The lanes are infinitely long. That means that there will be cars going into the second lane throughout the entire lane length. And that means that ALL cars will become completely free to go faster and faster, because no matter where a car moved into the second lane, the cars ahead had another car further up also move into the second lane.
If I am reading his restriction on passing correctly, cars in lane 2 can never increase their speed because they are not allowed to move up. Only cars in lane 1 can move up.
If this is not so then this statement makes no sense:
Moving to lane 2 traps the car at a certain speed with no possibility of gaining speed.
James, they can’t go infinitly faster, because the phenomena has never been observed before. We have road systems all over the place, the result isn’t ever faster traffic, but ever slower traffic. The conditions for Carleas theory must pop up all the time, but all we get are traffic jams.
Its because the theory is lacking elements to it, hence the caterpillar effect. Its atrocious mathematics, not to be taken seriously. We all have experience of it’s failures, not much of it’s success. Its why road rage happens.
No. because every car has another car further ahead that is moving into the second lane. No matter where you are, you know that someone ahead is moving into the second lane, allowing the lane that you are in to increase speed. So when you change lanes, you cannot catch up to the car that you were behind.
The question is why you would bother to change lanes. And the reason is that by changing lanes, all cars get to go faster including you whereas if you don’t change lanes, no one goes faster.
I am guessing that the “noise” being referenced is the noise of decision making such that some stay and some change.
His scenario is not real or realistic. So of course you have never observed it.
This is a false statement. Why can they not increase their speed to Z? If the lead car increases its speed to Z, so can’t everyone else.
This is another false statement. Adding the lane would make no difference if the lead car had the same behavoirs…All it would do is compress the cars in lane 1 by removing some cars and putting them in lane 2.
This whole entire thesis is false, it does not prove how noise improves a signal, it only proves that if you had more spatial bandwidth that you can compress a signal on one axis.
James wrote what I think is a good response to Phyllo while I was typing this, but I’ll post this anyway in case it’s useful.
Take a line of cars, moving to the right (towards car E).
1: A B C D E
2: _ _ _ _ _
If Car B move into lane 2, it is still stuck behind Car C:
1: A _ C D E
2: _ B _ _ _
Car B can’t move forward (thus it satisfies the condition that changing its strategy won’t lead to a better result). Car A can move up, but Car B doesn’t care about Car A:
1: _ A C D E
2: _ B _ _ _
But what if Car B and car D both move into lane 2:
1: A _ C _ E
2: _ B _ D _
Now, Car C moves up, and that allow car B to move up:
1: _ _ A C E
2: _ _ B D _
Car B hasn’t violated any rule, and it’s been able to move forward one car-length, exactly the same as Car C. In an infinite line of cars, Car D would be in the same position, accelerating as the car that it isn’t allowed to pass accelerates.
The noise is in the application of strategy to outcome. An equilibrium strategy is for everyone to stay in lane 1. No individual car can improve its speed by changing strategy. The noise is in the connection between strategy and action: if a car’s strategy doesn’t completely determine a car’s actions (in this toy example, the only ‘action’ it can take is changing lanes or staying in its lane), the signal (here, speed) is improved.
Though I note in writing that that I’ve used noise to refer to a random variation that is not in the signal itself, although it is noise in the factors that produce the signal. It is noise in that it carries no information, so I’d still say we’re introducing noise into the system and improving the output, which still seems unexpected to me, but perhaps I could do with some better labels as to what constitutes the noise and what constitutes the signal.
The added noise does increase the traffic volume, which we can see in the last diagram (again placing it into an infinite line, where cars behind move up to fill the newly freed spaces in lane 1).
There is no lead car in an infinite line of cars 
There is no “lead car”.
The post I made dealt with a 1 way infinite system with a lead car, this post deals with a 2 way infinite system with no lead car.
If the operation was only applied once, it would double the speed, not make it infinite.
And you are right, his thesis has little to do with noise theory, it is more to do with how increasing the bandwidth can improve a signal.
Well, I would disagree except that you specified “an individual”. If every individual has the choice, rather than merely one, then flipping a coin to choose whether to change lanes would increase the average speed. I guess that is the “noise” factor. And thus, each individual has an opportunity to improve the situation.
Not in the scenario that I explained.
Absolutely impossible, Caterpillar effect. You live in DC, the area is choke full of military and ex-military personal. Your theory is completely and totally voided. I know you won’t accept it coming from me, go ask them, they will knuckle your head and tell your why your wrong. It doesn’t work. Period. If there was a better way, we would of found it by now, we’ve been experimenting with this one since Ancient Assyria. The caterpillar effect dooms your system to absurdity. Your not able to see it, but consider it next time your stuck in a traffic jam trying to get to Baltimore. Traffic should be moving, it just was… but isn’t now. Why?
You need more lanes open, not just two, for small packs to form, if it’s high density like your suggesting. Two lines work if cars are rare, not consistent.
If that is the case, then a logical strategy is for car D to move to lane 2 as soon he sees that car B is in lane 2, because he has the reasonable expectation that car F will do the same as soon as he sees that car D has done it.
