I cannot understand Sobel & Crawford 1982 (cheap talk) I am a master student, currently reading this paper for fun during the break. I got lost during the part $U^S(\bar y, \bar m, b) = \max_{y \in Y} U^S(y,\bar m, b)$ on page 5. I dont understand why they wrote this without any explanation. Then I got lost again during proof of lemma 1. Should I read other paper first before approaching this?${{}}$
$U^S(\bar y,\bar m,b)=\max_{y\in Y}U^S(y,\bar m,b)$ is standard notation that says $\bar y$ is the action (taken by receiver) that would maximize sender's utility given message $\bar m$ and bias parameter $b$. In other words, sender would prefer receiver to choose $\bar y$ when he sends message $\bar m$. Sender's maximized utility is $U^S(\bar y,\bar m,b)$.
For a simpler introduction to Crawford and Sobel, you can read Chapter 18 of Tadelis's Game Theory textbook on cheap talk games.