Game Theory Optimal vs. Exploitative Play in Poker. Poker is a game of probability, or so most people think. Given a hand, there is a certain probability of your opponent having a better hand. As a result, there is always an optimal decision to make that maximizes expected value. This style of play is called GTO (game theory optimal). Play Optimal Poker shatters the myth that game theory is only for elite poker players. Renowned poker pro and coach Andrew Brokos takes you step-by-step through the fundamentals, explaining core game theory principles and how to apply them in real poker situations.
As new depths of poker strategy continued to be discovered, Texas holdem
tables sound more like science labs than the scene of a simple card game.
Thinking players in today's game casually toss out references to balancing or
merging their hand ranges, applying an 'exploitative' approach to take advantage
of 'suboptimal' strategies, and of course, integrating 'game theory optimal'
plays into their arsenal.
Perhaps more than any other advanced strategy concept, the notion of game
theory optimal play – better known as GTO – has seeped into the mainstream poker
consciousness. Players of every skill level have at least familiarized
themselves with the idea of making their own game GTO, but as with any other
ubiquitous term, the exact definition of poker's newest buzzword differs
depending on who you ask.
The steady advancement in the way players tackle Texas holdem problems is
only natural, as the Poker Boom of 2003 to 2006 prompted millions of thoughtful,
intelligent, and analytical individuals to take their talents from the classroom
to the card room. The merits of that choice are debatable on the individual
level, but what can't be disputed is how the new generation of poker students
ultimately became masters of the field.
Eschewing traditional advice about 'playing the man, not the cards,' young
poker players today focus their minds on the mathematical underpinnings of Texas
holdem gameplay. By using hand distribution to equity calculators like the Poker
Stove product, basing every possible decision on the all important variable
known as expected value EV, and scaling back the standard opening bet from three
times the big blind, modern Texas holdem experts have fundamentally altered the
game's very foundation.
For beginners just now entering the world of Texas holdem, or even old hands
who simply struggled to keep up with the game's accelerating advancement,
hearing smart and savvy opponents reference ideas that sound more like calculus
homework than a card game can be quite intimidating. It's hard enough figuring
out what to do when you get four bet holding pocket jacks, so the thought of
learning about intricate game theory constructs and the higher level reasoning
behind GTO plays can be daunting to say the least.
With this page, we don't purport to be PhD holders or even Texas holdem
experts, but rather recreational players like yourself who simply wanted to
learn more about game theory as it applies to poker. In keeping with the
instructional theme, we'll offer a syllabus of sorts, starting out with a basic
glossary of the key terms and concepts you'll hear repeatedly during any
discussion on GTO play. Next up you'll find a section detailing several common
examples, written from the perspective of a poker player, that help to
illustrate the technical terms described earlier. From there you'll find a list
of applicable resources – written or developed by successful high stakes
professional players and game theory experts – through which you can pursue an
advanced education.
Glossary of Game Theory Terms
Before we move on to the descriptions, it's important to discuss what the
concept of game theory really means.
According to Roger B. Myerson, whose introductory textbook titled 'Game
Theory: Analysis of Conflict' was published by the Harvard University Press in
1991, game theory can be defined as 'the study of mathematical models of
conflict and cooperation between intelligent rational decision makers.'
As you can see, this definition doesn't mention anything at all about poker
or Texas holdem. That's because game theory is applicable to any game or contest
which involves decision making on the part of players combined with access to
partial information. Additionally, the ideas put forth by game theory experts
have also been co opted for use by economists, political scientists, biologists,
and several other fields of study. Thus, while the study of game theory is
predicated on the various rules and procedures used to govern classic games like
Texas holdem, the ideas that emerge from game theory investigation are widely
applicable across a diverse range of subjects.
Although game theory wasn't codified as a field of study until the 1920s,
evidence of GTO approaches to basic card games can be found dating back to the
early 1700s.
In 1711, Charles Waldegrave wrote a letter to his brother outlining a
'minimax mixed strategy' to the simply two player card game known as Le Her.
In 1913 a German mathematician named Ernst Zermelo developed 'Zermelo's
Theorem,' which states that
'In any finite two person game of perfect
information in which the players move alternatingly, and in which chance does
not affect the decision making process, if the game cannot end in a draw, then
one of the two players must have a winning strategy.'
As you might suspect, this
long passage was used by Zermelo to describe chess, which he successfully proved
to be a 'strictly determined' game from a strategic sense.
Throughout the 20th century, mathematicians and logicians like John von
Neumann, Oskar Morgenstern, Merrill M. Flood, Melvin Dresher, and John Nash each
contributed fundamental theories and postulations to the field of game theory
study.
For poker players with an educational background in advanced mathematics – of
whom there seemed to be an endless supply during the Poker Boom – learning the
lingo of game theory and applying it to their favorite game proved to be a
highly beneficial proposition. These players were able to expand their lines of
thinking beyond the most basic constructs – what do I have or need, what does my
opponent have or need, etc. – to turn a seemingly simple poker hand into an
exercise in statistical modelling and probability based prediction.
But for the rest of us, the laymen at the table who haven't memorized reams
of mathematical formulas, diving deeper into the subject of game theory study
can present a firm barrier. The average person can only take so many
abbreviations and hypotheticals before their head begins to ache, so breaking
things down to their basic meaning is a helpful way to begin.
Take a look below for a comprehensive glossary of essential terms and
concepts used within the world of game theory:
Any strategy that offers a reduced expected value EV,
compared to GTO strategy, when playing against an exploitive strategy. Any non
game theory optimal GTO strategy is, by definition, an exploitable strategy.
Any strategy that offers an increased expected value EV
than a game theory optimal GTO strategy, when playing against any particular
strategy. Any non GTO strategy that counters an exploitable strategy better than
a strictly GTO approach is, by definition, and exploitive strategy.
The strategy that offers the highest possible
expected value EV when an opponent always applies an optimal counter strategy.
