Figure 1: Probability Distributions for Three Different Dice Pools (Unadjusted)
Perhaps this post will be useful for any part-time game designers out there. Clayton at Kill It With Fire put up a link to his new retro-clone game, Kill It With Fire, which uses a combination of target DCs and dice pools to resolve skill checks and attacks; basically, the player rolls a number of D6, sums the result, adds bonuses, and then tries to get over a given DC. Dice can be added to the pool for various factors, and there are bonuses that apply based on level, skill training, etc. See, e.g., this paragraph:
Example: Lothar the barbarian is attacking a ghost with his magical sword. The sword’s magic grants an extra die to attacks that targets ghosts. His attack’s description says he uses his prowess trait as a bonus, and he has the ghost cornered, adding a circumstantial bonus that the referee says is worth one more die to the attack roll. So his usual three attack dice and his two bonus dice are rolled, and their result is added to his prowess and number of hit dice to see if he matched the ghost’s defense number.
In a follow-up post, Clayton asks a few questions about the mathematics, and particularly how the probabilities change with dice pools and target DCs. The nub of the matter is in this paragraph:
Actually, what I am concerned about with at the moment is math. To hit a target, I wrote that you add 10+current hit dice (which fluxuate throughout the session)+3d6.
3d6 average out to 10 themself, if I remember my Arcana Unearthed 3.5 info correctly. Meaning you may be rolling well over 20 eight times out ten when you attack. Seems a bit high.
Various suggestions have been given in comments, but since I had half a day free I thought I’d explore this in detail, so I made a spreadsheet that calculates success probabilities for 3, 4 or 5d6 dice pools against a range of target DCs, across a range of bonuses. Here I’ll present the results, and make some comments about dice pools and target numbers. Calculation details are given at the bottom of this post. First a few points about comments in Clayton’s post and the comments:
- 3d6 “average out” to 10.5; the most likely values to occur are 10 or 11. By “average out” here we are thinking of the expected value, that is the average value rolled over many rolls. Note that 10.5 is the same as the expected value for 1d20, which has a very different probability distribution to 3d6. By way of comparison, 4d6 average out to 14, and 5d6 to 17.5. This means that, on average, with no bonuses, rolling 3d6 you will beat a difficulty of 10.5. What this actually means in practice, I don’t know.
- Contra commenter Joshua, the central tendency doesn’t get stronger as you roll more dice. In fact, the probability of rolling any single value decreases, as the probability spreads over a wide range. I think here Joshua is referencing the Central Limit Theorem, which states that as the number of dice gets larger, the distribution of their sum tends to be normally distributed. This doesn’t mean that the distribution has to get sharper (which would be the requirement for a “stronger” central tendency); what exactly happens depends on the dice you’re rolling. See Figure 1 for the probability distribution of three dice pools without bonuses
- I approve of death spirals
In fact, it’s unlikely that these dice pools vary very much from just rolling 2d10. The crucial point is that they have a very different central tendency to 1d20, where any value has a 5% chance of occurring. Because the world is normally distributed, d20 is a terrible, terrible way of resolving probabilities of success in gaming (IMHO). Also, adding dice to a pool widens the range of outcomes, so if you rescale stats and modifiers accordingly, you get a better range of outcomes – 5d6 covers 25 possible values, while 3d6 covers 15 and 2d10 covers 18. But this is just a matter of nuance.
Success Probabilities for Given Bonuses and DCs
Figure 2 shows the probability of success for three different dice pools in the Kill it With Fire system, for a bonus of +8. That is, the PC has a bonus of 8, and the chart shows the probability that PC will be successful for DCs ranging from 8 to 37 (horizontal axis) for the different dice pools.
