How Making Something Better Can Make Everything Worse

February 19, 2018

On a recent commute, I had the pleasure to listen to a fantastic talk by Tim Roughgarden on game theory and algorithms. Check it out.

 

A highlight of the talk—for me, anyway—was the discussion of Braess’s Paradox.

 

Think of the following transportation network:

Here, we have two cities, A and B, and some roads connecting them.

 

There are two ways to get from A to B. If you take the upper route, travel time to Junction 1 depends on how many other people choose that route. If 50% of people living in City A take the upper route, it’s a half-hour drive to Junction 1. However, if everyone takes the route, the road becomes congested, and it takes a full hour. Travel time from Junction 1 to City B is always one hour: It’s a big highway that never gets too crowded. The lower route is similar, except that you first take the big highway and then the smaller, congestable road.

 

For this network, the unique Nash equilibrium is that half of the people will take the upper route, and the other half will take the lower route. That adds up to an expected travel time of 1.5 hours for everyone.

 

So far so good.

 

But now suppose a teleportation device is installed between Junctions 1 and 2:

 

Now comes the trick question: Will travel time decrease?

 

It turns out that the answer is no. Expected travel time will go up by half an hour. Everyone will now take the congestable road from City A to Junction 1, get on the teleportation device, and then follow the second congestable road. Although drivers are doing the best they can to reduce travel time, they are now all facing a longer commute.

 

In other words, making something better (adding a teleportation device) can make everything worse (travel time goes up for every driver). And who said that economics was dismal!

 

(Quick side note: An interesting aspect of the example is that the initial equilibrium was efficient. So Braess’s Paradox is not just the theory of the second best in disguise. Of course, that’s not to say that there are no parallels in economics. For example, increasing trading opportunities in financial markets may reduce welfare. This idea plays an important role in some of my own work.)

 

Anyway, fascinating stuff. If anything, Braess’s Paradox should at least make you a little bit suspicious that self-driving cars will solve all our commuting problems!

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