There is another similar "paradox" when you add a third, "central" path connecting A and B but not intersecting the other two paths. In this case, traffic can actually increase in ALL three paths. The idea is that total traffic is not fixed: there are some people that travel from A to B by other means: train, bike, on foot. However, if the third path opens, they might switch to car driving. In the short run this might not be so big an issue, but then some people that did not use any of the paths might buy a car and start using them. And traffic ends up going up everywhere! This is usually called the problem of "the supply generates its demand".
"ill guess that closing a main road that is overused will cause people to use alternate routes" Good so far. "that can handle the extra traffic." My question would be: if they can handle the extra traffic, why aren't they being used already? In the example in the video, it's because the other route is longer, which I guess is plausible because main roads usually take you fairly directly to where you want to go, while finding side roads that have a route all the way through to where you want to go may involve a less direct path. (On the other hand, if the main road has a higher speed limit, people may take a less direct path in order to use it.)
tifforo1 Time. The main route would normally be the quickest way to get to your destination, but when there is a backup on your normal route, taking a route that is a bit longer will still be quicker than waiting in the traffic.
if there are alternatives not all cities have alternatives for some roads. and this only works when people know the road will be closed and they opt out to use the car, in reality when a road is closed by accident? you have huge traffic jams. why in reality we get traffic jams? because the paradox is bulshit
Often times closing a road carries a tradeoff whereby the closure of the road does in fact increase travel time but with the tradeoff that traffic is more spread out among multiple routes.
+Jeremy Corson That's because they both come from the same area of mathematics: the study of logic and decisions known as game theory. It's not quite the same as the prisoner's dilemma, but you're on the right tracks and I see where you're coming from.
+SamBrev Actually I agree with Jeremy, the link is not just that they come from the same area of mathematics, but also that all drivers would actually be better off if they made a contractual agreement to not use the fast lane and to evenly divide their route. For the same reason the prisoners are better off contractually agreeing not to rat on one another... But as soon as they have agreed not to rat, by the rule of never playing a dominated strategy... You are better off ratting.
Interesting. Was reading an article about how GPS might become communal one day to optimise traffic flow for everyone. Never studied this area of maths, but a very interesting insight into how the computer might make some of the decisions!
this will be if everyone is not egoistic, if everyone split like before ok that every car will go in 30 minutes, but so a single person would say "wait, if i go here i will spend only 20 minutes by using that road" and so if everyone use this logic, that roads will become full again and the travel times become once more 40 minutes
Judging from the comments, I think someone needs to post an example up using Cities Skylines. This isn't just theoretical, you could make an actual road that has this behavior.
Can someone please explain me what the Zero time road is? I am from South America (3rd word) and we only have regular old school roads. I hope some day we will have NEAR zero time travel roads (speed o light) but ABSOLUTE ZERO time travel... damn guys, first world is awesome!
It means the picture is not to scale, and that road is very very short. It does not take zero minutes to travel, but very little time, a few seconds at most.
He would have done better to draw something that was almost an infinity symbol, then with the new road it would become the infinity symbol. The 'new road' would then have been a new junction linking the two different roads at the point they came nearest to each other in their middles, and said junction would then allow changing from one road to the other mid-journey.
I find it hard to believe it costs nothing to switch routes.. Just the turning and decelerating and accelerating increases your time, but you've built a road that's instantaneously teleports you on the other side.
A highway is not supposed to have jams but it most deffinitely does. Also, in this video, he is not talking about highway, he is talking about city streets.
"A true highway has no jams", but most of the highway that exists today can not be qualified as "true highways" I think... Highways on their own does not have jams, but the road network connected to them has, and the jams just extend onto the highway
The drivers are acting under game theory, where they all do what is best for them. The tragedy of the commons is a famous example of how this goes wrong. They do not act in the best interest of the group, only themselves, which makes everyone in the group worse off.
You can make a payoff matrix and come to the conclusion that leads to the tragedy of the commons, but if you make a second payoff matrix to decide whether you should use the first payoff matrix or not, you'll end up choosing the optimal route and avoid the link.
I don't quite get what you mean, DerLamer. How would a going another layer deep with a payoff matrix change your actions? It seems like it is still as fast or faster to take the link. Why would a rational driver not take it if they were only acting in their own self-interest?
Greg Erlandson It's about the realization that considering any of the newly opened options leads to a worse outcome for everyone. You can consider considering them, reject that and take the optimal solution, which is acting as if nothing had changed. In a world of truly perfect logicians, they'd realize this as well, realize that the others would also come to that conclusion, and everyone would choose to be blind to the link. These are perfect logicians operating at one level higher than ordinary perfect logicians who only use straightforward payoff matrices.
+DerLamer this is the tragedy of the commons, as I mentioned earlier. Although you're right that a perfectly rational logician would realize that they make travel times slower for everyone by taking the new route, they don't care. They only care about their own travel time. You're making the additional assumption that 1) they are perfect logicians and 2) they care about other people. In the situation presented in the video, the drivers would say "screw everyone else. I'm going to take the new route if it makes MY travel time even 1 second faster and it makes everyone else's total time an hour slower." If you make different assumptions, then you're solving a totally different problem.
This may sound simplistic but say you have intersected roads such as 42nd street and 5th avenue in new york city. If 42nd street is closed traffic on 5th avenue, which intersects 42nd street, will improve because drivers on 5th won't have conflicting movements with vehicles on 42nd street. When the signal is red for 5th (green for 42nd) drivers on 5th can proceed through red lights (assuming pedestrian conflicts are halted as well)
In the NY example, there are many other possible factors. Maybe people knew the road was going to be closed and took a different route completely, or didn't go where they normally would have gone. Or left their car at home that day. Or one of dozens of other factors.
i agree when in my city there will be a close of road,i know many people will not use their cars, and also try not to have business in the affected area. Which leads in amazing fewer cars not in the streets and the left roads can handle the traffic better than a normal day. and by the way when a road is closed by accident and not by plan where propel are informed, then you see huge traffic jams
The second example he put forward seems similar to the prisoners dilemma. If everyone cooperates and takes the original routes it will take 30 minutes. Because the cars are split evenly the time it takes to travel down each traffic related road is 10 minutes. Say one car sees that if they take just the traffic related roads it will only take them 20 minutes. All the other cars see it takes less time so they all take that route. Now the traffic on both is at a maximum so both roads take 20 minutes to travel down. This means every car takes 40 minutes total to get there. The problem is if every car cooperates and takes the original routes it's the best for the group, but any one car can do better by taking the traffic related routes. Every car knows this so they all "defect" leading to a worse time than they would have had if they all cooperated. Though acting with the group is better, the temptation to defect is always there.
The problem I have with this and other so-called Braess Paradox examples, they always postulate some fanciful set of travel times that could never exist in the real world, and thus manipulating them is illustrative of nothing. Nothing is more obvious than removing a road can improve travel times, just as adding a road can increase them. Consider your favorite interstate. Now add a cross street and a traffic light. Pretty disastrous, right?
ClavisRa Please read the references in the video description--this does exist in the real world. The Braess paradox can be physically demonstrated with springs in a physics experiment. It has also been experimentally found in New York City, London, and Boston.
The concept behind the so-called paradox exists. The example of the paradox is terrible, because there is no road that has a fixed commute time, and there is no road that scales commute times by load with straight linear correlation. As, I said, if you want to understand why adding roads can cause traffic problems all you have to do is imagine adding a stoplight to a highway. Braess is bad analysis masquerading as principle.
+ClavisRa The example is simple so that calculations are simple, and in that way one can intuitively understand why the Braess paradox can happen. That example is quite 'stable': you could tweak the functions of these roads a bit and you would still get the "paradox". So even if the 'constant time roads' were tweaked and replaced by roads which depend on traffic - but only moderately, reflecting that these roads have a large capacity - one could still form an example fulfilling the paradox but which is more realistic. So I think you are misguided. It _is_ a principle, and one needs to be aware of it in road design (as well as in other types of systems). However, in the context of traffic flow, I would say that it is also quite a rare phenomenon. You need this situation of two routes consisting of very different capacity roads. I imagine that in general, the real world versions of this paradox are more likely psychological on the part of the drivers, who don't exactly know with precision which route is best for them (although, as satnavs and traffic monitoring improve and become more ubiquitous, this is becoming less far from the truth).
+ClavisRa Play Cities: Skylines... it has very sophisticated and realistic traffic simulation, and you will see this happen. Adding new roads and intersections can easily make the traffic worse on any roads you are hooking together, and you will see noticeable delays in services getting to those areas. Also, there are several real life examples they mentioned in the video, in New York, in Korea... I mean the thing in new york is actually really a kind of perfect example because the streets there are a square grid with alternate one-way avenues going in opposite directions, and you can see how going around it, and going with the relatively non-traffic affected avenues for an extra block in one direction or the other depending on which way you were coming from, would split the traffic in half and actually improve overall flow, when otherwise all that traffic would go right down 42nd street.. because they all want to save the 2 extra blocks of driving on the avenues...
+ClavisRa Mhm, I found this as an issue as well. It also doesn't account for everyone knowing the absolute fastest route, which I imagine would actually have a huge flux on which road drivers decide to drive on.
This is like supermarket express lanes, which end up taking way more time because everyone assumes they're faster and they end up more crowded than regular ones.
I don't think adding a time there matters. The new road already makes the second choice slower. Adding a time only makes it even more slow. Also I live in a state where our highway system is fairly good. I can switch highways very quickly and move from one to the next. It's not instant but the switch time is negligable
I don't know about anyone else but I would always take the roads with the guaranteed times. Not having to deal with traffic at all is totally worth any time that could possibly be saved.
To update to modern times where this a real time app telling you the quickest route and possible future apps that would actually direct the traffic(telling each car which route to take), very close to optimal flow could be achieved thus destroying the paradox.
It is always the conflict between your own interest and the total interest. If everyone in busy city's would choose an alternative road every other day that will take as long or a bit longer, the busy road will be way more efficient. But you cant count on other people to not choose the short term best personal solution
While I think your position has some merit, you just have to play cities: skylines for a while to figure out that more roads is not always the answer. You end up with more intersections, more swapping lanes, etc. You also have to factor in the direction of travel on the roads as well.
the most glaring flaw in these trumped up scenarios is they all assume everyone is going to try to take what would ordinarily be the shortest route which therefore causes a tremendous bottleneck, I however on the other hand when sensing a bottleneck will then opt to take the longer distance route which will take me even less time because all the other boneheads have made the wrong decision and created a tremdous traffic jam while leaving the alternate routes free for me to sail along on my merry way. Classic example of shortsightedness, but with modern day apps like Waze and other traffic data router planning you should be able to move more cars more efficiently by KEEPING all of the roads and allocating the load to balance all routes accordingly. That would be really SMART planning. Closing roads is not the solution, using the roads more effectively is!
