[Math] Gaea's Blessing, with arbitrarily large number of Brain Freezes

[Moxlotus]

If you don't agree with Slow Play, how else will you resolve the situation in a way that is fair to both players and to everyone else in the tournament.

P.S., discussion of what policy changes you want should probably go to Customer Service that I posted above, but it's extremely unlikely to change.  Just saying.

So who again is getting the slow play penalty?

[Englishman_in_NH]

This a suggestion I have heard a few people make. I would not say it is foolproof, but is an interesting idea.

Take the number of cards in target players library and graveyard and divide by the number of cards milled per iteration of spell or ability ( 3 in the case of Brain Freeze and Laquatus.) Round down the number and resolve those copies as the last copies of Blessing. (This assumes a Blessing has just finished resolving and allows for the possibility of Blessing being in the bottom few cards, whilst also allowing for its shuffle effext to still come into play.

Again this is not foolproof, but allows for a potention quick resolution that should appease both players and then continue the match.

Example:  Player A has 40 cards in their library and 10 cards in their graveyard. 50 cards total after the Blesssing would trigger. 50/3 = 16.666... which would give 16 copies. Player B now resolves the final 16 copies of his spells/abilities and play continues from that point.

In my mind this seems the best way to give both players a chance at a resonable fascimile of what would have happened over time whilst staying true to the spirit of the effect.

Their are certainly arguments against this but it seems the best solution that I can come up with presently.

Oli

[Anusien]

If you don't agree with Slow Play, how else will you resolve the situation in a way that is fair to both players and to everyone else in the tournament.

P.S., discussion of what policy changes you want should probably go to Customer Service that I posted above, but it's extremely unlikely to change.  Just saying.

So who again is getting the slow play penalty?

Depends on who is playing Slowly, which players offered a draw, whether the Brain Freeze player knew his opponent had Blessing, who is recommending shortcuts, who is accepting shortcuts or not and why, stuff like that.

I love how you can have a whole post of logic and suggestions, and players latch onto the words "SLOW PLAY".

Example:  Player A has 40 cards in their library and 10 cards in their graveyard. 50 cards total after the Blesssing would trigger. 50/3 = 16.666... which would give 16 copies. Player B now resolves the final 16 copies of his spells/abilities and play continues from that point.

That's actually one of the shortcuts I mentioned that a player could suggest in my earlier post.

[Apollyon]

@Anusien - I don't think anyone here is arguing about what the official DCI policy is.  I believe the discussion here reflects the fact that people feel that policy is inadequate, as well as a desire to discuss better ways to handle these situations.  This is reflected in requesting input from mathematicians, statisticians, and Tournement Judges.  If this was just an exercise in looking up DCI policy, Jeff would not have spent the thought, time and energy that he did in looking at this problem. 

One large problem with the theoretical discussions about interactions is they assume perfect information.  The library, as well as players hands, not being public information creates some issues.  Does the player have one or two Blessings?  Do they have relevant plays in hand?  Are there flashback cards or other interactions that might come up?  These questions impact the situation greatly.  Both players do not necessarily have all the info, so a judge would certainly have to be involved.  This does not mean these issues could not be overcome with a judges discretion, but it does complicate things. 

I would like to discuss the possibility of seeing shortcuts to the endgame of the problems.  Dice could easily simulate game actions much more quickly.  In the example of an infinite mana'd Laquatus vs Blessings, the odds of blessing not ending up in the bottom 1, 2 or 3 cards could be determined to be of a certain value, like say much higher than 1,000,000,000,000 to one.  In this case, I would like to see 13 D10 rolled, and unless they are all 10s, the results hold up.  Even though this result would be more likely to fail than actually doing it out, it would allow players to fast forward and continue the game.  Other shortcuts like these could be reasonable ways to resolve these issues. 

One step forward would be to create a rule regarding arbitrarily large numbers.  With rules regarding a certain upper limit in place, this would allow people to proceed with predictable results.  People would know if they go with X= 1,000,000,000,000 instead of X=1000, they would be dealing with these new rules, and should make their choices accordingly.  If the DCI would create certain judge rulings on given situations where the variable is said to be of a minimum size, we could eventually handle many of the plausible tournament situations in a way that would reflect the outcome if cards actually played out. 

In the instance of Brainfreeze with Ancestral vs Gaea's Blessing, I propose the following.  Continue the loop until there is no graveyard, with the entire library shuffled up.  With the multiple Brainfreezes still on the stack, roll dice to determine where in the deck the Gaea's Blessing is.  This could be done with 2d10 and rerolling numbers higher than the number of cards in the deck.  A 1 means the top card, 2 the second an so on.  Eventually, there will be an instance where the Blessing is in the bottom few cards.  Players can then proceed from that point once the correct roll comes up.  The judge would roll, and keep the results to himself, so the players would not know where in the bottom cards the Blessing is. 

I believe all these situations could be handled in a similar fashion.  It is messy, but it seems better than the current system. 

One thing I am unclear about in these examples is who is actually "slow playing?"  To me, the only player who is taking any actions is the player targeted by Brainfreeze and controling the Gaea's Blessing triggers.  The Brainfreeze player made plays that are commonly seen, without any penalty or consideration, in every Type 1 event.  The player who is being targeted is the one doing all the shuffling and milling, while the controller of the Brainfreeze is powerless to do anything but sit and watch.  It's not like the player controlling the Brainfreeze copies is able to do anything, and there are no rules against how high the storm can get, nor should there be. 

You also missed one of the most important points that I stated to diceman: Judge calls are about customer service. Answering rules questions is good customer service. Making sure that the round turnover is prompt is good customer service. Clearing trash off tables and pushing in chairs is good customer service. Enforcing penalties on players that violate the rules is good customer service. A judge who answers a question "This is how it is and that's that." is performing poor customer service. If they can't explain their rulings, that's poor customer service. Some calls might require a statement of: "This is my ruling, if you want to learn more about it, talk to me after you are done your match." That's good customer service. You answer their call without possibly delaying the tournament too much.

Your idea of using dice to equate the likelihood of the Blessing situation is not a bad one, except that it requires judges to have a math degree. Forcing judges to do complex math to figure out situations is poor customer service. You are talking about having someone go over to a computer and start crunching numbers. That's poor customer service. Or worse yet, doing it out on paper. So while they are busy doing math, someone else could have a situation come up that requires their attention (assuming 1 judge per event, because of the small size of the tournament), thus causing a delay.

