dGeo

[The Hive Custodian]

dGeo

Here's the latest version, taken from my planned starship test, hopefully soon to be running. The base is now 10^-0.1, or about 0.794.

This is the primary die that will be used. In the idealized form, it forms a geometric distribution with base 10^-0.1. Coded on a computer, it would look something like this:

int result = 0;
while (Math.random() < pow(10.0d, -0.1d)) result++;
return result;
It can be approximated on a table of 1d80, starting from 0 (which can be rolled as 1d8 x 10 + 1d10, treating the top number of each die as a 0):

d80 Result
00-15: 0
16-29: 1
30-39: 2
40-47: 3
48-54: 4
55-59: 5
60-63: 6
64-66: 7
67-69: 8
70-71: 9
72-79: 10+
"10+" means roll again and add 10 to the result. Keep rolling as long as you get a result of "10+", adding 10 each time.


A dGeo is almost always rolled against another dGeo; the higher roller is the winner. Ties are broken by flipping a coin (or equivalent).

[BigRedRod]

The name is indeed horrid, but I'm unable to think of anything suitable at the moment.

I am not sold on the only having one character roll in any contest, it just doesn't sit well that me for some reason I can't quite explain.

Also, is the Laplace distribution really the best choice? The ability to reach infinity is good, I like exploding dice as much as the next man (possibly a little more) but a 35% chance to roll a zero is as upsetting to me as possibly not getting to even a roll a dice in the first place.

This sounds all rather negative, I don't meant to be. What you have described seems to work, I just don't like it.

[LuneMoonshadow]

With bonuses being relatively close together, how does only having one person rolling (and quite realistically winning because they are the only roller) remain balanced? I'm no math major and won't pretend to be, so this little question is based purely on intuition.

[Noocytx]

It does seem to be an interesting concept, but I think I feel similar to BRR on this concept.

The idea of only having one person roll seems very unbalanced as well. Especially since a person has a fifty percent chance of winning if their modifier is only 1 higher than the other person.
Ex. A person is trying to bluff their way into the city. Their bluff modifier is +3, while the guard's sense motive check is +4. The guard would be guarenteed to win at least fifty percent of the time just because of the coin flip.

I like the idea of having modifiers mean more, and make the dice roll less of a factor. If I wanted to go that way though, I would focus more on the bell curve rolls (http://www.d20srd.org/srd/variant/adventuring/bellCurveRolls.htm) though. The idea of infinite rolls also has potential though.

[The Hive Custodian]

A couple minor points before I address what seems to be the main point:

Also, is the Laplace distribution really the best choice? The ability to reach infinity is good, I like exploding dice as much as the next man (possibly a little more) but a 35% chance to roll a zero is as upsetting to me as possibly not getting to even a roll a dice in the first place.

Perhaps I should explain why I designed the die like this. The die was designed to approximate a geometric distribution with base 10^-0.2. The reason for this is that this means that if the prices of bonuses are exponential with base 10^0.1 (decibels, as it were), then having a bonus much larger than your target is as inefficient as having a bonus much lower (by the same amount) than that of your target. (On a side note, the amount of money that a Purchase DC in d20 Modern represents is exponential with base 10^0.125. I don't know why they decided to use this base instead of decibels.) For every point your bonus is less than that of your target, you pay a factor 10^0.1 less, but your effectiveness drops by a factor of 10^0.2, so overall your efficiency (effectiveness / cost) drops by a factor of 10^0.1. If your bonus is much higher than your target, then your odds of success are pretty much constant at 1; however, each point of bonus costs you a factor of 10^0.1 more, so your efficiency drops by 10^0.1 here as well.

In a more qualitiative sense, massing kobolds with slingshots is an inefficient way of destroying a tank, and using a positron rifle is an inefficient way of cutting down a tree, and I wanted to design something that would represent this. In truth, adapting it as a replacement for d20 was something I did just out of curiosity of how well it would work.

Unfortunately, 10^0.2 means that a large proportion of the sides on the die end up being the lowest number. I could price things with base 10^0.05 and use a geometric distribution with base 10^-0.1 for the die, which would cut down on the proportion of sides with the lowest number... but unfortunately, as far as I am able to tell, this requires a die with a greater number of sides to do while maintaining a good approximation of the geometric distribution.

Hmm. That probably wasn't too helpful, but hopefully you can make at least a little sense of it.

I like the idea of having modifiers mean more, and make the dice roll less of a factor. If I wanted to go that way though, I would focus more on the bell curve rolls though.

A nitpick here: having the modifiers mean more compared to the die roll is a function of the variance of the die roll. The simplest (though probably not best) way to make the modifiers mean more is to decrease the die size; for example, using a d12 instead of a d20. The distribution still has the same shape; it's just tighter. The bell curve roll described changes the shape of the distribution as well as decreasing its variance. However, one could easily imagine a bell curve roll that instead increased the variance (e.g., 2d100).

