Signaling Games for Log-Concave Distributions: Number of Bins and Properties of Equilibria

Abstract

We investigate the equilibrium behavior for the decentralized quadratic cheap talk problem in which an encoder and a decoder have misaligned objective functions. In prior work, it has been shown that the number of bins under any equilibrium has to be at most 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 there exist unique equilibria for any number of bins. In contrast, for sources with semi-unbounded support, there may be an upper bound on the number of bins depending on certain conditions stated explicitly. Moreover, we show that for log-concave sources, an equilibrium with more bins is preferred by the encoder and decoder as it leads to a better cost for both of them. Furthermore, for strictly log-concave sources with two-sided unbounded support, we prove convergence to the unique equilibrium under best response dynamics, which makes a connection with the classical theory of optimal quantization and convergence results of Lloyd's method. In addition, we consider 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.