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 Post subject: The MTGO Shuffler Thread
PostPosted: Sat Sep 26, 2015 2:10 pm 
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This thread is an institution, there is no way I can let it die with the WotC Forums.

First, the [O]fficial explaination of how the MTGO shuffler works, originally provided by Chris Green, president of Leaping Lizards (who developed the thing), preserved by muriban and passed on to Vitalogy1994 who posted it in the last incarnation of this thread:
ChrisG wrote:
MTGO's Shuffle Algorithm...get the technical low down... March 15, 2002 by Chris Green A technical description of Magic Online's shuffler and random number generator. The core random number generator used is "Algorithm A", from Knuth's "Art of Computer Programming", sec 3.2.2. This is a fast, easy to implement random number generator. Knuth states that it has "consistently produced reliable results, in extensive tests since its invention in 1958." I first implemented this generator in 6502 assembly code in 1981 or so, and it has never failed me. The implementation of this generator used in our libraries uses the standard constants (24,55). Because this is somewhat fewer than the number of bits required to produce all possible hands, it was augmented with another generator using the constants (33,68). This yields a total state size of 3936 bits. Both generators were combined so that the random number calls used in our library could still return the same sequence of numbers when initiated by our old programs (never know when we might have to rebuild a new version of Centipede3D for the Dreamcast :-) ). In MTGO, random numbers are initialized by the game servers. When a new game is started, the random number state is seeded via /dev/random, which uses hardware delays for a source of true random data. In addition, whenever a packet is received from a user by the game server, the lower order bits of the system CPU's clock cycle counter are added into the random state. Shuffling is performed by swapping every card in the deck with a random other card in the deck. This is algorithm "P" for shuffling from Knuth. The book contains a formal analysis of its randomness. The 32 bit random values returned by the basic random number function are mapped into the appropriate range by fixed point multiplication. One of our programmers, Sergey, was not satisfied that the random number generator wasn't mana-screwing him, and so performed the following test: The shuffler has no idea what is a land and is not a land, so if there is any unnatural clumping of lands, it must be based upon the initial ordering of the deck. So he performed the following test: Create in memory a virtual deck of 20 "1"s, representing lands, and 40 "0"s representing non-lands. Put all the "lands" first and then all the "nonlands". Apply the shuffler. Perform the same test, except with lands and nonlands interleaved before shuffling. Perform each test multiple millions of times. After each test, count the sizes of land/non-land clusters and keep a running total of each. Compare the results from the millions of runs with the deck ordered with all lands together versus the interleaved one. The results were the same to within a minuscule fraction of a percent. In addition, he wished to verify that shuffling extra times would have no effect. If it did have an effect that would mean that the shuffle was insufficiently random. He performed this test and got the same statistics from one shuffle as from many.


We have no guarantee the shuffler is identical to the original incarnation of MTGO, but there also doesn't seem to be a reason they would change it.

The primary reasons people think the shuffler isn't random:
Most people are under the misconception that a random distribution means an even distribution. Random distributions have natural clustering.

Experience doesn't match what they are used to in paper; there are many poor (human) shufflers out there. When confronted with *actual* randomness of the computer shuffler, they blame the computer instead of their own faulty shuffling practices. In reality these people have been unknowingly cheating for a long time.

People tend to remember significant events more vividly then normal occurrences. They remember the times when they were horrifically flooded but forget all the normal games in between.

