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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?
A mutation rate controls how often a newly created candidate will be mutated. The mutation is done immediately after creation via crossover. The mutation impact controls how much a candidate is mutated, in terms of percentage of its cells that will be randomly changed.
All three tables hard hands, soft hands and pairs are mutated the same percentage. It is clear that mutation does not help for this problem — the more candidates are affected by mutation, the worse the results.
Knowing when to quit a genetic algorithm can be tricky. Some situations call for a fixed number of generations, but for this problem the solution was to look for stagnation — in other words, the genetic algorithm stops when it detects that the candidates are no longer improving.
Genetic algorithms are a powerful technique for solving complex problems, and they have the benefit of being easy to understand.
For problems with huge solution spaces due to combinatorial factors, they are extremely effective. For more information about GA, please start with this Wikipedia article or the PluralSight paid course I wrote that covers the topic in far greater detail.
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Greg Sommerville. Results Using a GA One of the cool things about GAs is simply watching them evolve a solution.
Combinatorial Implications As impressive as the resulting strategy is, we need to put it into context by thinking about the scope of the problem.
Testing Fitness Genetic algorithms are essentially driven by fitness functions. Genetic Algorithm Configurations 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: Population Size Selection Method Mutation Rate and Impact Termination Conditions Varying each of these gives different results.
Written by Greg Sommerville. Computer programmer with extensive experience in front end development,. NET, and machine learning.
Check your inbox Medium sent you an email at to complete your subscription. 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.
The reason splitting is the better strategy with DAS is because if you split, say, a pair of 2s and draw a 9 giving you an 11, or an 8, giving you a 10, you would be able to bet more money by doubling down in a very favorable situation.
Note: There is one exception to the above rule: If you are playing a single-deck game, you should always split a pair 2s when the dealer shows a 3 upcard, even if the game is NDAS.
The reason is because the payoff for the insurance bet 2 to 1 is less than the odds that the dealer will have a blackjack, making it a sucker bet.
Although this strategy is correct, you can improve your playing accuracy by taking into account whether your 16 is a multi-card In the latter case e.
Historically, a blackjack hand has always been paid at 3 to 2 odds. The house edge increases by about 1.
Your best strategy is to play only blackjack games that offer a 3 to 2 blackjack payoff. He is the author of the Ultimate Blackjack Strategy Guide, and Blackjack: Take The Money and Run.A, stand on 17 vs. The process of finding good candidates for crossover is called selection, and there are a number of Haarnetz Dm to do it. The soft hand and pairs tables are getting more refined:. 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.