The Stochastic Lake Game: A Numerical Solution

In this paper, we numerically solve a stochastic dynamic programming problem for the solution of a stochastic dynamic game for which there is a potential function. The players select a mean level of control. The state transition dynamics is 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. The control variables are the levels of phosphorus discharged (typically by farmers) into the watershed of the lake, and the random shock is the rainfall that washes the phosphorus into the lake. The state of the system is the accumulated level of phosphorus in the lake. The system dynamics are sufficiently nonlinear so that there can be two Nash equilibria. A Skiba–like point can be present in the optimal control solution. We analyze (numerically) how the dynamics and the Skiba–like point change as the variance of the noise (the rain) increases. The numerical analysis uses a result of Dechert (1978) to construct a potential function for the dynamic game. This greatly reduces the computational burden in finding Nash equilibria solutions for the dynamic game.