As for the specific freeway/back roads distinction, I think the analogy can go either way. The bigger point is that local incentives can result in global inefficiencies–termed “the price of anarchy.”

So what is one to do in practice? I think commuters would tend to minimize risk. That is, they don’t just take the path of expected risk, but one that has a reasonable travel time with low variance. Just my guess

As for the problem in its beginning form… I like to think of it this way. The “T/25″ roads are freeways, which are faster and often times more direct, while the “50″ roads are “back roads” ie, slower, but due to speed restrictions, they are safer and congestion has a negligible effect on drive times. It’s just a different way of looking at it, but it gives the same result–if a new freeway is built which bypasses back roads, then pretty much everybody will use it (especially in winter here in Michigan), and drive times will slow for everyone.

Mike: By social planning I meant “planning of better designed systems.” In Illinois, for instance, when the government plans “congestion relief” it almost always amounts building more roads–which as we see can backfire if you do things the wrong way! I hope they will be more creative in the future…

Hinheckle: One possible answer: imagine all the roads are freeways but the “narrow” ones are ones with lanes closed for construction. With few cars traffic flows fine, but with many everyone has to slow down for merging.

Kalid: Great point–this does seem like a prisoner’s dilemma where each party has a dominant strategy that leads to a suboptimal outcome.

Scott: You’re right–it is the precise locations of the roads that determine if the system will fail. I also was thinking that widening roads would seem to reduce traffic times, but often times it is not possible to widen some roads (businesses or rivers might be in the way).

gilltots: Yes, the roads were somewhat hazardous to drive so I was saving time since the roads were empty…Good point about adding traffic lights and signs.

Scott: I’ll have to check your math but that’s a very interesting idea how flexibility can help in the presence of accidents.

Accidents can slow the roads in unusual ways. There is another phenomenon called “gaper’s delay” in which a traffic jam is caused not by obstruction but instead by drivers slowing down to see the damage of the accident. Happens all the time on Illinois highways.

Let’s say an accident on a path increases the time by A. Using a population of 1000 and going with the original scenario (no A-B road), an accident on a single road alters the equilibrium.

The people on the accident-free road will be 12.5*A + 500.
The people on the acciident road will be 500 – 12.5*A.

The time for each path will be A/2 + 70.

Let’s call the drive time when there is an accident Ta.

Now, lets say an accident occurs every N times you take a path. Your overall average drive time is:

(70*(N-1) + Ta)/N

Now let’s plug all this in to the second scenario. Like before we divide it into two parts. For simplicity we will say the accident only affects travel on the half it occurs. So for the half where the accident doesn’t occur, everyone will take the 40 minute route.

So all we have to do is calculate any new equilibriums.

If the accident occurs on the flat 50 minute route, people will still take the 40 minute route, which will not change the overall travel time of 80 minutes.

If the accident occurs on the 40 minute route, then this depends on the value of A. If A > 10 minutes (which isn’t unreasonable), then people will go to the 50 minute route for an overall travel time of 90 minutes.

If an accident is as likely to happen on one side as the other then the average time only changes every other accident (on average). This makes the overall average:

(80*(2N-1) + 90)/2N

The purpose of this is to see under what conditions the second scenario is, on average, better than the first. So we see when the overall average time of the first scenario is less than the overall average time of the second scenario:

(80*(2N-1) + 90)/2N < (70*(N-1) + Ta)/N
(80*(2N-1) + 90) < 2*(70*(N-1) + Ta)
160N – 80 + 90 < 2*(70N – 70 + Ta)
160N + 10 < 140N – 140 + 2Ta
20N – 2Ta < -150

If we recall, Ta = A/2 + 70
20N – 2(A/2 + 70) < -150
20N – A – 140 < -150
20N – A 10

So when the increased time due to an accident, subtracted by 20 times the number of days between each accident is greater than 10, the second scenario is better. Let’s try it out. Let’s say an accident happens every other day and increases the travel time by 51. That makes A – 20N = 11.

The average travel time of the scecond scenario becomes:

(80*(2N-1) + 90)/2N = 82.5 minutes.

The average travel time of the first scenario becomes:

A/2 + 70 = Ta
95.5 = Ta
(70*(N-1) + Ta)/N = 82.75

So on this accident ridden road, adding A-B would increase your average travel time by a quarter of a minute. Now, this is not much of an improvement and hour-long accidents happening every other day is a bit extreme, but the scenario is meant to be an abstraction and shows that, under certain circumstances, being able to switch routes may make the situation better.