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 “sub-optimal” 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 game-play. 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 “mini max 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 alternatively, 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 hypothetical's 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:

Exploitable Strategy

Any strategy that offers a reduced expected value EV, compared to GTO strategy, when playing against an exploitative strategy. Any non game theory optimal GTO strategy is, by definition, an exploitable strategy.

Exploitative 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 exploitative strategy.

Game Theory Optimal GTO

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.

Optimal Exploitative Strategy

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 exploitative 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.텍사스홀덤사이트

Sub-optimal Strategy

Any strategy that offers a lower expected value EV than the optimal exploitative 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 sub-optimal strategy at best.