Blackjack Calculator: Optimal and Personalized Strategies

Hello everyone, I'm Lucas, a researcher from the Blackjack Theorem team. After more than 3 years of research, development, and continuous feedback from the community, we're thrilled to officially announce the launch of our site:

bjtheorem

What have we developed?
As a result of our research “Optimal Blackjack Betting Strategies Through Dynamic Programming and Expected Utility Theory,” our team has created over 300,000 mathematically optimal strategies for playing Blackjack. These strategies determine both the ideal bet size for each round and the best in-game decision (hit, stand, double, split). They are divided into two clearly distinct modes:

Live Strategies (In-person Play)
• Based on the True Count and your current bankroll
• Determine the optimal bet before each round using a precise mathematical formula
• Apply basic strategy during each round — ideal for physical casinos

Online Strategies
• Based on the exact composition of the deck and your current bankroll
• Calculate both the optimal bet and decision in real time using our specialized calculator
• Ideal for online Blackjack, taking full advantage of the exact deck composition

How do our strategies work?
Each user can personalize their strategy based on their individual risk profile and specific goals by adjusting three key parameters:
• Absolute Risk Inclination (ARI)
• Relative Risk Inclination (RRI)
• Target Return (TR) for the session

What kind of performance can you expect?
Every strategy has been rigorously tested in both simulations and real-world conditions to ensure consistent and reliable outcomes. In the Simulations section of our website, you’ll find interactive charts and detailed tables showcasing each strategy’s performance across entire betting sessions.

How can you get started?
You can access our calculator and instantly explore the performance of each strategy through a weekly or monthly subscription. To celebrate our launch, we’re offering a limited-time 90% discount on all plans.

We invite you to try our platform and discover how to consistently and personally maximize your winnings in Blackjack.

Looking forward to your feedback and questions — and thank you for being part of this launch!

— Lucas, Blackjack Theorem

Looks like an interesting project.

Best of luck to your team!
-D

For this forum, I think a better answer to your question, “What kind of performance can you expect?,” can be found in the abstract of your research paper:
“betting strategies based on the exact composition of the deck slightly outperform the Hi-Lo counting system.” If you offered free access to your commercialized product for some members of this site you might get more meaningful criticism than the pier review of your paper can provide.

Great work, Lucas! My advice would be to keep the platform as beginner-friendly as possible maybe add a simple guide or video walkthrough for new users. The math is impressive, but making it easy to understand will really help more people get value from it.

There are a few new definitions and acronyms here:

Expected Utility Theory
Absolute Risk Inclination (ARI)
Relative Risk Inclination (RRI)
Target Return (TR) for the session

However, there is no risk-of-ruin (RoR), which has been referenced many times.

Lucas,

I noticed in your work on your website that you use the term "convenient" where we would say "advantageous", as in your phrase "convenient decks" rather than "advantageous decks".

In terms of brick-and-mortar play, what would your website give that I could not obtain from, say, CVData or MGP's Combinatorial Analysis program?

Dog Hand

Quote: BleedingChipsSlowly

For this forum, I think a better answer to your question, “What kind of performance can you expect?,” can be found in the abstract of your research paper:
“betting strategies based on the exact composition of the deck slightly outperform the Hi-Lo counting system.” If you offered free access to your commercialized product for some members of this site you might get more meaningful criticism than the pier review of your paper can provide.
link to original post

Hello friend, thank you for your reply. The paper does not cover all the work we have done. On the site itself in the SIMULATIONS section we show extremely clearly and explicitly the performance of all our strategies (you cannot reduce the comparison between the performance of 2 strategies to one being “better than the other”). What you say about giving free access is true and is a good idea, and we would do it with members who are referents within the community. Anyway we are currently with 90% discount, and the price is really low, just looking for people to know the product. Greetings my friend.

Quote: mmmoretti11

Great work, Lucas! My advice would be to keep the platform as beginner-friendly as possible maybe add a simple guide or video walkthrough for new users. The math is impressive, but making it easy to understand will really help more people get value from it.
link to original post

Thank you for your good energies my friend. What you mentioned was a key point of the work. Indeed, we worked hard to make the site targeted to a general audience, making sure that the way the strategies are presented and their performance is easy for anyone to understand. We have text tutorials on how to use the calculator within the site (it computes the strategies), and the calculator itself has been optimized to be intuitive, although of course there is still room for improvement. I will be happy to help you with any questions.

Quote: aceside

There are a few new definitions and acronyms here:

Expected Utility Theory
Absolute Risk Inclination (ARI)
Relative Risk Inclination (RRI)
Target Return (TR) for the session

However, there is no risk-of-ruin (RoR), which has been referenced many times.
link to original post

Hello my friend. I will try to be clear. We have covered several rules that define variants of the game (ENHC, soft17, double, etc). Then, a strategy is defined by 5 degrees of freedom: counting method on the basis of which optimal bets are made (true count or exact composition of the deck), decision method during each round (basic strategy or optimal composition dependent decision), ARI, RRI and Target return. ARI, RRI and TR configure the risk appetite of the strategy and the target return. The ROR (risk of ruin) of the strategy is implicit. That is, once a strategy is configured under a specific rule environment, you can observe its performance in the simulations section. Here you can see explicitly how the strategy performs during a betting session. Within the information associated with the strategy performance, you can observe the ROR of the strategy throughout the session. I hope I helped you my friend!

