Moderator: Flannel Jesus
Carleas wrote:(This is sort of policy-ish, but I'm posting here because I'm interested in the math side of it.)
In the US, passing on the right is generally prohibited. On the highway, that means that the left-ward lanes are 'fast lanes' and the rightward lanes are 'slow lanes' (they all have the same speed limit, but the leftmost lane will tend to be 5-10 mph over the speed limit, while the rightmost will be at or below the speed limit.
Riding on a two lane highway the other day, I experienced a situation in which this policy led to an inefficient use of the road. Traffic was going at or below the speed limit, but moving consistently. Everyone on the road seemed to want to faster, so they moved into the left lane. But since each person could go no faster than the car in front of them, this led to a line of traffic going less than the speed limit.
The solution to the problem was to pass on the right: passing on the right would have allowed half the cars to move into the right lane and go as fast as they could. This would necessarily increase the speed because it would double the throughput of the highway if cars approached each other to the same distance they were approaching each other when there was a solid line of traffic in the left lane!
Carleas wrote:The math is in the paradox that when everyone wants to go faster, no one can. It's a case of Nash equilibrium: no driver can go faster by changing her strategy, even though the result is suboptimal for all drivers (again leaving aside policy considerations; if passing on the right would increase the number of accidents, the paradoxical situation could well be optimal).
Carleas wrote:The math is in the paradox that when everyone wants to go faster, no one can. It's a case of Nash equilibrium: no driver can go faster by changing her strategy, even though the result is suboptimal for all drivers (again leaving aside policy considerations; if passing on the right would increase the number of accidents, the paradoxical situation could well be optimal).
James S Saint wrote:Given that no two cars can be going at exactly the same speed
James S Saint wrote:I'm not sure where the "paradox" comes into this
Lev Muishkin wrote:The perceived problem is not solved by allowing 'undertaking', as that has to be used for cars entering and leaving the motorway.
Lev Muishkin wrote:The system would works perfectly, where no speed limit applies as in the Autobahn's in Germany.
Carleas wrote:Lev Muishkin wrote:The perceived problem is not solved by allowing 'undertaking', as that has to be used for cars entering and leaving the motorway.
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.
I think this might be faulty logic, mainly because you are not acknowledging the problem of undertaking the "right-most" lane.Lev Muishkin wrote:The system would works perfectly, where no speed limit applies as in the Autobahn's in Germany.
This problem does not depend on the speed limit.
You have not thought it through. Consider why it is that the "overtaking lane" is blocked in the first place. This is often due to morons not wanting to exceed the speed limit. overtaking and rigidly sticking to what ever dumb speed the US still imposes blocks the "fast lane" is what often causes the blockage.
What is the Speed limit on US roads these days, pray tell!
WIth no limit, fast cars 120-130 mph often push the slow coaches out because people tend to be more reluctant to block the 'fast lane".
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 care 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.
Lev Muishkin wrote:you are not acknowledging the problem of undertaking the "right-most" lane
Lev Muishkin wrote:You have not thought it through. Consider why it is that the "overtaking lane" is blocked in the first place.
Lev Muishkin wrote: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.
Carleas wrote:Lev Muishkin wrote:you are not acknowledging the problem of undertaking the "right-most" lane
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.Lev Muishkin wrote:You have not thought it through. Consider why it is that the "overtaking lane" is blocked in the first place.
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.
This is often due to morons not wanting to exceed the speed limit. overtaking and rigidly sticking to what ever dumb speed the US still imposes blocks the "fast lane" is what often causes the blockage.
With no limit, fast cars 120-130 mph often push the slow coaches out because people tend to be more reluctant to block the 'fast lane".[/color]Lev Muishkin wrote: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.
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.
Carleas wrote: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.
Carleas wrote: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.
Lev Muishkin wrote:Maths is not relevant when unpredictable humans are at stake.
You can't model for ignorance.
James S Saint wrote:Could you explain that quoted part in more detail?
Carleas wrote: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.
Carleas wrote:Lev Muishkin wrote:Maths is not relevant when unpredictable humans are at stake.
You can't model for ignorance.
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?
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James S Saint wrote:If that is a premise to the scenario then you have dictated the outcome. What is there to discuss?
Lev Muishkin wrote:Yes, social science is not predictive, only descriptive.
Lev Muishkin wrote:The trouble with what you are saying is that you have already rejected the human factor in favour of maths.
Carleas wrote:James S Saint wrote:If that is a premise to the scenario then you have dictated the outcome. What is there to discuss?
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.
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