Consider a set of firms producing comparable goods. Each firm aims to strategically choose the selling price of their products in the market in order to maximize their expected income. However the income of each firm depends not only on his own chosen price but also on the chosen price by the other competitors; in fact, buyers choose the best quality-price ratio. In this context, how can one describe the optimization problem faced by any firm? Is there any equilibrium situation for this system? How do we define it and characterize it?
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. – John von Neumann
Games Theory and Optimal Control Theory provide powerful tools to tackle these kinds of questions in a satisfactory way. These tools are not specific to the example presented above and can be used for any system of interacting intelligent agents (e.g. crowds dynamics; banking system; financial markets, etc).
One of the key ideas in Games Theory is the notion of Nash equilibrium. It was named after the mathematician John Forbes Nash and earned him the Nobel Prize in economics in 1994. We say that a system is in Nash equilibrium if no player regrets his choice (he could not have done better) in view of the choice of others. In the example described at the beginning of this article, a Nash equilibrium is a situation where each firm correctly predicts the choice of its competitors and maximizes its gain given this prediction.
In many situations, it is possible to provide a characterization of Nash equilibrium by a system of partial differential equations (PDEs) using tools from optimization theory. That is, the resolution of this system of equations enables one to exactly compute the Nash equilibrium. However, the analysis and simulation of these equations are, in general, quite difficult, especially when the number of interacting agents becomes very large.
To address this, a new idea was introduced in 2007, independently by Jean-Michel Lasry and Pierre-Louis Lions on the one hand, and by Minyi Huang and Roland P. Malhamé on the other, to slightly simplify the interactions between the players by considering that the agent determines his optimal strategy by considering the evolution of a continuum of players rather than considering the evolution of each of the players. This new theory gave birth to a vast field of research which has strongly expanded in recent years. In particular, and in many situations, this new approach has considerably simplified the problem of Nash equilibrium computation, since in many cases, the resolution of the “simplified system” (with a continuum of players) enables one to provide a good approximation of the Nash equilibrium for the initial game with finitely many players.
In a recent work, we use this new modeling approach, called Mean-Field Games (MFG for short) to explore situations with “less rational” agents. In contrast to classical MFGs where players anticipate the evolution of the crowd, we consider a system of agents (called “myopic”) who do not anticipate anything; they only notice the changes in their environments and act according to them. This could model panic situations, for instance. In this article, we propose a model to this situation, and we explain in a rigorous way the link between the MFG model and the corresponding game with finitely many players.
Moreover, we have discovered a surprising behavior of this system: we show, under certain conditions, that such an irrational system of agents can self-organize and converge towards a highly rational equilibrium! More precisely, this system becomes more “intelligent” by reaching (or learning) an equilibrium configuration that corresponds to a very intelligent (non-myopic) choice of agents. In practice, this result can help to mathematically understand situations where the wisdom of the crowd, or the collective intelligence, surpasses the intelligence of individuals within the group.
These findings are described in the article entitled On Quasi-stationary Mean Field Games Models, recently published in the journal Applied Mathematics & Optimization. This work was conducted by Charafeddine Mouzouni from École Centrale de Lyon.