This paper studies distributed multi-relay selection in energy-harvesting cooperative wireless networks and models it as an Indian Buffet Game (IBG). Particularly, the IBG is utilized to model the multi-relay selection decisions of network source nodes, while accounting for negative network externality. Two scenarios are considered: (1) constrained selections (CS), and (2) unconstrained selections (US). In the former scenario, each source is constrained to a maximum number of relay selections; while in the latter scenario, the source nodes can select as many relays as possible. Since the relays are energy-harvesting—and thus intermittently harvest random amounts of energy—the accumulated energy at each relay is unknown to the source nodes, leading to uncertain relays’ energy states. In turn, a non-Bayesian learning (NBL) algorithm is devised for the source nodes to learn the relays’ energy states. After that, two distributed best-response (BR) recursive algorithms, namely BR-CS and BR-US, are proposed to allow the source nodes to make multi-relay selection decisions, while guaranteeing subgame perfect Nash equilibrium. Simulation results are presented to verify the efficacy of the proposed distributed NBL and multi-relay selection algorithms. Specifically, the NBL is shown to efficiently learn the true relays’ energy states. More importantly, the BR-CS algorithm is shown to be comparable to the centralized multi-relay selection—and superior to other relay selection schemes—in terms of network sum-rate improvement (and utility). Lastly, the number of relay selections of the BR-CS algorithm must be constrained to the minimum so as to reduce complexity and fully exploit diversity gains.