The final table for the WSOP Main Event has now been set, and we’ve got three days off until it resumes. I’d like to take that time to take a look back at some of the more interesting moments from the first seven days of action. For the first of these, we have to go way back in the tournament, so far back in fact that some players hadn’t yet taken their seats.

Vanessa Selbst had an extremely abbreviated Main Event this year, going broke in the first level of flight 1B when her Aces full proved no good against Gaelle Baumann’s quad Sevens. Massive coolers like this naturally get the poker fans chattering, especially when they happen in a tournament as important as the Main Event, and between players as well-known as Selbst and Baumann. What interests me, however, is not the improbability of a collision between such monster hands, but the way the conversation surrounding the hand highlights one of the weird paradoxes of tournament poker.

What I’m talking about is the idea that sometimes, the best play from a theoretical standpoint is not the best play from a practical one. To some extent, this is quantifiable: For instance, ICM (Independent Chip Model) is a well-understood and, in most cases, close-enough approximation for how the relationship between chip stack and prize pool equity becomes non-linear as the tournament approaches a pay jump. ICM is just the surface, however, and the rabbit hole goes very deep indeed.

Beyond ICM

Even with ICM, there’s disagreement about how universally applicable it is, and whether there are higher-order effects that need to be taken into account. For instance, ICM tells us that with any payout structure other than winner-take-all, doubling one’s stack produces less than a doubling of equity; that is, each chip won is worth less than the one before. At the same time, though, ICM tells us that the deeper-stacked player can take advantage of a shallower-stacked one when a bubble or important pay jump approaches. This is because the chips used by the deeper-stacked player to bluff with are worth less to him than those the shallower-stacked player would have to use to call with.

The paradox there should be obvious: If having the deepest stack at the table produces a strategic advantage resulting in more opportunities to win even more chips, how do we reconcile this with the idea that larger stacks suffer from diminishing returns in terms of tournament equity? Most good players will agree in hand-waving terms that during low-ICM periods of a tournament, it’s important to be aggressive in order to build a stack; chips are needed so that one can exploit short stacks and not be exploited by deep stacks during the higher-ICM periods to come. But ask someone how to quantify that advantage and calculate exactly how much it’s reasonable to gamble, and you’re likely to get an extremely vague response.

The Selbst-Baumann hand

Before I get any more deeply into the paradox, let’s take a look at the crucial details of the hand itself. Selbst, holding two Aces – Spades and Diamonds – opened in middle position for a standard raise to 400. Baumann, with two red Sevens, called on the Button, and Noah Schwartz came along in the big blind.

The flop came out Ace-Seven-Five, all Clubs, giving both women a set. The only thing preventing the hand from being a guaranteed cooler for someone at this juncture is the monotone flop. Selbst bet 700, just a hair over half pot, Schwartz folded, and Baumann flatted.

The turn was the Seven of Spades; seemingly a dream card for Selbst, but soon to be revealed to be a nightmare. No longer needing to worry about losing to a flush, Selbst checked to trap, and Baumann elected not to slowplay her quads. She bet 1700 into 2750 and Selbst, sensing strength, followed through with her plan to check-raise to 5,800. Baumann wisely elected to just call and leave Selbst enough rope to hang herself.

The river was a blank and Selbst went for maximum value, betting 16,200 into 14,275. Baumann moved all in, and Selbst realized that she had very likely walked straight into the nuts. With the bet sizings she’d chosen, however, she was now only being asked to call 20,300 more, with 46,675 already in the pot. In order to fold, she’d need to assume that Baumann held two Sevens at least – appropriately enough – 77% of the time.

Numerological coincidences notwithstanding, it’s never easy to say an opponent will turn over one specific combination of cards that often. Yet, Selbst said she thought there were just two: the two Sevens, or Ace-Seven of Hearts. She felt that she could have folded if she’d been holding both red Aces; most people commenting on the hand agree with this, while some argue that Baumann should or would not be shoving with Ace-Seven. Baumann herself said at the time that Selbst was correct to assume that Ace-Seven suited was in her shoving range, but later reconsidered and said she might have just called with even a hand that strong.

A fundamental contradiction

There’s a fundamental contradiction here, stemming from the idea that both women are, or should be, playing a pure strategy on the river; that is, always playing a given hand a certain way, rather than occasionally playing unpredictably. Specifically, it’s with the idea, stated by Selbst and others, that Baumann has “zero bluffs” in this spot.

If we assume that Baumann has no bluffs and that Selbst is calling with pocket Aces and folding anything worse, then obviously Ace-Seven should just be a call for Baumann, as she’d be getting called only by better and not profiting more from anything worse. But if Baumann is only going all-in with the nuts, then Selbst, holding the second nuts, should clearly fold.

But if Selbst is folding pocket Aces, and those are a big part of her range (as is also being assumed), then Ace-Seven, and indeed any hand containing a Seven, should be a shove at least some of the time, albeit not for value. Rather, if Baumann’s value range consists only of quads, then any time she’s holding a Seven, she should consider turning her hand into a bluff, as it blocks Selbst from having the hand she’d be representing.

There’s an interesting generalization to make here, which is that the only way – from a game theory standpoint – for a range to contain zero bluffs if for it to contain only the nuts; put another way, any range containing a non-nut hand contains at least one bluff.