The classic GTO strategy example concerns the zero sum hand game known as 'Rock,
Paper, Scissors.' In this game, the GTO approach involves selecting randomly
between rock, paper, and scissors while using an equal distribution. This
strategy provides the highest level of EV, at 0.50 percent equity, against any
opponent strategy that consists of all rock, all paper, or all scissors.
The strategy that offers the highest possible
expected value EV against any opponent strategy. Returning to the Rock, Paper,
Scissors example, in a game where you know your opponent's strategy was to throw
rock on every game, the optimal exploitive strategy would be to counter with
paper every game – because this would create an EV of 100 percent. And should
your opponent modulate to a strategy based on using rock on 50 percent of games,
paper on 25 percent, and scissors on the other 25 percent, the optimal
exploitative strategy would also be to throw paper on every game – because you'd
create a scenario in which you'd win or tie on 75 percent of games, while losing
only 25 percent of the time.
Any strategy that offers a lower expected value EV than
the optimal exploitive strategy. Back to that Rock, Paper, Scissors game, where
your opponent's strategy is to throw rock on each game, you could opt for a 50
percent paper and 50 percent rock blend of moves. And while this would still be
a winning strategy, because you'd only win or tie, it's performance can't match
that of the optimal exploitative strategy throwing paper every time – making it
a suboptimal strategy at best.
Game Types
Cooperative / Non Cooperative GameIn a cooperative game, players are
permitted, encouraged, or even required to form binding agreements with fellow
players. A game like Monopoly, in which players can negotiate the price of a
mortgage on property deeds among other agreements, is a classic cooperative
game.
A non cooperative game, on the other hand, forbids players from making
similar arrangements among themselves. Technically speaking, poker variants like
Texas holdem are non cooperative games, because the rules preclude collusion and
other forms of explicit cooperation. Even so, as you'll learn in the next
section featuring examples of Texas holdem game theory in action, many
situations in the game compel players to form implicit agreements to achieve a
certain effect stalling on the bubble, big stacks avoiding one another with
pending pay jumps, etc.
From a practical standpoint, any game involving human
players must be a finitely long game – or one that has a fixed endpoint. Whether
that means attaining a certain score, satisfying a series of conditions, or
otherwise defeating your opponent, a finitely long game has a beginning – and a
definitive end. And even in a game designed to stretch on into perpetuity, the
limits of human endurance, and indeed lifespan, prevent it from being truly
infinite.
But game theorists have no such limitations on their work, and in the process
of investigating mathematical proofs, they've created the concept of infinitely
long games that are never forced to end. These games are devised, in part, to
study the relative strengths and weaknesses of dueling strategies which adapt
based on one another's actions.
For poker players, every cash game or tournament session has a start and an
end. But as any experienced poker pro knows quite well, judging the results of
any particular session provides an inconsistent appraisal, and the truth is best
discovered by examining results over the long run. That long run can encompass
years, decades, or even a player's entire lifetime on the felt – making Texas
holdem and other poker formats an infinitely long game in spirit.
For game theorists, a 'meta game' means something entirely
different, but as a poker player, you'll hear this expression used largely to
describe the multitude of external factors that conspire to influence every
action, hand, and session.
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These factors can span the spectrum from personal history between particular
players, the relative importance of pending prize money to opponents of
different means, the impact of physical fatigue and diminished stamina, and even
the presence of television cameras or a similar spotlight.
Some players can dominate a large tournament field until reaching the final
table, where the change of setting from anonymous area on the floor to ringed
off feature table can jar their nerves. Experienced players use their knowledge
of this meta game to apply increased pressure and make things uncomfortable for
less experienced foes.
For the most part though, when a poker player mentions the meta game
affecting their decision making, they're referring to prior history between
themselves and an opponent. Perhaps the other player has shown a propensity for
checking back strong hands in position, so you may begin using flop checks more
often to clarify his range. Or maybe you were a tournament victor to their
runner up twice before, and you know they'll be looking to knock you out of the
final table earlier to prevent another heads up match, so you widen your range
in anticipation of them playing back light.
The concept of meta game at the poker table can go as deep as a thinking
player prefers to take it, but in many cases, if your opponent isn't a thinking
player in their own right, the advantages gained simply aren't all that
effective. An oblivious opponent who doesn't even realize that they've played
dozens of pots with you before can't really be exploited based on that meta
game, as they aren't even aware that it exists.
A perfect information game
is one in which both players have full knowledge of each other's previous moves
or actions. The classic example of a perfect information game is chess, as both
players begin with identical piece alignments and witness all subsequent moves.
An imperfect information game is one in which both players are limited to an
obscured view of the full game conditions. In blackjack, for example, you know
your own hole cards, but not that of the dealer, leading to a situation in which
making educated guesses is the only way to proceed. Texas holdem is another
imperfect information game, because even though all players can see the same
community cards on board, and their own hole cards, the hole cards of every
other opponent remain concealed until the showdown round is reached.
A zero sum game is one in which the amount of
'available resources' in play can never be changed. Poker is the standard zero
sum game, because leaving aside the house's rake in cash games, every pot that
is played results in an equal transfer of chips. If you win 12,000 chips in a
pot, a player or players at the table must have lost 12,000 chips as a result.
A poker tournament is a perfect encapsulation of a zero sum game, as every
chip put in play throughout the proceedings will wind up in the eventual
winner's stack. Players will transfer chips back and forth throughout the
tournament, stacks will grow, shrink, and disappear, but when it's all said and
done, the same amount of chips will be present and accounted for when the final
two competitors begin heads up play.
Conversely, a non zero sum game is one in which the amount of available
resources in play can be changed. While playing Monopoly, for example, everybody
begins with a set amount of dollars in their bank, but factors like Chance cards
and other features can add dollars into the game's economy without transferring
them from one player to another.
