Networked Digital Public Goods Games with Heterogeneous Players and Convex Costs
Yukun Cheng, Xiaotie Deng, Yunxuan Ma
TL;DR
This work analyzes networked public goods games with heterogeneous players and convex costs, formalizing each player's payoff as $u_i(\bm{x}) = f_i(k_i) - c_i(x_i)$ with $k_i = \sum_j w_{ij} x_j$ and $w_{ii}=1$. It develops a rigorous framework for existence and uniqueness of Nash Equilibria, leveraging an $\alpha$-convex modification and Brouwer's fixed-point theorem for existence, and a $(\bm{\gamma},\Sigma)$-near-potential/diagonal-strict-concavity approach for uniqueness, including contraction arguments for exponential convergence of the pseudo-gradient dynamics. The paper also analyzes welfare via gradient-flow dynamics, establishes comparative statics under money redistribution using a high-dimensional implicit-function theorem, and provides a case-study with quadratic and homogeneous approximations to illustrate practical implications and policy relevance for digital networks. A key contribution is the introduction of game equivalence and near-potential conditions that extend NE-uniqueness results to broader classes of networked public goods with heterogeneity and convex costs. Overall, the results deepen the understanding of strategic interactions in networked digital public goods and offer tools for designing welfare-enhancing interventions in internet economies and social networks.
Abstract
In the digital age, resources such as open-source software and publicly accessible databases form a crucial category of digital public goods, providing extensive benefits for Internet. This paper investigates networked public goods games involving heterogeneous players and convex costs, focusing on the characterization of Nash Equilibrium (NE). In these games, each player can choose her effort level, representing her contributions to public goods. Network structures are employed to model the interactions among participants. Each player's utility consists of a concave value component, influenced by the collective efforts of all players, and a convex cost component, determined solely by the individual's own effort. To the best of our knowledge, this study is the first to explore the networked public goods game with convex costs. Our research begins by examining welfare solutions aimed at maximizing social welfare and ensuring the convergence of pseudo-gradient ascent dynamics. We establish the presence of NE in this model and provide an in-depth analysis of the conditions under which NE is unique. We also delve into comparative statics, an essential tool in economics, to evaluate how slight modifications in the model--interpreted as monetary redistribution--affect player utilities. In addition, we analyze a particular scenario with a predefined game structure, illustrating the practical relevance of our theoretical insights. Overall, our research enhances the broader understanding of strategic interactions and structural dynamics in networked public goods games, with significant implications for policy design in internet economic and social networks.