Dropping one route can boost a network鈥檚 overall performance by emphasising better options (Image: Bruno Barbey/Magnum)
Bench your best player to win the series. Close roads to get everyone home faster. Can we harness the power of Braess鈥檚 paradox?
IT IS the second game of the 1999 US National Basketball Association play-offs 鈥� the New York Knicks vs the Indiana Pacers. The eighth-seeded Knicks are holding their own against the number 2 seeds when their best player, Patrick Ewing, tears his Achilles tendon. All seems lost with the Pacers heavily favoured for the rest of the series. Yet against all odds, the Knicks go on to win the series 4-2 and qualify for the finals.
Advertisement
The Knicks鈥檚 success against the Pacers was so unexpected that the story behind it has since become a legend, even gaining its own name. The so-called 鈥淓wing effect鈥� has been evoked by pundits to explain sporting victories in which an underdog inexplicably triumphs.
The question is why. Surely science has little to say about such tales. After all, it鈥檚 only to be expected that occasionally the underdogs should win, through simple luck. Or perhaps there are psychological factors that strengthen the resolve of teammates who have lost a colleague or that weaken the determination of the opposition, who expect to triumph easily.
But there may be more to it. According to the emerging science of networks, there are good reasons why some systems perform better in seemingly disadvantageous conditions. It鈥檚 just a natural property of certain kinds of networks, albeit a paradoxical one. Could this explain why teams suddenly missing their best players somehow do better?
It鈥檚 an intriguing idea and one that could have broad implications. Since our world is increasingly tied together with complex networks, physicists are using the same network-style approach to make all kinds of similarly counter-intuitive predictions about other systems. Their studies show that everything from road, power and to food webs and the metabolic systems behind disease demonstrate similar properties. Theorists say that if we鈥檙e careful, it may be possible to exploit these properties to reduce traffic jams, prevent power outages and even fight disease in new ways.
The key to understanding this weirdness comes from the work of Dietrich Braess, a mathematician at Ruhr University in Bochum, Germany. In the late 1960s, he developed a fascination for traffic modelling, and it was while working on ways to find the optimal solution for traffic flow through a network of roads that he made a surprising discovery: adding an extra street to a simple network can actually increase overall travel times. It left him puzzled. 鈥淚 wanted to understand what was going on,鈥� he says.
Imagine two roads connecting A and B. The longer route is a highway and always takes 10 minutes, regardless of how many cars it carries. The shorter route is narrow and becomes congested as traffic increases. This route takes 1 minute for one car, 2 minutes for two cars, 3 minutes for three cars, and so on.
What is the best route if there are 10 cars? If all drivers choose the shorter, narrow route, travel time is 10 minutes for everyone. Taking the longer route offers no benefit since that always takes 10 minutes.
In this situation, no one driver has anything to gain by changing route. In game theory, this solution is known as the Nash equilibrium, named after Nobel prizewinning mathematician John Nash. Each driver is making the best choice possible, given everyone else鈥檚 decisions.
Yet there could be a better outcome. Suppose five drivers choose the 10-minute journey on the highway. The other five take the shorter route giving them a 5-minute commute. Now the average journey time is 7.5 minutes 鈥� the shortest possible average.
That鈥檚 a significant improvement but it is not a stable solution, since the five drivers with the 10-minute commute would be tempted to switch routes, returning the system to a Nash equilibrium. The only way to enforce the shorter average commute would be with some kind of centralised traffic god that controls routing.
More routes, less speed
This idea, that the Nash equilibrium may not equate with the best overall flow through a network, is the basis for Braess鈥檚 discovery. And it leads to some extraordinary paradoxes.
Imagine that there is only a single route between A and B: the highway. If 20 cars travel this route, their journey time is 10 minutes.
Now add the second route which everybody can see is shorter 鈥� one car can make the journey in 1 minute, two cars in 2 minutes, and so on. If everybody takes this shortcut, the average journey time doubles to 20 minutes. So adding a shortcut has dramatically increased the travel time for everyone (see diagram). That鈥檚 the paradox.