Examples of Game Theory You Already Use in Texas Holdem
After perusing the scholarly definitions listed in the Glossary section, some
readers may be thinking that game theory is a bridge too far in terms of what
they're willing to learn. Poker is supposed to be a fun game after all, and most
of us aren't trained in upper level mathematics anyway, so can game theory
approaches really help the recreational player?
They can, and they already do. In fact, if you've spent any serious time at
the Texas holdem tables, whether in tournament play or cash games, chances are
high that you already apply game theory concepts without even knowing it.
Strategies that rely on unspoken acknowledgement of certain factors, deviations
from the norm decided on when competing against certain players – these plays
that seem instinctual are actually demonstrations of game theory in action.
We'll run through a laundry list of commonly encountered Texas holdem
scenarios below, covering both the No Limit and Limit versions of the game, to
show you a few different ways game theory principles are routinely put into use
by beginners:
Checking It Down to Eliminate a Short Stack
Imagine yourself playing a $55 Sit and Go tournament at your local casino.
You wind up playing your way into the final four out of nine players – but only
three players will earn a payout.
The next one to be eliminated will take home nothing for their efforts, an
ignoble end to a long tournament, but you don't really have to worry too much
about that at this point. You sit with 7,000 chips, another has 6,800, while two
short stacks are clinging to 1,200 and 1,000 respectively.
In the big blind position, with 400 chips already committed, you watch the
shortest stack shove all in for his last 1,000. The small blind player, who is
your fellow big stack, makes the call to put the shorty at risk. You look down
at Kc 10c – a decent hand to try and bust the next player with – so you call as
well, creating a heads up side pot while the all in player sweats the action.
Game Theory Optimal Poker Software
The flop comes down 10s 9h 7h, and the small blind checks it over to you. In
most spots, firing out with top pair on a textured flop would be advisable, as
to prevent opponents from backing into a straight or flush on the turn. But you
shoot your heads up opponent a quick look and knock the felt with your fist,
signaling a check.
The turn comes a blank with the 2d, but this time the small blind is checking
as the baby card falls. You check back quickly, and the process repeats itself
on the Kh river. Upon the showdown, you show your top two pair, but the short
stacked player flips up his J 8 with a smile, knowing his straight has beaten
one of the two hands it'll need to fade.
But the small blind turns his 5h 3h face up on the felt, and the flush is
good enough to bust the short stacked player in fourth place. You, the small
blind, and the other shorty have each made the money – and all because you never
bet to force out the small blind's ragged flush draw.
In this case, even though poker is a non cooperative game by rule, you and
the small blind recognized a prime opportunity to cooperate. By checking down
through all three streets, you and the small blind effectively ensured that two
hands, rather than one, would have a chance to eliminate the fourth place player
and burst the bubble.
In the game of Texas holdem, communicating your intent to join forces with
the small blind would represent a violation of the rules against collusion.
Players know this all too well, so when a check down situation like the one
described here presents itself and it invariably will on any tournament bubble
or pay jump, the agreement to check it down remains unspoken. This isn't
cheating by any stretch, but rather an effective application of communicative
game theory strategy to ensure a higher likelihood of an optimal outcome taking
place.
Keeping Short Stacks Alive to Apply Bubble Pressure
Expanding on the previous example, let's imagine you're enjoying every poker
player's dream: sitting on a big stack deep in a multi table tournament. In this
case, you have a stack of 850,000 when the average is only 300,000 – and 29
players remain with the final 27 getting paid.
A look around your table presents a beautiful sight, as four of the six
opponents across from you are riding short stacks of 100,000 or under. With the
big blind set at 10,000, they'll be forced to move in with marginal hands, which
should make them ripe for the picking as you attempt to burst the bubble with
your big stack.
You begin raising several hands in a row, hoping to get involved with a short
stacked player and take your chances on a gamble. This tournament is a big one
for your bankroll, and landing a cash on your Hendon Mob record at this level
would be quite the accomplishment, so you're intent on busting one of the
shorties in short order.
The vulnerable players aren't biting though, and your raises are receiving
nothing but folds around the table. The four short stacks are playing it snug,
each hoping to keep enough chips in hand to sneak into the money. As for the two
bigger stacks, they remain fearful of entering pots against you – the only
player who can bust them.
As you relentlessly raise, and the table keeps passively folding, you keep
dragging seemingly small pots which contain the 5,000 chip small blind, 10,000
chips in the big blind, and another 7,000 in antes at 1,000 chips each. That's
22,000 chips each time you raise and take, and before you know it, your stack is
approaching the coveted 1 million chip mark.
Online roulette games for free. Then, a funny thought occurs to you: having this particular dynamic at the
table may actually be better than bursting the bubble. In pure cash terms,
knocking out two more players from the tournament will return double your buy in
amount at the least. But in terms of chips, keeping the field at 29 players –
and out of the money – presents the best opportunity to build an even bigger
stack. And as we all know, tournament chips equate to cash on an escalating
level, so as you progress later in a tournament, having more chips can mean the
difference between a min cash or winning life changing money.
Suddenly, a short stacked player shoves all in from the hijack position and
the action folds around to you in the big blind. A look over at the other table
confirms that only 28 players remain, so this is your shot to burst the bubble
and ensure a cash. You peek down to see Ad 9d, certainly good enough to warrant
a call, and probably the better hand against a short shover.
But instead of calling to put the player at risk, you shake your head as if
it's the 9 2 instead, and toss the cards into the muck. The short stacked player
lives to fight another day, and you go back to raising on the very next hand –
and many more after that – ravaging your fearful opponents for every last chip
before the money bubble finally does break.
Had you called and eliminated the shorty with ace high, you'd have added a
paltry 20,000 to your stack – or just two big blinds. But by folding, and
keeping them in the game for a few more orbits, you were able to use the raise
and take method to add another 120,000 or so without facing any serious fight.
Those extra chips prove to be quite useful too, and instead of the usual 8th
or 9th place exit you wind up with on deep runs, you go on to win the entire
tournament for your first big time score.
