In this paper, we investigate the impact of path additions to transport networks with optimised traffic routing. In particular, we study the behaviour of total travel time, and consider both self-interested routing paradigms, such as User Equilibrium (UE) routing, as well as cooperative paradigms, such as classic Multi-Commodity (MC) network flow and System Optimal (SO) routing. This work aims to provide an analysis and categorization of the properties of objective functions for transport network design, with the purpose of informing algorithm (and also network) designers. Among our results, we prove, via counterexample, that total travel time, under both cooperative and self-interested routing, is not supermodular with respect to path additions.
