Texas Holdem Game Theory

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:

Game Types

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.

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.

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

• Any “Sticky” Thread on the 2+2 Poker Theory Forum
• 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

• Finite Mathematics Utility: Game Theory Tool
• Gambit Open Source Game Theory Software
• GameTheory.net
• Game Theory Explorer
• Game Theory Society

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?