Industrial symbiosis is promoted as a sustainable way to convert production residues into high added-value products. The by-product synergy is a particular configuration of an industrial symbiosis system, where the by-products generated by a production unit are used as raw materials by another production unit. The by-product exchange takes place between two or several related or autonomous companies, which requires to align lot sizing decisions of each involved actor. To cope with this joint production planning problem, we focus on the framing of symbiotic partnerships within a by-product synergy network, involving one supplier and one receiver of by-products. The collaboration policies can differ from one industrial symbiosis to another. In this paper, we investigate collaboration policies for one-sided asymmetric information sharing designed by using game theory. A careful analysis is performed to evaluate the impact of the estimation quality of the unknown parameter. Based on the obtained results, the relevance of the game theoretic-based collaborations policies is discussed with respect to the baseline collaboration policies under full and no information sharing.