# Traffic lane selection problem

Hi! I've been dealing with a situation like this for a couple of years now and it made me start thinking...

A city is populated entirely by infinitely logical drivers (I know, it's hypothetical :) .

This city has two major N-lane highways (N > 1) which intersect each other at a traditional interchange (there's one exit lane at the right side of the road). This is a busy intersection and a backup of some length is expected on all lanes.

At any given time, a ratio X (0<X<1) has to transfer from one highway to the other. They have to therefore get into the rightmost lane to exit the highway.

The question is this. Assume that the lanes are numbered from right to left (the exit lane is 1, the one adjacent to it is 2, and so forth up to N).

If you are an infinitely logical driver, which lane do you choose to get to your destination the fastest? There are two cases: (1) you are planning to take the exit ramp, and (2) you are planning to not take the exit ramp.

You may change lanes until you reach the traffic jam, at which point you're more or less stuck in whichever lane you choose (unless you're forced at some point to get into the rightmost lane to exit the highway).

This is trickier than it sounds. Suppose you decide that since everyone is going to be trying to merge into the lane 1 to exit the highway, you want to get to lane N, as far away as possible. The net result is that everyone will jam up in lane N.

What about randomization? Randomization may work, except that you'll need a weighted randomization algorithm to avoid lane 1 or something like that if you're trying to stay on the highway.

For people getting off the highway, an obvious solution would be to use lane 2 and drive by all the people expected to be stalled on 1 until the last minute. Of course, since everyone is logical, then lane 2 will back up.

Any ideas?

ACG

I'm not sure what you mean by "infinitely logical." In my mind, you could say that the "logical" drivers left home with sufficient time to get to their destination without becoming aggravated by the inevitable traffic jam. Or, with sufficient time to pose the question in mathematical terms while they were stuck in traffic.

More seriously, I think what's missing from your problem is the fact that each driver will observe the situation they are presented with. That's why you don't see "everyone" getting into lane N or lane 2.

Interesting problem, I wonder if there are algorithms used in roadway design or traffic control studies. Seems like it's part math, and part psychology.

Changing the time at which the driver leaves may not do much either. Remember that all of the other drivers are logical. If they all realize that the traffic at 8:30 jams up, they'll figure to go at different times. So they'll all go at 8:15 and the traffic will jam up at 8:15 instead :)

I've actually seen this happen. Supposedly everyone is so nervous about trying to go down to Cape Cod on the Friday before a long weekend that people are starting to go down Thursday and backing up traffic then too to the point that Friday may be easier than Thursday.

In practice, if Lane N is clogged and Lane N-1 is relatively free people will start changing lanes in a way which will very likely make all the jams the same length on each lane. So that part can't actually happen in real life. However, suppose that once you choose a lane (and do so far from the exit) you're committed to it (in least in the case of the cars who are not getting off at the exit).

ACG

i think the situation has to be changed so that all of the drivers do not have the same information, because if they do, then they will all make the same choice.
another possibility is that the cars in front decide first, and then the new information is used by the remaining drivers.
sounds very hard to do from a computation standpoint.

Really, you need to define what you mean by "infinitely logical." Do you mean, "makes the best lane selection"? What's the best? Gets you there fastest? Gets you there soonest? Gets you there in the most relaxed state of mind? And if it is about getting there, where is 'there'? Is it the same for all drivers? etc etc...