One-to-many non-cooperative matching games

We study a strategic model of wage negotiations between firms and workers. First, we define the stability of an allocation in an environment where firms can employ more than one worker. Secondly, we develop a one-to-many non-cooperative matching game, which is an extension of Kamecke’s one-to-one non-cooperative matching game. The main result shows the equivalence between the stable allocations and the outcomes of the subgame equilibria in the matching game: for any stable allocation in this game there is a subgame perfect equilibrium which induces the allocation on the equilibrium path, and every subgame perfect equilibrium induces a stable allocation on the equilibrium path. Furthermore, as for the existence of a stable allocation, we argue that a stable allocation, as with a subgame perfect equilibrium, does not always exist, but it exists under some conditions, using Kelso and Crawford’s modelling.