Tuesday, August 2, 2011

Serial Vs. Parallel Dice Rolling

by SandWyrm


Mike Brandt wrote something yesterday as a comment to my post on dice over-rolling in 40K that I thought deserved a detailed reply. Since he's made the same comment before when I was discussing the Random-ness of Orks. Where I think that he is confusing the benefits of rolling dice in Parallel (all together at once) with rolling dice in Series (one after the other).

Mike Wrote:
This is old now, but, a greater # of dice roll sets between beginning and ending the determination of an attack, with generally better odds for each roll, is infinitely superior in terms of establishing reliability than having single or fewer dice rolls per attack, with worse odds.

That's to say, if a 2/3 chance to hit and 2/3 chance to wound and 2/3 chance to save is equivalent to 6/27, that's ALWAYS a better approach to take for a tactical game than it would be to roll a single "9-sided dice" or equivalent with 2/9 odds.

That's kind of the important thing ... the games like warmachine and others where they basically roll a single dice per attack per model, with a target # to do damage or not, are BAD in terms of reliable tactical games. Less #'s of dice rolls lead to lower statistical reliability, and thus a streakier game, and one harder to balance, etc. etc.

Buckets o' dice = good.

SandWyrm's Reply:

Define "bucket".

As a rule, I personally enjoy rolling between 10 and 15 dice most of the time. 20 maximum. If you go much above 20, then you either have to start rolling with a cup, or you have to start rolling your dice in batches. Which is the instant that the fun becomes tedium.

Can you fit 35-40 Chessex dice in your hands in a pinch? Sure! But then you're just dropping them (with a zero-rotation shake-rattle or two), resulting in the lack of even a pretense of rolling them randomly. Whatever they were when you picked them up is what you'll end up "rolling".

Are you having fun double-checking your opponent's math when he rolls 18, 18, and then 12 dice for his attacks (with re-rolls)? I'm not. Half the time I could be cheated blind because I just take their word for the results. Do most players narrate their actions well enough that you can follow their split-rolling? No on that too. It's a skill that you have to learn.

So simply as a matter of playability, I consider 20 dice to be a hard limit that we really shouldn't go over except in the most odd of exceptions. Probably not even then. Rolling more dice than that slows the game down and contributes only marginally to the "fun".

As to your other point...


Rolling Dice In Series Does Not Increase Or Decrease Reliability 


Mike Said:
...a greater # of dice roll sets between beginning and ending the determination of an attack, with generally better odds for each roll, is infinitely superior in terms of establishing reliability than having single or fewer dice rolls per attack, with worse odds.

That's to say, if a 2/3 chance to hit and 2/3 chance to wound and 2/3 chance to save is equivalent to 6/27, that's ALWAYS a better approach to take for a tactical game than it would be to roll a single "9-sided dice" or equivalent with 2/9 odds.
You need to do a better job of reasoning this out Mike. If the resulting odds would be the same rolling 2 dice as one, then why not just roll one? You can roll this example as many times as you want, but the odds will be the same no matter how often you do. The only reason to roll more dice is if one or more of those dice can have modifiers applied to it. Or if you want to curve the outcome so that certain results are more likely to happen.

Because there are downsides to rolling multiple dice in series. Especially with an auto-pass/auto-fail system.


Rolling Multiple Dice In A Series Narrows Your Success Range!

Let's look at the example of some poor BS1 Grot shooting a S2 pop gun at a Terminator. He needs a 6 to hit, a 6 to wound, and the Terminator will save on a 2+.

1/6 * 1/6 * 1/6 = 1/216  (or  0.46%)

That's a pretty small chance, almost non-existent in fact. Is it even worth rolling for?

Now let's consider what would seem to be the inverse: Dante's chances of frying one of the new Sisters of Battle (6+ Invul) with his Infernus Pistol...

5/6 * 5/6 * 5/6 = 125/216  (or  57.87%)

What? Really?... Yep.

That's not much of a top range. Dante is just short of 60% in his likelihood to kill that Space Nun. Under GW's system, a 1 is always a failure. So even if Dante is shooting his Infernus Pistol at an unarmored grot point-blank, his chances can't possibly get any better than...

