One day we must stop building new roads. When?

This simple example shows how adding a road to a network can slow everyone down, even with the same number of cars using it.

Figure 1 shows a simple road network between two points *A* and *B*. The cost of travelling down a road is shown as a function of *x*, the number of cars using it. (If you like, you can imagine that the formula *5x+1* between *B* and *C* is a road of length *1* with a width of *1/5*.)

Figure 1. Simple road network.

Suppose there are six cars which want to travel from *A* to *B*. In true free market fashion, everyone chooses the route that is best for them, i.e. the route with the lowest cost.

- It doesn't matter what the first car chooses. Let's suppose it goes
*ACB*. - The second car will choose
*ADB*because there is a car on*ACB*. - It doesn't matter which route the third car chooses because the network is evenly loaded. Let's imagine it chooses
*ACB*. - The fourth car will choose
*ADB*. - The fifth car could choose either route (see 3), but let's say it chooses
*ACB*. - The sixth car will choose
*ADB*.

The road network is evenly balanced, and we can calculate that the cost for each car is *5x+1 + x+25* with *x = 3* because there are three cars on each road. The cost to everyone is therefore 44.

This is the best solution, and it is *stable*. Adding more cars will have an obvious effect. If you add a car and dictate its initial root, the other cars will divert until this same solution is reached.

Now let's add a road between *C* and *D* so that people can switch routes to choose the less congested one. Let's also be generous and make the cost of this road always 1. (In other words, this road is very short and can take any number of cars without getting congested.) Figure 2 shows the new improved network.

Figure 2. Network with new road.

The same six cars want to travel from *A* to *B* as before.

- The first car will chosse
*ADCB*because it has a cost of*5x+1 + 1 + 5x+1 = 13*. This is much better than before! - The second car will do the same.
- The third car will do the same.
- The fourth car will do the same.
- The fifth car will notice that
*ACB*is now a better option, because*x+25 + 5x+1 = 52*which is less than the congested*ADCB*route which now costs*5x+1 + 1 + 5x+1 = 53*. - The sixth car will choose
*ADB*for similar reasons.

So, we have four cars using *ADCB* at a cost of 53 each, one car using *ACB* and one using *ADB* at a cost of 52. But they were all 44 before, so everyone's costs have gone up!

This solution is also stable. In fact, it doesn't matter which routes the cars start with. If each car changes its route selfishly you'll end up with this solution anyway.

Why doesn't everyone just ignore the new road? Well, everyone is choosing what's best for them, and that leads to a sub-optimal solution. In order to improve things they'd have to get together and impose some regulations on themselves. The free market simply doesn't optimise the network. (This is a classic *prisoner's dilemma* problem, by the way.)

The network above wasn't carefully chosen (except to be simple enough to explain) and neither were the formulae for the road costs. Try some of your own. This sort of thing happens all the time, and it's hard to believe because it seems so counter-intuitive that more roads mean less efficiency.

Last modified 1997-05-10. Copyright © 1997 Richard Brooksby.