I'll grant that Braess shows that closing and building new roads won't have a simple predictable outcome, but you should at least mention, if not mathematically figure in, the obvious other factors: 1) drivers won't all enter overly-busy-looking roads even if those roads often have given faster transit in the past. Once the T/10 roads in your example start to look clogged, drivers will start choosing the 20-minute alternate roads, thus speeding up the T/10's. 2) some drivers stick with roads they know, or deliberately like the scenery, or who knows what, so they violate the obvious choices. Lastly, how did you get 10 minutes for the T/10 routes in your 200-car example? There's no way to know how many cars were already on the various segments when the 200 new cars began to enter at the far left of the diagram. Seems to me the logical choice is to assume 50-50-50-50 for the new-car allocation on the 4 segments, and assume there are already about 50-50-50-50 other cars on those segments at the start. I.e., there will be about 50 cars on each segment when the first of the 200 cars enters from the left, then those 200 will spread out so that there are about 50 on each segment when the last of the 200 enters, etc. But even if not, what possible reason could there be to assume, as you seem to, that all 100 on the upper route will bunch together on its T/10 segment?
The New York example is not a fair example. The event was advertised and people had additional incentive to avoid the area as a result. However, if the road were permanently closed and people adjusted their driving habits and THEN the impact of the closure was evaluated....then you may have a case. but not before.
+Mike Smeding the idea is that with less roads means less intersections and less traffic lights which improves flow. Jams are caused more so by other roads traffic than that of the road you're driving on.
less roads with the same cars, means more packed road, means more time even if you have less intersections you will still have more packed roads, so more travel time. this is observed in scenarios when the people know that the road will be closed and many will otp to use the car that days in reality when a road is closed because of an accident do you see a better flow in the city? or worse?
I feel as though the point could be better displayed by making the constant road 21 minutes, that way taking T/10 is always better and the less than or equal to thing is less fuzzy, and it still causes the worse outcome for everyone.
Well if you close just about all the roads that lead into a city, then you are obviously going to improve traffic because although the few routes left open will be jammed up with traffic, the city will only have a slow trickle of cars within it so the actual city roads will fair much better. Also when you close roads that lead into the city people will have to take alternative options like working from home, public tranport, walking etc, which may enjoyed less but is ultimately better for everyone.
There are a few problems with this 1) you just removed lights from every street intersecting 42nd street 2) your case assumes that only 200 people will take the street that if there is only one car on the fixed street then it will be 20 min however one car on the variable street will take only 6 seconds 3) if a vertical street was added it will lower traffic 4) the same number of people don't travel the streets every day and if people knew that a main road was being closed may have taken a day off
the only problems i have with this is the assumption that only enough cars would use the t/10 road to make it equal, so of course it hurt and the assumption taking an entirely extra road would add 0 time.
A paradox is a statement that is self-contradictory because it often contains two statements that are both true, but in general both cannot be true at the same time. There is no need to discuss paradoxes in a factual discussion. Identifying paradoxical statements is what matters.
Love this paradox, but what if we consider that people may also want to travel between the 2 points of the new road? What if we factor that in as well?
2:43 or we could send 1 car at a time down the roads so 1 \ 10 = 0.1, so I can send 1 car down each road every 10 seconds so that there is always one traffic on the t/10 road. All cars would be done in 121.1 minutes though.
While the traffic *flow* probably did improve, I'm skeptical that the travel *times* improved. Additionally, the real world and game theory scenario are measuring two different (but potentially related) metrics. I suspect why traffic improved was because (as this game theory explanation does point out) the traffic will be more distributed; however, what this game theory explanation (and what I suspect was not taken into account) was ease of access to where you want to get. I suspect, that since people had to take detours. Let us remember, there were likely people who took wrong turns which could easily increase their time on the road. Further, I suspect these errors in navigation are the REAL reason why the volume of traffic would increase. Usually, when you make a wrong turn or miss a turn, you simply keep driving until you find a new easy place to turn and recalculate. Therefore, drivers who make wrong turns typically take the path of least resistance, where as drivers who have a set destination are willing to wait in quite a long queue (that you probably wouldn't be willing to wait in if you were simply wanting to turn around). Lets remember, what was said was that traffic *FLOW* was increased; but this is quite a meaningless statistic. The better statistic (and ironically, the statistic that the game theory scenario uses) is travel time on the road. Yet, this was NOT the statistic used in the real world scenario.
Yeah, that's what I thought as well. I was expecting the theoretical example to explain increased traffic flow by pointing out that if there is one best road, everyone takes that one. But if there are two equal roads, the traffic splits up.
Some might argue, (some have:) that the oversimplified example doesn't reflect reality. Indeed, there are many unrealistic details in the example, such as a '0' travel time for the connecting road, or the linear "weight" function of T/10 on some road segments. But ultimately, even if we could tweak the constants and functions to reflect ideal conditions, this would not change the fact that under certain circumstances ADDING a road will slow down traffic rather than speed it up. Let's assume that we have the same configuration as above, that the weight (travel time) function F1 we use instead of T/10 is realistic, and so is the weight/function for the connecting road, then: ...IF the minimum time for 1 car to travel the F1-dependent segment is still 0.1, AND the max time for that function is still 20, AND bear in mind that the alternative segment is a constant 20, THEN: 1) As long as the travel time X on the connecting road is less than 19.9, no (sensible:) driver would ever choose to take the 20 min. road connected to point A, (top 1st segment from A to B), even if some may take the one connected to B, (bottom 2nd segment from A to B), depending on the value of X. 2) A travel time greater then 19.9 would cause the connecting road to be unused, effectively removing it from the graph/network. Which illustrates the point: EVEN IN A NON-SIMPLIFIED, REALISTIC SCENARIO, ADDING A ROAD IN THIS CONFIGURATION SLOWS DOWN TRAFFIC, AND REMOVING IT SPEEDS IT UP.
Keep in mind that you're showing that adding a middle road increases travel time when using the parameters of a problem designed to show that adding a middle road increases travel time. This video is purely circular logic. The real-world examples mistake traffic flow with travel time, as well (that is, they don't explain how long it took for each car to get to its destination, which could have been longer than usual thanks to taking a longer route), and don't incorporate potentially mitigating circumstances (i.e. how many drivers were on the roads when the road was closed down, vs how many would be on the roads if there was no closed road; chances are at least some people decided to stay at home because of the necessary detour).
Having played Cities Skylines, I've learned first hand that more roads doesn't mean better traffic. It's not about road width or capacity, it's about FLOW. Grids are terrible for flow because every intersection requires half of the traffic to slow down or stop. Slowing down/stopping = traffic. Moving = not traffic. Fewer roads means fewer intersections means faster flowing traffic.
T/10 doesn't make sense, it means that if there's 0 traffic, it takes 0 seconds to travel it? If there's one car, it takes 6 seconds? It doesn't make sense. Here's how it should be: First off there should be an initial value, which would be the minimum time it takes to travel on it at max speed (e.g. 10 minutes, so => T/10 + 10). Then, we must realize that the correlation of traffic isn't directly proportional with time lost. The fonction should probably look more like a straight line on the initial value axis, and then half a sine wave, followed by another straight line, capped at a certain point. Very little cars on the road shouldn't increase traffic AT ALL, which is the straight line parallel to x axis. Then, at a point, the cars will start generating traffic, but not so fast, so the curve shouldn't go up fast at the start. Then, from a certain point, traffic escalates faster, the more cars we add the more traffic we get... And then, with even more cars, the traffic time shouldn't go up anymore, because there simply can't be more cars in front of you than what the road can take, so the cars added to the function would be BEHIND you, which wouldn't influence you AT ALL.
+Philippe Poulin You don't need to overthink it; the point was to simplify it so people could understand. Not everyone knows what things like sine waves are. Of course real roads are more complex, but this is intended to be a simple demonstration.
+Eugenio Garza That is not a valid comparison. What we are talking about here is simplifying a relatively complex function into a linear function so that people can understand it better, not using a completely different scenario.
MrHatoi I understand, but as for making it simpler, it was taken TOO far. Maybe there should be a middle ground as a PLAUSIBLE example that is simpler than the rest of the examples, instead of a mathtopia example. There are no roads that have a fixed time of transport, and there are no roads that have a t/x time of transport, and there are no roads that take 0 time to travel. Simple as that.
so if i put in T=1000 then what? no. people would now take 1000/10+20=120 minutes now they would rather use the connector to switch from the north road 20 to south road 20 taking only 40 minutes so in my case where T is overwhelmingly large the connector seems to cut the traffic time.isn't that correct?
+Vaibhav Shah That's only if the fixed routes can accommodate the higher traffic level. They were only set at 20 for simplicity. If you're increasing the sample size that much, you have to adjust the rest of the problem to match a realistic situation.
+Vaibhav Shah That's excactly the point of this video. Building more roads generally decreases traffic, but in some situations, that extra road can make matters worse.
so it's an occurence that happens only when the traffic is in proportion to the travel time.. a special case as i predicted (or retrodicted) earlier. thanks .
The only problem I have with this example is that it assumes there is some sort of teleporter exists at the midway point. If there is a teleporter than can instantly take one driver from one point to another, why not just have one going straight from A to B and cut everyone the extra 30-40 minutes! Problem solved! :)
Redraw the map more like a figure-8 on its side, with a bridge where the two roads cross. It's now the situation that building an interchange at the bridge will make things worse, not better.
you should of cited the chinese example where they closed down that enormous stretch of highway and turn it into miles of park, expecting the surrounding roads to become inundated with traffic a strange thing happened the traffic disappeared as if it was never there to begin with.
Consider more than 200 cars. Let's say 400. now the travel time in the first example is 40 minutes for both roads. The travel time in the second example is...40 minutes. What about 800 cars? That's 60 minutes for both roads in the first example and...40 minutes in the second example? This continues for any N number of cars. The travel time of of the first scenario is 20 + T/20 (since we're assuming half of T goes each way). The travel time in the second is actually a piecewise function where Travel time = min{T/5, 40} What has occurred is that the time for any given person has increased for traffic between 133.3... and 400 but reduced travel time for less than 133.3... and more than 400. Simply saying that traffic is now worse is naive as the flow of cars per hour has now been astronomically increased. What do I mean by that? Consider for a moment that the number cars on any route from A to B is dependent on what each driver expects to spend on the road waiting. This is a cost. If the value of going from A to B is less than how much they value their time, a driver simply won't drive. In the first example, the total number of cars on the road is equal to the equilibrium point of people willing to wait in traffic at a given number proportional to their willingness to wait. While this equilibrium heavily relies on this exact distribution, let's just say that we can describe it with this function: T(wait.time) = 2,000 - 20*wait.time We now set this function equal the inverse function of scenario 1: T=20*(wait.time-20) Solving for this gives us wait.time=60 and 800 people on the road (400 each way.) Now consider the second scenario where you will spend at most 40 minutes in traffic. How many people are willing to wait 40 minutes in traffic? 1,200. In the first example, 800 people were flowing through. In this second scenario, 1,200 were now flowing through. But that's unrealistic as equilibrium here works just like in economics. Let's consider another pair of scenarios where the equilibrium point was 200 cars and 30 minute traffic. Then the distribution could look something like this: 200 = 5000 - 160*30 -> T(wait.time) = 5000 - 60*wait.time Using the math from before, 200 people under the second scenario will lead to 40 minute wait times. But plugging in 40 minutes leads to a negative number. Thus, less than 200 people will be on the road. How many is the equilibrium point? We can form a composite function to find out. wait.time = T(wait.time)/5 = (5000 - 60*wait.time) / 5 -> w = 1000/33 ~ 30.3030... We can see that the total traffic now is slightly less and the total wait time is slightly more. Let's consider one more example where demand is significantly more inelastic: T(wait.time) = 260 - 2*wait.time Then: wait.time = (260 - 2*wait.time)/5 -> w = 260/7 ~ 37.14 What's going on here? In both scenarios, the wait time is greater, one is significantly greater. Well, it's simple. More people are competing for the same road. Competition between consumers drives up prices. Let's consider one final scenario where the routes have the same wait.time functions as before but they cannot communicate i.e. no mid road. Then the two routes will be: 20 + T/10, T/5 If we set them equal to each other with the highly elastic distribution, we get: 20 + (5000 - 160*wait.time)/10 = (5000 - 160*wait.time)/5 wait.time = 30 We then conclude with more substitution that the number of cars will become 250 (100 left and 150 right). 50 more cars for the same cost of travel time. The simple fact that the routes competed with each other and overlapped is what drove not only the wait time up but the number of cars down. The conclusion of this all should be obvious: The way to make an iphone cheaper is not to force consumers to fight over them; the company needs to make them cheaper. The way to maximize flow while minimizing time in traffic is to improve already existing roads, not make ones that force drivers to compete.
