Mean Field Games, Information Design and Evolutionary Finance

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Mean field games (MFGs) provide a useful approximation for large population stochastic games in which the players are coupled through their empirical distribution. Many financial and economic models that were once intractable due to high dimensionality can now be analyzed using the MFG approach. Part of the research proposal studies two MFG problems arisen in mathematical finance and economics. The first one is a continuation of the principal investigator's previous work on rank-based games and optimal reward design, where we propose to look at a multi-stage mean field competition with a reward stream that depends on the ranking of the completion time of each stage or the running rank of the progress process. The model applies to situations where many firms or individuals compete to be the first to achieve a goal with multiple milestones. The second one is a MFG of optimal stopping where the interaction among players is neither through the state process nor the cost structure, at least not in a direct way, but through the belief or information revealed from the action of stopping. The optimal stopping problems proposed include the quickest detection of the occurrence of a financial event and the sequential testing of the value of an unknown market variable. By specifying how each agent process the public information, one may be able to analyze the effect of conformity to the public opinion on people's decision making, and whether it benefits people individually as well as collectively. The proposal also contains two new lines of research. One is concerned with stochastic control and games where the control variable is the information, modelled by sigma-algebras, and the performance criteria involves the conditional expectation of a random variable given that information. Such a problem is known as Bayesian persuasion or information design which has seen a variety of applications in finance, politic science, law enforcement, medical testing and so on. Our goal is to develop the mathematical theory and numerical algorithm for both static and dynamic information design problems, possibly with information constraints, and static information games. The other line of research incorporates biological models of evolution and learning based on the principle of selection and mutation into the financial market. Both topics bring interesting problems and ideas from other fields such as economics and biology to the stochastic control and mathematical finance communities, which will open doors for new theories and applications.