Frostgrave - Alternative Probability Approaches and Results (Or, "Don't Let Me Design Games!")

As I previously mentioned in my Frostgrave first impressions post, Mike and I decided to try the system out with a 2d10 approach versus the official d20 system.  For me, this idea was introduced by the Meeples and Miniatures podcast (I think) having mentioned using 3d6 instead of a d20.  I assume their reasoning, like ours, was the appeal of a Gaussian distribution that multiple dice give versus a straight uniform distribution.  See below for what I mean (being lazy, I didn't feel like doing probability math so I just simulated results instead):

From left to right, the probability of dice roll:  1d20, 2d10, 3d6

As you can see from the 1d20 graph, it results in an equal probability of any potential result, whereas the 2d10 and 3d6 methods are weighted toward the middle.  Seemed like a great idea to me and it works very well for Warmachine.  Now, if you are smarter than me you already know I was pretty much wrong (or at least not very right).  If you don't know why then keep reading but the key is the fact that attack rolls are "opposed" rolls in Frostgrave (attacker and defender both roll, as opposed to Warmachine where the attacker rolls against a Target Number).

Assumptions/Definitions:

• For the simulation, I used 100,000 "random" rolls for each attack and defense roll, for each of the proposed distributions to to calculate the statistics for:  1d20, 2d10 and 3d6.  From above you can see it was more than enough to guarantee a good sample size by how uniform the graphs are.  
• To provide the absolute most clarity in the interpretation of the results I assumed the attacker and defender had the same pertinent stat lines with:  Fight +0, Armor 10, Health 10.  
• %HIT:  In order for a attack to "hit" the attackers roll + fight must be greater than the defenders roll + fight.  
• %DMG:  To inflict damage, you must score a "hit" and using the original attacker's roll + fight + modifier (some weapons grant extra damage or reduce damage, 0 in this exercise) - defender's armor must be greater than 0.  
• %H&D:  The combined chance to not only hit the defender but to damage the defender (%HIT * %DMG).
• AVG DMG:  The average amount of damage per hit.
• HITS2KILL:  Given the average damage how many hits to kill the defender
• SWINGS2KILL:  Given the %H&D, how many swings (attacks) to kill the defender

As you can see from the table above, the alternative approaches only have a marginal impact on the chance to hit (%HIT) but interestingly lowers the chance.  The follow on chance to damage, if you do hit (%DMG) is a little more interesting for a couple of reason.  Firstly, I expected this number to be more along the lines of 50% since (with armor 10) the attack roll would have to be greater than that, but I "think" the fact that it is an opposed roll is pushing %HIT rolls higher in the distribution (I ran this a couple of different ways with the same result, but I need to go back and look at the distributions to make sure).  The second is that the 2d10 method results in almost a 5% gain in chance to damage - I think this is because you've maintained the same top end range of possible results (a roll of 20) while reducing the lower end range (ie, you can't roll a 1).  Despite the 5% advantage in %DMG though, everything evens out much more closely if you look at the combined chance to hit and damage (%H&D).  Anyway, given how close all these numbers are it is a bit of a push which method to use until you get to looking at average damage and the related hits to kill and swings to kill.  Pretty significant impacts here based on the different distributions, 3d6 looks like it would really draw the game out.

Now, in the case of casting spells things are a little different and end up more as intended (since they are compared to a Target Number).  There is definitely a non-marginal impact and in retrospect, looking at the data, a possible good reason to stick with the as published 1d20 system.  Below is a graphs showing the probability curves for each method vs the spell casting Target Number.

Probability vs Target Number

You can see at the lower end of the TN, the alternative approaches give a better likelihood of success but the higher TNs successes are less likely.  

In both the case of fighting and spell casting, an argument could be made to utilize an alternative approach to tailor the game to the players desire.  Want your solider's to live longer, use 2d10 or even 3d6.  Maybe you want out of school casting to be harder, use an alternate.  I initially thought that the alternatives could be used to make your game more like "low" magic but they actually make a portion of your spells even easier to cast, so can't really do that.

Anyway, the reality is if you really wanted these things you could do it an easier way (IMO).  Want your solider's to live longer, bump the base armor stats for everyone up by +2 (or whatever you desire).  Bump the armor down if you want them to die quicker.  Want "low magic", bump all school TN modifiers up by +2.  "High" magic, bump them down.  Want just the out of school magic to be harder, bump those specific modifies up.

Well, I think I will keep my toes out of game design (or modification) from here on out.  To a degree.  Maybe.  At least for awhile...