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A Stochastic Lake Game


  • Sharon I. O'Donnell
  • W. Davis Dechert


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.

Suggested Citation

  • Sharon I. O'Donnell & W. Davis Dechert, 2004. "A Stochastic Lake Game," Computing in Economics and Finance 2004 104, Society for Computational Economics.
  • Handle: RePEc:sce:scecf4:104

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    More about this item


    stochastic dynamic programming; dynamic game; potential function;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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