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Subject: Utilizing law of large numbers to mitigate huge swings in single die mechanics rss

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Mika R.
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I have been thinking about game systems that are somewhat marred by huge swings of luck due to using single die mechanic for resolving action outcomes. As an example, think about the recent D&D board game systems with d20 roll vs. opponent Armor Class attack resolution. It is common to throw long sequences of highs or lows and therefore produce huge swings in the game balance.

Now, law of large numbers state that by increasing the sampling of a stochastic process your accumulated numbers start converging towards mean. A simple average of a say, three d20 throws, will give you Normal distribution according to CLT and therefore less likelihood for consequent high/low rolls over several sessions.

But calculating a simple average is tedious for every action resolution. Therefore I have thought of an alternative: to use median values.

The mechanic to mitigate luck of a single die roll would be following (by the example of D&D attack roll):

For a single attack, roll d20 three times and select middle value as your attack value.
So for example +2 attack against Armor Class 14 would be resolved from the numbers 1,7,16 by selecting the number 7 (a failure).

Motivation for the mechanic:
Using median is more straightforward than calculating average and it gives broader standard deviation than Normal distribution from the average of three values, so the high/low values would be more common than with averaging but less common than with a single die roll. This would give more significance to initial modifiers and hopefully reduce chaotic feel for a game.

Has any game used this type of mechanic before and what do you think about using it in existing games where stochastic sequences from single die roll seem to dominate the outcome of the game too much over player choice?

EDIT:

Here are some computed characteristics of median wrt other systems:

probability to succeed (roll same or higher) with 1:20 target numbers (this not a probability distribution)


simulated distributions of different systems. 3d6 means the sum of three d6 values.
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Tim
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I like a good dose of luck in my games, but I've always been immediately put off by games that rely on a single die roll of any kind, especially D&D. Sure my THAC0 (Yeah, I went old school) is a nice 4, but on any given die roll that doesn't mean anything.

Back in the day, I ditched D&D for the old West End Games d6 Star Wars system. Your character's skill number was the number of d6 you'd roll to use that skill, and you could do multiple things on your turn, at a cumulative loss of 1d6 per action beyond the first. It was a great system with a pinch of push-your-luck.
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Mika R.
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tofarley wrote:
I like a good dose of luck in my games, but I've always been immediately put off by games that rely on a single die roll of any kind, especially D&D. Sure my THAC0 (Yeah, I went old school) is a nice 4, but on any given die roll that doesn't mean anything.

Back in the day, I ditched D&D for the old West End Games d6 Star Wars system. Your character's skill number was the number of d6 you'd roll to use that skill, and you could do multiple things on your turn, at a cumulative loss of 1d6 per action beyond the first. It was a great system with a pinch of push-your-luck.


Sounds neat. How did you use the values of multiple d6 rolls in the Star Wars skill system?
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Dan Cassar
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eikka wrote:
How did you use the values of multiple d6 rolls in the Star Wars skill system?


You added them up and compare them to target numbers usually in multiples of 5. 5 was something easy, 15 was something hard, 30 was something very tough.
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S. Turner
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I ran a quick Matlab simulation to test out your idea that the median of 3 D20 rolls is a good approximation of the median of 3 D20 rolls.

I set the simulation to run 10,000 sets of 3D20 rolls and compute the percentage of times the mean and median are within 1, 2, 3, 4, 5 of each other, as well as the percentage of times they are >5 apart. I also ran the simulation multiple times to make sure that the results were reasonable over multiple runs.

Here's what I got for one of the runs, other runs were similar:
Difference between mean and median / Percentage
Within 1: 43.55%
Within 2: 68.76%
Within 3: 85.14%
Within 4: 94.61%
Within 5: 98.82%
More than 5: 1.18%

Keep in mind that these are the percentages of the difference between the median of a 3D20 roll and the average of a 3D20 roll. Once you start doing 3D20 instead of 1D20, you'll have a lot more difficult time getting a 1 or 20 than a 10 or 11.
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Sean Westberg
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Three dice for every "attack" is a lot of rolling. I'm also not convinced that it actually achieves anything. Each die roll is independent of all the others. If you roll a 2 and a 20, you still have 3-19 as your "median" roll, which is almost as much swing as just rolling a straight D20.