Quote: DogHand

Lucas,

I noticed in your work on your website that you use the term "convenient" where we would say "advantageous", as in your phrase "convenient decks" rather than "advantageous decks".

In terms of brick-and-mortar play, what would your website give that I could not obtain from, say, CVData or MGP's Combinatorial Analysis program?

Dog Hand
link to original post

Hello friend, thanks for the language comment, we will certainly update “convenient” to “advantageous”! Regarding your question, I could answer it precisely if you tell me clearly what you get from these programs, as I have not used them personally and I only have an idea of what they offer. However, I'll go ahead and say that we offer extremely flexible strategies, definable by 2 risk parameters (ARI and RRI) and a Target Return of withdrawal for the betting session. This flexibility allows you to adjust a strategy precisely to your interests. So, let's say you start your betting session with $1000. Once you define a strategy (ari, rri, tr), we give you an exact formula to calculate the size of the bet before each round according to 2 variables: the value of the true count, and the value of your current bankroll (the bet is not the same if you have $500, $1300, $1000, etc). In addition to providing the formula, the way in which we present the performance associated with this strategy (SIMULATIONS section) is extremely clear and detailed, through the explicit distribution of your bankroll over time and tables with values (you can also simulate betting sessions in real time and visualize the evolution of your equity). I recommend the web version of the site ! I look forward to your comments my friend.

When betting, several Blackjack authors cite the Kelly criterion, which is a formula for sizing a sequence of bets by maximizing the long-term expected value (EV) of your bankroll. I’m learning and this is my understanding of Kelly. Your wager amount should be proportional to the current EV but inversely proportional to the current standard deviation (SD) , that is,

Wager = constant x EV / SD.

I also believe wager should be proportional to your current bankroll. With these two relationships, a betting sequence is well defined. All we need is these two, EV and SD (square root of variance). Why do we need your “5 degrees of freedom” (counting method, strategy, ARI, RRI, and TR) to size our bet?

Quote: aceside

When betting, several Blackjack authors cite the Kelly criterion, which is a formula for sizing a sequence of bets by maximizing the long-term expected value (EV) of your bankroll. I’m learning and this is my understanding of Kelly. Your wager amount should be proportional to the current EV but inversely proportional to the current standard deviation (SD) , that is,

Wager = constant x EV / SD.

I also believe wager should be proportional to your current bankroll. With these two relationships, a betting sequence is well defined. All we need is these two, EV and SD (square root of variance). Why do we need your “5 degrees of freedom” (counting method, strategy, ARI, RRI, and TR) to size our bet?
link to original post

Thank you for your excellent comment. To recap, the Kelly criterion does not maximize expected return in the long run—it maximizes the median of your returns over time. The formula you presented is roughly correct and can be expressed as:

Wager = (EV / STD) * (rri * current_bankroll)

Here, "rri" stands for Relative Risk Inclination, which corresponds to the constant you mentioned but is named this way because it scales proportionally with the current bankroll. It's not that the five degrees of freedom I mentioned are strictly "necessary," but rather that they offer a richer and more complete way to define a betting strategy. Let me explain.

Decision Method (DM)
The Decision Method refers to the strategy you follow in each round (e.g., composition-dependent or basic strategy). This method determines the behavior of your return per round, specifically the EV (Expected Value) and STD (Standard Deviation) of your return. For example, composition-dependent strategy yields the maximum possible EV.

Counting Method (CM)
The Counting Method is a system for deriving information from the deck. Based on the count value (Hi-Lo, exact composition, etc.), and in conjunction with a chosen Decision Method, we can calculate the EV and STD of the return for the upcoming round. Together, the CM and DM allow us to compute the following:

EV(count, DM): The expected value of the next round, given the current count and decision method.

STD(count, DM): The standard deviation of the return for the next round, given the current count and decision method.

ARI, RRI, and TR
With EV(count, DM) and STD(count, DM) established, we can now formulate a flexible betting function tailored to a user’s risk preferences:

Wager(count, DM, ari, rri, tr) = (EV(count, DM) / STD(count, DM)) * (ari * initial_bankroll + rri * current_bankroll)

Here:

ARI (Absolute Risk Inclination) is tied to the initial bankroll, making it an absolute measure.

RRI (Relative Risk Inclination) is tied to the current bankroll, making it a relative measure.

TR (Target Return) modifies the wager as the player nears their target profit. For example, if TR = 50%, the strategy will bet according to the formula above, but will reduce bet sizes as gains approach 50%, and eventually stop once the target is reached.

This is the most efficient way to reach a desired return, given the CM, DM, and selected risk parameters (ARI and RRI).

On our website, you can configure strategies by choosing:

CM (Composition Dependent or True Count)

DM (Composition Dependent or Basic Strategy)

ARI-RRI values (from 0 to 20)

Various Target Returns (25%, 50%, 100%, ... up to 500%)

You can then visualize performance for each configured strategy in the SIMULATIONS section, and finally, access the explicit strategy using our live calculator.

I hope this helped clarify things. I’m happy to continue answering your excellent questions!