Modified Rule of Six, ...and "always having a shot" |
Modified Rule of Six, ...and "always having a shot" |
Apr 7 2005, 11:12 PM
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#9
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
Just to give a bit more substance, I went and modified my SR4 success calculator to show what would happen with the Rule of Six I outlined. Here is a sample of the old results, which doesn't use the new Rule of Six:
Here's the results for rolling with the new Rule of Six:
For any interested, here's the code. First the old method sans reroll:
Now, the new method with rerolls:
Note, r is a random number generator. In Java, the NextInt(x) function of Random gets a number from 0 (inclusive) to x (exclusive), hence why result equals NextInt(6)+1. Also, I limited the number of rerolls to 20. |
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Apr 7 2005, 11:15 PM
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#10
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
Out of interest, why do you sample rolls? It seems to me it would be better (both to avoid statistical anomalies and to avoid any weaknesses in your RNG) to just calculate the theoretical expected successes. There are visible anomalies creeping in there all over the place (look at expectation for one die sans NRo6, for instance; you've been rolling high).
~J |
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Apr 7 2005, 11:17 PM
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#11
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
The results that I feel are most important to compare are for 6 dice and 12 dice. 6 dice represents SR4's average ability, and the successes on a 6D6 roll represent what an average man on the street with intermediate training can pull off. 12 represents the most exceptional ability in SR4 from chargen and without enhancements from areas other than attributes and skills. The successes from a 12D6 roll represent what an exceptional individual with expert training can accomplish. I contend a Shadowrunner will roll dice closer to 12 than to 6 in most cases, but for establishing a global norm, 6 dice results are important.
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Apr 7 2005, 11:22 PM
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#12
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
I sample the rolls because just doing formulaic analysis, at least without the rerolls, is a matter of just dividing the dice rolled by 3. I wanted something closer to what actually rolling the dice would get. As for the reroll method, I lack the mathematical ability to calculate expected averages for it. What do you mean an anomoly on one die? Look at the results sans rerolling: one die averages 0.33368 successes, which is very close to 1/3, which is what we expect. I'm not sure I follow you. |
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Apr 7 2005, 11:23 PM
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#13
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
You just noted it yourself, it's off by over three millionths. You're not, paradoxically, going to get closer to actually rolling the dice by actually rolling the dice, at least not while your dierolling doesn't approach infinity.
~J |
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Apr 7 2005, 11:24 PM
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#14
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
Um... so?
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Apr 7 2005, 11:25 PM
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#15
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
So it's wrong, and for at least that portion it's easily corrected. I'll see if I can figure the formula for the with-reroll side; it shouldn't be difficult, once I get some caffeine in me.
Actually, I think what bugs me isn't that it's wrong, but that it was far more work to do what you did than to get the precise value. I'm really not sure what the point of it was. ~J |
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Apr 7 2005, 11:25 PM
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#16
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
Look, I'm doing this for fun, because I want to make guesstimates about a P&P RPG. I'm not building a bridge here. Lives aren't on the line. Is there some reason why we need accuracy to the millionth degree?
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Apr 7 2005, 11:26 PM
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#17
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
As my ninja-edit states, the part that gets to me is that it was extra work for reduced accuracy.
~J |
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Apr 7 2005, 11:27 PM
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#18
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
Well, let's see. I'm not a math/stats wizard. I am a (fairly, IMHO) competant programmer. I enjoy programming, further, more than doing statistical analysis, and it's easier for me. Were I able to do the analysis you suggest, I would have, probably. But I am not able to do it. Are you starting to get the picture? Maybe we could discuss something relevent now? Like the 6D6 and 12D6 results perhaps? |
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Apr 7 2005, 11:36 PM
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#19
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
Fair enough.
