## When Noise Improves the Signal

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### When Noise Improves the Signal

It seems there are times where noise improves the signal. To see what I mean by this, let me return to a thought experiment I described before:
Picture an infinite [single lane] road with an infinite line of cars. Each car is traveling at a distance X behind the car in front of it, and no car will approach the car in front of it closer than X. Each car is driving at speed Y, but wishes to travel at speed Z>Y. It's clear here that no car can increase their speed to Z, no matter what the speed limit.

To that, add another lane. Let's say the cars are all in lane 1, and we add lane 2 with the condition that a car in lane 2 cannot pass a car in lane 1. For simplicity, let X = 0, and let 'passing' mean moving into the space in lane 2 adjacent to a car in lane 1. Also, though perhaps it should go without saying, let each car have a length L>0.

In this case, no car in lane 1 would change to lane 2 in order to increase their speed, because changing to lane 2 would not result in an increase of speed: they could move no further ahead (again, given that X = 0 and our definition of 'pass').

This appears to be a Nash equilibrium (it meets the conditions as listed in that article).

But if the cars changed lanes randomly at some probability, the sub-optimal equilibrium goes away: some percentage of cars move into lane 2, and though they can't pass the car in front of them, every car behind them can increase their speed. And cars in both lane 1 and 2 can increase their speed when cars ahead change from lane 1 to lane 2.

Assuming the foregoing is all correct, the outcome is unexpected: 'noisy' execution of a strategy can lead to a better result across the system, and a more optimal equilibrium for all players. Is there a similar fix for all Nash equilibria?

I also note that this is related to Taleb's idea of antifragility, where noise improves the system, though I don't know if Taleb discusses Nash equilibria.
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Carleas
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### Re: When Noise Improves the Signal

But if the cars changed lanes randomly at some probability, the sub-optimal equilibrium goes away: some percentage of cars move into lane 2, and though they can't pass the car in front of them, every car behind them can increase their speed. And cars in both lane 1 and 2 can increase their speed when cars ahead change from lane 1 to lane 2.
When a car moves to lane 2, the cars in lane 1 increase their speed and close the gap which was created by that car. Therefore, one car will come along side of the car which is in lane 2.

Why can't the car in lane 2 move along side of the car in lane 1 which was ahead of it?
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### Re: When Noise Improves the Signal

This thread shows a good example of my other thread about ultimate reality and explaining concepts. My RAM memory is not that good so its difficult for me to render this thing all in my head. However, if I rendered it all in a computer program and made this into a video, I'd understand this in 5 seconds. Viola.
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### Re: When Noise Improves the Signal

phyllo wrote:Why can't the car in lane 2 move along side of the car in lane 1 which was ahead of it?

Carleas wrote:let 'passing' mean moving into the space in lane 2 adjacent to a car in lane 1

By that I meant moving forward into the space in lane two adjacent to a car in lane 1. You are correct that the cars in lane 1 will increase their speed to fill the gap left by a car that moves into lane 2. Indeed, that is what allows a general speed increase under the condition where cars randomly move into lane 2.
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Carleas
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### Re: When Noise Improves the Signal

Carleas wrote:
phyllo wrote:Why can't the car in lane 2 move along side of the car in lane 1 which was ahead of it?

Carleas wrote:let 'passing' mean moving into the space in lane 2 adjacent to a car in lane 1

By that I meant moving forward into the space in lane two adjacent to a car in lane 1. You are correct that the cars in lane 1 will increase their speed to fill the gap left by a car that moves into lane 2. Indeed, that is what allows a general speed increase under the condition where cars randomly move into lane 2.

That would cause the average speed to increase to infinity.

This seems only mildly related to noise. You significantly changed the order. That is different than merely adding noise. If you originally had two lanes fully occupied, as was the first, there would be no change possible.

Adding noise would be more like:
All of the people understood and obeyed the speed limit laws, never going faster nor slower than permitted. During high traffic hours, the highway traffic moved at a particular volume. Then one day, people began acting chaotically, ignoring the laws and traveled at random speeds.

Did the "added noise" increase the traffic volume?
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It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

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### Re: When Noise Improves the Signal

Carleas wrote:
phyllo wrote:Why can't the car in lane 2 move along side of the car in lane 1 which was ahead of it?

Carleas wrote:let 'passing' mean moving into the space in lane 2 adjacent to a car in lane 1

By that I meant moving forward into the space in lane two adjacent to a car in lane 1. You are correct that the cars in lane 1 will increase their speed to fill the gap left by a car that moves into lane 2. Indeed, that is what allows a general speed increase under the condition where cars randomly move into lane 2.