This kind of effect has been seen in the real world. In 1990, New York City鈥檚 transport commissioner decided to close 42nd Street, one of the city鈥檚 busiest roads, for a day. Everyone expected chaos, but instead . Indeed traffic planners and researchers have found evidence of Braess鈥檚 paradox applying to road networks in many other cities. One study identified six roads in Boston, 12 in Manhattan and seven in central London that could .
鈥淓veryone expected chaos after 42nd Street was closed, but traffic was better than usual鈥�
In 2009, Braess鈥檚 paradox caught the attention of physicist in Minneapolis, who has a passion for basketball. To his mind, the paradoxical behaviour of traffic flow bore a curious resemblance to the behaviour of basketball teams whose big-name line-ups ought to perform well, but which fail to reach their potential in competition. 鈥淚 had these two ideas going round in my head and wondered whether they might be connected,鈥� he says.
Skinner began by thinking of a basketball team as if it were a network. The players were nodes and the ball was like a car navigating between them. A team has a certain number of plays 鈥� each essentially a pre-planned combination of moves and passes 鈥� which usually revolve around a single playmaker, often the most important player on the team. Another way to think about this is that the basketball has certain 鈥減referred鈥� routes through the network.
The team also knows that a given play using the best player has, say, a 60 per cent chance of scoring. So it鈥檚 natural for the team to choose that play over one with a success rate of 50 per cent. The trouble is, opposing teams get wise if these plays are used too often.
When best isn鈥檛 best
Skinner realised that this was . Choosing the 60 per cent play with the best player was like commuters choosing the shortcut but running into congestion. This is a Nash equilibrium.
Skinner used a network model to show that the Nash equilibrium is not always in the team鈥檚 best interest. Instead, he found a global optimum that can produce better results even though it requires the team to sometimes pass the ball in a play that is less likely to score. 鈥淭here is a clear incentive to have a balance between the 鈥榖est鈥� plays and the not so good plays,鈥� says Skinner.
In 2009, he explained his ideas to a conference on sports analytics. Afterwards several analysts from NBA teams expressed interest in his ideas. 鈥淚鈥檝e had people say it has changed the way they think about game strategy,鈥� he says. Indeed, Skinner thinks there has even been a change in the way teams play. 鈥淏efore, I felt like the majority of possession amounted to giving the ball to one of your best players and attempting to score,鈥� he says. 鈥淣ow teams are becoming more evenly distributed about the way they do things,鈥� he says. 鈥淚 don鈥檛 dare take credit for this, though.鈥�
If Skinner is right, Braess鈥檚 paradox could explain the extraordinary performance of the New York Knicks in 1999 without their best player. It may also be responsible for the performance of who, against the odds, have pulled victory from the jaws of defeat.
Although there is a persuasive logic behind the ideas, the science is far from conclusive. 鈥淔or physicists, it鈥檚 a little wishy-washy,鈥� Skinner admits. 鈥淚t鈥檚 not easy to support with experimental data and it鈥檚 hard to falsify.鈥�
That鈥檚 less of a problem in disciplines where data is easier to come by, though. Take electricity generation, for example.
The move to generate more energy from renewable resources is changing electricity networks around the world. Instead of having a few large power stations in a grid, many nations are investing in lots of small generators distributed around the country. The modern challenge is how to maximise the reliability of these decentralised networks.
Working with colleagues at the Max Planck Institute for Dynamics and Self-Organization in G枚ttingen, Germany, physicist Dirk Witthaut has studied how best to stabilise the grid. In one experiment, Witthaut came up against Braess鈥檚 paradox. He modelled the backbone connections in the UK power grid and found that adding an extra link in one of two positions destabilised the network, actually , just as a new road or bridge can increase congestion. 鈥淭his is a very general phenomenon,鈥� says Witthaut. 鈥淲e have analysed very abstract, simple networks and we always find it.鈥�
With many grids operating increasingly close to their capacity, the question of where to add new connections is a key issue. According to Witthaut, a take-home rule of thumb is that when a power line is operating close to capacity, a new line should always be added in parallel. Any attempt to relieve the pressure by adding a new transmission line elsewhere in the network could be like building a big bridge over a river without the infrastructure on each side to cope with extra traffic it will attract. He says that network engineers already have a sense of this. 鈥淭hey tell you that nobody would ever build a network like this,鈥� he says.
But is that always true? Simulating the entire grid in every possible power generating state is a time-consuming and difficult task. What鈥檚 needed are general rules that engineers can apply when altering grids.