Game Theorists are the funniest math folks to listen to. Do people know the functional time constraints on each pathway? And also you made the cross road instantaneous. It seems overly idealized. However the idea that a group of people can make individual decisions that add up to a bad result for them all is very intriguing and quite possible. It seems to contradict some basic premises of economics wherein it is claimed that self-interest always finds the most efficient solution.
+Larry Cornell Nash Equilibrium covers this as well, and it applies to investing. Selfish choices by everyone often causes less than optimal results, particularly in economics. It's not as basic as just supply and demand, which is all what most of us ever learn in high school. Unfortunately, since many people assume economics is that simple, they are often swayed by fast talking heads on news programs or politicians making a bummer of a deal sound good.
It is likely that as, fascinating as game theory is, it also does not explain everything. I knew from the first day in my first economics course that supply and demand were only part of the issue. Have you ever mapped the first and second differences of oil price? I was able to predict the oil price spike of '08 doing this. I'm not dissing on my the post or poster, I meant to say funnest but auto-correct beat me to it. The problem is that at times we do present to the masses things that are overly simplified, but these simplifications rob the listener. And on the other hand most of modern mathematics makes almost no effort to clearly explain its work. This is a sad thing because most papers are coming off at some high-level of abstraction that would be understood by more people if someone took the time to give good examples which drill down to the foundation. Abstract algebra and topology are probably the perfect examples of this. I keep my hopes high that the advanced sciences will invest some time and money in actually clarifying step by step how to apply some of the great knowledge that is being obtained. I am afraid that instead of raising the human race to a higher intellectual state we are creating silos of information which will not be conducive to the stated ideal of educating the masses. In short, I think we need to do a far better job of using the Internet to create knowledge trees and rich examples. So in that I applaud Efforts like these to put interesting mathematics out there. I also wish we in science were less self-centered, because just as with the marketplace where self-interest on Wall Street is leaving the world a burned out husk, self-interest in academia is surely making it harder for people to understand and enjoy mathematics. I don't believe that only a person with a Phd in mathematics can understand elliptic curves, it's just that almost no good popularizing descriptions exist. Maybe we disagree on this but I think there should be a field in math matins, a degree focus called math popularization that works hard to take all results and make them as easy to understand as possible, disseminating that information through multimedia content on the web. It almost seems to me we should admire physics for making truly impressive efforts to make its modern discoveries a part of the everyday language. The word "quantum" is now a meme. Just some thoughts.
sevret313 i can see why one would say it and i suppose it depends on where you live. only brought it up cause there's been "recent" headlines about the competition between two of the major companies.
What I don´t get here is: If they now for a fact that closing certain roads would improve the overall traffic, why don´t they close them permanently for anyone that isn´t i.e. City Maintenance, Law Enforcement, Fire Departement, Ambulance, Delivery etc.?
"This logic holds true for every driver, and therefore we can conclude that all 200 drivers take the same route." There are so many problems with that statement--the biggest one of which is assuming that all 200 drivers are capable of using logic. Anyone who has driven ANYWHERE knows that no one uses logic when driving.
But what if I'm driving on the example roads and there are no cars? Would using the traffic dependent roads be instant? Also, why is the middle road automatically instant?
+Matthew “MadMatt” Younce that's my issue with the problem. it's foundation is based on the fact that a) the traffic is "constant" and b) drivers know the given traffic route. what i thik is going on is that in the original part, each route is taken up by a constant time and variable time. and in the end since both routes are even, traffic is then divided evenly. in the second set, you have 2 steps. travel time to the junction and then to the next city. and what is then created is you have a path a-a'-b. and it's set so that either path to a' is the same. 20 fixed or (200)/10 = 20. and then you repeat. so i claim this isn't a paradox in that when you change the rules of the game you can't compare the before and after.
+Matthew “MadMatt” Younce the roads with set time (20 mins) are supposed to be high traffic roads (multiple lanes to minimise eliminate traffics influence) the other roads which are influenced by traffic (T/10) are supposed to be urban roads so there would almost always be traffic (like his example new york city, you could get from top to bottom in minutes but because of traffic it's more likely that it would take at least an hour and it's the traffic on intersecting roads not the traffic on the road you are on. The link road was given a value of 0 just for demonstrative purposes he could have put a value onto it but it would have made explaining the paradox more complicated. If you wanted to look into it properly you would have to take junctions into account because they are what causes the jamms. removing roads also removes junctions which means there's less traffic lights to hinder flow.
James Mulchrone Well, you really know how to get a point across. And I see your point. But still, let's say hypothetically that I'm driving on one of the theoretical traffic dependent roads and there are no cars for some reason. Would my travel time be instant?
Matthew Younce Thank you, I try to be as coherent as possible. It wouldn't be instant because if you were driving on the road there's at least one car to be added to the value of traffic (yours) and again it's a massive simplification just to help people understand the concept.
+Nick Reeser Think of it as two parallel (or at least non-intersecting) highways with exit/entrance ramps crossing over one another (like you might find in many major cities in the U.S.). The difference in time it takes to travel over the ramp versus continuing on the same highway is negligible.
The problem with traffic is that many people think it acts like a liquid: if you give it more roads, it will spread out more thinly over that area. The reality is that traffic is like a gas: it will expand to fill all spaces it is given. Consider the following: in an old country village with zero roads, how much traffic will there be? That's right: zero! because nobody will drive anywhere.
What kinda shitty logic is that? There will be dirt roads because every one travels. This paradox doesn't apply every where. In most cases, a new road will improve the flow.
There are so many problems with your example. First off, one reason traffic improves with road closures is because it removes a set of traffic lights, and, assuming New York uses one way systems, encourages people to take the one way systems up and down to get to their destination, rather than up and down, left and right, down 42nd street. Okay, now as for your example - first of all, no road is going to be guaranteed a travel time. Even if you can average it out, t/10 doesn't make sense, as if you're the only car, it takes only 6 seconds to travel it, which is inaccurate. In fact, the travel time will be, in general, the time it takes to travel the road at the speed limit, up to capacity, after which traffic will slow down exponentially with each new car entering the road. Secondly, unless the two routes are the same, no connecting road will take zero minutes, it will take some amount of time. Thirdly, people will take the route they think will be the shortest time. When they can see traffic on the routes, they will take the route that has the shortest time with the amount of traffic estimated to be on it. So if at point A, a driver sees all 200 cars taking the southern route, they will take the northern route.
You do realize most of your criticisms arise because the narrator wanted to keep the example simple and decided to use ideal conditions. He could have made it several times more complex and even factor in the likely road choice of drivers depending on the traffic. That would be cool and all but it's suppose to be a simple example not something you want a computer to process.
It does it's job quite fine. I've seen a lot of science that starts from simple formulas and then they add in more variables to make the problem more realistic.
You do realize that this paradox doesn't happen every time you add a new road, right? there are certain situations, combination of roads that leads to this. May be more people make left or right turns, which in some way makes the traffic worse.. I'm sure every situation has a different reason.
I was hoping they were going to be specific showing where cars actually ended up driving instead rather than modeling a hypothetical based on assumptions.
Sort of related, why does flow through an intersection often improve when the traffic lights fail? Assume a cross intersection. I *think* it's because drivers giving way to each other and trying to be fair make more efficient use of the intersection and can pack more cars into the intersection than is allowed by the lights system. Peak transit velocity is lower but vehicles per second is higher due to close interleaving. Alternatively a car at an intersection with lights spends say 1 minute waiting and then 5 seconds transiting. i.e. 65 seconds to cross the intersection. With no lights the car spends say 10 seconds waiting for a gap and 10 seconds crossing or 20 seconds to transit.
Im not sure if this "often" is actually verified. But just using logic, it would be because traffic lights can not perfectly respond to incoming traffic. Just as sometimes lights are green with no traffic while packed cross streets have to wait. At a broken light it becomes alternating cars go, meaning that some of the congestion will be constantly relieved. However, while congestion may diminish, travel times go up, because you have to come to a complete stop at every intersection. Traffic doesn't always equate to longer travel times, long as there is enough space for every car and buffer zones, it could be as fast as no traffic at all (ex: interstates)
@@jaojintalonis92 also at a traffic light, often 5-10 people in line get through on green. The OP forgets to factor that in when he’s talking about everyone waiting their turn at a broken light. Dunning Krueger strikes again.
Sorry I'm late to the party. Yes, I agree that this one case is theoretically faster. Just because of that doesn't make it a good model that explains why these real world cases happened. The model relies on the maximum value of T/10 being the same as the constant travel time of 20. If you change that to 10, there's no more paradox: Even split of traffic (T=100) 10 + T/10 = 20 T/10 + 10 = 20 All inner (T=200) 10 + T/10 = 30 T/10 + 10 = 30 T/10 + T/10 = 40 10 + 10 = 20 Since the T/10 + T/10 route is no longer the fastest, no one would take that, and everyone would take the 10 + 10 route, which would then be the highest travel time possible, with lower times possible as people take the 10 + 10 route, thus reducing the T count on other roads. But a much better model of the real world examples would be to insert not a cross road, but a third road going directly from A to B. It is an interesting paradox, but it requires contrivance of the numbers and the direction of the roads.
This example is ideal. He could have factored in other variables based off of real roads but that would get complex fast and is not necessary to understand the paradox. In the real world these roads are not necessarily straight and may have other roads feeding into them. Secondly you cannot always make a straight line between two points so your line from A to B isn't realistic. Thirdly because road networks can get complex over time a single new road designed to improve flow between two points may limit flow for other points. Fourthly you said it is interesting but it involves contrivance of the numbers and the directions of the roads. Those two things are how roads actually work so they cause the paradox, they do not take away from it. Finally It is a paradox of basic human reasoning not math.