Also, with things like THAC0 and D&D in general it's surprisingly easy to break the RNG by either having impossible target numbers or no-fail target numbers with all the modifiers. D20s sound like a large range of results, but it's actually not that difficult as soon as you're talking about target numbers and modifiers to drastically impact the RNG.
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Franz Kafka
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TheFlatline wrote:
Three dice for every "attack" is a lot of rolling. I'm also not convinced that it actually achieves anything. Each die roll is independent of all the others. If you roll a 2 and a 20, you still have 3-19 as your "median" roll, which is almost as much swing as just rolling a straight D20.


Yes, but your roll of 3d20 will contain at least one 2 and at least one 20 only 1.425% of the time (someone check me?). I think it's reasonable that there's still such a big swing in that instance.
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Ian Hedberg
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Isn't the uncertainty part of the excitement of the die rolls? The roll of 1 that turns what was going to be a assured victory into a sudden defeat, the 20 that comes out of nowhere to save you at the last second? Taking the median gets rid of those heart-pounding extreme results.
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John "Omega" Williams
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In combat its all about chance so D&Ds system works well and in general as you progressed in levels chances of missing diminished noticibly. Least in the AD&D era. Toss in a magic weapon with some + bonus and you were even better set.

In my own books I used a percentile system. Lots less rolling and easier to calculate target difficulties. Combat though was alot more brutal.

Something to keep in mind is that the more you eliminate the random element, the more your going to have to either finagle with modifiers to keep things interesting, or the more bland combats going to get untill it may come to the point that even having a roll is somewhat pointless.

Dice "pools" are a interesting approach and a couple of games have tried this track.

Currently though Im considering applying an "express" style system to a RPG type adventure/explorer game.
 
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Oh my God They Banned Kenny
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eikka wrote:
Now, law of large numbers state that by increasing the sampling of a stochastic process your accumulated numbers start converging towards mean.


Close. The average of your sample converges towards the mean.

eikka wrote:
A simple average of a say, three d20 throws, will give you Normal distribution according to CLT...


No, not even close. The distribution of a sample mean will converge to a Normal distribution, if the sample is sufficiently large and if the underlying distribution satisfies certain conditions. The average of only 3 rolls from a discrete uniform distribution (0,19) will not be particularly close to having a normal distribution, although it will be rather more 'mound shaped' than the uniform distribution.


eikka wrote:
Has any game used this type of mechanic before and what do you think about using it in existing games where stochastic sequences from single die roll seem to dominate the outcome of the game too much over player choice?


Don't recall any game that used quite that mechanism, although having 'rerolls' is fairly common. I think the thing is in this case, if I understand correctly, all you are really accomplishing is changing the probability. Again, correct me if my understand is wrong, however, if the results are indeed a matter of either 'hit' or 'miss', then there aren't really 'extremes' to be averaged out. If a really low roll meant instant death for me, and conversely a high roll was instant death for you and rolls between the extremes meant varying amounts of 'damage' suffered by one or the other of us, then that makes more sense. The usual objective in 'averaging' (or in your case, taking the 'middle value') is to make the extreme die rolls, and therefore results, less likely. But again, that is in the context of an extreme die roll corresponding to an extreme result.
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Bill Parker
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For what you are trying to accomplish you could also consider rolling multiple dice and using the sum (appropriately modified) so that you've got more of a bell shape to your probabilities. I've used three six sided dice for this kind of thing before.

If you absolutely insist on recreating 1 - 20 then you could do something like rolling 2d8 & 1d6 and then subtracting 2 from the result. The fringes of this curve are a little too rare for my liking but YMMV...

01 - 1/384 or 0.26%
02 - 3/384 or 0.78%
03 - 6/384 or 1.56%
04 - 10/384 or 2.60%
05 - 15/384 or 4.69%
06 - 21/384 or 5.47%
07 - 26/384 or 6.77%
08 - 33/384 or 8.59%
09 - 38/384 or 9.90%
10 - 39/384 or 10.16%
with 11 - 20 ramping down more or less the same (its not completely symmetric because we used 2 different types of dice). You end up with a 20 sided die that rolls 8 - 13 more than half the time and 6 - 15 80% of the time but still gets out to the extremes occasionally.