So Witthaut is turning to nature. He plans to study the way vascular networks in plant leaves change as they grow. First, he will inject dye into leaves to reveal the network and photograph it. Then he will simulate the network to find out how various changes influence its capacity. Finally, he will go back to the plant to see how the real network evolves as the leaves grow. If evolution has found a way to avoid the phenomena Braess鈥檚 paradox creates 鈥� or a way to exploit them 鈥� that could be hugely useful. 鈥淲e plan to start this work this year,鈥� says Witthaut.
In the meantime, phenomena reminiscent of Braess鈥檚 paradox are emerging in many other networks, with implications for and . In 2013, Israeli scientists . Their models show that adding new technology to transmitters 鈥� giving them the ability to adjust the power of transmissions or cancel interference, say 鈥� can reduce the average capacity of the network to carry information. Some key questions remain, they say: do other wireless technologies exhibit this behaviour, and if so, how might we spot them in advance?
杏吧原创s have even found similar patterns in natural networks such as food webs. Ecologists have long known that a change in one part of a food web can ripple through the entire network. When one species becomes extinct, for example, this can have dramatic effects on other species in the food web. Indeed, extinctions can cascade through a food web like fires through a forest.
According to Adilson Motter at Northeastern University in Evanston, Illinois, this is a clear network effect, and he points to an interesting example. The number of large sharks in the northern Atlantic has dropped dramatically in the last 40 years. Since these sharks prey on cownose rays, the rays have flourished and their numbers have risen significantly. But the rays eat scallops so these have become all but extinct in some areas. The drop in shark numbers has cascaded through the web to influence the scallop population.
An interesting question is how to stop the spread of extinctions. Motter has a provocative answer. By simulating a food network involving 33 species in Chesapeake Bay on the US east coast, he has shown that it is possible to halt the spread of extinctions by cutting out other parts of the network, just as a firebreak can halt a forest fire (, ).
In certain circumstances, he argues, the early removal of a species that would otherwise eventually go extinct anyway can prevent all secondary extinctions and improve the entire system鈥檚 viability. That鈥檚 an analogue of Braess鈥檚 paradox, says Motter.
This kind of thinking could have a profound influence in other areas. Many aspects of biological function are being increasingly recognised as network phenomena. The human body is made of vascular networks, neural networks, gene regulatory networks and so on. Motter has studied metabolic networks which determine the biochemical properties of cells. These often stop working correctly when a gene is damaged or missing, and this has led researchers to investigate repairing or replacing the missing gene, using a treatment known as gene therapy.
Motter suggests that an entirely different approach may work just as well or even better. His idea is to restore network function by cutting out parts of the network. Just as closing a road can sometimes improve the flow of traffic, the removal of certain genes can .
鈥淚t is the exact opposite of what traditional gene therapy would suggest,鈥� says Motter. There, the usual idea is to replace damaged genes in cells. Motter鈥檚 heretical suggestion is that it ought to be possible to repair such cells by knocking out other genes instead of targeting the mutated genes.
鈥淲e could repair cells with damaged genes by knocking out other genes instead鈥�
Metabolic diseases are only the beginning. Motter says that other conditions that involve networks could also be targeted in this way, such as certain types of cancer. Witthaut agrees that this kind of paradoxical thinking could have important applications. 鈥淢otter鈥檚 work is impressive,鈥� he says.
This approach is controversial but is gaining traction, partly because of the increasing number of examples in the body that we are finding. Motter points to neural networks where Braess-paradox-like phenomena are also at work. For example, damage to a specific part of one brain hemisphere can significantly impair our ability to focus on visual stimuli. However, says Motter, the creation of a lesion in the other hemisphere following a stroke, for example, has been found to lead to partial restoration of the lost function. Exactly how this occurs isn鈥檛 known, but it clearly has links to Braess鈥檚 paradox.
It is very early days in our understanding of biological networks. But if Motter, Witthaut and others have their way, counter-intuitive network effects may have a much more significant role to play in future. It may even be the key that helps the Knicks to another NBA play-off, and perhaps even reunites them with the championship trophy that has long eluded them.
This article appeared in print under the headline 鈥淟ess is more鈥�
