strange attractors: future earnings

The first order of business is to issue an apology to all folk of mathematical ilk for sensationalising these attractors by applying the ‘strange’ adjective. Originally labelled ‘attractors’, a fellow on-line poker-pundit, who always has an eye for a catchy title, inadvertently coerced me into the name-change.

Though not directly defined, the skill-attractor is vaunted across the forums, often argued to pass up small but risky edges since the skill-factor (the skill-attractors' pull) is viewed as sufficient compensation to not run the risk of ruin: they’ll get the chips later. The stack-attractor, or hitherto strange attractor, though, is not as commonly identified. Playing the big stack adeptly clearly requires know-how and as such the skill-attractor must in part be a function of stack-size, however, it is contended certain aspects of growth attributable to a player’s holding require little more skill than the basics, for all intended purposes, independent of it. Chief stack-attractor contributors are: the stack-size corresponding to the table/field; the table-position (relative to other stacks); the tournament state (blinds, antes, proximity to payouts, payout differentials).

Poker blogger, Big Dave D, observes astutely: ‘perhaps it’s not so much that some players play the big stack well, but that the big stack plays them well’.

So, refreshing the first article, it follows, somewhat inevitably, if skill-independent growth occurs, so must recession of the same nature: attractors exist to induce decline as well as growth. It follows these forces will resolve to form an equilibrium point, or, for practical purposes, a range of stack sizes, a 'neutral zone', where the attractors roughly balance out.

In the movie-classic, ‘As Good as it Gets’, Jack Nicholson’s surprising love-life quandary receives no betterment from his flat-mate; a disgruntled Nicholson complains: ‘I’m drowning here, and you’re describing the water’. ‘Describing the water’ is a rather apt reflection of the progress thus far; unfortunately, it is beyond my experience to right any wrongs of tournament decision-making, though perhaps inferences can be drawn.

It’s hand T and you’re in the neutral zone with 14k and a coin-flip to double up or bust-out: assuming attractors are in play, should you go for it? Rejecting the opportunity leaves an unscathed stack of 14k, for the next hand, T+1. A neutral holding implies neither attractor effectively, well, attracts; so you ‘expect’ to maintain those chips in 50 hand’s time (@T+50) [1]. Therefore passing @T yields the same chip-EV for both concerns, hands T+1 and T+50, respectively.

The alternative, of course, is to gamble - it’s 50:50 to double-up or bust out. So, at hand T+1 your expected holding is 14k (50% of 0 + 50% 28k) - the same as passing. What, though, will it be at hand T+50? Well, it should be more.

Winning will steer the stack out of the neutral zone, leaving it in the welcome clutches of the positive attractor. The benefit (or implied value) of which will naturally not be realised immediately, but very likely after, say, 50 hands (@ T+50) and beyond. So by attempting to include the expected stack-growth attributable to attractors (skill or stack-based), one should, arguably, be capable of a more informed judgement on the merits of either folding or calling @T.

Suppose with a stack of 28k - the double up - under the influence of a strong positive attractor, one optimistically ‘expects’ to turn 28k into 40k by hand T+50. Now @ T+50, the expected value in chips (chip-EV) of a call at T, the coin-flip, is 20k (0.5 * 40 + 0.5 * 0), as opposed to 14k @ T+50 with a pass @ T. So, in other words, calling the coin-flip at T generates an expected stack of 20k in 50 hands time, where a pass projects only 14k. If, though, the values of the choices at T are estimated by considering the holdings after only one hand (@ T+1), then passing and calling appear identical with respect to chip-EV.


Disclaimer: Don’t be seduced by the numbers! They shouldn’t be taken as evidence to support the claim; instead they are an illustration or translation of what might happen, in a special case, if the attractors are bona fide.

The example asserts decisions should be undertaken with consideration of the expected increase/decline subsequent in all outcomes; naturally, this growth can't be ascertained looking forward just one hand. And, predicated by the existence of these attractors, it shows that choices sharing the same hand-EV attribute aren't necessarily, or indeed likely, to be matched with a similar growth-EV attribute.

Logic dictates situations will arise where decisions with lower hand-EV, but higher growth-EV, than alternate options will be preferable w.r.t maximising tournament chips.

Note: Decisions to maximise chips often conflict with those for maximal tournament-reward. Hence, above, the information is insufficient to conclude gambling to be the right decision.

Experienced players of multi-table-tournaments (MTT’s) and, especially, those of single-stable-tournaments (STT’s) [2] will be very aware of this tournament nuance - it has been discussed at great length; however, nothing written here disputes that wisdom. The decision eliciting the highest return in chips should yield to the one offering the greater fiscal reward, and it in turn should be forsaken for utility. However, by factoring in the growth, or decline, effected by these attractors we expect a more informed decision on all of these measurements. It is, of course, by comparison, facile to illustrate the impact of attractors on the decision-maker’s stack than upon the complex and subjective metric that is utility.

Perhaps the value of this insight lies in heightening our awareness of opportunities resulting from substantial stack-increass, as well as, of course, to the short-stacked perils particular to the current game-dynamics. The latter consideration might instigate the rejection of a normally rewarding gamble if losing suffers added penalties winning can’t cover, or indeed, induce seemingly premature shots at doubling up or blind-stealing when conditions shift out of favour.

If attractors do impact materially then ignoring the growth-attribute of options available at crucial decision-points woud be foolhardy.

Next article: strange attractors: money goes to money

[1] This assumption is possibly a little shaky, but we're only painting pictures.

[2] STT players are very likely tuned into the impact of stack-based attractors, since regular participants are often in the thick of it, feeling the pressure, when the field is down to 4 or 5 runners.