Shapes and games

Doctoral thesis, 2024

This thesis summarizes the four articles “Decisions and disease”, “Diversity strengthens competing teams”, “Shape analysis via gradient flows on diffeomorphism groups” and “Team game adaptive dynamics”. Each article has a different context, but the mathematical aims are unified as we prove optimality of solutions or well-posedness of dynamics. Moreover, the modelling perspective is central to each investigation. In “Decisions and disease” we combine the classic SIR and SIS models from epidemiology with the prisoner’s dilemma game. Here, the steady state solutions are interpreted in terms of cooperation during a pandemic. Another game is studied in “Diversity strengthens competing teams”, namely the so-called Game of Teams, for which all Nash equilibria are found. The optimal solutions, i.e., the Nash equilibria, are characterized by teams with maximal diversity in the sense that the successful teams have as different members as possible. Gradient flows are explored next, with a focus on an efficient method for image matching. We prove well-posedness of a gradient flow that is regularized by the deformation of the Riemannian metric of the manifold which the images are defined on. Lastly, the adaptive dynamics framework is applied to the Game of Teams. This model of evolution pushes the strategies of the game in the direction of the selection gradient. We have analyzed the well-posedness of the adaptive dynamics equations and answered questions about the stationary solutions, that is which solutions that do not display any dynamics despite the selection pressure that the selection gradient forces on them. It is found that the stationary solutions agree with the Nash equilibria.