Plus, you also need to remember that rulings are meant to be consistent. Requiring lots of math doesn't help consistency of rulings. Judges come from all backgrounds and all areas of expertise. If the judge isn't outstanding at math, then you might get a different answer than if the judge is.

Rulings based upon large numbers are both arbitrary and, at best, inconvenient. What's the cutoff point for "large enough that something will happen"? And where do you store those rulings? In a giant Box of Rulings section of the Wizards site? And then what happens if you have That Number - 1 samples? And do you want judges to remember them? How about players? It's poor customer service for the judges to go "All right, let me go check if that number is big enough." *2 minutes elapse* "Sorry guys, you are several short, looks like we wasted 2 minutes to just get you to do it out. Have fun."

[Vegeta2711]

One thing I am unclear about in these examples is who is actually "slow playing?"  To me, the only player who is taking any actions is the player targeted by Brainfreeze and controling the Gaea's Blessing triggers.  The Brainfreeze player made plays that are commonly seen, without any penalty or consideration, in every Type 1 event.  The player who is being targeted is the one doing all the shuffling and milling, while the controller of the Brainfreeze is powerless to do anything but sit and watch.  It's not like the player controlling the Brainfreeze copies is able to do anything, and there are no rules against how high the storm can get, nor should there be. 

I am also confused.  The storm player unknowingly created a bad situation so i don't understand why he would ge the slow play penalty.  The player targeted by Brainfreeze is only doing the repeated motions that the opponent told him to (and the player target w/ BF would clearly rather not do).  Who is getting the slow play penalty?

Once again, "You chose to cast storm for a million of something, it's still on you to resolve each copy until the game is ended in an appropriate manner." You can choose to shortcut the likely course of events to a smaller number as Kevin and Englishman pointed out or you'll just have to justify how storming for some absurd number was done for any reason other than to stall.

[Moxlotus]

If you don't agree with Slow Play, how else will you resolve the situation in a way that is fair to both players and to everyone else in the tournament.

P.S., discussion of what policy changes you want should probably go to Customer Service that I posted above, but it's extremely unlikely to change.  Just saying.

So who again is getting the slow play penalty?

Depends on who is playing Slowly, which players offered a draw, whether the Brain Freeze player knew his opponent had Blessing, who is recommending shortcuts, who is accepting shortcuts or not and why, stuff like that.

I love how you can have a whole post of logic and suggestions, and players latch onto the words "SLOW PLAY".
 

I'm sorry, I didn't understand part of the post about who "slow play" was applying to.  No need to be condescending.

[Matt]

pi:1-pi?

Well, I'm pretty sure the odds aren't ever going to be (pi):(1-pi)...;P

I don't like how the current policy punishes the milling player for trying to win. At least in the finite-Freeze case, no one can say that NONE of the Blessing loops will let the Freeze player win, and no one can say that one of the Blessing loops MUST let the Freeze player win. Since the only way to find out if one of those loops will work is to actually play them out, the right thing to do would be to let the Freeze player keep going until time is called. If there were no untimed additional turns, I'd call any match still going a draw and be done with it.

Now, consider this: Blessing player has won game 1, and Freeze player does his infinite thing, trying to win game 2, but he doesn't want to take the entire round to try, and wants a shortcut. What to do? Well, what OUGHT to be passable is this: compute not the odds that his large-N copies will do the trick, but rather the odds that he'll pull off the trick in the amount of time left. If there are 30 minutes left in the round, and on average it takes 1.2 minutes to mill, find a Blessing, and reshuffle, then he should get an average of 25 Blessing loops before time's up. Calculate the probability that he'll get his optimal arrangement of cards in that, much lower, number of tries, and perform a random event with an outcome of that probability. If he wins, then declare that one of his 25 loops was successful, and move on to game 3; if not, declare that he "took all the remaining time" trying, but failed to get the setup he needed, so the game is a draw, and the Blessing player wins the match 1-0.

[Harlequin]

Hey, just as an update, I was able to solve the endstate probability using Markov chains.   had to dust off my advanced probabilty text to do it.

Here's the basic scoop:

Prob (D'=d) = d / Sum(all possible d's)

So for a 52 card library the possible value of d = 52, 49, 47, ... , 4, 1   therefor the sum of all d's = 18 + 3(17*18/2) = 477
So using the formula above:
Prob( D'=52) = 52/477
Prob( D'=49) = 49/477
Prob( D'=47) = 47/477
...
Prob( D'=4) = 4/477
Prob( D'=1) = 1/477

This is really interesting because the probability density function is linear!  This is good because you can actually reasonably shortcut to the end ... given litteraly any size deck rather quickly.  I'll post more about the details tomorrow.

[Godder]

At its core, Magic is an exercise in Probability, particularly where constructing decks is concerned. Trying to claim that Probability has no place in Magic rules is inconsistent with the deck construction rules, for example, which are based on Probability.

"Because the rules say so" is viewed as a poor answer, but sometimes, it's the only answer, and that's all there is to it.

Incidentally, what number is considered statistically insignificant from 1? I've seen it before, but I've forgotten the answer.

[Harlequin]

First to answer Godder's question... that is what you call "Alpha."   For most practical everyday examples you generally use alpha 0.01 or 0.05.  Meaning there is a 1 in 100 chance that the distribution I think we're working with is wrong. 

If your talking about the laquatic example, it is litterally anything you can think of.  For example, let's say you want to know the mana required to mill such that the odds of failure are less than the odds of winning every lottery, in every state, every week, of a persons life from age 18 to 95, then while on retirement in Florida they got struck by lighting - survived - but where then immediately killed by a shark atack.....   It would be a BEAR to calculate that - but given enough computing power (which may or may not even exist) you could calculate that amount of mana.



Ok, buckle up this is going to be a math-filled ride.

So.. a quick lesson on Markov chains and thier application in state-based probability.

Here is the classic example: 
If it's rainny today, then there is a 30% chance it will rain tomrrow.
If it's sunny today, there is a 10% chance it will rain tomorrow.

