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Comment by seanhunter | Hacker News Reader

I know of people who have written things like online casino games (in unregulated jurisdictions) where the shuffle was deliberately skewed in order to achieve certain configurable winning percentages from the games. ie it's not done by accident but design.

That said people do sometimes unintentionally write something like this (terrible and also buggy) algo

   for idx in size(deck):
      new_spot = int(random() * size(deck))
      swap(deck[idx], deck[new_spot])

...which won't give an even distribution of all possible outcomes. If shuffling, look up a real algo (like Fisher-Yates for instance) and implement that. Or better yet, use a library implementation.

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The correct Fisher-Yates shuffle is this (from Wikipedia):

  -- To shuffle an array a of n elements (indices 0..n-1):
  for i from 0 to n−2 do
     j ← random integer such that i ≤ j < n
     exchange a[i] and a[j]

For those interested, I wrote a small article on the various was of writing incorrect Fisher-Yates [0]. The post also covers generating single cycle permutations.

Fisher-Yates is pretty obvious but weirdly subtle in some ways. For someone not paying attention, it would be easy to get wrong, as the grandparent response points out.

[0] https://mechaelephant.com/dev/Fisher-Yates-Shuffle.html

Interestingly the method used to deal bridge hands for top tournaments doesn’t shuffle at all, it generates a random number between 1 and 52!/(13!^4) and maps that number to a corresponding deal of the cards.

Are you sure they do bijective mapping instead of simply running a RNG with the given seed?

Yes - see the “Making bridge deals” section of:

https://sater.home.xs4all.nl/doc.html

Right. The subtle difference being that because you only sample from position i to n you only ever exchange each card exactly once. In my algo you sample from 0 to n-1 so a card exchanged in a previous round can be exchanged again. This generalises into cards early in the shuffle being exchanged more than later cards, which changes the distribution of possible outcomes.

Nerd sidenote: Because generating random numbers tends to be expensive, I had an idea once where you generate the full permutation with only a single random sample. Basically you could use something which fully enumerates a space like Gray codes[1] (you want one with a closed form solution which can arbitrarily you the nth code) then you generate one random number, multiply by the number of combinations, look up that code and decode that into the sequence of cards. I never actually got around to implementing it, but I suspect if your code lookup was fast it would be much faster than Fisher-Yates.

The use case was if you wanted to make a Monte Carlo algorithm to evaluate strategies in a card game for instance.

[1] https://en.wikipedia.org/wiki/Gray_code

I wrote that code for fun 15-20 years ago (code definitely lost in the mists of time).

You can think of it as rolling one random number up to n!, then use x%n to pick the first element and then x/n to pick the next (n-1) elements (so the next step is then to pick the second element with (x/n)%(n-1) and use (x/n)/(n-1) for the remaining (n-2) elements).

Though as a sibling comment points out, since you need a random number up to n!, even 64 bits of randomness only get you so far (I think up to 20, but didn't check my math).

You’ll find that the number of ways to shuffle a card pack (52!) is about 8 × 10⁶⁷, so even if your RNG returns 64 bits, you can’t use a single call.

Also, you’ll have to make sure you pick each permutation is picked with equal probability. random() % factorial(52) likely won’t work because your RNG won’t return integers in a range k × 52! for some k > 1 (https://stackoverflow.com/questions/10984974/why-do-people-s...)

I've wondered about the same thing of generating a permutation with a minimum number of samples. I'm not sure how Gray codes helps you here, and the multiplication sounds wrong since many outputs aren't reachable from any input.

The idea is just a closed form solution that gets you from a number in 1...nPermutations to a single specific permutation. I was using Gray codes as an example of a sequence that exhausts all permutations with a closed form that gets you to a specific point in the sequence.

Might not work - as I say I never actually got around to implementing it.

I think you're looking for a factorial number system [1] and/or Lehmer code [2].

[1] https://en.wikipedia.org/wiki/Factorial_number_system [2] https://en.wikipedia.org/wiki/Lehmer_code

Thanks!

The n-2 bit is unnecessary and misleading because we've already established that an index can be swapped with itself.

for i in [0..n-1] swap(a[i], a[rand in [i..n-1])

where rand in [I..n-1] coud be implemented as i + (rand() % (n-i))

essentially start with your finger at 0, pick a card from the unpicked set which is everything on and after your finger, swap that card with your finger, move your finger up one card

Of course, the question is why would this be the case in Arena. You play against other players, not the casino.

The theory I’ve always heard about why the shuffler might be biased is that they’re trying to reduce land flood/screw. Drawing too few or too many lands is a commonly cited cause of frustration with players and if you’re reasonably assuming Arena wants to do all the usual monetised-game things of increasing playtime and MAU whatnot, this conclusion seems one of the first people come to.

They make money by selling the cards right? I would think that might change their incentives somewhat. I don’t really know the game.

Players who aren’t having fun won’t buy cards or will stop buying cards when they quit.

The better "experience" they transmit to the players, the more engaged they will be and the more cards/packs they will sell

Yes, but a frustrated player is more likely to buy event entries, aka gems, to draft more, and blame the poor randomization on bad luck, rather than intentional screw, because that is what you do at real life events. It's an insidious dopamine loop strategy, but it absolutely works.

> If shuffling, look up a real algo (like Fisher-Yates for instance) and implement that. Or better yet, use a library implementation.

If using a library implementation, you do still need to validate that it's using a proper algorithm. And that it doesn't change to something else during an update. Which probably requires you to understand the algorithm, and looking at wikipedia, it's a handful of lines of code to do it right and know that it doesn't change because it's your code. Maybe not worth the time to inspect and reinspect from a library.

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