In this case, you've successfully applied an exploitative strategy, because
you recognized that deviating from the normal optimal strategy of eliminating
short players worked to increase your EV. Given a normal tournament scenario,
calling with a huge chip advantage to bust a short stacked player from the field
would be the proverbial no brainer. Here, however, savvy players are able to
intuitively realize that a more productive strategy exists, one which involves
the antithesis of standard poker strategy: ensuring an opponent's survival.
Big Stacks Avoiding One Another on the Bubble
Back to the bubble phase of a tournament, which seems to provide more
opportunities for GTO maneuvers, picture yourself playing a big stack of 525,000
at the 1,000/2,000 blind levels. The average stack at the moment sits at
175,000, and while you have more than seven players at the table, one actually
has you covered with a 600,000 stack.
The field has dwindled to 39 players and the final 36 will make the money.
You've been playing relatively snug of late, as has your fellow big stack,
when action folds around to you on the button. With Ah Qs in the hole, you're
ready to roll with a standard opening raise, so you make it 4,200 to go. The
other big stack is on the small blind and next to act, and rather than keep the
pot small, he splashes three of the pink 5,000 chips into the middle for a three
bet to 15,000.
For beginners just now entering the world of Texas holdem, or even old hands
who simply struggled to keep up with the game's accelerating advancement,
hearing smart and savvy opponents reference ideas that sound more like calculus
homework than a card game can be quite intimidating. It's hard enough figuring
out what to do when you get four bet holding pocket jacks, so the thought of
learning about intricate game theory constructs and the higher level reasoning
behind GTO plays can be daunting to say the least.
With this page, we don't purport to be PhD holders or even Texas holdem
experts, but rather recreational players like yourself who simply wanted to
learn more about game theory as it applies to poker. In keeping with the
instructional theme, we'll offer a syllabus of sorts, starting out with a basic
glossary of the key terms and concepts you'll hear repeatedly during any
discussion on GTO play. Next up you'll find a section detailing several common
examples, written from the perspective of a poker player, that help to
illustrate the technical terms described earlier. From there you'll find a list
of applicable resources – written or developed by successful high stakes
professional players and game theory experts – through which you can pursue an
advanced education.
Glossary of Game Theory Terms
Before we move on to the descriptions, it's important to discuss what the
concept of game theory really means.
According to Roger B. Myerson, whose introductory textbook titled 'Game
Theory: Analysis of Conflict' was published by the Harvard University Press in
1991, game theory can be defined as 'the study of mathematical models of
conflict and cooperation between intelligent rational decision makers.'
As you can see, this definition doesn't mention anything at all about poker
or Texas holdem. That's because game theory is applicable to any game or contest
which involves decision making on the part of players combined with access to
partial information. Additionally, the ideas put forth by game theory experts
have also been co opted for use by economists, political scientists, biologists,
and several other fields of study. Thus, while the study of game theory is
predicated on the various rules and procedures used to govern classic games like
Texas holdem, the ideas that emerge from game theory investigation are widely
applicable across a diverse range of subjects.
Although game theory wasn't codified as a field of study until the 1920s,
evidence of GTO approaches to basic card games can be found dating back to the
early 1700s.
In 1711, Charles Waldegrave wrote a letter to his brother outlining a
'minimax mixed strategy' to the simply two player card game known as Le Her.
In 1913 a German mathematician named Ernst Zermelo developed 'Zermelo's
Theorem,' which states that
'In any finite two person game of perfect
information in which the players move alternatingly, and in which chance does
not affect the decision making process, if the game cannot end in a draw, then
one of the two players must have a winning strategy.'
As you might suspect, this
long passage was used by Zermelo to describe chess, which he successfully proved
to be a 'strictly determined' game from a strategic sense.
Throughout the 20th century, mathematicians and logicians like John von
Neumann, Oskar Morgenstern, Merrill M. Flood, Melvin Dresher, and John Nash each
contributed fundamental theories and postulations to the field of game theory
study.
For poker players with an educational background in advanced mathematics – of
whom there seemed to be an endless supply during the Poker Boom – learning the
lingo of game theory and applying it to their favorite game proved to be a
highly beneficial proposition. These players were able to expand their lines of
thinking beyond the most basic constructs – what do I have or need, what does my
opponent have or need, etc. – to turn a seemingly simple poker hand into an
exercise in statistical modelling and probability based prediction.
But for the rest of us, the laymen at the table who haven't memorized reams
of mathematical formulas, diving deeper into the subject of game theory study
can present a firm barrier. The average person can only take so many
abbreviations and hypotheticals before their head begins to ache, so breaking
things down to their basic meaning is a helpful way to begin.
Take a look below for a comprehensive glossary of essential terms and
concepts used within the world of game theory:
Any strategy that offers a reduced expected value EV,
compared to GTO strategy, when playing against an exploitive strategy. Any non
game theory optimal GTO strategy is, by definition, an exploitable strategy.
Any strategy that offers an increased expected value EV
than a game theory optimal GTO strategy, when playing against any particular
strategy. Any non GTO strategy that counters an exploitable strategy better than
a strictly GTO approach is, by definition, and exploitive strategy.
The strategy that offers the highest possible
expected value EV when an opponent always applies an optimal counter strategy.
The classic GTO strategy example concerns the zero sum hand game known as 'Rock,
Paper, Scissors.' In this game, the GTO approach involves selecting randomly
between rock, paper, and scissors while using an equal distribution. This
strategy provides the highest level of EV, at 0.50 percent equity, against any
opponent strategy that consists of all rock, all paper, or all scissors.
The strategy that offers the highest possible
expected value EV against any opponent strategy. Returning to the Rock, Paper,
Scissors example, in a game where you know your opponent's strategy was to throw
rock on every game, the optimal exploitive strategy would be to counter with
paper every game – because this would create an EV of 100 percent. And should
your opponent modulate to a strategy based on using rock on 50 percent of games,
paper on 25 percent, and scissors on the other 25 percent, the optimal
exploitative strategy would also be to throw paper on every game – because you'd
create a scenario in which you'd win or tie on 75 percent of games, while losing
only 25 percent of the time.
