Stable source connection and assignment problems as multi-period shortest path problems

We extend the familiar shortest path problem by supposing that agents have demands over multiple periods. This potentially allows agents to combine their paths if their demands are complementary; for instance if one agent only needs a connection to the source in the summer while the other requires it only in the winter. We not only show that the resulting cost sharing problem always generates a totally balanced game, regardless of the number of agents and periods, the cost structure or the demand profile, but that all totally balanced games are representable as multi-period shortest path problems. We then exploit the fact that the model encompasses many well-studied problems to obtain or reobtain balancedness and total balancedness results for source-connection problems, assignment problems and minimum coloring problems.