The following equilibrium solutions show an unexploitable strategy in heads-up play with 20 BB and 15 BB effective stacks. Since we generally recommend a push/fold strategy only with less than 10 BB or 12 BB, these additional solutions allow you to take advantage of further actions, such as limping or making standard raises.
In all of these solutions, the button was allowed to initially fold, limp, raise to 2 BB, 2.5 BB, 3 BB, or push all-in. After the initial action, both players are allowed to make 0.5x-pot, 1.0x-pot, or 1.5x-pot raises, in addition to being able to jam at any time.
These tables use the same notation as in Chapter 13: For example, A9o says C90J, which means call 90% of the time and jam 10%. Later on, we’ll see more complicated entries, like F40 B30 p30, which means fold 40%, make a big (1.5x-pot) raise, and be willing to get it all-in 30%, and finally make a pot-sized raise, but fold to a re-raise the final 30%.
The equilibrium solution doesn’t open-raise as often as the authors usually do. We open-raise on the button at least 40% of the time, and maybe up to 100% if our opponent is tight. The equilibrium prefers to limp re-raise with several of its strong hands in an effort to better protect its weak limps. Although the solution was allowed to open raise to 2.5 BB or 3 BB, it always elects to min-raise, except for the rare cases where it open jams. Why does the equilibrium solution limp so often? I think there are a couple reasons:
• If you limp with some weak hands, you have to limp with strong hands too, which is part of balancing and protecting your limps. So your decision is, do you limp a lot or raise a lot with both?
• If you raise, how big? We’ll see later that the BB never folds to a min-raise. Raising to 2.5 BB is a bit more effective, but risks more if you’re raising with trash. The computer abandons the larger raise and goes with 2 BB consistently. But that means no fold equity, which means playing a bigger pot with all the weak hands if you raise a lot. The computer would rather not raise as much, but it limp-raises a bunch to make up for it.
• When your stack is 15-20 BB, raising on the button cuts a lot of your positional advantage. He can push over your raise and eliminate your position. Or if he flat-calls, stacks aren’t that deep, so your position counts less. If you limp with a good hand and he checks behind, you still have 3 streets to play in position with a deeper stack. Plus, limping makes it overkill for him to push. If you solved the equilibrium strategy for much deeper stacks (say 50 BB), I think you would see a lot more button raising, since it builds a pot in position and doesn’t cut down the deep stacks as much. But those deeper-stack solutions are very difficult to solve4.
Here’s the situation where we just limped and the BB makes a min-raise. We never fold, since we’re in position and getting 3-to-1 pot odds. Notice that there are limp re-raise bluffs, such as T2o.
If we open with a raise and he 3-bets, there are very few bluff 4-bets. That’s because stacks are fairly short and we don’t have too much fold equity over a 3-bet. If we make it 2 BB and he makes it 4 BB, there are a few bluffs, mostly with weak suited aces and kings.
Now we’re looking at the strategy from the BB after the button just limped. There are a few complicated entries, because the follow-ups don’t always fit nicely into a compact table like this. The most complicated example is probably T6s. With T6s, the solution makes it 2 BB 80% of the time, checks 10% of the time, and makes it 3 BB 10%. After both of those raises, it’s willing to call a half-pot 3-bet, but nothing bigger. A lower- case “p” without an asterisk means that it will fold to any sized 3-bet.
Most weak hands are simply checked behind. The bluffs tend to be hands with one high card that could potentially hit a top pair and take the lead. Suited connectors are less valuable due to the short stacks (smaller implied odds), as well as the fact that he’s less likely to flop a big hand. If
The big blind never folds when the button min-raises. Aces always 3-bet and there are also a few bluffs thrown in for balance (good suited and offsuit connectors, plus some garbage hands like T2o).
Notice that the solution for calling an open jam for 20 BB here isn’t the same as you might find in a Nash equilibrium solution on the web or in another book. That’s because that other solution is calling a push when the button was forced to only push or fold. In this case, our button has many other options besides push/fold, so his pushing distribution is different. Our button pushes occasionally with 22 and A9o, plus a few other hands that push less than 5%, so they don’t show up on the table. So our push-calling strategy is heavily weighted to counter those hands, which is why you see odd strategies, like preferring to call with QTo over QJo (QT actually does better against those hands!). In practice, calling or folding with both of those hands is equally fine and you shouldn’t worry about little nuances like this.
Now we’ll repeat all the same tables, but with 15 BB effective stacks.
Open pushing is more common at this shorter effective stack; in fact, we rarely make the smaller raise. Limping is still prevalent for the reasons outlined earlier and pocket aces now join the list of limp-trappers.
The 15 BB solution after the button limps is very similar to the 20 BB solution. There is still a variety of bet sizes—the solution freely uses all of the bet-size options it was given (2 BB, 3 BB, 4 BB, and jamming to 15 BB).
Here’s a difference from the 20 BB solution: The BB will occasionally fold to a min-raise at 15 BB stacks. A shorter stack is an overall advantage to the BB, since on average he’ll be out of position for fewer betting decisions. But this advantage mostly helps his strong hands. Weak hands are actually worse with shorter stacks; you have worse implied odds and you can’t bluff as much post-flop (less fold equity). So while over all hands the BB is happier with a short stack, his weak hands are less desirable, so they fold despite getting the 3-to-1 pot odds.
Again, this solution of calling an open jam isn’t the same as an equilibrium solution when the button is forced to jam or fold.
Although these solutions don’t take advantage of weaknesses in your opponent’s play, they’re very strong and sound strategies to use without deviation. These strategies have been field-tested in actual play and have proved to be extremely profitable. You can’t really go wrong to play “by the book” in these cases.