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

本文主要讨论广义纳什博弈的概念，解决解的存在性问题，寻找这些解的计算方法，以及这个建模概念在种群应用问题中的作用。广义纳什博弈是在50年代的S提出的，它代表了玩家之间的不合作行为模型，他们的策略集和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受欢迎程度并不像通常的纳什游戏那样广泛。这是因为求解GNG带来了非常复杂的数学困难，解的存在理论和计算方法都因所研究的GNG的子类而异。我的研究计划包括理论部分和模型部分。*1)在理论方面，我计划推广最近发展的个人结果和计算方法，以回答没有共享约束的GNG的广义纳什均衡(GNE)的存在性问题。GNG最耐人寻味的特征之一是它们的解决方案集通常非常大。我自己在这个主题上的工作计划全面探索基于变分不等和进化算法的方法来描述GNG的整个解集。*此外，我计划将进化稳定状态(ESS)的概念与GNG联系起来，并调查这个概念是否可以与(略有修改的)复制者动力学联系起来。如果是这样的话，我想研究在一般情况下，ESS状态和Nash游戏之间的经典关系/结果的对应关系。*2)在建模方向，我将专注于开发有意义的群体行为模型，其中对玩家选择的约束的增加是有机发生的。我感兴趣的模型分为两类：健康模型和社会经济模型。我想研究医疗生产者的单一支付者预算限制，以及它对为人群中特定年龄段的公共覆盖治疗分配的影响(例如带状疱疹、流感或艾滋病毒的预防性疫苗)。在社会经济领域，可将资源共享或规范制定等制约因素纳入人口群体或个人的决策，从而形成GNG框架。这类模式是经济体或地区之间的总量管制与交易环境协议，或者是对希望抵御针对其市场的网络攻击的生产商的资源汇集限制。*我有兴趣调查在GNG框架中查看手头的特定应用问题的重要性，了解/计算GNE状态的好处及其在给定模型中的特殊意义。我计划在各自的应用社区(人口健康、运筹学、经济学)中传播这些成果。

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).