Author: Julian Glenesk, Research assistant, RAND Europe
Ever-present internet access and the ubiquity of smart phones have allowed real-time traffic apps like Waze and Google Maps to thrive. However, news headlines reveal some unintended consequences emerging due to the prevalence of these apps.
Communities in Los Angeles and New Jersey are finding problematic increases in traffic on their residential streets as a result of these wayfinding apps directing traffic through their previously quiet neighbourhoods. Drivers might be tempted to shrug off these reports as a case of nimbyism however there is an unintuitive phenomenon behind inefficiencies resulting from traffic flow ‘improvements’.
In game theory, it is shown that in a competitive environment, individuals acting to rationally maximize their own benefit can lead to suboptimal outcomes (the Prisoner’s Dilemma is perhaps the most famous example of this). Such scenarios lead to a Nash equilibrium, where no individual would be better off by changing their behaviour to another alternative.
The Wardrop principles of equilibrium take these game theory principles as applied to route choice decisions in transport networks. Wardrop’s first principle of route choice is known as ‘user equilibrium’, which is a Nash equilibrium achieved where each road user chooses the fastest route available to them.
In contrast, Wardrop’s second principle of route choice is known as ‘system optimal’ or ‘social equilibrium’, where average journey time for all road users is minimized. In the latter case, all users behave cooperatively to achieve the most efficient use of the network.
The ratio of system optimal to user equilibrium conditions is known as the ‘Price of Anarchy’, as it represents the efficiencies lost as a result of individually selfish yet rational decisions. These principles not only apply to traffic flows, but any domain that can be represented by a network such as the internet, power grids, food chains and even basketball tactics. It is closely related but distinct from The Tragedy of the Commons, where the greedy use of a shared-resource depletes that resource for the whole.
In the road traffic context, the price of anarchy can be illustrated in the simplest scenario known as Pigou’s example. Imagine a transport network with 10 vehicles and 2 roads connecting origin O with destination D. Road 1 is a highway that always takes 10 minutes to get from O to D, while Road 2 is a narrow alleyway that takes 1 minute of travel time for each vehicle using it (i.e. as more people use the alleyway, the slower it becomes).
In the system optimal scenario, every driver would be directed to the roadway that minimizes the average travel time of the whole system. In this scenario, the minimum average travel time is 7.5 minutes, with 5 drivers taking Road 1 (the highway) at 10 minutes travel time while 5 drivers take Road 2 (the alleyway) at 5 minutes travel time.
In a user equilibrium where every driver tries to minimize their personal travel time (i.e. everyone is using navigation app!), each driver on Road 1 (the highway) would switch to Road 2 (the alleyway), until the last driver on the highway makes the switch to the alleyway and consequently degrades the alleyway’s travel time to 10 minutes for all users.
In this example, the highway remains empty while the alleyway is no longer a shortcut that saves anyone any travel time. In this case the price of anarchy is 33%, calculated as the ratio between system optimal (10 minutes) and anarchy (7.5 minutes) travel times. This is the highest known upper bound of user equilibrium inefficiency and is a worst-case example of anarchy in a network.
Pigou’s example is analogous to communities complaining of detrimental traffic being routed (via navigation apps) through their neighbourhoods while drivers lose their ‘secret’ shortcuts because they are no longer effective with so many other drivers exploiting it. Despite a technological ‘efficiency’ and information gain, drivers are worse off on average than they were before.
Even more counterintuitive scenarios can arise such as adding a new road increasing travel times or removing a seemingly vital transport link decreasing travel times. This is known as Braess’ paradox – where a new roadway simply attracts new road users rather than alleviating conditions for the existing road users and it can occur in road networks regardless of the effects of induced demand.
There are numerous real world examples of Braess’ paradox such as in New York, Seoul, Stuttgart and Paris, and there are likely to be many more undocumented cases as discovering it requires deliberate longitudinal analyses.
Both technology and policy play a vital role in establishing a balance somewhere between user equilibrium and social optimum. In our navigation app example, if the optimisation solution is set to minimize each individual user’s travel time, we may be worse off with this technology than before. If the optimisation solution is centralised to minimize average travel times by a central body – such as in a road network comprised of 100% autonomous vehicles and a government entity directing traffic – then we could find up to 33% greater efficiency in our road networks.
The main proposed solution to getting such efficiency out of our road networks is with road pricing. While often fiercely opposed, road pricing is proven to induce travel behaviour changes that can leave everyone better off, including those that cannot afford to pay tolls or peak charges. It is therefore the responsibility of policymakers to leverage technology such as GPS navigation apps, traffic information systems and policies such as road pricing to nudge behaviour from an inefficient competitive anarchy towards a more efficient social optimum.