Thursday, July 22, 2004

"Roll em For TJ!"
HDouble to me at this years WSOP, at the craps table

Uber post pending. Hope this lil essay holds you over.

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Confessions of a math guy, part 2

In part 1, I focused on exploitive play, how it relates to optimal play, and
why non-math folks can apparently do it fairly well. In this part, I will
discuss features of optimal play in more detail.

Simply put, you are playing optimally when your opponent can't (correctly)
figure out a way to exploit you. For example, you aren't calling too much
or too little, so he can't decide whether it is profitable to try to bluff
you or not. I need to emphasize that the opponent may THINK that he can
exploit you (perhaps he thinks you call too often and therefore doesn't
bluff), but in fact you are playing in such a way that despite what he
thinks, he is not exploiting you.

[begin digression]
In my opinion, there is far too much criticism of optimal play in poker
because of the lack of a single equilibrium solution for multi-handed play.
Bringing this up is usually a straw man argument to dismiss entirely the
usefulness of studying game theory for use in poker. There are three
reasons I think this is overdone. The first is that even many-handed poker
hands often dwindle to heads-up before the showdown, and though the hand
sets of the players in the early multi-player rounds of betting are
affected, a decent approximation of a solution for late round heads up play
can be found. The second reason is that equilibria even for multi-way games
can be found when one makes simplifying assumptions about the opponents (in
particular, how they implicitly or explicitly collude with each other or
with you). And finally, when it comes to the practical use of game theory,
the margin for error for an individual's implementation is probably
significantly larger than the error in the approximations derived.
[end digression]

Of course, in playing in this optimal fashion, you eschew the option of
exploiting the play of your opponent as well. Many people are of the
mistaken opinion that this means optimal play is purely defensive, and is
therefore not profitable as a general strategy. I think there are several
reasons for this belief. The three that come to mind first are:

1. Someone who knows the basics of optimal play in game theory has seen many
examples where an opponent gets the same return no matter what he does when
you play optimally. For example, if you randomize between the three choices
in roshambo (rock/paper/scissors), your opponent can never beat you in the
long run - but you can't beat him either.
2. There is a macho "outwit and outplay" image associated with poker that
makes it unthinkable that some geek could come along and solve it like it
was tic-tac-toe. It's eat or be eaten, and if you resort to game theory you
are avoiding getting eaten, so it can't possibly allow you to eat.
3. Just as everyone seems to believe they are better-than-average drivers,
poker players all like to think they are winning players. When you believe
this, and you (and every other winning player you know) have been playing
exploitively, you refuse to believe that there is some "play by numbers"
method out there that works at least as well and probably much better than
what you have been doing (and which will without a doubt beat you if used
against you).

Well it turns out that poker is quite a bit more complicated than roshambo,
and there exist very, very common situations where if one player plays
optimally the opponent must do so as well, or he will play with -ev. All
the tricky, instinctive, fancy plays in the world won't help. For a more
detailed discussion of this, see my RGP experiment from a few years ago:


All of this naturally leads to the question, "So who will win more money,
someone who plays exploitively or someone who plays optimally?" This is a
very hard question to answer, because perfect optimal play is so difficult
to work out that it is impossible to achieve. And of course there are many
degrees of exploitive play as well. But here is my feeling...

As players get better at the game, their "basic strategy", if you will,
gravitates towards optimal. Therefore the tougher the game gets, the less
profitable exploitive play becomes. But in very weak games, optimal play
does not get all that it can from opponents' mistakes. The irony here is
that the people that "graduated" to the toughest games did so by being
better exploiters than their opponents in the softer (presumably
lower-limit) games, so they are widely of the opinion that you can't
possibly win in the big game unless you have that special exploitive knack,
when in fact an optimal player who doesn't try to do anything in an
exploitive manner is going to actually fare the best.

I have yet to explain how knowledge of this subject can be used in real
life, and I will save that for part 3. But what I will say is that the math
that has to be performed (by hand or by computer) is not trivial - it is
much more than calculating outs and pot odds like so many people think is
the extent of mathematics in poker.

Tom Weideman

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