In the dLaplace case, I wanted to have the modifiers mean roughly the same as they do in d20 (they will be somewhat different, since d20 is a uniform distribution as opposed to a Laplace distribution), which is why I recommended dividing modifiers by 3 during the conversion from d20 to dLaplace.

Also on the subject of bell curves: I did consider trying to approximate a normal distribution (including infinite "tails") using dice. However, I was unable to come up with a simple scheme for doing so.

With bonuses being relatively close together, how does only having one person rolling (and quite realistically winning because they are the only roller) remain balanced? I'm no math major and won't pretend to be, so this little question is based purely on intuition.
The idea of only having one person roll seems very unbalanced as well. Especially since a person has a fifty percent chance of winning if their modifier is only 1 higher than the other person.
Ex. A person is trying to bluff their way into the city. Their bluff modifier is +3, while the guard's sense motive check is +4. The guard would be guarenteed to win at least fifty percent of the time just because of the coin flip.

When it comes to probability, intuition can be both your best friend and your worst enemy. Unfortunately, in this case, it is closer to the latter. I'll take Noocytx's example, assuming that the modifiers have been already been divided by 3 as detailed in d20 Conversion in the original post and earlier in this post. In other words, the modifiers were something like +9 and +12 in d20, and +3 and +4 in dLaplace.

In d20, the bluffer has a 34% chance of succeeding, while the guard has a 66% chance of succeeding.

Now consider what happens in dLaplace. If the guard wins the coin flip, they win automatically, as you said. So that's a 50% chance of the guard winning already. Now, let's look at the other 50%, where the bluffer wins the coin flip. The bluffer then has to roll at least a 1 to succeed (since the roller wins ties). This happens 65% of the time, given that the bluffer is the roller. Since the bluffer is the roller only half the time, the bluffer succeeds 32.5% of the time overall, while the guard sees through the bluff 67.5% of the time. A little different than the d20 version; after all, the probabilities curves have different shapes.

Picture break!

Laplace distribution:
http://upload.wikimedia.org/wikipedia/commons/thumb/8/89/Laplace_distribution_pdf.png/325px-Laplace_distribution_pdf.png

Uniform distribution:
http://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Uniform_distribution_PDF.png/350px-Uniform_distribution_PDF.png

Normal distribution:
http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/325px-Normal_distribution_pdf.png

Still, the probabilites are still reasonably close to each other.

However, probability is not the only important thing here, which brings us to the main point:

I am not sold on the only having one character roll in any contest, it just doesn't sit well that me for some reason I can't quite explain.
It does seem to be an interesting concept, but I think I feel similar to BRR on this concept.

Ah, yes, the aesthetic angle. It seems that I did not give as much weight to it as I should have. However, I believe I have a solution that, while preserving the same distribution, will be more palatable to you. According to Wikipedia,

The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution...
As such, if we have each character roll the die introduced before (which I will call dGeo, since it approximates a geometric distribution) and add their bonuses, with the higher number winning and a coin flip breaking ties, I believe it can be shown that the exact same distribution is produced. However, now both characters are rolling, which makes the scheme seem more fair, which is perhaps as important as the scheme actually being fair. The tradeoff is that there tend to be a few more die rolls and additions involved. (Another side note: I had considered this way of doing things before, but I had thought that it would produce a different distribution, and I wanted to keep the number of die rolls down. However, after looking over it again, it seems that the distribution is, in fact, the same, and the drawback of extra die rolls is outweighed by the aesthetic angle of the whole thing.)

I've edited the original post to reflect this.

There's also another of what I consider quirks of d20, which is that character vs. character conflicts are often resolved with two dice (one for each character), but character vs. obstacle conflicts are resolved with only one (since the obstacle often has a static DC). Of course, there are many exceptions, such as Escape Artist vs. Use Rope, attack rolls vs. AC, and so forth. Personally, I think they should either use one die for everything or two for everything. But that's a topic for another thread.

[SinbadEV]

Sorry to be the peanut gallery but your die reminds me of this one created for a fantasy setting called Battal DΦ (http://elothtes.pbwiki.com/ELotH%3A%20TES%20-%20Pen%20and%20Paper%20RPG%20Dice#D%C3%8E%C2%A6)

[The Hive Custodian]

Not the most flattering comparison perhaps, but an apt one. They are both designed to approximate a geometric distribution. Even their bases are similar: my dGeo has a base of 10^-0.2 = 0.630957344..., while dPhi has a base of sqrt(5)/2 - 0.5 = 0.618033989...

[nightinverse]

Why couldn't you bring this up while we were sharing quarters?

This is rather different. I assume you stumbled upon something which interested you, and it ballooned into this? Hurm... I like dGeo - could use it where I'm getting bored of normal die systems, though I might just go free.

[The Hive Custodian]

I've come up with a new version of dGeo, this time with twice as much precision (among other things, this means that the chance to roll a zero is only about 20%); however, it now requires a d80 to simulate. I've edited it into the first post. I hope to be using it in an upcoming test.

[SinbadEV]

Hurray!