对数凹分布的信号博弈：仓数与均衡的性质
Signaling Games for Log-Concave Distributions: Number of Bins and Properties of Equilibria
摘要：
我们研究了分散式廉价谈话问题的均衡行为，其中编码器和解码器具有不对齐的目标函数的实随机变量和二次成本标准。在以前的工作中，它已被证明，在任何平衡箱的数量必须是可数的，推广了一个经典的结果，由于克劳福德和索贝尔谁认为与密度支持$[0,1]$的来源。在本文中，我们首先完善这一结果的上下文中的对数凹源。对于双边无界支撑的源，我们证明了，对于任何有限数量的箱，存在唯一的平衡。相反，对于具有半无界支撑的源，取决于明确规定的某些条件，平衡中的箱的数量可能存在有限的上限。此外，我们证明了对数凹源，编码器和解码器在平衡的预期成本减少的箱子的数量增加。此外，严格的对数凹源与双边无界支持，我们证明了收敛到唯一的平衡下的最佳响应动力学开始与一个给定数量的箱，使连接与经典理论的最佳量化和Lloyd的方法的收敛结果。此外，我们考虑更一般的源，满足某些假设的尾部（S）的分布，我们表明，存在无限多箱的平衡源与双边无界支持。进一步明确的特征提供了指数，高斯和紧支持的概率分布的源。
英文摘要：
We investigate the equilibrium behavior for the decentralized cheap talk problem for real random variables and quadratic cost criteria in which an encoder and a decoder have misaligned objective functions. In prior work, it has been shown that the number of bins in any equilibrium has to be countable, generalizing a classical result due to Crawford and Sobel who considered sources with density supported on $[0,1]$. In this paper, we first refine this result in the context of log-concave sources. For sources with two-sided unbounded support, we prove that, for any finite number of bins, there exists a unique equilibrium. In contrast, for sources with semi-unbounded support, there may be a finite upper bound on the number of bins in equilibrium depending on certain conditions stated explicitly. Moreover, we prove that for log-concave sources, the expected costs of the encoder and the decoder in equilibrium decrease as the number of bins increases. Furthermore, for strictly log-concave sources with two-sided unbounded support, we prove convergence to the unique equilibrium under best response dynamics which starts with a given number of bins, making a connection with the classical theory of optimal quantization and convergence results of Lloyd's method. In addition, we consider more general sources which satisfy certain assumptions on the tail(s) of the distribution and we show that there exist equilibria with infinitely many bins for sources with two-sided unbounded support. Further explicit characterizations are provided for sources with exponential, Gaussian, and compactly-supported probability distributions.
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