Now, some required reading before you post if you want any hope of people caring what you say:

Stats 101: Am I Shuffling Enough - Or Correctly, For That Matter? - A very good explaination of what a randomized deck *should* look like; it's far different than many think. [Provided by Vitalogy]

The Wikipedia Entry on Knuth Shuffling. Here's a code snipet using the method [adapted from code posted by many]:
Code:
deck = new Card[n]          ; an array of Card objects n long (syntax: first element is element zero)
do i=0,n-1
    swap = Random(i,n-1)    ; random integer between i and n-1 (inclusive) generated as described by ChrisG
    temp = deck[swap]       ; store the card to swap into current position for a moment
    deck[swap] = deck[i]    ; move the current card where the swapped card was
    deck[i] = temp          ; move the swapped card where the current card was delete(temp) ; cleanup
    delete(swap)            ; cleanup
end do

Actual OpenBSD code for a strong random number generator using hardware noise. Includes underlying theory [pointed out by slpalmer]

Article describing similar complaints in bridge since they switched to computer-generated deals 30+ years ago. Turns out (suprise!) people weren't shuffling correctly so their expectations were wildly different than should occur with truely random deals. Also discusses how many shuffles are necessary to randomize a deck. [Provided by me]

Math Magic Made Easy - Excel™ spreadsheet I created that computes various Magic math situations. One of the pages produces the probability of drawing so many cards of a certain type (listed as lands) out of N cards drawn from a deck of your choice. This should be useful if you want to try and disprove the shuffler by drawing a set number of cards from your deck instead of comparing the number of land clumps in the full deck as SCG does. [Shameless self-plug]

If you want to try and disprove the randomness of the shuffler, here are some things you MUST do:

Use EVERY trial you do, don't pick-and-choose 'bad' ones and note them.

Run a BARE MINIMUM of 1000 trials. Hundreds of thousands would be better.

Read up on statistical confidence intervals and learn how to conduct a T-Test, Z-Test or similar.

Xtofyr recorded data for several thousand hands. The initial results appeared skewed and prompted more thorough testing by elf. These later tests show the shuffler to behave as expected. Unfortunately, the full discussion thread and results have been lost.

The probability for common MTG cases:

Seven Cards (opening hand):
#land drawn|16/40|17/40|18/40|23/60|24/60|25/60
0|01.86%|01.31%|00.91%|02.67%|02.16%|01.74%
1|11.55%|09.20%|07.20%|13.84%|12.10%|10.51%
2|27.36%|24.55%|21.61%|28.56%|26.94%|25.22%
3|31.92%|32.30%|32.02%|30.29%|30.87%|31.18%
4|19.76%|22.61%|25.28%|17.82%|19.64%|21.44%
5|06.47%|08.40%|10.62%|05.80%|06.93%|08.19%
6|01.03%|01.53%|02.19%|00.97%|01.25%|01.60%
7|00.06%|00.10%|00.17%|00.06%|00.09%|00.12%
2-5|85.50%|87.85%|89.52%|82.46%|84.39%|86.02%


Six Cards (mulligan):
# land drawn|16/40|17/40|18/40|23/60|24/60|25/60
0|03.51%|02.63%|01.97%|04.64%|03.89%|03.24%
1|17.72%|14.90%|12.35%|20.03%|18.07%|16.21%
2|33.22%|31.37%|29.16%|33.38%|32.47%|31.37%
3|29.53%|31.37%|32.74%|27.49%|28.87%|30.07%
4|13.09%|15.69%|18.42%|11.78%|13.37%|15.03%
5|02.73%|03.71%|04.91%|02.49%|03.07%|03.71%
6|00.21%|00.32%|00.48%|00.20%|00.27%|00.35%
2-4|75.84%|78.44%|80.31%|72.64%|74.71%|76.48%


