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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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