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.

翻译：Logit动态是描述多参与者行为均衡转移的演化方程。我们在连续行为空间中利用广义指数函数构建了一种配对Logit动态，称之为广义配对Logit动态，该动态由空间非局域的新型演化方程描述。我们证明了广义配对Logit动态的适定性与可逼近性，表明其具备计算可实现性。同时揭示了该动态与受控纯跳跃过程的平均场博弈存在显式关联，从而可在统一框架下理解这两种不同的数学模型。特别地，我们证明广义配对Logit动态可作为相应平均场博弈的短视版本推导得出，且两者保证唯一解存在的条件各不相同。该推导过程的关键在于基于Logit函数构建平均场博弈中待优化的目标函数。效用函数的单调性对广义配对Logit动态并非必要，但对平均场博弈至关重要。最后，我们展示了这两种方法在渔业管理问题中的实际应用，并使用了采集数据进行验证。