1. My point is that as the model gets more complex (and thus more like real life), it's harder for the model to exhibit this paradox because the numbers are contrived. That doesn't mean it can't happen, just that this phenomenon is uncommon. And absolutely, the title of the video is "can speed up traffic," so I don't have any problem with that. 2. I was adding a point to the model, but in reality we're talking about removing an already existing road, so to provide the real world example of 42nd street, 42nd street would be the direct route from A to B, and we're comparing 42nd street open to 42nd street closed. 3. Yes, it may limit flow, but I doubt it's likely. A couple documented cases in history exist, but I'm sure all of us experience road closures often, and I have yet to see a scenario that has reduced the length of my trip. 4. In my example, I tweaked the numbers just a little bit, and the whole thing fell apart. If you're arguing how the depicted traffic flow is more realistic than mine, please tell me why you think so. I believe my traffic flow is more realistic because the only kind of road that would produce a constant time of travel is one with many lanes and no traffic lights. A highway completely independent of traffic (still not feasible, but theoretical). That would never ever take as long as the (T=200)/10 maximum traffic delay scenario, that, based on the arrangement of the variables given, would have to be the same mileage as the opposite side of the parallelogram. 5. I agree here. "Paradox" sounds better than "Counter-intuitive," however.
If 100 cars would travel straight the north path and 100 cars would travel the south path then switch to the north path, then 100 cars will have a travel time of 10 mins., while the other hundred will have a travel time of 30 mins. The 100 cars that took the south path then switched to the north path will arrive to B as the 100 cars that took the north path finishes the 20 min path and into the T/10 path
If the new road were built directly from A to B though, then the volume of traffic per road would be reduced, and thus travel time would have been cut.
So in a less-mathy terms, before the free road, both routes were equally attractive, so the traffic load was shared evenly between the two routes. But the addition of the free road made one route more attractive to the drivers, so that route became overloaded and thus became slower.
Why would drivers switch from the southern route to the northern route if switching from one route to the other would take additional time. This example is flawed.
+NTG The drivers "selfishly" do what is best for themselves as individuals; they don't team up and come up with the best distribution amongst themselves before departing.
Was this not first discovered by Sam Schwartz (AKA Gridlock Sam) in the 1970's? Perhaps the theory wasn't fully developed at the time, but I'm quite sure he was shutting down roads in New York City in order to improve traffic flow
I was thinking you put a line in the center from A to B it would take 30 minutes to get across and all 200 drivers would take it. if you were to shut that down there would be the other 2 roads where there are 100 on each road and it would still take 30 minutes. It doesn't matter about the time because this is about Traffic not effective time
Maybe just me, but I thought this video was going to be about how closing a road can speed traffic up, not how creating a new road can slow travel time.
Not saying it's wrong, just don't 100% "get" why this is the answer to this particular situation. This shows how closing "a" road can possibly make traffic better, but do these real-life roads fit this model? I don't see how, but maybe. I would kind of think one explanation could be along the lies of: "If you have one clearly optimal, straight-shot road that gets you from A to B (and let's say A to B is the heaviest traffic flow around the city at this given time -- lots of people in the area of A, and many of them wanting to get to B), then practically everyone at A is going to take that road, and they will become jammed. If that one single "perfect" road is closed, then everyone has to decide among several roads that are a bit off-course, but none significantly shorter or better than the others. Odds are, they will split up evenly among those routes, and won't get jammed as badly, if at all."
+Ace Diamond Your explanation is correct. The second part of your explanation, with the main route closed, is very similar to the first portion of the video. The beginning of your explanation is similar to the second portion of the video, after the new road has been added.
I suppose that makes sense. I guess I was just expecting more for them to add a third road that was a straight shot from A to B. But the more I think about it, yeah you're right, it's essentially the same thing.
The road which depends on traffic will always cause more time for more value of "T".. so why not people prefer the constant time road if they know the traffic is more...🤔
sarcasmo57 Have systems set up that aid in decelerating and accelerating traffic in unison, which could also be coupled with a system that keeps a shorter distance between cars. It is getting the system to work more like a train or a rope being pulled rather than a staggered and choppy stop starts and stops for those in the middle and end of a traffic jam(especially). Also, if absolute stops were needed, there would be a system put in place to park cars more closely together on the roadway (not to extend the traffic jam any further back then necessary). Although too close together could also slow traffic starting back up in unison, but could be aided some by one car moving in rapid succession after one another after a minimal distance is reached between 2 successive cars. Now there would need to be systems put in place to offset lane mergers and on ramps. Speed and distance would be locally adjusted to efficiently allow vehicles to merge, for instance. Also, there would be communication between cars so that if say a large object fell off of an overpass or off of something being towed that all cars would move away from it in both directions (utilizing shoulders if necessary) in unison like a school of fish moving from a shark attacking the center, and then would regroup on the 'other side'. It could also alert cars further back to be adjusting speed and moving to different lanes if necessary well before the road hazard would be visible or brake lights would typically be seen by cars in front of you. An occurrence like this would normally cause a rapid stop and long delays, but could be optimized to only cause minimal slow down. This is all theoretical and would need to be adjusted for cars with different accelerating and decelerating and handling capabilities, but would work optimally if all vehicles had a certain minimum acceleration, deceleration and handling capabilities. The biggest threat to this on an interstate would be large trucks that accelerate much slower. A partial solution for this would be to have one truck lane added 2nd from right that would allow for easier merging from onramps and allow if stopped, trucks to keep a certain minimum distance from each other at the stop, and then all slowly begin to accelerate in unison together, while allowing lanes left of them to move freely at normal rates and allow those entering from the right to possibly accelerate past the trucks and otherwise there would be slight increases or decreases of speed to allow cars to enter between trucks and gain access to the other lanes. The 'truck lanes' would be used minimally by cars and could possibly be opened up if few trucks were present(although that would complicate things) and allow cars to smoothly be able to merge into other lanes once approaching a truck a certain distance away. Ideal number of lanes would be 4 or 5, but if say only 3, trucks would mainly be in the middle lane with a larger gap between them that would allow for both slower stopping and also allow for cars to pass between the 1st and 3rd and vice versa.
That relies on every driver knowing what every other driver is doing and also taking a decision for the benefit of the whole group and not for themselves. It would never happen.
Your first example assumes that the theoretical travelers follow an even distribution on purpose because they know it will work out. Your second example assumes every traveler makes an informed mathematical decision based on traffic flow. The last time I was in traffic, a sunburnt man in a wifebeater and a mullet was trying to balance a soda can on his knee. I understand that this is a simplified example that assumes perfect logic, but... that concept is nonexistent in real life.
that new road has a travel time of 0? what kind of magic is this? it's also not a paradox that trying to force all traffic into a single path will make things worse.
You compared 100 cars to 200 cars. The variable of traffic flow was still the only part that made a difference. The third road is irrelevant in your comparison I think, unless I missed something. I'm no genius either so it's very possible I did miss whatever makes this a "paradox".
When is the last time anyone built s road with *_ZERO_* travel time? Your mid connector would add 10 min travel time to all who use it, therefore few will choose that route.
+Insanity cubed Indeed, if the time you need to travel from A to B depends on the amount of cars on it, the ideal way of course is to open the road again that connects the two halfway points and let everyone only walk to their destination.
The last statement is the only one of merit on a video about traffic. Roadways and roadway system are extremely complicated and take into account much more then just simple capacity analysis... If that's what you could even call your paradox. As a civil engineer who spends time studying and designing roadways, travel time itself is usual not a big factor. Safety is the most important. So when lanes or roadways are added for capacity, it is because the existing system was overtaxed, which is the main cause of accidents. Most design standards in our state allow for a level of service D, in peak hours (that is-obstructed flow, and a decreased speed from free-flow). Interesting theoretical math problem none the less, but not practical.
That.... doesn't seem like a paradox. That doesn't seem like much of anything to me... Put an uneeded road in and it can mess some things up. I can get that (intersections and whatnot) But it say's nothing about closing down a road increasing traffic.
No duh. There are metered ramps to freeways in attempt to prevent freeway clogging to the point of crawling and stopping all traffic, including all traffic on all on ramps. Metering an on ramp is stopping traffic there for the overall good. Clearly any purposeful stoppage, like stop signs and traffic lights, as long as they're set up well, fit into this overall idea that stopping some flows helps overall flows.
At first T/10 is not valid time road can take for a driver and therefore it is not valid example because roads are based on throughput so drivers that join the road do not affect drivers that are on the road wich means that when the driver decide if he join the top one or the bottom one he will choose the final time as waiting time + time the road take if it's empty So... make a meaningful example
RUclips comments are so math illiterate it's not funny. Thanks for these vids! I really enjoy them. I'm also a fan of Numberphile. I appreciate the no nonsense approach.
underwhelming, as a fan of "city-building" type computer games this paradox barely scratches the surface of what it takes to lay out a road system ( In a game! ) That said i've often times looked at satellite footage from google earth for inspiration, In this sense- i find this example particularly amusing :)
There is another similar "paradox" when you add a third, "central" path connecting A and B but not intersecting the other two paths. In this case, traffic can actually increase in ALL three paths. The idea is that total traffic is not fixed: there are some people that travel from A to B by other means: train, bike, on foot. However, if the third path opens, they might switch to car driving. In the short run this might not be so big an issue, but then some people that did not use any of the paths might buy a car and start using them. And traffic ends up going up everywhere! This is usually called the problem of "the supply generates its demand".
ill guess that closing a main road that is overused will cause people to use alternate routes that can handle the extra traffic.
why can't we upvote on phones?
Reading this was much quicker than the video.
"ill guess that closing a main road that is overused will cause people to use alternate routes"
Good so far.
"that can handle the extra traffic."
My question would be: if they can handle the extra traffic, why aren't they being used already? In the example in the video, it's because the other route is longer, which I guess is plausible because main roads usually take you fairly directly to where you want to go, while finding side roads that have a route all the way through to where you want to go may involve a less direct path. (On the other hand, if the main road has a higher speed limit, people may take a less direct path in order to use it.)
tifforo1 Time. The main route would normally be the quickest way to get to your destination, but when there is a backup on your normal route, taking a route that is a bit longer will still be quicker than waiting in the traffic.
if there are alternatives
not all cities have alternatives for some roads.
and this only works when people know the road will be closed and they opt out to use the car, in reality when a road is closed by accident? you have huge traffic jams. why in reality we get traffic jams?
because the paradox is bulshit
Often times closing a road carries a tradeoff whereby the closure of the road does in fact increase travel time but with the tradeoff that traffic is more spread out among multiple routes.
This reminds me of the Prisoner's Dilemma, where everyone is taking what seems like the best choice for themselves and everyone ends up worse off.
That's the tragedy of the commons, not the prisoners' dilemma.
That's because it involves the same principle.
Braess' paradox is a fancy prisoner's dilemma
+Jeremy Corson That's because they both come from the same area of mathematics: the study of logic and decisions known as game theory. It's not quite the same as the prisoner's dilemma, but you're on the right tracks and I see where you're coming from.
+SamBrev Actually I agree with Jeremy, the link is not just that they come from the same area of mathematics, but also that all drivers would actually be better off if they made a contractual agreement to not use the fast lane and to evenly divide their route. For the same reason the prisoners are better off contractually agreeing not to rat on one another... But as soon as they have agreed not to rat, by the rule of never playing a dominated strategy... You are better off ratting.
Specifically, it concerns the route that the prison bus takes between facilities.
+tibschris Pah oh very good ;)
tibschris hahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha! VEEEEERY GOOD!
Interesting. Was reading an article about how GPS might become communal one day to optimise traffic flow for everyone. Never studied this area of maths, but a very interesting insight into how the computer might make some of the decisions!
if everyone had perfect logic like this suggests then nobody would switch routes and keep travel times between a and b at 30 minutes
Conor Bracken no, because the cars cant communicate
this will be if everyone is not egoistic, if everyone split like before ok that every car will go in 30 minutes, but so a single person would say "wait, if i go here i will spend only 20 minutes by using that road" and so if everyone use this logic, that roads will become full again and the travel times become once more 40 minutes
My first thought was "Well if there's no road to have traffic on, there'll be no traffic at all!"
Judging from the comments, I think someone needs to post an example up using Cities Skylines. This isn't just theoretical, you could make an actual road that has this behavior.
Can someone please explain me what the Zero time road is? I am from South America (3rd word) and we only have regular old school roads. I hope some day we will have NEAR zero time travel roads (speed o light) but ABSOLUTE ZERO time travel... damn guys, first world is awesome!
It means the picture is not to scale, and that road is very very short. It does not take zero minutes to travel, but very little time, a few seconds at most.
He would have done better to draw something that was almost an infinity symbol, then with the new road it would become the infinity symbol. The 'new road' would then have been a new junction linking the two different roads at the point they came nearest to each other in their middles, and said junction would then allow changing from one road to the other mid-journey.
I find it hard to believe it costs nothing to switch routes.. Just the turning and decelerating and accelerating increases your time, but you've built a road that's instantaneously teleports you on the other side.
A lot of assumptions were made in this example. Why does one road have a fixed time, while the other has a variable time?
consider the 1st as a very wide road without lights that cars never make jams there
highway is a very wide road without lights but it surely has jams
a true highway has no jams; if it has then the country/contructors did it wrong
A highway is not supposed to have jams but it most deffinitely does. Also, in this video, he is not talking about highway, he is talking about city streets.
"A true highway has no jams", but most of the highway that exists today can not be qualified as "true highways" I think... Highways on their own does not have jams, but the road network connected to them has, and the jams just extend onto the highway
In this world with perfect logicians, they should theoretically see the problem and thus, never use the pass through road
The drivers are acting under game theory, where they all do what is best for them. The tragedy of the commons is a famous example of how this goes wrong. They do not act in the best interest of the group, only themselves, which makes everyone in the group worse off.
You can make a payoff matrix and come to the conclusion that leads to the tragedy of the commons, but if you make a second payoff matrix to decide whether you should use the first payoff matrix or not, you'll end up choosing the optimal route and avoid the link.
I don't quite get what you mean, DerLamer. How would a going another layer deep with a payoff matrix change your actions?
It seems like it is still as fast or faster to take the link. Why would a rational driver not take it if they were only acting in their own self-interest?
Greg Erlandson
It's about the realization that considering any of the newly opened options leads to a worse outcome for everyone. You can consider considering them, reject that and take the optimal solution, which is acting as if nothing had changed. In a world of truly perfect logicians, they'd realize this as well, realize that the others would also come to that conclusion, and everyone would choose to be blind to the link. These are perfect logicians operating at one level higher than ordinary perfect logicians who only use straightforward payoff matrices.
+DerLamer this is the tragedy of the commons, as I mentioned earlier. Although you're right that a perfectly rational logician would realize that they make travel times slower for everyone by taking the new route, they don't care. They only care about their own travel time.
You're making the additional assumption that 1) they are perfect logicians and 2) they care about other people.
In the situation presented in the video, the drivers would say "screw everyone else. I'm going to take the new route if it makes MY travel time even 1 second faster and it makes everyone else's total time an hour slower."
If you make different assumptions, then you're solving a totally different problem.
This may sound simplistic but say you have intersected roads such as 42nd street and 5th avenue in new york city. If 42nd street is closed traffic on 5th avenue, which intersects 42nd street, will improve because drivers on 5th won't have conflicting movements with vehicles on 42nd street. When the signal is red for 5th (green for 42nd) drivers on 5th can proceed through red lights (assuming pedestrian conflicts are halted as well)
In the NY example, there are many other possible factors. Maybe people knew the road was going to be closed and took a different route completely, or didn't go where they normally would have gone. Or left their car at home that day. Or one of dozens of other factors.
Or they simply were divided all throughout the other streets instead of all gathering to the big one that covers 3 big points.
i agree when in my city there will be a close of road,i know many people will not use their cars, and also try not to have business in the affected area. Which leads in amazing fewer cars not in the streets and the left roads can handle the traffic better than a normal day.
and by the way when a road is closed by accident and not by plan where propel are informed, then you see huge traffic jams
The second example he put forward seems similar to the prisoners dilemma. If everyone cooperates and takes the original routes it will take 30 minutes. Because the cars are split evenly the time it takes to travel down each traffic related road is 10 minutes. Say one car sees that if they take just the traffic related roads it will only take them 20 minutes. All the other cars see it takes less time so they all take that route. Now the traffic on both is at a maximum so both roads take 20 minutes to travel down. This means every car takes 40 minutes total to get there. The problem is if every car cooperates and takes the original routes it's the best for the group, but any one car can do better by taking the traffic related routes. Every car knows this so they all "defect" leading to a worse time than they would have had if they all cooperated. Though acting with the group is better, the temptation to defect is always there.
The problem I have with this and other so-called Braess Paradox examples, they always postulate some fanciful set of travel times that could never exist in the real world, and thus manipulating them is illustrative of nothing.
Nothing is more obvious than removing a road can improve travel times, just as adding a road can increase them. Consider your favorite interstate. Now add a cross street and a traffic light. Pretty disastrous, right?
ClavisRa Please read the references in the video description--this does exist in the real world. The Braess paradox can be physically demonstrated with springs in a physics experiment. It has also been experimentally found in New York City, London, and Boston.
The concept behind the so-called paradox exists. The example of the paradox is terrible, because there is no road that has a fixed commute time, and there is no road that scales commute times by load with straight linear correlation. As, I said, if you want to understand why adding roads can cause traffic problems all you have to do is imagine adding a stoplight to a highway. Braess is bad analysis masquerading as principle.
+ClavisRa The example is simple so that calculations are simple, and in that way one can intuitively understand why the Braess paradox can happen.
That example is quite 'stable': you could tweak the functions of these roads a bit and you would still get the "paradox". So even if the 'constant time roads' were tweaked and replaced by roads which depend on traffic - but only moderately, reflecting that these roads have a large capacity - one could still form an example fulfilling the paradox but which is more realistic.
So I think you are misguided. It _is_ a principle, and one needs to be aware of it in road design (as well as in other types of systems). However, in the context of traffic flow, I would say that it is also quite a rare phenomenon. You need this situation of two routes consisting of very different capacity roads. I imagine that in general, the real world versions of this paradox are more likely psychological on the part of the drivers, who don't exactly know with precision which route is best for them (although, as satnavs and traffic monitoring improve and become more ubiquitous, this is becoming less far from the truth).
+ClavisRa Play Cities: Skylines... it has very sophisticated and realistic traffic simulation, and you will see this happen. Adding new roads and intersections can easily make the traffic worse on any roads you are hooking together, and you will see noticeable delays in services getting to those areas. Also, there are several real life examples they mentioned in the video, in New York, in Korea... I mean the thing in new york is actually really a kind of perfect example because the streets there are a square grid with alternate one-way avenues going in opposite directions, and you can see how going around it, and going with the relatively non-traffic affected avenues for an extra block in one direction or the other depending on which way you were coming from, would split the traffic in half and actually improve overall flow, when otherwise all that traffic would go right down 42nd street.. because they all want to save the 2 extra blocks of driving on the avenues...
+ClavisRa Mhm, I found this as an issue as well. It also doesn't account for everyone knowing the absolute fastest route, which I imagine would actually have a huge flux on which road drivers decide to drive on.
This is like supermarket express lanes, which end up taking way more time because everyone assumes they're faster and they end up more crowded than regular ones.
Why wasn't a time added for the new road? Or is it a light speed transportation point?
Better than lightspeed.
I don't think adding a time there matters. The new road already makes the second choice slower. Adding a time only makes it even more slow.
Also I live in a state where our highway system is fairly good. I can switch highways very quickly and move from one to the next. It's not instant but the switch time is negligable
I don't know about anyone else but I would always take the roads with the guaranteed times. Not having to deal with traffic at all is totally worth any time that could possibly be saved.
To update to modern times where this a real time app telling you the quickest route and possible future apps that would actually direct the traffic(telling each car which route to take), very close to optimal flow could be achieved thus destroying the paradox.
It is always the conflict between your own interest and the total interest. If everyone in busy city's would choose an alternative road every other day that will take as long or a bit longer, the busy road will be way more efficient. But you cant count on other people to not choose the short term best personal solution
While I think your position has some merit, you just have to play cities: skylines for a while to figure out that more roads is not always the answer. You end up with more intersections, more swapping lanes, etc. You also have to factor in the direction of travel on the roads as well.
the most glaring flaw in these trumped up scenarios is they all assume everyone is going to try to take what would ordinarily be the shortest route which therefore causes a tremendous bottleneck, I however on the other hand when sensing a bottleneck will then opt to take the longer distance route which will take me even less time because all the other boneheads have made the wrong decision and created a tremdous traffic jam while leaving the alternate routes free for me to sail along on my merry way. Classic example of shortsightedness, but with modern day apps like Waze and other traffic data router planning you should be able to move more cars more efficiently by KEEPING all of the roads and allocating the load to balance all routes accordingly. That would be really SMART planning. Closing roads is not the solution, using the roads more effectively is!
I'll grant that Braess shows that closing and building new roads won't have a simple predictable outcome, but you should at least mention, if not mathematically figure in, the obvious other factors: 1) drivers won't all enter overly-busy-looking roads even if those roads often have given faster transit in the past. Once the T/10 roads in your example start to look clogged, drivers will start choosing the 20-minute alternate roads, thus speeding up the T/10's. 2) some drivers stick with roads they know, or deliberately like the scenery, or who knows what, so they violate the obvious choices.
Lastly, how did you get 10 minutes for the T/10 routes in your 200-car example? There's no way to know how many cars were already on the various segments when the 200 new cars began to enter at the far left of the diagram. Seems to me the logical choice is to assume 50-50-50-50 for the new-car allocation on the 4 segments, and assume there are already about 50-50-50-50 other cars on those segments at the start. I.e., there will be about 50 cars on each segment when the first of the 200 cars enters from the left, then those 200 will spread out so that there are about 50 on each segment when the last of the 200 enters, etc. But even if not, what possible reason could there be to assume, as you seem to, that all 100 on the upper route will bunch together on its T/10 segment?
The New York example is not a fair example. The event was advertised and people had additional incentive to avoid the area as a result. However, if the road were permanently closed and people adjusted their driving habits and THEN the impact of the closure was evaluated....then you may have a case. but not before.
Logical!!!
+Mike Smeding the idea is that with less roads means less intersections and less traffic lights which improves flow. Jams are caused more so by other roads traffic than that of the road you're driving on.
less roads with the same cars, means more packed road, means more time
even if you have less intersections you will still have more packed roads, so more travel time.
this is observed in scenarios when the people know that the road will be closed and many will otp to use the car that days
in reality when a road is closed because of an accident
do you see a better flow in the city? or worse?
