The joint exploitation of a productive asset: a game-theoretic approach

In the present paper we explore the set of equilibria in a game-theoretic model in which players can jointly exploit a productive asset. As in repeated games, we find that under certain circumstances there may be efficient as well as inefficient equilibria. In the model we study, efficient trigger-strategy equilibria may exist from some starting states (stocks of assets) but not others. More precisely, there is a stock level, say y′, such that an efficient trigger-strategy equilibrium exists from starting stocks greater than or equal to y′, but not from those strictly less than y′. (This statement is meant to include the cases in which y′ is zero or infinite.) Under some circumstances, there may exist a new kind of equilibrium, which we call a switching equilibrium. We show that, in our model, whenever y′ is positive (and finite), there is an open interval I with upper endpoint y′ such that, from any starting stock in I there is an equilibrium of the dynamic game with the following structure: the players follow an inefficient but growing path until the stock reaches the level y′, and then follow an (efficient) trigger strategy after that. The use of a continuous-time model enables us to conveniently decouple the delay of information from the time interval between decisions.