Asymptotic Consensus with Coalitions on Hybrid Mean Field Game and Its Financial Applications

Description

Large population dynamical multi-agent phenomena occur in a wide range of designed and natural settings and they also underlie much economic and financial behavior. Analysis of such systems with even a moderate number of agents is regarded as being extremely difficult using the finite population game theoretic methods which were developed over several decades for multi-agent control systems. Recently, Mean Field Game (MFG) theory, as a part of the modern control theory, was introduced in a series of founding works by [Lasry and Lions] and by [Huang, Caines, and Malhame] in 2006, and it has been recognized useful to provide an effective solution through asymptotic analysis in the continuum limit for large systems.Along another line, Hybrid System (HS) has been developed rapidly in a variety of fields since the 1998 Special Issue on Hybrid Systems of the IEEE Transactions on Automatic control. Hybridness is a characteristic which exists almost every system by exhibiting continuous state process and discrete regime switching process of hierarchical systems. The mathematical treatment of hybrid systems is interesting in that it builds on the preceding framework of system design, but its mathematics is qualitatively distinct from the mathematics of purely discrete or purely continuous phenomena.Due to the editorial works for the NAHS (Nonlinear Analysis: Hybrid System), PI (Associate Editor) had many discussions on these two rapidly evolving fields with Professor Caines (Senior editor). Surprisingly, it has been recognized that there had been very limited literatures on the study of the combination of these two emerging fields of MFG and HS (HS-MFG) to date, while there exist wide variety of important applications and interesting mathematical challenges. After more intensive discussions during our attendance of MFG workshops held in Hong Kong and Los Angeles, Professor Caines is particularly interested in, and willing to collaborate as Co-I on the topic related to Event Triggered Hybrid Mean Field Games, among many other possible directions.The reasons to investigate this particular topic are summarized as following: (1) It is new both in Hybrid System and Mean Field Game theory. Consequently, it may bring fresh insights for large systems on top of well-established (single-player) models. (2) It will be a mathematically challenging but feasible project based on Co-I and PI’s experience. Motivated from an explicit counter-example on the existence, we first identified that it’s not a trivial extension of the existing theory. To overcome this difficulty, we will design a novel methodology to identify a sufficient condition for the characterization of Nash Equillibrium. (3) There would be many other potential ramifications upon the completeness of the current project.