(OK, so it is symmetric anyways.)
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Paul Dale
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The exact probabilities for rolling each of the numbers 1 - 20 on the median of 3d20 are:

1 0.725%
2 2.075%
3 3.275%
4 4.325%
5 5.225%
6 5.975%
7 6.575%
8 7.025%
9 7.325%
10 7.475%
11 7.475%
12 7.325%
13 7.025%
14 6.575%
15 5.975%
16 5.225%
17 4.325%
18 3.275%
19 2.075%
20 0.725%



I'll let you figure out the cumulative values.


- Pauli
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David
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eikka wrote:

Has any game used this type of mechanic before


If the basic mechanic is to get a bell shaped curve, plenty of games use multiple dice (especially 2d6) to resolve conflict. Hero System (Champions), Illuminati, Squad Leader, and Car Wars come to mind.

Quote:
and what do you think about using it in existing games where stochastic sequences from single die roll seem to dominate the outcome of the game too much over player choice?


That's a matter of opinion. 3rd and 4th edition D&D put a lot of thought into balance and they came up with what they thought works well. If you were to use your system it would break standard encounters because PCs are more powerful than encounter-appropriate enemies. A horde of kobolds that need a 19-20 to hit would now connect 3% of the time instead of 10%.

That might be better for your style -- but it's important to keep in mind.

I went through something similar in my 2nd edition days playing with a group that used a horrible crits and fumbles table. It was d20 based but had things like, "behead nearest ally". I sneakily changed the fumble chart to be 2d10 instead of 1d20 and it helped a lot.
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Mika R.
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Here are two sample statistical distributions for the outcomes median of three d20 rolls (above) and mean of three d20 rolls (below) repeated 50 times.




The shapes of the distributions vary between iterations but the lower (mean_distr) show smaller STD than above. Of course the N here is very small but is somewhat representative of a situation in games.

The increase of sample size distributions shows the differences in standard deviation more clearly. Median value gives a distribution that is between the average and single die systems. It does give broader swings than average but has tendency towards mean unlike single die rolls.
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Mika R.
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TheFlatline wrote:
Three dice for every "attack" is a lot of rolling. I'm also not convinced that it actually achieves anything. Each die roll is independent of all the others. If you roll a 2 and a 20, you still have 3-19 as your "median" roll, which is almost as much swing as just rolling a straight D20.


I disagree, there is clearly a bell shape forming in the probability chart of median values. When N=1 anything can happen, though.

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Mika R.
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The Loneliest Banana wrote:
Isn't the uncertainty part of the excitement of the die rolls? The roll of 1 that turns what was going to be a assured victory into a sudden defeat, the 20 that comes out of nowhere to save you at the last second? Taking the median gets rid of those heart-pounding extreme results.


Uncertainty is always there when you are observing stochastic processes. The difference in behavior emerges over several repeats.
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Mika R.
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deadkenny wrote:
Close. The average of your sample converges towards the mean.


Right, sloppy writing of me.

Quote:
No, not even close. The distribution of a sample mean will converge to a Normal distribution, if the sample is sufficiently large and if the underlying distribution satisfies certain conditions. The average of only 3 rolls from a discrete uniform distribution (0,19) will not be particularly close to having a normal distribution, although it will be rather more 'mound shaped' than the uniform distribution.


Based on some simulations with 50 repeats, the bell starts to take rough shapes, as it does even with 2d20 averages (in that case, very rough though).

Quote:
If a really low roll meant instant death for me, and conversely a high roll was instant death for you and rolls between the extremes meant varying amounts of 'damage' suffered by one or the other of us, then that makes more sense. The usual objective in 'averaging' (or in your case, taking the 'middle value') is to make the extreme die rolls, and therefore results, less likely. But again, that is in the context of an extreme die roll corresponding to an extreme result.


That's how it works with many of the board game dungeon crawls/adventure systems. Opponents are typically having 1-2 hit points meaning that most of them are meant for shooting once and moving on. If you get bad luck streaks with your encounters, you are often drained by the system without control. And other times you may sweep through the obstacles without breaking any sweat.

