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.

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

steps: an about turn

After my last post it seems an appropriate time to slip in an article on the purported trappings of party’s steps, the essence of which was written and posted some time ago.

Single-table-tournaments (STT’s) typically consist of 10-players with a payout structure of 5:3:2 of the buy-in for the first three places. Cashing out in the steps though is more challenging. The goal is to win a place in the $1000+$65 STT [I] where prizes of $4500, $2500, $1800, $1200 are awarded to first through to fourth respectively. Entry levels ranging from a $10+1 STT right up to a direct buy-in ensure the dream is alive for all punters; each level guarantees two or three take the elevator to the next floor, some lucky-losers are afforded another crack at the same or lower entry-level.

Naturally, as is standard with STTs, rake is charged with a direct buy-in at each level. Finishing 3rd, for example, in a $50 + 5 entry-level (Step 2) merits a fresh shot at the same stage – an effective payout of $55. Unfortunately the prize fund is boosted by only $50 since an additional $5 is raked; so the site appears to ‘tax’ players on buy-in and payout. Uproar ensued, the value police alerted: this was a con to rake to death anyone misfortunate enough caught cycling the steps. Not being a prolific STT player, I only became aware of the steps product courtesy of a link posted to this excellent mathematical analysis.

The worst example was the $10 entry-level; the analysis claimed the effective levy for this step stood at just over 50%. Many marvelled how these players with no sense of value would need to wrestle a ridiculous rake to profit. Bowled over by the analysis, I rounded on Party too - another rip-off scheme, after all one needs to beat a 50% rake.

Now there was irony here, as those very people who boast how easily others are fooled and have no understanding of value, were themselves completely sucked in, as was I. In fact the problem wasn’t thought about deeply enough before determining it a mug’s game to buy-in at a low level. There were still angles from Party, but the principle issue should have been greeted with a ‘So what?’ rather than: “Shock, Horror!”. The rake is no big deal: it is the deal [ii]. A player decides to buy-in to a $200+15 STT with payouts of $1000; $600 and $200 respectively. Conveniently, STT’s with the following buy-in’s are also available: $370+30; $560+40; 950+50. The said player is committed to reinvest any returns direct into one of the higher buy-in STT’s. Assuming he has an average chance of locking in a prize in all the events the expected rake paid from the $215 investment is calculated as follows: $

15 + 0.1 x $30 + 0.1 x $40 + 0.1 x $50 = 27

Adopting the analysis linked to earlier: $188 equity; $27 rake: 14.4%

Sure it looks bad, however, this is what players are doing day in day out in all forms of poker; unless driven to cash out permanently, winnings will always be raked.

Example: chaospoker.com offer a deal: deposit $200 and play only $20 STT’s at $1 rake. However, you are precluded from withdrawing any funds until you amass $1000. This bears some resemblance to the step structure: a player buys in for a fixed amount, potentially plays an indefinite number of games but can’t cash until reaching a certain goal. You’re an above average player and expect to profit $1 from each STT. Now you’d anticipate enduring 800 tournies at a whacking cost of $800 in rake to attain cashout status. What would the advice be here: ‘400% rake on investment! Stick to the 10 % rake STT’s!’? Of course it is nonsense and perfectly transparent that winners will gain, some losers become winners, and others lose less or assume increased longevity through signing up. Although, no contract is ever drawn up people commit themselves to just this arrangement all the time.

If you still need convincing imagine Party choose not to pay out in full to anyone placing in an STT. Instead they pay cash plus a credit into another STT. While it infringes on liberties, it has no impact on value for the regular player: you were going to play again anyway. With some hocus pocus, though, the accusations could fly:

How much rake would an average player pay in a $50+5 event?

The expected rake is 5 + 0.3*5 + 0.3^2 * 5 + 0.3 ^ 3 * 5 +...... + 0.3^n*5 (I)

= $5 * 1/1-0.3 = $7.14 (summing an infinite series)

Rake on equity is 14.9%: it appears through the analysis to increase by nearly 50%, yet the game is evidently no harder to beat. This analysis doesn’t illustrate how demanding it is beat the ‘party steps’, it emphasises how tough it is to beat a raked game period, for the majority of people. It’s quite simply the law of diminishing returns applied to average on-line poker Joe; the step structure demonstrates perfectly what players are doing in all forms of the game day in day out.