This is a simple two state model.  Using this, you can build a matrix that has the starting states represented by rows, and the end states represented by cols.  So Matrix A would be:

Code:

     R     S
R [ 0.3 , 0.7 ]
S [ 0.1 , 0.9 ]   ~> A

A has some really need properties.
If you matrix multiply  A*A you get a matrix that we will call A2

If you take a close look at A2, it represents the "2-step" probability.  This will give you the probability that "If it's known to be Rainny (or Sunny) on Monday, what is the chance of rain on Wednessday?"

Now again multiply   A2*A = A3

A3 as you may be able to guess, is the three step probability.  This will break down If you know the weather on Monday, this is the probability it will be Raining on Thursday.

Now... Generally, there exists a Matrix Z  Such that the following is true:
Z * A = Z
Now when you look at that, you think ... wait that can't be true!  How can you have ZA=Z if A isn't an identity.  Well, your thinking too much about algebra and not enough about what it is saying.  It's saying that you can create a state Z, where the next step probability is also Z.  Meaning as you move from step to step, using A... there's no change in probability distribution.  This is sometimes known as the Stable State Matrix, or Horizon Matrix. 
Considering Step Ai, as i goes to infinity Ai approches Z.  This was the missing peace to solving the problem.  So....

Let's consider we have a situation where there are 9 cards between the deck and library.  Let's construct A

Code:

    9    6   3
9[ 1/3, 2/3, 0 ]
6[ 1/2, 0 , 1/2]
3[  1 , 0 , 0  ]

So looking at that, it's basically saying... ok if you have 9 cards in the library, you have a 1/3 chance to hit blessing and go to 9 cards again, and a 2/3's chance to not have blessing at instead go to 6 cards left.  The next row tells us that if you have 6 cards, you have a 50:50 chance to either hit blessing and go back to 9, or not hit blessing and go to 3.  The last row shows us that if you have 3 cards left in the deck, the only possability is hitting blessing and going back to 9.

Now lets look at A2

Code:

    9    6   3
9[ 4/9, 2/9 , 1/3]
6[ 2/3, 1/3 , 0  ]  = A2 = A*A
3[ 1/3, 2/3 , 0  ]

So again, this is saying that if you start with 9 cards in the deck and resolve 2 brainfreezes the TOTAL probability of ending on 9 is 4/9, ending on 6 is 2/9 and ending on 3 is 1/3.  And the other rows represent the other possible two step probabilites given different starting points.

Now consider the matrix:

Code:

9[ 1/2, 1/3 , 1/6]
6[ 1/2, 1/3 , 1/6] ~> Z
3[ 1/2, 1/3 , 1/6]

Now, using our example above... do the matrix multiplication Z * A. I'll bet you could have guessed from the notation I used that you will end up with Z again!  So this is basically saying that A(1,000,000) is aproxamatily Z.  Another interesting property of Z is that all the rows are the same.  This says that each of the starting states are the same... meaning if you start with 9 cards in the library on brainfreeze #1, you'll get the same distribution as if you had started with 6 cards on brainfreeze #1.

So, the conclusion is that if we were to resolve 1,000,000 brainfreezes on a 9 card library with 1 blessing, we would see the following outcomes:
9 cards about 50% of the time.
6 cards about 33 1/3% of the time.
1 card  about 16 2/3% of the time.

This would be insignifgantly differant from doing 1,000,001 brainfreezes, and 1,000,002 and 1,000,003 ... etc.  Because each of these probabilites is close to the matrix Z above (accurate out to about 100,000 decimals I would immagine).

so to follow the formula in the last post...
Prob(9) = 9 / (9+6+3)
Prob(6) = 6 / (9+6+3)
Prob(3) = 3 / (9+6+3)
Simple!

//////////////////////////////////////////////////////

Probability Simulation #1:

Ok, so I brainfreeze you with 1,000,000 copies on the stack.  And ut oh you have 1 gaea's blessing in the deck.  You have 5 cards in play, 3 cards in hand, 1 card RFGed and a 60 card registered deck.  So 60-5-3-1 = 50. 

Step 1, build a table of values that goes like this:
50 = 50
47 = 97  {50+47}
44 = 141 {97+44}
41 = 182
38 = 220
35 = 255
32 = 287
29 = 316
26 = 342
23 = 365
20 = 385
17 = 402
14 = 416
11 = 427
8 = 435
5 = 440
2 = 442

Step 2 - Roll 3 unique D10 (so a D1000) and reroll any numbers that are bigger than 442 (the last value on the table).  With tripple 10's couting as 1000.... not 0. 

Whatever number your roll is Equal To or immediately Less than is your end state. 

So for example I roll 575 -> reroll.  Then I roll 226.  I look at the chart.. 226 is bigger than 38 but smaller than 35.  So my end state is 35. 

Step 3 - Take my library and fish out the Gaea's Blessing.  Now shuffle the library face down.  Separate out 34 cards face down, and add the gaea's blessing to that stack giving me my magical 35 cards!  Now the remaining cards become the graveyard.
Or if you prefer 50 - 35 = 15.  So with the gaea's blessing removed, mill 15 cards.  Then after that recombine the remaining deck with the blessing and shuffle.  Either way, the 226 roll has dictated that you should end with 35 cards including blessing in your deck, with 15 cards in the yard.

----------------------------------------------------------------

The next task for me is to determin the "Minimum Number" of brainfreezes for each size deck such that the differance in probability falls within a few selected alpha values of signifigance.  If I can do that then we will have a 'magic number' (or a few magic numbers based on deck size) that will give us a number for which the probability is basically the same as 1,000,000 storm. 

[Godder]

The reason I ask is because without having something that can be agreed upon as close enough to 1 to be good enough, you've got nothing. Saying that it's essentially user-defined is worthless because that's subjective which is precisely why the ruling is what it is now. If there's a figure that's acceptable for scientific experiments, for example, that works because that's from an objective source.

[diopter]

The reason I ask is because without having something that can be agreed upon as close enough to 1 to be good enough, you've got nothing. Saying that it's essentially user-defined is worthless because that's subjective which is precisely why the ruling is what it is now. If there's a figure that's acceptable for scientific experiments, for example, that works because that's from an objective source.

Why not make it so that the loop is only legal if you can make alpha as close to 1 as you want to? That way for anybody who might say "no I'm not satisfied with a probability of 0.9999999999999999" or something, you can show them that you can make the probability higher than that, and higher than that, and higher than that, etc.

