poker dynamics

the skill factor

2008-07-21T05:54:00.000-07:00

This is a slightly modified article written a few years back. It’ll be the last one for the foreseeable future.

Which version of poker requires the greater skill? We’re all biassed one way or another; prone, perhaps to vouch for which we know, or succeed at – whether it's because we witness greater depth or are driven to service our ego. Still, for others it will be the converse, the games foreign seem more complex, leaving us unsure underfoot. This article will at best only vaguely inform, on what is, a rather ambiguous and somewhat intractable question.

Poker, like most games, is a combination of both luck and skill. It is, it seems, entirely rational to state a game more dependent on luck to be a less skillful game. Classifications, of say, ‘80% luck, 20% skill’ are tempting, and arguably, provide some insight; however, it is not meaningful to qualify a game’s skill-level this way. Our personal ranking order of skillful games doesn’t, nor shouldn’t, faithfully follow some simplistic percentage-luck metric: compare snap and tic-tac-toe to poker, backgammon and bridge. Are the skillful games more or less reliant on luck? There are qualitative and quantitative attributes of skill in a game.

Those who’ve undertaken some basic measure of computer programming will likely recall being tasked to code a sorting algorithm. A common poser is to orgnaise a group of numbers into numerical order, and or, possibly, the more taxing version, to sort a list of words alphabetically. Each rinse of the program creates a more ordered list [1], until, the task is completed – within a determinable maximum number of loops.

One might argue a poker tournament serves as one cycle of a sorting algorithm for players, albeit a somewhat stochastic one: the list might easily become less entropic after one tournament loop. However, you’d figure, eventually, given enough tournaments to arrive at a perfect ordering of ability for 100 fixed-skilled players. Though, unlike, the word and number sorting algorithms there is no upper-bound on the number of loops guaranteeing a completed task. In the real world of poker, though, the very thing (skill levels) set out to be ordered in the first tournment, no longer exist at the start of the second: abilities change.

It appears quite apparent, in tournament poker, Limit-Hold'em (LH) orders better than No-Limit (NL) : skilful Limit-Holdem players should, like for like, expect to outperform their NL counterparts. Still, even were such an assertion shown to hold true, it in no way testifies to a more skilful game, or skilled players. What it would evidence, is limit-holdem discriminating better between players, than NL. Thus one’d anticipate the limit version of holdem to establish rank more readily than NL in our iteration of poker tournaments.

Perhaps one way of attempting to model Limit-Holdem would be to identify it as a series of multiple-choice questions; NL might also be approximated similarly, but of course with a greater number of choices, albeit approximated.

Now, it is without modesty, that I can currently claim to hold both a better vocabulary, mathematical insight than my 7-year old neice. However, it is, clearly, trivial to set 50 mutiple-choice mathematical questions, or tests of vocabulary, to which I could fail to hazard any sort of meaningful guess. Such questions might tax the knowledgeable, or brilliant, but they’d fail to discriminate between our respective abilities, or knowledge.

A continual lowering of standard, will, eventually, yield the odd problem I’d reckon on tackling, or guess at educatedly. Yet in spite of this edge, I could easily lose, since in only a handful of instances are my responses an improvement on outright guesses. But as the quality is relaxed still further, a point should eventually arrive, hopefully, where I’d anticipate confidently answering all 50 questions, and where she is still forced to guess at all of them. At this level the test discriminates very well (but measures poorly) between our respective knowledge of the subject.

Continued simplification will witness a juncture where we are both able to answer all 50 questions: once again the test fails to discriminate. Clearly, the level of difficulty mostly likely to discriminate between candidates isn’t the one offering the superior challenge or requiring the greatest degree of skill or understanding, or naturally, the simplest one either.

Bidding to analogise more meaningfully with NL/Limit hold'em tournaments debate, we might choose to spice up the multi-choice. Suppose in a discriminating multiple-choice test we implenent a scoring system tolerating 3 mistakes; the score awarded is the running score when the third error occurs. Then run it with, say, with 6 strikes, or one strike: which of the tests is likely to reveal a more representative order of ability?

While the questions are the same in both cases the penalty for failure isn’t. One could easily design a test that would both discriminate and challenge more than another on merit alone, but if the penalty system employed was stern enough, it’d be expected to do less well at sorting out the participants in order of ability: higher variance. It certainly appears NL hold’em has a higher penalty system for poor decision-making*: the limit version yields more lives. That said, employing a heavy penalty system could serve as a better discriminator too. In life, for example, one person may get more day to day life-decisions right than another person, but may have a tendency of getting the important ones wrong. Which of them will be the happier? But if a set of tennis were settled by the first unforced error from either player, what odds for Nadal at the French Open?

The weighting or importance assigned to different decisions or outcomes is crucial when trying to ascertain an individual's effectiveness, ability or, indeed, to establish a proper order of things. Calling down a possible bluff on the river in NL often holds a far greater penalty for failure than such a call in Limit-Hold’em. It might appear that the ‘weights’ are a little kinder to the limit player.

So, in conclusion: the skill factor in a game is measured in terms of quality as well as quantity; a game that more readily discriminates between players isn’t necessarily more challenging or more skilful; heavier penalties for poor decision-making can level the playing field; it is not a paradoxical to state a game to be both more luck and skill dependent than another.

*Note, by suggesting a game has a heavier penalty system one is not implying it is more reliant on luck. A game can be deterministic, free from luck and uncertainty, yet still employ a penalty system. One might consider the luck aspect of a game to be the extent to which poor decisions can be rewarded and, consequently, good ones penalised or indeed how much of the game is free from decision-making all together.

dreaded delays

2008-05-05T16:15:00.000-07:00

A couple of years ago I ran across the following article in a chance purchase of a scientific magazine:

Dread lights up like a pain in your brain

The first trial apparently is designed to verify the presence of dread and infer a relationship with time. Subjects are offered a choice between: a quantity of physical pain with a short delay; the same quantity of pain and longer delay.

p1 + d1 > p2 + d2 where p1=p2 and d1>d2

Nurturing a basic assumption on dread, it’s a no-brainer: the former dominates the latter. Except, for those, 16%, with unorthodox value systems. The second trial, more interestingly, sets out to determine if dread and pain are tradeable currencies. The subjects choose between:

p1 + d1 and p2 + d2, where p1>p2, a1 < a2.

Neither option dominates, so subjects are now forced, non-trivially, to consult their value systems [2] to trade-off the delay and shock differences expressed by the two options; less pleasantly, but comparable, as to how one might negotiate choosing between a car of superior drivability but inferior fuel efficiency, to another.

The second trial might elicit, or bias, perceived rational-choice, rather than value or preference: it appears irrational to opt for additional pain, at the expense of waiting. Dread is a state-of-mind – it’s going to happen, why worry about it: dread is internal. Whereas physical pain is real, validated, somewhat externally (by the electrodes!), thus a more plausible cost. So, perhaps, some of the subjects are predisposed to answer no to an increase in pain, rather, than attempt to trade-off, two very different and so difficult to trade, metrics. Though, arguably, such rationale might advance more readily on the philosophical thought-experiment plane, than, well, when wired up experiencing both forms of anxiety.

Asymmetry: if waiting for pain is painful, shouldn’t waiting for pleasure be comparably pleasurable? Nature is cruel; still what would our ancestors have got done?

Hyperbolic discounting is the term ascribed by economists to the preference for low-early rewards, over higher-late ones: conferred to psychologists, one assumes, to ascertain why. One might conjecture it is as a consequence of our, mostly, innate inability to delay gratification; or, an intrinsic (related) tendency to discount value to our future selves.

A friend winning £5000, one hopes, will evoke cheer in us; mirror neurons fire-up our, sharing, in part, the experience too, if not the cash. Of course our responses are seldom so clean; however, preference for the pal’s windfall, over a personal gain of, say, £50 should fail to surface in but a few. Exchange £50 for £1000, though, what then?

The altruistic gene, extent of friendship, of course, will influence the almost inevitably discounted value we store on someone else’s gain over an equivalent personal one. Arguably a similar sense of discounted empathy is present when contemplating rewarding our future selves. Confirmation, or knowledge, of future rewards doesn’t benefit the present, as it does the future-present. The nearer the reward, typically, the greater the anticipation, the better the ‘now-experience’ - the less we discount it. As with the linked-example a similar present-satisfaction level is predicted, now, at the prospect of the equivalent fixed-rewards in either 5 or 6 years’ time. Thus they’re comparably discounted, valued similarly. Inevitably, therefore, doubling the latter date’s reward, as in the linked example, trivialises the choice. So in other words, the sooner the reward, the more “me” is benefiting, the further away, the more it is someone else’s gain: the future-me. Somewhat off-track, and besides, there are other discounting drivers such as circumstance, risk.

