Finding a Nash equilibrium and an optimal sharing policy for multiagent network expansion game

In this work, a multiagent network flow problem is addressed, aiming at characterizing the properties of stable flows and allowing their computation. Two types of agents are considered: transportation‐agents, that carry a flow of products on a given network and another agent, either a producer or a customer, who is willing to ship (receive, respectively), products. Every transportation‐agent controls a set of arcs, each having a capacity that can be increased up to a certain point at a given cost. The other agent (i.e., the customer/producer) is interested in maximizing the flow transshipped through the network. To this aim, we assume it offers the transportation‐agents a reward that is proportional to the realized flow value. This particular multiagent framework is referred to as a Multiagent network expansion game. We characterize and find particular stable strategies (i.e., Nash equilibria) that are of interest for this game. We particularly focus on the problem of finding a Nash Equilibrium and a sharing policy that maximize the value of the total flow. We prove that this problem is NP‐hard in the strong sense and show how such a strategy can be characterized considering paths in specific auxiliary graphs. We also provide a mixed integer linear programming formulation to solve the problem. Computational experiments are provided to prove the effectiveness of our approach and derive some insights for practitioners.