I think if this would be implemented, one way to do it is to have a subset of combos that judges would be notified as being "legal". The rules themselves could explicitly explain the confusing "any alpha you want" principle, but in reality there are so few interactions like Laquatus/Gaea's Blessing.

[Anusien]

Why not make it so that the loop is only legal if you can make alpha as close to 1 as you want to? That way for anybody who might say "no I'm not satisfied with a probability of 0.9999999999999999" or something, you can show them that you can make the probability higher than that, and higher than that, and higher than that, etc.

There is no accepted probability in the scientific community, for what it's worth.

As a player, I wouldn't be happy with any probability less than 1.

[diopter]

Why not make it so that the loop is only legal if you can make alpha as close to 1 as you want to? That way for anybody who might say "no I'm not satisfied with a probability of 0.9999999999999999" or something, you can show them that you can make the probability higher than that, and higher than that, and higher than that, etc.

There is no accepted probability in the scientific community, for what it's worth.

As a player, I wouldn't be happy with any probability less than 1.

The fact that an infinite combo can approach a probability limit of 1 should be good enough. The difference from the probability of winning and 1 can be made smaller than the probability of all of the atoms in your cards disappearing through the table and the floor via quantum mechanics, thus granting you a game loss for having a deck of less than 60/40 cards - is that probability good enough for you???

Honestly I think you're protesting for the sake of protesting.

[Anusien]

The fact that an infinite combo can approach a probability limit of 1 should be good enough. The difference from the probability of winning and 1 can be made smaller than the probability of all of the atoms in your cards disappearing through the table and the floor via quantum mechanics, thus granting you a game loss for having a deck of less than 60/40 cards - is that probability good enough for you???

Honestly, no.  If I paid entry for a tournament, and you prove to me that the probability is LESS than 1, that's not going to be good enough for me.  Even if you can prove it's 1, especially if this is game 2 or 3, I'm likely to say "Prove it".  Let's say I won game 1.  I would be within my rights as a player to make you play it out.  Even if the probability approaches 1, it doesn't do so in finite time.  All of this which would have to be in your explanation.

(What's more, even if you can prove that it's guaranteed, it still doesn't reach the definition of a deterministic loop since it doesn't do so in a regular way.)

Edit: As a player, I would also be hugely uncomfortable with rolling a die to determine the outcome of a match.  Then again, I'm not sure why as a player I should believe that even my opponent can't win before the universe ends, that I should just let them win anyway.

Honestly I think you're protesting for the sake of protesting.

That's actually rather insulting, especially as I did my best to stay out of it for a while.  I thought I was trying to be rational and logical, both presenting the official viewpoint and giving a way to try and get it changed (as well as to be frank about the chances), as well as trying to present the opponent's point of view.  I thought this was supposed to be a discussion, not a chance for you to flex your numbers brain and insult me.

As an aside:

Trying to claim that Probability has no place in Magic rules is inconsistent with the deck construction rules, for example, which are based on Probability.

This is disturbing because it flies in the face of the whole "Judge calls need to be consistent" thing Apollyon and I have been harping on.

[diopter]

The fact that an infinite combo can approach a probability limit of 1 should be good enough. The difference from the probability of winning and 1 can be made smaller than the probability of all of the atoms in your cards disappearing through the table and the floor via quantum mechanics, thus granting you a game loss for having a deck of less than 60/40 cards - is that probability good enough for you???

Honestly I think you're protesting for the sake of protesting.

Honestly, no.  If I paid entry for a tournament, and you prove to me that the probability is LESS than 1, that's not going to be good enough for me.  Even if you can prove it's 1, especially if this is game 2 or 3, I'm likely to say "Prove it".  Let's say I won game 1.  I would be within my rights as a player to make you play it out.  Even if the probability approaches 1, it doesn't do so in finite time.  All of this which would have to be in your explanation.

(What's more, even if you can prove that it's guaranteed, it still doesn't reach the definition of a deterministic loop since it doesn't do so in a regular way.)

Obviously it's not a deterministic loop, that's what the thread is about, exploring whether or not a deterministic loop is the only kind of infinite loop that you can shortcut.

So let's say you come to the table and face me, and you tell me that you want me to mathematically prove to you that some card that is in my backpack won't, through the funniness of quantum mechanics, pass through my backpack and my body and end up in my hand; if I don't prove this, you believe I am breaking the rules and you want me to concede to me - because the probability of such an event happening is not precisely 1. What you are telling me is that you would be unsportsmanlike enough to do this.

That's actually rather insulting, especially as I did my best to stay out of it for a while.  I thought I was trying to be rational and logical, both presenting the official viewpoint and giving a way to try and get it changed (as well as to be frank about the chances), as well as trying to present the opponent's point of view.  I thought this was supposed to be a discussion, not a chance for you to flex your numbers brain and insult me.

Frankly you've been rather confrontational and non-helpful in this thread. This is an exercise in numbers and the possibility of changing a rule that is *actually* unfair to somem players with a very signicant non-zero frequency of occurence. In the face of the numbers you basically ignored them and said "nope not listening, this is the way it is and I'm going to have my probability of 1".

[Anusien]

So let's say you come to the table and face me, and you tell me that you want me to mathematically prove to you that some card that is in my backpack won't, through the funniness of quantum mechanics, pass through my backpack and my body and end up in my hand; if I don't prove this, you believe I am breaking the rules and you want me to concede to me - because the probability of such an event happening is not precisely 1. What you are telling me is that you would be unsportsmanlike enough to do this.

Quantum mechanics doesn't work that way, last time I checked.

But no, I wouldn't say that.  I'd only get upset with the possibility of a card from your backpack making its way into your hand if it actually happened.  And if you were actually breaking the rules, I would not ask you to concede to me; I would call a judge and explain to them that I thought you were cheating.

Edit to answer diopter's phantom edit:

That's actually rather insulting, especially as I did my best to stay out of it for a while.  I thought I was trying to be rational and logical, both presenting the official viewpoint and giving a way to try and get it changed (as well as to be frank about the chances), as well as trying to present the opponent's point of view.  I thought this was supposed to be a discussion, not a chance for you to flex your numbers brain and insult me.