Waiting for pain seems unequivocally bad; pleasure delays appear somewhat nebulous - particularly when pleasure banishes a negative state. However, it seems, in general, we choose instant gratification - which often won’t bind to value, or arguably preference. Rewarding or not, delaying pleasure, holds less impact than dread-induced pain-delay.

The poker tie-in is an obvious one. Players predisposed to inflict protracted delays on their adversaries before fixing on a decision often qualify their actions as ones geared to elicit information, or to guarantee a measured decision; often, though, it is designed solely to factor in dread into competitor’s future decision-making. However, waiting on an opponent’s play is in theory a mixture of both states: anticipatory pleasure, anticipatory pain. Uncertainty, though, a cost not mentioned so far, will typically mitigate pleasure and augment pain. For example, in the above shock experiment one could easily imagine a shock ‘sometime in the next minute’ to be less preferable than one at precisely 30 seconds. Indeed, one might expect uncertainty to be traded out for shock increases, as delays were.

The game or hand-specifics, of course will determine the mix of anticipatory emotions. Anticipatory pleasure appears drowned by anticipatory dread (& ~ regret) when enduring the uncertainty accompanying a big bluff. It makes sense: the bluff-state is, typically, emotionally, net-negative, since to not bluff is to suffer losing. As such the design of bluffing is often to gain a less (expected) net-negative state, rather than move to a positive one – so it should seldom feel great. In reality of course the effect of our bluffing on our emotions is not so calculated. We are often inclined to bluff for the wrong reasons, notably, loss aversion, which in fact would lead to a poorer (expected) state. Anticipatory-pleasure should perhaps surface, or even dominate, in free-roll situations or on occasion of a pot wrapped up. Unfortunately, it rarely feels so tangible: it seems we bank the gain, and unfulfilled gains result in disappointment. The lack of emotional-symmetry between gain/loss states (biasing anticipatory dread over pleasure), the discomfort of uncertainty, testify to a tough time waiting.

If one accepts a player is damaged though dread, damaged by more than just the fact, but of its anticipation then ethical questions of a sort are bound to arise. The claim that to bear, tolerate, endure, mask, even mutate such suffering, is a requisite part of players’ armour, (as to inflict is of the arsenal) is legitimate. However, it is trivial, and often, cost free to do so. In live play it is mostly unregulated; players are generally only restricted within the hand. While some target sufferance selectively, others initiate a catchall strategy: sweating a guy out at every opportunity, guarantees sufferance under any weak position. While admittedly, those allocating delays selectively will inflict more angst, since the threat is weightier, the distorted perceptions under such circumstances doubtlessly leads one to conclude, and experience, the catchall play to elicit the greater net anticipatory dread. Anticipatory dread, of course, is the design of this, at times, tedium.

In blackjack, circumstances frequently surface where emotions bait the knowledgeable player into falling out of line; however, the prevailing sense of ‘but I know this is the right decision’, will usually suffice for all but the most marginal of decisions. Hence placating emotions via a compromised strategy is seldom facilitated: the two remained partitioned. [1]. In poker, such defences are easily breached. Situations are unique, measurements subjective, hands can be played in a mixture of ways. Consequently, without a definite rebuttal to hand, emotions gain easy access to decision-making, and so in seeking out plausible strategies to reduce, or minimise, the emotional cost, they corrupt the decision-making process. Anticipatory dread is just one, rather powerful emotion, the mind is eager to dump anyway it can.

Once again, though it is rightly viewed as a specific instance, of the far greater challenge of managing our emotions in poker, the test while appropriate should be even-handed. For purposes of practicality, fairness and skill, the resource should be restricted over an interval, with individuals left to consider how best to allocate their resource (as it is on-line).

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On-line poker rooms should be fully tuned in to the cost, or value of inserting time delays.

Life for the multi-tabler can be exhaustive, keeping track of all stacks at any given time, is extremely tough: it is easy to be a pot or two out on an estimate on any one table. Years ago while playing regularly at both and party and stars, I observed a marked difference between the sites' respective feel-good factors. No, not the softness differential, but rather, the emotional reflex upon winning a pot. Glancing post-hand on party I’d be gratified by seeing the cash-total update, on stars it’d already updated. Naturally, I, the Pavlov dog, half-salivated when the bell rung: partly expecting the stack size to remain unchanged, partly to increment. Of course, I knew which site was which, but not in time to catch the reflex emotion. Needless to say, I’d experience a fleeting burst of pleasure/disappointment depending when the roll was updated. Significantly, no flipside or downside exists to the inserted delay – neither site deducted losses at the end of the hand, so no complementary, deflating, ‘delayed loss’ at party.

In the early years, PokerStars’ reputation for tournaments and big bet cash-games soared, as did their notoriety for killer rivers: the luckless rapidly coined the site, RiverStars.

Defeat snatched from the jaws of victory is the bitterest pill: virulent when administered by no-limit poker. Death in limit comes by a thousand cuts, a mere handful of meaty blows will suffice to slay the NL-victim; as such, those beats persist in the memory, not recollected as some hazy nightmare, as is often the case with disastrous limit sessions.

In a bid, it seems, to retain the authenticity and drama of live poker, stars, when no more action could take place in a hand, turn the cards on their backs. In addition, they injected a palpable, dramatic delay, between each chapter - every street. In limit-cash, at the time, the mainstay of other sites, this wouldn’t occur, players were seldom all-in, and when they were, (certainly now) hands were only revealed, or mucked, once the river was revealed, concluding the hand. These delays augment the torments of counterfactual regret and anticipatory dread: emotional investment implicit to waiting, knowing what needs to be avoided for just that card, is considerable, defeat, therefore, is all the more crushing. With cards face-down, the intensity, dread of the river is generally less – seldom are miracle rivers apparent in ignorance of an adversaries holding – your aces are vulnerable to every street.

When cards are dealt in one swoop, however, there is little time for anticipation, the beginning, middle and end are fuzzily defined; as such, an abrupt defeat engenders a less vivid counterfactual reality of winning - with all hurdles to be vaulted at once – in contrast, say, to one street and four outs standing before victory. In addition, of course, there is the aforementioned anticipatory dread: lengthier delays increase dread. The purported lack of symmetry between experiences in delays of favourable and unfavourable events, allied to general emotional-aversion to uncertainty, infers such injections unwise: unnecessary accretion of negative experiences hardly value-adds to the brand.

So in short, stars, it seemed, implicitly ensured PokerStars rivers scarred the most.

Next article

[1] Incidentally, delays from the croupier in blackjack frustrate the majority of punters, accompanied losses seem to torment the most.

mindset (ii)

2008-02-11T06:27:00.000-08:00

London buses indeed.

Consider once again the performance drivers: playing well; playing to win [1]. On inspection, one might view them interchangeable, synonymous: a player driven to play well will win, anyone driven to win must play well.

Logical cracks surface, though, under light analysis: winning is a defined, unambiguous state; ‘playing well’ is non-absolute, typically subjective and often relative. Winning is an effect; decisions are causes.

The drivers, while correlated, are distinct - it is not a purely causal relationship. Winning is a state often achieved playing poorly; moreover, playing well doesn’t guarantee winning. Variance is the chief culprit, but also both in the subjective and absolute ways performance is typically measured. Ordinarily, playing your best game (the subjective) or even playing a ‘great game’ (the absolute) will be sub-standard versus top-class opposition. Conversely, antithetic, contrasting, performances might suffice against weak adversaries.

Additionally, winning, or being a winning player, in like passing an exam, is qualitative: standard-of-play metrics, however, confer grading. Consequently, the win-driver relents, runs out of steam, somewhat, when attaining the win-state; the play-well driver, though, is the duracell-bunny.

Of course it is reckless to pay heed only to the playing-well driver: playing better, than our opponents, is pretty important too. Since the subjectivity inherent in measuring personal performance requires intermittent, at least, objective validation, we inevitably become sucked in by the winning-driver: it’s not profitable playing well with superior opponents. We must care about winning too [1].

The following hypothetical gambits contrast the mindsets.

A bursary of $2000 is awarded for attaining some goal over a six-hour session of $5-10 limit hold’em. In the first instance only a profitable session is required; in the second, the sum is awarded if the performance is deemed accomplished over the duration.

In the latter case the focus is clear: play great poker. Against the backdrop of a $2000 bonus, who cares if you win or lose? The object is to execute credit-earning decisions; perhaps a check-fold, bluff check-raise on the river, slow-playing AA pre-flop etc. Sure decision-making is tough, but there is only one agenda; it’s about as uncluttered as poker gets.

