This article discusses the card game blackjack as played in the casinos of Las Vegas. The basic rules for the game are described in detail. The player's strategicâ€‹.

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This article discusses the card game blackjack as played in the casinos of Las Vegas. The basic rules for the game are described in detail. The player's strategicâ€‹.

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#4: ALWAYS HIT A HARD 12 AGAINST A DEALER'S 2 OR 3 UPCARD.

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If you're an active player that number will add up over time. The best (and only correct) mathematical strategy for achieving optimal play is to use a blackjack chart.

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#2: ALWAYS SPLIT A PAIR OF 8s AND ACES.

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If you're an active player that number will add up over time. The best (and only correct) mathematical strategy for achieving optimal play is to use a blackjack chart.

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Here Is All Of Basic Strategy In 30 Simple Phrases: In case you're not a visual learner these simple phrases might help you commit these rules to memory. Theâ€‹.

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Here Is All Of Basic Strategy In 30 Simple Phrases: In case you're not a visual learner these simple phrases might help you commit these rules to memory. Theâ€‹.

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This article discusses the card game blackjack as played in the casinos of Las Vegas. The basic rules for the game are described in detail. The player's strategicâ€‹.

Enjoy!

Optimal Blackjack Strategy. The rules of play. 1. The number of players. At most blackjack tables, there is one dealer and from one to six players. The player to.

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This is the very best solution based on fitness score from candidates in generation 0 the first, random generation :. The first thing to notice is that the two smallest populations having only and candidates respectively, shown in blue and orange performed the worst of all sizes. One of the unusual aspects to working with a GA is that it has so many settings that need to be configured. By measuring the standard deviation of the set of scores we get a sense of how much variability we have across the set for a test of N hands. The first generation is populated with completely random solutions. One simple approach is called Tournament Selection , and it works by picking N random candidates from the population and using the one with the best fitness score. It works by using a population of potential solutions to a problem, repeatedly selecting and breeding the most successful candidates until the ultimate solution emerges after a number of generations. Imagine a pie chart with three wedges of size 1, 2, and 5. As impressive as the resulting strategy is, we need to put it into context by thinking about the scope of the problem. As it turns out, you need to play a lot of hands with a strategy to determine its quality. Since the parents were selected with an eye to fitness, the goal is to pass on the successful elements from both parents. Once an effective fitness function is created, the next decision when using a GA is how to do selection. A cell in the child is populated by choosing the corresponding cell from one of the two parents. Given those findings, the fitness function for a strategy will need to play at least , hands of Blackjack, using the following rules common in real-world casinos :. Basic concepts get developed first with GAs, with the details coming in later generations. Neural networks are great for finding patterns in data, resulting in predictive capabilities that are truly impressive. Varying each of these gives different results. One of the cool things about GAs is simply watching them evolve a solution. But how many hands is enough? Once two parents are selected, they are crossed over to form a child. To use the tables, a player would first determine if they have a pair, soft hand or hard hand, then look in the appropriate table using the row corresponding to their hand holding, and the column corresponding to the dealer upcard. A higher fitness score for a strategy merely means it lost less money than others might have. The solution is to use Ranked Selection , which works by sorting the candidates by fitness, then giving the worst candidate a score of 1, the next worse a score of 2, and so forth, all the way up to the best candidate, which receives a score equal to the population size. The source code for the software that produced these images is open source. If you play long enough, you will lose money. Each candidate has a fitness score that indicates how good it is. Finally, the best solution found over generations:. In the case of a Blackjack strategy, the fitness score is pretty straightforward: if you play N hands of Blackjack using the strategy, how much money do you have when done? By generation 33, things are starting to become clear:. Could we run with , or more hands per test? Clearly, having a large enough population to ensure genetic diversity is important. The chart here that demonstrates how the variability shrinks as we play more hands:. In fact, the coefficient of variation for , hands is 0. The variations from run to run for the same strategy will reveal how much variability there is, which is driven in part by the number of hands tested. If, by luck, there are a couple of candidates that have fitness scores far higher than the others, they may be disproportionately selected, which reduces genetic diversity. Roulette Wheel Selection selects candidates proportionate to their fitness scores. Genetic algorithms are essentially driven by fitness functions. The soft hand and pairs tables are getting more refined:. There will be large swings in fitness scores reported for the same strategy at these levels. Reinforcement learning uses rewards-based concepts, improving over time. Tournament selection has already been covered. The following items can be configured for a run:. Oftentimes, crossover is done proportional to the relative fitness scores, so one parent could end up contributing many more table cells than the other if they had a significantly better fitness score. And then the final generations are used to refine the strategies. One of the problems with that selection method is that sometimes certain candidates will have such a small fitness score that they never get selected. The other hints of quality in the strategy are the hard 11 and hard 10 holdings. The lack of genetic diversity in those small populations results in poor final fitness scores, along with a slower process of finding a solution. That optimal strategy looks something like this:. Back in the s, a mathematician named Edward O. The best way to settle on values for these settings is simply to experiment. First, testing with only 5, or 10, hands is not sufficient. Here are two other approaches:. The columns along the tops of the three tables are for the dealer upcard, which influences strategy. As you might imagine, Blackjack has been studied by mathematicians and computer scientists for a long, long time. That gives us something called the coefficient of variation , which can be compared to other test values, regardless of the number of hands played. By generation 12, some things are starting to take shape:. We solve this by dividing the standard deviation by the average fitness score for each of the test values the number of hands played, that is.