Game theory assumes that the opponent knows the player’s strategy, but not their holdings. Regardless of whether a player’s range is polarized or merged, the opponent should be folding hands which don’t beat any part of the range; ergo, the bottommost hands in a range must be considered bluffs, as they should never be called by worse, but might get better hands to fold. This is true even if they’re not that different in absolute value from those at the top of the range. So, here, if a range consists only of the nuts and the third nuts with a blocker to the nuts, then the latter is a bluff, targeting specifically the second nuts.

But if Baumann is sometimes bluffing, then our fundamental assumptions about the situation are wrong. This is an example of the “principle of explosion” in logic, which states that if you begin with mutually contradictory assumptions, you can derive any conclusion you like.

The difference between practice and theory

From a practical standpoint, you can resolve the contradiction by postulating that both players are making specific mistakes, they both know that they’re making mistakes, and they both know that their opponent is making mistakes.

That is, Baumann knows that she should theoretically bluff sometimes with a Seven blocker, but won’t do so because she knows that Selbst will call all the time with pocket Aces, even if she shouldn’t. Selbst, meanwhile, knows that she should sometimes fold her Aces, but feels she has to “keep her opponents honest,” or more likely, can’t tolerate the angst that would ensue if she were to find out she folded the second nuts getting over 3-1 and was somehow wrong.

This is a fairly realistic assessment of what probably went on, but it’s philosophically dissatisfying, because it means that top-level players are knowingly making mistakes for emotional or, at best, long-game image reasons, and doing so predictably enough that their opponents can also play “incorrectly” in important spots, counting on them to make these mistakes.

The live tournament metagame

With a tournament like this, however, there’s an added layer of complexity to the paradox, because of the opportunity the Main Event represents and the rarity of that opportunity. It’s by far the softest $10,000 buy-in tournament in the world, it runs only once per year, and re-entries are not permitted.

Many tournament professionals estimate that their long term expected ROI in the Main Event would be 200% or more. Put another way, that means that they would need to be offered at least $20,000 simply not to play. Put yet another way, that means that these players should be assigning a value of $30,000, not $10,000 to their starting stack at the beginning of the first hand. This tripling of value isn’t continued throughout the tournament, however, as the field gets tougher as the player runs deeper; a typical professional at a typical main event final table might have close to zero edge.

That creates a separate ICM-like effect, as busting out surrenders the player’s entire edge, while chips accumulated beyond what are likely be needed in the early going are worth less, as they’re only likely to come into play later on, when that edge is smaller.

So the conventional wisdom goes, at any rate, but there’s something a little self-contradictory about this way of thinking. As we discussed at the beginning of this article, being in a high-ICM spot is a disadvantage, in that it forces one to take less profitable lines when more profitable but higher-variance ones are possible. Here, it’s similar, except that it’s not the payout structure but rather the assumption that the player has an edge that handcuffs her. Consider, however, that a lot of the edge held by professional players over the field comes from being willing to tolerate risk and be aggressive in ways that amateurs are not. If that assumption forces the player into a tighter and more passive style, the actual edge may not be as large as she guesses.

Cutting Gord’s Knot

Of course, I don’t have the answer to this riddle any more than I know how to calculate the exact value of accumulating chips early in a tournament in order to abuse the bubble later. It’s very difficult to predict what the chip stack and skill level distributions will be like at one’s table later in the tournament, and a probabilistic approach to quantifying these effects would be “non-trivial,” to use a classic mathematical understatement. I’m also definitely a worse poker player than either Selbst or Baumann, and probably than many of the people discussing the hand.

That said, I’m very good at games in general, and I think there’s a lot to be learned from studying those elements of strategy which don’t depend on the rules of any specific game. Here’s the one I think applies here: If you’re analyzing a decision and all options seem bad, then you’re analyzing the wrong decision and the real mistake came earlier.

Whatever the game, if it’s complex and strategic enough that people spend time doing post hoc analysis, you’ll find people wasting huge amounts of time trying to find the winning move in a lost position. If you find yourself doing this, it’s a good idea to go back one node on the decision tree and see if there’s a better line stemming from there. It’s a natural reflex to want to find the last point at which a game could have been saved, but in practice it’s often better to find the last point at which it could have been made easy.

I’m going to go out on a limb and assert that folding the second nuts when getting better than 3-1 is never good, at least not in No-Limit Hold’em. Perhaps we can accept that it’s the least-bad option for Selbst at the final decision point as played, but it still can’t be good. That being the case, if we want answers, we’re looking at the wrong decision.

There are two likely places that Selbst might have made her mistake, in my eyes: Either she should have simply led the turn instead of check-raising, or at least one of her bet sizings on the turn and river was wrong. Played differently, she could have perhaps made the last bet herself, or given Baumann a spot in which she’d be willing to bluff sometimes, or even left herself the correct odds to make a hero fold.

Which of those is best is beyond me, and beyond the scope of this article in any case. My takeaway from this hand, however, is that if players are going to dramatically change their play in big pots on the basis of it being early in the Main Event, then those changes have to factored into decision-making on earlier streets as well. Starting to agonize over these “meta-ICM” paradoxes only after the pot becomes huge is a recipe for finding oneself in the sort of nightmare spot experienced this year by Selbst.

Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Dartmouth, Nova Scotia, Canada.