In the first scenario, right from the off, we sweat the balance sheet. Winning attracts a preservation-mindset, losing switches on hunt-mode: the former leads to conservative plays, the latter to lines of high variance. At times, rightly so: an increased chance of landing the 2k prize might easily compensate expected losses suffered during a hand. Inevitably, though, added complexity and pressure leads to overly conservative/risky strategies – especially when time runs out on an uncertain outcome.

But so what, we’re discussing a hypothetical, unrealistic, more complex and seemingly pointless problem. The scenarios, though, albeit hypothetically, inject reward, value into the decision-making process - it just so happens to be fiscal. Which, in theory, facilitates analysis: resulting, in the first case, to purported technical adjustments.

Unfortunately, added incentives are not fiscal. Daniel Kahneman and Nobel prize winner Amos Tversky reasoned people were, typically, loss-averse. They were not merely affirming the barefaced truth the populous dislike losing; rather, they, inferred the loss-state bore an additional cost, beyond the loss itself - derived from countless examples of consistent economically irrational decision-making. Most of us for example, experience a greater emotional differential between $50 win/lose states, than between, say, gains of $50 and $150; unless the sums involved are critical, in some way, this appears irrational.

Loss-states aren’t necessarily zero-sum: gain-states frequently occur in life without a complimentary loss-state (& vice versa); in poker, few feel as replenished in victory as they do damaged by defeat [2]. Poker, especially, tournament-poker, seductively potrays an image of fair competition; losing, therefore, inevitably baits underachieving sentiments - a natural, impulsive but somewhat imprudent response, since no two players sit the same exam. Another candidate meta-cost is the losing process; in poker, unlike most gambling, when you lose, for example, you lose to someone[3]. So we are averse to the tangible-loss, the loss-state, competitive-failure, the losing process.

We continuously credit and debit these and other mental accounts, though, they trade in different currencies - if play-well is in credit, so what if the ego-account is in the red? That's fine: as long as we don't hold the exchange rates. Trade-off, though, as is done routinely in day-to-day multi-criteria decisions, we compromise our play. Money is quite meaningless without emotion; a prospective purchase is seldom contemplated without assimilating the emotional pay-off. So unfortunately, the exchange-rates may already be in place, or could be sensibly extrapolated. That said, decisions in poker frequently surface under a cloud of emotion, not typical, seldom present in day to day financially-oriented decisions.

Next article

[1] In this context ‘playing well’ represents a drive to execute good decisions, not to put up a ‘decent performance’. So the driver could easily be to ‘play perfectly’, ‘your A game’ etc

[2] That is w.r.t symmetric states (win/lose $500); rather than, naturally, in tournaments.

[3] which you’d suspect may in part, be offset by winning off someone, but, arguably, not equitably.

mindset (i)

2008-02-08T04:56:00.000-08:00

Consider the following two attitudes:

to win;

to play well;

Sports’ coaches and their ilk would reckon to steer a truck between the pair. The tradition of fair play allied with putting up ‘a spirited fight’, supposedly endemic to the British psyche, is routinely cited as the anathema of the nation’s sport.

‘Winning ugly’ was doubtless schooled into hard-knocks poker grads long before the intense Brad Gilbert coined the phrase. An object lesson in the above mindset-distinction should seldom be required in poker: few competitive environments exist where the glorious failure oxymoron is less self-evident. Nevertheless, some players – most of us to an extent – will opt to lose, or certainly profit less, than adopt a counter-machismo style.

In a sport as tennis a winning strategy, in theory, is self-similar at all levels. In other words, in order to maximise the likelihood of winning a set, one must aspire to win every game; likewise, the requirement is on each point to win games. Naturally, meta-issues surface: injury or fatigue will occasionally necessitate the submissive relinquishing of a forlorn set, or game; shots deemed ‘low percentage’ will serve as loss leaders from time to time. Those issues aside, the game is in structure, strategically self-similar. However, many sports witness combatants purposefully adopt localised losing-strategies in order to protect leads or chase victories (and so not follow self-similarity) [1].

In poker, the seductive strategies tend not to be self-similar: opting to maximise potential profit in a hand will not maximise the year’s potential profit [2]; minimising short-term risk, will not minimise long-term risk [3]; lines optimising the chances of winning individual pots, will rarely return the best chance of winning over a clutch of hands let alone a session, or year. Material meta-issues aside, a suitably bankrolled guy maximising EV from each pot, though, will maximise EV over the session, a year [4]. So what’s the problem with EV?

Now, with the gymnast there’s no reward for deviation, no punishment for accuracy. There is determinism: better execution will reflect in better marks; improve one element - improve the whole (ceteris paribus). However, in such sports as golf, darts and snooker the penalisation of superior accuracy and/or technique, abounds. But not sufficient to confuse, mask their overall benefit [5]. Nevertheless, as a consequence, the random walks of these sporting-events witness wrong steps move in the right direction and vice versa.

As we move to more game-theoretic and high-impact low-probability event-driven sports, confusion and uncertainty often rein free [6]. As such stakeholders (interested parties) seek frameworks within which to operate, rules to follow, conventions to reinforce, superstitions to suffer and so on. All in a bid, it seems to: bring order; certainty; mitigate regret, in particular, counterfactual regret. The ease with which one can visualise or imagine not losing in the manner lost, will determine to a large extent the level of regret experienced in losing said way. As such stakeholders, decision-makers, will anticipate contrasting levels of counterfactual regret with candidate options, and so attach to them varying levels of anticipatory regret (ex ante); this regret will doubtlessly filter (or attempt to) into the decision-making process.

Put simply, if doing that which would’ve won the game was something seldom done, there should be little cause for regret; conversely, if failing to do the very thing which is done routinely causes losing, well, you’ll be immersed in it. Naturally, this is (typically) known in advance of the decision being taken, or event occurring, and so the emotions are anticipated and so, inevitably, affect. It is therefore, arguably a strong mind for which a great number of decisions are in-play (Jose Mourinho, tactical substitutions after 20 minutes).

In this class of sport, any (interesting) two situations are seldom the same and similar ones will often be too infrequent to draw meaningful conclusions. As such learning is difficult; a move towards a better strategy might not realise immediate or obvious benefit (expected or otherwise). If the strategy breaks convention, and, or, upon failure, increases the level of counterfactual regret suffered by stakeholders, the expected gain could well be overshadowed by the personal risk now incumbent on the decision-maker.

Revered ex-England cricket captain, Mike Brearly, observed a marked difference in the recrimination imparted on a captain’s retrospectively poor toss-decision by the type of failed decision [7]. When the batting team fails in the first innings they are seen as disadvantaged but very much in the game: they could still win. However when the batting side succeeds - the bowling team concedes many runs, but a few wickets at stumps on day 1 - the bowling side are deemed to be batted out of the game: they can hope for but a draw. So, it appears, to suffer a disadvantage having electing to bowl is more lamentable than a comparable fate after opting to bat.

This intuitive feel-good logic is attractive and rational, but hardly rigorously analyzed. It is somewhat akin to such poker rationale as: ‘if you’re not in the game on day 2 of the WSOP, you can’t win: make sure you’re there - you got to be in it, to win it’. Nice sentiment, but optimal? No, not likely. Batting first effectively puts a cap on how bad things are at the close of play; but does not to testify as to how good things, or importantly, how things are likely, or expected, to be. So the cricket captain battles both convention and anticipatory regret [8].

A footballer’s (soccer) missed scoring opportunity is typically less regrettable when ‘working the keeper’ than on occasion of missing outright; so a strategy, or adjustment, boasting a marginal improvement in scoring, but material hike in misses, could, under the sufferance of criticism and/or self-recrimination, become distorted and feel less, not more, effective. Indeed, for some, it would be self-fulfilling, as confidence wanes under the blame-burden. Still even with a clear mind, issues of personal-risk, aversion to criticism and blame might adjudicate maintaining the status quo the judicious choice.

Coaches often, or are inclined to, alter tactics, personnel if the game-state is undesirable. Once again there is a regret issue, to not change, is to not try (“do something!”); few upon failure, will grumble ‘you’ll never know what would have happened had you not made those changes’ [9]. Of course the game-dynamic is typically different and so change is legitimate. However, if, post-change, the team dominates possession, creates a multitude of chances, but fails in the end-goal, one might legitimately state winning to have been a ‘close-call counterfactual reality’: they nearly did it.

