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Once two parents are selected, they are crossed over to form a child. This works just like regular sexual reproduction — genetic material from both parents are combined.
Since the parents were selected with an eye to fitness, the goal is to pass on the successful elements from both parents. A cell in the child is populated by choosing the corresponding cell from one of the two parents.
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.
Populations that are too small or too homogenous always perform worse than bigger and more diverse populations.
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.
One of the cool things about GAs is simply watching them evolve a solution. The first generation is populated with completely random solutions.
This is the very best solution based on fitness score from candidates in generation 0 the first, random generation :.
By generation 12, some things are starting to take shape:. With only 12 generations experience, the 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.
Basic concepts get developed first with GAs, with the details coming in later generations. The other hints of quality in the strategy are the hard 11 and hard 10 holdings.
The pairs and soft hand tables develop last because those hands happen so infrequently. By generation 33, things are starting to become clear:.
The soft hand and pairs tables are getting more refined:. And then the final generations are used to refine the strategies. Finally, the best solution found over generations:.
The hard hands in particular the table on the left are almost exactly correct. The source code for the software that produced these images is open source.
As impressive as the resulting strategy is, we need to put it into context by thinking about the scope of the problem.
Running on a standard desktop computer, it took about 75 minutes. During that run, about , strategies were evaluated.
Genetic algorithms are essentially driven by fitness functions. The idea of a fitness function is simple. Even though we may not know the optimal solution to a problem, we do have a way to measure potential solutions against each other.
The fitness function reflects the relative fitness levels of the candidates passed to it, so the scores can effectively be used for selection.
But how many hands is enough? As it turns out, you need to play a lot of hands with a strategy to determine its quality. Because of the innate randomness of a deck of cards, many hands need to be played so the randomness evens out across the candidates.
Using a single strategy, multiple tests are run, resulting in a set of fitness scores. 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.
The more hands played, the smaller the variations will be. 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.
Standard deviation is scaled to the underlying data. 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.
That gives us something called the coefficient of variation , which can be compared to other test values, regardless of the number of hands played.
The chart here that demonstrates how the variability shrinks as we play more hands:. There are a couple of observations from the chart.
First, testing with only 5, or 10, hands is not sufficient. There will be large swings in fitness scores reported for the same strategy at these levels.
In fact, it looks like a minimum of , hands is probably reasonable, because that is the point at which the variability starts to flatten out.
Could we run with , or more hands per test? Of course. It reduces variability and increases the accuracy of the fitness function.
In fact, the coefficient of variation for , hands is 0. 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.
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 :.
One of the unusual aspects to working with a GA is that it has so many settings that need to be configured.
The following items can be configured for a run:. Varying each of these gives different results. The best way to settle on values for these settings is simply to experiment.
Population Size. The X axis of this chart is the generation number with a maximum of , and the Y axis is the average fitness score per generation.
The flat white line along the top of the chart is the fitness score for the known, optimal baseline strategy. 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.
The lack of genetic diversity in those small populations results in poor final fitness scores, along with a slower process of finding a solution. Clearly, having a large enough population to ensure genetic diversity is important.
The process of finding good candidates for crossover is called selection, and there are a number of ways to do it. Tournament selection has already been covered.
Here are two other approaches:. Roulette Wheel Selection selects candidates proportionate to their fitness scores. Imagine a pie chart with three wedges of size 1, 2, and 5.
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.
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.
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.
Once this fitness score adjustment is complete, Roulette Wheel selection is used. As you can see, tourney selection converges on an optimal solution very quickly — in fact, the bigger the tourney size, the faster the average fitness score improves.
Even though it had the fastest initial improvement, Tourney 7 ends up producing the worst results. That makes sense, because although a big tourney size results in rapid improvement, it also limits the genetic pool to only the best.
The best performers look to be Tourney 2, Tourney 3, and Tourney 4. Given a population of , these numbers provide good long-term results.
After that is done, normal crossover begins. This chart shows the effects of four different elitism rates later generations only, to show the details.
You might think that deliberately including the best from each generation would speed things up, but in fact it looks like using only crossed-over candidates gives the best results, and is also the fastest.
Keeping genetic diversity high is important, and mutation is an easy way to introduce that. There are two factors relating to mutation: how often does it happen, and how much of an impact does it have when it does happen?
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Note: The following strategies can be used in all games unless stated otherwise. The strategies discussed below also assume you are a recreational player, and not a card counter.
If you are playing a multi-deck game where the rules specify the dealer must stand on soft 17, you are slightly better off hitting against a dealer Ace rather than doubling down.
They reason that because they are a big underdog in this situation, why bet more money by splitting? In other words, you will lose less money in the long run by splitting 8s against a 9, 10, or Ace than by hitting hard Note: If surrender is offered and you are playing in a multi-deck game with H17, or a double-deck game with H17 and NDAS, your best strategy is to surrender the pair of 8s vs.
A pair of 5s is also a hard 10 and you are always better off taking a one-or-more-card draw to a 10 than splitting the 5s and playing two hands, each starting with a 5.
Even though splitting 10s is much more often than not a winning play, keeping them together as 20 is an even greater winning play in all circumstances.
This is a situation where most players chicken out and stand on their 12 because they fear busting. The facts are these. This is because if you draw a small card e.
And if you draw any of the four ten-valued cards, you do no harm to the hand. Bottom line: Your best strategy is to always hit A-7 when the dealer shows a 9, 10, or Ace with a goal of getting to either a soft 19—21 or a hard 17 through If you are playing an H17 game, the above are the three doubling strategy changes you should make vs.
Note: There are surrender strategy changes as well. In all games, you stand to win more if you always double down an A-2 through A-7 i.
Where they fumble the ball is when the dealer shows a 7. One way to remember this best strategy is as follows. There is a good chance that the dealer will have a ten in the hole since there are four times as many ten-value cards in a deck than other ranks.
Your pair of 9s, which is an 18, would beat her potential 17, which makes standing the better play vs. A hard 15 and 16 are two of the worst hands in blackjack , especially when the dealer is showing a strong upcard e.
Surrender is your best strategy simply because it saves you money in the long run. Note: Depending on the number of decks of cards being used and the blackjack rules , there are other hands where surrender is the best strategy.
In double- and multi-deck games, you never double down with a two-card 8; however, in a single-deck game, the odds of blackjack shift to make doubling down the superior strategy over hitting.
Note: The above best strategy includes a pair of 4s, with one exception. If the rules are DAS, you should split a pair of 4s instead of doubling down.