Boltzmann-type and mean-field models in socio-economic applications

Our interactions and behavioral patterns not only impact our personal environment, but also effect the progress and development of our society. Understanding how the exchange of information and knowledge between individuals impacts the productivity of an economy and which incentives promote life long learning and prolonged productivity is of great importance and has stimulated research in the field of social sciences, economics and more recently applied mathematics.

A main challenge corresponds to the problem of deriving mathematical models for macroscopic quantities, for example the productivity of a society, from microscopic interactions. A related problem was studied by Ludwig Boltzmann, who wanted to describe macroscopic quantities of a gas, such as the pressure or the temperature, based on the microscopic collisions of gas molecules. He proposed that the macroscopic behavior can be deduced by studying the average behavior of the system, laying the foundations for kinetic theory. Similar ideas can be adapted to understand how individual interactions affect the overall behavior of a large group. But human interactions are rather complex, often based on personal preferences and/or rational decisions. Rational decision making corresponds to considering pros and cons, while trying to find the optimal strategy. This problem is well known in mean-field game theory, which studies strategic decision making in large groups of interacting individuals.

In this project we aim to develop mathematical tools to describe and analyse trading mechanisms and their impact on the dynamics of the price, using ideas proposed in the field of kinetic theory. A second goal corresponds to the coupling of kinetic and mean-field game theory to understand how strategic decisions influence the overall behavior of large groups. This approach will be used for example to analyse the effect of knowledge exchange and dissemination on the overall productivity of an economy. The developed techniques and results allow us to identify driving factors for sustained economic growth. As a part of this project we will develop numerical algorithms, which shall support the theoretical results but also give implications about the dynamic behavior in more complex situations.

The research will be carried out at the Mathematics Institute at the University of Warwick by the PI working with collaborators from prestigious overseas and national universities. These experts in the field of nonlinear partial differential equation theory, mathematical finance as well as economics will be actively involved in the project, and will help to develop and disseminate the obtained results.

Understanding how complex economical and behavioral interactions influence the dynamics of a price or economic growth are questions which have challenged researchers in the fields of economics, social sciences and more recently applied mathematics. The main goals of this research project address these challenges, by developing a mathematical framework which shall give novel insights into the driving forces of these complex processes. The proposed project outcome will benefit the following groups:

*) Researchers in the field of kinetic and nonlinear PDE theory
*) Researchers from financial mathematics studying price dynamics or the distribution of wealth
*) Economists working on multi-agent systems
*) Researchers in social sciences analysing opinion dynamics or consumer behavior
*) On the longer time-scale opinion polling and development aid policy

The impact on researcher in kinetic and nonlinear PDE theory is dercibed in the Section 'Academic beneficiaries'. Boltzmann-type models have been used successfully to understand the distribution of wealth in a society or the dynamics of a price traded in an economic market. These models can be understood as building blocks to address more complex problems and provide versatile mathematical tools which can be used to analyse similar problems in socio-economic applications. The proposed coupling to mean-field game models will impact research on the dynamics of multi-agent systems in economics and social sciences. It provides the necessary mathematical tools for the development of quantitative models,which allow to identify the main driving forces of complex processes. Our collaborators are among the leading experts in the field, which will help to disseminate our work to the target audience and promote these models in their respective communities.

The results of the project can be put into use to understand complex dynamics in our society in the future. For example by analysing the impact of personal communication and information transfer on opinion dynamics or consumer choices.