However, it is a leap to conclude the team were benefitted (in an expected sense or otherwise) since the performance improved post-intervention. Perhaps the decision was fortuitous, rather than astute; or indeed, the game-state suggested any change would trigger improvement [10]; in addition, games can and do progress organically, so the non-interventionist counterfactual reality may outperform the interventionist one. Nevertheless, informed decisions should correlate with favourable “factuals” (outcomes), as they should with positive close-call counterfactuals (nearly good outcomes), and, naturally, poor decisions map to negative close-call counterfactuals (& factuals). So the decision-maker, in theory, is better equipped, more advised, when cognisant of both: the factual and the close-call counter-factual.

In such environments marginal changes do not have marginal effects - where it matters. Which, as suggested, hinders learning; however, at least the football coach is availed discriminating metrics, beyond the score line, with which to measure the impact of change.

Instead consider the poker player pondering the big-bet call-down. There are no close-call counterfactuals here; it’s binary – miss or monster. No bluff-percentages accompany the showed hand, nothing to help realign a wayward strategy, save a potentially misleading showdown. Decisions typically feel a good deal less marginal ex-post, than they do ex-ante (assuming hand disclosure); which rather worryingly suggests a somewhat selective or perhaps suppressed world-view, either before or after the fact.

So learning through EV is tough: decisions are rarely awarded the amounts ‘expected’ (the feedback (profit or loss), therefore, invariably distorts); meta-issues abound - the value, or cost radiates over a multitude of hands; scenarios exist without the edifying close call counterfactuals.

The idea is, of course, to learn over the long haul. Unfortunately, brain-accountancy is typically shoddy, the database unreliable and lacking in the necessary detail. What’s more, lessons are scattered, not learnt in chunks. Akin, perhaps, to hopping from one language to the next, after each new word, facet of grammar understood.

So, we budding players have a torrid time; mind you, not that I’d swap the monitor and mouse, for the high-bars and rings.

Next article: part ii

[1] Say, the very defensive golf-shot of the tournament leader at the 18th; the advancing goalkeeper of a trailing football (soccer) side.

[2] Note this is to survey the possible lines and select the option corresponding to the greatest attainable reward (eg raising, rather than a sensible call hoping, by chance, to be called by a weaker hand). While potential for a mother-of-all-runs could technically be viewed as technically self-similar, it is a somewhat facile point.

[4] Over-cautious strategies mitigating the risk of busting out of a game, may, through loss of EV, increase the chances of busting the bankroll.

[5] Although of course, there is often considerable uncertainty over which technique generates the best results: a golfer reinventing a swing. So there is technique and execution; which applies to poker too, at a high level.

[6] So for example, in football (soccer) a goal is a very infrequent event during a match, compared, to say, a pass, or a free-kick, but of course has very high impact since goals absolutely determine success or failure.

[7] Cricket, specifically, is a game played over up to 5 days, where a team must bowl out 10 batsmen from the opposition twice in order to win; if both teams achieve this then the team scoring the highest runs wins; if neither are able to do so, the game is a draw. Therefore, it’s hard to envisage a side losing if they’ve scored a lot of runs for the loss of a few wickets (batsmen) during their first innings on day 1.

[8] They are clearly not independent, and doubtless reinforce each other.

[9] Except, when of course changes were instigated from a favourable, winning, position.

[10] A so-called secondary counterfactual – at least in a qualitative sense. Although the specifics of the improvement effected could only occur with that explicit change, it might be argued, a number of alternate alterations would have advanced matters uniquely, too. So improvement upon change was virtually inevitable.

sklansky's theorem: fundamentally flawed (iv)

2008-02-07T15:46:00.000-08:00

Though entirely proper to test, disprove, any theorem, perhaps, the most pertinent inquiry of Skalnsky’s is to ask: is it useful?

The general interpretation or use of the theorem appears in the validation of decisions through hand playbacks with opponents’ cards face up: if performed the same way, the boy done good.

I’ve personally found the process to placate rather than inform. After bad-beats, I’ve consoled myself with: ‘well I’d have played the same if I’d known what he had’. Such thinking is quixotic, since, although the strategy, face-up, might be seen as perfect; face-down, it could be dire. As poker players, we seldom deal with certainties.

On occasions where a decision is marked retrospectively correct in this contrived manner, but on the (judgement-based) balance of probabilities at the time, deemed wrong, applying hindsight-analysis will likely mislead and so hinder learning.

A conflict, of sorts, arises when we witness a respected, more informed player deviate from our selected path or option. We are, instinctively, compelled to re-evaluate, coaxed, perhaps, into presuming we were in error and tempted to mimic (or migrate towards) the play [1]. However, this can resemble the above trap - his balance of probabilities, worldview (not to mention image) contrasts our own; a superior player is found, typically, to be better informed, in the same sense, but to a lesser to degree, as someone privy to hole cards: if our tools, skills, are different then so must be our models, and our answers [2].

Consider an amateur weatherman using an historical 3-day forecast algorithm [3]; light showers are predicted, at a low-confidence level, for the following morning. The amateur weatherman baulks at the prediction and forecasts a dry day. Later that evening, after submitting his prediction, he listens out for the Markov Amateur-Weathermen Society’s forecast, which utilises a sophisticated 10-day algorithm: a clear day is predicted – with a high level of confidence. The day was indeed free of rain. So the amateur's forecast was correct, but the decision? With better tools, more information, the weatherman will, perhaps, claim to have reached such a forecast anyway; however, contradicting his basic algorithm, whimsically, is clearly gambling against the odds, which, ultimately, is a strategy, destined to under-perform.

Of course, inevitably, as we develop, the cart will, on occasion, precede the horse - we are seduced into reproducing the effects (plays) of better models, so our model resembles it, appears similar - to kid ourselves we’re improving; nevertheless, in the poker world, one might argue extrapolating this way, persisting in (based on the current model) erroneous decisions, may render a speedier, albeit costlier, progression - at least for some.

To conclude, it is, surely, advantageous to measure our decisions based on our interpretation of the information gathered, not what wasn’t, or couldn’t. The value in ascertaining the correctness of a decision based on a near utopian-level of awareness is far from apparent.

Next article

[1] Which of course can edify (& so develop the model), but also regress if applied without the necessary insight. E.g. value-betting weak hands.

[2] Obviously, not on every occasion.

[3] i.e. it uses the previous 3-days weather to predict the next.

sklansky's theorem: fundamentally flawed (iii)

2007-04-25T14:53:00.000-07:00

‘The Fundamental Theorem applies universally when a hand has been reduced to a contest between you and a single opponent’ - Sklansky.

Though hardly churlish to contest on utility grounds, statements free from individual preference-states often illuminate and edify while those suffocated with caveats at times confuse.

There appears little value in generating chip-theories, so ‘gain’ is reasoned to mean the bottom-line, cash-EV (EV). Sklansky’s statement, though, is naïve in its universal acclamation as it is predicated on EV flowing only between active players, which is patently untrue: chips do, but not always EV.

Players found enduring tournaments will invariably experience material emotions in the decisions, or outcomes, of hands in which they appear uninvolved. Unsurprising, since the odds of securing any given payout fluctuate, wildly at times, for all players, with every decision or turn of a card (pot-active or not). Consequently, tournaments-hands should seldom, if ever, be considered zero-sum between only those active in the pot, not even, as Sklansky claims, when reduced (the pot) to headsup (HU). At each junction, EV might flow in or out of the hand: it isn't contained. Therefore, where tournament-dynamics permit, the still pot-active players could each lose EV were a specific action engaged (or passively gain when avoided).

Two big and two small stacks, only, survive the tournament. The minnows give deference and fold; the chipped-up small-blind elects to push against his rival’s big-blind. Now, said rival has a read; he adjudges, with certainty, his adversary to hold either JJ or AK. Glancing down he discovers QQ. Gulp. The short-stacks, inevitably, are not passive on-lookers: they long for a calling big-blind. While ongoing events will impact on neither’s chance of winning outright (materially), the opportunity of securing a higher prize would present itself should these stacks collide.

The big-blind is aware that calling will see them both shed EV to the now salivating small stacks. However, the Queen's are a favourite over each of the small-blind's potential holdings - they hold an edge over the likelier AK, but dominate JJ. After due consideration, the (expected) pay-off from the SB is deemed to be sufficient compensation for his contribution to the EV-drain: he's calling.

Just as he committs, his opponent’s cards are accidentally disclosed: AK is shown. Despite still gaining EV from the SB, now without the luxury of a chance domination of JJ the the SB's payout no longer adequately covers his loss to the small stacks. So he passes.

Now since he, rightly, changes his mind, by Sklansky’s Theorem he will gain. Also, the small-blind, by definition of the theorem, must lose-out from this redress. Except he doesn’t, he gains, in fact, more so than the QQ since he was losing EV to everyone. It is the small stacks who lose out; Sklansky’s Fundamental theorem does not hold.

