In this paper we solve a stochastic dynamic programming problem for the solution to a dynamic game in which the players select a mean level of control. The state transition dynamics are a function of the current state of the system and a multiplicative noise factor on the control variables of the players. The particular application is to lake water usage. (A deterministic version of the model was analyzed by Brock and Dechert, "The Lakegame.") The control variables are the levels of phosphorus discharged (typically by farmers) into the water shed of the lake, and the random shock is the level of rainfall that washes the phosphorus in to the lake. The state of the system is the accumulated level of phosphorus in the lake. The system dynamics are sufficiently non-linear so that there can be two Nash equilibria, and hence a Skiba point can be present in the optimal control solution. In the paper we analyze (numerically) how the dynamics and the Skiba point change as the variance of the noise (the rain) increases. The numerical analysis uses a result of Dechert (JET 1978) that allows us to construct a potential function for the dynamic game. This greatly reduces the computational burden in finding Nash equilibrium solutions for the dynamic game.
