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.
翻译:我们用真实随机变量和二次成本标准来调查分散式廉价谈话问题的平衡行为, 其中, 编码器和解码器具有不匹配的客观功能。 在先前的工作中, 已经显示, 任何均衡中的文件夹数量都必须可以计算, 并基于某些明确规定的条件, 将Crawford 和 Sobel 的经典结果加以概括。 在本文中, 我们首先根据对密度为 $[0, 1$] 支持的源进行修改这一结果。 对于有双向无约束支持的源, 我们证明, 对于任何数量有限的垃圾桶来说, 我们有一个独特的平衡。 相反, 对于有半默认支持的源, 任何均衡中的垃圾箱数量可能有一个有限的上限。 此外, 我们证明, 对于日志源而言, 编码器的预期成本和平衡减少的预期值随着垃圾桶数量的增加而得到改进。 此外, 对于有双面支持的纯正正向支持的源, 我们证明我们与在最佳响应动态下的独特平衡。 开始于一个给定的正统的正统值源,, 与某种正统的正统值和正统的正统的值连接连接联系联系, 。