Random motion is not required since there is a logical way to increase speed without it. Car D facilitates the acceleration of car B, car F facilitates the acceleration of car D, …
Although the situation is being artificially restrictive because car A can come along car B while car B can’t come along car C.
Isn’t the logical strategy to yield to telelogy, and epistemeolical constraints?
Your describing a logic based on Ideo-Kenetic Apraxia. Its not logical at all. Very illogical, has too few command operations to succeed for long before it collapses into oblivion.
Its a deeply flawed, invalid math. Quit treating it like it’s going anywhere but a dead end. Motorcycles stagger on a similar system, but I haven’t seen a successful long weekend warrior motorcycle gang, they fall apart into packs after a while. Especially silly in Hawaii, there were longer attempts at packs than highways and they would just drive around like fools around each other in another neighborhood each week, occupying every stop sign in a four way intersection, looping and juxtapositioning, trying to act like a badass rebel in a confined cage.
These dreams aren’t possible. Look outside, what do you see? Traffic operates in groups… little pacts, or individually, or stalled dead traffic. Thats it. Its because of the availability of lanes overcome frictions, people usually have the ability to completely overcome a obstruction of slow moving or stalled cars… then they get behind someone, and accept their speed.
This backfires in Alaska, during the winter, roads aren’t salted, so cars can pile up big time in slow motion, and will get ticketed even if people plea it was unavoidable. Carleas logic is dangerous in such places, it can get people killed. Usually, just guarantees you’ll be late, if you get there at all.
It seems that here we are again with the blue eyed problem.
The Nash Equilibrium specifies that they all know the same things and all know that they all know that they all know. And if that is the case, then they all know that if another lanes opens up, they each flip a coin to make the decision. They all know the same and they all do the same thing (flip the coin). But by doing so, the speed increases. Thus it is not an Nash equilibrium lock.
Have you ever seen Idiocracy Carleas? Your doing the “Brawndo has Electrolytes” argument. Only Iambigious is allowed to do that on this forum with his existentialist apologetics, as a Administrator, your held yo a higher standard.
Let’s take a similar problem… you literally get a $1 million dollar prize for it.
Navier–Stokes Equation
en.m.wikipedia.org/wiki/Navier–Stokes_equations
Its solvable in both systems, but they want to know if both systems are compatible, or if there is some weird Schrodinger’s Cat phenomena going on between the two maths, and that they don’t causually coincide in tracking… so the prize also exists to show if the two systems are incompatible… if they essentially fall apart, can’t be reconciled.
Its a massive pain in the ass headache, and paper after paper has been proposes explaining mere aspects of the phenomena, it’s closely related to your assumption.
In your case, your formula is simply put, to primitive… it lacks the plasticity to organize non-linear binary behavior, which gets remarkably complex. A couple of rule fixes won’t overcome the Ideo-kenetic Apraxia issues of coordination, humans too have equal if not superior knowledge, given we possess rear view mirrors, traffic rules, knowledge of complex traffic laws, GPS warnings, windows to see, usually knowledge of the law of the ground.
What we do have is Clausewitz friction. Your system falls apart in the real world. You know this from experience, I doubt your so autistic you can escape the reality presented in your own experiences on the road… you’ve been stuck in traffic a few times.
Perception and Apperception isn’t the same. The ability to move with elegance is controlled outside of the factors listed.
en.m.wikipedia.org/wiki/Neurosc … _free_will
en.m.wikipedia.org/wiki/Ideomotor_apraxia
If your theory isn’t applicable, at all… is a inherent menace in fact, to real world traffic, what is it good for?
Can it be used in near absolute zero superconductors? I dunno… I think non-linear dynamics still does wobbly stuff at this level, but maybe it will work per for given elements, I suspect we are a long way from having the engineering and manufacturing precision to notice if your wrong on this level of size for… who knows how long.
What about synchronization of artificial neural networks? I suspect usually failure, but I would rule out the impossibility for occasional success in custom made systems.
For human traffic, even lead by AI, never.
en.m.wikipedia.org/wiki/Alien_hand_syndrome
en.m.wikipedia.org/wiki/Environ … y_Syndrome
And that last link brings up modes of navigation issues, we use two separate kinds of navigation in our mind.
m.pnas.org/content/107/32/14466.abstract
How well does your theory balance out in the two separate Hippocampus regions? When we pull off a lane change, both in deciding to as a act of free will, and actually doing so, which one dominates?
This by default has to effect your math, by default, if the underlining logic in your math is to make sense.
If nothing else, your going to gain a good ability to fight your own tickets in traffic court from this reading up.
Sorry James, I misunderstood your earlier question about traffic volume as directed at my scenario. Although, under some conditions of noise, your scenario may still lead to increased traffic volume, although the definition of the system I described isn’t complete enough to say for sure.
To translate your scenario into the defined terms I used initially: the desired speed Z of the cars changes randomly such that sometimes it is less than Y, the speed a car is currently traveling.
