Partial Differential Equation Methods for Mean Field Games

平均场游戏是大量个体需要做出与许多其他个体相关的决策的问题的模型。平均场游戏的许多应用都源于经济学。一个要处理的例子是对家庭在储蓄和消费之间分配收入的决定进行建模；一个家庭的决定的好处取决于其他家庭所做的决定，因为这个例子中的利率是通过所有家庭的决定的集合来确定的。这项研究将试图理解这些模型的数学理论；这包括证明模型有解，以及理解这些解如何依赖于模型中存在的参数。发展平均场博弈理论可以对经济预测和经济决策的质量产生影响。几名研究生和本科生将通过参与这项研究项目进行培训。平均场博弈模型的解的存在性和正则性理论将被发展，并将研究重要的渐近问题。正则性理论包括证明解的解析正则性和Gevrey正则性，这些正则性已被证明存在有限的Sobolev正则性。渐近机制包括严格地研究当玩家数量趋于无穷时有限个玩家的微分对策的极限，当扩散参数为零时研究极限，以及当时间范围趋于无穷时研究极限。将强调不可分哈密顿量的问题和应用中出现的问题，如家庭储蓄问题。平均场游戏系统采用初始和最终数据，而不仅仅是初始数据；流体动力学等领域的初值问题的工具将适应平均场游戏的向前向后设置。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Mean field games are models for problems in which a large number of individuals need to make decisions which are related to those of many other individuals. Many applications of mean field games arise from economics. One example to be treated is modeling the decision of households to allocate their income between savings and consumption; the benefits of a household's decision depend on the decisions that the other households make, since the interest rate in this example is determined through the aggregation of the decisions of all the households. This research will seek to understand mathematical theory of these models; this includes proving that the models have solutions and understanding how these solutions depend upon parameters present in the models. Developing the mean field games theory can have impact on the quality of economic forecasts and economic decision-making. Several graduate and undergraduate students will be trained through participation in this research projectExistence and regularity theory will be developed for solutions of mean field games models, and important asymptotic problems will be investigated. The regularity theory includes demonstrating analytic and Gevrey regularity for solutions which have previously been proved to exist with finite Sobolev regularity. Asymptotic regimes include rigorously studying the limit of differential games with finitely many players as the number of players tends to infinity, studying the limit as diffusion parameters vanish, and studying the limit as the time horizon goes to infinity. Problems with nonseparable Hamiltonian and problems arising from applications, such as the household savings problem, will be emphasized. Mean field games systems are taken with initial and terminal data rather than just initial data; tools from initial value problems in areas such as fluid dynamics will be adapted to the forward-backward setting of mean field games.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.