People and their Paths 5: How to get people to behave optimally
October 18, 2013
At the beginning of this series, we described Wardrop’s First Principle, of User Equilibrium. He also had a second principle, of System Optimality, which says: “At equilibrium the average journey time is minimum.” To achieve this requires every traveler to act in accordance with society’s best interest, which as we noted in part 2, is not generally calculable by an individual. This ratio of the total system travel time associated with a user equilibrium traffic pattern and the system optimal travel pattern has been dubbed “The Price of Anarchy” by Tim Roughgarden, who has applied this to computer networks. This number indicates the inefficiency of autonomous (or selfish) control in a system, compared to a theoretically best central control.
When choosing a route, selfish users see the costs they incur, but not the costs they impose on others. This is analogous to the difference between average and social marginal costs in economics. If we somehow persuaded travelers to make route decisions considering the cost they impose on others, their marginal cost, we could achieve a minimal total cost for the system. In economics, the classic theoretical mechanism for this is called a Pigouvian Tax, which charges the polluter for the negative externalities imposed on the pollutee (the difference between the social marginal cost and social average cost). In this case the externality is congestion, or travel time imposed by a vehicle on all other vehicles in excess of what would be borne in the vehicle’s absence. The Pigouvian Tax gains its name from Arthur Pigou, a British economist from the 1920s, who discusses the idea in his text The Economics of Welfare.
Travelers facing travel times and a Pigouvian Tax might choose a route that satisfies both of Wardrop’s Principles. The User Equilibrium (UE) solution would equal the Social Optimal (SO).
Using traffic assignment models we compared system optimal and user equilibrium flows and travel times for the Minneapolis – Saint Paul regional planning network, assuming total traffic flow between origins and destinations were fixed (i.e. unaffected by our distortion of route prices). We found the SO assignment had a 1.7% overall time savings, and a slightly higher average speed (63.2 km/h vs. 61.8 km/h). Perhaps surprisingly, it also had somewhat more total vehicle kilometers traveled (9.37M vs. 9.33M), as drivers had to take longer routes to avoid imposing congestion on others.
While the SO result is better than the UE (it cannot be worse), we might ask “SO What?”. The price of anarchy, letting drivers choose their own routes rather than being centrally directed, is relatively small, under 2 percent. It turns out it is much more important to get people to choose an efficient time of day than to worry about micro-managing which route they select.
We could post time-varying prices (just like the HOT lanes of Part 3, or many transit systems which have peak and off-peak fares) to discourage demand when it is highest, and encourage demand at off-peak periods. This is done on some toll facilities now, and other schemes, like the London Congestion Charge, have two prices: free or tolled, depending on time of day. But this can be as refined as we want it, with prices changing every hour, every five minutes, or even continuously. The prices might change in real-time, or change according to a fixed and posted schedule.
Nobel-winning Economist William Vickery developed the first version of the bottleneck model, which showed how varying prices would allow people to trade-off being on-time (at a higher toll) or being early or late (at a lower toll, but a higher cost in what transportation researchers call “schedule delay”).
The simplest version of this has two players1. Imagine two boats racing for a canal lock, or, as in the image, two weightlifters trying to get through a narrow door on the London Underground. When they arrive at the same time, only one can make it through first, the other has to wait. The one who makes it through imposed schedule delay on the one who waited. But if they arrived at different times, there would be no direct schedule delay, though one might not get into the canal (or through the door) at their preferred time. If we appropriately price simultaneous arrivals, we will discourage them. While with two players this may be feasible to coordinate with direct communication by saying don’t arrive when the other guy arrives, and negotiating, for 2000 people instead of 2, coordination is better through posted price signals than conversation and negotiation. Prices varying by time-of-day is what congestion pricing is about, putting a higher price on times which are most desired, and lower prices on the less desired times.
