Card Shuffle Inspector

A program designed to check if a shuffle is efficient -- by Stéphane Kastenbaum

Introduction

Since a discussion I had with my friend I wanted to know if there is a way to compute how efficient is a shuffle. I don't want to do to much of statistics so I'm gonna bruteforce a little bit.

The goal is to program to compare different shuffling method. Idealy, in the end I could implement every kind of shuffling method.

Representing a shuffle

A shuffle is a permutation. If I take the first card and put it to the end of my deck, this is a simple cycle permutation. Yet when one shuffle a deck it usualy involves randomness ("Take the first card and put it whether at the top or at the bottom of the deck") So I need something better.

My idea is to use matrices (because matrices are cool) where Mij is the probability to put the j card at the i place. So I (the identity matrix) represent no shuffling, and for a deck of three card the matrix

0 0 1
0 1 0
1 0 0

represent the inversion of the first and the third card.

Also we can represent the chance of card being at a place or at another. For example the matrix

1/3 1/3 1/3
1/3 1/3 1/3
1/3 1/3 1/3

would be the perfect shuffling where every card have equal probability to be at every position.

It is important to notice that since two cards cannot be at the same place and a card cannot be at two different place, the sum of the probabilites over a line and over a column equals to 1.

With this method we do not differenciate a shuffle method and the state of a deck of cards, this is convenient since the state of a deck is the same as "the shuffle applied to the identity matrix"

Computing the shuffle matrix

In the real world people often choose a shuffle method and apply it multiple time on the same deck. It can be represented on my method by finding the matrix of a single shuffle, and multiple the matrix by itself.

Finding the single shuffle matrix

For some kind of shuffle method the matrix can be found by hand, but I think this could be hard and boring to find the right matrix for some shuffling method, the faro shuffle for example.

So what I'm gonna do, is creating a function which will do the said shuffle on a list and then call it many times and finding the mean of the shuffled states.

##Implementation To do this I will use :

• Python 3
• Numpy to represent the matrices
• random to generate randomness

Note on the represntation of the matrices

Since every value of the matrices is inferior to 1 and approximation on small number are quite bad in python I decided that I will miltiply the matricies by the size of the deck. So with n the size of the deck n * I will be the deck where each card is at its initial position and the matrix with 1 everywhere will be the uniformely distributed one.

TODO list

• Do a function that compute the "efficienty" of the shuffle matrix
• Do a function that given a random shuffle function return a approximation of the corresponding shuffle matrix
• Add a way to handle a deck with the same card multiple times.
• Clarify readme
• Implement Faro shuffle
• Add other types of shuffle

State of the project

Date of this edit : 2018-03-17

The project is now abandoned because I found a major flaw on the representation of the shuffles. I will keep this project on GitHub as is, even though it will never work as intended. I think it can be a source of inspiration for others, or myself if I ever want to work on this again.

The flaw

The flaw is quite obvious actually, it can be seen on the very first example of shuffle I made.

The shuffle is : "Take the first card and put it whether at the top or at the bottom of the deck"

This is a very bad shuffle, because it doesn't actually mess with the order of the cards. Indeed if we have a deck of 6 cards the state of the deck after a very large number of the top_bottom shuffle will be one of theses:

1 2 3 4 5 6 (The starting state)
2 3 4 5 6 1
3 4 5 6 1 2
4 5 6 1 2 3
5 6 1 2 3 4
6 1 2 3 4 5

which mean that if we look at the first card of the deck, we know the state of the deck.

Now, if we try to use my method to see if it's a good shuffle, the result will be:

first_top_bottom,  200  times, with  6  cards.
[[ 1.  1.  1.  1.  1.  1.]
 [ 1.  1.  1.  1.  1.  1.]
 [ 1.  1.  1.  1.  1.  1.]
 [ 1.  1.  1.  1.  1.  1.]
 [ 1.  1.  1.  1.  1.  1.]
 [ 1.  1.  1.  1.  1.  1.]]

Because each card as an equal probability of being at any position in the deck.

So my method consider this shuffle to be good (or good enough), yet the shuffle is very bad. This, in my opinion, makes my method not-usable.

This flaw comes from the fact that I didn't take into account the influence each card has on the others. The probability of each card to be at any position is not enough to judge the "randomness" of a shuffle.

Well, too bad. I should look up on wikipedia how to evaluate shuffles.