there is a good, clear demo of the spring experiment at https://www.youtube.com/watch?v=ekd2MeDBV8s

I’m not sure how this is different from the concept of induced demand? BTW, the ”See also” section is awesome. A serious time sink.

Induced demand is when more people drive, or they drive more often, or longer routes because of added roads.

This is when the same amount of traffic is distributed less optimally because of adding new edges on the graph.

Ah, right. Thanks!

I think that some changes in traffic aim to optimize the instantaneous speed, while it should aim to minimize the door-to-door time. Driving 1km@10km/h takes 6 minutes, while driving 5km@30km/h takes 12 minutes, and you might end up driving 5km@1km/h at rush hour because the longer distance keeps you in traffic for more time.

This actually happened in my home city, some areas implemented "improvements" that made it very difficult to reach some destinations, you need to follow long detours intended to "improve flow", they might improve flow for whoever is just passing by, but traffic that originates or terminates in that area, it is hell.

This might not be the exact manifestation of the paradox, but I think it's at least a supporting anecdote.

The latest Sim City had its problems, but I definitely experienced this phenomenon in its traffic simulation. Adding more roads created more intersections which -- using a computer analogy -- created more context switches between competing cars (threads). Intersections would become clogged, potentially backing up nearby intersections. The best traffic systems in Sim City minimized intersections just as much as -- if not more than -- throughput and mean travel distance.

If anyone is interested in topics like this and wants to delve some more into the mathematics, I’d highly recommend Tim Roughgarden’s lecture series on Algortihmic Game Theory:


Palo Alto Central Express. Cut off side streets. Traffic will improve, and kids will be safe in the street.

At the limit, if you remove all capacity (or leave a nominal capacity for a single commuter that works beside their home to remove infinites), the individual transit time goes to zero! It follows we should remove all roads!!!

It seems to me that the average transit time is the wrong metric to optimize for.

No. In Braess' paradox both the situation with and without the extra road have the exact same total flow from A to B. Removing flow capacity is not allowed.

"In 1983, Steinberg and Zangwill provided, under reasonable assumptions, the necessary and sufficient conditions for Braess' paradox to occur in a general transportation network when a new route is added."

Anyone know what those conditions are, offhand?

The biggest is that the drivers have adequate information about road capacity and traffic conditions and are acting rationally on that information.

I actually believe that Waze has contributed to a lot of urban traffic problems as of late, because it ensures that more people contribute to braess' paradox instances that might have gone unrealized in the past.

While I don't have an in-depth technical understanding of exactly why this happens in every instance, I have definitely observed this while playing Cities: Skylines, a SimCity-like game which emulates traffic flow.

However, note that the traffic flow simulation in C:S is rather simplistic for performance reasons: specifically the pathfinder doesn't take dynamic traffic conditions into account at all! A traveler never chooses an alternative route even if the primary one is congested. But yeah, adding routes usually means also adding choke points (that is, intersections) which naturally slows down traffic.

Whoa, I was just reading about this yesterday! Did I consent to tracking when I created an account here? ;-)


Your harsh tone and eagerness to dismiss anything with which you disagree even remotely is probably why you're shadowbanned. I would suggest rethinking how you frame your arguments.

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