Braess' paradox: adding a new road to a city can slow down traffic
en.wikipedia.org
en.wikipedia.org
Not surprisingly, the flow isn't large. The electricity takes many path through the circuit.
Here, too, having fewer paths would result in large flow.
See: http://physics.stackexchange.com/questions/2072/on-this-infi...
I wish I had a video of this demonstration, it really hammered in the point (for me) that this is a real phenomenon and not just mathematical trickery or electrical weirdness.
I presume in the demonstration you're describing the central rope was not partially elasticated as the other two? (as demonstrated in the two videos)
Does Braess' paradox apply if the central rope is elasticated? Does it apply for a parallel route also or just for a route that joins to existing series routes?
I think if it's just a joining link then the effect could be modelled in signal theory as the result of partial feedback. Still no idea how the experiment works though :-)
In this setup, cutting the link essentially converts the two springs from being in series to parallel: http://ibin.co/2JEsCKEYRisb
If the central string was elastic, this would cause the weight to initially hang lower, but have no effect on the position reached after it is cut (assuming that it does not stretch so much that the other strings stop being slack). The effect would thus appear larger.
* The user is aware of the new roadway and utilizes the roadway.
* The user knows that the old route was 10 minutes more and was the current ideal.
* Other users utilize the route under the same 10 minute saving condition, driving up the amount of traffic over time, even when the new route ends up adding travel time to the original time.
* Users do not consider going back to the old route, even though it may be better now as the system builders had declared this route to be the best. There is also an internal feeling that if the new route was like this, what will the old route look like.
* When talking to people who drive, once a known route has been established, it takes a lot to get them to change. That is why accepting Waze provides such a great opportunity for balancing traffic and utilizing roadway capacity.
In other words, it is very unlikely to simultaneously minimize both the expected commute cost for the group and for each individual.
However, in the continuous case, you can fix this using taxes, tolls or incentives (in theory - in practice I don't know. )
Blanchet and Carlier have some nice mathematical articles including
https://www.ceremade.dauphine.fr/~carlier/blanchetcarlierfin...
The paper does however note the traffic problem in passing: "the variational approach we develop presents some similarities with the variational approach to Wardrop equilibria on congested networks and in both cases equilibria are socially inefficient."
What typically happens when a new roadway opens or a congested route is widened is that travel times first fall, as expected; but soon the reduced travel times attract drivers who previously had used other routes, or who had commuted at off-peak times, or who had used public transit. Thus the improved roadway ends up with significantly more traffic than the old roadway, and the price (in travel time) of the increased capacity is bid back up near its previous level.
Added roadway capacity can lead to a welfare improvements, but it mostly comes in the form of fewer people shifting their schedules to beat rush-hour traffic or deigning to take public transit, and not through reductions in travel times. In actual fact, as we've seen throughout the era of American suburbanization, vast new roadways have probably substantially increased average travel times [1] by markedly changing patterns of land use (i.e., by giving rise to the modern low-density, car-dependent suburb), albeit while also engendering some welfare gains (e.g., increased suburban living space).
If you want to prevent induced demand from gobbling up expected improvements in travel times when you build new roads, you pretty much either have to (a) decrease demand by deliberately increasing the financial cost of travel (e.g., through tolls or congestion charges), or (b) over-build roadway capacity in the extreme, to the extent that just about everyone in the region can fit on the roads at once, alone in his or her own car (as is the situation in cities like mine).
0. https://en.m.wikipedia.org/wiki/Induced_demand
1. I mean in comparison to a counterfactual alternative history in which the urban-suburban commuter freeway systems typical of American cities went unbuilt. Commute times in European metropolises, which mostly lack the extensive commuter roadways and concomitant suburban sprawl of their American counterparts, are broadly lower than in American metros of similar sizes, for example.
You cannot solve the problem of an external source of new traffic, be it a new stadium at the edge of town or new machine added to the edge of a network, by not building in any new capacity. Well, you could if you simply chose not to serve that new node. Not running any new roads to a new airport is a great way to get that airport to close, just as not serving that new customer on your network is a great way to make that customer go elsewhere.
The other flipside to the pardox is that it really is applicable to any network. Sometimes the best way to improve the efficiency of mass transit is to cull bus/train routes. But try selling that in a modern city.
Also same thing applies to this paradox. Even if individual travel time is reduced for some, I doubt that overall traffic throughput is reduced. At worst case scenario it should stay the same.
It's one of the big shifts in planning thought to get away from a simple engineering problem of moving more cars faster and try to integrate different modes and give land use a more careful impact assessment.
Not necessarily for the people on the bus. Efficiency in terms of capacity represents one of several possible optimization criteria; the distribution of transit times to various destinations represents another.
The paradox still applies when we consider the Tokyo rail network. There's lots of rail in Tokyo - it transports most of the commuters. Major lines are quite literally packed during rush hour. [0] Adding more rail will make the system even more popular, so the problem won't get better.
OTOH consider congestion charges. When interviewed[1], individuals do not believe they were significantly affected by the introduction of a charge, even though overall traffic levels go down substantially, by one fifth in the example used of Stockholm.
