Poker Game Theory – How You Should Think About the Game
Poker has evolved rapidly in recent years. So much so that many strategy resources including some books, videos and digital content have become out-dated.
The most obvious change is that old-school players made their millions from exploitative play, whereas almost all players making millions now depend on poker game theory — with exploitative play mixed in — to take their game to the next level.
In this article, we’ll discuss:
- The basics of game theory and poker
- Why you should use a game theory-influenced strategy
- An example from Doug Polk that highlights game theory’s importance
- 4 benefits of using game theory influenced strategy
Let’s get to it!
Game Theory and Poker
John Nash developed game theory as a branch of mathematics at Princeton University around 1950. As poker has become more popular over the last 15 years or so, players have improved dramatically, to the point where it’s very difficult to consistently to beat the game without game theory knowledge in your corner.
Understood mathematically, poker’s complexity runs deep — from the particular hand you open from each position, to a seemingly unimportant check on the river in a small pot, every decision influences your win-rate as a poker player. This can be measured by expected value (EV). If a decision is profitable it is said to be +EV, and if it is not profitable, it is -EV.
A very simple example of using a theoretically balanced strategy is when a player uses a range of open-raising hands. Below is an example of a typical opening range for a player UTG (first to act).
It is obviously a profitable play to raise our very strong hands UTG, but playing only those hands would be too predictable. By raising with some less profitable hands–like 9s8s or 6h6c–we balance our opening range, making us harder to play against. This way it’s still possible for us to have hit a very strong hand when the flop falls low or middling, like the one below
Why use a GTO poker strategy?
You might wonder why it’s important to play a game theory-influenced strategy when most of your money will be made by exploiting weaker players, or players who simply aren’t paying attention.
There are two main reasons:
- With a balanced, GTO-based strategy, you will win money in the long run regardless of how skillful your opponents are.
- Making adjustments to counter your opponents is easier if you have a baseline strategy from which to adjust (more on this later).
From a GTO point of view, your hand review sessions should involve analyses of how hands played out objectively. From that point of view you can decide if you played your range in a balanced way. Moreover, from a GTO standpoint you should know what you would do with any holding in any particular situation, not just the 2 cards you were in fact dealt. So, during review sessions you ought to ask yourself what you might have done with different holdings.
If you are betting for value in certain situations, you should also be betting as a bluff with other hands in your range so that your opponent is unsure whether you are betting for value or as a bluff. If you are only betting value hands on a certain river, your opponent can fold profitably every time knowing you have the goods. On the other hand, if you are bluffing too much in certain situations, then your opponent can call profitably every time knowing that you are less likely to have a strong hand.
If you’re still not convinced that a GTO-influenced strategy is the way to go, these hypothetical examples from Doug Polk should help:
Game theory poker examples
On the river you bet $100 into a pot of $100, so your opponent must call $100 to win $200. Thus, your opponent is getting 2-to-1 pot odds and needs to win at least 33% of the time to break even (learn how to calculate pot odds here).
This quick calculation reveals the optimal proportion of bluffs in your betting range on the river: 33% (one bluff for every two value bets). This frequency is optimal because it allows you to win the pot most often without the possibility of being countered.
Let’s test 4 different bluff-to-value bet scenarios so that you can see why a range with 33% bluffs and 66% value bets is the best option from a GTO standpoint, and there’s nothing you opponent can do about it.
(For simplicity, assume that we always win when our value bets are called, and always lose when our bluffs are called.)
Scenario 1 – Bluff 0%, Value Bet 100%:
Your opponent can fold 100% of the time. This results in you winning $100 with your betting range.
Scenario 2 – Bluff 100%, Value Bet 0%:
Your opponent can call 100% of the time. This results in you losing $100 with your betting range.
Scenario 3 – Bluff 50%, Value Bet 50%:
If your opponent calls 100% of the time, you win $200 when you are value betting and lose $100 when bluffing. This results in you winning $50 with your betting range only if your opponent always calls (50% * -$100 = -$50; 50% * $200 = $100. $100 – $50 = $50).
This scenario shows that not bluffing at all is actually more profitable than bluffing 50% of the time.
Scenario 4 – Bluff 33%, Value Bet 67%:
If your opponent always calls, you again win $200 when you are value betting and lose $100 when bluffing. This time, though, you lose $100 only 33% of the time and win $200 67% of the time, which comes out to a net profit of $100 (33% * $100 = -$33; 67% * $200 = $133. $133 – $33 = $100).
The bluff-to-value bet ratio used in this scenario is optimal because:
- You win $100 if your opponent always calls
- You win $100 if your opponent always folds
You make a profit of $100 regardless of whether your opponent calls or folds. This win-win scenario is only achievable with a perfectly balanced range. Your opponent is indifferent to calling and folding because no matter which option she chooses, your range profits the same amount.
