广义成对 logit 动态及其与平均场博弈的联系：聚焦资源管理的理论与计算研究
Generalized pair-wise logit dynamic and its connection to a mean field game: theoretical and computational investigations focusing on resource management
AI总结：
本文提出连续行动空间中的广义成对 logit 动态，证明其适定性与可计算性，并揭示其作为受控纯跳过程平均场博弈短视版本的联系及唯一性条件差异，且应用于渔业资源管理。
摘要：
Logit 动态是描述众多参与者之间行动向均衡转变的演化方程。我们在连续行动空间中利用广义指数函数构建了一种成对 logit 动态，称之为广义成对 logit 动态，并由一个新的空间非局部演化方程加以刻画。我们证明了该广义成对 logit 动态的适定性与可逼近性，以表明其在计算上是可实现的。我们还表明，这种动态与受控纯跳过程的平均场博弈具有明确联系，借助这一联系，两种不同的数学模型能够以统一方式被理解。特别地，我们表明广义成对 logit 动态可作为相应平均场博弈的短视版本导出，而且保证二者唯一解存在的条件彼此不同。这一过程的关键在于基于 logit 函数找到平均场博弈中待优化的目标函数。效用的单调性对于广义成对 logit 动态并非必要，但对平均场博弈却至关重要。最后，我们结合所收集的数据，展示了这两种方法在渔业管理问题中的应用。
英文摘要：
Logit dynamics are evolution equations that describe transitions to equilibria of actions among many players. We formulate a pair-wise logit dynamic in a continuous action space with a generalized exponential function, which we call a generalized pair-wise logit dynamic, depicted by a new evolution equation nonlocal in space. We prove the well-posedness and approximability of the generalized pair-wise logit dynamic to show that it is computationally implementable. We also show that this dynamic has an explicit connection to a mean field game of a controlled pure-jump process, with which the two different mathematical models can be understood in a unified way. Particularly, we show that the generalized pair-wise logit dynamic is derived as a myopic version of the corresponding mean field game, and that the conditions to guarantee the existence of unique solutions are different from each other. The key in this procedure is to find the objective function to be optimized in the mean field game based on the logit function. The monotonicity of the utility is unnecessary for the generalized pair-wise logit dynamic but crucial for the mean field game. Finally, we present applications of the two approaches to fisheries management problems with collected data.
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