Post by Ziphion on Aug 7, 2014 11:55:44 GMT -8
Are you ready for some math?
So, I've come up with a way of analyzing these data such that we can objectively compare the performance of one stat vs. another. If we examine the "stat vs dummy" curves (that is, increment a stat by one, pit that character against a "dummy" character who stays at constant stats, write down the first character's win rate, repeat), we can see that they roughly follow an exponential curve similar to what you'd see if you plotted "Number of coin flips vs. Chance of at least one of them coming up heads". This is that equation:
1-0.5^(1+x) (for x>0, of course).
Not all of our curves will look exactly like that, though, so we need to add another variable. Let's call it "a", and let's insert it right here:
1-0.5^(1+a*x)
That way, the curve still goes through 50% when x = 0, but as "a" increases, the curve approaches 100% faster, and as "a" decreases, the curve approaches 100% slower. a = 1 for coin flips. You can play with this function here. With me so far?
Ok. So I got a bunch of data for Strength and Agility vs. a dummy character, and used gnuplot to curve-fit the above equation to the data. It doesn't always fit perfectly, but close enough to get a value for "a" so that we can compare apples to apples. Here are the curves for the original derived stat scheme (LP = build + resolve + 3), where each player has a blue (0) weapon:
a = 0.626 for Agility, a = 0.382 for Strength. That means Agility is 1.64x better than Strength in the original system, under these combat conditions. You'd have to multiply your Strength score by 1.64 in order to match Agility's performance. This isn't surprising, for reasons I mentioned previously.
I suggested earlier that the problem could be fixed simply by using Strength to calculate Life Points. Now, here's the thing about that. Build + (Strength or Resolve) + 3 is a very small number of level 1 LPs (3 or 4, maybe 5), considering a character with blue attack and a blue weapon can deal approximately 2.25 damage per successful attack (more if you don't count hitting with a strike). This is a somewhat attack-heavy game, since critical hits pierce armor. A level 1 character can easily one-shot another level 1 character, so an extra Life Point goes a LONG way. Therefore, adding your Strength to your Life Point total actually gives you a bigger advantage than Agility had in the original system when dealing with such low LPs. BUT! Here's the wonderful thing: The problem disappears entirely for larger LP totals.
Derek, how would you feel about changing the Life Point calculation to Build + (Strength or Resolve) + 5? Downside: Battles may take slightly longer, but probably not since you don't have to apply this to opponent creatures. Two benefits: 1) Characters don't get knocked out from full LP as often, reducing frustration, and 2) You get beautifully balanced curves like this:
1.034 for Agi, 1.183 for Str. That means Str is 1.14x better than Agi under these conditions. Much nicer, and unnoticeable to the players at the table!
Side note: paulooshun, that last curve is for you; it represents a character who evenly splits their points between Str and Wis. For "1", I averaged the results for a character with 1 point in Wisdom, and the results for a character with 1 point in Strength. For "2", the results for a character with 1 Wis and 1 Str. For "3", averaged 2 Wis 1 Str with 2 Str 1 Wis. And so on. You can see that the Str-Wis guy has a disadvantage in combat (0.776 means he's only 66% as good as pure Strength guy), but he also has other benefits both in and out of combat, as I mentioned before. So I think that's nicely balanced.
What's that? You want more plots? Well, I shouldn't disappoint:
That was for both characters (test character and dummy) at Build = 1. Amazing.
Build = 1 for both, and they also both have a green (1) weapon. Look how Agility actually takes the lead here, slightly.
For that one, both test character and dummy had a blue weapon AND blue armor (Build = 0).
Here's what I was talking about with LP being too low. If LP = Build + (Strength or Resolve) + 4, Strength has the edge because it takes two or three hits to bring him down, instead of one or two, and Agility's extra defense isn't enough to make up for it. In the bizarre case where you had two characters facing off who BOTH had -1 Build for some reason, you'd see an imbalance, but I think that's unlikely since character card stats start at 0.
I also want to make the point that even if Strength has a very slight advantage over Agility in this new derived stat scheme, I personally don't have a huge problem with that. At least players can still look at the character card and see that there's a bonus benefit for going melee (LP) or ranged (Defense). And as Derek mentioned in a previous post, Strength-based characters are already at a slight disadvantage because they have to move into melee range to attack (yes, ranged characters need to reload, but that only costs a move action, which I don't think is enough to counteract the intrinsic advantage of attacking at long range). So I think it evens out this way.