It seems that you are placing a very artificial restriction on the cars in lane 2. The cars in lane 1 can move up and be adjacent to a car in lane 2.
But cars in lane 2 can't move up and be adjacent to a car in lane 1.

It also seems that cars in lane 2 can never get back into lane 1. They can never increase their speed. So, the randomization forces cars into a disadvantageous position. Cars in lane 1 benefit at the expense of any car unlucky enough to be assigned to lane 2.

Can this really be called "noise"?
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phyllo
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### Re: When Noise Improves the Signal

Nope.

Its called "the caterpillar effect". Armies have been confounded by it since the dawn of time.

Lets say you have unit 1, marching..... unit 2 is behind it, similarly marching, unit 3 marching, unit 4...... unit 500 is way in the back..... not marching. Its been standing for a hour, confused why it was forced to stop marching by the unit in front of it. Suddenly, unit 499 starts screaming, and rushes off, and Unit 500 screams "oh shit" in return, takes off running even faster to make up the gap. 499 stops suddenly, 500 crashes into them, 501 really rams them hard.

Why?

All sorts of damn reasons, but primarily this.... linear coordination is very difficult for humans (annually migrating animals seem better). The units up front, especially the first unit, has the lead, and does what it wants, and doesn't have to look back. The slightest bit of fatigue or distraction from the pace, and unit 2 is slowed, as well as 3, 4, etc. The gaps may decrease between units. Some units stop. Some individual soldiers might struggle, or drop something, need to pee, etc.

So the rules of disciplined March, such as cadenced with marching step, leadership running through units, filling the gap, drumming/buggkes, etc are very important, and they only work sometimes, and next to never when everyone is trying to move faster..... it has to be exact, or else it will break the bulk of the army off, and it's logistical flow will open up tidal gaps..... seriously insane gaps, in the middle and ends. The troops up front will only care for themselves, arrive just fine. The guts in the middle and back spend minutes to even a hour doing nothing, then sprinting.like crazy, only to stop confused. It breaks them physically.

By adding two lanes, it helps somewhat, but doesn't help too much.... it turns into a game of leapfrogging, especially if the route is a circuit favoring a inner track, giving it a shorter length.

Its one of the problems of non-linear dynamics occurring in a linear system. Any soldier on the planet can tell you this. Squad runs..... great. Platoon runs, OK. Company runs..... getting a little gay. Battalion runs.... nightmare. Brigade runs..... looking for a doctors excuse to get out of it. Division level runs, I'm going AWOL. Seriously, I'm not dealing with that crap.

One of the advantages of the rural highway system in the US is we push for straight lines past or over city grids, rarely through. Its mostly straight, and we have too many lanes, so everyone can pick their own speed, relative to the speed limit..... right faster, left slower.

Your theory ignores Clausewitz concept of Friction. Your not considering a Entropy of Purpose arising from mismanagement. Anarchism is a fine theory, so long as it doesn't come into contact with reality. I wouldn't want a Anarchist to be a electrical engineer, or head of the department of transportation.

Seriously, just don't. I don't envy Google trying to figure out how to mass coordinate the inevitable clusterfuck of AI lead highways. It has the idea of slow moving packs down, but haven't seen it move beyond this, likely because all it's,programming is based off Army driver less studies, and the army likely hasn't allowed testing beyond a few dozen vehicles at once, knowing damn well from experience (division level runs) that the system will just collapse, resulting in one hell of a goatfuck as AI vehicles proceed to honk at one another in traffic jams, not knowing why everything is stopped, or how to get it going again. I can't imagine a line of 2000 army trucks being driven by AIs under real world conditions ever shoeing up. 10, 20, maybe 50. Its just going to snap eventually.

You'll want to read Onasander's writings on the psychology of subjective troop movements.... he was a Platonic philosopher of the Principate, I've nearly finished with a illustrated translation of his work, or Asclepciodotus, who was a Stoic under Posidendonus, who write a mathematical treatise on hypothetical unit movements. Onasander shoes you the pitfalls, while Asclepciodotus demonstrates a pure mathematical approach, with most formations collapsing on simply turning movements if anyone was ever stupid enough to try them.

The Caterpillar Effect fucks all theories that ignores non-linear behavior in linear logistics.
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### Re: When Noise Improves the Signal

phyllo wrote:It also seems that cars in lane 2 can never get back into lane 1. They can never increase their speed. So, the randomization forces cars into a disadvantageous position. Cars in lane 1 benefit at the expense of any car unlucky enough to be assigned to lane 2.