The example is, admittedly, extreme; however, whenever it is possible for EV to flow out of the hand, which is headsup, Sklansky’s claim is under threat since it isn’t a ‘zero-sum game’ w.r.t. the two combatants.

In tournament poker, passive gains and losses abound from other tables - a skilful player is at risk, a short stack created. While on the player's own table, a collision, say, resulting in the emergence of a threatening stack to the right, at the expense of one to the left, is typically advantageous. Gains, passive or otherwise, must accrue from somewhere, vacate someone's EV, whether it be it from active-players or, indeed, from those passively, negatively, affected (e.g. the player now at risk from the big stack on his left). Nevertheless, when one or more inactive players gain passively during a HU pot, both active players might, in theory, net-lose on the decision, to foot the passive-gain bill. Consequently, should either become cognisant of the other’s holding, both might benefit from a revised decision. So, the theorem doesn’t apply universally 'when a hand has been reduced to a contest between you and a single opponent'.

Passive gains exist in cash games too; furthermore, once gains w.r.t utility are included the failure-space increases. Sklansky’s claim holds only if measuring gain w.r.t chips and if the future-impact of their redistribution is excluded, in so doing, supporting uninformed and potentially erroneous decisions.

Part (iv) to follow.

sklansky's theorem: fundamentally flawed (ii)

2007-02-11T16:34:00.000-08:00

The late Andy Morton cleverly disproved Sklansky’s theorem in the late 90’s. Morton’s Theorem, as it became known, is markedly more analytic and conventional than the softer rebuttals expressed in part (i). In multi-way pots, he showed, situations occur often where a player undertakes a decision benefiting both himself and an opponent, contradicting Sklansky’s Theorem. Were a player, say, to gain, in the EV sense, from you but leak a greater sum to another, he’d be right to fold: and you‘d want him too.

Example:

Limit Holdem: The board shows: 8h, 10h, 4c, 6s:

Player 1 holds Ad-Ah
Player 2 holds: Jh-Qh
Player 3 holds: Kc-10c

Transparency necessitates an assumption of zero implied odds with the best hand winning on the river. Player 2’s (P2) chances of winning can’t be diminished from a call by Player 3 (P3), as such he can only benefit through winning, potentially, a greater pot. Player1 (P1), though, is ambivalent over this decision, since, like P2 when he wins he gains an extra bet; however, unlike P2 his chances of securing victory are reduced by such a call: there is trade-off.

Suppose P3 receives the precise odds to call. Our, model player 3 should, thus, be indifferent between folding and calling. However, P2 gains from, and thus hopes for his call; so since P3 is indifferent to his own call, then by deduction it is the Aces accounting for, and so bearing the cost of, P2’s potentially improved position. Although not strictly illustrating Morton’s case, since P3 didn’t gain by folding, it does for all intended purposes - Sklansky’s theorem suggests if your opponent is indifferent to calling, then so are you. Clearly, that’s not always the case.

Now, for a slightly more numerical approach: as P3 deliberates, one might view the pot as jointly owned by his adversaries already committed to the river. The dilemma for player 3 is whether or not to join the party. Assume P1 owns 70% and P2 30% of a pot currently standing at $800. For simplicity assume forty cards remain, of which, just four land player 3 the spoils. So, evidently, he is a 9-1 shot receiving odds of only 8-1. In this case, as above, P2’s prospects of winning are unaffected by his successor’s decision, still 30%. Should P3 elect to call, he and the front-runner, P1, could, through an award of 30% of the pot, legitimately settle-up with player 2 and proceed to strip-out or void P2’s 12 outs, with the river allocating the remainder.

With P3 folding, P1 can expect a return of 70% of $800: $560. But with a call, and P2 settling out-of-river, what now P1’s reward? The depleted deck holds just 28 cards from which P1 draws to all but 4 for the reduced pot of $630 ($900-$270). Player 3’s expected return from the $100 turn-investment is 4/28 * $630 = $90. As expected, a losing investment: he’d have been better off folding. Player 1’s equity is also reduced, by $20 to $540 ($630-$90), thus both P1 and P3 are better off if P3 elects to fold.

Unfortunately, the final example testifies, somewhat inevitably, to the existence of subtle and almost improvable degrees of collusion in our game. Here, P3’s $10 loss is the result of a $20 credit from P1 and a $30 debit to P2; of course, as colluders, the drain on EV to P2 is an illusion. Naturally, no such explicit case will occur, seldom will the two cohorts be certain of their single foe’s holding; however, clear folds become marginal ones, marginal folds become clear calls and so on. All the while there is little hint of cheating: simply a localised increase in the frequency of bad-beats by weak(ish) calls.

For a fuller and more mathematical treatment it is fitting to visit Morton’s original post.

Morton’s contends occurrences are more frequent than Sklansky's ‘rare exceptions’ ; which appears rational, given the non-exceptional situations described.

Still Sklansky affords no concession for Heads-Up pots (one-on-one), stating: ‘The Fundamental Theorem applies universally when a hand has been reduced to a contest between you and a single opponent’.

Next article: part (iii)

sklansky's theorem: fundamentally flawed (i)

2006-10-27T15:47:00.000-07:00

David Sklansky’s book, The Theory of Poker, is highly revered but, though I own it, I have read but a little. However, I am familiar with his Fundamental Theorem of Poker (FToP), which states:

‘Every time you play a hand differently from the way you would have played it if you could see all of your opponent’s cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, you gain. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.’

The first criticism isn’t technical, but one of communication, it lies in the implication of ‘gain’. The usage is ambiguous: the intended interpretation is reliant on knowledge not requisite for engaging poker or much of its material. In reality you wouldn’t gain every time you executed a perfectly informed decision - that’s poker. So we must assume Sklansky attempts to communicate gain in the EV [1], or average, sense. However, even accepting this interpretation of ‘gain’ the theorem merits a challenge. It is easy to construct a hand where the same strategies would be employed by players whether they are all covertly aware of the each other’s cards or not. Going all zero-sum, are they all gaining, as the theorem would suggest? Well not in an EV sense (ordinarily).

Perhaps the theorem could fence off criticism under such contrived circumstances by claiming careful interpretation suggests they are both losing and gaining thus facilitating a neutral outcome; well, perhaps. The theorem might seem more robust - certainly clearer – if, for example, Sklansky replaced ‘lose’ with ‘can’t gain’ – in the EV sense, of course. Clearer, robust, perhaps: less memorable, definitely.

Moving on, the theorem appears to advance or at least encourage the notion that resolving the optimal play requires only knowledge of your opponents’ cards; after all, you know his holding, what more should you demand? Simple: his strategy. We’ve all borne the frustration of discovering our foe’s failure to act as we’d anticipated (or as pro’s are prone to lament, act as they should) despite correctly establishing their hand. Knowing a player’s holding could lead us to check expecting a bet, bet expecting a fold or raise only to be disappointed. Thus, exploiting this information would inevitably, on occasion, direct one to an incorrect play, where ignorance would not. If we are to reason our opponent’s decision ultimately determinable then our judgement is culpable at some level in such instances, and therefore the theorem won’t hold.

Alternatively, a non-deterministic view of our opponents’ strategy could on occasion see misjudgement become the scapegoat for misfortune. It might be considered unlucky to check a monster to find our foe choosing to take this opportunity not to value bet top pair. Despite not explicitly gaining, the theorem, under a typical interpretation, might stand up because check-raising could be the best play against his mixed strategy; in the EV sense, we’d gain every time. However, it doesn’t automatically hold should we misjudge his strategy, mixed or not. If we do so to such a degree as to render our current decision sub-optimal, then we are losing out, contradicting the theorem. So whether our opponent’s decision is theoretically determinable or not, judgemental failure can always be the theorem’s undoing.

Advocates of the FToP could respond by drilling down further into EV argument by claiming players would be more informed of the right strategy, and thus more likely to make a better decision, and so gain. So where are we heading: if you see your opponent’s cards you’ll probably make a better decision than if you hadn’t? That’s sounds quite fundamental, and pointless.

One must assume the failure to condition the theorem on strategy to be an oversight, since it could easily be shown on a hand-by-hand basis to fail in many instances.

