strange attractors

One compelling feature of Internet-tournament poker is the dynamic chip-count; with near perfection a player can ascertain his or her tournament-state at any time, and thus, enable more informed decision-making. In so doing the temptation exists to monitor chip-leaders' movements, though seldom serving any purpose. One trend appeared to emerge from this compulsion: the relentless progress of the chip-leaders.

Of course it's not a trait exclusive to Internet-poker, though it might excaerbate the tendency, the Internent does, however, readily facilitate its observation. It seemed possible tournament-poker supported attractors beyond those of skill, which in fact mostly were free from it; specifically, functions of stack-size. If a player’s holding fell below a certain level, say, an equilibrium point or range, then the negative-attractor (located at zero) pressures the stack, climb above it and a high, positive-attractor prevails. Needless to say, the pull exerted by the dominant attractor increases with remoteness from the equilibrium point(s). At least that’s the theory.

As the tournament progresses the stack-size required to maintain pole-position will generally rise, culminating with all chips residing with one player, the winner. There appeared, though, no precept stating those in the lead should witness growth simply by virtue of an above normal holding. Convention suggests only that a good player’s stack will head skywards, and a bad one’s will, well, eventually experience that earthly feeling.

Naturally, some of poker’s losers are great front-runners but should a below average front-runner expect to accrue chips too? Will doubling up a large stack more than double the chances of winning outright? Through acquiring a substantial holding is one at times propelled forward, in some cases caught in a jet stream, almost irrespective of skill? Well, it sure seemed that way.

Using a rather loose and crude definition of an average player, one might, with conventional wisdom, expect such a player with, say, 5 times the chip-average to occupy the same number in an hour’s time - since he is deemed neither good enough to make chips, nor sufficiently poor to lose them. But it was my experience, or perhaps belief, that an average or marginally bad player (the margin depending on the game-state) expects more chips, not the same, one hour hence, if powerfully placed.

As a rule, I am extremely loathed to trust my intuitive, or experience-based, perception of probability where the outcome matters; in Blackjack, for example, whenever the house produced an Ace it was odds-on the dealer would make ‘Blackjack’, especially if I held a decent total or, even worse, refused insurance - an incredible display of poor judgement from someone who'd played out that scenario many, many times.

Although, one is often better off when the big stacks advance, there is inevitably a sense of foreboding as one falls behind in the race. So perhaps, with one eye on the trophy, I feared the runaway chip leader and thus my judgement was distorted, my accounting false. Perhaps, but I wasn’t convinced.

Now, one could get all Bayesian and argue those with big stacks are often the gamblers, aggressive players who are thus (generally) more accomplished front-runners. So the ‘random-walks’ embarked upon by these guys are likely to stagger forwards not backwards; in other words, they expect to accumulate chips. It is a consideration providing, at least in part, a rational explanation for such observations and apparent tournament trends. Nevertheless, it was still my opinion that a holding above some dynamic critical level or range is attracted upwards; and, conversely, one below it would experience an undercurrent moving in the opposite direction, to zero. Not that there aren’t other more significant factors in play which ride roughshod over ‘inherent’ stack-gains or losses - like decisions!

It was an extreme situation in a rare tournament excursion a couple of years ago, which focussed matters, a $500 freezeout at a major on-line site. When the $1k bubble appeared on the distant horizon the usual time-consuming antics ensued. As the first feeding station approached the short stacks set about shedding chips at a disturbing rate. Particular, perhaps, to my table it seemed anything less than 8-9k would see you caught in a downward spiral; if you lay between 10-12k you’d just have to wait to see whether you were fish or foul; above say 14k and, if fate were fair, you were home-free. Of course, the bigger the stack, the more opportunities to actively earn or passively gain, and hence the greater the attraction.

Unfortunately for me, the bar was raised a little higher; the chosen land of the small stacks lay beyond two much larger ones, not a gauntlet I favoured running. So, with opportunities thin on the ground, it was clear I'd be dragged into the mire sooner than were I positioned just two spots to the left. Consequently, after significant attrition and still wishing to maintain some hope of landing a major prize, I dubiously moved all-in on the button with A9o (blinds 1-2k) for around 11k, only to run into a pensive small blind, holding Jacks.

At the time, I’d have bought any large stack with an option to sell half an hour down the line similarly, over the same period I'd have sold almost any short-stack with a buy-back option. In this tournament-state the large stacks were to grow alarmingly, the short stacks to fall, almost irretrievably.

It takes little skill but a big stack to raise late with garbage and expect the small stacks to pass; it requires no skill to post the big blind and drag a pot no-one has bid you to contest. Yes the more skilful, aggressive or possibly crazy you are, the greater the inclination to fold to your blind, pass to your bet or raise. However, the elicitation of these and other favourable responses are assisted by a weighty chip presence, especially at times when size matters most. For some, or in certain conditions, especially with low blinds, the reverse can be true: big stacks represent opportunity as well as danger. Exceptions, though, break rules, not trends.

So it seemed these not-so-strange attractors were certainly in-play, but do extreme conditions augment or create this aspect of the game? Even if these attractors are prone to reverse polarity for some players or under certain conditions[1], I suspect they are mostly on, even if the amplification is set fairly low at times, less noticeable against the influence of luck and, hopefully, skill.

Next article: 26th July: strange attactors: future earnings

[1] In satellites, for example