Mathematics' hot topic: Mean Field Games

Imagine it is summertime and we are off to enjoy a beautiful day at the beach, says French mathematician Professor Pierre-Louis Lions. “We need to decide where to put our towel...there are points of interest and being French, that means places where you can eat and drink, but there are also other factors to consider such as how crowded a particular stretch of beach is.”

One way of solving the ‘problem’ is to invoke the use of a mathematical theory called Mean Field Games, says Professor Lions from Collège de France. Professor Lions, who was in Singapore for the Global Young Scientists Summit held from 20 to 25 Jan, says that it isn’t so much what the individual beach-goer is going to do, but rather the collective behaviour of all beach-goers. “It’s not to teach you where to put your towel — you know how to solve that. What the model tries to do is, if we all decide in a rational fashion where we are going to put our towels, can we predict the outcome of towel distribution?”

Beach towels may seem like a trivial notion, admits Professor Lions, but it’s a fun and easy way to illustrate the potential applications of Mean Field Game Theory, and the power that lies in its use. The mathematical model can be applied in the study of how large populations make strategic decisions, whereby the interactions of individual agents or players who comprise the population have a very small impact on the overall outcome. It is used to analyse the workings of large systems such as financial markets, crowd behaviour, and social networks.

“It looks like a silly question,” says Professor Lions. “But it’s really to think about: ‘Where do I want to put a new antenna for a mobile communication network?’ or ‘Where do I want to open a new store?’ It’s exactly the same problem.”

From two to many players

Mean Field Games traces its roots back to 2006, when two groups of mathematicians separately conceived the idea. Professor Lions, together with fellow mathematician and compatriot Jean-Michel Lasry, was one of its creators. “The goal was to create and use new mathematical models for situations and solutions involving many players and externalities,” he said.

The model builds upon Classical Game Theory, especially the notion of Nash equilibria, taking it one step further. “You cannot do much with game theory” because you can only use it to analyse a small number of agents (typically two), Professor Lions explains. “As soon as you have more players, you cannot track them.”

Mean Field Games, on the other hand, works best with large numbers, capitalising on the smoothing effect of large numbers. Classical Game Theory looks at how one individual interacts with another, a scenario that can get incredibly complicated once more players are introduced. In comparison, Mean Field Games concerns itself more with how a single individual interacts with the mass — or in other words, the mean or average — of others. “So, it’s the average behaviour of small agents,” says Professor Lions. “Mean Field Games is a mixture of statistical physics ideas and game theory.”

More than a decade ago, Professor Lions and Lasry embarked on creating a new mathematical model as a project for French energy provider EDF and Green Bank, an organisation dedicated to finding clean energy solutions. “The title was Finance and Sustainable Development and it was about how to use financial tools in order to improve the sustainability of economics,” recalls Professor Lions.

As one of the pioneers in a nascent field, he says that they “really had no clue where to start, what to study, and so on.” Eventually, the pair developed processes to help guide and improve decision-making processes of the firm, and later other firms with their industrial, financial and technological innovations. They went on to establish mfg labs in 2009, working to help companies transform the data they have to “improve their decision-making, automate their processes and create new services.”

“Progressively the subject got more organised...now it’s even fashionable,” says Professor Lions. In a sign of how Mean Field Games quickly became the “in topic,” he recalls coming across a paper describing its use in Olympic sailing strategies. “I really laughed when I saw it in the paper because there is no way it can be used in Olympic sailing...one has to be serious in its applications.”

To that extent, Mean Field Games has been used in a variety of scenarios, ranging from analysing how people would evacuate a high-rise building to finding out how energy production can be optimised to studying swarm intelligence of shoals of fish navigating together.

Of his creation, Professor Lions says: “It’s a mathematical toolbox.”