I feel as though the point could be better displayed by making the constant road 21 minutes, that way taking T/10 is always better and the less than or equal to thing is less fuzzy, and it still causes the worse outcome for everyone.
So basically when you add an extra road, you are turning 2 different routes into a single route, so it is slower
Only because the differences in the roads are artificial and illogical to begin with….
Well if you close just about all the roads that lead into a city, then you are obviously going to improve traffic because although the few routes left open will be jammed up with traffic, the city will only have a slow trickle of cars within it so the actual city roads will fair much better.
Also when you close roads that lead into the city people will have to take alternative options like working from home, public tranport, walking etc, which may enjoyed less but is ultimately better for everyone.
There are a few problems with this
1) you just removed lights from every street intersecting 42nd street
2) your case assumes that only 200 people will take the street that if there is only one car on the fixed street then it will be 20 min however one car on the variable street will take only 6 seconds
3) if a vertical street was added it will lower traffic
4) the same number of people don't travel the streets every day and if people knew that a main road was being closed may have taken a day off
the only problems i have with this is the assumption that only enough cars would use the t/10 road to make it equal, so of course it hurt and the assumption taking an entirely extra road would add 0 time.
A paradox is a statement that is self-contradictory because it often contains two statements that are both true, but in general both cannot be true at the same time. There is no need to discuss paradoxes in a factual discussion. Identifying paradoxical statements is what matters.
Love this paradox, but what if we consider that people may also want to travel between the 2 points of the new road? What if we factor that in as well?
2:43 or we could send 1 car at a time down the roads so 1 \ 10 = 0.1, so I can send 1 car down each road every 10 seconds so that there is always one traffic on the t/10 road. All cars would be done in 121.1 minutes though.
While the traffic *flow* probably did improve, I'm skeptical that the travel *times* improved. Additionally, the real world and game theory scenario are measuring two different (but potentially related) metrics.
I suspect why traffic improved was because (as this game theory explanation does point out) the traffic will be more distributed; however, what this game theory explanation (and what I suspect was not taken into account) was ease of access to where you want to get.
I suspect, that since people had to take detours. Let us remember, there were likely people who took wrong turns which could easily increase their time on the road. Further, I suspect these errors in navigation are the REAL reason why the volume of traffic would increase. Usually, when you make a wrong turn or miss a turn, you simply keep driving until you find a new easy place to turn and recalculate. Therefore, drivers who make wrong turns typically take the path of least resistance, where as drivers who have a set destination are willing to wait in quite a long queue (that you probably wouldn't be willing to wait in if you were simply wanting to turn around).
Lets remember, what was said was that traffic *FLOW* was increased; but this is quite a meaningless statistic. The better statistic (and ironically, the statistic that the game theory scenario uses) is travel time on the road. Yet, this was NOT the statistic used in the real world scenario.
Yeah, that's what I thought as well. I was expecting the theoretical example to explain increased traffic flow by pointing out that if there is one best road, everyone takes that one. But if there are two equal roads, the traffic splits up.
Some might argue, (some have:) that the oversimplified example doesn't reflect reality.
Indeed, there are many unrealistic details in the example, such as a '0' travel time for the connecting road, or the linear "weight" function of T/10 on some road segments.
But ultimately, even if we could tweak the constants and functions to reflect ideal conditions, this would not change the fact that under certain circumstances ADDING a road will slow down traffic rather than speed it up.
Let's assume that we have the same configuration as above, that the weight (travel time) function F1 we use instead of T/10 is realistic, and so is the weight/function for the connecting road, then:
...IF the minimum time for 1 car to travel the F1-dependent segment is still 0.1,
AND the max time for that function is still 20,
AND bear in mind that the alternative segment is a constant 20,
THEN:
1) As long as the travel time X on the connecting road is less than 19.9, no (sensible:) driver would ever choose to take the 20 min. road connected to point A, (top 1st segment from A to B), even if some may take the one connected to B, (bottom 2nd segment from A to B), depending on the value of X.
2) A travel time greater then 19.9 would cause the connecting road to be unused, effectively removing it from the graph/network.
Which illustrates the point: EVEN IN A NON-SIMPLIFIED, REALISTIC SCENARIO, ADDING A ROAD IN THIS CONFIGURATION SLOWS DOWN TRAFFIC, AND REMOVING IT SPEEDS IT UP.
Keep in mind that you're showing that adding a middle road increases travel time when using the parameters of a problem designed to show that adding a middle road increases travel time. This video is purely circular logic.
The real-world examples mistake traffic flow with travel time, as well (that is, they don't explain how long it took for each car to get to its destination, which could have been longer than usual thanks to taking a longer route), and don't incorporate potentially mitigating circumstances (i.e. how many drivers were on the roads when the road was closed down, vs how many would be on the roads if there was no closed road; chances are at least some people decided to stay at home because of the necessary detour).
Having played Cities Skylines, I've learned first hand that more roads doesn't mean better traffic. It's not about road width or capacity, it's about FLOW. Grids are terrible for flow because every intersection requires half of the traffic to slow down or stop. Slowing down/stopping = traffic. Moving = not traffic. Fewer roads means fewer intersections means faster flowing traffic.
T/10 doesn't make sense, it means that if there's 0 traffic, it takes 0 seconds to travel it? If there's one car, it takes 6 seconds? It doesn't make sense. Here's how it should be:
First off there should be an initial value, which would be the minimum time it takes to travel on it at max speed (e.g. 10 minutes, so => T/10 + 10).
Then, we must realize that the correlation of traffic isn't directly proportional with time lost. The fonction should probably look more like a straight line on the initial value axis, and then half a sine wave, followed by another straight line, capped at a certain point.
Very little cars on the road shouldn't increase traffic AT ALL, which is the straight line parallel to x axis. Then, at a point, the cars will start generating traffic, but not so fast, so the curve shouldn't go up fast at the start. Then, from a certain point, traffic escalates faster, the more cars we add the more traffic we get... And then, with even more cars, the traffic time shouldn't go up anymore, because there simply can't be more cars in front of you than what the road can take, so the cars added to the function would be BEHIND you, which wouldn't influence you AT ALL.
+Philippe Poulin You don't need to overthink it; the point was to simplify it so people could understand. Not everyone knows what things like sine waves are. Of course real roads are more complex, but this is intended to be a simple demonstration.
+MrHatoi it's like explaining Einstein's relativity theory with an apple. It's just not possible!
+Eugenio Garza That is not a valid comparison. What we are talking about here is simplifying a relatively complex function into a linear function so that people can understand it better, not using a completely different scenario.
MrHatoi I understand, but as for making it simpler, it was taken TOO far. Maybe there should be a middle ground as a PLAUSIBLE example that is simpler than the rest of the examples, instead of a mathtopia example. There are no roads that have a fixed time of transport, and there are no roads that have a t/x time of transport, and there are no roads that take 0 time to travel. Simple as that.
The point wasn't to simulate road traffic realistic, it was to show what the Braess paradox is.
so if i put in T=1000 then what?
no. people would now take 1000/10+20=120 minutes
now they would rather use the connector to switch from the north road 20 to south road 20 taking only 40 minutes
so in my case where T is overwhelmingly large the connector seems to cut the traffic time.isn't that correct?
+Vaibhav Shah That's only if the fixed routes can accommodate the higher traffic level. They were only set at 20 for simplicity. If you're increasing the sample size that much, you have to adjust the rest of the problem to match a realistic situation.
+Vaibhav Shah That's excactly the point of this video. Building more roads generally decreases traffic, but in some situations, that extra road can make matters worse.
so it's an occurence that happens only when the traffic is in proportion to the travel time..
a special case as i predicted (or retrodicted) earlier.
thanks .
The only problem I have with this example is that it assumes there is some sort of teleporter exists at the midway point. If there is a teleporter than can instantly take one driver from one point to another, why not just have one going straight from A to B and cut everyone the extra 30-40 minutes! Problem solved! :)
The lines in that graph do not represent real distances.
Redraw the map more like a figure-8 on its side, with a bridge where the two roads cross. It's now the situation that building an interchange at the bridge will make things worse, not better.
MrArcticShadow Look at it as two paralel roads like in an American city.
you should of cited the chinese example where they closed down that enormous stretch of highway and turn it into miles of park, expecting the surrounding roads to become inundated with traffic a strange thing happened the traffic disappeared as if it was never there to begin with.
Consider more than 200 cars. Let's say 400. now the travel time in the first example is 40 minutes for both roads. The travel time in the second example is...40 minutes. What about 800 cars? That's 60 minutes for both roads in the first example and...40 minutes in the second example?
This continues for any N number of cars. The travel time of of the first scenario is 20 + T/20 (since we're assuming half of T goes each way). The travel time in the second is actually a piecewise function where Travel time = min{T/5, 40}
What has occurred is that the time for any given person has increased for traffic between 133.3... and 400 but reduced travel time for less than 133.3... and more than 400.
Simply saying that traffic is now worse is naive as the flow of cars per hour has now been astronomically increased. What do I mean by that?
Consider for a moment that the number cars on any route from A to B is dependent on what each driver expects to spend on the road waiting. This is a cost. If the value of going from A to B is less than how much they value their time, a driver simply won't drive. In the first example, the total number of cars on the road is equal to the equilibrium point of people willing to wait in traffic at a given number proportional to their willingness to wait. While this equilibrium heavily relies on this exact distribution, let's just say that we can describe it with this function:
T(wait.time) = 2,000 - 20*wait.time
We now set this function equal the inverse function of scenario 1: T=20*(wait.time-20)
Solving for this gives us wait.time=60 and 800 people on the road (400 each way.)
Now consider the second scenario where you will spend at most 40 minutes in traffic. How many people are willing to wait 40 minutes in traffic? 1,200. In the first example, 800 people were flowing through. In this second scenario, 1,200 were now flowing through. But that's unrealistic as equilibrium here works just like in economics. Let's consider another pair of scenarios where the equilibrium point was 200 cars and 30 minute traffic. Then the distribution could look something like this:
200 = 5000 - 160*30 -> T(wait.time) = 5000 - 60*wait.time
Using the math from before, 200 people under the second scenario will lead to 40 minute wait times.
But plugging in 40 minutes leads to a negative number. Thus, less than 200 people will be on the road. How many is the equilibrium point? We can form a composite function to find out.
wait.time = T(wait.time)/5 = (5000 - 60*wait.time) / 5 -> w = 1000/33 ~ 30.3030...
We can see that the total traffic now is slightly less and the total wait time is slightly more. Let's consider one more example where demand is significantly more inelastic:
T(wait.time) = 260 - 2*wait.time
Then:
wait.time = (260 - 2*wait.time)/5 -> w = 260/7 ~ 37.14
What's going on here? In both scenarios, the wait time is greater, one is significantly greater. Well, it's simple. More people are competing for the same road. Competition between consumers drives up prices. Let's consider one final scenario where the routes have the same wait.time functions as before but they cannot communicate i.e. no mid road.