5/6 * 5/6  = 25/36 (or  69.44%)

Dante, Chapter Master of the Blood Angels and the oldest Loyalist Marine still alive, can only wound a defenseless Grot about 70% of the time? What's up with that Grandpa?

Well, it's all about the compounded effects of rolling dice in series. The more dice you roll with auto-fail results, the lower the probability of your best shots. Let's imagine that the Sister of Battle in the example above also somehow got a 6+ Feel-No-Pain roll...

5/6 * 5/6 * 5/6 * 5/6 = 625/1296  (or  48.23%)

And we wonder why we sometimes feel so unlucky in this game. It's because the mathematics of series rolling dictate that it's always easier to whiff than to wound.


The chart above shows the results if we were to take this phenomenon to it's ridiculous extreme. At 20 rolls in a series, we would have only a 3% chance of success even if each individual roll succeeded on a 5+.

Additional Explain'n for Lantz:

This chart shows the probability of passing a 5+ check for a series of die rolls. So up at the top left, the chance of passing a 5+ on 1 die is 5/6 or .8224 (82.24%). For two dice, the number is 5/6 * 5/6, or 25/36. Which is .6945 (69.45%). For 3 dice, your chances of keeping up your 5+ streak is 5/6 * 5/6 * 5/6 (or 5/6 ^ 3). That result is 125/216, or .5787 (57.87%).

The point being that adding more dice in a series with fail-on-a-1 has a detriment. It means that even your best characters can't hit the broad side of a barn in a 3 or 4 roll series. Even under optimal conditions. It also means that for most troops, failure is always far more likely than success.


Rolling Dice In Parallel Does Increase Reliability

What will increase your reliability is rolling multiple dice in Parallel. That is, rolling more attacks at once.

This chart illustrates it nicely:


The probability never reaches 100%, it just gets closer and closer to zero on a curve that gets flatter and flatter.


Conclusion

I hope that I've illustrated the fallacy that rolling lots of dice is always more reliable. HOW you roll those dice, whether in parallel or in series is just as important to the outcome.

I hope this also illustrates the stupidity of having automatic passes on a 6 and automatic fails on a 1. The former produces results so fine grained that we waste time rolling buckets of dice for the smallest of chances. While the latter compresses our odds of success into just a fraction of the range that a D6 system can offer.

27 comments:

  1. While I understand your contesting Mike Brandt's point on whether or not rolling more dice in series is "more reliable," I believe you misunderstood his point. Granted, he did not come right out and say it, so it's understandable that it could be misinterpreted. It is not necessarily that the probability changes with rolling multiple dice in series. It is the balance of deviation and variance from the expected probable outcome that becomes important. If you consolidate things down to a single dice roll, the likelyhood of an "unlikely event" like rolling 4 for 4 on 6+ to wound a Trygon with Bolters is not balanced out by other "unlikely events" further down the line.

    In other words: probability and statistics only truly begins to work accordingly with a large enough sample size. Ever experience games where it seemed every gun you fired caused devastating explosions across your enemy's lines? Or those games where the dice have "betrayed" you and your meltaguns can roll nothing but aces? These examples just go to show that, even with as many dice rolls as we have in the game currently, the variance and deviation from the "probable" outcome is still significant enough that you can have entire games streaked with "improbable events."

    The larger the sample size, the closer your outcomes will adhere to statistics.

    Or, the more dice you roll, the better =)

    This principle can also be applied to the shift we've seen at the most competitive levels of the metagame: the shift towards Autocannons from Lascannons as tank-busters. The shift from deathstar to MSU.

    And, of course, why that damn vehicle damage chart can be so infuriating.

    Long story short, when you compress a certain important event in the game down to a single die roll that has been modified by different characteristics as you've proposed, these "improbable outcomes" have an even more critical impact on the game. It increases the "luck" factor if you will, because it allows for no possible counterbalance to a lucky roll. I wound with 10 of 12 bolter shots against another marine unit? Pretty lucky it sounds like. The other guy then has a possibility to "get lucky" and roll up 9 of 10 saves or something similar. Is it probable? No, as statistics will tell us. Is it possible? Absolutely, and I'm sure many gamers can cite different instances when such hilarity has occurred. It is a source of balance, and of entertainment in a game we love.

