In this study, multi-owner maximum flow problems defined under grey uncertainty are investigated within the framework of cooperative game theory. In this context, egalitarian distribution approaches are comprehensively addressed by considering the flow and risk allocation dimensions in flow problems. Grey interval numbers are used to model situations where arc capacities cannot be known precisely in real-life logistics and infrastructure networks. In this context, we aim to overcome the limitations of both deterministic and probabilistic approaches.

Based on a single-source, single-sink and multi-owner network structure, where arc capacities are expressed by grey interval numbers, two separate games are formulated for all possible coalitions: the grey maximum flow game  and the grey risk game . The grey characteristic function of each coalition is obtained using the BWC algorithm and the Grey REILP method. Through this method, a quantitative balance is established between risk and system return.

Four Equal Surplus Sharing Approaches, namely the Grey Banzhaf value, Grey CIS-value (𝒢CIS), Grey ENSC-value (𝒢ENSC), and Grey ED-value (𝒢ED) are applied to both games. The results are evaluated comparatively in terms of efficiency, individual rationality, and coalitional rationality criteria. The findings demonstrate that the 𝒢ENSC-value lies within the grey core and ensures coalition stability. The results provide significant contributions to cooperative models for the design of fair, stable, and risk-sensitive collaboration in infrastructure and logistics networks under grey uncertainty.