What we are changing when we change capacities is not speed or comfort, but a preference for what kind of congestion we get and how cheaply it will be filled. If our only goal is congestion reduction we should not be looking at the roads at all.
The San Francisco area has plenty of car traffic, but more recently, within the past decade or so, the public transit has become extremely popular. This is not because the transit has gotten substantially better by quality or speed metrics. It is because the population is making more trips and longer ones, and the public transit systems were the last resort for capacity. Demand for new trips is in turn caused by available housing being located distantly from workplaces. If newcomers were able to live where they worked, congestion would drop significantly.
[0] https://www.youtube.com/watch?v=pRBLnth4oSg [1] https://www.youtube.com/watch?v=wC33HAq--x8
Sure, there always are some cases where fiddling with lanes, transport types and other relatively cheap means of changes makes traffic better, but in the end simply more roads are needed.
But there's something right in what you say. When you add the extra road you make travel time worse at low usage and better at high usage.
Suppose your network has an upper path with cost for x cars 1 on the left and x on the right, and a lower path with cost x on the left and 1 on the right, and the optional vertical road is free at any capacity. Then, if I've done the calculations right, average travel time with t cars is 1+t/2 without the extra road, and min(2,2t) with. So at t=1, which is the case usually used to demonstrate the paradox, the extra road increases the travel time from 1.5 to 2, but beyond that point increased traffic is free for the enlarged network but not for the original one. So, e.g., if t=10 then the extra road makes things 3x faster. The crossover point is at t=2, where both networks have average travel time 2.
So the extra road is a win if you expect t >= 2, and a loss if you expect t <= 2.
(If we measure throughput as "max traffic with such-and-such a limit to average travel time", then if the limit is <= 2 then both have finite throughput and the one with the extra road is worse, but if the limit is >= 2 then the network with the extra road has infinite bandwidth.)
On the other hand, with socially optimal traffic patterns there would be induced demand and more drivers would come on the roads and potentially use Waze.
It's actually difficult to know what would make Waze better off, providing socially optimal routes or individually optimal routes.
Waze might also behave like many companies and ask for subsidies from local governments for turning on the social optimization, and the local governments are ill-equiped to know how much to pay.
If it optimizes for each individual traveller, then Braess' paradox still applies even with Waze because all the travelers in Braess' paradox make the individually optimal choice.
If it optimizes for the socially optimal distribution, then it does solve the problem. But this sounds like an optimization problem that would be really hard to formulate and solve in practice, so I'd be surprised if Waze (or any other networked traffic app) did it right now. Maybe eventually?
The catch is no one knows in advance who is going to go the bar. It can be shown that no pure strategy can work for EFB.
Whether traffic moves faster or slower pollution is bound to increase.
At Nash equilibrium, the highway is over-utilised while the road is under-utilised. So stripped to the core, this is an example of tragedy of the commons where if every single individual works for their sole interest, they will all lose out and yet nobody has an incentive to make a change.
1. http://www.americanscientist.org/libraries/documents/2015617...
I wonder how that would change if the actual cost (in time to that driver) of each potential route were known; even if that is in current time costs? (IE how would they operate if they had real time traffic enabled GPS?)
Also, their refusal to bypass turning sims is inane.
That's not why it took so long to rebuild. The nine year process was the result of the usual corruption, insider jockeying, incompetence, bureaucracy, and lack of urgency from Bay Area government. The objections of CDOT and local residents that ardently opposed the freeway in their neighborhood were ignored and steamrollered as usual. Of course, the local officials, contractors, and CDOT shared and enjoyed the lucre from the $1,200,000,000 we all paid for the three mile connector.
Meanwhile the 1994 earthquake in LA triggered a special exception to the usual legal process where Gov Wilson could take personal responsibility for selecting a design, a contractor, a price, a schedule, and contract terms for rebuilding several segments of LA highway. They were all delivered on time and well under expected budget. Some were rebuilt in a couple months. No one had time to figure out if those could be beneficially abandoned.
And today Oakland and CDOT still cannot make the simple decision to admit error and close the 880 connector.
https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws
It is not as if cars are equipped with some superior intelligence when compared to electrons in a circuit, the behaviour is identical and the normal laws of physics apply.
Or do you have an example of a situation in which Kirchhoff's laws predict that adding a wire to a circuit decreases the total current?
The simple example I was thinking about is a "digital potentiometer". Think about the audio output of your motherboard, which's amplification you can control digitally.
Kirkhoff's circuit laws are extremely simplifying.
An interesting exercise would be to prove the statement we were talking about in the fictional world of "Kirkhoff circuits".
Network frames don't "selfishly choose their route" and are entirely at the whims of the Routers they pass through.
Because of complex routing and suboptimal peering/routing, a "shortcut" could be added that makes it quicker to get from A to X1 but makes it take much longer to get from A to X2 Additional routes (as in "directing packets the right way", not physical routes) would fix this and get the best of both worlds.