GTO Open Index vs. Effective Stack

This is the fifth in a series of articles introducing my GTO Holy Grail Algorithms. These articles are based on my newest book, Lectures on Poker, Volume 1: Small Stack Tournament Strategy.

Arthur Koestler quote

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GTO open opportunities

Suppose the action folds to us in the cutoff with 20 BB effective stacks. I wrote Part 3 and Part 4 as if our only options were shoving or folding, which was the way those game theory models were constructed. But we actually have two additional options; we can also limp or open.

Arguably, game theory doesn’t provide for unexploitable limping. If limping is our most profitable play, it’s likely because we have an exploitive opportunity against our specific opponents. Many pundits claim we should rarely limp at all, especially when our stacks are short.

That leaves us with opening (raising when no one has yet entered the pot) as a theoretical alternative to shoving or folding. Yet, we still must decide on our “optimal” bet size. Modern tournament pundits seem to prefer open sizes of 2.0-2.5 BBs, but some still prefer much larger bets. This makes a game theory model more complex for opening than for shoving.

I usually prefer smallish bets, so I will create a baseline open strategy based on a 2.5 BB open size. This is a convenient choice since it doubles the size of the pot when we have a Big Blind Ante structure. It also creates a reasonable baseline from which can make a modest adjustment when we want to make a 2.0 BB or 3.0 BB open.

ICMIZER3 Analysis

We can use ICMIZER3 to get a handle on this problem. Figure 1 depicts the model setup for a 2.5 BB open with 16 BB effective stacks (after posting) and a BBA equivalent. Note that every player has exactly 16 BBs after posting. Then we set the action so that everyone folds to the button.

 

ICMIZER3 for button open ranges

Figure 1. The ICMIZER3 setup for a 2.5 BB button open range.

Clicking the ICMIZE button displays the button’s opening range, which is 41% for this model. Clicking on the “41%” button displays the chart in Figure 2. The shaded cells include the 49.32% of all combos which are +EV to for a 2.5 BB open.

button open range chart

Figure 2. 2.5 BB Opening from the button with 16 BB stacks. The EVs (in BBs) are for opening vs. folding.

However, this isn’t the GTO opening range, which is the “40.87%” value displayed at the bottom-left of this graphic. You can see the GTO range by clicking on the Edit button, which displays the chart shown in Figure 3. The shaded combos are those we should open at least 50% of the time — the GTO range. Notice that some combos are +EV but aren’t in the GTO opening range. The default ICMIZER3 analysis is based on this 40.87% GTO range, so this is what I use for my analysis.

gto opening ranges

Figure 3. 40.87% GTO opening range for a 2.5 BB open.

Creating a GTO open index

We can repeat this exercise for various stack sizes and from every table position. We then convert each GTO range percentage into an open index, just as we did for open shoving. The result of this procedure is plotted in Figure 4, below.

GTO Open Index vs. Effective Stack

Figure 4. GTO Open Index vs. Effective Stack for each open position.

These curves are based on a 2.5 BB open ICMIZER3 model, which allows betting or folding, but not shoving. So, although opening may be +EV in a particular situation, jamming may be a better option. Similarly, not having a jamming hand doesn’t mean we should fold; opening may be a GTO option.

The holy grail open index

We can see that these curves are quasi-parallel and shift with table position. This provides an opportunity for simplification.

First, we can ignore the region of these curves where they bend downward below about 15 BBs. This model allows only opening or folding, but we will normally be shoving in this region rather than opening. Notice that the slopes of these curves are nearly flat between 15 and 30 BBs, with similar slopes that average about -0.12. This negative slope suggests that we can open with a gradually wider range as the effective stacks get larger, which is what we should expect.

We can simplify our model by assuming that these curves are flat above 15 BBs, which creates an error of less than ±1 index unit across this region. This has the advantage of being simpler to use at the table while generating only small errors.

So I propose the simple holy grail open indexes summarized in Figure 5, below:

GTO Open Indexes and Ranges

Figure 5. The Holy Grail 2.5 Open Indexes and Ranges.