Frankly you've been rather confrontational and non-helpful in this thread. This is an exercise in numbers and the possibility of changing a rule that is *actually* unfair to somem players with a very signicant non-zero frequency of occurence. In the face of the numbers you basically ignored them and said "nope not listening, this is the way it is and I'm going to have my probability of 1".

The rule is unfair to all players equally; it just affects some players more because of the decks they choose to play.  I'd argue that changing the rule back negatively affects the players who are going to start losing games they didn't lose under the same system.
You might have missed the point earlier; the possibility of changing the rule doesn't stem from this thread.  It stems from reaching out to Wizards of the Coast; customer service would have the best way to do this which is why I linked it.  I'm only here to point out that this rule has been confirmed three or four times and is unlikely to change.  One might even say the probability of it changing is nonzero but very significantly close to 0.  Would that mean that it's never going to change?

In the face of the numbers you basically ignored them and said "nope not listening, this is the way it is and I'm going to have my probability of 1".

It actually IS the way it is, and it's unlikely to change.  If individual TOs want to have it work differently in their unsanctioned tournaments, that's fine, and it's a point that's been raised before.  But it is a deviation from Wizards policy, and important to note.  Wizards of the Coast and the DCI have shown a willingness for consistency in rulings, and that's unlikely to change.

But yes, as a player, I'm unlikely to accept being cheated out of ANY chance to win, however small it is.  This is ignoring that the judge is doing something for you you can't do in the time it takes for the entire universe to end.

[Harlequin]

But yes, as a player, I'm unlikely to accept being cheated out of ANY chance to win, however small it is.  This is ignoring that the judge is doing something for you you can't do in the time it takes for the entire universe to end.

Isn't that exactly what you are doing??  You have one player with a 99.99.. % chance of winning and you are saying that player should not be able to win?  You think that game should be a draw right?

Now I'm really confused about what you think the outcome should be...

[dicemanx]

So you have a high level of intolerance for any probability less than 1, and yet it is acceptable for judges to make subjective rulings such as assessing slow play? Why not make a likewise subjective ruling for Brainfreeze/Laquatus + Blessing by adopting an arbitrary cut off where you essentially equate a non-1 probability to 1?

[Apollyon]

Anusien is correct. You have a finite amount of time, due to the whole "customer service" aspect of tournament management. Imagine a tournament where round 3 starts over 4 hours after the tournament does. Now, instead of it being a TO issue, it's because of a game that's gone on far too long, and is still going on. What part of that is fair to the other players in the tournament? You complain about the current ruling being "unfair" to you. How is your likelihood of winning, given infinite time, fair to your opponent and fair to the rest of the tournament?

I'm not disputing that you can have extremely overwhelming odds of winning. That's not the reasoning behind the current ruling.

Let me state this again: I agree with you, you should probably win. But you need to do so in the context of the rules. The rules say that if it's not a deterministic loop, you can't shortcut repeated iterations of that loop.

Complaining that it's not fair and that you are hosed because of this is analogous to saying that it's not fair and that you are hosed because of Wrath when you play Elves.

[Harlequin]

But in the same vien, its the differance between "losing to wrath" and the rules/judge robbing you of 49.99999% and awarding it to your opponent.   A better example would be like saying:  "You play elves, and about 75% of your elves die to EE on 2... so that means if your opponent casts EE for 2, we'll just wipe your board becaue it takes too long to look at casting costs.  Sucks to be you, you shouldn't play elves."   In that example you are overpower the EE as a valid solution to elves.  An even better example would be to say that well EE at zero kill elf tokens, and even though your board is less than 1% elves we'll just say that EE wipes your board to save time.

You have one person with 99.999% chance of vitory and one with 0.001% chance of victory.  By declaring the game in favor of the 99% you are essentially cheating the losing player by 0.001%.  By declaring the game a draw, you cheat the winning player by 49.999%.   How can you sit back and say the ruling is fair?

[Shock Wave]

Why not make a likewise subjective ruling for Brainfreeze/Laquatus + Blessing by adopting an arbitrary cut off where you essentially equate a non-1 probability to 1?

I'm not sure I agree with this solution. Even if the probability approaches 1 such that it is more likely that God appears and starts playing Magic with you as opposed to an event not occurring, there is still no guarantee of eventuation. Creating a game state where it is likely that you'll win does not equate to actually winning, regardless of probability.

If I have seven cards in hand and my opponent has none, and no permanents in play, does that mean my opponent should concede in light of the probability that they'll recover being extremely slim? Assume my opponent is on Turn 5 of extra turns, has no way of winning, has no cards in hand, and no permanents. Does that mean that they should concede to me?

Another issue I see with the Laquatus/Blessing scenario is that while continuing the loop indefinitely creates a likelihood of arriving at a certain game state, the time constraints of a match are not indefinite. Therefore, it seems very unreasonable to posit that because a circumstance is probable given infinite time, that this resulting circumstance is applicable to a match with a 50 minute time limit.

The right solution seems to be to just allow a player to continue going through loop iterations or create 1,000,000 Brain Freeze copies. I'm of the opinion that since they are actually attempting to win, that this is acceptable. Yes, it is tedious, and will likely result in the match being a draw, but at least the circumstances are fair for both players. That is, unless I'm totally overlooking something, and I'm sure that's also possible.

[Apollyon]

Simple. Because the "solutions" that you stated aren't legal, according to the rules. As such, they aren't consistent among the judging community. In fact, the EE situations are what happens when you shortcut things that you shouldn't shortcut.

I can say that it's fair because you need to win within the constraints of the tournament. If you can't win given the frame of "50 minute time limits, plus a few extra turns, if necessary" and "no slow playing", then you don't deserve to win. Arguing "I will most likely win, given a violation of slow play rules" doesn't really stand up as an argument as to why you should be awarded a win.

[Harlequin]

Code:

alpha-> 0.1% -- 0.01% -- 0.001% - 0.000 1% - 0.000 000 1%
40 ----- 36 ----- 51 ----- 65 ----- 80 ----- 126
50 ----- 45 ----- 62 ----- 79 ----- 99 ----- 155
61 ----- 52 ----- 74 ----- 95 ----- 118 ---- 187

Ok, back to the Blessing Storm example (let give Laquatus a break for a min). 