Figure 2: Success Probabilities for Three Dice Pools
As can be seen, with a dice pool of 3d6 and a +8 bonus, the PC has a probability of 50% of beating a target DC of about 19 (actually, from my spreadsheet this is an exact value). At 4d6, this probability becomes 84%, and at 5d6 it is 97%. If we suppose that +8 is about right for a 1st level fighter, then we need to construct our system so that a first level fighter presents a target DC of about 19 if we want a 1st level fighter to hit a 1st level fighter about 50% of the time. A few other points:
- If you think of bonuses as shifting a PC along the curve for a given dice pool, then a +1 bonus will tend to have a smaller effect as the dice pool increases in size. A +1 increase in the bonus will essentially improve a PCs chances of success by about 12% for a 3d6 dice pool, by 7% for a 4d6 pool, and by about 3% for a 5d6 dice pool
- On the other hand, increasing the dice pool by 1 has a large effect on success probability. It increases the probability of success for any given DC by between 20 and 30%
- Furthermore, the largest effect is in the first additional die. For example, the chance of beating a DC of 20 is 37.5% for a 3d6 pool, 76% for a 4d6 pool, and 94% for a 5d6 pool. So the first additional die doubles the chance of success, while the second one increases it by only another 20%.
- In terms of odds, the odds ratio for success is 5.3 times higher going from 3d6 to 4d6, and 5 times higher again going from 4d6 to 5d6. For a shift from a bonus of 8 to a bonus of 9, the odds ratio is 1.7.
- This effect of dice pools is huge for small bonuses – the odds of success in going from 3d6 to 4d6 is 10 times greater for a PC with only a +4
- Thus, additional dice are a powerful circumstantial modifier, and should be balanced carefully against bonuses
This makes a dice pool mechanism very successful, but I think Clayton might have been thinking to use the dice pool changes more than bonus adjustments to reflect circumstances. My suggestion would be that those additional dice be reserved for extreme circumstances (opponent is stunned, backstabbed, etc.) and smaller bonuses for things like magic weapons.
A Few Thoughts on Dice Pools and Target Number Mechanisms
The main benefit of Dice Pools as far as I can tell is that they give you a normally distributed random variate. Changing the number of dice will significantly increase the chance of success against the same DC, but also makes the random variate more normally distributed. Alternative mechanisms – like changing the dice type – will affect the parameters (mean and variance) describing the approximate normality of the random variate, but they’re in principle no different. So when you compare a dice pool result to a target number you’re not varying, fundamentally, from the method of 3rd edition D&D, all you’re doing is changing the relative balance of outliers and central values. I moved to 2d10 in 3rd Edition D&D to reduce the chance of criticals (and then dropped the second critical resolution roll), but you can do this without changing any of the bonuses and modifiers. Adding flexibility to the dice pool size gives the advantage of large steps in probability of success, but also gives the GM almost infinite flexibility to break the encounter by throwing in an excessive dice pool modification (as Figure 2 shows). In my opinion, D&D 3rd Edition was fundamentally flawed in using d20s, but otherwise the roll-and-beat-the-target mechanism is simple and useful. Changing dice pool sizes simply adds flexibility to the probability distribution underlying this mechanism.
Unless your game system is mainly story-telling, the probability structure of the underlying task resolution mechanism is going to be a strong defining aspect of the mechanics of play. Hopefully if anyone is designing their own system with a dice pool/target mechanism, the material I’ve put here (or the spreadsheet itself) will help them in establishing the parameters of their task resolution mechanism, and avoiding accidental game-breaking mechanics.
Calculation Methods
The formula for the probability of any outcome of a given number of dice is not pretty, but it can be obtained from this website, which gives an analytic solution from The Theory of Gambling and Statistical Logic by Richard A. Epstein, formula 5-14. This is relatively easy to implement in Excel using a visual basic function, or at least it would be if visual basic included a combinatorial function (how useless can a programming language be?) Once I’d figured out the details of that, it was fairly easy to implement the formula in a basic spreadsheet, which anyone who is interested in is welcome to ask me for. The formula can be extended to other dice pools (e.g. d10s, d8s), though my spreadsheet isn’t that flexible (I would have to change a few details of the function, which I’m willing to do if a reader needs it). Just leave a comment here if you want me to send it to you – but note I’ll only send it on one condition: that you have a RPG-related blog. Otherwise, perhaps one day some pesky university student will trick me into handing them the solution to their class assignment.
