Date: May 3, 2021
Time: 9:00 to 12:00

Zoom link: https://hecmontreal.zoom.us/s/83409640472

Jury members:
President: Dena Firoozi
Supervisor: Georges Zaccour
Member: Ngo van Long
External member: Herbert Dawid

Abstract: This thesis consists of four essays on deterministic finite-horizon two-player nonzero- sum differential games where one player continuously controls the state while the other player strategically intervenes at certain (discrete) time instants to shift the state from one level to another. In contrast to classical differential games where all players take actions continuously, the impulse player also decides when to intervene during the game in addition to determining the level of the interventions. Impulse control models constitute a natural approach for analyzing strategic behavior of agents in many contexts such as investment in product quality, environmental regulation and cybersecurity. However, to solve practical problems, we need to address a series of theoretical and computational challenges that are due to the endogeneity of the timing of actions. This is the general topic of this work. In this thesis, we characterize the Nash equilibria under all the three information structures that have been considered in the differential game theory literature, namely, open-loop, feedback and sampled-data information structures. Further, we show that the timing of impulses can be obtained as a solution of a constrained non-linear optimization problem in the case of linear-quadratic differential games with impulse controls. To analytically characterize the equilibrium number, timing, and magnitude of impulses, we introduce canonical linear-state game models and compare the equilibrium behavior of players across different information structures. In each essay, we also illustrate the theoretical results using a game problem between two players, one of whom prefers a higher state value while the other aims to lower the state, a situation that arises in many regulation and cybersecurity applications.

Keywords: Differential games, impulse control, open-loop Nash equilibrium, feedback Nash equilibrium, sampled-data Nash equilibrium, quasi-variational inequalities, cybersecurity, regulation