If you've played low-limit poker at all, you've probably
heard someone complain, “I can't beat this game—the
players are all too bad!” Maybe you've even
said something like this yourself. The idea is that,
since a) the bad player will play almost any two cards,
and b) the bad player will stay in the hand longer than
is reasonable, it is hard to gauge when your hand is beat,
as you will often be beaten in the most unimaginable ways.
Typically,
this complaint is written off as being a myth, a short-term
statistical illusion. I mean, how could a bad player
really be tougher to beat than a good player?
It just doesn't make any sense. Either the complainer
isn't as good as he thinks he is, or he's just seen some
short run bad luck, right? This is how most people
think, and the typical response one hears is something along
the lines of “Hey, you want that guy in your
game!” or “You gotta love it!”
Something
about this easy dismissal didn't sit right with me.
Of course, it made sense—after all, conventional wisdom
always does. But one thing I've learned over the years
is that, if a bunch of gamblers who have put in thousands
and thousands of hours all believe in the same illogical
superstition, it probably warrants further investigation.
There may very well be some mathematical principle at work
that causes this pattern to recur so frequently. And
indeed, there is.
Let's
begin by defining what “bad players” and “good
players” are. We will define “bad players”
as being those players who will play any hand they want
to play, even if the pot has been raised. We will
define “good players” as those who will only
play a raising hand when the pot has already been raised—but
will never fold that hand before the flop. (This is
in contrast to “great players,” who will frequently
fold a raising hand if the raiser is someone who deserves
respect, but may reraise with the exact same hand if the
raiser has low raising standards. I think there are
very few people who believe that bad players are harder
to beat than great players; as a result, they are not a
part of this study.)
Okay,
so how can it be possible that someone who plays all kinds
of bad hands can be harder to beat than someone
who only plays good hands? Let's do a little math.
Let's say there is a raise from middle position—the
type of hand a good player might call with would be something
like AQ or AJ offsuit, or maybe even KQ suited. A
bad player, on the other hand, would probably call with
hands like JT offsuit, pocket 5s, or even 87 suited (and
a ton of other hands too—these are just a few examples).
Let's compare how these hands do versus a common raising
hand—AK offsuit.
Of
the three choices given, which of the good player's hands
would you rather have? Conventional wisdom would say
AQ offsuit. But before the flop, AQ will beat AK just
26.0% of the time, while AJ will beat AK 26.8% of the time.
That's right—if you are up against AK, AJ is slightly
better than AQ. And KQ suited—a hand with which
some good players wouldn't even call a raise—is the
best of them all, beating AK 30.3% of the time.
Okay,
what about the bad hands, you ask? Most people know
a small pair is a slight favorite to two overcards, and
indeed pocket 5s beats AK 55.0% of the time. But JT
offsuit beats AK 37.4% of the time—much better than
any of the good player's calling hands—and 87 suited
beats AK a whopping 41.8% of the time. It turns out
it's actually much more likely that one of these “bad”
hands will take down AK than one of the “good”
hands.
Of
course, there is more to the hand than calling before the
flop, but that is often where the “good players”
place the blame. “How could you call a raise
with a hand like that?” they ask. Hmm, probably
because they do better with hands like that.
But let's look at what can happen after the flop.
Let's
look at a good flop for AJ: A 7 4 rainbow. Chances
are, if a good player hits a flop like this with AJ, he's
stuck in the hand until the end. But which hand do
you think has a better chance of catching up with AK—AJ
or 87 suited? It should be easy to see that 87 suited
is in a much better position here: this player has
three 8s and two 7s to improve his hand, as well as the
backdoor straight and flush draws. By contrast, the
player with AJ only has three cards to improve his hand.
Overall, the 87 suited will catch AK 24.3% of the time,
while AJ will only win 12.7% of the time.
What
about a hand like pocket 5s? Let's compare it to AJ
on the same flop—A 7 4 rainbow. Only a lunatic
would call a bet on the flop with a hand like pocket 5s,
right? Actually, pocket 5s will beat AK on this flop
13.6% of the time, compared to 12.7% for AJ. The backdoor
straight draw, combined with the two other 5s, turns out
to be a better draw than AJ's three other jacks. Yet,
if pocket 5s misses on the turn, he likely dumps the hand,
saving the bets on the turn and river, but if AJ misses
on the turn, it doesn't matter—he's calling bets on
the turn and river no matter what. The worse draw
loses the maximum, while the better draw saves money when
they miss.
So
indeed, there are times when a bad player is tougher to
beat than a good player. But doesn't the fact that
he is called a “bad player” imply that he's
going to lose more money than a “good player,”
since that's what “bad” and “good”
typically mean in a poker context? And if so, how
does that reconcile with what we've just seen? Yes,
it does mean that the bad player will lose the most money.
Figuring out how to beat the bad player, though, requires
one to approach the problem from a different angle.
Next time, we'll do just that.
Martell
can be reached at martell@babblog.com.
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