The problem with the initial definition of the problem is that we don’t know what will happen to a car in lane 2 if a car in lane 1 slows down (because in the original case, that never happened). “Moving into the space in lane 2 adjacent to a car in lane 1” may or may not describe a car in lane 2 at constant velocity as a car in lane 1 decelerates. One way to read it would requires that the car in lane 2 should slow down to match (car in lane 2 is ‘moving’, so if it enters the space it breaks the rule); another is that a car at constant velocity wouldn’t be ‘moving’ into the space, rather the car in lane 1 would be moving the space, (it is, after all, car 1 that is acting and car 2 that is remaining constant).
Under the latter interpretation, somewhat noisy speeds could increase traffic volume, because cars in Lane 1 would have some reason to believe that they could improve their outcome by changing to lane 2. Depending on how much speed fluctuated, if B wants to go faster and C is starting to slow down, B could move into lane 2, maintain constant speed, and end up moving faster. Shifting lanes becomes a strategy to improve payoff. Thus, both lanes will be used, and depending on the minimum speed Z of the slowest car, the total volume could increase if the use of the second lane increases road use to more than half (which it is in the case where every car is in lane 1 and there is no space between the cars in lane 1).
I don’t follow this. Why wouldn’t Car E move into lane 2 upon seeing Car B move into lane 2, since E has a reasonable expectation that Car H will move into lane 2? And if E would, why wouldn’t every car move into lane 2 upon seeing any car move into lane 2? And if every car would, wouldn’t a car seeking so maximize its speed prefer to stay in the soon to be open lane 1? And if a car would, wouldn’t every car prefer to stay in lane 1 and thus not move at all?
In short, there are two problems here. First, car B has no reason to move into lane 2, car B would have no expected increase in payoff in changing lanes. Second, car B’s actions do not provide a reason for any other car to change lanes, since every car individually can expect to go no faster by changing lanes. Randomness provides for the changing of lanes despite that there’s no good reason to do it.
If there is an agreement that everyone will flip a coin and change lanes if it turns up heads, then I agree, there is no Nash equilibrium. If there is no such agreement, there is a Nash equilibrium. No one has a reason to flip a coin and move simply because there is a Nash equilibrium. After all, a situation where every car but 1 flipped a coin would also resolve the Nash equilibrium. A situation where every car but some finite number C of cars flipped a coin would also resolve the equilibrium. A situation where every other car, or every third car, or every 1/C cars, without flipping a coin, suddenly changed lanes would also resolve the Nash equilibrium.
Again you interpret common knowledge to be tantamount to coordination. That is not the case. But I’m sure if you can rigorously demonstrate that Nash equilibria are self-defeating, there’s a Nobel prize in it for you.
Yes. An infinite line of cars on an infinite road is difficult to sustain in the real world.
It’s not about traffic, it’s about signals and noise.
One could equally well point out that Nash equilibria aren’t observable, because flawless execution, complete commitment to increasing payoff, and (as James correctly reminds us) perfect common knowledge don’t exist in the real world. Nevertheless as a model it is valuable. Similarly, the theory of the relationship between signal and noise is worth understanding, even if in the real world it’s much more complicated.
2op
On an infinite line there would be an infinite amount of points in the spaces between anything traversing it. So a car could continually speed up forever and not reach the one in front of it. Any z is achievable.
The notion as you further expounded,still stands though, and we must assume there are finite distances even on an infinite road. Which means its infinite in that the raod can continue forever ~ potentially, but is not actually infinite [cannot be divided into finite segments].
This seems wrong in multiple ways. There can be two points a finite distance apart on an infinite line. For example, 4 and 5 are a finite distance apart, even though the line of real numbers is infinite (and even though there are infinitely many real numbers between them). Your objection feels something like Zeno’s paradox, which calculus resolves (infinite points can be traversed in finite time).
And, in any case, in the setup it is given that the starting distance between the cars is 0 (in fact, I think it must be 0 for the equilibrium to exist).
It isn’t a matter of agreement. It is the logical conclusion to make by every “perfect logician”, if such had been specified. The Nash Equilibrium requires that no player CAN benefit by any other choice and thus does nothing. That doesn’t apply to this scenario because merely by every driver knowing the situation, they all know that by flipping a coin, they all benefit.
Adding the noise of flipping a coin, or any ordered understanding/agreement to establish who changes and who doesn’t, immediately improves the situation.
It is like Zeno’s paradox, because that is dealing with infinities and in a sense making the contrast to our reality not being like that. Start at 1 on an infinite line, then where do you draw 2?.. you have to travel from point a – b which with an infinite line you cannot arrive at the next cardinal point without taking an infinity to do so. If there were no movement we can imagine as in calculus that we can just jump to an infinite amount of finite divisions [which is imho a contradiction] along an infinite line. The integers there are in meta[physical] position ~ there would be nothing to denote fixed points/cardinality. Start moving and then your car has to actually traverse between two infinities at finite speed!
As pure thought experiment and mathematically, your solution is fine though.