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.
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.
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.
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.
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.
It seems that you do not understand calculus. Calculus is about comparing infinities. When you compare two infinities (the infinite number of points and the infinitesimal time it takes to traverse each of them), you can have a finite ratio, a finite velocity. Zeno has no actual paradoxes.
No i do understand the philosophical basis of calculus, but understanding how the universe can be infinite and finite is to me the most important question of our time. I dont think calculus answers it so i am taking it as something [and combining the meaning in this thread] to build off of ~ to ask further questions.
Consider that in real terms we get to relativity and qm before we can think of potential particles in infinite metaposition via calculus. So already the cardinality has broken down beyond maths as we understand it. It seems a tad strange or illogical even, to then denote singularity to the cardinality of what lies beyond qm/r. The answer wont be points and spaces, but more like an elastic band which can be stretched infinitely - probably to the point of transparecy.
Ergo I am attempting to further the debate rather than simply reiterating the given/‘known’.
There is no assumption that the drivers are perfect logicians.
No car benefits itself by changing lanes, it only benefits the cars behind it.
Each driver knows that if the drivers in front of them flip coins, they benefit. No driver has to participate in the coin flipping in order to benefit. Indeed, they have no reason to participate, as their participation does not in any way affect their own outcome.
You are adding noise by adding coin flipping, which is exactly my point: if we add noise, the outcome improves.
I’m not interested in talking about the Blue Eye Problem. If you want to, please take it up in one of the threads where we’ve discussed that problem, I’m happy to continue our conversation in an appropriate thread. Your objection to Nash equilibria on the basis that all Nash equilibria are self-defeating because they rely on common knowledge is also beyond the scope of this discussion; if it helps, I’m happy to acknowledge that my argument here is dependent on the assumption that Nash equilibria are not a self-defeating concept.
Amorphos, I believe the outcome I describe still results if we restrict ourselves to discrete states and discrete math. Say the cars are composed of pixels, and time ticks, and the cars jump ahead Y pixels per tick, but would like to jump ahead Z>Y pixels per tick. I think the result should be the same: if the cars randomly jump to lane 2 with some probability, the cars will be able to jump ahead Z pixels per tick.
The Nash Equilibrium requires it by specifying that there is no better decision possible and that every member knows it. Only perfect logicians could know that.
That is in a quasi-“yes and no” area. No car detracts from itself by changing lanes. It would be a zero-loss, zero-gain except by choosing to make the decision by a random method, all cars that are behind every car benefits. Logically, that inherently includes every driver.
What do you call it when there is zero risk of loss and zero prospect of gain … unless some choose to do X at which time it becomes zero risk of loss and 100% chance of gain?
By more than one understanding that game scenario, they know to give themselves the chance of gain because there is no risk and the alternative is zero gain.
The Nash Equilibrium requires that they all already know that. And to me, that means that they would all choose to flip the coin, or something similar.
For the above reason, that isn’t really true.
By “participating”, every knows that everyone gains.
It is similar to the situation of the solder. In an army, every solder shares a risk and thus allows for every solder to win the war. From the perspective of each individual, there is great risk and no benefit by fighting in a war. There is potential maximum loss (death) by fighting and no direct gain other than a paycheck that could have been gained by other means. By every solder being an altruist and participating in the risk, the risk is reduced, potentially to zero, and the gain is the shared booty (whatever the war was about) acquired by team work.
I agree. And I stated that … twice. Why are you arguing with me?
If the question is “will a driver benefit by changing lanes?”, the answer is a solid “no” where doing so is zero-gain.
And again, even in your hypothesized solution, the expected gain for an individual driver in changing lanes is zero. For driver D, the difference between everyone flipping a coin and everyone-but-D flipping a coin is zero.
It’s true for any driver D that, if all the drivers ahead of D changed their strategy, D would benefit. But that does not entail that D benefits by changing strategies. We know that D does not.
If we assume that D wants to both go fast and be altruistic, we’re just changing the meaning of ‘benefit’: we’re adding a new value that drivers are seeking to maximize, and so it isn’t surprising that we’re destroying the equilibrium.
They are all required to be the same. They all know the same scenario and all know that they all know it. There is no “D” acting differently than the others. So whatever D does in the way of decision making, the others also do.
This is not a given and does not follow from the givens. Nothing about the construction of the scenario requires that if one car has a thought, they all have the thought. They aren’t clones or computer programs (they aren’t people either, they’re cars). They’re players in a game-theoretic scenario, and their motivations are fully specified. It’s perfectly consistent for them to be different in innumerable ways so long as they otherwise satisfy the criteria in the scenario.
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