ISOTACE (Interacting Systems and Optimal Transportation, Applications to Computational Economics) is a french ANR-12-MONU-0013 (2012-2016) funded consortium.
Public summary of the proposal :
The use of partial differential equations (PDE) in Economics and more generally social sciences is a relatively young field of research. Optimal Transportation (OT) and Mean field Games (MFG) theories have, in this context, recently received a lot of attention. The “Computational Fluid Dynamic” formulation of OT is intimately linked to the more general PDE structure of MFG. Both OT and MFG mathematically underlie our proposal. The corresponding research effort to produce numerical analysis and algorithms remains scat- tered and largely academic. Meanwhile, the number of modeling issues and associated mathe- matical and numerical analysis questions is expanding at a rapid pace. The design of efficient solvers is a bottleneck that can be addressed only by numerical analysts and specialists of scienific computing together with economists and mathematicians; without such experimental and numerical development, it will be difficult for these numerical models to become industrial simulation tools. Numerical simulations are also a key ingredient in proposing and studying new academic models. The proposed consortium brings together economists, applied mathematicians and specialists of scientific computing in an attempt to bridge the gap between theory and numerics, academic and industrial applications. Producing reusable code is a common approach for numerical methods targeting “hard” science applications. It is known that making public domain implementation available stimulates research and helps technology transfer to the industry. We therefore plan to deliver two libraries of solvers dedicated respectively to MFG/Dynamic OT and Static OT.
MFG and OT based models for crowd motion under congestion are mature enough to propose a multi-solver/multi model software development task. It will be a “real wold” test for the integration of solver libraries mentioned above. The software objectives are both academic to understand/compare these modern congestion models and also to be used as a prototype for industrial collaborations. With a different perspective the academic environment of the consortium will help/support the development by our industrial partner ZELIADE of a software with application to model-free bounds in derivative pricing. It is a completely new application of OT and is expected to have a direct industrial impact in finance.
In a more academically oriented approach, we finally propose to explore a set of new models in economy where the MFG and OT libraries will provide useful base for numerical simulations. In particular :
– Many relevant markets are markets of indivisible goods characterized by a certain quality: houses, jobs, marriages… On the theoretical side, finding equilibria in such markets is equivalent to solving a certain optimal transport problem (where the cost function depends on the sellers and buyers preferences). On the empirical side, this allows for trying to recover information on the preferences from observed matching; this is an inverse problem.
– The central principal-agent model addresses the problem of a principal minimizing cost under the incentive constraint faced by an agent with multivariate type in the case of a simple bilinear
utility. The problem can be reduced to the maximization of an integral functional subject to a convexity constraint. Convexity/well-posedness for more general utilities has been recent obtained using OT techniques.
