The problem is people often automatically (and incorrectly) apply MOP's "bluff 1/(P+1)" formula (in the case of limit poker; s/(1+s) for variable bet sizes, where s=1/P, the bet size in pots) to any and all bluffing situations, partially because it was presented that way (i.e. the distribution of A, K, and Q is always even).

Yeah

Can't say about that. Alpha relates the amount of bluffing and value combos and holds in your game as well as in the original. If the value of alpha is directly used to something unrelated, like "how many of my nuts I should fold" or "how many of my bluff catchers I should bet" or "my range is AAKKKQQQQQ, what percentage of my Q's should I bet", I'm sure ranges get wacky.

Alpha says the percentage of the time you should bluff when you hold the Q. But it only works in my game when x=z because only then does x/z(P+1)=1/(P+1), the value of alpha given in MOP. In other words, my value for alpha generalizes the result given in MOP and takes more information into account, thus applying to more situations.

I don't find that in MoP, can you give an example? They come up with b by doing the math (without using alpha), and yes, in their game it turns out to be the same as the value of alpha. Maybe unfortunately, after solving for b=1/(1+P) on page 146, they do state that b=alpha. Is that what's bugging you?

"assume b is Y's bluffing frequency with queens" (middle of p. 145).

What bugs me is what I explain in greater detail in the thread that I linked to earlier, that alpha does not apply to most situations; in most situations it is not the case that x=z. MOP does not make this clarification, and as a result (at least partially due to this) people have treated alpha as if it is the one and only GTO bluffing frequency, which applies to all situations (i.e. situations in which x≠z).

On page 145, they start with 0 assumptions and the reduced matrix. b or c are in no way tied to alpha at this point. After the sentence you quoted, they solve for b and c. They don't use alpha to solve for b or c. They do arrive at the same value for b and alpha, though, in this case.

They have defined alpha as 1/(P+1) in the limit betting case. That value can be directly used to get the ratio of bluffing hands to value hands, or how often should one fold when holding a hand that beats a bluff. Nothing less, nothing more. Again, all that still works in your game.

They're not using it directly to get some other results, at least as far as I can see. If you want to find the ratio of your air you want to bluff with, or any other number that can't be derived from pot and bet sizes only, I suggest picking another letter and giving alpha a break.

I see what you're saying. Alpha says the percentage of the time you should bluff when you hold the Q, but not generally – this just happens to be the case in the example on p. 145. The actual definition is on p. 113 and is as you stated it.

It doesn't seem appropriate to assign another Greek letter to my equations, though, because they have 2 more variable than just P.

Really no, it doesn't. Once again, where do you see this in MoP, other than them (uselessly, imo) mentioning on page 146 that "b=alpha", after having solved for b already? They don't plug in alpha for b anywhere after that, so the statement "b=alpha" is never used anywhere, I guess it's there just for the heck of it, showing a curious coincidence of the solution for this particular game.

Even as is, "alpha says the percentage of the time you should bluff when you hold the Q" on that page is not wrong, they say "b is the percentage of the time you should bluff when you hold the Q" and mention that alpha seems to equal b for the game.

Again, if that particular line on page 146 is what's bringing this thread up, I agree, it should probably not be there without some caveats.

If it's another place where they claim b can be directly substituded with alpha, let me know.

lol I'm in agreement with you, pasita. Read my previous post again and you'll see that the point was that I don't see this in MOP other than on p. 146, which is why I say, "this just happens to be the case in the example on p. 145."

Anyways, what I am critiquing is how people misapply alpha.

Ok, they might. Still can't say about that.

After I missed the "need to be a mixed strategy" constraint on the AKQ game for the solution you gave in OP, I did more than my fair share of trainwrecking this thread. I guess this was a good demo in having a point and not really looking at the anwers, I'm sure I've learned something here. Good thing we got the train back on track again

As I mention in the other thread, this discussion with you helped me clarify a misunderstanding I had about alpha (namely, it's definition), so thanks for that.