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:

  1. 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.
  2. 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
  3. 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.

 

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