Incidentally, unless I messed up my math, you should expect exactly .4 successes per die with exploding sixes (which jibes with your findings). So what about 6 and 12 dice are we looking at? We've got 2.4 and 4.8 successes, so that suggests that any 3+ Threshhold test is fairly hard, any 5+ very difficult indeed. Doesn't seem like a lot of modifier wiggle-room for Threshold… ~J |
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Apr 7 2005, 11:38 PM
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#20
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
My intention here is to show that while exploding sixes allow the theoretical chance to pull anything off, they don't grant a huge number of average greater successes. 6D6: w/o reroll avg = about 2, w/ reroll avg = about 2.4 12D6: w/o reroll avg = about 4, w/ reroll avg = about 4.8 It's not a huge deal, and it allows for something that I at least think is important, as do several others. EDIT:
The way I see it, they are very much close enough for my purposes. |
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Apr 7 2005, 11:40 PM
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#21
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
I see that you consider them important, but I'm not seeing what about them we're discussing. They've got an expected number of successes. Are we debating whether they should have more? Less?
~J |
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Apr 7 2005, 11:43 PM
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#22
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
Er, see my latest edit. I was deciding how I wanted to phrase what I said.
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Apr 7 2005, 11:45 PM
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#23
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
I pity anyone who has to read this after all of the edits :)
I think it's a better solution than doing nothing, certainly, but I'm up in the air as to whether or not I like it as a solution in general. I'll have to run some numbers. ~J |
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Apr 7 2005, 11:49 PM
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#24
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
Seriously, what a mess...
Well, the idea behind the nRo6 is to allow a neat thing for "mechanical flavor" without breaking the game. An average 0.06666... extra successes doesn't seem like that big a deal; it more or less preserves the statistics that we'd expect for the most common rolls and preserves that "lucky shot" feeling everyone loves from the Rule of Six in SR3. |
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Apr 7 2005, 11:53 PM
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#25
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Manus Celer Dei Group: Dumpshocked Posts: 17,006 Joined: 30-December 02 From: Boston Member No.: 3,802 |
It's true, but providing that extra successes still mean a better overall quality of success, it provides a chance for a better lucky shot, as far as I can tell. I'm going to have to spend some time with Mathematica before I rightly figure out whether or not I care, though.
~J |
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Apr 8 2005, 12:10 AM
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#26
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
True. True. I just think that we can fairly modify the word "better" here with either "negligably" or "insignifigantly in most cases" when it comes to the nRo6. |
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Apr 8 2005, 01:18 AM
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#27
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Moving Target Group: Members Posts: 778 Joined: 6-April 05 Member No.: 7,298 |
It is better to do this analytically, but it's not quite trivial.
The probability of getting at least N successes on one roll is > q(N) (1/3)*(1/6)^(N-1) because you have to roll a 6 (N-1) times and then you can roll a 5 or 6 to finish off. Therefore, the chance of getting exactly N successes is > p(N) = q(N)-q(N+1) = (1/3)*(1/6)^(N-1) - (1/3)*(1/6)^N = (1/3)*(5/6)*(1/6)^(N-1). Therefore the expected number of successes E you will get is sum( N*p(N) , N=1..infinity), which we can rewrite as > E = (1/3)*(5/6)*( 1 + 2*(1/6) + 3*(1/6)^2 + ... + N*(1/6)^(N-1) + ... ). This is a standard infinite series of the for n*x^(n-1). If we note that dx^n/dx = n*x^(n-1), we can rewrite as (1/3)*(5/6)*( sum( d(x^n)/dx ) )|x=(1/6). Pull out the derivative to get > E = (1/3)*(5/6)* (d/dx)[ sum(x^n,n=1..inf) ] | x=(1/6) we then note that sum(x^n,n=1..inf) = 1/(1-x) - 1, and that (d/dx)[ 1/(1-x) ] is 1/(1-x)^2, and we have > E = (1/3)*(5/6)/(5/6)^2 = (1/3)/(5/6) = 2/5 = 40%. Since each die is independent, the expected number of successes for n dice is > E(n) = 2*n/5 In contrast, for the non-exploding case, the expected number F(n) is > F(n) = n / 3 You can also compute the probability distributions of each method to compute the probability of getting a certain number of successes with or without exploding dice. The scope of that computation is beyond what I want to explain here right now, but I'm happy to produce tables upon request. Just as an example, if you're rolling four dice, the probabilities are (without and with exploding dice) > 0 successes: 19.75% 19.75% > 1 success: 39.51% 32.92% > 2 successes: 29.63% 26.06% > 3 successes: 9.86% 13.49% > 4 successes: 1.23% 5.32% > 5 successes: 0% 1.76% > 6 successes: 0% 0.51% > 7 successes: 0% 0.14% > 8+ successes: 0% 0.05% |
