Four Essays on Dynamic Games Theory and Economic Applications

This thesis focuses on the study of differential games and their applications. In the first part, we describe the classical solution concepts of Nash and Stackelberg equilibria. To obtain results, we need tools coming from optimal control (OC), such as the Pontryagin Maximum Principle (PMP) and the Dynamic Programming (DP) approach. The aim is to study situations related to Stackelberg games. Whereas the theory of fixed final time hierarchical games is well defined, the variable time situation misses a proper formulation in the literature. We formalise and describe the solution procedure for this class of games by looking for an open-loop Stackelberg equilibrium (OLSE), where the final time becomes a decision variable for the leader. As typical for open-loop strategies, we face the time consistency issue by providing a couple of counterexamples. Then, we propose an applied model for infinite time horizon hierarchical games. Assume that a firm wants to invest in standard advertising and sponsored content on a media outlet highly concerned with its credibility. Since sponsored advertising may damage the media credibility, we allow the media to control its flow. We formalise the problem as a Stackelberg differential game where the media acts as the leader. We characterise a time-consistent OLSE despite the leader’s upper bound on the follower’s sponsored content control function. The introduction of this bound represents an innovation for this class of problems. The second part of the thesis is devoted to Markov chain-based models. We begin by proving that for the class of binary state continuous-time problems, the classical Hamilton–Jacobi–Bellman (HJB) equation can be substituted by an ordinary differential equation (ODE). This expedient allows us to obtain solutions for complex analytical situations. Later, we propose an application with infinitely many symmetric players based on offshoring and reshoring phenomena. We present a two-country model where the South country has lower production costs and the North country provides a huge tax discount to induce reshoring. We study the problem by considering a representative player. By employing the HJB to ODE transformation, we can compute a feedback solution for the problem and understand how national subsidies affect the process. Eventually, we modify the offshoring-reshoring model by making the incentives proportional to the amount of offshored businesses. As a result, the problem evolution turns out to be defined by coupled equations. The new formulation leads to a mean-field game (MFG) with no easy analytical solution. We obtain insights by numerical simulations.