Thanks for the reply, Andrew, and for writing the book in the first place. I almost didn't buy it because of a couple of Amazon reviews that said it wasn't about "real poker", but of all the books I've started reading since I got interested in poker three months ago, this is the one I feel is helping me most at figuring out how to think about the game.
Thanks for the book recommendations, too, I've already bought POP2, and I'm looking forward to reading it once I've finished POP1 (it's a slow read if you try to work through the questions yourself before reading the answers). I just read the introduction to The Mathematics of Poker, looks like that's right up my alley, too.
As for my second question, I'm not sure I get it. I'll try to go through my assumptions and conclusions step by step, and I'd be grateful if you could tell me at which step I'm wrong:
- You're saying that at equilibrium Ivan's frequency of bluffing with a Q is the same as his frequency of bluff-catching with a K.
- "At equilibrium" means that the frequency at which Ivan bluffs with a Q makes Opal indifferent to calling with a K, and the frequency at which Ivan bluff-catches with a K makes Opal indifferent to bluffing with a Q.
- Both of those frequencies depend on which bet size is chosen for this toy game, which is an arbitrary choice. We could choose, say, 1/3 pot instead of 1/2 pot as the only possible bet size for this toy game, which would still result in an equilibrium, but at different frequencies.
- If the only allowed bet size is 1/3 pot, Ivan has to have 1/5 bluffs in his betting range to make Opal indifferent to calling with a K. Having 1/5 bluffs in his betting range means that he will bluff with 1/4 of his Qs.
- If the only allowed bet size is 1/3 pot, Ivan has to call with 3/4 of his range to make Opal indifferent to bluffing with a Q. Calling with 3/4 of his range means that he will call with 1/2 of his Ks.
- Ivan bluffs with 1/4 of his Qs and calls with 1/2 of his Ks at equilibrium. Those frequencies are not the same.
Addendum to my last post: The formulas I came up with have sign errors and are switched around, which doesn't really change anything, but I'd like to correct that—I can't edit the post though, does anyone know why? Am I too new a member to be granted those rights?