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Morton's Theorem

This article originally appeared in Card Player Magazine.

One of the enduring pleasures of owning a computer and connecting to the Internet is the ease of finding a community of people for just about any interest you might have - including poker. If you're not wired up and dialed in, you may not know about poker's primary community. It's a newsgroup called rec.gambling.poker. Newsgroups are locations in cyberspace where you can read, post, and respond to messages about issues of mutual interest and concern.

Some messages are banal, others boring, and a few are simply news items about events of interest. Recantations of poker road trips abound, as do debates that occasionally escalate into flame wars. But real gems sometimes work their way through poker's newsgroup, and when that happens, the level of dialogue is elevated to a higher order. When all has been said and done, those who took the time to follow the debate usually learn something that measurably improves their game.

Morton's Theorem is one of those gems. It was posted to the newsgroup by Andy Morton, a Los Angeles-based poker player who concluded that David Sklansky's Fundamental Theorem of Poker does not usually apply in multihanded situations. Upon reading Morton's Theorem for the first time, Mike Caro was quoted as saying: "It is simply one of the most interesting and unexpectedly thoughtful pieces about poker I've read recently."

Sklansky's Fundamental Theorem of Poker postulates that whenever your opponents make mistakes, you gain, and when they play correctly, you lose. "In Hold'em," said Morton, "I always assumed that if all of those calling stations want to chase with five-out draws to make trips or two pair when I flop top pair with the best kicker - and the pot odds don't warrant it - that sounds like a good situation to me."

At first, Morton thought that players who complained about getting drawn out on simply did not understand the increased variance of playing in loose games with lots of callers. In one sense, however, Morton realized that these players were right: The large number of calling stations, combined with a raise or two early in a hand, created pots that were very large, relative to the bet size.

"This," he wrote, "has the effect of reducing the magnitude of errors made by individual callers at each individual decision point. A pot can get so big that callers ought to chase." Morton coined a name for this behavior on the part of the fish who called too often. He referred to it as schooling. "Still," he thought, "tight-aggressive players who enter multiway pots with hands like top pair and best kicker should have the best of it against each of the long-shot draws - like second pair and random kicker."

Morton observed that the schooling phenomenon increased the variance of the player who flops top pair with A-K, and he theorized that it also increased his expectation in the long run - particularly when compared to games in which opponents correctly fold their weak draws. What he found, however, was that when betting the best hand against two or more opponents with more cards to come, you make more money when opponents fold, "... even if they are folding correctly, and would be making a mistake to call your bet."

"You want opponents to fold correctly," he wrote, "because their mistaken chasing will cost money in the long run." He offered the following example: "Suppose that you hold Ad Kc and the flop is Ks 9h 3h, giving you top pair with the best kicker. When the betting on the flop is complete, you have two opponents remaining, one of whom you know has the nut-flush draw (Ah 10h, for example - with nine outs), and you believe that the other holds second pair, random kicker (Qc 9c -- four outs). All of the remaining cards in the deck are your outs. The turn card is a blank. When you bet the turn, the player holding the flush draw is sure to call and has correct pot odds to do so. Once the flush draw calls, the player holding Qc 9c must decide whether to call or fold."

The mathematics of Morton's Theorem shows that there is a range of pot sizes (in this particular case, between 6.25 and 8.5 big bets) when it is correct for the player holding second pair to fold, and you make more money when he plays correctly and folds than you will when he chases.

This apparent inconsistency with the Fundamental Theorem of Poker stems from the fact that the pot is not heads up, but multiway. When the pot size is in the middle region, a player holding second pair is paying too high a price for his weak draw, but you are no longer the sole beneficiary of his mistake. The player with the flush draw will take some of that money whenever he hits his hand.

According to Morton, situations like this come up all the time in Hold'em, both on the flop and on the turn. And it's in this middle range of pot sizes that you want some of your opponents to fold correctly. This explains the standard poker strategy of thinning the field as much as possible when you have the best hand. Even players with incorrect draws cost you money when they call your bets, because part of their calls end up in the stacks of other players drawing against you.

To the degree that Morton's theory applies, there is a myriad of strategic inferences that can be drawn from it, not the least of which are the increased value of suited cards, and the need to raise - or reraise - in order to thin the field. Put another way, selectively aggressive play is worth its weight in gold in multiway schooling pots, all of those eager fish notwithstanding, their fingers firmly on their chips, aching to call, and all you can do now is hope that they will fold.

Life, like cards, occasionally deals some unbelievably bad beats, and on July 17, 1998 at 6:45 a.m., high in the Colorado Rockies, Andy Morton was killed when his motorcycle collided head-on with a pickup truck that pulled out into oncoming traffic to pass a slow-moving car. Morton, who loved motorcycles as much as poker, recently earned his Ph.D. in biochemistry, although he preferred earning his living playing Hold'em to holding a post in academia.

I regularly played poker with Morton. He was among the most decent human beings I've come in contact with, inside or outside of card casinos. He had a zest for life, and was inquisitive about almost everything. His energy, intelligence, and personal warmth affected everyone he met. He was 34 years old, and had his entire life ahead of him.

Morton was a regular contributor to rec.gambling.poker, and always worked diligently to raise the level of debate within the newsgroup. He bequeathed poker players everywhere one legacy: Morton's Theorem. This is my personal tribute to him: to introduce his work to a wider audience than the newsgroup. The world, and the poker table, was a better place with Andy Morton in it. It's much sadder without him.

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