Jam-or-fold: (near)optimal for stacks up to 10 or 11BB?

[Jerrod Ankenman]

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My intuition must be failing me. I find it difficult to believe that for the full game the equity of the SB on the button can drop to negative values. A hand-waving argument for this is that when the SB would follow a (non-optimal) strategy that for each hand starts with limping, the result is that the BB and SB have contributed symmetrically to the pot, and that the BB is first to make a move. Effectively, the SB would have position on each betting round (including the pre-flop betting). Could this ever turn into a -EV proposition for the SB?

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Well, this game isn't symmetrical, right? If the big blind checks after the small blind limps, the small blind doesn't get an action. If he did, I would cheerfully concede that the game must be positive for the SB. Instead, however, the BB has the option of raising or checking and if he checks, there's a flop immediately. This is a pretty big advantage.

Let's talk about the small blind's equity in the jof game at a stack size of eight. We know that his equity in the game is around zero, because we've calculated it. At 8, he jams with like around 65% of hands (i dunno what the exact number is, it's not that important).

We know that:

EV(game) = EV(jamming hands) + (-.5)(% fold) = 0

right? He loses .5 units on the hands when he folds (35%), or .175 units. So the EV of jamming with all those jamming hands must be about .175.

Now suppose that the BB decides to play this strategy after the button limps. He's going to pretend that he's the small blind in a jam-or-fold game, and jam all the 65% of hands that the small blind would jam. But now instead of folding the other 35%, he'll just check and take a flop. So to the .175 he gains from jamming his good hands, he adds some fraction of the pot. As long as his total equity from the asymmetrical game where he has his bottom 35% of hands and the SB has his whole set of hands is at least .5 units (out of the two unit pot), he has positive equity playing this game.

A quick pokerstove of some 35% crosssection that I just picked out of the air shows him having about 40% allin equity, which is a good 15% higher than he needs. Maybe he loses that much postflop, maybe he doesn't. But on the other hand, he can additionally jam with fewer hands preflop, especially the marginal ones, in order to strengthen his postflop distribution, etc.

Anyhow, I think the point is that with stacks of eight units, position on the later streets is worth hardly anything at all. Hence the dude on the button can have negative equity.