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.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
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### Re: When Noise Improves the Signal

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:
In this case, no car in lane 1 would change to lane 2 in order to increase their speed, because changing to lane 2 would not result in an increase of speed: they could move no further ahead (again, given that X = 0 and our definition of 'pass').

Moving to lane 2 traps the car at a certain speed with no possibility of gaining speed.
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### Re: When Noise Improves the Signal

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.
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### Re: When Noise Improves the Signal

phyllo wrote:
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:
In this case, no car in lane 1 would change to lane 2 in order to increase their speed, because changing to lane 2 would not result in an increase of speed: they could move no further ahead (again, given that X = 0 and our definition of 'pass').

Moving to lane 2 traps the car at a certain speed with no possibility of gaining speed.

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.

Turd Ferguson wrote: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.

His scenario is not real or realistic. So of course you have never observed it.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
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### Re: When Noise Improves the Signal

Carleas wrote:Picture an infinite [single lane] road with an infinite line of cars. Each car is traveling at a distance X behind the car in front of it, and no car will approach the car in front of it closer than X. Each car is driving at speed Y, but wishes to travel at speed Z>Y. It's clear here that no car can increase their speed to Z, no matter what the speed limit.

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.

Carleas wrote:But if the cars changed lanes randomly at some probability, the sub-optimal equilibrium goes away: some percentage of cars move into lane 2, and though they can't pass the car in front of them, every car behind them can increase their speed. And cars in both lane 1 and 2 can increase their speed when cars ahead change from lane 1 to lane 2.

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.
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### Re: When Noise Improves the Signal

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.
phyllo wrote: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.

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.

James S Saint wrote:This seems only mildly related to noise.

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.

James S Saint wrote:Did the "added noise" increase the traffic volume?

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).

Ultimate Philosophy 1001 wrote:Why can they not increase their speed to Z? If the lead car increases its speed to Z, so can't everyone else.

There is no lead car in an infinite line of cars
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Carleas
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### Re: When Noise Improves the Signal

Ultimate Philosophy 1001 wrote:
Carleas wrote:Picture an infinite [single lane] road with an infinite line of cars. Each car is traveling at a distance X behind the car in front of it, and no car will approach the car in front of it closer than X. Each car is driving at speed Y, but wishes to travel at speed Z>Y. It's clear here that no car can increase their speed to Z, no matter what the speed limit.

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.

There is no "lead car".
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
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### Re: When Noise Improves the Signal

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.

James S Saint wrote:That would cause the average speed to increase to infinity.

This seems only mildly related to noise. You significantly changed the order. That is different than merely adding noise. If you originally had two lanes fully occupied, as was the first, there would be no change possible.

Adding noise would be more like:
All of the people understood and obeyed the speed limit laws, never going faster nor slower than permitted. During high traffic hours, the highway traffic moved at a particular volume. Then one day, people began acting chaotically, ignoring the laws and traveled at random speeds.

Did the "added noise" increase the traffic volume?

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.
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### Re: When Noise Improves the Signal

Carleas wrote: 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.

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.

Carleas wrote:
James S Saint wrote:Did the "added noise" increase the traffic volume?

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).

Not in the scenario that I explained.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
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Posts: 25760
Joined: Sun Apr 18, 2010 8:05 pm

### Re: When Noise Improves the 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.

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.
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### Re: When Noise Improves the Signal

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 _
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.
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### Re: When Noise Improves the Signal

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.
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### Re: When Noise Improves the Signal

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.
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### Re: When Noise Improves the Signal

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
https://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.

https://en.m.wikipedia.org/wiki/Neurosc ... _free_will
https://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.
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### Re: When Noise Improves the Signal

https://en.m.wikipedia.org/wiki/Alien_hand_syndrome
https://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.

http://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.
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### Re: When Noise Improves the Signal

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).

phyllo wrote: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.

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.

James S Saint wrote: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.

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.

Turd Ferguson wrote:Your system falls apart in the real world.

Yes. An infinite line of cars on an infinite road is difficult to sustain in the real world.
Turd Ferguson wrote:If your theory isn't applicable, at all..... is a inherent menace in fact, to real world traffic, what is it good for?

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.
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Carleas
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### Re: When Noise Improves the Signal

2op
Picture an infinite [single lane] road with an infinite line of cars. Each car is traveling at a distance X behind the car in front of it, and no car will approach the car in front of it closer than X. Each car is driving at speed Y, but wishes to travel at speed Z>Y. It's clear here that no car can increase their speed to Z, no matter what the speed limit.

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].
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### Re: When Noise Improves the Signal

Amorphos wrote: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.

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).
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