sklansky's theorem: fundamentally flawed (iii)

‘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.