Networked Public Good Games : Equilibria and Best Response Dynamics

This thesis investigates game theoretic models for public good provision. We consider network games defined on weighted and directed graphs, where agents are heterogeneous with respect to income level, preference structure, and effort cost. Public goods are here studied in their local form: the benefits of one agent’s contributions are shared only among her direct out-neighbors. Each player allocates her income between private consumption and contributions to the public good. While private consumption provides purely individual benefits, public contributions yield payoffs that also depend on the allocations of neighboring agents, scaled by the intensity of their network connections. Initially, we formalize the model, derive best response functions, and establish the existence of Nash equilibria. In addition, we provide the characterization of equilibrium profiles, with a focus on internal equilibria, where all agents contribute positively, and specialized equilibria, where only a subset contributes while others free-ride. The main results are threefold. First, we derive a sufficient condition for the uniqueness of Nash equilibrium in the general setting; provided the necessary assumptions, this condition reduces to a bound on the lowest eigenvalue of the symmetrized, per-row rescaled adjacency matrix. Second, we establish a Lipschitz condition ensuring contractivity of the synchronous best response function when the dominant eigenvalue of the adjacency matrix, suitably rescaled, is lower than one. This sufficient condition implies the contractivity of the discrete-time best response dynamics, which results in both the uniqueness of the Nash equilibrium and its global asymptotic stability for the discrete- and continuous-time dynamics. Moreover, under proper hypothesis, we characterize the stability of internal equilibria. Third, we study the stability of the continuous-time best response dynamics within an appropriate framework where contractivity may not hold; we prove that the trajectories globally converge to the set of Nash equilibria. In addition, we characterize locally asymptotically stable equilibria as local maxima of an associated functional and, if the uniqueness condition holds, establish that the global maximizer of such functional is globally asymptotically stable. Finally, we investigate possible efficiency metrics to evaluate equilibrium outcomes and explore intervention strategies aimed at improving welfare. We propose a preliminary study of a subsidy mechanism that incentivizes higher contributions and of a redistribution policy in which an external planner reallocates income to steer the system toward socially optimal equilibria.