On the fairness and cooperation among free-riders

We consider the free-rider problem in peer-to-peer (P2P) systems, where agents enjoy the group effort without contributing their share. We introduce the Free-Rider Game (FRG), a non-cooperative game incorporating a fairness-aware profit allocation rule based on the Robin Hood index. We show that FRG admits strong structural properties. First, making a non-zero contribution is a dominant strategy for any player. Second, a player contributes positively whenever at least one other player does so. Third, FRG admits a unique Nash equilibrium in which each player contributes the fullest, eliminating free riding. Fourth, equilibrium outcomes are proportionally fair, ensuring balanced allocation across agents. Finally, FRG guarantees full participation by embedding fairness directly into the payoff structure, differentiating it from classical public goods games, which often yield zero or partial contributions.