  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.