There are perhaps other ways to achieve this end. On most roads, it is assumed no one owns the travel time, and so we get congestion. If there were some kind of property in the right to travel at a given time, we could auction off this right to the highest bidder, and similarly avoid congestion. This would more closely follow a strategy of establishing property rights to avoid externalities, as suggested by British-American economist Ronald Coase (who is still talking about economics at the age of 102). In the transportation literature, this has come to be known as reservation pricing. Just as one does not expect to be seated when showing up unexpectedly at a popular restaurant that takes reservations, one should not expect to use a high-demand bottleneck facility on the transportation network without making arrangements in advance. Of course it is much more complicated with a real-time system like transportation, and to maximize throughput, it is likely that some queueing is required. This queue ensures there is someone waiting to take advantage of the next gap that opens. The alternative would be that the facility remains under-utilized for part of the time, which has its own costs. Even restaurants that reserve tables sometimes make you wait a little bit, for their immediate convenience, not yours, maximizing the productivity of their staff.
Unfortunately congestion pricing in any form remains more in the realm of theory than practice. While there are a few Congestion Charging programs: notably Singapore, London, and Stockholm, they are not over a large enough area, or variable enough in prices, to produce an end to congestion. Once many of these are implemented, I expect many cities will look at their peers and copy them, and it will become standard in all large metropolitan areas. But to date, the cases are fairly exceptional: Central Singapore, a city-state governed by a strong Prime Minister, whose family has been in power for five decades; Central London, a city governed at the time by “Red Ken” Livingstone, a radical thinker who was willing to take the political heat for the decision; and Stockholm, which conducted a trial experiment before holding an election to allow residents to vote up or down. Technically the systems all work well, and certainly do reduce congestion compared to the unpriced alternative. Politically they have been difficult to emulate. New York City tried and failed2, and no other US city has been willing to do something quite so radical.
Another possible deployment path for congestion pricing is through what is variously called a Vehicle Mileage Tax, or a Mileage-based User Fee. Gas tax revenues, which provide a large share of road funding, have been declining for a long time in the US, both due to leveling off of demand for driving, as well as better fuel economy. The simplest solution is to raise the gas tax, which solves an immediate problem, but not the longer term one. While hybrid gasoline-electric vehicles (like the Toyota Prius) still pay some gas tax, plug-in electrics (like the Tesla, Chevy Volt, or Nissan Leaf) pay almost none. Yet they still use the roads. Although they are presently a small share of the market, that share is likely to grow. Some states are beginning to think about how to charge EVs for the use of roads, just as gasoline-powered vehicles are charged based on a gas tax. Once a device is placed in cars tracking miles traveled (basically just the odometer, though possibly with some locational data to allow prices to vary by location (urban vs. rural) (although it technically feasible to ensure privacy, by not tracking which specific miles are traveled, no one will believe government protestations isn’t tracking them anymore, anyway), that device can also track when those miles are traveled, and vary the rate by time-of-day. The State of Washington now taxes EVs $100 per year to offset the lack of gas tax revenue. Oregon is conducting a large scale test of the Vehicle Mileage Tax, allowing 5000 volunteers to pay by the mile and have their gas tax rebated.
We are getting to the point where we can provide incentives and disincentives to efficiently manage road use. The technology exists, it is probably accurate enough. The cost of collecting a new road fee is non-trivial (especially compared with the gas tax, which simply requires an annual check of refinery sales), but the costs should drop with widespread deployment. The benefits are a significant improvement in the management of road use, so that drivers who do not need to travel when roads are congested, will have incentives to avoid those times.
If applied correctly, the resulting changes in route choices it will reveal where roads are overbuilt, and where demand, even after pricing, is sufficient to justify new capacity. The most cost-effective thing we can do in transportation is to get the prices right, all else will follow. This requires above all else, field experiments where different strategies are tested and evaluated, and deployment by replicating the successful experiments.
- Levinson, David (2005) Micro-foundations of Congestion and Pricing: A Game Theory Perspective. Transportation Research part A Volume 39, Issues 7-9 , August-November 2005, Pages 691-704.
- Zou, Xi and David Levinson (2006) A Multi-Agent Congestion and Pricing Model. Transportmetrica Vol.2, No.3, 2006 pp.237-249.
- Schaller, Bruce (2005) New York City’s congestion pricing experience and implications for road pricing acceptance in the United States. Transport Policy 17(4) 266-273.