Then the two routes will be:
20 + T/10, T/5
If we set them equal to each other with the highly elastic distribution, we get:
20 + (5000 - 160*wait.time)/10 = (5000 - 160*wait.time)/5
wait.time = 30
We then conclude with more substitution that the number of cars will become 250 (100 left and 150 right). 50 more cars for the same cost of travel time. The simple fact that the routes competed with each other and overlapped is what drove not only the wait time up but the number of cars down. The conclusion of this all should be obvious:
The way to make an iphone cheaper is not to force consumers to fight over them; the company needs to make them cheaper. The way to maximize flow while minimizing time in traffic is to improve already existing roads, not make ones that force drivers to compete.
Mind = Blown... I can only understand the conclusion, but nothing else above
On your Boston map, that is the Longfellow Bridge, not Main Street. It connects Broadway on the Cambridge side to Cambridge Street on the Boston side.
Game Theorists are the funniest math folks to listen to. Do people know the functional time constraints on each pathway? And also you made the cross road instantaneous. It seems overly idealized. However the idea that a group of people can make individual decisions that add up to a bad result for them all is very intriguing and quite possible. It seems to contradict some basic premises of economics wherein it is claimed that self-interest always finds the most efficient solution.
+Larry Cornell Nash Equilibrium covers this as well, and it applies to investing. Selfish choices by everyone often causes less than optimal results, particularly in economics. It's not as basic as just supply and demand, which is all what most of us ever learn in high school. Unfortunately, since many people assume economics is that simple, they are often swayed by fast talking heads on news programs or politicians making a bummer of a deal sound good.
It is likely that as, fascinating as game theory is, it also does not explain everything. I knew from the first day in my first economics course that supply and demand were only part of the issue. Have you ever mapped the first and second differences of oil price? I was able to predict the oil price spike of '08 doing this. I'm not dissing on my the post or poster, I meant to say funnest but auto-correct beat me to it. The problem is that at times we do present to the masses things that are overly simplified, but these simplifications rob the listener. And on the other hand most of modern mathematics makes almost no effort to clearly explain its work. This is a sad thing because most papers are coming off at some high-level of abstraction that would be understood by more people if someone took the time to give good examples which drill down to the foundation. Abstract algebra and topology are probably the perfect examples of this. I keep my hopes high that the advanced sciences will invest some time and money in actually clarifying step by step how to apply some of the great knowledge that is being obtained. I am afraid that instead of raising the human race to a higher intellectual state we are creating silos of information which will not be conducive to the stated ideal of educating the masses. In short, I think we need to do a far better job of using the Internet to create knowledge trees and rich examples. So in that I applaud Efforts like these to put interesting mathematics out there. I also wish we in science were less self-centered, because just as with the marketplace where self-interest on Wall Street is leaving the world a burned out husk, self-interest in academia is surely making it harder for people to understand and enjoy mathematics. I don't believe that only a person with a Phd in mathematics can understand elliptic curves, it's just that almost no good popularizing descriptions exist. Maybe we disagree on this but I think there should be a field in math matins, a degree focus called math popularization that works hard to take all results and make them as easy to understand as possible, disseminating that information through multimedia content on the web. It almost seems to me we should admire physics for making truly impressive efforts to make its modern discoveries a part of the everyday language. The word "quantum" is now a meme. Just some thoughts.
Yet another reason why we need network controlled computer drivers instead of individual ones.
+Timothy Swan or a healthy competition between public transport focused corporations:-)
"Healthy competition" is an oxymoron.
+Timothy Swan
How so? Please elaborate.
+90hijacked Public transportation is a natural monopoly, so you cannot have true competition there.
sevret313 i can see why one would say it and i suppose it depends on where you live.
only brought it up cause there's been "recent" headlines about the competition between two of the major companies.
What I don´t get here is: If they now for a fact that closing certain roads would improve the overall traffic, why don´t they close them permanently for anyone that isn´t i.e. City Maintenance, Law Enforcement, Fire Departement, Ambulance, Delivery etc.?
"This logic holds true for every driver, and therefore we can conclude that all 200 drivers take the same route."
There are so many problems with that statement--the biggest one of which is assuming that all 200 drivers are capable of using logic. Anyone who has driven ANYWHERE knows that no one uses logic when driving.
But what if I'm driving on the example roads and there are no cars? Would using the traffic dependent roads be instant? Also, why is the middle road automatically instant?
+Matthew “MadMatt” Younce that's my issue with the problem. it's foundation is based on the fact that a) the traffic is "constant" and b) drivers know the given traffic route.
what i thik is going on is that in the original part, each route is taken up by a constant time and variable time. and in the end since both routes are even, traffic is then divided evenly.
in the second set, you have 2 steps. travel time to the junction and then to the next city. and what is then created is you have a path a-a'-b. and it's set so that either path to a' is the same. 20 fixed or (200)/10 = 20. and then you repeat.
so i claim this isn't a paradox in that when you change the rules of the game you can't compare the before and after.
Yeah, why not just build one of teleporting roads directly from A to B? Then everybody's travel time is 0.
+Matthew “MadMatt” Younce the roads with set time (20 mins) are supposed to be high traffic roads (multiple lanes to minimise eliminate traffics influence) the other roads which are influenced by traffic (T/10) are supposed to be urban roads so there would almost always be traffic (like his example new york city, you could get from top to bottom in minutes but because of traffic it's more likely that it would take at least an hour and it's the traffic on intersecting roads not the traffic on the road you are on.
The link road was given a value of 0 just for demonstrative purposes he could have put a value onto it but it would have made explaining the paradox more complicated. If you wanted to look into it properly you would have to take junctions into account because they are what causes the jamms. removing roads also removes junctions which means there's less traffic lights to hinder flow.
James Mulchrone Well, you really know how to get a point across. And I see your point. But still, let's say hypothetically that I'm driving on one of the theoretical traffic dependent roads and there are no cars for some reason. Would my travel time be instant?
Matthew Younce Thank you, I try to be as coherent as possible. It wouldn't be instant because if you were driving on the road there's at least one car to be added to the value of traffic (yours) and again it's a massive simplification just to help people understand the concept.
So the new road takes litterally no time to travel?
+Nick Reeser Think of it as two parallel (or at least non-intersecting) highways with exit/entrance ramps crossing over one another (like you might find in many major cities in the U.S.). The difference in time it takes to travel over the ramp versus continuing on the same highway is negligible.
+VioletTheGeek But still existent.
Nick Reeser It's a highly simplified example. Negligible = 0.
Why would the travel time on some roads depend on traffic, and on other roads be fixed? And how can the connector take no time to traverse?
The problem with traffic is that many people think it acts like a liquid: if you give it more roads, it will spread out more thinly over that area. The reality is that traffic is like a gas: it will expand to fill all spaces it is given.
Consider the following: in an old country village with zero roads, how much traffic will there be? That's right: zero! because nobody will drive anywhere.
What kinda shitty logic is that? There will be dirt roads because every one travels. This paradox doesn't apply every where. In most cases, a new road will improve the flow.
How would travel time ever be 20 mins no matter what? Wouldn't it always depend on traffic?
A little more volume please.
There are so many problems with your example. First off, one reason traffic improves with road closures is because it removes a set of traffic lights, and, assuming New York uses one way systems, encourages people to take the one way systems up and down to get to their destination, rather than up and down, left and right, down 42nd street.
Okay, now as for your example - first of all, no road is going to be guaranteed a travel time. Even if you can average it out, t/10 doesn't make sense, as if you're the only car, it takes only 6 seconds to travel it, which is inaccurate. In fact, the travel time will be, in general, the time it takes to travel the road at the speed limit, up to capacity, after which traffic will slow down exponentially with each new car entering the road.
Secondly, unless the two routes are the same, no connecting road will take zero minutes, it will take some amount of time.
Thirdly, people will take the route they think will be the shortest time. When they can see traffic on the routes, they will take the route that has the shortest time with the amount of traffic estimated to be on it. So if at point A, a driver sees all 200 cars taking the southern route, they will take the northern route.
You do realize most of your criticisms arise because the narrator wanted to keep the example simple and decided to use ideal conditions. He could have made it several times more complex and even factor in the likely road choice of drivers depending on the traffic. That would be cool and all but it's suppose to be a simple example not something you want a computer to process.
bwoy12345 My point was, his example is so far removed from actual traffic, that it does nothing to explain why it happens.
It does it's job quite fine. I've seen a lot of science that starts from simple formulas and then they add in more variables to make the problem more realistic.
You do realize that this paradox doesn't happen every time you add a new road, right? there are certain situations, combination of roads that leads to this. May be more people make left or right turns, which in some way makes the traffic worse.. I'm sure every situation has a different reason.
That's Main St in Cambridge, and Cambridge St in Boston, for the record.
I was hoping they were going to be specific showing where cars actually ended up driving instead rather than modeling a hypothetical based on assumptions.
Or could it just be that many people decided not to drive on days they were forewarned of major road closures ?
Sort of related, why does flow through an intersection often improve when the traffic lights fail? Assume a cross intersection.
I *think* it's because drivers giving way to each other and trying to be fair make more efficient use of the intersection and can pack more cars into the intersection than is allowed by the lights system. Peak transit velocity is lower but vehicles per second is higher due to close interleaving.
Alternatively a car at an intersection with lights spends say 1 minute waiting and then 5 seconds transiting. i.e. 65 seconds to cross the intersection. With no lights the car spends say 10 seconds waiting for a gap and 10 seconds crossing or 20 seconds to transit.
Im not sure if this "often" is actually verified. But just using logic, it would be because traffic lights can not perfectly respond to incoming traffic. Just as sometimes lights are green with no traffic while packed cross streets have to wait. At a broken light it becomes alternating cars go, meaning that some of the congestion will be constantly relieved. However, while congestion may diminish, travel times go up, because you have to come to a complete stop at every intersection. Traffic doesn't always equate to longer travel times, long as there is enough space for every car and buffer zones, it could be as fast as no traffic at all (ex: interstates)
@@jaojintalonis92 also at a traffic light, often 5-10 people in line get through on green. The OP forgets to factor that in when he’s talking about everyone waiting their turn at a broken light. Dunning Krueger strikes again.
Sorry I'm late to the party.
Yes, I agree that this one case is theoretically faster. Just because of that doesn't make it a good model that explains why these real world cases happened.
The model relies on the maximum value of T/10 being the same as the constant travel time of 20. If you change that to 10, there's no more paradox:
Even split of traffic (T=100)
10 + T/10 = 20
T/10 + 10 = 20
All inner (T=200)
10 + T/10 = 30
T/10 + 10 = 30
T/10 + T/10 = 40
10 + 10 = 20
Since the T/10 + T/10 route is no longer the fastest, no one would take that, and everyone would take the 10 + 10 route, which would then be the highest travel time possible, with lower times possible as people take the 10 + 10 route, thus reducing the T count on other roads.
But a much better model of the real world examples would be to insert not a cross road, but a third road going directly from A to B.
It is an interesting paradox, but it requires contrivance of the numbers and the direction of the roads.
This example is ideal. He could have factored in other variables based off of real roads but that would get complex fast and is not necessary to understand the paradox. In the real world these roads are not necessarily straight and may have other roads feeding into them. Secondly you cannot always make a straight line between two points so your line from A to B isn't realistic. Thirdly because road networks can get complex over time a single new road designed to improve flow between two points may limit flow for other points. Fourthly you said it is interesting but it involves contrivance of the numbers and the directions of the roads. Those two things are how roads actually work so they cause the paradox, they do not take away from it. Finally It is a paradox of basic human reasoning not math.