Compromise and Conceit 
January 6, 2012 at 8:05 pm
Excellent info! Since I made it, I’ve been thinking of doing a system where instead of the usual RPG type +1 weapons and the like, bonuses are in terms of extra dice (and then it would be really easy to see what bonus to add just by counting checks above hit dice on the character sheet… it’s kinda hard to explain without a sample). Now I see I have to temper the inclination to give out bonus dice like they were candy though. The example you quoted is almost definitely giving too many bonus dice. On the other hand, for a player, missing sucks a lot, so I think hitting 80% of the time may be the ideal in my system. The monsters don’t have to hit so well though.
January 6, 2012 at 10:52 pm
Bonus dice could be very useful for handling things like surprise attacks, backstabs, stunned opponents, being knocked prone, etc. I think the combination of dice pool and target number is a good idea, but you need to fiddle carefully with the relative values of each to get the right balance of effect from adding dice to the pool An obvious example would be to use d4s as a die pool (but no one wants to do that!) What you would need though is a defining principle for the difference between a bonus and a bonus die. Something to do with a big, but potentially random, benefit that the PC receives…?
January 6, 2012 at 11:51 pm
This may be a question more for the game designer than the statistician, but isn’t adding an extra die pretty much equivalent to giving a +3.5 bonus to a roll? The mean result will be the same, but the variance will be much lower. It seems like a similar question to “does a 20 on a d20 always hit?” For “normal” DCs, your chance of success should be nearly the same, but for high DCs, you can’t possibly hit the target number if you use the +3.5 bonus method.
January 7, 2012 at 12:00 am
On average, yes it would be. But the difference is all in that variance – adding a d6 enables your PC to hit targets well beyond what the +3.5 bonus would. I personally think the effect of being able to hit higher numbers (opening up a wider range of possible achievements) is important, but it may not be worth the additional effect on the average. A partial way around that might be “exploding” dice (e.g. on a 6 you can reroll and add) but the mathematics of exploding dice can be hideous.
January 9, 2012 at 8:46 am
For simple modelling (i.e. what you actually do when playing the game) treating d6 as equivalent to +3.5 should be a sufficiently good rule of thumb. While the dice roll has a wider variance, that variance is as likely to be against the player as for them. That means that the variance only really helps when a +3.5 bonus simply can’t reach the target number.
The other element that needs to be considered is that as the dice pool size goes up it’ll trend towards the mean. That probably doesn’t matter much in dice pools of 3 to 6 dice, but in Exalted size pools of 20+ does tend to make assumptions about your results more accurate (though more exciting when the assumptions are wrong).
March 4, 2012 at 5:52 pm
lately it occured to me that the fact the three 1s in your pool’s results equal a fail and three 3s equal a crit could justify some of the extra dice being given out. High stakes and all that. I wonder where those probabilities lie on the spreadsheet.
March 4, 2012 at 9:41 pm
I’m not sure what you mean … three 3s?
March 5, 2012 at 12:16 am
sorry, three 6s. I would edit my comment if I could. Three 6s in your die results make a critical. Three 1s make a critical failure.
March 5, 2012 at 10:44 pm
Yes Clatyonian, a downside of dice pools that you add together is that as the dice pool increases, critical results become less and less common unless you fiddle with the rules for a critical. I think this is usually handled by including exploding dice (roll-again-and-add type scenarios). I approached the probability distribution for these a while back but they’re nasty. I’ve always wanted to try the distribution for Exalted (why start simple, after all?) Maybe I will…
July 18, 2012 at 12:22 am
Thanks for the write up. I’ve been doing a lot of reading on dice probability and combat mechanics as I also felt that the d20 system is flawed. After looking at a lot of probability sites, running formulas in Excel, and using anydice.com, I decided to go with 3d6 for my players to use in my new system. Now I’m working on the combat mechanics for to hit and came across your blog. I’d love to look at your Excel sheet since you offered to share it. While I don’t have a blog per se, I do have a gaming site that my gamers use to chronicle their adventures even though it has been a while since I’ve ran a campaign. Thanks again!
