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：对数线性动态是描述多个参与者行为均衡转变的演化方程。我们在连续行动空间中引入带广义指数函数的配对对数线性动态，称其为广义配对对数线性动态，并通过空间非局域的新演化方程加以刻画。我们证明了广义配对对数线性动态的适定性和可逼近性，表明其具有计算可实现性。同时，我们揭示了该动态与受控纯跳跃过程的平均场博弈存在显式联系，从而能够以统一方式理解这两种不同的数学模型。特别地，我们证明广义配对对数线性动态可推导为对应平均场博弈的短视版本，且保证各自唯一解存在的条件有所差异。这一过程的关键在于基于对数线性函数找到平均场博弈中需优化的目标函数。效用函数的单调性对于广义配对对数线性动态并非必要，但对平均场博弈至关重要。最后，我们展示了两种方法在基于采集数据的渔业管理问题中的应用。