sklansky's theorem: fundamentally flawed (ii)

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)