structually sound?

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

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

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

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

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

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

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

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

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

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

Next Article: 13th July

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

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

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

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