There could be for example a "concentrate" action that allows you to use median of three instead of single die roll. The outcome would still be uncertain and exciting but you would have better control of your odds.

It is also less biased than rerolling, which slants the probabilities significantly.

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Mika R.
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Menace wrote:
That's a matter of opinion. 3rd and 4th edition D&D put a lot of thought into balance and they came up with what they thought works well. If you were to use your system it would break standard encounters because PCs are more powerful than encounter-appropriate enemies. A horde of kobolds that need a 19-20 to hit would now connect 3% of the time instead of 10%.


That is a good point for the RPG system. But the balance is designed differently for the board game versions in various dungeon crawlers. Even meager enemies are more likely to hit and drain resources than their RPG counterparts. So yes, I am approaching the problem from the board game perspective.
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Mika R.
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Omega2064 wrote:
Currently though Im considering applying an "express" style system to a RPG type adventure/explorer game.


By "express" you mean a Yahtzee style game?
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David Fisher
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A problem with using either a median or average of three rolls is that it makes extreme values much less likely (see paulidale's list of probabilities above). If the aim was to roll an 18-20 (or a 1-3), instead of a 15% chance, there would be just over a 6% chance of success. Seems a bit harsh ...

If the idea is to counteract streaks of bad rolls, maybe something like this could help instead:

* Every time a roll results in failure, the player gets a token.

* Add the number of tokens to their next roll (or 2x the number of tokens to get a quicker effect).

* As soon as a roll results in success, the player loses all their tokens -- their "luck" resets again.

Alternatively, the player could "spend" a token (or two) to reroll the die.
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Mika R.
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davidf wrote:
A problem with using either a median or average of three rolls is that it makes extreme values much less likely (see paulidale's list of probabilities above). If the aim was to roll an 18-20 (or a 1-3), instead of a 15% chance, there would be just over a 6% chance of success. Seems a bit harsh ...


This is true but depends also on what style of game play is to be created.

This may be a bit philosophical and certainly depends on a personal taste: If a game relies on multiple one-shot single die rolls (f.ex. crawling through a dungeon) to meet the final goal of the game, wouldn't it be important to use mechanics that allow players with good modifiers to reach the final obstacle more reliably than what a single die dictates? As the low/high sequences are more rare, they should also taste that much sweeter when they happen (but they would not be critically needed for the game experience).

Your proposal of "anti-luck tokens" to mitigate bad luck is certainly another feasible way to do it.
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David Fisher
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eikka wrote:
This may be a bit philosophical and certainly depends on a personal taste: If a game relies on multiple one-shot single die rolls (f.ex. crawling through a dungeon) to meet the final goal of the game, wouldn't it be important to use mechanics that allow players with good modifiers to reach the final obstacle more reliably than what a single die dictates?

Yes, it's helpful in cases where the chance of success is already high; but if the chance is low (less than 50%), using the average or median actually decreases the chance of success ...

Is the aim to counteract the "unfair" feeling of someone having a streak of bad rolls, or to amplify the chance of success (making high chances even higher and low ones even lower)?


 
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Stijn
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It's why we don't play basic D&D anymore. We take D&D settings like Dragonlance or Forgotten Realms, and combine them with 3d6 based rulebooks.

Much more chance to roll average, almost impossible to roll either critical failures or critical hits. To make up for the much tinier chance at crits we also make them more powerful though. arrrh
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Sturv Tafvherd
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why not just roll 3d6 and take the sum to replace 1d20? You get to eliminate outliers and get results closer to the mean


edit: Stijn Van Hove beat me by less than a minute ...
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Mika R.
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davidf wrote:
Yes, it's helpful in cases where the chance of success is already high; but if the chance is low (less than 50%), using the average or median actually decreases the chance of success ...

Is the aim to counteract the "unfair" feeling of someone having a streak of bad rolls, or to amplify the chance of success (making high chances even higher and low ones even lower)?


I would say the latter (if I understood your options correctly). High risk endeavors would necessarily become more difficult to succeed but would be more "fair" (roll 16 or higher with 0 modifier). Low risk endeavors would be lower (roll 16 or higher with +4 modifier). But I think that would lead to better fairness because the outcomes would reflect the initial position better.
 
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