Eric Cavalcanti writes regarding
http://plus.maths.org/issue17/features/traffic/index.html:

> I agree with the general conclusion:
> "when we randomly select a driver and ask her
> whether she thinks the next lane is faster, more
> often than not we will have selected a driver from
> the lane which is in fact slower and more densely
> packed."
> ...
> From another perspective, I have just arrived at the
> road and there was no particular reason for me to 
> initially choose lane A or lane B, so that I could just
> as well have started on the faster lane, and changing
> would be undesirable. From this perspective, there
> is no gain in changing lanes, on average.

That's a good question.  One thing I would note is that if everyone
entering the road chose between the two lanes with equal probability,
and stayed in their lane, then neither lane would be more crowded
than the other.  So to some extent your premises are contradictory.
If everyone behaved like this, one lane wouldn't be faster than the other.

> Extending the argument, suppose I drive for a couple
> of miles, and get to another point where I want to decide
> if I should change lanes. Since I had no reason to
> change lanes a couple of miles ago, I still have no reason
> to do so now. Unless, of course, I can clearly see that
> the next lane is faster, but adding that assumption changes
> the problem completely.

I think this is true as well, assuming you have not changed lanes yet.

Let's go on and suppose that you drive for a while and change lanes
occasionally based on how the traffic seems to be moving at that moment,
and that you are a typical driver in this regard.  Then your alarm clock
rings and you ask yourself, am I more likely to be in the more crowded
lane.  I think you will agree that in that case, the answer is yes.

Does this resolve the paradox?

Hal Finney

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