Eleven Cards (5th turn play/4th turn draw):
# land drawn| 16/40| 17/40| 18/40| 23/60| 24/60| 25/60
0|00.1080%|00.0585%|00.0305%|00.2495%|00.1753%|00.1217%
1|01.3574%|00.8413%|00.5035%|02.3378%|01.7801%|01.3392%
2|06.7869%|04.8074%|03.2920%|09.1842%|07.5820%|06.1810%
3|17.8157%|14.4222%|11.2870%|19.9518%|17.8719%|15.7958%
4|27.2476%|25.2389%|22.5739%|26.6024%|25.8834%|24.8220%
5|25.4310%|27.0205%|27.6531%|22.8266%|24.1578%|25.1644%
6|14.7232%|18.0137%|21.1465%|12.8399%|14.8064%|16.7763%
7|05.2583%|07.4493%|10.0697%|04.7247%|05.9490%|07.3445%
8|01.1268%|01.8623%|02.9149%|01.1117%|01.5323%|02.0656%
9|00.1366%|00.2660%|00.4858%|00.1588%|00.2404%|00.3547%
10|<0.0001%|00.0193%|00.0416%|00.0124%|00.0206%|00.0334%
11|<0.0001%|<0.0001%|<0.0001%|<0.0001%|<0.0001%|<0.0001%


NB: booster/tournament packs are NOT random, nor are they intended to be. They are created using card listings called print runs.

If there is anything important you think I missed, let me know.

Credits: Everyone who has contributed their time to the rational discussion of the shuffler in various threads over the years.
Edit (23 Nov 05 19:05 UTC): Added 'Sweet-spot' total probabilities (at least 2 lands + 2 spells in opening hand) and probabilities for a mulliganed hand.
Edit (10 Jul 06 05:53 UTC): Added link to data provided by Xtofyr and Elf.
Edit (18 Jan 08 20:10 UTC): Updated FAQ reference
Edit (15 Mar 10 19:50 UTC): Reformatted with html tags
Edit (26 Sep 15 18:47 UTC): Port to NGA Board

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PostPosted: Sat Sep 26, 2015 2:31 pm 
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It hasn't changed a bit--it's still just as beautiful as I remember. And just in time for the new scry-mulligan rule to change the classic expected land calculations and ruin everything. :-P I'm not sure when MTGO's going to implement it, but when it does the probability tables might warrant changing up to take it into account.

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PostPosted: Sat Sep 26, 2015 2:35 pm 
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It hasn't changed a bit--it's still just as beautiful as I remember. And just in time for the new scry-mulligan rule to change the classic expected land calculations and ruin everything. :-P I'm not sure when MTGO's going to implement it, but when it does the probability tables might warrant changing up to take it into account.

It should look the same, I just copy/pasted and made spot fixes.

I've been away from Magic for a couple years, so ping me when the Scry change happens so I can update.

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PostPosted: Sat Sep 26, 2015 3:40 pm 
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I really need to carry this thread around and slap people over the head with it.


Good thread.

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PostPosted: Sat Sep 26, 2015 5:08 pm 
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this is a nice thread

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PostPosted: Sun Sep 27, 2015 11:37 pm 
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yeah, when they say that "computer random number generators" aren't random, they mean that they might repeat a sequence after 100s of billions of generations. No one plays MTGO that much.

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PostPosted: Sun Sep 27, 2015 11:38 pm 
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True randomness should repeat a sequence after 100s of billions of generations to be fair.

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PostPosted: Sun Sep 27, 2015 11:46 pm 
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True Randomness is any organization of cards that makes me win and my opponent lose.

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PostPosted: Mon Sep 28, 2015 6:25 pm 
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Wow, when I heard people mentioning the "MTGO Shuffler Thread" I expected something full of confirmation bias and misconceptions about probability. This is 100% the opposite of that. Well done.

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PostPosted: Tue Sep 29, 2015 12:24 am 
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**** ****


Last edited by GobO_Althalus on Tue Sep 29, 2015 1:01 am, edited 1 time in total.
Edited to correct partial censoring--please either censor completely or just let the filter catch it.


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PostPosted: Tue Sep 29, 2015 8:42 am 
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I still wonder why the test shuffler in the client code is not the same as the server shuffler? At least that is what they are saying when the test shuffler was found to not be random enough.

:D


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PostPosted: Tue Sep 29, 2015 12:42 pm 
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Wow, when I heard people mentioning the "MTGO Shuffler Thread" I expected something full of confirmation bias and misconceptions about probability. This is 100% the opposite of that. Well done.
If this thread is like the original, those will appear quite often.


