Fuzzy on the statistics... « Thread Started on Feb 14, 2012, 12:05am »
So the next campaign is going to use a home-brew variant drawing on Metamorphosis Alpha. To this end I am trying to make combat focus a lot more on what weapons you have and how you use them than just on level (introducing the weapon class system, etc.).
So one of my players had a suggestion and it is just too late at night for me to work through the statistics on it. Hopefully some of y'all kind folk can help. Basically the ideas was this: rather than tinkering with + and -, fighting with two weapons allows you to roll the attack roll twice and take the highest, fighting with a shield forces your opponent to roll the attack twice and take the lowest (with two weapon vs. shield effectively canceling), and fighting with a two handed weapon allows you to roll damage twice and take the highest.
Now, I get the general notion that this means that two weapons is much better when you have a lower chance to hit and that a two handed weapon is much better when you have a high chance to hit. But I am not sure at exactly what point it crosses over and whether or not there is a similar range of utility for each fighting style or if two weapon fighting will consistently eclipse two handed. Any advice?
P.S. if it matters, the weapon class vs. armor class table is set up so that the target numbers range from 20-2 so there is no 'negative' AC or guaranteed hits.
Finarvyn Administrator Dungeon Master member is offline
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Re: Fuzzy on the statistics... « Reply #1 on Feb 14, 2012, 6:03am »
I don't think it's an easy "statistics" problem to solve since there are so many variables, but it's a cool idea and that's what "old school" gaming is all about.
As you've noted, having two attempts to roll an attack (two weapons) certainly improves your odds of hiding a bad roll. Most RPGs are designed where a typical person has roughly 50% chance to hit a typical foe, so we can approximate it with flipping coins. Heads hit, tails miss. With one coin you hit half the time (H, T -- miss on T) but with two coins you hit more like 3/4 of the time (HH, HT, TH, TT -- miss on TT). With the shield forcing a character to take the low roll, it's more like having to hit twice so you only hit 1/4 of the time (HH, HT, TH, TT -- hit on HH). This is an oversimlification, but you get the idea. In reality, with variable AC types the hit-miss thing isn't as cut-and-dried as 50%. My guess is that you'd get similar effects, however.
Advantage: it's a simple rule. Disadvantage: it might overbalance the effect of shields and/or two weapons.
I'd say give it a playtest and see how it works in a game. Then report back!
Marv / Finarvyn DCC playtester (2011) C&C playtester (2003) I'm partly responsible for the S&W WhiteBox Builder of the TrollBridge Master of Mutants; MA since 1976 OD&D Player since 1975
"Don't ask me what you need to hit. Just roll the die and I will let you know!" - Dave Arneson
Joined: Dec 2010 Gender: Male Posts: 347 Location: Waukegan, IL Karma: 87
Re: Fuzzy on the statistics... « Reply #2 on Feb 14, 2012, 9:59am »
Sounds like a fun way to do it. Like Marv said, try it out. If it doesn't work out, you can always fall back on the old +1 hit/dmg/AC (or whatever else floats your boat).
Joined: Jan 2008 Gender: Male Posts: 2,330 Location: New Hope, MN Karma: 93
Re: Fuzzy on the statistics... « Reply #3 on Feb 14, 2012, 1:40pm »
I like it---though I do have to ask what happens when a two-weaponed fighter is in combat with a shielded opponent...
You might want to consider making a DEX requirement for two-weapon fighting, and perhaps a STR requirement for shield use, otherwise you may have every single character fighting with two weapons...
Re: Fuzzy on the statistics... « Reply #4 on Feb 14, 2012, 3:07pm »
To compare these two techniques you can calculate damage per round:
dpr = % chance to hit x average damage
Let's assume you're an OD&D 4th level fighter needing a 10 to hit leather (AC 7). That's 55% chance. Now if you roll two dice and take the best, that's the same as saying 'What is my chance of hitting at least once on two dice?'.
This is easier to calculate if you reverse the question: 'What is the chance that I not hit with two dice?'. This is %miss x %miss = 45% x 45% = 20.25%
So your chance to hit is 1 - 0.2025 = 0.7975
And average damage on a d6 is 3.5 so:
DPR for roll atk twice = 0.7975 x 3.5 = 2.79125 dmg per round
Now for the other case, average value for roll 2d6 and take highest is trickier to calculate. I basically used a spreadsheet to get the value 4.47.
DPR for roll dmg twice = 0.55 x 4.47 = 2.4585
Hopefully that gives you enough info to build out a spreadsheet to analyze things the way you want.
Re: Fuzzy on the statistics... « Reply #5 on Feb 14, 2012, 3:18pm »
I'm slow on the uptake, so just to clarify,
The extra attack roll increases the probability of hitting (and thusly scoring damage) to an extent that it is superior to a roll 2 take the best damage 'advantage'?
FWIW, while I have not tried to get as detailed into weapons and combat emulation as the OP, one of the things I have done to help my Holmes based game is to use a tweaked weapon vs Armor table.
I find it helps a great deal. I also give shields a +2 AC adjustment vs all missile attacks (obviously not from the flank/rear etc) This makes bow fire/hurled weapons much less useful against shield equipped enemies. I do not allow bucklers to be used in such a fashion, however.
Anyway, just an idea shield wise.
