If the pot is 2 units, and any size bet is B, then the value to SB in the bet or check game (symmetric) is:
V(sb) = B/(B+4)(B+1)
This has a maximum at B=P=2
This is the solution to Von Neumann poker, and is pretty basic in poker game theory.
Google “von neumann poker solution”
If this book has not provided you the method for this part, it may be unlikely to provide you with much more, since asymmetric is much more difficult. Iirc, the volume one only adds a nuts vs air hot take and then volume 2 blasts off from there.
Solutions for section 7.3.2 are obtained computationally by solving the indifference equations, google sbbet-or-checkgame.pdf. What you can do is try to replicate the answers, maybe in another program, and find out why they would differ from those obtained by simplifying the indifference equations.
Thank you for answering, I'm working on it. If I tell you that the equity of SB's worst valuebetting hand against BB's calling range has to be 50% --that's to say that the worst valuebetting hand has to win half the time to be indifferent between checking and betting (neglecting the times the hand ties)-- do you agree with that? To clarify the point I'm not saying the worst valuebetting hand in SB's range has to have at least 50% equity, I'm saying it has to have exactly 50% equity, independently of the equity of that same hand when checked back (i.e. the equity of that hand against the entire BB's range getting to the river). Let me know if that makes sense to you
I am currently studying this book as well. What do you think about having our own discussion? I am thinking it is going to be difficult to find any other forum that is active. You can private message me if you are interested.