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Apr 8 2005, 01:32 AM
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#28
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Moving Target Group: Members Posts: 669 Joined: 25-May 03 Member No.: 4,634 |
That would be pretty rocking, actually. |
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Apr 8 2005, 02:32 AM
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#29
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Moving Target Group: Members Posts: 778 Joined: 6-April 05 Member No.: 7,298 |
Aha, there are code tags. That will make this much prettier. How big do you want the tables? I'm doing 12x8 for now, just to keep things in a reasonable amount of space, and truncating to three decimal places. All values are in percentages, of course. Chances of getting at least N successes, no exploding dice:
Chance of getting at least N successes, exploding dice:
Chance of getting exactly N successes, no exploding dice:
Chance of getting exactly N successes, exploding dice:
From the previous analysis, we know that with exploding dice we expect 0.4 successes per die, and without we expect 0.3333.... This means that the "12" line for non-exploding dice should average 4 successes, and the "10" line for exploding dice should also average 4 successes. Keep this in mind when comparing rolls--it's fairer to compare sets of dice with the same average. |
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Apr 8 2005, 04:14 AM
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#30
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Target Group: Members Posts: 81 Joined: 12-April 02 From: the shadows.... Member No.: 2,548 |
AH!! For the love of Nuyen!! Let's get back to the topic of the post....
Actually this idea is not new at all. Remember it being stated that the new SR mechanic is based on the nWoD dice rolling mechanic except that SR will use d6s? Well in nWoD they have the 10 again rule, which states 10s count for their success and then the player gets to roll them again. Each 10 that comes up counts again and gets a reroll again.... For SR, "6 again" might not be a bad idea. But I don't know (as few do) if the devs will want to make the SR mechanic so similar to the nWoD mechanic (not that that's a bad thing). Rather I see a player being able to get dice from a convention SR3 already has called a "Karma Pool". Often times a player only had 1 or 2 dice and really needed to get more than 1 or 2 successes. By spending Karma they could increase their dice pool.... ...At least I hope Karma Pool is still in the game and works the same way... |
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Apr 8 2005, 04:42 AM
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#31
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Moving Target Group: Members Posts: 778 Joined: 6-April 05 Member No.: 7,298 |
The above stuff shows how much of a shot someone has under different conditions. How is this not exactly the topic of the post?
Anyway, no, it's not really new, but then again, it's not new in nWoD because SR was doing it before (albeit structured slightly differently). I'd bet there was some game doing it before SR was doing it. Heck, there are even entries like "20 - roll again twice" in 1st ed. AD&D treasure tables. Maybe we can find some classic dice games where you get rerolls to rack up success.... Regardless of how old the idea is and who thought of it first, it still makes sense here, I think. |
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Apr 8 2005, 05:54 AM
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#32
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Freelance Elf Group: Dumpshocked Posts: 7,324 Joined: 30-September 04 From: Texas Member No.: 6,714 |
Dammit. Who got her started? Who was it?! You've doomed us all! The math-speak, it consumes my soul !!
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Apr 8 2005, 06:03 AM
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#33
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Immortal Elf Group: Members Posts: 11,410 Joined: 1-October 03 From: Pittsburgh Member No.: 5,670 |
she blinded me with advance probability maths.
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