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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Dec 29, 2021


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... (read more)

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Braess's paradox with springs and strings! 

I'd be hesitant to conclude from prices -naturally- skyrocketing that welfare is lower. "Reasoning from a price change" as Scott Sumner would say. If you have a shortage due to supply constraints, and innovation eases the supply constraint and unlocks complementarity value in other products, that'll be reflected in their prices and does not necessarily mean people are worse off.

I like your positioning of Braess's paradox as an externality. It's a special case in that it isn't the participation in the system that exerts a social cost but the particular pathway of participation that does. I suppose because the traditional economic analysis assumes homogeneity in lots of dimensions of the problem (for the most obvious example, there aren't multiple pathways to participate in the market, just one - a non-descript exchange at a strike price) that it would be challenging to characterize a multi-path system like this as a typical economic market.

Perhaps as you mention with innovation, we could approach this from one step up from the market at resource-based and production-function-based generation of supply. Although we might say there is one non-descript type of "exchange" that is participating in the market, before you get to that node, there are other nodes you could take, which facilitates modeling this as having multiple pathways. Consider the decision to use greenhouse gas scrubbers in production or not (assume no regulations). For most companies, "not" is the dominant option to maximize profit, which constitutes the vast majority of their utility. For other companies choosing to maximize their own slightly different utility function, using scrubbers is desirable. Then one introduces a new scrubber that is way cheaper and becomes adopted by some marginal firms (reducing the total externality, reduction on net) who then can increase their production to get their total emissions back to the previous aggregate level they were comfortable with but at their lower emission/unit level (increasing the total externality, no change on net), becomes adopted by prior "green" firms (reducing the total externality, reduction on net) who can increase their production to profit a bit more while keeping some of the environmental gain intact (increasing the total externality, slight reduction on net), and prompts entry by marginal would-be firms (increasing the total externality, the net depends on the sizes of all the margins). What if the short-term net effect is helpful, but if down the line this leads to the old scrubber producer exiting, demand overwhelming the new scrubber producer, price increases leading to insufficient adoption or even to disadoption, and ultimately the total externality sneaking back up and over where it was originally?

As with most paradoxes, they are largely a function of an ill-defined problem, an analysis at a "different level" than needed, or neglecting relevant complexities. In this case, it's specifically failure to consider all the margins.

Another example could be in the supply chain itself, which is great for the overloading aspect. Your firm promises 30 day delivery because your models say your company could do 28 days at the current level of demand. You innovate that down to 25 days and start promising 28 day delivery (you even net a day of safety, right?!), but new demand overloads your node (or even one specific upstream node that is now part of your innovative process), and you can't even deliver in 30 days now (maybe this is really, thanks Wikipedia for the name to it, Jevons paradox, and perhaps so is my emissions example, but I think either way the point is that there is a "missing margin" that planners didn't see, leading to overload from an "improvement").