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PostPosted: Tue Sep 29, 2015 1:17 pm 
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pcjr wrote:
Wow, when I heard people mentioning the "MTGO Shuffler Thread" I expected something full of confirmation bias and misconceptions about probability. This is 100% the opposite of that. Well done.
If this thread is like the original, those will appear quite often.

Indeed. The whole point of the shuffler thread is to (1) consolidate all that idiocy into one thread (any other threads on the topic were promptly locked, and I hope the same will be done here) and (2) preemptively rebut the complaints so we don't have to post the same response repeatedly.

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PostPosted: Tue Sep 29, 2015 1:40 pm 
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Are you planning on bringing over the history of magic online as well or is that so out of date or too much work?


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PostPosted: Tue Sep 29, 2015 8:21 pm 
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Are you planning on bringing over the history of magic online as well or is that so out of date or too much work?

I was thinking about it, but I didn't want to be responsible for maintaining it. It would be better if someone who wants to maintain it bring it over. I'll do it if no one steps up.

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PostPosted: Tue Sep 29, 2015 9:57 pm 
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Ironically (or not, i have no idea what irony is), DotP has this thread too. It was one of my favourite reads and I imagine it is much of what the old thread contained.

viewtopic.php?f=38&t=11283

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PostPosted: Tue Sep 29, 2015 10:16 pm 
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PostPosted: Fri Oct 02, 2015 5:47 pm 
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Telir wrote:
Peldone wrote:

Angrypossum wrote:

Peldone wrote:
The problem with the shuffler is because it is random. Card games do not have random distribution of cards. If MTGO is meant to replicate a real world card game it should replicate what a real deck shuffled by real players would look like. It is ridiculous to claim that players that use physical cards are bad shufflers because their shuffles don't match a randomised deck of cards. Nobody shuffles their decks by pulling cards out randomly and swapping them with each other thousands of times to create a truly random deck. A deck that is randomized is going to mana screw or flood a person with far more frequency than a physical shuffled card game. Losing or winning because of mana issues is NOT fun. Magic should be a game based on skill and not the luck of a randomizer. The way real world players put together their decks is to randomly mix their lands and other cards in two separate piles. Then they put lands into their deck every 1-3 cards to create a base deck. Then the deck is split into two groups between 40%-60% of the whole deck. Cards are then interleaved every 1-6 cards with the opposing groups 5-10 times. In a tourament the opponent interleaves the cards another 5-10 times. This is a real world shuffle and will provide more competitive and skill intensive games than a randomized deck of cards. And because real world players shuffle instead of randomize their decks everyone is on the same playing field. Switching MTGO to a true shuffle would decrease mana screw and flood for all players making playing online more fun and less frustrating.






I'm gonna really miss these forums when they shut down. How else will I get to enjoy such off-the-wall insanity as this?


I was just creating a general example that covers how most people shuffle their decks. Some people do a table shuffle and then interleave the cards 5-10 times (frequently more than 5-10 times on the initial creation of the deck. Everytime a deck is played it will need a complete randomization because cards become clumped on the playing board with creatures together and lands together. Players mix the played cards before shuffling them back into the deck. Anybody that believes that their shuffling sufficiently randomizes their deck would not do this. The truth is that MTGO does not accurately reflect the real world of playing magic. They are taking the easy way by creating a randomized deck instead of trying to figure out how to recreate the real world game. Random is easy when programming a computer but determining how the game is actually played and recreating that experience for everybody is hard. Playing magic online is less fun than playing in real life because everybody gets mana screwed/flooded more which leads to less competitive games.

