Finite State Graphon Games with Applications to Epidemics

Area of Science:

• Game Theory
• Network Science
• Mathematical Epidemiology

Background:

• Heterogeneous interactions in large populations are complex to model.
• Graphons offer a way to represent interactions in a continuum of players.
• Understanding player behavior in dynamic systems is crucial.

Purpose of the Study:

• To develop a rigorous mathematical framework for games with a continuum of non-identical players.
• To analyze Nash equilibria in such systems.
• To propose and validate a machine learning-based numerical method for these games.

Main Methods:

• Utilized graphon theory to model player interactions.
• Developed a mathematical framework for analyzing Nash equilibria.
• Applied machine learning techniques for numerical simulations.
• Tested the approach on compartmental models in epidemiology.

Main Results:

• Established a sufficient condition for the existence of Nash equilibria.
• Proved the existence of solutions for continuum of fully coupled forward-backward ordinary differential equations.
• Demonstrated the efficacy of the machine learning approach through experimental results.

Conclusions:

• The proposed framework provides a robust method for analyzing complex games with continuum players.
• The machine learning approach offers a practical tool for solving such games, particularly in epidemiological contexts.
• This work bridges theoretical game theory with applied mathematical modeling.