The everyday environment in which we live is made up of encounters. Some are recurrent: every day, or almost every day, we share a meal with family members or work colleagues; we regularly greet our neighbors. And then there are those encounters that are purely coincidental, like the motorist we pass at a red light, or even beyond the visible, when all of us, inhabitants of the same planet, consume the same energy, the same food or share the same eco-system. In physics, these encounters could be called interactions. How can we describe them mathematically when they occur on a large scale? How can we account for the fact that, on a day-to-day basis, each of us seeks to improve our daily lives, based on quantities (such as the price of oil or wheat) that depend on the daily lives of billions of other similar people? How can we predict the evolution of such patterns, and above all, what happens when this gigantic network through which we are all connected begins to oscillate?
In mathematics and physics, models that reproduce the behavior of a multitude of individuals who only see each other through aggregated quantities (also known as macroscopic quantities), the result of each individual's actions, are called mean-field models. For example, the number of people visiting cafés and restaurants on a Friday or Saturday evening is a collective indicator resulting from individual decisions. This collective indicator simply summarizes overall behavior, without knowing the behavior of each individual. It simply aims to describe the statistical behavior of the population.
Mathematically, mean-field phenomena are precisely those in which an individual sees others only in terms of statistical quantities, summarizing the overall state of the population. These models become particularly subtle when each individual making up the gigantic population in which she or he evolves can make choices. If everyone decides to go to a restaurant on a Saturday evening, available tables will certainly run out and menu prices will soar. In fact, a balance is formed between those who insist on going out (for example, because it is an occasion not to be missed) and those who prefer to stay at home. This equilibrium is the basis for restaurant visits. In game theory, this equilibrium is known as a Nash equilibrium.
Mean-field game theory was introduced almost twenty years ago by P.L. Lions and J.M. Lasry, with the aim of answering the following questions: when can an equilibrium be found, and how does it vary with population size? Other questions remain, however. For example, can several equilibria be found in the mathematical sense, and if so, does one make more practical sense than another? Mathematically, the existence of several equilibria corresponds to the presence of abrupt changes in the population's state when one of the model's parameters is moved. One of the hypotheses of the ERC ELISA project is that these abrupt state changes can be erased by forcing the entire population to oscillate under the effect of small random pulses, and then averaging the observed states over each of these small pulses. To validate this hypothesis, we need to construct suitable random pulses. The difficulty lies in the fact that these impulses aim to modify the statistical state of the population as a whole: in effect, we are talking about moving a very large object.