The table above shows my results of the markov chain itteration.  Here is in layman's terms what it is saying:

The number on hte left hand side of the table (40, 50, and 61) are how many cards are in the deck after the first blessing resolves.  So its the number of cards in the deck + the inital GY.  I chose 61 as my upper limit of how many cards would realistically be a maximum number of cards.  I could run the loop for 255 if we wanted... but the results would be ugly and seriously who brings battle of wits WITH gaea's blessing!!  Anyway.  Lets assume a 50 card remaining deck durring brainfreeze.

So we look at the row for 50.  What that number indicates in the first column is that once you've reached storm 45, you are within 0.1% (or 1/1,000) of the distribution for 46, 47, 48, 100, 1000, 100000, 10^50.  The point is that no matter how much storm is out there, the end state distribution will not devate by more than 0.1%.  So that means suppose you did 1,000,000 simulations of storm 45 and then a 1,000,000 simulations of storm 500,000.  It's saying that out of a million simulatios only 1,000 would possibly favor another outcome (or 1 in 1000).  Now keep in mind that the 1 time out of 1,000 is just saying some other outcome is reached - wether this favors the caster or the blessing holder is unknown, and likely won't favor one player or the other (it will be something like the differance between milling 15 cards and 18 cards).  So right there that first alpha value I would consider "fair."  Its saying that 1 out of a 1,000 times there will be a slightly differant outcome in the short cut that the full storm - and I would say maybe 1 in 100 of those 1 in 1000 times it will be signifigant in the outcome of the game - and probably half the time it will hurt the owner of bless and half the time it will help them.   So no big loss.

But for those of you who can't handle that, I provided different alpha levels.  So going left to right you have 1/1,000 ; 1/10,000 ; 1/100,000 ; and 1/100,000,000.   So basically the final column is saying that with a 50 card deck, storm 155 will yield a disproportionate outcome from storm 1,000,000 only once out of one hundred million simulations.



So how does this help us.  Suppose we agree on Alpha 0.01%.  This means that for a 61 card deck any amount of storm greater than 74 will be essentailly the same as storm 74.  So to resolve a storm greater than 74, you can basically agree to resolve 74 storm instead, and call the stack empty.  You can even choose some number greater than 74 and it won't effect the outcome.  this is why you can apply 74 to a 60 card deck, or a 55 card deck, or basically any deck lower than 61... because the same distrubution is the same distrubution (and lowering the deck size will only inevitably lower the require blessings to meet the same alpha). 

Resolveing 74 storm isn't so bad either.  You can do it quickly if both players agree to it.  so back to the 61 card deck... it would take 21 storm to mill completely so if storm is greater than 20, you WILL hit blessing before the stack resolves.  So here's what you do. 
- We start with 74.  Shuffle up the deck, then just flip it over face up and find the blessing.
- Count from the top down in sets of 3 cards.  So lets say its #8 from the top, you count down, 1 (3 cards), 2 (6 cards), 3 (9 cards) and hit the blessing.  Now you subtract 3 from 74 and shuffle the deck agian
- now we have 71,   71 is greater than 20 so we can do that again.  Shuffle the deck, fan it out face up and locate blessing.  Lets say it's 42 cards down.  42/13 = 14.  71-14 is 57.  Shuffle up with 57 more storm to resolve.
- 57 > 20, so we can do it again.  and again and again... until the storm get's to 20 or less.

- Now lets say the storm is 16.  ok, so we shuffle up the deck and leave it face down this time.  Now just resolve 16 more brainfreezes normally.  Not that hard.

All in all even with 74 storm I'll bet it would only take about 4-6 mins to completely resolve using this method.

----------------------------------------------------------------------------------------------------------

A final point to all us Brainfreeze players:  There's really no reason to cast brainfreeze for any number over 50 storm (milling 150 cards).  At ELDs tournement when I had the chance to go infinite I said "I generate up to 50 storm and brainfreeze you."  Every time my opponent just scooped.   This is the responsible thing to do, because there's basically no way they would survive a storm 50 and not a storm 1,000,000 (packing 35 counterspells??) But in the event they do have Blessing, we could resolve the stack.

[Anusien]

So you have a high level of intolerance for any probability less than 1, and yet it is acceptable for judges to make subjective rulings such as assessing slow play? Why not make a likewise subjective ruling for Brainfreeze/Laquatus + Blessing by adopting an arbitrary cut off where you essentially equate a non-1 probability to 1?

Easy.  Slow play is a binary thing; either you are slow playing or you aren't, in the eyes of the tournament.  In fact, it's rarely the case that the player was not Slow Playing but got called for Slow Play; more often it is that a lot of judges hesitate and would wait too long to give a Slow Play infraction when the player is actually slow playing.
Judging is inherently subjective, that's why we're called Judges.  Part of the importance of the DCI and the Penalty Guidelines and such is to make LESS subjectivity.  In the case of slow play, there are things like mentoring, large tournaments and judge articles to help encourage an even standard of decision making so that what is Slow Play in the Ukraine will be Slow Play in New Zealand or in the States.
So arguing that the rules need to be changed to increase subjectivity is probably not the way to go.

Isn't that exactly what you are doing??  You have one player with a 99.99.. % chance of winning and you are saying that player should not be able to win?  You think that game should be a draw right?

Now I'm really confused about what you think the outcome should be...

You have one person with 99.999% chance of vitory and one with 0.001% chance of victory.  By declaring the game in favor of the 99% you are essentially cheating the losing player by 0.001%.  By declaring the game a draw, you cheat the winning player by 49.999%.   How can you sit back and say the ruling is fair?

I don't think I've ever gone and stated how I think the game state should be resolved.  I've just gone and stated how it IS ruled, and ways to make the game state playable again after it is wrecked by the casting of Brain Freeze.  I've certainly never said arbitrarily, "The game is a draw" or "The Brain Freeze player loses."
And it's not fair to say the player has a 99.99999% chance of winning.  Given enough time to play out every Brain Freeze copy, that may be their chance of winning.  I would guess that given even the entire 50 minutes it's not nearly that high..

I'm not sure I agree with this solution. Even if the probability approaches 1 such that it is more likely that God appears and starts playing Magic with you as opposed to an event not occurring, there is still no guarantee of eventuation. Creating a game state where it is likely that you'll win does not equate to actually winning, regardless of probability.