Another apparent implicit assumption is that perfect decision-making follows from complete or sufficient information: it doesn’t. It takes knowledge, skill and discipline to turn information into effectiveness: it’s not a given. Proficiency in probability is requisite to determine the right decision; in addition, execution of the correct strategy is a non-trivial task in itself, requiring material discipline - players will often weigh up what they want to do as well as what they should do. Despite casinos allowing us to see ‘all their cards’ many a shrewd winning gambler can be found propping up a house game. In the same way such savvy punters knowingly brave the odds and execute losing bets at roulette, I suspect all poker players will, at some time, take on draws they know aren’t justified. In fact, given this trait, it is easy to illustrate how knowing an opponent’s cards could wittingly guide to a poorer decision.

It’s the turn: hero faces a bet and the certainty a gutshot is his only hope. The 3-flush board of this strongly played hand leads him to deduce one of his outs risks handing his opponent a flush and accept he may already be drawing dead; so the awful 9-1 offer to complete the hand is wisely declined. However, had he gleaned his opponent’s cards and observed his outs to be clean – he’d call, wilfully accepting the marginally poor odds [2]. The -EV decision could easily be vindicated from a number of meta-game [3] positions, but, of course, it won’t always. And when it isn’t the FToP fails since the foe gains when hero pursues a course of action he’d only undertake in light of his adversary’s holding and loses out when he doesn’t; when hero folds. So even though the theorem is not openly predicated on rational decision-making, even if it were such decisions shouldn’t be presumed irrational or indeed losing, particularly when taking a systemic view of a poker player [4].

One could surmise FToP reasonably assumes some requisite level of poker-competency. However, these competencies are non-trivial; in fact, strict satisfaction would be very untypical. Only a minority of players are cognisant of, say, the precise pot odds required to call when holding 3-6o against an all-in k-10. Moreover, even with complete information, still fewer will know, or can calculate dynamically, the exact odds of every situation arising; I’d fail on the first and naturally the second. Furthermore, there is still the process of determining the odds you’re currently getting against the odds required; not hard, but mistakes happen (especially counting the chips live). Finally, there is the requirement to execute the decision your poker-brain knows to be right. Who hasn’t failed this task? Most of us, routinely. So it appears nigh on the entire poker population fails the requisite competency-levels expected for this fundamental theorem to hold.

Sklansky ambitiously and mistakenly, in my opinion, attempts to craft a theorem, and a fundamental one no less, out of how people, the reader, make decisions and would respond to information. It would remain contentious and flawed, but more forgiving, were it to purport how people should, not would, behave and so benefit.

A fundamental theorem on individual human behaviour: Nobel Prize material indeed.

Next Article: part (ii), Friday 24th (edit) November

[1] EV – Expected Value, an unfortunate term as it often fails to represent value and is seldom expected.

[2] Ignoring implied odds.

[3] meta game: the impact of the current decision on future hands.

[4] The theorem would fail in some cases even when negative EV decisions are justified w.r.t. meta-issues; since, invariably, at least some of hero’s meta-gain won’t be endured by this opponent.

move on, come rain or shine

2006-10-18T07:49:00.000-07:00

A change of tact delayed this month’s posting - apologies. Perusing the high stakes forum at 2+2, I ran across this interesting post. Though he admits to not following his own advice, the author concludes:

Anyway - when you running very bad or very hot - it's harder to get the maximum EV from this situation compared to when you running normal, cause stats of some players changed instantly (and PT can't catch this) and it takes time to find the players who really start taking shots/start to made too many folds against you.

So it's easier to change a table or take a break quitting the complicated analysis of players...

The poster advocates quitting when significantly winning or losing because the context generated by either position forces one to deviate from a more typical playing strategy: more extrapolation, less interpolation. As such our judgement is warped, decision-making is tougher. Undoubtedly every player has seen routine decisions transform into real quandaries under the climate of running at extremes.

However, even though our decision-making becomes less efficient, it can increase in effectiveness – even with a greater error-count [1]; despite acting less optimally, we might still expect to gain. Conditions change for everyone: success is to adapt quickly and advantageously, not to match or improve on performance metrics achieved under normal conditions. It’s a different game.

The presence of asymmetric information, the culprit of adverse selection, appears to hatch the belief, here, that winning or losing reference-states are penalised. It is common-place in, and to some degree particular to, on-line poker, it seems, to be largely unaware of an adversary's observations and thus ignorant (at least initially) of any factoring in of your reference-state into his strategy; however, he will be perfectly aware of any adjustments, or lack of them. A normal strategy would be deployed against one oblivious to your standing, to an opponent aware of it, an adapted one. Since we know not if our rival is fish or fowl, we must compromise or risk being compromised: not an issue, or at least less of one, under routine conditions. So, comparatively, we are likely to under-perform and execute more errors; your opponent appears to have you at a disadvantage. The solution, easily affordable on the internet, is to quit, reinitialise the variables and start elsewhere under the (likely) realisation we are observed as neither winning nor losing.

However, there is an oversight: the advantage of asymmetric information is not held solely by your opponent. Your foe, for example, isn’t aware if you are planning on his game to be normal, adaptive or some compromise: you are. We can reflect dilemmas to and fro indefinitely: has he presumed your response to be normal or adapted? You don’t know; he does. And so on. It seems reasonable to presume each dilemma (or item of information) is weighted and so in any given exchange summing these weighted advantages should determine who net gains from the context driving these imbalances.

On a more practical note, suppose against his veiled strategy you deliver your standard game; a player unaware of your reference-state is likely to adopt his typical playing style (for you) and so maintain the status quo. Were, though, he to be conscious of your winning or losing state and adopt, in his eyes, an appropriate adaptive-strategy then the chances are your strategy will be less efficient (or sub-optimal for you). However, he too is inappropriately applying a strategy - an adaptive game to a normal one. On this superficial evidence it is not apparent who has the upper-hand. Who gains: who knows? At a guess, whichever strategy is more robust, and, naturally, whoever adapts to their opponent’s true strategy the quickest, and most effectively[2].

Of course individual reasons exist to quit when winning or losing; however, increases in sub-optimal play typical in a change of climate, or indeed, the generation of additional asymmetric information, aren’t necessarily among them.

Next article: Friday Oct 27th: Sklansky's theorem: fundamentally flawed (i).

[1] Error count is a poor metric since errors vary with significance; however, the arguments hold even if we view ‘error-count’ as ‘weighted-error count’ or some measure of optimality.

[2] you'd certainly expect the guy playing less tables to to be advantaged.

strange attractors: money goes to money

2006-08-03T22:20:00.000-07:00

It is often asserted those with deep pockets unfairly blight under-bankrolled tournament players, especially in rebuy events. Some poker pundits dispel this contention, reasoning re-buys are only tickets in a raffle: if you buy more tickets, you win more prizes. If every buy-in is profitable, though, the more the better; that isn’t to infer rebuys should be actively sought but if, oddly, fate deals a heavy dose of misfortune, the talented but well-heeled poker player will have secured many potentially lucrative, all-be-them luckless, opportunities. Hopefully that doesn’t sound too paradoxical.

Take a pro with a dozen rebuys lining his wallet; although, say, the tenth life is seldom called upon, when it is, (s)he’d expect to make good on the investment. It seems reasonable to approximate the worth of this reserve buy-in to be the product of its expected profit when played and the chances of it being needed. So we may determine the expected reward from this venture to be the sum of these calculations for each of the potential buy-ins held. Then, clearly, for two identical winning players, the one holding more cash will, by virtue of summing the equities for all reserve buy-in's, expect more from the tournament.

An illustration. To keep it simple (ignoring add-ons, tournament-dynamics etc) a return-on-investment (ROI) of 15% for each buy-in/rebuy is assumed. Below, some arbitrary percentages for buy-in's used for two players (Rebuy Heavy, Rebuy Light) in a £100 tournament.

Rebuy Heavy: Initial buy-in, 100%; 1st rebuy, 75%; 2nd, 45%; 3rd, 25%; 4th 12%; 5th, 5%. Rebuy Light: Initial buy-in, 100%; 1st rebuy, 75%.

And so the bottom line projections are ...

Expected Tournament ROI (Rebuy Heavy): £15 (1 + 0.75 + 0.45 + 0.25 + 0.12 + 0.05) = £39.3

Expected Tournament ROI (Rebuy Light): £15 (1 + 0.75) = £26.25

Additional benefit might be gained from the deeper rebuys above their ‘in-play’ value: they act as enablers for buy-in's further up the order. In other words, they facilitate a greater freedom of play for early buy-ins during the rebuy period. Word of warning: it's not always to the player’s advantage!

This all appears to add up to an advantage for the cash-rich over the rebuy-rationed; and naturally, that gain is someone's loss. However, there is an upside - the flush few procuring the loss making, negative-EV, tickets. Surely, this redress, albeit perhaps partial, should appease those restricted by funds. Possibly, except it is often claimed those less skilled, the bad gamblers, profit the most from the re-buy structure.