Any strategy that offers a lower expected value EV than
the optimal exploitive strategy. Back to that Rock, Paper, Scissors game, where
your opponent's strategy is to throw rock on each game, you could opt for a 50
percent paper and 50 percent rock blend of moves. And while this would still be
a winning strategy, because you'd only win or tie, it's performance can't match
that of the optimal exploitative strategy throwing paper every time – making it
a suboptimal strategy at best.
Game Types
Cooperative / Non Cooperative GameIn a cooperative game, players are
permitted, encouraged, or even required to form binding agreements with fellow
players. A game like Monopoly, in which players can negotiate the price of a
mortgage on property deeds among other agreements, is a classic cooperative
game.
A non cooperative game, on the other hand, forbids players from making
similar arrangements among themselves. Technically speaking, poker variants like
Texas holdem are non cooperative games, because the rules preclude collusion and
other forms of explicit cooperation. Even so, as you'll learn in the next
section featuring examples of Texas holdem game theory in action, many
situations in the game compel players to form implicit agreements to achieve a
certain effect stalling on the bubble, big stacks avoiding one another with
pending pay jumps, etc.
From a practical standpoint, any game involving human
players must be a finitely long game – or one that has a fixed endpoint. Whether
that means attaining a certain score, satisfying a series of conditions, or
otherwise defeating your opponent, a finitely long game has a beginning – and a
definitive end. And even in a game designed to stretch on into perpetuity, the
limits of human endurance, and indeed lifespan, prevent it from being truly
infinite.
But game theorists have no such limitations on their work, and in the process
of investigating mathematical proofs, they've created the concept of infinitely
long games that are never forced to end. These games are devised, in part, to
study the relative strengths and weaknesses of dueling strategies which adapt
based on one another's actions.
For poker players, every cash game or tournament session has a start and an
end. But as any experienced poker pro knows quite well, judging the results of
any particular session provides an inconsistent appraisal, and the truth is best
discovered by examining results over the long run. That long run can encompass
years, decades, or even a player's entire lifetime on the felt – making Texas
holdem and other poker formats an infinitely long game in spirit.
For game theorists, a 'meta game' means something entirely
different, but as a poker player, you'll hear this expression used largely to
describe the multitude of external factors that conspire to influence every
action, hand, and session.
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These factors can span the spectrum from personal history between particular
players, the relative importance of pending prize money to opponents of
different means, the impact of physical fatigue and diminished stamina, and even
the presence of television cameras or a similar spotlight.
Some players can dominate a large tournament field until reaching the final
table, where the change of setting from anonymous area on the floor to ringed
off feature table can jar their nerves. Experienced players use their knowledge
of this meta game to apply increased pressure and make things uncomfortable for
less experienced foes.
For the most part though, when a poker player mentions the meta game
affecting their decision making, they're referring to prior history between
themselves and an opponent. Perhaps the other player has shown a propensity for
checking back strong hands in position, so you may begin using flop checks more
often to clarify his range. Or maybe you were a tournament victor to their
runner up twice before, and you know they'll be looking to knock you out of the
final table earlier to prevent another heads up match, so you widen your range
in anticipation of them playing back light.
The concept of meta game at the poker table can go as deep as a thinking
player prefers to take it, but in many cases, if your opponent isn't a thinking
player in their own right, the advantages gained simply aren't all that
effective. An oblivious opponent who doesn't even realize that they've played
dozens of pots with you before can't really be exploited based on that meta
game, as they aren't even aware that it exists.
A perfect information game
is one in which both players have full knowledge of each other's previous moves
or actions. The classic example of a perfect information game is chess, as both
players begin with identical piece alignments and witness all subsequent moves.
An imperfect information game is one in which both players are limited to an
obscured view of the full game conditions. In blackjack, for example, you know
your own hole cards, but not that of the dealer, leading to a situation in which
making educated guesses is the only way to proceed. Texas holdem is another
imperfect information game, because even though all players can see the same
community cards on board, and their own hole cards, the hole cards of every
other opponent remain concealed until the showdown round is reached.
A zero sum game is one in which the amount of
'available resources' in play can never be changed. Poker is the standard zero
sum game, because leaving aside the house's rake in cash games, every pot that
is played results in an equal transfer of chips. If you win 12,000 chips in a
pot, a player or players at the table must have lost 12,000 chips as a result.
A poker tournament is a perfect encapsulation of a zero sum game, as every
chip put in play throughout the proceedings will wind up in the eventual
winner's stack. Players will transfer chips back and forth throughout the
tournament, stacks will grow, shrink, and disappear, but when it's all said and
done, the same amount of chips will be present and accounted for when the final
two competitors begin heads up play.
Conversely, a non zero sum game is one in which the amount of available
resources in play can be changed. While playing Monopoly, for example, everybody
begins with a set amount of dollars in their bank, but factors like Chance cards
and other features can add dollars into the game's economy without transferring
them from one player to another.
Examples of Game Theory You Already Use in Texas Holdem
After perusing the scholarly definitions listed in the Glossary section, some
readers may be thinking that game theory is a bridge too far in terms of what
they're willing to learn. Poker is supposed to be a fun game after all, and most
of us aren't trained in upper level mathematics anyway, so can game theory
approaches really help the recreational player?
They can, and they already do. In fact, if you've spent any serious time at
the Texas holdem tables, whether in tournament play or cash games, chances are
high that you already apply game theory concepts without even knowing it.
Strategies that rely on unspoken acknowledgement of certain factors, deviations
from the norm decided on when competing against certain players – these plays
that seem instinctual are actually demonstrations of game theory in action.
We'll run through a laundry list of commonly encountered Texas holdem
scenarios below, covering both the No Limit and Limit versions of the game, to
show you a few different ways game theory principles are routinely put into use
by beginners:
Checking It Down to Eliminate a Short Stack
Imagine yourself playing a $55 Sit and Go tournament at your local casino.
