Mean Field Games and Optimal Contracts

平均场博弈和最优契约

• 批准号: 1714607
• 负责人: Yuchong Zhang
• 金额: $ 14.82万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714607&HistoricalAwards=false
• 关键词: Mean; Field; Games; Optimal; Contracts

Many questions in finance, economics, and engineering involve competition or interaction among a large number of agents. Large-population games have traditionally been difficult to analyze due to the large size of the system and the game nature of the interaction. The theory of mean field games provides a useful way to approximate these complex systems, and the theory helps to improve understanding of the effects of interactions and how populations react to different compensation schemes or public policies. On the other hand, the 2008 financial crisis revealed the fragility of financial models. Since then, the urge to develop financial theories that take into account model risk has grown tremendously. This research project explores mathematical questions in mean field game theory and robust finance. The first part of the project analyzes a dynamic competition involving a large number of players, where the interaction among players is through the ranking of the completion time of their respective projects. The model applies to situations where many firms or individuals compete to be the first to achieve a goal. The objective is to understand the equilibrium and design a reward scheme that encourages early project completion given that the organizer has a limited budget, or to minimize the budget given a desired rate of completion. The second part of the project is concerned with the accuracy of the mean field game approximation, when in many applications, the typical size of competition is only modestly large. The objective is to study the fluctuation around the hydrodynamic limit of the mean field game approximation so as to improve accuracy. The third part of the project studies a mean field game of optimal stopping when 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 last part of the project considers pricing and hedging of contingent claims in a financial market under both transaction costs and model uncertainty, where model uncertainty is described by a collection of probability measures. In general the collection need not have a reference measure that dominates every other measure, and therefore, standard tools from functional analysis cannot be applied, and new techniques are called for.

金融、经济和工程领域的许多问题都涉及大量主体之间的竞争或互动。由于系统规模大和交互的游戏性质，大规模游戏传统上很难分析。平均场博弈理论提供了一种近似这些复杂系统的有用方法，该理论有助于提高对相互作用的影响以及人们对不同补偿方案或公共政策的反应的理解。另一方面，2008年的金融危机暴露了金融模式的脆弱性。从那时起，开发考虑模型风险的金融理论的愿望急剧增长。该研究项目探讨平均场博弈论和稳健金融中的数学问题。项目的第一部分分析了一场涉及大量玩家的动态竞争，玩家之间的互动是通过各自项目完成时间的排名来进行的。该模型适用于许多公司或个人争夺第一个实现目标的情况。目标是了解平衡并设计一个奖励计划，在组织者预算有限的情况下鼓励项目尽早完成，或者在给定期望的完成率的情况下最大限度地减少预算。该项目的第二部分涉及平均场博弈近似的准确性，在许多应用中，竞争的典型规模只是适度大。目的是研究平均场博弈近似的流体动力极限周围的波动，以提高精度。该项目的第三部分研究了最优停止的平均场博弈，此时参与者之间的互动既不是通过状态过程，也不是成本结构，至少不是直接的方式，而是通过停止行为所揭示的信念或信息。该项目的最后一部分考虑了交易成本和模型不确定性下金融市场中或有债权的定价和对冲，其中模型不确定性由概率度量的集合来描述。一般来说，集合不需要有一个主导所有其他测量的参考测量，因此，不能应用功能分析的标准工具，并且需要新技术。