strange attractors: money goes to money

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?