You wind up playing your way into the final four out of nine players – but only
three players will earn a payout.
The next one to be eliminated will take home nothing for their efforts, an
ignoble end to a long tournament, but you don't really have to worry too much
about that at this point. You sit with 7,000 chips, another has 6,800, while two
short stacks are clinging to 1,200 and 1,000 respectively.
In the big blind position, with 400 chips already committed, you watch the
shortest stack shove all in for his last 1,000. The small blind player, who is
your fellow big stack, makes the call to put the shorty at risk. You look down
at Kc 10c – a decent hand to try and bust the next player with – so you call as
well, creating a heads up side pot while the all in player sweats the action.
Game Theory Optimal Poker Software
The flop comes down 10s 9h 7h, and the small blind checks it over to you. In
most spots, firing out with top pair on a textured flop would be advisable, as
to prevent opponents from backing into a straight or flush on the turn. But you
shoot your heads up opponent a quick look and knock the felt with your fist,
signaling a check.
The turn comes a blank with the 2d, but this time the small blind is checking
as the baby card falls. You check back quickly, and the process repeats itself
on the Kh river. Upon the showdown, you show your top two pair, but the short
stacked player flips up his J 8 with a smile, knowing his straight has beaten
one of the two hands it'll need to fade.
But the small blind turns his 5h 3h face up on the felt, and the flush is
good enough to bust the short stacked player in fourth place. You, the small
blind, and the other shorty have each made the money – and all because you never
bet to force out the small blind's ragged flush draw.
In this case, even though poker is a non cooperative game by rule, you and
the small blind recognized a prime opportunity to cooperate. By checking down
through all three streets, you and the small blind effectively ensured that two
hands, rather than one, would have a chance to eliminate the fourth place player
and burst the bubble.
In the game of Texas holdem, communicating your intent to join forces with
the small blind would represent a violation of the rules against collusion.
Players know this all too well, so when a check down situation like the one
described here presents itself and it invariably will on any tournament bubble
or pay jump, the agreement to check it down remains unspoken. This isn't
cheating by any stretch, but rather an effective application of communicative
game theory strategy to ensure a higher likelihood of an optimal outcome taking
place.
Keeping Short Stacks Alive to Apply Bubble Pressure
Expanding on the previous example, let's imagine you're enjoying every poker
player's dream: sitting on a big stack deep in a multi table tournament. In this
case, you have a stack of 850,000 when the average is only 300,000 – and 29
players remain with the final 27 getting paid.
A look around your table presents a beautiful sight, as four of the six
opponents across from you are riding short stacks of 100,000 or under. With the
big blind set at 10,000, they'll be forced to move in with marginal hands, which
should make them ripe for the picking as you attempt to burst the bubble with
your big stack.
You begin raising several hands in a row, hoping to get involved with a short
stacked player and take your chances on a gamble. This tournament is a big one
for your bankroll, and landing a cash on your Hendon Mob record at this level
would be quite the accomplishment, so you're intent on busting one of the
shorties in short order.
The vulnerable players aren't biting though, and your raises are receiving
nothing but folds around the table. The four short stacks are playing it snug,
each hoping to keep enough chips in hand to sneak into the money. As for the two
bigger stacks, they remain fearful of entering pots against you – the only
player who can bust them.
As you relentlessly raise, and the table keeps passively folding, you keep
dragging seemingly small pots which contain the 5,000 chip small blind, 10,000
chips in the big blind, and another 7,000 in antes at 1,000 chips each. That's
22,000 chips each time you raise and take, and before you know it, your stack is
approaching the coveted 1 million chip mark.
Online roulette games for free. Then, a funny thought occurs to you: having this particular dynamic at the
table may actually be better than bursting the bubble. In pure cash terms,
knocking out two more players from the tournament will return double your buy in
amount at the least. But in terms of chips, keeping the field at 29 players –
and out of the money – presents the best opportunity to build an even bigger
stack. And as we all know, tournament chips equate to cash on an escalating
level, so as you progress later in a tournament, having more chips can mean the
difference between a min cash or winning life changing money.
Suddenly, a short stacked player shoves all in from the hijack position and
the action folds around to you in the big blind. A look over at the other table
confirms that only 28 players remain, so this is your shot to burst the bubble
and ensure a cash. You peek down to see Ad 9d, certainly good enough to warrant
a call, and probably the better hand against a short shover.
But instead of calling to put the player at risk, you shake your head as if
it's the 9 2 instead, and toss the cards into the muck. The short stacked player
lives to fight another day, and you go back to raising on the very next hand –
and many more after that – ravaging your fearful opponents for every last chip
before the money bubble finally does break.
Had you called and eliminated the shorty with ace high, you'd have added a
paltry 20,000 to your stack – or just two big blinds. But by folding, and
keeping them in the game for a few more orbits, you were able to use the raise
and take method to add another 120,000 or so without facing any serious fight.
Those extra chips prove to be quite useful too, and instead of the usual 8th
or 9th place exit you wind up with on deep runs, you go on to win the entire
tournament for your first big time score.
In this case, you've successfully applied an exploitative strategy, because
you recognized that deviating from the normal optimal strategy of eliminating
short players worked to increase your EV. Given a normal tournament scenario,
calling with a huge chip advantage to bust a short stacked player from the field
would be the proverbial no brainer. Here, however, savvy players are able to
intuitively realize that a more productive strategy exists, one which involves
the antithesis of standard poker strategy: ensuring an opponent's survival.
Big Stacks Avoiding One Another on the Bubble
Back to the bubble phase of a tournament, which seems to provide more
opportunities for GTO maneuvers, picture yourself playing a big stack of 525,000
at the 1,000/2,000 blind levels. The average stack at the moment sits at
175,000, and while you have more than seven players at the table, one actually
has you covered with a 600,000 stack.