    Sorry for the short story here, but I wanted to beat that horse into a bloody pulp to abbreviate explanation in future comments =)

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  2. The game is meant to be heroic, which is why it's set up the way it is. A bunch of grots killing a bunch of terminators in CC isn't likely, but wow are you going to remember it. That is what the game is about, not about predictability.

    That said, probability math is a lot of fun:)

    Also I thought that Björn the Space Wolf Dreadnought was the oldest Space Marine. Hard to grasp all the inconsistencies sometimes :)

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  3. I agree with Sandwyrm

    1. The amount of dice rolled in a game of 40k, esp. with special/heavy weapons, elite armies or powerful&costly HQ models is still far far away from anything that pleases the "law of large numbers". At least that's what I remember from my 2weeks course on statistical analyse back in my first semester of at the U.

    I'm mean just take a look at a squad of Longfangs: with 4 ML/LC and 6 rounds you end up rolling something between 2(hit, wound, no save( and 4 dice per model. That's 8-16 dice per round, 48-96 dice per game. Assuming that you manage to play 6 rounds, keep the whole squad alive and not run out of targets.

    And don't get me started on the numbers you roll with a single gun tank.

    So I don't think cutting down the number of dice you roll in a serie and moving towards a "to hit/wound"-chart that allows stuff like a carnifex wound&kill a grot w/o the player having to roll and hoping that no 1 shows up, won't screw the game...because it is far far away from producing reliable numbers anyway.

    2. The "other games are worse" argument is also wrong, because WM for example uses a 2d5+value and you need 2 tests to hit&remove a model. Sure, WM also has a auto.fail mechanism but the chances to roll snake eyes are 1/36 for a normal check and 1/216 for a boosted 3dice attack.

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  4. I've got big hands, always shake well for a few seconds before dropping with a built in roll just so no matter how many dice I've got (up to a point, dont play IG or Nids!), I'll give them a good random roll. Large number of dice are not an issue.
    Small number of dice (Deathwing) always get a good in-air spin.
    I would ALWAYS prefer to roll more dice to achieve the same probability than fewer dice, mainly due to... Technical wurds like wot Ahrimaneus sed.
    Just went back to your 'reinventing 40k' post, and quite like the new X vs Y table. What I don't like is the Mephiston waltzes through combats auto killing. This game is about thinking and working the odds. As a DW player, I can roll an incredible number of invul saves, then roll three 1's in five wounds from a flamer. If you're trying to remove this sort of thing from hapenning, fine, but Ol'glittery vampire lord has, can, and will roll six 1's.

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  5. God I love these posts. Though I didn't understand the first unlabeled chart in the least. Can you clarify what it represents?

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  6. Ahrimaneus hit the point on the head.

    I get what you're saying, I just don't think I was as clear in a brief comment with what I'm saying.

    The enjoyment of a game decreases (For me), and the "feel" of random increases when there are fewer sets of rolls between targeted outcome and outcome. Again, Ahr hit it well.

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  7. PS - Anything is "bad" if taken to an extreme, which is why a single roll to determine the entirety of hit/damage/save/etc. is bad, and why tons and tons of rolls to get there is bad. You have to find a sweet spot, where both players are involved in the rolling, and where follow-up rolls have an opportunity to better balance "streaky" rolls.

    Also, 40k is not a skirmish sized heroic game really ... heroic scale, but it's not about Mephiston auto-ganking entire squads ... at all. It's about grunts and tanks and guns and swords and all of them coming together in big ole full scale conflicts.

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  8. @ Ahrimaneus I think you're falling into the same trap Mike did.

    Rather than use the terms "series" and "parallel" I would use "dependent" and "independent".

    If you make multiple independent rolls then you will smooth your results thanks to averages. An Ork's "bucket-o-dice" is a great example of this. Because none of the rolls affect each other the goofy outliers are essentially drowned out and made irrelevant by the sea of average rolls.

    If you roll the same number of dice but have them all dependent on each other then you're basically just making one "super" roll. Those goofy outliers can't be ignored because a single whiff will scupper the whole result.

    A chain of dependent rolls adds nothing for the stability of the final outcome.

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  9. @Ahrimaneus

    "…Or, the more dice you roll, the better =)…"

    I understood Mike's point (and yours) just fine. What I'm saying is that both of you are incorectly applying a general truism to a case where it doesn't work.