Adjusting this ratio in order to exploit weak players can be even more profitable, but that requires careful and correct adjustments based on reliable evidence. If you want to move up in stakes and really crush the game long-term, understanding a GTO-influenced strategy is essential.
4 Benefits of Using GTO
Finally, take a look at these 4 general benefits that result from using solid GTO strategy.
1. Avoid Circular Thinking
A relic of poker training from the 90’s is trying to understand what “level” players are playing at.
- Level-based thinking starts with you considering only your own hand.
- Then it moves to thinking about what your opponent may have.
- Then it moves to what your opponent thinks that you have.
- Then it moves to what you think that your opponent thinks that you think that he has
- And so on.
Ideally, you’d somehow determine where this “leveling” process should end up—that is, you’d determine on what level your opponent is playing and then adjust accordingly. But the reality is that this process is unreliable against weak players. And against more experienced players it could, in theory, repeat until the end of time, with both players trying to outdo each other’s level of thinking.
We can avoid ending up in this sort of situation by using a GTO-influenced bluffing strategy, which keeps us from confusing ourselves and getting into leveling wars on the flop with no equity.
2. Avoid Making Assumptions
Another benefit of a GTO-based approach to poker is that it forestalls potentially incorrect assumptions of other players. Of course, certain assumptions can be made if you’ve played against someone over a sufficient sample of hands, but making very general assumptions can be costly.
For example, it isn’t wise to say things like, “this is NEVER a bluff,” or, “he ALWAYS has it here.” Similarly, you shouldn’t assume an unknown opponent cannot have a certain hand in their range, or that they open very wide or very tight.
A well-constructed GTO strategy eliminates confusion, and helps you make the long-term profitable play.
3. Objective Analysis
Many players incorrectly judge how they played a hand by its outcome. But the further a player progresses into her poker career, the more she’ll come to realize that she’s not in the business of looking at results in a vacuum.
Yet, thinking objectively can be tough, especially when the result of a hand is either really bad or really good. Just because you hit your full-house on the river and stacked your opponent doesn’t mean that calling twice was the right play.
Once you’ve worked out what the correct GTO strategy was in a certain spot, apply it to your post session analysis to see if you made the long-term profitable play with your range, not just your 2 hole cards.
Every successful poker player knows that admitting mistakes to one’s self is absolutely crucial to consistently playing well. Game theory provides a foundation for discerning mistakes more easily.
4. It makes adjusting easier
Why is theory important when it comes to making badass adjustments in your strategy? To find out let’s play a little game.
Suppose you just forgot everything you know about poker strategy, except the rudimentary knowledge of the game, and you are just about to play your first hand ever.
Live $1/$2. $200 Effective Stacks.
Hero is dealt A♦ 9♦ in the big blind
folds to btn. BTN raises to $7. sb folds. Hero calls.
Flop ($14) A♠ T♦ 3♥
Hero checks. BTN bets $9. Hero calls.
Turn ($32) J♣
Hero checks. BTN bets $21. Hero calls.
River ($74) 9♣
Hero checks. BTN bets $50. Hero calls.
BTN shows A♥2♣. Hero wins $174 with two pair.
What do you make of the button’s aggressive line with his weak top pair? How could you adjust to exploit him in the future? Well, without understanding the theoretically correct way of playing his specific hand, you would not know where to start.
On the other hand, if you know the theoretically correct way to play A2o in the button’s situation, then you know how he is deviating from it. This knowledge makes it easy to deduce ways to exploit this opponent.
Here are the specific adjustments we can make to crush this player’s aggressive thin value betting strategy:
- The small exploit: call down a bit lighter when he bets and barrels (but not too much).
- The big exploit: relentlessly attack his check back range–which is apparently very weak–with big bets for thin value mixed with the appropriate amount of bluffs.
Very often, understanding the theoretically optimal way to play a hand makes exploiting your opponents’ easier because you know exactly how they are deviating from optimal. When you don’t know what is right, it’s nearly impossible understand what is wrong?
Conclusion
Striving for a perfect GTO strategy might seem like the logical conclusion, but the truth is nobody plays an entirely game theory optimal strategy. Poker has yet to be solved by man or machine, but we still highly recommend using game theory to influence your strategy as much as possible. As always, this means working on your game both on and off the felt.
This article is a very basic summary of how game theory is applied in poker, but hopefully you’ve gained something from it, or at least become more curious about how to develop your game in-line with game theory concepts.
For more on this topic, check out Doug Polk’s article on GTO vs Exploitative Play.
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