If I have seven cards in hand and my opponent has none, and no permanents in play, does that mean my opponent should concede in light of the probability that they'll recover being extremely slim? Assume my opponent is on Turn 5 of extra turns, has no way of winning, has no cards in hand, and no permanents. Does that mean that they should concede to me?

Another issue I see with the Laquatus/Blessing scenario is that while continuing the loop indefinitely creates a likelihood of arriving at a certain game state, the time constraints of a match are not indefinite. Therefore, it seems very unreasonable to posit that because a circumstance is probable given infinite time, that this resulting circumstance is applicable to a match with a 50 minute time limit.

The right solution seems to be to just allow a player to continue going through loop iterations or create 1,000,000 Brain Freeze copies. I'm of the opinion that since they are actually attempting to win, that this is acceptable. Yes, it is tedious, and will likely result in the match being a draw, but at least the circumstances are fair for both players. That is, unless I'm totally overlooking something, and I'm sure that's also possible.

I think you're definitely on the right page, and a lot of judges have said, "If both players want to keep resolving Brain Freeze copies until time runs out, let them as long as they do it at a reasonable pace."  The problem arises when they hit time and still have, say, 10 minutes worth of Brain Freeze copies to resolve.  In this case one of the judge's primary responsibilities is to keep the tournament running smoothly and make sure it doesn't get held up for everyone because of one match; this is why Slow Play infractions are so important in general.
(I also bolded what I feel are the really strong points in your argument to bring them out more).

A final point to all us Brainfreeze players:  There's really no reason to cast brainfreeze for any number over 50 storm (milling 150 cards).  At ELDs tournement when I had the chance to go infinite I said "I generate up to 50 storm and brainfreeze you."  Every time my opponent just scooped.   This is the responsible thing to do, because there's basically no way they would survive a storm 50 and not a storm 1,000,000 (packing 35 counterspells??) But in the event they do have Blessing, we could resolve the stack.

This is a good point.  I tried to make it a few times earlier, but it bears repeating.

[dicemanx]

Easy.  Slow play is a binary thing; either you are slow playing or you aren't, in the eyes of the tournament.  In fact, it's rarely the case that the player was not Slow Playing but got called for Slow Play; more often it is that a lot of judges hesitate and would wait too long to give a Slow Play infraction when the player is actually slow playing.

You didn't answer my question. I don't care if the decision is binary or not; my point is that it is still subjective.

Judging is inherently subjective, that's why we're called Judges.  Part of the importance of the DCI and the Penalty Guidelines and such is to make LESS subjectivity.  In the case of slow play, there are things like mentoring, large tournaments and judge articles to help encourage an even standard of decision making so that what is Slow Play in the Ukraine will be Slow Play in New Zealand or in the States.
So arguing that the rules need to be changed to increase subjectivity is probably not the way to go.

If the idea is to decrease subjectivity, then you do so by setting an arbitrary cut off for the probability to equate it to a probability of 1. This is because according to the current rules, we have a non-subjective decision (probability must=1) paving the way to very subjective decisions (judges having to deal with the fall-out of Blessing-Laquatus/Brainfreeze) such as assessing slow play. For instance, let's say that I abide by the current rules with my Laquatus, but decide that it is a least worth a shot to try as many loops as I can get away with to leave Blessing as the final card in my attempt to win. I am not intending to stall, I am intending to make progress and win. I try the first time, and hit a Blessing; I try again, and hit the Blessing again. At which point does the judge step in and say that I can no longer keep trying? After the 3rd time? The 5th? The 10th? Notice that this decision is entirely subjective, and not covered by any DCI rulings.

Now, we could just argue that the Laquatus player should have the right to continue doing this until time runs out. This is Rich's proposed solution:

The right solution seems to be to just allow a player to continue going through loop iterations or create 1,000,000 Brain Freeze copies. I'm of the opinion that since they are actually attempting to win, that this is acceptable. Yes, it is tedious, and will likely result in the match being a draw, but at least the circumstances are fair for both players. That is, unless I'm totally overlooking something, and I'm sure that's also possible.

Now, the problem with this solution is that the WGD player (or Brainfreeze player) is actually given the option to stall out the match if he chooses, because he can claim that he is actually trying to win. Once he enters the Laquatus loop or creates Brainfreeze copies, the match is immediately over, the result being either a win or a drawn game. Apart from a direct admission by the player, you will never know their true motivations (whether or not they are trying to stall). Now we have a further problem. If he doesn't win, how exactly does the game become a draw? You never get to complete five additional turns, because you're stuck continuously looping within one turn. Does the judge swoop in and declare the game a draw 5 minutes after time is called? After 10? After 30? This is a second arbitrary decision.

It seems to me that in pursuing an objective decision (enforcing the requirement that P must equal 1) you create the need to make 1 of 2 highly subjective decisions instead. Is that really the best decision then? It should be intuitively obvious that the probability of the judge making the correct decision from an objective standpoint is not even remotely close to the probability of stacking the Blessing as the final card in X number of cycles (X approaches infinity). So according to Kevin's argument (the desire to decrease subjectivity) it is a mistake to not change the rules on this matter.

[Harlequin]

Isn't that exactly what you are doing??  You have one player with a 99.99.. % chance of winning and you are saying that player should not be able to win?  You think that game should be a draw right?

Now I'm really confused about what you think the outcome should be...

You have one person with 99.999% chance of vitory and one with 0.001% chance of victory.  By declaring the game in favor of the 99% you are essentially cheating the losing player by 0.001%.  By declaring the game a draw, you cheat the winning player by 49.999%.   How can you sit back and say the ruling is fair?

I don't think I've ever gone and stated how I think the game state should be resolved.  I've just gone and stated how it IS ruled, and ways to make the game state playable again after it is wrecked by the casting of Brain Freeze.  I've certainly never said arbitrarily, "The game is a draw" or "The Brain Freeze player loses."
And it's not fair to say the player has a 99.99999% chance of winning.  Given enough time to play out every Brain Freeze copy, that may be their chance of winning.  I would guess that given even the entire 50 minutes it's not nearly that high..

Then that's slow play for the person with gaea's blessing.  The player with blessing in thier deck is intentionally making thier opponent play it out, hopeing it takes them longer than 50 mins to resolve.  This is Identical to doing the tutor->tutor->tutor->tutor->draw->fetch plan to run out the clock.  

