Quote from: casinobitco on June 01, 2016, 08:03:12 PM

OK, I follow that. Let's talk practical --- are there really 2^31 outcomes that are good for the house (only) in Blackjack, Roulette, Video Poker?

Still would love seeing some real proof where I can change the client seed to anything I want, and still get one of those 'rigged' shuffles; otherwise this is just a thread with a ton of people (not you, or the OP) chiming in who have no understanding of how math or provably fair works.

Just checking in with you, casinobitco, if RHavar provided enough information for you to examine shufflepuff in greater detail.

A Provably Unfair Blueprint

Here's what a malicious casino would do if they wanted to cheat.

#1: Take your standard roulette wheel and letter encoding.

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36
0	1	2	3	4	5	6	7	8	9	a	b	c	d	e	f	g	h	i	j	k	l	m	n	o	p	q	r	s	t	u	v	w	x	y	z	A

#2: Convert the wheel to a point deck based on the desired optimization.

Here, we represent the wheel as a set of points. For illustration, we'll keep it simple and use a zero (0) as neutral and a one (1) for the wheel positions that favor the house. You could use weighted values, like -1 and 1 to show the exact deviation in favor of the house or player. I prefer 0s and 1s to stay aware of the count within a space (just a math thing).

Examples:

Optimize for Color (red/black)
0 000000 000000 000000 111111 111111 111111

Optimize for Green
1 000000 000000 000000 000000 000000 000000

Optimize for Third
0 111111 111111 000000 000000 000000 000000

For this example, I'll choose to optimize for color.

#3: Run shufflepuff with the desired seed space.

First, let's compare the seed space (2³² = 4,294,967,296) and the arrangement space (37! / 19! / 18! = 17,672,631,900). The seed space covers about 24.3% of the arrangement space (4,294,967,296 / 17,672,631,900). What does this mean? It means that there are some final shuffles that are unattainable if the house decides to optimize the deck. In roulette, we're drawing one "card" (number). To optimize, we want to push as many favorable shuffles for the house into the seed space, and push as many favorable shuffles for the player outside of the seed space (into the unattainable space). This seed space covers a relatively "large" portion of the arrangement space, so one would expect there to be optimization, but smaller optimization than in other games, such as blackjack.

Someone might say, "Wait, there are actually 37! possible initial decks in roulette." That's true, but since we're optimizing for color, we can cut down our search space by ignoring the order of the numbers and focus on the position of the colors. You'll see why in a bit.

As a toy example, let's reduce the seed space to something small, like 40. Fire up shufflepuff and you'll see this:

0000000000000000000111111111111111111: 16
0000000000000000001101111111111111111: 17
0000000000000000001111011111111111111: 18
0000000000000001000111011111111111111: 19
0000000000000001001101011111111111111: 20
0000000000000001001111011101111111111: 21
0000000000001001001101011101111111111: 22
0000000000001001001111011100111111111: 23
0000000000011001001111011100110111111: 24
0000000000111001001111011100110011111: 25
0000000001001001001101011100111111111: 26
0000000001001001001111011100110111111: 27
0000000001011001001111011100110011111: 28
0000000001111001001111011100110001111: 29
0000000011111001001111011100110001110: 30
...
0000111001001001001111011100110001110: 35

The last line indicates the arrangement and the number of seeds (out of 40) that are beaten. So, in this example, if we optimize for the house, the house will win 35/40, or 87.5% of the time.

#4: Partition the roulette encoding scheme and randomize.

For this instance, I'll optimize for black, meaning I want black to appear 87.5% of the time. So, I'll partition the encoding into red + green, and black:

Red + Green: "013579cegijlnpruwyA"
Black: "2468abdfhkmoqstvxz"

Now we randomize each encoding:

Red + Green: "Awp1yejr5c3ngi07u9l"
Black: "davhk4mtz82sbof6qx"

#5: Zip the shuffled encoding onto the optimized arrangement.

0000111001001001001111011100110001110 =
Awp1 ye jr 5c 3n g i0 7u9 l
dav h k 4 mtz8 2sb of 6qx

Result: Awp1davyehjrk5c43nmtz8g2sbi0of7u96qxl

You can keep randomizing to create 19! * 18! = 7.788 x 10³² cold decks. Using this deck, or any under this optimization, results in black appearing 87.5% of the time, despite an expectation of 18/37 = 48.65%. There is no way for the client to randomize their seed enough to beat this optimization.

Granted, with a 2³² seed space, you're not going to get such a dramatic result. Additionally, even if you found a high optimization, you probably wouldn't play it often (tuck it away as your "nuclear" option).

Here's the big thing: You also have all the arrangements > 20 that also work well. They can also be randomized and used against the player. Again, the casino doesn't necessarily want to find the best deck, just a better deck.

It's also easy to optimize for red now. Just flip the red and black (leave green alone).

Let me know if you have more questions. Thanks for reading!

Addendum
Remember, the shufflepuff code does not adhere perfectly to any one particular casino, so while it will give you optimized decks for its configuration, the decks will fail if implemented. This is intentional.