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Posted by Zorglub on July 22, 2014 – 8:05 am

Every day, a large number of people commute from city A to D by going through B and C. There are 4 roads
and the travel times are
A-B 40 minutes
C-D 40 minutes
The travel times on the two other roads depend on the proportion of traffic that uses them. Let p be the proportion of the commuters that use a segment.
A-C 30*p minutes
B-D 30*p minutes
For example, if 90% of the traffic goes A-C then it takes them 27 minutes.

An equilibria is reached with p = 50%, and each commute requires a total of 40 + 15 = 55 minutes on both paths A-B-D and A-C-D.

Here is the question: Construction a new road can only help with the traffic flow, right ? What happens after that a road joining B and C is constructed ?
The travel time is
B-C 5 minutes.

This post is under “Tom” and has 6 respond so far.

### 6 Responds so far- Add one»

1. 1. Chris Said：

Aargh, I totally deleted my write up. So here’s a short version.

While I find it hard to believe that no-one would be tempted by the (nearly) empty 40 minute roads, I see no real choice, everyone would take the 65 minute ACBD route. This is most likely when everyone knows the full situation. If they were quite ignorant of the situation, they would fair better.

When there is no BC link, then if the flow was better down one road than the other, some drivers would switch to it, and that eventually would bring about the optimum 45 minute time for everyone. When the flows were equal, there would be no preference for either route.

I can’t picture the scene well enough to determine what I think might really happen.

2. 2. Karl Sharman Said：

Well Chris, I have studied this problem in great detail. No.1 In England, the road opening would be delayed due to the wrong type of leaves on the road, or weather conditions.
No.2 The moment the road opens, one of the other roads would close for re-surfacing, taking this road out of the equation for x months.
No.3 Should all 4 roads be operational at the same time – there would be restricted flow on some roads for some of the time to allow for maintenance.
No.4 Speed cameras would pop up, thereby affecting traffic flow.
No.5 As can be shown from the M25 round London – any circular route around a town or city will, de facto, become a parking lot.

Taking into consideration the above outlined variables, this problem becomes too difficult to solve… but nonetheless I shall try.

3. 3. Karl Sharman Said：

Just to make sure I am reading this right…
To get from A to D you have two choices, before the new bypass is built – ABD or ACD
After the bypass is built you have some additional choices
Original – ABD
Original – ACD
New – ABCD
New – ACBD

There is no correlation between the percentage use of AC and BD, but rather ACD and ABD

Thus it leads me to think that if you know the time it takes to travel AC you know the time it would take to travel BD, and would swap if appropriate by taking CB, but then so would everyone else….

If 0% of the traffic is using AC or BD, would that mean that travel is instantaneous? Although would my car represent 1%, or more or less?

I am beginning to sound like Vissini vs The Dread Pirate Robert in Princess Bride. I’ll shut up.

My best guess – traffic volume would even out, and ACBD would win out.

4. 4. Chris Said：

Hi Karl. I can’t see why the traffic would even out (as per your final statement). Except for some unusual (either unaware or aware and selfless) drivers, only ACBD would be used. Assuming that’s approximately true, then ACBD takes 65 mins, ACD and ABD take 70 mins, and ABCD takes 85 mins. Only a nutter would take ABCD though.

5. 5. Zorglub Said：

No one goes AB because it requires more time than ACB.
No one goes CD because it requires more time than CBD.

Thus, everyone will end up going ACBD for a total of 65 minutes.
Yet, when there were fewer roads, it took everyone 55 minutes.
To was very counter-intuitive (at least for me).

This is knows as Braess’s Paradox, and is nicely explained here: