Speaker
Description
As argued by Harsányi and Selten, selecting one Nash equilibrium from the set of all equilibria is one of the most fundamental questions in game theory. Introduced around two decades ago, Mean Fields Games (MFGs) aim to model symmetric large population dynamic games, where the agents are interacting via the distribution of their states and controls. While there is a very good theory for MFGs under various monotonicity conditions on the data (leading to uniqueness of equilibria), non-monotone MFGs are poorly understood. Multiple equilibria give rise to the breakdown of classical solutions to the so-called master equation — an infinite dimensional nonlinear and nonlocal PDE on the Wasserstein space of probability measures — as well. In certain cases such master equations boil down to ‘simpler’ PDEs, that can be seen as infinite dimensional conservation laws. In such situations selection principles are intimately linked to entropy solutions to these PDEs. However, building a satisfactory weak solution theory for these conservation laws is a very challenging task, because of the lack of a canonical measures (replacing the Lebesgue measure) on the Wasserstein space. In this talk we show some potential first steps towards building an entropy solution theory for a class of conservation laws on the Wasserstein space. The talk will be based on joint works with Jameson Graber (Baylor University).