Well …
I’m not sure where the “paradox” comes into this, but the solution is option (C);
A) the trivial math
B) the infinite math
C) how far do you want to go.
I can explicate each case, but I’m not going to bother detailing the cases.
The trivial case b[/b] is the fact that you are merely proposing two lanes of flow versus one. This is an issue of energy compression, the formation of inertia, and relates to why the universe exists. But I suspect that isn’t the part of it you had an interest in (I can work RM:AO into anything … [size=85]snicker[/size]).
The math for the two lanes vs one is simply that the slowest car in any lane dictates the speed in that lane. Given that no two cars can be going at exactly the same speed, the slowest in one lane will be higher than the slowest in the other lane. The overall traffic flow is a result of the slowest car(s). The one lane scenario dictates the the slowest car sets the flow rate. The two lane scenario averages the slowest of each lane into a flow rate that is necessarily higher than the slowest car on the road. Thus two lanes will always yield a higher flow rate (as if everyone didn’t already know that).
Case b[/b] involves the more realistic variances of particulate flow of conscious entities, the perception of hopes and threats underlying all gaming scenarios. Many issues come into play such as their; perceptions of speed related hope and danger, perception of compression danger, worse case memory (any one traumatized driver affects all others), resonance harmonics and spacing (traffic flow is a spring model in mechanics), the mean, best, and worse case reaction times, religious mix (“cooperative gaming”), the information network (who is being informed what as the scenario progresses), and so on.
So the correct answer is option b[/b]; “How far do you want to go?”
The perceived problem is not solved by allowing ‘undertaking’, as that has to be used for cars entering and leaving the motorway.
In my experience this problem is caused by people 1) adhering to the speed limit, or in some few cases 2) by people not knowing the correct protocol of the passing rule: that movement into the second and subsequent lanes should only be done when passing is required. Where no passing is possible, cars should remain in the lane they are in. But overall, the problem is the result of traffic volume - not solvable by removing overtaking rules.
The system would works perfectly, where no speed limit applies as in the Autobahn’s in Germany.
This was not given, and is not a necessary premise.
It is paradoxical in that, even though the result is suboptimal, and there is a way to change strategies so that everyone benefits, no one has any rational reason to change strategies. In some sense, it’s obvious what everyone should do, but it’s also clear that no one has any individual incentive to do so.
Even if it were truly the case that no cars can travel in lanes in which cars are entering and leaving, undertaking still solves the problem for most sections of highway, i.e. all sections where there is not an onramp or exit. Sections where cars are entering or exiting can be seen as any other section of highway where the number of lanes is reduced to 1, and then it’s trivial to say that undertaking won’t solve the problem on a 1 lane road (on which there is no lane in which to undertake).
This problem does not depend on the speed limit.
Picture an infinite 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. However, undertaking solves the problem by giving people an incentive to change strategy. If undertaking is not allowed, a person’s maximum expected speed if they change lanes is Y, the speed of the car already in front of them. If undertaking is allowed, their expected speed is greater than Y.
I think the speed limit it the fastest you’re supposed to go. Sometimes, when I’m in the middle lane and going the speed limit and someone passes on the right, I scream at them…“if you want to break the law do it in the fast lane like a man you coward!” Other times, I hit my right blinkers and pretend like I don’t see them, then swerve in front of them a bit to condition them to pass appropriately. I got an insurance check once from a trucking company where this guy was passing on the right. I was hydroplaning and clipped him and spun into the median. They had to pay me the bucks. Good times.
When the distances are infinite, all distances are infinite. Your model is impossible. You have asked me to consider an infinitely long piece of rope, then you have asked me to divide the rope. What do you get? Two infinitely long ropes.
But to play your game, if people actually obeyed the protocol and moved back after overtaking as they should then the “fast lane” would only be blocked when the “slow lanes” are already at total capacity, and no undertaking would ever be possible, in any event.
it has always appeared to me that the jams are either caused by people who are ignorant of the protocol, or by those who “road-hog” at or just above the speed limit.
I’ve passed driving tests in the UK and the US. The US test (State of California) is complete shit and poorly requires the driver to know as much as they ought to know. I doubt if half the people I saw going through the process know how to behave on the freeway.
I am saying that whether or not I acknowledge the problem, it is irrelevant. In sections where undertaking cannot be done, be it because there is only 1 lane or only 1 lane can safely be used, it is trivially true that undertaking won’t solve the problem. Those sections are the exceptions; here, we’re talking about stretches of road with two useable lanes.
This is irrelevant. I’m talking about a situation where all cars are in the overtaking lane, all want to go faster than they are going, and none can because they are as close to the car in front of them as they are willing to be.
Of course the model is impossible, but then it’s not a road, it’s a model of a road, and it suffices to show the principles at play. An infinitely long rope can be cut into sections, a finite section of rope can be cut from the middle of an infinite rope to leave two infinitely long ropes (but bounded infinities, because we’re holding the ends). None of this is impermissible in constructing a simple model. And in the American Midwest, it is not unreasonable to treat your average stretch of two-lane highway as an infinitely long, straight, and flat.