If you beat the game at every investment point, then you beat the game (although you can beat the structure without beating every investment point): at the lowest step it’s 10%. The devious and sly aspect to the steps format is the flatness of the payout structure, this makes it somewhat harder to beat – sometimes there are 9 prizes! Although, the structure does appear to make life tougher, it is certain as mentioned earlier, and although there is less variance within the payout of each step compared to an STT, ultimately it is quite high – someone’s going to win a sizeable lump of cash. Unless it is delivered to big cash player it is likely to be cashed out or inactive – either way it’s not earning rake: not good for the cardroom.

So while the analysis is sound the conclusions drawn were false. It would be easy to construct a step structure offering better value than the current STT’s (e.g. 5% rake) but still appear horrendous under the scrutiny of the type of analysis applied previously. The value police weren’t comparing like for like. Similar rake is paid per STT whether it be standalone or step, the difference is you play more STT’s for the initial outlay, per unit investment, and so also additional rake.

A further criticism comes from the belief that most who enter are nothing more shark food, as a consequence of the opportunity afforded for pros to buy-in and wait for the battle-wearied, less-able nervous players stagger into the business end of the structure. It is a claim hard to refute and certainly must be priced into the decision.

If party’s steps could be considered a fair game, then prima’s ‘rounders’ must be viewed as a fantastic deal*. The structure is similar to steps, except the rake is redistributed in the payout except in the top tier. This translates to a significant reduction in rake per unit investment. Additionally it is much less of a shark-trap.

Instead of being rewarded with entry to another or higher level, payment is made effectively in the form of rounder’s dollars, which can be used to contribute to a buy-in at any level. Although to cash out a player must still pull up a chair in the big game, in which he may disadvantaged, over a period of time he can choose to play much less of them than in the restrictive party structure. For instance, consider a marginal winning player at $50 – any higher and he’s a loser; over the course of a year he plays many events and wins $1000 just playing $50 stt’s. He finally sits down and plays the big one with his rounder’s dollars. Over the course of the year only $1000 is poorly invested, -EV: what amount though, had he been forced to move up after each successful sortie? And what’s more the only rake contribution occurred in the main event.


* At least it was when a friend drew my attention to the benefits of Prima’s steps equivalent.

Next Article: Strange Attractors, July 14th

structually sound?

In look after the pennies I mention an explicit occurrence of the tendency to reason monotonically cropping up in poker. During a discussion on the merits of the then fashionable party steps, one of the pillars of the prosecution’s case was the suggestion of a negative impact on value ingrained in flatter payout structures. The rebuttal to any sceptics was simple: if you keep flattening the payout structure the game becomes unbeatable.

So we are supposed to infer from this statement that increasing the flatness of a tournament produces a less yielding game. Although the structure doubtlessly served party’s interests, the argument forwarded by the dissenters was not prevailing, robust. The extreme case of a perfectly flat payout structure, one where all players are remunerated their expenditure, less the juice, appears quite reasonably to leverage the claim that increasing flatness lessens value. Clearly, inching towards an even payout will ultimately lead to unbeatable game; however, it would be false to conclude this implies any increase in flatness decreases value.

If, though, it could be reasoned as such, we’d rapidly conclude single payout (SP) events would be the design of skilful players (in theory) - though I’d expect to see few clambering to play them. The inherent sharp rise in variance induces money-management problems: maintaining an equivalent risk-of-ruin requires an increase in bankroll or a buy-in level reduction - the latter may negate any alleged gain. Also, crucially, weaker players would steer clear of such unforgiving events.