Notice that these Indexes are two or three points higher than those suggested by Figure 4. This makes them a bit conservative, so we will be opening with a range slightly tighter than GTO. On the other hand, they conform somewhat closer to what most pundits actually recommend.

The variation between these holy grail ranges and the various ranges suggested by the pundits can be explained by the fact that they may be opening to something different than 2.5 BBs. In many cases, they don’t even bother to specify the open bet size that applies to their range suggestion.

If we compare these open indexes with the holy grail position indexes introduced in Part 3, we create our holy grail open algorithm:

2.5 BB Open-Bet Index = (Position Index) + 8 points

We calculate our open index by simply adding eight points to the same position index we use for open-shoving, shown in Figure 6. (We can use six points if we wish to be closer to the GTO model and play more aggressively.)

position indexes

Figure 6. The Holy Grail Position Indexes.

If the combo power index of our hand is larger than our open index, our hand is within the GTO opening range. We need only memorize the position indexes and the eight-point rule.

Open or shove?

Figure 6 shows both the shoving and open indexes in one simple graphic. We can see that whenever our effective stack size is 16 BBs, the open-shove index and open index are identical. So, for a given starting hand, we might prefer shoving when the stacks are below 16 BBs, or we might prefer a 2.5 BB open when they are larger. That doesn’t mean we should always do this, but it becomes a reasonable baseline we can easily memorize.

(Note: this transition point would be 12 BBs if we use a six-point rule instead of our eight-point rule, which we are free to do.)

Adjustments

This algorithm is not strictly a GTO strategy since we are opening a bit more conservatively than the model recommends. If you prefer, we could call it a quasi-GTO strategy, just like my other holy grail algorithms. I’m not a stickler for the math; I want a system I can easily use at the table. This system is still far superior to no system at all, or one which we’re likely to imperfectly memorize after a thousand hours of study.

Suppose action is folded to us in the cutoff with 16 BB effective stacks. We have JTo, which is 37 points (22+10+5). Our open index is 38 points (30+8), which makes our hand a marginal fold. However, we know that our open range is conservative, and we expect that the players on our left won’t call with a GTO range. So we might decide to open anyway.

We also know that our model applies to a 2.5 BB open. If we were to repeat our ICMIZER3 analysis for a 2.2 BB open, we would find our borderline 2.5 BB fold has become a borderline 2.2 BB open. And if we detect a folding tell from one of the players on our left, opening becomes even more attractive.

This concept also applies when we have a strong hand. Suppose we have TT, which is 66 points (20+10+36). Since our hand isn’t close to borderline, we could choose to open with a larger 3.0 BB bet.

We should also base our decision on the skill we perceive in the opponents on our left. A highly skilled player is more likely to react to us with a GTO range, so we may prefer to play very close to our GTO optimum so we can’t be exploited. But if those on our left are weak and passive, we can play additional marginal combos.

Finally, we should take the bubble situation into account. If the effective stacks are 16 BBs and we have 40 BBs, for example, players on our left will be reluctant to risk their cash without a premium hand. So we can exploit them by opening even wider than our open index would justify.

Summary

Our GTO holy grail open and shoving strategies can be reliably implemented when we have an open opportunity since we don’t need to worry about the style of the players who have already folded. This allows us to use a simple and easily memorized fold/open/shove strategy. Next time, I’ll introduce the holy grail algorithm for re-shoving, which is shoving over an open-raise.


Steve Selbrede is the author of seven poker books: The Statistics of Poker, Beat the Donks, Donkey Poker Volume 1: Preflop, Donkey Poker Volume 2: Postflop, Donkey Poker Volume 3: Hand Reading, Tournament Poker for the Rest of Us, and Lectures on Poker, Vol. 1: Small Stack Tournament Strategy.

Written by

Steve Selbrede

Retired Silicon Valley engineer, vigorous poker researcher and author of seven poker books, including his latest, Lectures on Poker, Volume 1: Small Stack GTO Tournament Strategy (The Holy Grail Algorithms).

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