    Rolling dice in parallel? YES, more is better. Up to a point.

    Rolling dice in Series? More doesn't increase reliability. Because the sample size is still one. You DO get more finely grained results, but those results are not linearly distributed. You get extremely fine-grained results at the low-end (failure), and extremely coarse results at the high end (success).

    Lastly, the statistical abberations you mention are mostly a product of the auto-pass-on-6 and auto-fail-on-1 system that we play with. Going with a auto-pass/auto-fail system (no 1 or 6 charity) on 2 series dice would expand the range of results we play in, save time, and still allow us to have our hero moments by using a different mechanic such as a single "Weight of Fire" roll for an otherwise useless unit's firing that always wounds on a 6.

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  10. @Flekkzo

    There are other ways to ensure Hero moments that don't gimp the game. The weight of fire idea, for example. If your 10 lasguns can't wound that Terminator (because the chart doesn't allow it) you can roll a single die per 10 guys and wound a single Terminator on a 6.

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  11. @Lantz

    I've added some more explanation just after that chart.

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  12. Korona has it.

    @Mike

    If we could design a game that did away with the bucket-o-dice (anything over 20) and yet still felt less random than 40K, while reflecting the fluff better; would you be interested in playing? How about all of that in a 1 hour game?

    That's what I think that we can accomplish here with a series-of-2 mechanic (instead of 40K's series of 3/4) and a few "hero" rolls to give us those shining moments of valor without all the tedium.

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  13. When you roll dice in sequence you do generate a probability for that sequence, but each individual die roll is still an independent result on its own.

    So if you roll 10 dice at one time, or 10 dice in a row, the given probability of seeing any one result on any one die is still 1/6.

    What Mike is saying, and I agree with, Is that, if you have 2 events or 3 events that work out to the same average probability, 3 is the better choice. It's better because 3 independent events that are interpreted together in a sequence are slightly less susceptible to random chance than two.

    You can't predict the results of a single d6, but if your roll enough dice you will get an even distribution of all the results. Whether I roll dice in row or at the same time, that doesn't effect the distributions of all the results. If you reduce the total number of dice you roll in a game from 200 to 100, you are that much more likely to see a total distribution of results that skews to one subset of results. So while your method controls the probabilities for single events within the game, it would effect the overall game because less dice means you are less likely to see an average distribution of results.

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  14. The alternative is to use 2d6 together and get yourself a nice Gaussian normal distribution. Then you can effectively get a way with fewer overall dice rolls because those rolls are more predictable.

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  15. One more example to make it clear.

    Lets say that instead of rolling dice for a game of 40K we used a random number generator to generate n random values between 1 and 6. Every time we need a die result we just pop the top value from the list. Notice that if you pop 4 results in a row versus 4 dice at one time, you still get the same results.

    Say we also played a really short game that used only 10 die rolls under the current system, and then 6 die rolls under your system. Here's the random set of 10 dice: [6,6,6,3,5,2,4,1,1,1] That's a average distribution over the ten dice, but if you only look at the first 6, you rolled one hell of a game.

    So if you total sample size of random events is smaller, there's a higher chance of having a un-average sample set.

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  16. @CaulynDarr

    You aren't following what I'm trying to say.

    Yes, rolling a bucket of dice, say 200, for your to-hit rolls (one throw) will even out your luck such that it approaches a distribution that looks like a bell curve. This is statistics 101. I don't dispute that.

    What I'm disputing is the taking of this principle and trying to say that it applies to dice rolled in a sequence (one after the other). It doesn't. That's lazy thinking.

    Different principles apply to dice rolled in a sequence/series than to dice rolled in parallel. You get a different sort of distribution curve and an overall lowering of your granularity at the success end the longer the sequence. Which is what I'm trying to show in this post.

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  17. I get what you're saying about how we roll too many dice as things are written right now (glance at Green Tide), and I don't really have a problem with that. The only thing I really have a problem with is that stat chart from your previous article with Terminators being T7. This would mean that nothing that's S4 or lower (on your other chart) could hurt it. Period. Which would make armies that can take Termies as Troops (you know who you are) nearly impossible to kill, since (most) base weapons are S4. This would, more than likely, break the game. Now, I'm not saying it's wrong to want the game to reflect the fluff, I'm just saying that you have to be careful when doing so, and draw a line in the sand somewhere. Because if the game accurately reflected the fluff perfectly, Draigo would be a single-model army.
    (Braces for rebukes)

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  18. There is no distribution curve on a d6. It's flat.