The "I've never said the game is a draw" arguement doesn't work... If I say: "I'll kill you if you don't give me all your money" is it not theft because they chose to give me the money?  I mean, who are invissioning playing the game? Jonhy 5?

How is it not "intentially running out the clock." if you say "Well I know I have a better chance of winning the lottery in every state.. but hey I don't think you can beat me in the time alotted so lets play it out..."  or perhapse better worded as " you probably can't beat me so I intend to let the clock to run out."

[Apollyon]

Isn't that exactly what you are doing??  You have one player with a 99.99.. % chance of winning and you are saying that player should not be able to win?  You think that game should be a draw right?

Now I'm really confused about what you think the outcome should be...

You have one person with 99.999% chance of vitory and one with 0.001% chance of victory.  By declaring the game in favor of the 99% you are essentially cheating the losing player by 0.001%.  By declaring the game a draw, you cheat the winning player by 49.999%.   How can you sit back and say the ruling is fair?

I don't think I've ever gone and stated how I think the game state should be resolved.  I've just gone and stated how it IS ruled, and ways to make the game state playable again after it is wrecked by the casting of Brain Freeze.  I've certainly never said arbitrarily, "The game is a draw" or "The Brain Freeze player loses."
And it's not fair to say the player has a 99.99999% chance of winning.  Given enough time to play out every Brain Freeze copy, that may be their chance of winning.  I would guess that given even the entire 50 minutes it's not nearly that high..

Then that's slow play for the person with gaea's blessing.  The player with blessing in thier deck is intentionally making thier opponent play it out, hopeing it takes them longer than 50 mins to resolve.  This is Identical to doing the tutor->tutor->tutor->tutor->draw->fetch plan to run out the clock.  

The "I've never said the game is a draw" arguement doesn't work... If I say: "I'll kill you if you don't give me all your money" is it not theft because they chose to give me the money?  I mean, who are invissioning playing the game? Jonhy 5?

How is it not "intentially running out the clock." if you say "Well I know I have a better chance of winning the lottery in every state.. but hey I don't think you can beat me in the time alotted so lets play it out..."  or perhapse better worded as " you probably can't beat me so I intend to let the clock to run out."

How is the Blessing player stalling because he has a card in his deck? I didn't realize that merely having cards that were legal in the format in your deck and registered in your decklist was an infraction of the rules. It's not his fault that his opponent cast Brain Freeze or that his opponent failed to get rid of Blessing before casting Brain Freeze.

Is it stalling because he makes them play it out? You need to win the game, not make your opponent lose. If CF sits down across from Timmy Mc12-year-old, he can't just say "you are probably going to lose, so just scoop now." He needs to sit down, and play the games.

[Anusien]

DA: What if we say, "The game is a win if P(Win) > 0.9999999999 and somebody puts the wrong number of copies of Freeze onto the stack resulting in a probability of 0.9999999998 (maybe they didn't read this thread).  What then?  Yes, there is always going to be subjectivity in judging.  Slow play and non-deterministic loops are really in two different classes, unless you want to start giving probability seminars to judges.  You might or might not get a penalty for USC or Slow Play depending on if the judge notices it, that doesn't mean we should ignore these infractions.  Yes, it's a sticky situation, and none of your fixes eliminate the stickiness; in fact they're going to make it worse.  What if there was a loop that was similar to the infinite mill + Blessing situation?  Now we have told judges that you can rule on a non-deterministic loop only in this one case, and not in any others.  How are you going to explain that this new case is different?
For examples of such a case:
Thermopod + Narcomeoba + Shiva Hellkite + Shuko + Cephalid Illusionist + Gaea's Blessing
Infinite red mana, Viashino Sandskimmer + Chance Encounter

Harlequin: It's not automatically slow play if you make your opponent play it out.  If I have Academy Ruins + Mindslaver and you don't scoop, it's not automatically slow play.  If I have Solitary Confinement + Squee, Goblin Nabob and you don't scoop with 0 outs in your deck, it's not automatically slow play.  You're at an obligation to play at a reasonable pace, but you're never required to concede a game.
"tutor->tutor->tutor->tutor->draw->fetch" is not necessarily slow play.  In fact, I could probably play out that chain in an incredibly reasonable pace.  If you stop and think in between each one without any intervening changes, it might be.  But if you do it slowly in order to run out the clock, the phrase you're looking for isn't Slow Play, it's Stalling.  Incidentally, it's Stalling for the Brain Freeze player to put 5 million Brain Freezes on the stack knowing the other guy has Blessing in order to run the clock out and win 1-0.  It is definitely NOT slow play making your opponent go through the motions of their combo to see how it works, to see if they know how it works, or for any other reason.  If you have a win on board but you can't win yet, say attacking with Mishra's Factories, if I don't concede, it's not automatically slow play.

The "I've never said the game is a draw" arguement doesn't work... If I say: "I'll kill you if you don't give me all your money" is it not theft because they chose to give me the money?  I mean, who are invissioning playing the game? Jonhy 5?

Please leave the gaping logical fallacies out of the thread.  Unless I'm mistaken, I never said how I would rule on the situation in particular, and there's definitely a large number of ways the game could proceed from that point.  It could end for either player, it could be a draw, or it could be shortcut through the redundant Blessings.  I never said the game was automatically a draw.

[Harlequin]

I'm not going to get into this any further, if you wish to not tell us how you would rule it, then fine.  Anything else I have to say on this subject is just going to basically be a flame, and I don't want to get this thread derailed or locked.

I'd like to bring the discussion back to the post about the magic number of brainfreezes, and how people feel about that. (see a few posts up.

Code:

alpha-> 0.1% -- 0.01% -- 0.001% - 0.000 1% - 0.000 000 1%
40 ----- 36 ----- 51 ----- 65 ----- 80 ----- 126
50 ----- 45 ----- 62 ----- 79 ----- 99 ----- 155
61 ----- 52 ----- 74 ----- 95 ----- 118 ---- 187

Any questions, concerns or snide remarks about that approach?  How would people rule (or not rule...) on this type of short-cutting the stack?  Do people feel comfortable with the 0.1%? or should we go after a narrower fit to the distribution?