What they were getting at above is the description you gave is the exact definition of a type of shuffle that is frowned upon. I believe such shuffles would leave your deck insufficiently random. Which is the interesting issue you raised via your opening that the shuffler is too random. That might be the case. I would not disagree that no matter how one shuffles irl you will not get the bubble sort shuffle that a computer can accomplish in microseconds. But most people shuffle their decks to sufficient randomness that it would be hard to pick apart a pattern to how their cards are aligned after finishing. And that is what the MTGO shuffler should be doing. And despite tons and tons of annecdotal evidence to the contrary no one has proved that the MTGO shuffler is not doing that. Because all the many pages of statistical arguments back and forth boil down to "It takes a heck a lot of data to prove such a thing."

As to whether it is more or less desirable to have mana flood/mana famine that is a question of skill analysis. The better a player you are the less of an impact such variance should have on your play results. Hence pros get to platinum status and the rest struggle. And there are a lot of tiers below that level where one tier of skill deals with a bad shuffle better than the tiers below it. And then we get into "To Mull or Not To Mull: the eternal question." Anyway I recommend you bring this conversation to that link that the guys provided if you want to continue this because I probably won't be posting again in this thread.


I appreciate the reasonable nature of your remarks. I am unclear on what, about the shuffle, I described would be considered a poor shuffle. According to researchers 7 riffle shuffles are enough to mix a deck of cards thoroughly and that more than 7 does not improve the mixing. http://www.nytimes.com/1990/01/09/scien ... umber.html . I listed 5-10 shuffles being done by each player which would mean that if each player shuffled their deck 5 times and their opponent shuffled their deck 5 times each deck would get shuffled 10 times. If the opponent declined to shuffle after you only shuffled 5 times it could be shuffled too little. Since riffle shuffling 7 times is enough to mix the cards thoroughly it shouldn't matter if you stack all your lands on top first but I think that most people will mana weave their deck first and according to Wizards mana weaving is not illegal if it is followed by other forms of shuffling http://archive.wizards.com/Magic/magazi ... /20060707a .

Since we have the statistics of what a random deck should do posted at the top of the thread; 2 or less lands by 11 cards (9.5347%) and 7 or more lands by 11 cards (7.74231%) for a combined percent of mana screw/mana flood of 17.277% for each player. One thing that I plan on doing now to collect data to compare results of shuffling verses MTGO is to count the number of hands, from a 60 card 24 land deck, that have only 2 lands by the eleventh card (mana screw) and those that have 7 or more lands by the 11th card (mana flood). Since it has been proven that 7 riffle shuffles are enough to put the cards in a random order I will always shuffle 7 or more times. Anybody that wants to contribute to the data can keep track of their own games as well. The percentages should become close to these numbers the higher the number of games that are counted if real world shuffling matches a randomized deck.


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PostPosted: Fri Oct 02, 2015 6:13 pm 
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My friend refuses to shuffle, but rather cuts his deck 2 times, because 2 is closest to the golden number. When I showed him this, he said he thinks it's **** because he doesn't understand math. How should I help him?


Last edited by GobO_Althalus on Fri Oct 02, 2015 6:50 pm, edited 1 time in total.
Edited to correct partial censoring--please either censor completely or just let the filter catch it.


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PostPosted: Fri Oct 02, 2015 7:07 pm 
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My point re: Cheating on the WotC forums is that weaving your cards makes your shuffle non-random. Not merely insufficiently random. If a judge catches you doing that in tourney you will at least get a game loss and maybe more severe penalties. Don't mana weave. 5-10 is a big range. I would say 7-8 is a better range of shuffles. Of course sleeves make it more complicated to shuffle thoroughly without ruining your cards for trading purposes so learning how to do that well is important.

When you see pros shuffling you may see something that LOOKS like mana weaving because they often pile shuffle first but they are not sorting their lands into separate piles. Pile shuffling IS acceptable as long as you are not doing so in such a manner as to create a discernible pattern. Yes there are also cheaters on the Pro Tour and when they get caught it is a big deal and they get huge bans. But those guys are (perhaps slowly) getting weeded out and are not the norm.

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