Another is that I allow for the shield-push style maneuver as well as a clubbing strike with a shield inspired by 300. The TV show Deadliest Warrior (which is occasionally downright laughable, granted) did measure the force of a shield-edge strike to the head and let me tell you, it is nothing to sneeze at. If I were in the unfortunate position of being cornered by a few orcs, I think I;d rather have the versatility of a shield over a offhand weapon TBH.
Re: Fuzzy on the statistics... « Reply #6 on Feb 14, 2012, 3:30pm »
... and I just kept going... It looks like the crossover point is a to-hit value of 8 which means two-weapon fighting is better at lower levels and against heavier armored foes.
Edit - the horizontal axis is your 'to hit' number, and vertical is damage per round.
Marv / Finarvyn DCC playtester (2011) C&C playtester (2003) I'm partly responsible for the S&W WhiteBox Builder of the TrollBridge Master of Mutants; MA since 1976 OD&D Player since 1975
"Don't ask me what you need to hit. Just roll the die and I will let you know!" - Dave Arneson
The extra attack roll increases the probability of hitting (and thusly scoring damage) to an extent that it is superior to a roll 2 take the best damage 'advantage'?
In general, rolling multiple dice where only one has to beat a target number greatly increases your odds. As you saw in my example, it jumped from 55% to 79% with just one die.
The other thing is that your chance to hit always goes up, and as it gets higher, the chance of hitting grows exponentially when you roll two dice. Whereas your rolling of two dice for damage doesn't increase in damage as you level up.
... and I just kept going... It looks like the crossover point is a to-hit value of 8 which means two-weapon fighting is better at lower levels and against heavier armored foes.
To keep some perspective on this, we're talking about less than 1/2 point of damage per round difference between the two. So really it won't make much difference in an OD&D combat.
'Feel' wise, I think it captures the 'light fighter' vs. 'heavy fighter' really well.
As it stands if I were faced with the three weapon options I would choose two-weapons fighter not because it does more damage, but because it partially negates a shield opponent's advantage. I would suggest you make the shield advantage apply regardless of opponent type.
Now it would be interesting to see how a shielded opponent would do...
Joined: Jan 2008 Gender: Male Posts: 2,330 Location: New Hope, MN Karma: 93
Re: Fuzzy on the statistics... « Reply #10 on Feb 14, 2012, 4:22pm »
Quote:
The other thing is that your chance to hit always goes up, and as it gets higher, the chance of hitting grows exponentially when you roll two dice. Whereas your rolling of two dice for damage doesn't increase in damage as you level up.
That is a variable I'd never considered before with this sort of combat adjustment. Fascinating!
The other thing is that your chance to hit always goes up, and as it gets higher, the chance of hitting grows exponentially when you roll two dice. Whereas your rolling of two dice for damage doesn't increase in damage as you level up.
To be clear, one of the reasons that I invoked Metamorphosis Alpha in my first post is because in my system your to-hit value is not based on level, but rather based on the make and condition of the weapons you are using. HD goes up with level, to-hit goes up when you find or forge a shiny new toy.
average value for roll 2d6 and take highest is trickier to calculate. I basically used a spreadsheet
For the damage from rolling 2 six-sided dice and taking the max, you can find an exact formula, rather than using a spreadsheet to enumerate all possible outcomes. This kind of probability distribution is called an "order statistic." Take a look at the following Wikipedia article, especially the section for discrete random variables:
You apply these formulas to this particular case as follows:
n = 2 dice iid X1,X2 ~ DU(6,1,1) each die is discrete uniform distribution with probability range {1,...,6} p1 = (x-1)/6 is the probability of rolling less than x on one die, x in {1,...,6} of course p2 = 1/6 is the probability of rolling x on 1d6
Order statistics are just the random variables in the sample sorted in non-decreasing order from minimum to maximum. Therefore, when n = 2, X(2) = max {X1, X2} is the second order statistic or maximum of the two random variables. Notice the parentheses around the numeral two in the subscript: this differentiates the second (unsorted) random variable X2 in the sample from the (sorted) second order statistic X(2).
The CDF P(X(2) <= x) and PMF P(X(2) = x) of the second order statistic can now be computed by plugging-and-chugging p1 and p2 into the formulas in the encyclopedia article:
Specifically, the probability mass function (PMF) is P(X(2) = x) = (-1 + 2 x)/36 for x in {1,...,6}, 0 otherwise
Therefore the mean (expected value of X(2)) is E(X(2)) = Sum[x P(X(2) = x), {x,1,6}] = Sum[x (-1 + 2 x)/36, {x,1,6}] = 161/36 (exact) = 4.472222 (2 repeats forever)
You don't even have to compute this sum yourself: Just plug it into WolframAlpha.
Notice the value of the mean 4.472 from the formula is the same value of the mean you got from your spreadsheet. But now you also have a nice expression for the probability of getting any value of x from 1 to 6, namely (-1 + 2 x)/36, without using a spreadsheet.
Re: Fuzzy on the statistics... « Reply #13 on Feb 15, 2012, 6:33am »
Thanks for that aher!
I was never a huge fan of stats (more a calculus guy). Since I've restarted playing D&D (quit when I was 12) I have been reteaching myself using first principles. I'll add your formula to my bag of tricks, but I'll need to derive it myself to truly grok it.
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