Let’s first understand the shuffle algorithm. The Fisher-Yates Shuffle, also known as the Knuth Shuffle, is one of the most efficient and widely used algorithms for randomizing the order of elements in a list.
How the Fisher-Yates Shuffle Works
Here’s a step-by-step explanation of how the algorithm works:
- Initialize the Deck: Start with an array (or list) of n elements. For instance, consider a standard deck of 52 playing cards.
- Iterate from the Last Element to the First: The algorithm begins at the last element of the array and iterates backwards to the first element.
- Select a Random Element: For each position i (starting from the last element and moving towards the first), select a random index j such that 0 ≤ j ≤ i. This index j determines the position of the element to be swapped with the element at position i.
- Swap the Elements: Swap the elements at positions i and j. This step ensures that the element currently at position i is placed in a random position within the subarray of the first i+1 elements.
- Repeat: Continue this process for each element until you reach the first element of the array.
By the end of these steps, the elements of the array will be in a thoroughly randomized order. The key to the Fisher-Yates Shuffle’s fairness is that each of the n! (n factorial) possible permutations of the array is equally likely.
Verifying the Fairness
A “fair shuffle” means that all possible permutations of the cards appear with equal probability. To verify the fairness of a shuffle algorithm, we can use probability theory and statistical tests.
1. Probability Theory
Verify that the probability of each arrangement is 1/n!, where n! is the number of all possible permutations. For example, in a standard deck of 52 cards, the probability of each specific order should be 1/52! ≈ 1.2 x 10^-68.
2. Statistical Tests
Perform a large number of shuffles, such as 100,000 times, and analyze the actual shuffle results to see if they match the theoretical distribution.
Methods to verify the fairness
Chi-Square Test
You shuffle a deck many times, record how often each card order appears, and see if these match the expected counts.
Monte Carlo Simulation
In a Monte Carlo Simulation, you shuffle the deck a huge number of times and look at the results. You compare these results to what you’d expect from a fair shuffle.
Kolmogorov-Smirnov Test
The Kolmogorov-Smirnov Test compares the shuffled results to the ideal fair results by looking at the differences in their distributions. If the differences are small, the shuffle is fair.
Anderson-Darling Test
The Anderson-Darling Test is like the Kolmogorov-Smirnov Test but focuses more on the extremes (like very rare card orders).
Examples
Now that you understand how the shuffle algorithm works and how to test it, let’s actually run the algorithm. We will run both a flawed shuffle algorithm and the Fisher-Yates algorithm, which we use at Basepoker.
Flawed implementation
The following chart is an example of the results when the shuffle algorithm is incorrectly implemented. In this chart, clear patterns are visible, indicating that the shuffle is not performed randomly.
Right implementation
The following chart represents the results of a fair shuffle algorithm. Here, the distribution of consecutive card pairs appears uniform.
In conclusion, the Fisher-Yates shuffle algorithm was confirmed to be an effective method for meeting the fairness required in card games. However, implementing, verifying, and continuously monitoring the algorithm are essential processes for providing a reliable gaming environment.