So, I've come up with a way of analyzing these data such that we can objectively compare the performance of one stat vs. another. If we examine the "stat vs dummy" curves (that is, increment a stat by one, pit that character against a "dummy" character who stays at constant stats, write down the first character's win rate, repeat), we can see that they roughly follow an exponential curve similar to what you'd see if you plotted "Number of coin flips vs. Chance of at least one of them coming up heads". This is that equation:
1-0.5^(1+x) (for x>0, of course).
Not all of our curves will look exactly like that, though, so we need to add another variable. Let's call it "a", and let's insert it right here:
1-0.5^(1+a*x)
That way, the curve still goes through 50% when x = 0, but as "a" increases, the curve approaches 100% faster, and as "a" decreases, the curve approaches 100% slower. a = 1 for coin flips. You can play with this function here. With me so far?
Ok. So I got a bunch of data for Strength and Agility vs. a dummy character, and used gnuplot to curve-fit the above equation to the data. It doesn't always fit perfectly, but close enough to get a value for "a" so that we can compare apples to apples. Here are the curves for the original derived stat scheme (LP = build + resolve + 3), where each player has a blue (0) weapon:
a = 0.626 for Agility, a = 0.382 for Strength. That means Agility is 1.64x better than Strength in the original system, under these combat conditions. You'd have to multiply your Strength score by 1.64 in order to match Agility's performance. This isn't surprising, for reasons I mentioned previously.
I suggested earlier that the problem could be fixed simply by using Strength to calculate Life Points. Now, here's the thing about that. Build + (Strength or Resolve) + 3 is a very small number of level 1 LPs (3 or 4, maybe 5), considering a character with blue attack and a blue weapon can deal approximately 2.25 damage per successful attack (more if you don't count hitting with a strike). This is a somewhat attack-heavy game, since critical hits pierce armor. A level 1 character can easily one-shot another level 1 character, so an extra Life Point goes a LONG way. Therefore, adding your Strength to your Life Point total actually gives you a bigger advantage than Agility had in the original system when dealing with such low LPs. BUT! Here's the wonderful thing: The problem disappears entirely for larger LP totals.
Derek, how would you feel about changing the Life Point calculation to Build + (Strength or Resolve) + 5? Downside: Battles may take slightly longer, but probably not since you don't have to apply this to opponent creatures. Two benefits: 1) Characters don't get knocked out from full LP as often, reducing frustration, and 2) You get beautifully balanced curves like this:
1.034 for Agi, 1.183 for Str. That means Str is 1.14x better than Agi under these conditions. Much nicer, and unnoticeable to the players at the table!
Side note: paulooshun, that last curve is for you; it represents a character who evenly splits their points between Str and Wis. For "1", I averaged the results for a character with 1 point in Wisdom, and the results for a character with 1 point in Strength. For "2", the results for a character with 1 Wis and 1 Str. For "3", averaged 2 Wis 1 Str with 2 Str 1 Wis. And so on. You can see that the Str-Wis guy has a disadvantage in combat (0.776 means he's only 66% as good as pure Strength guy), but he also has other benefits both in and out of combat, as I mentioned before. So I think that's nicely balanced.
What's that? You want more plots? Well, I shouldn't disappoint:
That was for both characters (test character and dummy) at Build = 1. Amazing.
Build = 1 for both, and they also both have a green (1) weapon. Look how Agility actually takes the lead here, slightly.
For that one, both test character and dummy had a blue weapon AND blue armor (Build = 0).
Here's what I was talking about with LP being too low. If LP = Build + (Strength or Resolve) + 4, Strength has the edge because it takes two or three hits to bring him down, instead of one or two, and Agility's extra defense isn't enough to make up for it. In the bizarre case where you had two characters facing off who BOTH had -1 Build for some reason, you'd see an imbalance, but I think that's unlikely since character card stats start at 0.
I also want to make the point that even if Strength has a very slight advantage over Agility in this new derived stat scheme, I personally don't have a huge problem with that. At least players can still look at the character card and see that there's a bonus benefit for going melee (LP) or ranged (Defense). And as Derek mentioned in a previous post, Strength-based characters are already at a slight disadvantage because they have to move into melee range to attack (yes, ranged characters need to reload, but that only costs a move action, which I don't think is enough to counteract the intrinsic advantage of attacking at long range). So I think it evens out this way.
