Sampled-Data Nash Equilibria in Differential Games with Impulse Controls
We study a class of deterministic two-player nonzero-sum differential games where one player uses piecewise-continuous controls to affect the continuously evolving state, while the other player uses impulse controls at certain discrete instants of time to shift the state from one level to another. The state measurements are made at some given instants of time, and players determine their strategies using the last measured state value. We provide necessary and sufficient conditions for the existence of sampled-data Nash equilibrium for a general class of differential games with impulse controls. We specialize our results to a scalar linear-quadratic differential game and show that the equilibrium impulse timing can be obtained by determining a fixed point of a Riccati-like system of differential equations with jumps coupled with a system of nonlinear equality constraints. By reformulating the problem as a constrained nonlinear optimization problem, we compute the equilibrium timing, and level of impulses. We find that the equilibrium piecewise continuous control and impulse control are linear functions of the last measured state value. Using a numerical example, we illustrate our results.
Nash equilibria in nonzero-sum differential games with impulse control
In this paper, we introduce a class of deterministic finite-horizon two-player nonzero-sum differential games where one player uses ordinary controls while the other player uses impulse controls. We use the word ‘ordinary’ to mean that Player 1 uses control strategies that are piecewise continuous functions of time. We formulate the necessary and sufficient conditions for the existence of an open-loop Nash equilibrium for this class of differential games. We specialize these results to linear-quadratic games, and show that the open-loop Nash equilibrium strategies can be computed by solving a constrained non-linear optimization problem. In particular, for the impulse player, the equilibrium timing and level of impulses can be obtained. Furthermore, for the special case of linear-state differential games, we obtain analytical characterization of equilibrium number, timing, and the level of impulse in terms of the problem data. We illustrate our results using numerical experiments.
Building efficient institutions: A two-stage differential game
We consider a two-stage dynamic game with a corrupt government and civil society as its players. We characterize open-loop Nash equilibria and an interior switching time from a regime with high government corruption which persists in the first stage (bad regime) to a free-corruption regime and greater institutional quality (good regime, second stage). We found that an increase of optimism (pessimism) in the society will lead the civil society to invest less (more) efforts to fight corruption whereas a corrupt government will invest more (less) efforts in repression policy. Overall, the numerical results show that the higher the efficiency of the civil monitoring effort, the efficiency of institutions and the lower the discount rate; the higher the inertia which will lead the economy to a much earlier switch to good regime with low corruption as the jump occurs early.