Comment by ars | Hacker News Reader

Comment by ars | Hacker News Reader

Parent Full thread

Maybe this paradox is real, I don't know, but it seems quite contrived. I don't think congestion really changes based on T/constant. It's almost certainly a log function.

And try this:

If just one person goes from start to end it will take him 4.8 seconds (.08 minutes).

So just line everyone up, and let them go one at a time. The last person in line will have to wait 80 minutes, but the average travel time is 40.04 minutes!

So adding a road does make it faster - much faster, contrary to what the paradox says. All you need is someone to police them and make them go one at a time.

And you are even waiting till they finish the trip before sending the next person. Send two at a time (one for each leg, since they are independent after all), and it's twice as fast.

I say putting in the road was a great idea! You cut the average travel time from 70 minutes to 20.02004 minutes. And the max from 70 to 40.08.

If I'm right, this does bust the paradox, but even more than that it shows that T/constant is the totally wrong formula for calculating congestion. Put in a correct formula and I bet this paradox won't exist.

I think your example is a little too limiting for this paradox to take effect. The paradox is based on people guessing what others are going to do given a set of options, where your example basically sends them through in the most efficient manner possible by suggesting that one person is going to be willing to wait 80 minutes when that person knows that others are able to do the trip in less than 10 seconds.

One of the more interesting highway behaviors I have seen was a huge backup on a 2 lane highway where truckers would block both lanes and prevent people from passing them. It's a slight cost the each trucker on the left but it significantly increased throughput.

In the blogs example one person 1 goes Start to A, then while person 1 goes B to end Person 2 goes start to A. etc. Worst case travel time is now faster after adding the new road and 2 agents can help this happen by limiting how fast people enter A.

Edit: with 2 people limiting traffic they can reduce their travel time by 25% (3x 1/2 vs 2 x 1) even if they are in the second group. 1/2 the cars do the first leg, then the other half do the first leg while the first set does the second leg.

PS: This is why we make traffic lights.

Exactly, not everyone will willingly comply. You (ars) have found an optimal solution, but it is not a nash equilibrium - cars will cut out of the line to make it through faster.

The constants in the example are arbitrary and were probably chosen for simplicity, a (thankfully) common thing to do in these sorts of examples.

> All you need is someone to police them [...] If I'm right, this does bust the paradox.

You completely misunderstand the nature of the paradox. Yes, if you have an authority (the "police") compute and enforce the social optimum, then more roads can only ever decrease the average travel time. That's obvious, and pointing that out didn't "bust" anything.

The nature of the paradox is that optimization at the individual level leads to a suboptimal result at the social level. That's in contrast to certain results from economics, where free markets can be proved to guarantee optimal resource allocation (in idealized circumstances).

Your complaint about the T/constant formula is downright silly; I'm convinced that an equivalent paradox can be created for every reasonable formula you agree to. (Actually, I think I can prove that.) The only thing that would change is that the "lines" on the equilibrium diagram would become "curves".

>You completely misunderstand the nature of the paradox.

You are right - I did, but then it was not said anywhere, and this is the first I'm hearing of this.

I understand that it's a mathematical/game theory paradox, but it's listed as a real world paradox - and it's not.

> Your complaint about the T/constant formula is downright silly; I'm convinced that an equivalent paradox can be created for every reasonable formula you agree to. (Actually, I think I can prove that.) The only thing that would change is that the "lines" on the equilibrium diagram would become "curves".

I think you can't. If you modeled real traffic congestion for this the paradox would not exist. Maybe I'm wrong, but I don't think so.

The "real world" examples listed on the page were not because of this type of paradox, but rather because in the real world it takes time to merge, so removing a merge point makes things faster.

But a road paradox structured like this one? I don't think you can. The whole paradox is because the T/x does nutty things as T changes (nutty as compared to the real world).

At the minimum I want to see f()+c. And f() should be an s curve.

Reply on news.ycombinator.com