The recent $102,000 WCOOP on Pokerstars featured many of the game's biggest names, such as Fedor Holz, Ike Haxton, and Jason Mercier. The winner was Bencb789, a much less well-known German pro who has been quietly crushing the highest-stakes SNGs. I recently had the chance to interview him and asked how he studied. In addition to the expected answers of working with GTO software like PioSolver, and talking hands over with other high-stakes pros, he also said he studied by tackling hands with pencil and paper.
Such an old-school approach takes much longer than software, and the types of hands that can be solved this way are comparatively limited. On the plus side, by devoting this extra time and seeing the mechanics underpinning the hand, you can take away deeper lessons each time.
Let's begin with a simple example.
Format: $27 Hyper-turbo MTT
Effective Stack: $9,700
Pre-flop: nine-handed, everyone folds to Hero in the small blind with 63s. The big blind is a strong reg that Hero has shoved into twice recently. Given choices of shoving or folding, is shoving profitable?
Analysis: In-game, Hero folded. When asked about the play, he said that he knew software advised shoving this hand, but that he felt uncomfortable shoving for the third time in a row with 6-high against a player he believed was looking to call him wide.
This player was right that it can be correct to deviate from software-generated equilibrium ranges. For example, a program like Holdem Resources might tell you that it's profitable to call a shove with a certain hand based on the Nash ranges, but your passive opponent is likely to be on a tight enough range that folding is correct. In situations like that, instead of just modifying opponent ranges and getting a new answer, it can be very helpful to look directly at the numbers.
The equation for the expected value of shoving is:
EV = P (Opponent Folds) x Gain/Loss (Fold) + P (Opponent Calls) x Gain/Loss (Call)
In the hyper-turbo 63s spot, suppose we think our opponent will call us very wide, say approximately the following range:
From my experience, it is relatively rare that even a tough reg will call us much wider than the range above.
If Villain is in fact calling us with the top 60% of hands, that simply means that he calls 60% often and folds the remaining 40% -- giving us the probability of him calling and folding in the expected value equation. We now need only the gain or loss associated with each outcome.
If we shove and our opponent folds, we gain $3,300 relative to folding.
How much do we gain or lose if he calls us? The way we can think about this question is as follows. If we were guaranteed for him to call, we would be paying our remaining $9,000 (after posting the ante and small blind) in exchange for the equity we had with 63s in the resulting all-in pot of $20,800. Equilab tells us that we have about 36.4% equity against this top 60% range. So this means that when he calls, we hand over our remaining $9,000 to the pot and get back $7,571 (36.4% of $20,800), for a loss of $1,429.
EV = 40% x $3300 + 60% x (-$1429) = $1320 - $857 = $463.
Rephrasing this answer:
Even though our opponent calls us the majority of the time, and we have low equity against his calling range, it is such a valuable outcome for us those times he folds that we still win almost half of a big blind in EV by shoving here.
Note that in this hand, we consider it “legal” to use Equilab to calculate our equity versus his range. If you do your equity-versus-range calculations by hand, then you are seriously hardcore.
In our next example, we'll similarly use software for the step of converting a stack distribution to dollar equities via ICM.
Format: $7 18-man SNG
Button (Villain): $10,000
Small Blind: $4,000
Big Blind (Hero): $7,000
Action: Villain, an aggressive reg, open-shoves. The small blind folds. Hero has AKs.
Analysis: Most 18-man regs will recognize this spot as a fold during post-game analysis, yet for some reason, many still call here. Even though they know if they put the hand in software it will tell them to fold, it can feel wrong to fold a premium hand against a player probably shoving any two cards. Conversely, it feels nice to put a wrench in the plans of a fellow reg by spite-calling his shove and costing him a lot of equity. So, is it worth it?
We can get better insight into this call by doing the math.
Suppose we fold. In that case, our equity will be 23.68% of the prize pool, or $27.49:
(Image credit to ICMpoker.com)
If we call and win, the same ICM calculator shows our equity increase to 32.93%, or $38.23.
If we call and lose, our equity falls to $0.
Assuming our opponent is shoving every hand, like he should be against most players in the blinds, we will win this pot 67% of the time with our AKs versus his range of all hands.
We therefore have two choices.
We can call, in which case 33% of the time our equity will fall to $0, and 67% of the time it will increase to $38.23. On average, our equity when we call will be $25.62 (67% of $38.23).
Or we can fold, in which case our equity will be $27.49 as we saw above.
Putting all this together, calling loses us $1.87, which is almost 27% of our $7 buy-in.
It is worth pointing out that we might consider taking a negative edge in this spot, for the reason that if we win we will be in a decent position to exploit the bubble. But an instant drop of almost $2 in equity in a $7 game is too difficult to overcome. We have to fold here.
It's one thing for software to calmly announce that it's a mistake to call with any unpaired hand in this 18-man spot. When we do the calculation by hand, however, we very clearly see the numbers behind our equity loss and are forced to focus on the fact that calling here turns $27.49 in equity into $25.62. The next time a similar situation comes up, you are more likely to recognize that doubling up represents only a modest equity gain, and so getting it in as a 2:1 favorite isn't worth it.
Both of the hands in this article pertained to preflop all-in poker. It is possible to do similar calculations for more complex hands, but ones involving future betting – such as the profitability of semi-bluff check-raising the flop with deep stacks – require many assumptions. Even so, start working on them by hand and see where they take you.Getting a better feeling for the numbers underpinning the decision-making process can be a valuable edge you have against the many online regs today who rely purely on software.