Dynamical inspection game with continuous strategies

Most contributions on the inspection game concern arm control and disarmament; recently some contributions have considered organizational settings. We consider an inspection game where a principal chooses to inspect or not inspect and an agent simultaneously can either work or shirk. Combined payoffs are maximized when the principal does not inspect and the agent works while the unique Nash equilibrium of the stage game is in mixed strategies with positive probabilities of inspecting/shirking. To overcome this difficulty we introduce a continuous action version of the inspection game which extends the original formulation and discuss the existence of the Nash equilibria in pure strategies depending on the convexity of the cost functions we consider. Then, as most of the interactions in organizations develop over time, we propose a dynamic model with adaptive adjustment. We address some characteristics of the dynamic behavior of the game and the bifurcations observed, through both analytical and numerical methods. For the dynamical game we determine the fixed points, and study their stability. Fixed points are related to the Nash equilibria of the continuous inspection game and the collectively optimal outcome is obtained as a fixed point that is just virtual. Our findings are interpreted in terms of stakeholders theory, relational contracts and negotiation.