# 31

Braess's paradox shows that in some cases, adding roads can make everyone's commute longer. (Pause here if you want to figure out on your own how that might be possible. The general situation is that everyone just tries to find the route that's quickest for them personally; and some roads take longer the more cars are on them.)

Roughly, an example of the paradox is as follows: everyone wants to get from one place to another place. There are two routes: road B followed by road B', and road C followed by road C'. Roads B and C' take a long time, which doesn't depend much on how many cars are there. Roads B' and C take a short time, given the current number of cars, but they take signficantly longer the more cars are there. Now, say you add a road D which is very short and connects the end of C to the beginning of B'. Since C, D, and B' take a short time, lots more cars take the new route C-D-B' rather than the old B-B' or C-C'. For some traffic-to-drive-time curves, the equilibrium of this ends up being that *everyone's* drive takes longer than before D was opened. (See wikipedia for details.)

What's even going on? We can rephrase the situation like this: each driver is presented with a choice. They can either take their old route, or they can take the new C-D-B' route, which is faster than their old route but causes everyone else a slightly longer drive. (I think for some settings of the numbers, this would be a sort of many-player Prisoner's Dilemma.) We could say, there's an unpriced externality of driving on C or B': you clog up the roads. If everyone disregards the externality, everyone ends up worse off.

This all seems like it should happen in lots of different contexts. Really, anything with lots of agents taking multiple pathways, some of which induce externalities. Wikipedia gives some examples, but doesn't give examples of this occurring in economic markets. So my question: are there examples of Braess's paradox in markets? In other words, are the examples where introducing a new good/service could make things broadly worse because of externalities from "overloading pathways"? Are there really big examples or ubiquitous classes of examples?

I'm not sure it's joint-carving to talk about Braess's paradox as separate from just externalities. The thing I have in mind is something like, specifically the externalities from raising the price on a good/service with an inelastic supply. In other words, we have the same situation sketched above, except now we have goods B and C' which are expensive but have an elastic supply (the production is expensive but can be extended to match demand); goods B' and C are cheap, but have a very inelastic supply (perhaps they rely on a fixed resource like a rare material or specialized knowledge, which at the moment is enough to meet demand); and many people very much want something you can get either by having B and B', or by having C and C'. Then, someone introduces good D, which is very cheap and supply-elastic, such that you can satisfy the same want by having B', C, and D. The inelastic supply is maxed out, and the price for B' and C skyrockets, leaving everyone worse off. Does this specific thing happen a lot? How elastic is supply, in general (a ridiculously broad and vague question, but still)?

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Vitor

### Dec 29, 2021

140

Perhaps this is a more theoretical answer than what you're looking for, but this kind of thing can be a problem in combinatorial auctions, even though they're designed to find a globally optimal allocation of goods.

Just think of the road segments as individual goods, with bidders (drivers) making bids on bundles of roads they intend to use. The auction then finds the best possible allocation of roads (maximizing the sum of bidders' reported utilities), and charges the winners some amount of money via a pricing rule (see below).

A pricing rules often used in auction is VCG (Vickrey-Clarke-Groves). Instead of paying your bid on the bundle that you won, you just have to pay the externality you impose on others by participating in the auction, which is always less than your winning bid. This is super nice because it makes the auction strategyproof, i.e., it's a dominant strategy to reveal your true values for each bundle of road segments.

Now, you would expect that when more goods are introduced, the competition for those goods must decrease, making prices cheaper on average. However, under VCG payments you don't have this kind of monotonicity: adding a road segment to the supply can increase the VCG payments while leaving the allocation unchanged. That's a recreation of the paradox, just in a different domain.

I'm just sticking with roads as the example, but this might in principle happen in any auction where goods are complements (= bidders might value a bundle more than the sum of the individual goods it's made of).

Thanks. I was hoping to get real-world examples, but yeah this is interesting and does seem structurally similar. Though, an auction sort of seems like it's structurally assuming "total inelasticity", like there's just a fixed pile of goods that you're auctioning off (and then external to that we can compare two auctions with different goods), contrasting to markets, which I imagine are mostly at least somewhat supply-elastic (though maybe that's a wrong imagination, and certainly on shorter time-scales there's lots of supply inelasticity). I can't immedia...