Generalized Nash games concepts: existence, tractability and applications to population models

这个建议的重点是广义纳什博弈（GNG）的概念，解决的问题的存在性，计算方法，寻找这些解决方案，和作用，这种建模概念可以发挥在人口中的应用问题。GNG是在50年代提出的，并表示模型的非合作行为的球员，其战略集，连同他们的支付函数，取决于其他玩家的策略选择。GNG作为一个建模框架的流行程度并不像通常的纳什博弈那样广泛。这是由于解决GNG带来了非常复杂的数学困难，并且解的存在理论和计算方法都取决于所研究的GNG的子类。我的研究计划包括理论部分和建模部分。1)在理论方向上，我计划扩展最近开发的个人成果和计算方法，以提供答案的问题存在的广义纳什均衡（GNE）GNG没有共享的约束。GNG最有趣的特性之一是它们的解集通常非常大。我自己的工作在这个主题上计划充分探索变分不等式为基础的和进化算法的方法来描述整个解决方案集的GNG.Further，我计划将进化稳定状态（ESS）的概念与GNG，并调查这个概念是否可以链接到（略有修改）复制动力学。如果是这样的话，我想在广义的背景下研究ESS状态和纳什博弈之间的经典关系/结果的对应关系。 在建模的方向，我将集中在开发有意义的人口行为模型，其中的球员的选择约束的上升发生有机。我感兴趣的模型分为两类：健康和社会经济。我想研究医疗生产者的单一付款人预算限制，以及它对特定年龄组人群公共覆盖治疗分配的影响（如带状疱疹，流感或艾滋病毒的预防性疫苗）。在社会经济领域，资源共享或规范建立等制约因素可以纳入人口群体或个人的决策，从而形成GNG框架。这类模型是经济体或地区之间的限额交易环境协定，或对希望抵御市场网络攻击的生产商的资源集中限制。 我感兴趣的是调查的重要性，看看手头的特定应用问题的GNG框架，了解/计算GNE状态的好处，以及它们在给定的模型中的特殊意义。我计划在各个应用领域（人口健康、业务研究、经济学）传播研究结果。

This proposal focuses on the notion of generalized Nash games (GNG), tackling questions of existence of solutions, computational methods for finding these solutions, and the role this modelling concept can play in applied problems in populations.GNG were introduced in the 50's, and represent models of noncooperative behaviour among players whose strategy sets, together with their payoff functions, depend on the strategy choices of other players. The popularity of GNG as a modelling framework is not by far as wide as that of usual Nash games. This is due to the fact that solving GNG poses very complex mathematical difficulties and both existence theory for solutions and computational methods vary depending on the subclasses of GNG under investigation. My research plans include both a theoretical component and a modelling one. 1) In the theoretical direction, I plan to extend recently developed personal results and computational methods to provide answers to the question of existence of generalized Nash equilibria (GNE) for GNG without shared constraints. One of the most intriguing features of a GNG is the fact that their solution sets are generally very large. My own work on this topic plans to fully explore variational inequality-based and evolutionary algorithmic approaches for describing entire solution sets of GNG.Further, I plan to relate the concept of evolutionary stable state (ESS) to a GNG, and investigate whether this concept can be linked to a (slightly modified) replicator dynamics. If so, I want to investigate the counterparts, in the generalized setting, of classic relations/results between ESS states and Nash games.2) In the modelling direction, I will concentrate on developing meaningful models of population behaviour where the rise of constraints on players' choices takes place organically. The models I am interested in developing fall into two categories: health and socio-economic. I want to study single-payer budget constraints across producers of medical treatments, and its impact on the allocation of publicly covered treatments for specific age groups in a population (such as prophylactic vaccines for shingles, influenza, or HIV). In the socio-economic realm, constraints such as resource sharing or norm establishment can be incorporated in population groups' or individuals' decision making, leading to a GNG framework. Such classes of models are the cap-and-trade environmental accords between economies or regions, or resource pooling constraints on producers who want to defend against cyber attacks on their markets. I am interested to investigate the importance of looking at the particular applied problem at hand in the GNG framework, the benefits of knowing/computing GNE states and their particular significance in the given model. I plan to disseminate the results in the respective applied communities (population health, operations research, economics).