Nash equilibrium in a singular stochastic game between two renewable power producers with price impact

Abstract

We study the singular stochastic game, formulated in Awerkin and Vargiolu (Decis. Econ. Finance 44(2), 2021), between two agents aiming at maximizing their profits by installing photovoltaic panels and selling the produced electricity, net of installation costs, in the case that their cumulative installations have an impact on power prices. We first solve explicitly the static, one-step, version of the game, and find that Nash equilibria divide the state-space into four regions: one where both players are idle, two where only one player installs new panels, and one where both players install. In some particular regimes, we find that the latter may not be uniquely distinguished from the previous two. We then consider the dynamic, continuous-time, problem. Led by the intuition garnered in the static case, we assume a free-boundary structure similar to that arising in the one-step game and provide a rigorous verification theorem for the corresponding system of free-boundary HJB equations, also taking into account the lack of smoothness of the value functions near the free boundaries. Finally, for each regular solution of the HJB system, we show that there exists a unique equilibrium strategy, which is obtained as the solution to the Skorokhod-type problem associated with the free boundary.