Some suggest judgement is distorted in rebuy events since those perpetually crying 'chips' seem to court the cashier, contend major prizes. Though they appear ahead of the game, in reality, they’re just buying more tickets in the raffle, it is claimed. Others protest, unsympathetically, that complaints emanate only from losing players unwilling to get ‘all introspective’ and face their failings. A gentler argument presents the skewed, risk-averse utility-functions intrinsic to the decision-making of under-bankrolled players as the cause of their demise: they sacrifice too much to secure lower prizes (stack-attractor culprit).

However, citing the usual suspects to explain away this malcontent may just leave us remiss; despite these hackneyed views accounting for much of the angst, there may exist more than a whisper of truth to counter the chorus of objections.

There is one conventional rebuff to the sceptics: deep pockets encourage, and often justify, chasing down thin edges. In a good game the rebuy player can seize risky opportunities without fear of relinquishing future earnings by busting-out or winding-up short-stacked - an almost ever-present concern of the freezeout player in prosperous times.

Consider how an offer of £100 to survive just one more hand would influence an all-in decision of (a) the rebuy (b) the freezeout player. Assume each were confronted with a 60:40 scenario for all their chips. Win, lose or pass the rebuy player will endure, and collect his £100 dividend: the bait changes nothing. Not so the freezeout player; by opting to pass he too receives the payment, but he will only make good on the £100 offer 60% of the time if electing to gamble - a £40 forfeit. For the freezeout player, passing becomes more attractive.

This survival bonus characterises opportunities the game dynamics throw up from time to time. When, for example, the EV is maximised by some requisite or critical holding for a particular decision, and as such, in that instance, excess chips are surplus to requirement [1]. Naturally, that isn’t to say an increase doesn’t create or improve other opportunities, just some are unaffected by further stack-augmentation. There is, still, perhaps another argument.

An average player enters two tournaments: one an unlimited re-buy, the other a freezeout. Despite the re-buy facility, he opts to play both as freezeouts. In the freezeout-proper, the short stack and the large stack are equal imposters - he is drawn to neither and expects to be averagely chipped. However, the re-buy event is different: the chip-average increases as the tournament progresses, not only to attrition - as is the case in freezeouts - but also with every subsequent cry of ‘chips’. As such, our non re-buying hero seems destined to lag further behind the average.

Now, if you’re of the ‘raffle-ticket’ school of thought this is not to his detriment (nor his benefit): he just holds a ticket in a bigger draw. However, it is detrimental if you subscribe to attractor-theory: most of his tournament life will be spent well below chip-average and seldom comfortably above it, indicating an occupation spent largely on resisting the negative attractor, as opposed to riding the positive one. This is in contrast to the freezeout tournament, where it's much tougher for the field to get away.

With this in mind, the benefits of bad decision-making attributed to many willing to take a gamble becomes apparent: through ‘losing chips’ with conventionally poor EV decision-making, including healthy-stack add-ons, an early premium is paid (effectively) to secure a strategic advantage deeper into the tournament, when attractors are very much in-play. At which time, it is contended their earlier chip-outlay is recompensed by seizing opportunities exclusive to big stacks.

The benefits, of course, don’t always outweigh the costs: there are lots of negatives inherently employed in an aggressive re-buy strategy. One might argue that, for some, this successful approach is owed more to luck than judgement: the strategy compliments gamblers’ needs. The mistakes fashioned by these risk-takers are forgiven in part by the structure of the tournament, and, as it progresses the emergence of the influential attractors.

Next Article: Early October.

[1] e.g. when attempting to steal a small pot, how much of an advantage can a bigger stack hold over a big one when it comes to stealing the small stack's blind?

strange attractors: future earnings

2006-07-28T01:06:00.000-07:00

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

2006-07-21T01:45:00.000-07:00

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

2006-07-13T14:41:00.000-07:00

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?

2006-07-06T11:56:00.000-07:00

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),

to show or not to show (iii)

2006-06-29T07:27:00.000-07:00

It is worth noting these mind games are not zero-sum between you and your would-be assailant. In other words your consideration is not simply whether you anticipate extracting benefit over the obvious foe; but if you net gain through future trades with all parties as a consequence of this revelation. An obvious point maybe, but I’ve certainly observed my attention to be rarely distributed, as it often should be, evenly across all players; creating an agenda of any sort seldom helps the cause.

Generally cards are disclosed in a bid to distort perceptions, however, it’s not always about the explicit second-guessing of each other’s strategy: it can alter or reinforce an individual's perspective/behaviour, or, indeed, your own.

In a tough game you’re getting run over and start to feel what you apparently are, a soft touch; an opportunity to advertise an unlikely bluff arises. Often, successfully executing the move is sufficient to restore confidence and give you that much needed shot in the arm; it smacks of self-indulgence to reveal a position just to let-on you’re no patsy or to become so enabled to metaphorically let rip a Lleyton Hewitt ‘Come on!’. However, the resultant change of impression effected in others may well favourably reform how you feel and see yourself, and perhaps enable you to feed off this change of image and liberate your game. Alternatively, as discussed in part ii, an opportunity emerges to mislead your opponents, since they’d likely empathise with your seemingly frustrated exploits.

An opponent enduring little thought to anyone’s play save his own will still be susceptible to timely insights into your game. Despite the lack of mental jousting between you both (showing your hand of course may change this), making public certain holdings could reinforce your adversary's favourable, but wavering position, or alternatively chip away at a stance currently handicapping your progress. Say your opponent is fed-up, in the falconry sense[i], and recently lost only a few chips but material pride, validating his decision through the disclosure of a strong hand may restore him to a stable state; if, of course, that’s desirable – a loose canon could be the order of the day.

Returning to scenario 1 (in part ii), we might view divulging/leaking a hand in this instance to be a futile act, especially against aware opponents, as we’d readily expect them to dismiss such an obvious ploy. However, disregarding a hand can be as demanding to a poker player as it is for a jury to strike off significant but inadmissible evidence or testimony: decision-making is seldom so clean and controlled. Ignoring, let alone utilising, such information is rarely easy especially when our decision-making, as I’m sure mine often does (particularly on-line), on occasions circumvents higher brain functions and races through to the amygdala. Apparently, it is the amygdala responsible for momentarily spooking us when, say, a misplaced glove on the sofa is reckoned for an instant to be a large spider. Decision-making is quite instinctive and unsophisticated at this level since the narrow bandwidth of information flow to it compared to the neo-cortex, leads to fuzzy rough and ready best matches for our circumstance. It’s difficult to believe this decision-maker could be so discriminatory as to rule out what another decision-maker deemed to be disinformation.

Bearing witness to junk hands would, it appears, distort a level of perception over which we have little control and so affect our judgement. Even allowing for instinctive reactions to dissipate it’s tough to gauge a sense for the true likelihood of a thing when we are witnessing (visually) such a biased sample, and where different emotional costs are inevitably attached to contrasting choices and outcomes. In many respects not seeing any hands can extend to us a clearer picture of someone’s game than just a few.

Consider a strategy requiring you to reveal every bluffed hand in a specific situation, say it's one in ten. Now suppose, without this wilful insight, your adversary correctly estimates your chances of bluffing to be as it is, 10%: will his judgement be affected by our controlled release of information? Well, with his rational hat he would now determine we are bluffing at least, not exactly, 1 in 10; without a dispassionate mindset the picture is more confusing. The set of hands shown in this scenario are those volunteered - the bluffs - and those called down, a mixture of bluffs and made hands; therefore the mind experiences and inevitably stores a far higher proportion of bluffs than are actually executed. Additionally, bluffs hurt more, and are consequentially weighted heavily: increasing the distortion. If we never show a bluff, we never afford our foes an opportunity to overweight them; that said, there is trade-off, as we’ll see alluded to in the next paragraph.

One clear disadvantage of an all-showing hand strategy is tactical: rivals would infer much from these exploits, and quickly. In addition, though, that EV makeweight curiosity vanishes. Another not unrelated makeweight, certainty[ii], is also distilled out of the decision-making process: these two bad-call biases would cease to influence your opponent should such an open strategy be adopted. And that’s generally not good: after all, how often would you have passed were you sure to be informed of the bettor’s hole cards, post-hand?

We might be inclined to wonder if exposing a fractional number of hands mitigates these biases and so enhance our opponent’s play. We should also be interested to learn ways , or specifically express hands, which instill and exaggerate these tendencies in others. Inevitably, one suspects it depends on which hands are shown and how we adapt to these biases that determines the effect generated. Certainly if you display superior hands, and so make passing more comfortable, then the call-candidate will be less in need of satisfying these desires. However, regularly exposing him to bluffs could easily foster[iii] call-biases, as well as taxing another emotion, pride. Of course we'd loath escalating these biases (or indeed more aggressive actions) in players who are typically under-calling, since as a consequence we may encourage them, albeit for the wrong reasons, to do the right things.