1. My point is that as the model gets more complex (and thus more like real life), it's harder for the model to exhibit this paradox because the numbers are contrived. That doesn't mean it can't happen, just that this phenomenon is uncommon. And absolutely, the title of the video is "can speed up traffic," so I don't have any problem with that.
2. I was adding a point to the model, but in reality we're talking about removing an already existing road, so to provide the real world example of 42nd street, 42nd street would be the direct route from A to B, and we're comparing 42nd street open to 42nd street closed.
3. Yes, it may limit flow, but I doubt it's likely. A couple documented cases in history exist, but I'm sure all of us experience road closures often, and I have yet to see a scenario that has reduced the length of my trip.
4. In my example, I tweaked the numbers just a little bit, and the whole thing fell apart. If you're arguing how the depicted traffic flow is more realistic than mine, please tell me why you think so.
I believe my traffic flow is more realistic because the only kind of road that would produce a constant time of travel is one with many lanes and no traffic lights. A highway completely independent of traffic (still not feasible, but theoretical). That would never ever take as long as the (T=200)/10 maximum traffic delay scenario, that, based on the arrangement of the variables given, would have to be the same mileage as the opposite side of the parallelogram.
5. I agree here. "Paradox" sounds better than "Counter-intuitive," however.
If 100 cars would travel straight the north path and 100 cars would travel the south path then switch to the north path, then 100 cars will have a travel time of 10 mins., while the other hundred will have a travel time of 30 mins. The 100 cars that took the south path then switched to the north path will arrive to B as the 100 cars that took the north path finishes the 20 min path and into the T/10 path
If the new road were built directly from A to B though, then the volume of traffic per road would be reduced, and thus travel time would have been cut.
So in a less-mathy terms, before the free road, both routes were equally attractive, so the traffic load was shared evenly between the two routes.
But the addition of the free road made one route more attractive to the drivers, so that route became overloaded and thus became slower.
The whole concept of one road that has a fixed time and the other road is set by traffic is a fallacy to start with. This is a poor presentation.
Why would drivers switch from the southern route to the northern route if switching from one route to the other would take additional time. This example is flawed.
+NTG The drivers "selfishly" do what is best for themselves as individuals; they don't team up and come up with the best distribution amongst themselves before departing.
+NTG that's why there was a zero at the new road.. the image ought to have reflected the negligible travel time, sure.
If it helps, pretend the 0 was a 5.
Before new road: 30
After (counting the 5): 45
Adding time to it makes it more right but also slightly wrong.
Was this not first discovered by Sam Schwartz (AKA Gridlock Sam) in the 1970's?
Perhaps the theory wasn't fully developed at the time, but I'm quite sure he was shutting down roads in New York City in order to improve traffic flow
put a toll on the new road and you make money and speed up the traffic
You sir are an evil genius (aka wannabe politician).
Really you want to put the tolls on the 'T/10' roads, since those are the ones where traffic causes negative externalities.
Hm, now I'm curious of this can be applied to business, particularly in the case of disruptive innovation...
I was thinking you put a line in the center from A to B it would take 30 minutes to get across and all 200 drivers would take it. if you were to shut that down there would be the other 2 roads where there are 100 on each road and it would still take 30 minutes. It doesn't matter about the time because this is about Traffic not effective time
Wait! What?
If there are 0 cars, it takes 0 minutes for second part or North & first part of South?!?
Maybe just me, but I thought this video was going to be about how closing a road can speed traffic up, not how creating a new road can slow travel time.
Same thing, just explained backwards. Consider the new road is closed, now we're back to the original example, which is faster.
Not saying it's wrong, just don't 100% "get" why this is the answer to this particular situation. This shows how closing "a" road can possibly make traffic better, but do these real-life roads fit this model? I don't see how, but maybe. I would kind of think one explanation could be along the lies of: "If you have one clearly optimal, straight-shot road that gets you from A to B (and let's say A to B is the heaviest traffic flow around the city at this given time -- lots of people in the area of A, and many of them wanting to get to B), then practically everyone at A is going to take that road, and they will become jammed. If that one single "perfect" road is closed, then everyone has to decide among several roads that are a bit off-course, but none significantly shorter or better than the others. Odds are, they will split up evenly among those routes, and won't get jammed as badly, if at all."
+Ace Diamond Your explanation is correct. The second part of your explanation, with the main route closed, is very similar to the first portion of the video. The beginning of your explanation is similar to the second portion of the video, after the new road has been added.
I suppose that makes sense. I guess I was just expecting more for them to add a third road that was a straight shot from A to B. But the more I think about it, yeah you're right, it's essentially the same thing.
The road which depends on traffic will always cause more time for more value of "T".. so why not people prefer the constant time road if they know the traffic is more...🤔
So how do we ease traffic?
sarcasmo57 Have systems set up that aid in decelerating and accelerating traffic in unison, which could also be coupled with a system that keeps a shorter distance between cars. It is getting the system to work more like a train or a rope being pulled rather than a staggered and choppy stop starts and stops for those in the middle and end of a traffic jam(especially). Also, if absolute stops were needed, there would be a system put in place to park cars more closely together on the roadway (not to extend the traffic jam any further back then necessary). Although too close together could also slow traffic starting back up in unison, but could be aided some by one car moving in rapid succession after one another after a minimal distance is reached between 2 successive cars. Now there would need to be systems put in place to offset lane mergers and on ramps. Speed and distance would be locally adjusted to efficiently allow vehicles to merge, for instance. Also, there would be communication between cars so that if say a large object fell off of an overpass or off of something being towed that all cars would move away from it in both directions (utilizing shoulders if necessary) in unison like a school of fish moving from a shark attacking the center, and then would regroup on the 'other side'. It could also alert cars further back to be adjusting speed and moving to different lanes if necessary well before the road hazard would be visible or brake lights would typically be seen by cars in front of you. An occurrence like this would normally cause a rapid stop and long delays, but could be optimized to only cause minimal slow down. This is all theoretical and would need to be adjusted for cars with different accelerating and decelerating and handling capabilities, but would work optimally if all vehicles had a certain minimum acceleration, deceleration and handling capabilities. The biggest threat to this on an interstate would be large trucks that accelerate much slower. A partial solution for this would be to have one truck lane added 2nd from right that would allow for easier merging from onramps and allow if stopped, trucks to keep a certain minimum distance from each other at the stop, and then all slowly begin to accelerate in unison together, while allowing lanes left of them to move freely at normal rates and allow those entering from the right to possibly accelerate past the trucks and otherwise there would be slight increases or decreases of speed to allow cars to enter between trucks and gain access to the other lanes. The 'truck lanes' would be used minimally by cars and could possibly be opened up if few trucks were present(although that would complicate things) and allow cars to smoothly be able to merge into other lanes once approaching a truck a certain distance away. Ideal number of lanes would be 4 or 5, but if say only 3, trucks would mainly be in the middle lane with a larger gap between them that would allow for both slower stopping and also allow for cars to pass between the 1st and 3rd and vice versa.
How does more traffic slow down time? If the rate if speed is constant it shouldn't matter.
How do you solve 2^x+3^x=1
Using algebra
I tried using it, I doesn't work?
its just zero?
nope 2^0+3^0=1+1=2
well my next guess would be to divide both sides by zero
In your second example, the car drivers should still do 100 and 100 split.
That relies on every driver knowing what every other driver is doing and also taking a decision for the benefit of the whole group and not for themselves. It would never happen.
And yet, people think socialism is supposed to work.
Where did this limit of 200 drivers come from?
Your first example assumes that the theoretical travelers follow an even distribution on purpose because they know it will work out. Your second example assumes every traveler makes an informed mathematical decision based on traffic flow.
The last time I was in traffic, a sunburnt man in a wifebeater and a mullet was trying to balance a soda can on his knee.
I understand that this is a simplified example that assumes perfect logic, but... that concept is nonexistent in real life.
is there zero travel time switching between the southern & northern routes?
Not in Toronto when they close the Gardiner and the DVP!
Someone with the resources should use a computer simulation to model this
that new road has a travel time of 0? what kind of magic is this? it's also not a paradox that trying to force all traffic into a single path will make things worse.
You compared 100 cars to 200 cars. The variable of traffic flow was still the only part that made a difference. The third road is irrelevant in your comparison I think, unless I missed something. I'm no genius either so it's very possible I did miss whatever makes this a "paradox".
Your videos are great, but the audio is too low. I'm with my pc and youtube on max volume and I stil struggle to listen
When is the last time anyone built s road with *_ZERO_* travel time? Your mid connector would add 10 min travel time to all who use it, therefore few will choose that route.
and Port Authority Bus Terminal!
I think assumption is not practical, it is not convincing how closing a main road did not create congestion.
if there are no cars on the road you break phisics
+Insanity cubed Indeed, if the time you need to travel from A to B depends on the amount of cars on it, the ideal way of course is to open the road again that connects the two halfway points and let everyone only walk to their destination.
Franz Luggin and since you go past it so fast there would only ever be 0 cars on it
So that's why there are always some roads closed in Budapest :D
Also people prefer the main road so that they don't get lost.
The last statement is the only one of merit on a video about traffic. Roadways and roadway system are extremely complicated and take into account much more then just simple capacity analysis... If that's what you could even call your paradox. As a civil engineer who spends time studying and designing roadways, travel time itself is usual not a big factor. Safety is the most important. So when lanes or roadways are added for capacity, it is because the existing system was overtaxed, which is the main cause of accidents. Most design standards in our state allow for a level of service D, in peak hours (that is-obstructed flow, and a decreased speed from free-flow). Interesting theoretical math problem none the less, but not practical.
Very interesting!
Volume? WTF, I can barely hear anything
That.... doesn't seem like a paradox. That doesn't seem like much of anything to me... Put an uneeded road in and it can mess some things up. I can get that (intersections and whatnot) But it say's nothing about closing down a road increasing traffic.
No duh. There are metered ramps to freeways in attempt to prevent freeway clogging to the point of crawling and stopping all traffic, including all traffic on all on ramps. Metering an on ramp is stopping traffic there for the overall good. Clearly any purposeful stoppage, like stop signs and traffic lights, as long as they're set up well, fit into this overall idea that stopping some flows helps overall flows.
At first T/10 is not valid time road can take for a driver and therefore it is not valid example because
roads are based on throughput so
drivers that join the road do not affect drivers that are on the road wich means that
when the driver decide if he join the top one or the bottom one he will choose the final time as waiting time + time the road take if it's empty
So... make a meaningful example
How about this: If the person in front does not start up on a green light within 1.25 seconds; you can legally shoot them.
RUclips comments are so math illiterate it's not funny. Thanks for these vids! I really enjoy them. I'm also a fan of Numberphile. I appreciate the no nonsense approach.
If you travel by yourself, you can get to the other town in 0.2sec.
underwhelming, as a fan of "city-building" type computer games this paradox barely scratches the surface of what it takes to lay out a road system ( In a game! )
That said i've often times looked at satellite footage from google earth for inspiration, In this sense- i find this example particularly amusing :)
There's no road that is isn't dependent on traffic.