We study general classes of discrete time dynamic optimization problems and dynamic games with feedback controls. In such problems, the solution is usually found by using the Bellman or Hamilton–Jacobi–Bellman equation for the value function in the case of dynamic optimization and a set of such coupled equations for dynamic games, which is not always possible accurately. We derive general rules stating what kind of errors in the calculation or computation of the value function do not result in errors in calculation or computation of an optimal control or a Nash equilibrium along the corresponding trajectory. This general result concerns not only errors resulting from using numerical methods but also errors resulting from some preliminary assumptions related to replacing the actual value functions by some a priori assumed constraints for them on certain subsets. We illustrate the results by a motivating example of the Fish Wars, with singularities in payoffs.

中文翻译：

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当价值函数的不正确性在最优和均衡中不传播时

我们研究离散时间动态优化问题和带有反馈控制的动态博弈的一般类别。在这样的问题中，通常在动态优化的情况下通过使用Bellman或Hamilton–Jacobi–Bellman方程作为值函数来找到解决方案，并为动态博弈使用一组此类耦合方程来找到解决方案，而这并非总是准确的。我们推导了一些通用规则，这些规则指出在值函数的计算或计算中哪种错误不会导致沿相应轨迹的最优控制或Nash平衡的计算或计算中的错误。该一般结果不仅涉及使用数值方法导致的误差，还涉及某些初步假设导致的误差，这些初步假设涉及在某些子集上用一些先验假定约束条件代替实际值函数。我们以具有启发性的“鱼类战争”为例来说明结果，其收益具有奇异之处。