[Jerrod Ankenman]

[ QUOTE ]
In reviewing an upcoming two plus two book, I have come across a statement that I believe to be incorrect (and if so will change). It has to do with how many hands are profitable raises on the button or small blind, given the tightness of the big blind. Playing Limit Holdem

Obviously, as a general rule, the tighter the big blind, the more hands you can profitably raise with. And vice versa. But what about if the blind is extremely loose? Is there some point where you can add back in some hands as profitable raising hands (assume you can't just call) that you would fold if he was a tad tighter? We can make this question a litter more explicit later on if need be.

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OK, well suppose we take the BB-on-the-button headsup case. I'll make the problem more explicit, as I suspect that arguments like "if he plays poorly on future streets, you can raise more hands," aren't what you're looking for. Hint: If the guy plays poorly postflop, you can raise every hand profitably.

Suppose that the BB is a limited nemesis; it will maximally exploit you, except that it is forced to vary its preflop strategy for the purposes of our problem. You can choose your preflop strategy as you wish, but starting from the subgame that begins with the flop, it will play optimally for the distributions and pot size that you have carried here.

Suppose that the BB plays the optimal preflop strategy. We don't know what this is, but your question seems to assume (and I agree) that it is right for the BB to fold at least a few hands, and it would be optimal for the SB to likewise fold at least a few hands on the button.

Suppose that the BB then moves his strategy toward extreme looseness by simply calling with all hands he would have folded previously, and you continue to play the full-game optimal strategy, ignoring his new strategy. How does this benefit your strategy? You gain value from your stronger hands - because he is calling with hands that are too weak.

Now consider the question of your borderline raise/fold hands. Can it be right in this new game to switch one from fold to raise? I maintain that it can't. Here's the argument:

Consider a hand X that is close to the border between raising on fold, but which is a fold in the full-game optimal strategy.

Suppose that the BB folds a set of hands Z playing the full-game optimal strategy. In order for this change to cause the SB to switch X from folding to raising, the new equity of raising X must become higher than the equity of folding it.

But we know that when the BB was folding with Z, he was giving the SB +1 unit each time he folded, and still raising wasn't profitable. Now in order for the SB to switch to raising, the increased equity from playing *postflop* against the widened distribution has to exceed the difference between the equity of playing out X, a marginal hand against the hands that the BB would fold.

This is virtually impossible because of the nemesis restriction on the postflop play. The SB can't make enough value from playing his weakest hand against an expanded postflop distribution to offset the +1 units he was getting from those hands that the BB was folding.

So then, no, there's no situation where you would back additional hands into your distribution as the BB tended toward extreme looseness. (this argument extends to situations where the BB plays only a few fewer hands as well.)

Jerrod