SO you don’t think it is evenrelevant that people don’t understand the protocol.
If you don’t understand the reason for the problem, nor wish to consider it, then you don’t have a snowball’s chance in hell of offering a solution; which at this moment seems to be to dismiss all rules.
I do not think that the rule is the problem.
But I think that having no rules is not a solution.
You might as well say; why do we have to drive on one side of the road at all. I’d be able to get by much better if I could drive on the left and the right!!!
The overtaking protocol is merely a simply extension of the rule to stay on one side of the road.
As I said in the OP, the reason I posted in this forum as opposed to SG&E is that I’m interested in the math side of this question, rather than the policy side. Which is why I’m doing things like positing infinite two lane roads with no onramps or exits. I’m not interested in solving a policy problem, but in analyzing a paradoxical suboptimal equilibrium. There are clearly certain policy implications for this analysis, but just as the prisoners dilemma can be analyzed apart from the practical consequences of the Geneva Convention, so too can an infinite road where all cars are in the fast lane and want to travel faster be analyzed with and without a rule against undertaking and without reference to road safety statistics.
Since you didn’t accept what I explained, apparently you mean something different that I am not seeing. Could you explain that quoted part in more detail?
The social sciences have been modeling human behavior for decades. And even a simple model can include ignorance as a premise, e.g. if there’s $500 hidden under a rock beside you but you don’t know it’s there, I have a robustly predictive model of how your net worth will or won’t change.
So I feel like I must be misunderstanding you, because as I understand it your statement is plainly false. Is there some nuance I’m missing?
The result is suboptimal, given (and this was given) that a higher speed would be more valuable to all participants. And a different set of strategies would achieve a better result: if half the cars moved into the other lane, the speed of traffic would increase (every driver wants to go faster, and is willing to come within X of the car in front of them, and if every other car moved into the slow lane, the distance would be 2X, so they would accelerate). However, no individual car would choose to switch lanes because the car in front of them that is limiting their speed would continue to limit their speed when they changed lanes.
The paradox is thus that everyone wants the same thing, there’s a set of strategies that would achieve it, but if people behave strictly rationally they won’t ever achieve the optimal set of strategies.
Carleas, I agree that speed limits are kinda dumb, but as an advocate for privacy who is aware of what the police typically consider probable cause cor a search, I feel like the smart thing to do is to stay under he speed limit. I dunno about you, but in my life, almost every time I’ve ever been pulled over, my car was searched. Nothing was found, ever. I felt raped. So now I’m the douche bag who obeys the speed limit. Even on long trips with hundreds of miles of interstate. I just don’t speed because I want my rights to be in effect at all times.
Yes, social science is not predictive, only descriptive.
The trouble with what you are saying is that you have already rejected the human factor in favour of maths.
What you can’t do is say how many are ignorant, drunk, changing channels on the radio, spilling their coffee, or getting a blow job.
I think it’s predictive, but in a limited way. I mean, it’d be hard to prove that not knowing the 500 bucks was under the rock would change your finances.
It isn’t a premise, it follows from a rule that stipulates you can only pass on the left. If you can’t pass on the right, there’s nothing to be gained from switching lanes, so there is no incentive to change strategies.
This also seems to be clearly false. There are many predictive models about how e.g. traffic will flow. That’s how the timing of traffic lights is established. There are a million such models that use math to model human behavior and make predictions about it.
Yes, I’ve rejected the human factor, just as we don’t complain when someone tells us “1+1=2” that that ignores that a man and woman can produce babies, so 1+1 can equal 4 if the condom breaks and there’s a family predisposition to twinning.
Again, the Prisoner’s Dilemma isn’t about prisoners, it’s about decisions and games. The “prisoners” are stand-ins that let us talk about expected outcomes and decision strategies and optimization in a certain kind of strategy game. Similarly, the “cars” here are ideal cars, whose sole ambition is to go faster than they are going, and whose constraints are the cars around them, the lack of additional leftward lanes, and the rule that prevents passing on the right.
Sorry for being so dim, but I’m still not getting it.
You said;
“no individual car would choose to switch lanes because the car in front…”
Did you mean that they typically won’t, which would be more realistic?
But even with that, I still don’t get any paradox issue out of this.
People could (if allowed) gain by changing to the slow lane (and many actually do that). And because some do, traffic flows a little bit faster, even if more dangerously so. But even if none of them ever did switched and thus didn’t gain anything, what is the “paradox” or even “issue”?
It’s a stupid road rule. You found a stupid road rule. Good for you.
If only none of the lanes were for going faster, it might actually make for less of a road jam. Having that rule in place makes drivers think they have the right to speed up, and that leads to even more danger on the road.
I would say break the rule, and you’re more likely to get where you want to go a lot faster.
People get a rush from following that rule, a rule that allows for extra SPEED. Breaking the habit would be life changing, and like I said, ironically, a faster way.
