Modeling the equilibrium of global shipping markets: A cyclic space-time supernetwork approach

International maritime shipping carries around 80% of global trade, and understanding long-run flow patterns is essential for analyzing competition in this market. This study develops a modeling framework for characterizing long-run equilibrated shipping flows under oligopolistic competition. We introduce a cyclic space-time supernetwork that captures spatial connectivity, seasonal demand variation, and vessel behaviors—including laden trips, ballast trips, and in-port holding—over an annual cycle. Within this network, each carrier operates as a Cournot-Nash (CN) player that allocates its fleet across cyclic paths to maximize yearly profit subject to cargo-demand and port-capacity constraints. We formally establish the existence of a CN equilibrium and prove that one such equilibrated shipping flow can be obtained by solving a system-optimal (SO) linear program defined over the same feasibility set. To solve the resulting large-scale SO problem, we develop a tailored column-generation algorithm that exploits the combinatorial structure of cyclic paths and incorporates problem-specific acceleration strategies. Numerical experiments based on real-world data from major ports demonstrate the computational efficiency of the algorithm and illustrate the managerial insights provided by the equilibrium solution.