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A
Little Game Theory
Talk
to poker theorists long enough and the concept of game theory
is bound to pop up. And it's not really a new idea either.
Nesmith C. Ankeny even wrote an entire book about it in 1981
called Poker Strategy: Winning With Game Theory. David
Sklansky also explained game theory in his seminal work, Winning
Poker, two years hence.
Game
theory, despite its name, isn't about Monopoly, Trivial Pursuit,
or other leisure-time diversions. It's a branch of mathematics,
dealing with decision-making, that has proven useful in fields
as diverse as economics, political science, operations research,
military science, and poker -- where the idea is to optimize
a decision, rather than to maximize or minimize any one of
a multitude of potential outcomes in certain situations.
That
sounds pretty theoretical, so here's a practical example.
After all the cards have been dealt, let's assume that just
you and one other player are contesting the pot. You don't
know a thing about your opponent's playing style. You've never
played against him before, and haven't picked up even the
slightest inkling of a tell. Just for the heck of it, pretend
you're playing against the invisible man. You don't have much
of a hand. In fact, you have nothing more than a busted flush,
and the only way you can win is by bluffing successfully.
We're also going to assume your opponent knows with absolute
certainty that you were on a flush draw. Although he cannot
beat a flush, his hand is strong enough to beat any busted
flush.
Here's
where game theory comes into play. Suppose you bet every flush
draw - whether you made it or not. What do you think would
happen? If your opponent was very cautious, he'd throw his
hand away most of the time and you'd win the pot whenever
he did. But if he were a decent player, he'd begin to suspect
you of stealing and call with increasing frequency. In fact,
if he knew you bluffed every time you failed to make your
hand, he'd call each and every time you come out betting.
Now the situation has reversed itself. Rather than winning
each time you came out betting, you'd lose most of the time.
While you'd win with your legitimate hands, each time you
tried to bluff, your opponent's call would capture the pot.
Since you'll miss those flush draws more often than not, compulsive
bluffing would cost you quite a bit of money.
Suppose
you took the opposite tack and never bluffed, but bet only
when you completed your draw. Just as he did in the case when
you bluffed too often, your opponents would soon get wise
to your habits. Once he gloms on to the fact that you never
bluff, he would adjust his strategy accordingly. Now he'll
fold when you bet, but he'll also show down the best had and
win the pot whenever you check.
Do
you see what's happening here? Not has your opponent wised
up to your pattern of always bluffing each time you had a
busted drawing hand, but more importantly, your results are
not a function of your actions. Instead, the results you achieve
are wholly dependent on the choices your opponent made. You
are no longer in charge, and that's a bad thing. Your playing
strategy has allowed the locus of control to pass to your
opponent, who, by virtue of his decisions about whether to
call or fold, is the one who determines how much you win or
lose.
It's
pretty clear from all of this that you can't be a one-dimensional
player -- always bluffing or never bluffing. And you didn't
have to know about an arcane branch of mathematics called
game theory to tell you that. Even if you bluffed once in
a blue moon, or refrained from bluffing only every once in
a while, you'd create opportunities for your opponent to make
errors by forcing him to decide whether or not you held a
legitimate hand. When you always bluffed or never bluffed,
your opponent was relieved of the responsibility for making
a decision. He knew that you bluffed all the time, or perhaps
realized you never bluffed at all, and either way it made
no difference at all. His strategy was easy and obvious, and
allowed your opponent to maximize his winnings as a result.
But
when you veered away from these polar extremes, your opponent
was put to the test by having to answer this question: Do
you or don't you have the goods? And you know what, if you
give your opponents a chance to make a mistake, he will make
some. He will, I will, and every player who's ever lived will
err in judgment. There's not a poker player alive who makes
the right decision all the time. Poker, after all, is a game
of incomplete information, and that means wrong decisions
will be made.
Game
theory gives one the wherewithal to optimize his play. When
you bluff properly -- not too often and not too infrequently
-- it makes no difference how your opponent responds. Game
theory allows you to control the outcome of your actions and
optimize your results.
Here's
how to bluff using game theory. Step one is to make sure the
odds against your bluff are equal to the odds your opponent
is getting from the pot. Confusing? Not really. Suppose by
betting on the river, you create a situation where your opponent
will be getting 4:1 from the pot. That's easy to imagine.
The pot contains $300, and by wagering $100, your opponent
stands to win $400 for his $100 wager.
Now,
let's say any one of eight available cards would have given
you the winning hand. If you bluff whenever two predetermined
cards come up in addition to the eight you need, you are bluffing
at a frequency that precludes your opponent from taking advantage
of your bluffing proclivities --regardless of what he chooses
to do.
How
easy is that to pull off? You can trigger your bluff versus
not-bluff decision by randomizing it with cards. Suppose you
are looking for either a seven or a queen to complete your
hand. Any one of those eight will do; it doesn't matter which
one pops out of the deck. Now suppose you tell yourself that
you will come out bluffing if your last card is a red deuce
instead of the hoped-for seven or queen. By giving yourself
two bluffing cards as well as eight winning cards, in the
4-to-1 ratio of winning cards to your opponent's pot odds
you've optimized your decision-making.
But
there is a rub. It's tough to make these kinds of calculations
in the heat of battle. Most players don't do this sort of
thing; trust me. But you can work out common drawing situations
in advance, just like we did here. And you don't even have
to be absolutely precise. Oh, sure, it's nice to be right
on the money, mathematically speaking. But as long as you
realize that winning poker requires making mistakes at the
polar extremes -- neither habitually bluffing nor always checking;
nor always calling your opponent's wager or folding every
time he bets -- to avoid making more costly mistakes in the
middle, you're on the right track.
When
all is said and done, you probably won't use game theory all
that often in the heat of battle. And the more you play and
the better you become at reading your opponents -- by putting
him on a hand or picking up tells -- the less you'll have
to rely on game theory. After all, you usually won't be playing
against the Invisible Man. Even if you play on line, where
your opponents actually are invisible, you can discover tells
about opponents and put them on hands based on their proclivities
for checking versus betting, and calling versus folding.
While
game theory is pretty cool stuff, and we all owe a debt of
gratitude to Ankeny and Sklansky for presenting it to us so
cogently two decades ago, its greatest value probably lies
in how well it serves as a parameter that can assist us in
learning how, as well as how often, to bluff versus bet for
value, or to fold in the face of a bet versus trying to pick
that bluff off.
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