Competitive uncapacitated lot-sizing game

We study the strategical behaviour of firms facing a lot-sizing problem with Cournot competition. Each player is a firm with her own production facility, modeled as an uncapacitated lot-sizing problem (i.e., production incurs set-up and variable costs and inventories are allowed). A Cournot competition is played in each time period (market) with each player deciding the quantity of product to place on it. The market price of that product in each time period depends on the total quantity placed in the market. We show that this is a potential game with possibly multiple pure Nash equilibria. We then investigate the plausibility of these equilibria to predict the game outcome by evaluating the difficulty of computing them. If the game has a single period, we prove that an equilibrium can be found in polynomial time, but it is weakly NP-hard to find an optimal pure Nash equilibrium (with respect to a given equilibrium refinement). If the game has no variable production and inventory costs, we prove that a pure Nash equilibrium can be computed in polynomial time.