# Dice Math – Mathematics of PURGE.bat

I recently found an in-development cyberpunk RPG called PURGE.bat. If you want to check it yourselves, you can find it here. The game itself is currently quite rough (completely understandable), but we can still explore the skill mechanic a little. The combat in the game uses a simple linear dice roll which is very easy to grasp, so I’ll be focusing on the skill system.

In short, the player has a certain rank in skill (plus or minus any modifiers?), and they roll that many d6s. The result of the roll is then the highest number rolled. Eg. if you get 4,5,1 and 6, your final result would be 6. I have very rarely seen this kind of resolution mechanic – although my experience isn’t maximally wide – and naturally I got curious.

Possible outcomes of the skill check are succeed (with result 6), succeed with a cost (result 4 or 5) or fail (result 3 or less). With one die, the results are exactly as you’d expect. You succeed with probability 1/6, succeed with a cost 1/3 of time and fail 1/2 of time. With added dice, the situation becomes more interesting. Now, in order to fail, you need to fail twice on two (or more) independent die rolls, meaning that the probability of failure becomes $$P(fail) = \left(\frac{1}{2}\right)^N$$ ,where N is the number of dice rolled. For two dice, you fail only 25% of time. For three, only 12.5% of time. Probability for succeeding with no strings attached is similarly simple$$P(succeed) = 1 - \left(\frac{5}{6}\right)^N$$ but succeed with a cost is a little more complicated:$$P(succeed w/cost) = 1 - P(fail) - P(succeed)$$$$= 1 - \left(\frac{1}{2}\right)^N - \left[ 1 - \left(\frac{5}{6}\right)^N\right]=\left( \frac{5}{6} \right)^N-\left(\frac{1}{2}\right)^N.$$

Or, to visualize, see the following graph.

What to make of this system? Immediately we notice that the probability of an outright failure goes down extremely fast. With 4 dice, you only fail 6.25% of time or once in 16 tests. However the chance to succeed with a cost doesn’t change much – It happens roughly between 30 to 40% of the time for pools between 1 and 6. For outright succeeding, we see a clear rise mirroring the drop in probability of failure.

The overarching theme of the rules seems to be ‘success comes with a cost’. For example, players can get bonus die by accepting more severe consequences or getting help from other players at the cost of ‘stress’ – a sort resource describing how much the character can take. The choice of skill test seems to follow this quite nicely, either by design or by fortuitous coincidence (I really do hope it’s by design!).

However, there is another consequence of this choice. There isn’t much room for different characters in the system. The range of skill ranks is only 1 to 5 which leaves limited room to model different levels of proficiency. Checking the plot, there are considerable differences in performance between each rank. Then again, PURGE.bat looks very much like a relatively rules light system, so perhaps this isn’t an issue. Keeping numbers small is good for resolution speed and it makes the game easier to comprehend.

Another concern might be that the effect of an additional die might not be clear without consulting e.g. the above graph. Then the players might not be able to make proper judgments about whether to burn stress or accept extra consequences for another die for the check in question. The same applies for the GM. How to feel about this is up to you, but it is still a feature of the chosen mechanic.

In any case, the game is very much in beta, and I haven’t looked through the “classes” thoroughly enough to say anything about them, but the skill mechanic was unique enough (at least for me), that I thought it would warrant a quick post. I’ll keep an eye on the game and see if I’ll return to the subject.