July 18, 2012 at 10:21 am
Hi Michael, thanks for commenting. I prefer 2d10 to 3d6 when I use the d20 system, because it gives a slightly wider range. It’s also close enough to the d20 distribution in its extrema that you don’t have to modify DCs. For 3d6 you might need to raise the DCs at the easy end and lower them at the hard end. I sometimes think 3d8 might be an excellent mechanism – slightly more normally distributed than 2d10 but with a wider range than the d20 (though only wider by 2). Alternatively one could completely rejig everything and use 3d10 with revised ability score bonuses and DCs, to get a very rich probability structure to one’s world.
I think at that point one might as well switch to dice pools, which I think are more fun to roll and sometimes easier for non-mathematically-inclined people to calculate.
I’m happy to send you the spreadsheet – I’ll mail it to you.
July 26, 2013 at 5:58 am
The best method in rpgs is either d20 or d%, period. Easy to calculate your chances. Multiple dice are the most terrible way to play a rpg. d20 is the best in my opinion because each number is 5% likely to come up, same as the others. This makes it easy to know your chance of success. Die pools are THE SUCK.
July 26, 2013 at 8:56 am
Thanks for commenting, chuckufarley. What do you make of the fact that d20s and d% are uniform distributions?
July 27, 2013 at 6:11 pm
Dice pools, especially with multiple types of dice, are FTW. Straight probability that is clearly visible to the players encourages purely gaming the system [1], while well designed dice pools forces the player to operate in an environment where the outcome (and even the chance) isn’t 100% certain.
[1] And leads at the extremes to D&D 4e where each encounter has an amount of gold it should pay out to get the right amount of gold for the level and the entire thing has been tuned until the soul falls out the bottom.
July 29, 2013 at 11:00 am
I agree with Paul, dice pools can be used to construct probability distributions that it’s hard for players to get familiar with and thus commensurately harder to game. On the other hand, the challenge of understanding hte probability distributions created can lead to confusion and bad game design.
July 29, 2013 at 12:32 pm
True, idfficult to calculate dice pools does make encounter balancing more difficult (as an example of poor game design) however this is the price paid for avoiding D&D style challenge ratings (which are themselves intensely misleading) and the associate predictability.
July 30, 2013 at 10:46 am
I like the idea of challenge ratings, but I seriously doubt that modern game companies have the time and money to run through all their monsters and assess them in enough detail to get those challenge ratings right. Certainly the ones in Warhammer 3 are useless. A wrong challenge rating is worse than none. I think challenge ratings are probably in any case dependent on the style of the GM, the type of adventures he or she sets up, and the attitude of the players. When you’re embarking on a new system, as a GM, I think it’s better to design adventures with escape routes, escalation options, and non-combat solutions to problems than to expect to be able to balance combats fairly. And absolutely I think it is the responsibility of the GM to help a group escape from a combat that is too challenging because the GM planned badly (or didn’t plan). GMs who punish groups for GM mistakes or lack of experience of a system are, in my opinion, very bad GMs. Which is part of the reason I think WFRP 3 is better for experienced GMs, because choosing monsters is hard and the solution to bad monster choice is experienced GMing.
July 30, 2013 at 12:15 pm
Choosing appropriate encounters is difficult in many gaming systems. For example, D&D encounters with stuff that requies magic weapons, especially when the DM expects the party to spot a magic weapon on the way to the encounter. Challenge ratings give a false sense of security when these sort of things aren’t properly reflected in them.
Some of this varies depending on whether you are using a sandbox play style. I prefer a sandbox with Gyaxian naturalism in my dungeons, which means the encounters can swing wildly in difficulty depending on all sorts of factors (i.e. spell usage, lighting, party order, unlucky saving throw). For the right group and mindset this can be good fun – you rolled badly, your character died but they may get rezz’ed if you mate is lucky.
For example, a Birthright game I started (and have to get back to when the kids are a bit older) is predicated on the idea that the players are really playing the family line/realm. If your character dies, too bad, but you’re not really out of the game.