    I do know that determining the probability of dependent events is different from the probability of independent events.

    What I'm saying is that a series of dependent events is made up of discrete independent events.

    The odds of rolling a six after rolling a six is still 1/6. The odds of forming a set of two rolls and having both of those rolls yield a six is 1/36. That still doesn't change the fact that each six was still a 1/6 chance. Rolling one die does not magically change the probability of the next roll.

    If you want a game that effectively manages random chance, you have to do one of two things. Roll enough dice(either in parallel or sequence) to average out the complete set of rolls, or use a dice mechanic that maps to a normal distribution(i.e. Warmachine).

    I'm not against games that use fewer die rolls if you can come up with a reasonably predictable dice mechanic. But if you reduce the number of die rolls with a dice mechanic that uses a flat distribution curve, you make random chance a more important factor in the results.

    You can manage probabilities all you want, but those are just estimations of possible results. Estimations that are less accurate over smaller sample sets.

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  19. We are arguing two separate points.

    You're saying that you can roll fewer dice and come up with the same probability.

    I agree. That is a fact.

    What I'm sating is that if you roll fewer dice you will see more variance in your results. You won't see the most likely outcome as much because you are taking from a smaller sample set.

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  20. @TheNeverThere: Well, that only works as long as you assume that there is no re-balancing of the army books.

    Can't remember it clearly, but I don't think Sandwyrm indicated to just throw in a different matrix and then go and change all the statlines in the game w/o overhauling the force allowance and point costs too. ^^

    @auto-hit/fail: I don't have a problem with a auto-hit/fail mechanism per se, but I don't think it works that well with a system thats build around single d6 checks, with a tendency of dependent strings of 3-4 dice rolls. I think the "chance to fail" is way too high and as long as the system keeps using the d6 core mechanism, I would rather see some 1000+ years old chapter master just waltz over 10 guardsmen in CC, than end with a system where a avatar of a war god with WS10 has a 45% chance to end up not hurting a grot or guardsmen. That's just plain silly...

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  21. idk. i kinda think doing graphs on dice rolling is taking the fun out of the game and moving it to an extreme. while i understand both points of view, i just dont know. why not just play some more games and increase your chances of winning by rolling more dice. lol

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  22. @TheNeverThere

    I keep repeating this idea, but nobody seems to be noticing.

    If your 10 lasguns can't possibly wound the target, then you get a weight-of-fire roll. Roll a single D6 for X number of models. On a 6 you cause a wound or a critical hit on a vehicle.

    If Draigo walks into your Grot squad and you can't hurt him, roll a D6 for that round of combat for X number of surviving models. On a 6 he takes a wound.

    It's a different mechanic that would produce similar results to now without all the extra rolling.

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  23. @Sandwyrm

    Sounds to me like the math behind this is far less important than the intent of what you suggest. A simpler system using other means to achieve certain aspects of a unit. GW generally add dice to "improve" a units chance of doing something. More attacks, a better chance for a die to "advance" to the next roll, etc. While you suggest fewer rolls and other mechanics.

    For instance the 10 lasguns. Why not say that for each time you half the number of shots you can up the strength by 1? Go from 10 shots to 5 shots but you can do more damage. It would simplify the game rather than complicate it (rolling 45+ dice is *not* simple).

    For the math inclined, I didn't even remotely do any math on my lasgun example. I'm sure it isn't fair, it's just an idea of how you can use a different design principle than "roll more dice".

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  24. @Flekkzo

    You've said it better than me. :)

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  25. A major thing with reducing the number of consecutive rolls is it makes the odds much easier to calculate. One or two dependent rolls is doable in your head whereas the chains of 4+ odds get fiendish to work out very fast!

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  26. Ah. I found where you made the "Lucky Shot" comment. It got lost somewhere in the middle of the other Dice topic. I also somehow missed the couple of references to it in earlier comments on this topic. Apologies for that.

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