If your RNG was perfect, additional shuffles would neither improve or degrade the randomness.

It seems that additional "shuffling" is required to prevent a permutation of the array where there is a correlation between items that are swapped.

It seems the "additional shuffling" is required.

Consider the shuffling of an array of three items without the so called "additional shuffling": Starting from the left:

Initial:   abc

1 move:    abc             bac(1/3)        cba(1/3)

2 move:    abc(1/6)  acb(1/6)
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I take issue with:

On the surface, each array entry has the appearance of being given a single opportunity at being swapped with another array entry. In practice, array entries nearer to the beginning of the array have additional opportunities of being swapped (as a result of later swaps), meaning that less shuffling happens at the end of the array than at the beginning.

One could argue in great length whether additional shuffling in the array improves the randomness, or whether it produces patterns in the longer term, but it isn't difficult to argue that regardless of which conclusion is valid, the fact that the entries nearer to the beginning have additional chances of being shuffled, while entries nearer the end have fewer chances, that *one* of these ends will be improved more than the other, therefore the algorithm is not optimal.

Which seems to say that the possible movement of an already moved item is a weakness of the algorithm. But an array cannot be randomly shuffled in place without allowing elements to be moved more than once.

The idea that the errors of a RNG will more adversely affect the items to which that RNG is more often applied is an argument that is more subtle. I reject it out of hand. If the RNG is incorrect the results will not be random and we are quibling about how obvious the error is. This distinction may be very important in practice and there are practical solutions to get a pseudo-randomness that is okay for varying definitions of okay.