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

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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).