One of the great things about machine learning is that there are so many different approaches to solving problems. During that run, aboutstrategies were evaluated. The three tables represent a complete strategy for playing Blackjack. The tall table on the left is for hard handsthe table in the upper right is for soft handsand the table in the lower right is for pairs.

But that improvement is definitely a case of diminishing returns: the number of tests had to be increased 5x just to get half the variability.

The hard hands in particular the table on the left are almost exactly correct. With only 12 generations experience, us blackjack sites most successful strategies are those that Stand with a hard 20, 19, 18, and possibly That part of the strategy develops first because it happens so often and it has a fairly unambiguous result.

Knowing the optimal solution to a problem like this is actually very blackjack optimal strategy. Populations that are too small or too homogenous always perform worse than bigger and more diverse populations. The pairs and soft hand tables develop last because those hands happen so infrequently.

Of course, in reality there is no winning strategy for Blackjack â€” the rules are set up so the house always blackjack optimal strategy an edge. The more hands blackjack optimal strategy, the smaller the variations will be.

The flat white line along the top of the chart is the fitness score for the known, optimal baseline strategy. A pair is self-explanatory, and a hard hand is basically everything else, reduced to a total hand value.

Because of the innate randomness blackjack optimal strategy a deck of cards, many hands need to be played so the randomness evens out across the candidates.

This works just like regular sexual reproduction â€” genetic material from both parents are combined. Standard deviation is scaled to the underlying data. A genetic algorithm GA uses principles from evolution to solve problems. Of course. Population Size.

The X axis of this chart is the generation number with a maximum ofand the Y axis is the average fitness score per generation.

Running on a standard desktop computer, it took about 75 minutes. Comparing the results from a GA to the known solution will demonstrate how effective the technique is. Blackjack optimal strategy such a strategy allows a player to stretch a bankroll as far as possible while hoping for a run of short-term good luck.

The idea of a fitness function is simple. There are a couple of observations from the chart. That means that if the same GA code is run twice in a row, two different results will be returned. The fitness function reflects the relative fitness levels of the candidates passed to it, so the scores can effectively be used for selection.

The goal is to find a strategy that is the very best possible, resulting in maximized winnings over time. In fact, it looks like a minimum ofhands is probably reasonable, because that is the point at which the variability starts to flatten see more. There are a number of different selection techniques to control how much a selection is driven by fitness score vs.

That score is calculated once per generation for all candidates, and can be used to compare them to each other. Using a single strategy, multiple tests are run, resulting in a set of fitness scores. Once this fitness score adjustment is complete, Roulette Wheel selection is used.

To avoid that problem, genetic algorithms sometimes use mutation the introduction of completely new genetic material to boost genetic diversity, although larger initial populations also help. Even though we may not know the optimal solution to a problem, we do have a way to measure potential solutions against each other. It reduces variability and increases the accuracy of the fitness function. The process of finding good candidates for crossover is called selection, and there are a number of ways to do it. Due to the house edge, all strategies will lose money, which means all fitness scores will be negative. That evolutionary process is driven by comparing candidate solutions. Knowing that, the best possible strategy is the one that minimizes losses.