The field has dwindled to 39 players and the final 36 will make the money.
You've been playing relatively snug of late, as has your fellow big stack,
when action folds around to you on the button. With Ah Qs in the hole, you're
ready to roll with a standard opening raise, so you make it 4,200 to go. The
other big stack is on the small blind and next to act, and rather than keep the
pot small, he splashes three of the pink 5,000 chips into the middle for a three
bet to 15,000.
The big blind gets out of the way and you're left to ponder, but the ace
queen is just too good of a hand to lay down to a little pressure, so you call
to see the flop come Ac 9s 2c. The big blind tosses out a continuation bet of
25,000, and as you reach for the calling chips, you stop and think for a second:
'The rest of the table is littered with shorter stacks, and I've only
invested 15,000 of my 525,000 so far. I like my hand of course, but it could
easily be beat by A K, so it's not like I have the nuts. Against any other
player at the table, somebody I had covered, I'd roll the dice and go with a big
raise here… but not this guy. It'd be foolish to go broke now, not with the
money one bustout away, so I'll toss this one away and let him have it.'
You fold the big hand and move on to the next, even though standard strategy
would suggest a flat call, or even a raise, might be in order. The issue here
was simple, as most players late in tournaments prefer to survive rather than
exploit fine edges. With the money right around the corner, playing big pots
against opponents who can eliminate you just isn't a logical move.
From a game theory perspective, this example illustrates a suboptimal
strategy, because a purely GTO approach to flopping a big ace on a dry board
would be to call or raise often, while folding almost never. This suboptimal
strategy may sacrifice a certain level of EV in terms of the actual hand, but
depending on a player's bankroll situation or personal finances, folding a big
hand against a big stacked opponent on the bubble is actually the correct play.
Technically speaking, 'soft play' of this nature isn't compatible with the
non cooperative nature of the game, but in Texas holdem circles, avoiding the
'risk of ruin' takes precedence in several crucial endgame scenarios. Big stacks
at the table late in tournaments often operate under an unspoken assumption,
taking turns bullying the smaller stacks, but seldom engaging in outright
aggression against one another.
And to take the game theory implications one step further, poker experts
generally agree that the optimal strategy as a big stack in this situation
should be based on exploitative play. If you know a fellow big stack is
practicing avoidance, you can push them around much more easily, and it's in
your best interest to do so in an attempt to build your stack even bigger while
eliminating your primary handicap to pure aggression.
Playing 'Bad' Hands Against Exploitable Opponents
For the most part Texas holdem hand ranges can be played in relatively
straightforward manner. Everyone gets creative with funky suited connectors and
baby card hands of course, but at the lower stakes, playing a snug game which
incorporates premiums and top tier hands from early and middle position will
prove to be profitable.
But every poker game is a different animal entirely, consisting of unique
playing styles and personalities, along with constantly fluctuating variables
like stack size, positions, and indeed, hole cards. In certain situations, you
may find yourself expanding your range drastically, playing basically every hand
in an attempt to exploit your post flop advantages over a weaker opponent.
Every casino and card room has its resident drunk, so imagine yours has just
stopped by with a full rack of red $5 chips and a rye bourbon on the rocks in
hand. She sits down in your usual $2/$5 cash game, and before her $500 in chips
is even unracked, she's stacked an unfortunate soul by going runner runner to a
straight. The inebriated gal's cards? The lowly 2d 4c.
She flopped nothing but a deuce on the Kc 6s 2h flop, called a bet to see the
3d fall on the turn, and called an even bigger wager to find her 5d gin card on
the river. Just like she drew it up.
At this point, your radar is blaring alarm bells and you realize that an
opponent demonstrating an extremely exploitable strategy has just arrived on the
scene. Rather than stick to the usual script and play your usual range of high
cards and pocket pairs, you begin to enter as many pots as possible, intending
to connect with any sort of hand that can run down the lady's loose play.
It takes an orbit or two, and you bleed a few chips along the way while she
continues to act as the table card rack, but eventually you call her opening
raise with the 10d 8d and catch a nice Jd 9c 2c flop. With an open ended
straight draw you call the woman's continuation bet on the flop, catching a Qc
on the turn to make the nut straight. The woman suddenly slows down with a tap
of the table, checking it over to you.
You fire out a wager and she comes over the top with a big all in raise,
which you happily snap off with your nut straight. She shows the Qd 2h for a
ragged two pair, having flopped bottom pair and made two pair on the river. Your
straight holds through the river blank and you take a hand like 10 8, which you
never would've played otherwise against a player using a standard strategy, to
score a massive double up.
In this instance, you've used an exploitative strategy – one which deviates
from normal GTO strategy – to take advantage of unique game conditions. Against
most opponents, hands like 10 8 may not be the most profitable holding over the
long run, but here you correctly identified an opponent who would allow you to
take the low risk, high reward shot.
And even better? The lady's propensity for aggression ensured that when the
right spot finally arrived, your big hand materialized into a massive payoff.
Using Push / Fold Charts to Dictate Short Stack Play
One of the more difficult phases of any tournament involves reaching the
endgame with very little chips to work with.
The excitement of running deep is counterbalanced by the desperation of
needing a quick double up. In this spot, most recreational players, and quite a
few professionals, are plagued by the problem of patience.
For some players, they're not patient enough, and the sight of any face card
is enough to warrant a reckless shove. For others, patience isn't really a
virtue, because as they wait for the perfect hand with which to make their
stand, their stack dwindles to the point of irrelevance.
To help solve this dilemma, poker focused game theorists have crunched the
various probabilities based on hand strength relative to current big blind
volume. In doing so, they've created reliable 'Push / Fold' charts that inform
players exactly which hands warrant a fold or an all in bet, based on how many
big blinds are currently in their possession. These charts function just like
the basic strategy charts used by blackjack players, providing clear and defined
guidelines on the optimal way to approach close situations.
