If you have a choice of two weapons.  One does 2D6 damage.  The other does 1D6 + Some Number.    What does that Number have to be for the two weapons to be equally interesting choices?

Is it 1D6+3 or 1D6+4?   Or it impossible to balance without some additional non-damage aspect to the weapon?  

I ask because the RPG I am working on will be using D20 for combat and only D6s for damage.  I wanted to see if I could replace +1D6 with some number and still maintain parity.

They can never be equal.  1d6 + "number" and 2d6 have different standard deviations, regardless of what the "number" is.

Quote from: Spinachcat;426011If you have a choice of two weapons.  One does 2D6 damage.  The other does 1D6 + Some Number.    What does that Number have to be for the two weapons to be equally interesting choices?

Is it 1D6+3 or 1D6+4?   Or it impossible to balance without some additional non-damage aspect to the weapon?  

I ask because the RPG I am working on will be using D20 for combat and only D6s for damage.  I wanted to see if I could replace +1D6 with some number and still maintain parity.

1d6+4 is my guess. The extra half a point of damage on average makes up for the lower maximum.

Quote from: Spinachcat;426011If you have a choice of two weapons.  One does 2D6 damage.  The other does 1D6 + Some Number.    What does that Number have to be for the two weapons to be equally interesting choices?

Is it 1D6+3 or 1D6+4?   Or it impossible to balance without some additional non-damage aspect to the weapon?  

I ask because the RPG I am working on will be using D20 for combat and only D6s for damage.  I wanted to see if I could replace +1D6 with some number and still maintain parity.

2d6 yields a wide bell curve from 2 to 12, with the midpoint at 7.

1d6+3 yields a flat curve with results from 4 to 9, all equally probable.

The first gives a centered result with a moderately wide fluctuation - more accuracy, less precision.

The second gives a uniformly varying result within narrower parameters - more precision, less accuracy.

Both are interesting in different ways.

-clash

If I can choose between 1d6+4 and 2d6 I'm going to take the former every single time.

Quote from: Benoist;426025If I can choose between 1d6+4 and 2d6 I'm going to take the former every single time.

Really? For me it would depend greatly on the character choosing, as I see different appeals in both.

-clash

2d6 gives a bell-curve Gaussian probability distribution with an average result of 7. Since it's a bell-curve, this also means 7 is the most likely result.

1d6+x, however, is a linear distribution. Which means that any result between 1+x and 6+x is just as likely. But if you assume fair, unbiased dice, regression towards the mean assures that there will be a trend towards the average damage of 3.5+x.

If you want to match average for average, it's dirt simple:

3.5+x=7
x=7-3.5
x=3.5

So, for both weapons to have equivalent "average" damage, 1d6+3 or 1d6+4 is what you're looking for.

But of course, IMHO, I feel that there should be more than flat numerical damage that makes weapon choice interesting...

Quote from: The Butcher;426029But of course, IMHO, I feel that there should be more than flat numerical damage that makes weapon choice interesting...

I think we all have added a mental "All other things being equal" there - except maybe Benoist, who seems very definite...

-clash

Quote from: Benoist;426025If I can choose between 1d6+4 and 2d6 I'm going to take the former every single time.

You know what, Benoist's on to something here. Since there's an underlying probability distribution, we're supposed to weigh the means with the relative probabilities for any given result. Am I making any sense? At all?

1d6+4 means equal probabilities of a result ranging between 5-10, resulting on a weighted mean of (5+6+7+8+9+10)/6=45/6=8.5

The weighted mean of 2d6 would be
[2+(2x3)+(3x4)+(4x5)+(5x6)+(6x7)+(5x8)+(4x9)+(3x10)+(2x11)+12]/36=
[2+6+12+20+30+42+40+36+30+22+12]=
252/36=7
which is the same as the average, of course, being a Gaussian distribution.

So 1d6+3, which would give a weighted mean of
4+5+6+7+8+9/6=39/6=6.5
might be a closer to what you want.

[my bad. I am not a mathematician, or engineer, or anything partciularly math-y]

Quote from: the butcher;4260341d6+4 means equal probabilities of a result ranging between 5-10, resulting on a weighted mean of (5+6+7+8+9+10)/6=45/6=8.5

45/6 = 7.5

Quote from: flyingmice;426027Really? For me it would depend greatly on the character choosing, as I see different appeals in both.

-clash

Ah if you put it that way, I completely agree! I was completely ignoring the concept of role playing here, to concentrate on the pure mathematical appeal between the two.

Going on with that ex-RP argument, It's nearly always better for me as a player to take a fixed value over a die, because then I can strategize better and randomness plays a lesser role in how I plan actions. This is why just comparing the average numbers of the two possibilities, 1d6+3 vs. 2d6, isn't actually an *equal* choice in my mind. I'll take 1d6+3 as well. Choosing between 1d6+2 and 2d6 makes the choice equal to me psychologically, though it is obviously not equal mathematically.

Quote from: ggroy;42603645/6 = 7.5

Told you I suck at math.

Quote from: Benoist;426042Ah if you put it that way, I completely agree! I was completely ignoring the concept of role playing here, to concentrate on the pure mathematical appeal between the two.

Ah - that's the game designer part of me. When I am designing things, statistical probability is extremely important to me. When I am a GM or player, I need to know this as well, but the choice would be based on the character, not what I like. Some characters would enjoy the uncertainty of 2d6, while others would prefer the certainty of 1d6+3, and yet others wouldn't know or care - close enough for hand-grenades, bocce, and head-bashing. That's why God didn't make everyone as handsome as Pundit - choice is important.

-clash

I'm with Benoist. 2d6 is not as good as 1d6+4. If we roll damage a million times, the guy with the 1d6+4 is going to win. For all the times I've knocked a bad guy down to 1 hit point, I think I'll take the higher average.

Quote from: flyingmice;426078Ah - that's the game designer part of me. When I am designing things, statistical probability is extremely important to me.

As it should be! I mean: I can hardly imagine how anyone could come up with some rules sub/system (whether as a house rule or printed material) and not care at all for statistical outcomes. That'd be the equivalent of walking in the dark, with the highest probability to crash into a wall at some point. Then you can't really be surprised you did.

Quote from: flyingmice;426078When I am a GM or player, I need to know this as well, but the choice would be based on the character, not what I like. Some characters would enjoy the uncertainty of 2d6, while others would prefer the certainty of 1d6+3, and yet others wouldn't know or care - close enough for hand-grenades, bocce, and head-bashing. That's why God didn't make everyone as handsome as Pundit - choice is important.

-clash

I agree on principle as well. I mean, personally, as a player, I would make choices based on both what I as a player want and what my character wants. Sometimes the two will be perfectly in tune, and sometimes, they are going to clash, with the character's wishes winning most of the time (I'm thinking suboptimal tactical choices like here, for instance), depending on the situation. Though there are times (I'm thinking deeply disruptive stuff, here) where I as a player need to step in and make a choice to play my character some other way, as to not wreck the game. I don't see how that could occur with the choice between the types of dice proposed, but there you go.