In a single-table-tournament (STT) with zero-flatness (an SP) there is naturally some degree of value for the above average player and also, trivially, in a flat event there is none; in fact, with an entry fee it’s negative. Does, though, value decrease monotonically with flatness? In other words, can value ever increase over some interval or will it decline for any incremental increase in flatness? So far we’ve mentioned two metrics: value and flatness. Consider a third: complexity of decision-making. Clearly, like value, a measure of complexity in decision-making can be found in single payout events, and, likewise, none in the flat structure.

In single-payout events the chip-cash relationship is close to linear. In other words, a percentage increase/decrease in chips precipitates a similar change in cash-value (although skill will affect this relationship). The translation of chip-ev into cash-ev is never easier than in such a simple structure; indeed, making decisions on chip-ev correlates with the right play far more frequently in SP’s than in any other style of tournament, close to cash-games.

The distinction between structures is striking and evident in the following scenario. A player calls all-in first hand against four opponents in a winner-takes-all 5:3:2 structured STT. For simplicity assume no rake/fee and the player wins precisely 1 in 5 and otherwise loses. On the occasion(s) of dragging this huge pot a win-rate of less than 1 in 2 will deem the call costly in a winner-takes-all event; however, bettering that ratio, given a 50% holding, seems a reasonable prospect. Incredibly, in the flatter, standard payout structure the player must win every time just to break even. This trapping of flatter structures doubtless causes many players to step out of line early on, even though such decisions could be in-line in other forms of the game.

Still, experienced players of STT’s regularly claim their advantage to be telling on the bubble[i], or close to the money. Not through many employing risk-averse strategies in a bid to secure a prize, which are too risk-averse, but because translating options into the bottom-line at this stage is extremely taxing and often counter-intuitive. In a single payout event, there are, perhaps, two high-level ingredients to decision making: skill bias of an option[ii], the chip EV of an option. In a multi-payout event there is a third: establishing an option’s chances of attaining each payout, and of course, implicitly weighting these chances with the prizes to evaluate the bottom-line.

One could argue that the third aspect replaces the chip-ev calculating phase of the decision-making process. The many-to-one relationship [iii] that exists between options and chip-EV matters much less in single payout events, especially for average players. The skill bias utilises the option in its raw form [iv] but is only likely to impact in the more extreme cases of SP events, and thus the condensed form of an option’s weighted-outcomes (w.r.t. chips), the chip-ev, is a pretty good benchmark for decision-making in such tournaments. Not so in structured payouts: ‘the how’ matters. So much so that you’d choose to throw away the calculation and leave it in its unedited mode: e.g. 20 % chance of 6000 & 80 % 4000 or, say, 55 % chance of 3500 &4 5% 0 - not in the form of chip-ev’s 4400 & 1925 respectively. This format should create a more representative picture of the merits of each option and lead to informed decision-making.

So, why this digression? Hopefully, it has been demonstrated decision-making initially becomes more complex as the payout flattens. We see, although a decision in a single-payout tournament has some measure of complexity, and one completely level has none, flattening an SP tournament does lead to an increase in the complexity of decision-making, even though it ultimately leads to none. So it might at least reasonably be contended value behaves similarly.

Value and complexity won’t always be comfortable bedfellows: if decisions become so tough neither player can make the judgement (i.e. the decision doesn’t discriminate), or indeed, the reward on offer is much less than for hard, but not so tough questions, then value & complexity move in opposite directions. In party’s steps this may well be the case; while it takes a skilful player to cover the angles integral to a flatter structure, the prize differentials may reward less than one which possesses a steeper-structure, but less testing game. However, in games of skill value & complexity typically correlate, and so perhaps it can be surmised they serve the same interests, for a while, in structured payouts.

Next Article: 13th July

[i] although, in an STT, the bubble typically refers to 4 handed

[ii] an option is a raise; fold; call; check etc

[iii] e.g. a Chip-EV of 5000 could be reduced from (0.75: 5600, 0.25:3200), (0.8:6250, 0.2:0), (1.0:5000),…

[iv] a skilful player should care if a chip-EV of 5000 comes from (0.75: 5600, 0.25:3200) or, say, (0.8:6250, 0.2:0),