Although it isn't explicitly evident that exposing a few well chosen hands will be beneficial, it certainly should be deemed false to assume showing some must negatively impact the bottom line since displaying many in all likelihood, would. It does, though, seem a reasonable proposition to expect a good exponent of the game to gain from, as many of the top professionals appear to, the release of selective information. I’m certainly no prolific hand shower; partly because it is somewhat discourteous, especially live, but often I simply don’t relish the distraction while multi-tabling and am far from convinced I’ll net gain. Trying to assimilate many variables is tough and often leads to poor judgement since we invariably overweight the importance of fresh factors. Nevertheless, it’s an interesting topic which merits greater discussion particularly, perhaps, because it doesn’t lend itself to the rigours of conventional analysis.

Next Article: 6th July

[i] “…one which has 'fed up' wants merely to sit still and digest its meal (ie it is totally unresponsive)”

[ii] certainty: the need for (whether you would’ve won or lost.)

[iii] by regularly showing bluffs (or any hands) you are it seems more likely to placate these needs; however, it could be contended the biases are generated, sustained and increased by showing bluffs in the first place.

to show or not to show (ii)

2006-06-15T09:37:00.000-07:00

Part (i) suggested never volunteering hole cards invariably beats the bottom line impact of constantly exposing them. Clearly, to transit from one state to the other requires us to escalate the flow of information; to show more hands. The sensible but simplistic deduction assumes the submission of any information to be detrimental, even just a little: so don’t give any.

Sure enough, correctly adding numbers on a sudoko or clues to a crossword invariably facilitates problem-solving; however, not always, though, the elicitation of a patient’s symptoms or, indeed, body language from a poker player. Sometimes, less is more; selective information misleads, too much invariably confuses.

At the table, and your adversary has you exactly where you are: he commits to you just the right amount of respect. Good on him: but what are you to do? Well, you have two options: either you attempt to alter his perception of your game, or you, stealthily, render this perception false. However, since game-adjustment isn’t always desirable, straightforward or globally good, sometimes we’re left with only the former. One way of achieving this is through disinformation or ‘the deliberate leakage of misleading information ’. Time, perhaps, for psyops.

The Information Exchange:

Voluntary disclosure, it is widely held, is to issue free information: however, it isn't free, information is exchanged. Once committed to the trade each player is required to exploit his or her angle with greater effectiveness.

There are instances where disinformation is delivered surreptitiously. In a ring game with unfamiliar players you raise very loosely with Q-10o:

Scenario1: Everyone passes. You show Q-10o.

Scenario 2: The Big Blind defends. The flop comes 9-K-J. You lead, your opponent passes. Once again, you show Q-10o.

Scenario3: The button cold calls. The A-rag flop is checked. The turn brings a 9, you secure the pot and then lay bare the bluff.

Opponents and the viewing gallery may discern only a very loose under-the-gun raise from all scenarios. What price, though, for this strategic insight?

The message sent is the one received in Scenario 1; the price you’ve charged is the intelligence that you know he knows. Evidently, you openly desired your adversaries to hold this knowledge, but they’d surely readily recognise your thinly veiled attempts to manufacture an image; after all, if Q-10o was, for you, a typical raising hand, why make a point of displaying it?

The motives for relinquishing the contents of your hand in Scenario 2 are considerably less obvious: are you being polite, showing off a great hand, saying ‘when I bet, I’ve got it’, or still ‘hey I’ve just raised utg with Q-10o’? Which seeds are you sowing?

Although, to him, your intentions are far from clear, neither to you will be his interpretation, or best guess, of them. Despite having sold him a ‘Q-10o utg raise’, quite what you’ve charged may not be apparent. However, that may not matter much, so long as you’re classified a loose early raiser.

In the third case, the ruse should bear greater success: the focus is on the turn where you appear to brag a bluff. So you know that he knows you’ve raised early with Q-10o, but he thinks you know that he knows you’ve just bluffed the turn!

Compare these events to an occasion where your adversary’s buddy, standing behind you, relays this information afterwards, without your knowledge. In this case you may not mind, but what price now? Nothing: unless we’re measuring ethics.

There are, naturally, subtler illustrations than those mentioned above*. For instance, say you commit an untypical (for you) but prevalent play on the flop leading to a successful bluff or completion of a hand, albeit uncalled, on a future street. Benevolent or vain your intentions to indulge in poker’s ‘show & tell’ appear honest, but in so doing you may just wrong-foot the observant opponent further down the line.

Next article: part (iii) 29th June – a consequence of my inability to finish part (ii) in time.

* Also, in big bet games such as no-limit there is an added dimension over many decisions in limit: quantity. So the potential misdirection implicit in hand-showing has extra depth in no-limit e.g. an opponent shows you a bluff on the river, but is trying to sell you an over-bet on the turn.

to show or not to show (i)

2006-05-25T10:04:00.000-07:00

Poker discussion seldom extends to the very last decision of a hand: whether to muck with or without showing. Hardly surprising; after all, it’s tough to analyse, a touch bland and after the fact, somewhat.

‘I’m not giving any more information out than I have to, if they want it they’ll have to pay for it’ is a not unpopular, if rather old school perspective, on the subject. Certainly the assumption of monotonic behaviour discussed previously could explain why, at least in part, this view is held.

The two extreme strategies of full and non-disclosure of hands will inevitably impact on players’ performances in different ways. And, forced to choose between them, the latter option would be preferred over the former by sentient poker players. Therefore one would deduce moving from a covert to an overt strategy to initiate a loss in EV, in value. A graphical representation of this statement would look like this:

What of the points between these two end-states; what would the curve joining them look like? The natural tendency is to approximate to a linear relationship, a straight line, as follows:

Even if the relationship isn’t linear, or strictly monotonically decreasing, it seems likely the next step will be a southerly one. Certainly, with little more to guide you than a couple of stones and a pair of crossed sticks (fig1), it seems a reasonable assumption. Although likely, it isn’t certain; what if it looks something like this?

Everyone’s plot would be different, of course, heavily dependent on which hands were shown, not simply how many. Without evidence, though, it is pure conjecture. Hopefully, in the next article, I will flesh out some fairly obvious reasons to support this view.

Next Article: Part (ii) June 15th

nb: the graphs were edited rather than constructed, so nothing should be read into where the axis intersect - in other words, that showing no hands has zero value and showing all has negative, the absolute values are pretty arbitrary, but not relatively.

look after the pennies…

2006-05-04T08:53:00.000-07:00

A frequent criticism levelled to the reluctant voter is: ‘ if we all took that attitude anyone could get into power…’ (and for want of a nail the kingdom is lost). ‘A no-vote is better than an uneducated vote’ would be a reasoned retort, or still, abstention precipitates improvement. Indeed, the disinterest in politics over recent years manifested in low turnouts has, apparently, done much to change the media coverage of politics, inevitably feeding back into the political system itself.

A common approach adopted when unable to rationalise the value or justification in a change, action or event is to consider the impact of a large sample or an extreme case, often justifiably so. At times we intuitively reason if a thing is detrimental at an extremity, then a fractional dosage or single occurrence is proportionately damaging. Of course we know this isn’t universally true, after all, our stomach is lined with acid. Generally, though, in life it seems we reason, if not act this way, until experience, logic or common sense dictates otherwise. Such a bias probably has a name; if not, it should certainly have one.

In maths there are classes of functions defined as monotonically increasing, monotonically decreasing. If a function is monotonically increasing (don’t run away) between two points, say, a and b, then as we inch along from one point to the other the function, or curve, always increases. Smoking as a function of physiological health is an example of such a decreasing function. Views citing indifference at either extreme of activity may well exist, but arguments purporting some amount of smoking to be good, or indeed any increase to be less damaging are non-existent (not, though, strictly satisfying the monotonic criteria). The same though is apparently not true of Bordeaux’s finest; while over indulgence is naturally to be avoided, research suggests a glass or two a week is beneficial. Perhaps gambling’s psychological effect traces a similar path to red wine’s physiological one. Certainly it is simplistic to reason a monotonic impact on those respective health-states since excess indulgence afflicts.

This, say, monotonicity bias, plays a supporting role in poker discussions from time to time too, both explicitly and implicitly. The first one I’ll mention is an implicit occurrence on the subject of whether or not to reveal hole cards.

Next Article: To show, or not to show. 25th May