Using a Push / Fold chart is an example of purely GTO play, because the
probabilities have been studied to reveal the exact dividing line between
positive and negative EV plays. When you use a Push / Fold chart to dictate your
decisions, you've effectively removed the human element from the game, letting
pure mathematics define the difference between the right and wrong move.
At one point Push / Fold charts were scoffed at by poker players, but today's
generation of game theory cognizant players has realized that any strategic tool
which makes GTO play easier is one worth incorporating into their overall game.
Essential Texts and Resources
As we mentioned earlier, the author of this page in no way, shape, or form
claims to be an expert in Texas holdem game theory. We can't teach you the proof
behind the Nash Equilibrium, the exact formula for a 'strategy space,' or
anything at all about the concept of 'Condorcet winners.'
What we can do, however, is provide a comprehensive list of books, websites,
and other resources that specialize in both game theory as a field of study, and
its practical applications for poker's most popular game. These resources were
written, or otherwise created, by experienced experts who have spent the long
hours in the classroom which are required to wield advanced game theory
knowledge.
With that said, everybody learns differently, so a dense textbook style
treatise on game theory may not work as well for you as it does for others.
Conversely, visual aids and interactive features are perfect for some students
who need a tangible grasp on abstractions, while more mathematically inclined
learners will have no need for the added clutter.
However you choose to learn about Texas holdem game theory though, the
following resources should provide a full fledged education on the topic, both
in a general sense and from the standpoint of GTO poker play:
Poker and Texas Holdem Game Theory Reference
Books
- Poker Strategy: Winning with Game Theory | Nesmith C. Ankeny | 1981
- The Theory of Poker: A Professional Poker Player Teaches You How To
Think Like One | David Sklansky | 1999 - The Mathematics of Poker | Bill Chen and Jerrod Ankenman | 2006
- No Limit holdem: Theory and Practice | David Sklansky and Ed Miller |
2006 - The Mental Game of Poker: Proven Strategies for Improving Tilt Control,
Confidence, Motivation, Coping with Variance, and More | Jared Tendler M.S.
and Barry Carter | 2011 - Jonathan Little on Live No Limit Cash Games: The Theory | Jonathan
Little | 2014 - Poker's 1%: The One Big Secret That Keeps Elite Players On Top | Ed
Miller | 2014 - Essential Poker Math: Fundamental No Limit holdem Mathematics You Need
To Know | Alton Hardin | 2015
Websites, Articles, and Other
- What Would Jesus Bet? | The New Yorker | 2009
- Poker Strategy With Reid Young: Game Theory vs. Pot Odds | CardPlayer |
Reid Young | - Game Theory Optimal Solutions and Poker: A Few Thoughts | Nikolai
Yakovenko | PokerNews | 2015 | - Got GTO? The Connection Between 'A Beautiful Mind' and Perfect Poker |
Robert Wooley | PokerNews | 2015 | - Game Theory Optimal GTO Play for Poker Explained | John Anhalt | 2015 |
- Decision Theory and Nash Equilibrium in Texas holdem Poker | Harry
Copeland and Sam Leclerc | 2015 | - 3 Ways GTO Can Help Improve Your Live Game | Red Chip Poker
- Poker Game Theory – 4 Ways Poker Game Theory Can Improve Your Play | Sit
and Go Planet - Push / Fold Chart App | Float the Turn |
- Game Theory Simplified, and Why Fixed Strategies Fail | DoyleBrunson.com
General Game Theory
Books
- Mathematics of Games and Gambling | Edward Packel | 1981
- The Compleat Strategyst: Being a Primer on the Theory of Games of
Strategy | J.D. Williams | 1986 - Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the
Bomb | William Poundstone | 1993 - Game Theory: A Nontechnical Introduction | Morton D. Davis | 1997
- The Art of Strategy: A Game Theorist's Guide to Success in Business and
Life | Avinash K. Dixi and Barry J. Nalebuff | 2010 - Game Theory 101: The Complete Textbook | William Spaniel | 2011
- The Complete Idiot's Guide to Game Theory | Edward C. Rosenthal Ph.D. |
2011 - The Joy of Game Theory: An Introduction to Strategic Thinking | Presh
Talwalkar | 2014 - Game Changer: Game Theory and the Art of Transforming Strategic
Situations | David McAdams | 2014
Websites, Articles, and Other
Conclusion
Texas holdem game theory is one of the subjects that scare off many players
before they give themselves a chance to learn about it and use it. If you didn't
take the time to read the entire page, set aside a few minutes and study it.
If you do, you'll realize that game theory isn't that hard and you can start
using it to improve your results while playing Texas holdem. The best players
use game theory whether they realize it or not, so why not learn more about it
and maximize your chances to win?
'Game Theory Optimal' — or 'GTO' — is a term often heard in poker these days. Often the term is used imprecisely as a general reference to the 'best' style or strategy in a given situation, though in truth it has a much more specific meaning than that.
To employ a 'GTO' style means to play in a way that is impossible to be exploited by an opponent. It means always making 'optimal' decisions that ensure you maximize your expected value in any situation. To play such a style perfectly would also mean that the very best your opponent could do against you would be to break even.
Such a style can be readily employed in simple games, but in more complicated ones like no-limit hold'em it is hard actually to play a strict 'GTO' style. Besides, in poker players almost always demonstrate certain tendencies that suggest sticking with a 'GTO' strategy isn't necessarily the best option — rather, an 'exploitative' style that responds to opponents' weaknesses can sometimes be a better choice.
Our Laura Cornelius caught up with a number of poker pros to ask them how frequently they themselves could be said to play a 'GTO' style. Internet cafe slots online. Their responses suggest the current 'GTO' debate in poker is in a way an updated version of the old 'math vs. feel' one.
Watch below and listen to reponses from Faraz Jaka, Mustapha Kanit, Luke Schwartz, Randy Lew, Fedor Holz, Jason Mercier, Celina Lin, Andreas Hoivold, and Bryn Kenney.
Game Theory Optimal Poker Pdf
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