Multi-sender cheap talk with restricted state spaces

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Pandering to Persuade (2013, methodology): “After presenting an illustrative example in Section 3, we turn in Section 4 to a general analysis of when one alternative 1Comparative cheap talk has been studied by Chakraborty and Harbaugh (2007, 2009).”
Spread of (mis)information in social networks (2009, background): “(4), x(k), converge to a consensus with probability one, i.e., x(k) → x̄e, where x̄ is a scalar random variable that depends on the initial beliefs and the random sequence of matrices {W (k)}, and e is the vector of all ones.”
How does honest costly signaling work (2014, background): “…models may often be referred to as examples of “cheap talk” (especially outside of biology, see, e.g., Farrell and Gibbons 1989; Battaglini 2002; Ambrus and Takahashi 2008) as there is often no need for a cost function to be associated with the signal as a requirement of maintaining honesty…”