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Originally Posted by yaqh
On pg 44, I said what I mean by "near-nut" hands -- a hand strong enough that we'll always try to get all-in, and whenever Villain has a better hand, he'll always try to get all-in too. Basically, it just means that whenever we have such a hand and Villain has better, we're guaranteed to go broke.
This applies to a full 1/3 of the BB's range in this game (in the symmetric distributions case, with B=S=P). The top 1/3 of BB's range is actually indifferent between leading and check-calling at equilibrium, and indieed, he sometimes leads and sometimes checks with hands in this region. And SB puts in the same amount of money, exactly, versus both actions.
In the way the solution is normally structured, BB leads with his strongest hands and check-calls some weaker ones, but that solution isn't actually unique. Other structures (which can involve check-calling with the nuts) are co-optimal. I talk about this and this game more in one of my videos.
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Originally Posted by pasita
Yep, I don't see how it applies. Can SB somehow exploit if BB always uses the structuring from MoP or A13.10?
No, co-optimal means multiple strategies are unexploitable.
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The co-optimality of the "check everything" from p44 and using the "bet-chcall-fold" regions came as a surprise to me.
I definitely don't say that checking his entire range is an unexploitable strategy of the BB's here. To be clear, it isn't.
If you don't see why, then let me know, but otherwise, I'll skip most of the text where you are assuming it is and confused because of it.
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Originally Posted by mmowgli
the best 1/3 of the BB range is indifferent between checking and betting ONLY if the SB is playing at equilibrium and it says to us that with that part of our range we must play a mixed strategy at equilibrium, BUT doing it with the wrong ratio bet:check would make us exploitable.
True if we remove the "ONLY".
This indifference is shown graphically in Figure 13.6.
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Originally Posted by pasita
Sorry, still does not compute. Which might be a problem at my end, don't get me wrong.
F13.10 describes the full street 1 psb symmetric game, same solution as in MoP. BB ALWAYS bets his top and bottom, x/c some and x/f some. This is an equilibrium to my understanding. I still don't see
a) why BB should sometimes (how often?) check (a part of ?) his top hands if he's at equilibrium already using the strategy from f13.10
The equilibrium strategy in Figure 13.10
does involve checking some of his top hands. There are other, co-optimal, strategies that also also involve checking and betting with some of his strongest hands, but not the exact same ones as in the strategy in the figure. These have the same EV at equilibrium.
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Originally Posted by mmowgli
a) In the figure 13.10 the BB is value-betting 1/6 of the time with his top hands and check-calling 2/6 of the time (50% minus 1/6 of the time). So between his 5/6th percentile and his 6/6th percentile he bets, and between his 4/6th percentile and his 5/6 percentile he checks. With the hands between 4/6th percentile and 6/6th percentile he is really indifferent at equilibrium because you are going to end all-in anyway vs the same number of hands better or worse than yours anyway.
yes
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b) I don't know why you think that the BB plays "check everything game", IMO this idea isn't in the book. The most profitable way to play the 0th percentile hand to the 1/12th percentile hand still would be betting as a bluff. And of course any variation in the structure of the BB ranges could have an exploitative response as you say if BB checks every hand, it's just the BB's top 1/3 hands would be always indiferent if SB doesn't change his strategy.
Actually, BB is (at least) indifferent to checking with each individual hand in his entire range at equilibrium (see Figure 13.6). His weakest 1/12 of hands, in particular, are indifferent between bluffing and check-folding. But that doesn't mean that going ahead and checking his entire range is unexploitable.
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Originally Posted by pasita
This from the Book (p.45):
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Why is it that Hero's GTO strategy cannot involve either always checking or always leading here with all his near-nut hands?
This from above:
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This applies to a full 1/3 of the BB's range in this game (in the symmetric distributions case, with B=S=P). The top 1/3 of BB's range is actually indifferent between leading and check-calling at equilibrium, and indieed, he sometimes leads and sometimes checks with hands in this region.
I thought those imply that BB sometimes plays the "check everything":
Tbh, it seems like that first quote explicitly says that he can't always check his near nuts, much less his whole range.
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If BB sometimes checks those top hands, how can he still be betting the bottom 1/12th every time?
He takes some of the good hands that used to check-call and bets with them instead. His overall value-leading frequency does not change.
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If Will would give us the strategy of how BB should check the top hands (always some? sometimes all? something different?), I could give a shot at the EVs. I'm saying "BB can always play the strategy of f13.10, no need to check those top hands". I'll also add "the game value for BB will be strictly worse if he checks some of the top 1/6 hands and announces his strategy to SB". Extremely happy if someone shows the math to prove those wrong.
Take the region of value-leading hands (which total 1/6 of his range) and move it to the left a bit. The strongest hands (which used to be leads) are now check-calls. Calculate EVs to verify that you're still at equilibrium.
And again, there's no
need to deviate from the strategy in Figure 13.10 -- the other co-optimal strategies have the same EV at equilibrium.
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BTW if anyone thinks this discussion should be moved to Theory to keep the book thread clean, I'm very happy with that too.
This thread seems fine to me.
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Originally Posted by pasita
I agree with all of those. But the book implies otherwise, and the author says otherwise in this thread. (That, or I have serious reading comprehension problems.) He says the equilibrium isn't unique, but I haven't seen his videos where he discusses this, nor any math anywhere supporting this. Even if it wasn't unique, I don't get why BB would _need_ to deviate.... it's still an equilibrium.
The equilibrium isn't unique, nowhere have I said that BB _needs_ to deviate, and the 65 minute video #6 in my (first and only) video pack is almost entirely dedicated to this model game.
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Originally Posted by pasita
Now wait a second. Does the book imply that you can bet ANY half of the top 2/6 range, i.e. 1/6 of total range, and still be at equilibrium? If that's what we're looking at, there might be infinite amount of equilibria there, yes. Times another infinite amount for choosing the bluffing range, as long as the frequencies stays right. It's just that those equilibria are as sensible as the GTO play of folding the nuts on the river at some branches.
There are an infinite number of co-optimal strategies.
A strategy involving folding the nuts on the river is dominated, i.e., there's some other strategy that's better (or at least as good as and sometimes better) than it, regardless of Villain's strategy. Those strategies can be equilibrium only if we never actually get to those river spots during equilibrium play.
I don't see how we can get any dominated equilibria by changing which of Hero's near-nuts check-call and which valuebet.
Last edited by yaqh; 06-05-2014 at 01:06 AM.