Alternate Accuracy Mechanics

[ggroy]

Update: Ignore this post. The math is done all wrong. Go down to post #15 instead.

The system I have now has a damage roll, where, depending on your power, you roll d4 to d12. Part of the issue with this one is that any system that relies on a "random chance" of happening, that can happen on the d4, is LESS likely to happen on the d12. If I say "Okay, if you roll maximum, it's a crit!" the d4 will crit 25% of the time, but the d12 only about 8% of the time. And I still want to involve the d4 in it. Also, any number that's going to come up on a d4 is going to be a LOT more common than any current crit is.

Here's one possible crit system.

If one rolls a 4 on a d4, then the d4 is rolled again and is added to the damage. The average value of the damage is:

(1 + 2 + 3 + 4)/4 + (5 + 6 + 7 + 8)/16
= 2.5 + 1 + 2.5/4
= 4.125

If one rolls a 10 on a d10, then the d10 is rolled again and is added to the damage. The average value of the damage is:

(1 + 2 + ... + 9 + 10)/10 + (11 + 12 + ... + 19 + 20)/100
= 5.5 + 1 + 5.5/10
= 7.05


In general, if one rolls N on a dN die, then the dN die is rolled again and is added to the damage. The average value of the damage is:

(1+1/N)(N+1)/2 + 1


EDIT: The dN die is only rolled again once.

EDIT2: An implicit assumption in this calculation, is that one can refuse to roll another dN die after rolling an N. (The calculation is much more complicated without this assumption).

 

[ggroy]

Update: Ignore this post. The math is done all wrong. Go down to post #15 instead.


(Removing the "roll dN die again once" assumption).

If one rolls an N on the dN die, one rolls the dN die again. (As many times if one continues rolling an N on the dN die). This is sort of like an "exploding dice" type of crit.


(Exercise for the reader).

In general, the average value of a dN "exploding dice" damage is:

(1+1/N + 1/N^2 + 1/N^3 + ...)(N+1)/2 + (1 + 2/N + 3/N^2 + ...)

which equal exactly to:

(N+1)/(2*[1-1/N]) + 1/[1-1/N]^2


For various dN die, we have the average "exploding dice" damage:

d4 -> 5.11
d6 -> 5.64
d8 -> 6.45
d10 -> 7.35
d12 -> 8.28
d20 -> 12.16

in contrast to the average damage of a single dN die (N+1)/2:

d4 -> 2.5
d6 -> 3.5
d8 -> 4.5
d10 -> 5.5
d12 -> 6.5
d20 -> 10.5

 

[Altamont Ravenard]

Then you don't want a relative scale, which changes depending on what die you roll. Assume that d6 is the smallest die. Instead of "max damage" or "minimum damage", just say that, for example, a miss occurs on a 1-3, a hit on a 4-9, a critical on a 10-12.

Allow circumstantial or power-based modifiers to the damage roll to increase the likelihood (or allow the possibility, in the case of d6 or d8 attacks) of a critical. Sneak attacks could work this way, for example.

If you were inclined, you could get really fiddly and have a different outcome for each result 1-12 (or higher, if you consider modifiers).

What about using the exploding die mechanic, but keeping the same crit value regardless of what die you are using? Say you crit when you roll a value of 13 or higher (lucky 13!)? So with a d4, you need to roll 4-4-4 (1 in 64) whereas with a d12, you only need to roll 12 (1 in 12).

AR

 

[Altamont Ravenard]

double trouble.

 

[ggroy]

OK. All the calculations in my previous two posts are wrong !!!

It turns out for a dN "exploding dice" scenario, in order for the probabilities to make sense (ie. the total probability of all dice roll possibilities have to add up to 1), it requires a crucial assumption.

When one rolls an "N" on a dN die, one immediately has to roll the dN die again and add the result to the previous result. This is a crucial assumption for the math to work.


Let's look at the case of a d4 "exploding dice".

If one rolls a 1, 2 or 3 on the d4, that's the damage.

If one rolls a "4" on the d4, then one rolls the d4 again and adds it to the previous result of 4, resulting in damage possibilities of 5, 6, 7, or 8. If one rolls another "4" on the second d4 roll, then one rolls the d4 again and adds it to the previous result of 8, resulting in damage possibilities of 9, 10, 11, or 12. (Ad infinitum).

So requiring the d4 to be rolled again after rolling a "4", this means the damage possibilities are:

- 1, 2, 3 -> each case individually with a probability 1/4
- 5, 6, 7 -> each case individually with a probability (1/4)^2
- 9, 10 , 11 -> each case individually with a probability (1/4)^3
- 13, 14, 15 -> each case individually with a probability (1/4)^4
- etc ...

In this d4 "exploding dice" system with the above mentioned crucial assumption, we can never roll a damage score of 4, 8, 12, 16, etc ...

If we add up all the probabilities, we have:

3*(1/4) + 3*(1/4)^2 + 3*(1/4)^3 + 3*(1/4)^4 + ...

Using the formula a/(1-x) = a + a x + a x^2 + a x^3 + a x^4 + ... for |x|< 1, this infinite sum adds up exactly equal to (3/4)/[1-1/4] = 1. (This is just a simple infinite geometric series).


This can be easily generalized to a dN "exploding dice" system, where one rolls the dN die again after rolling an "N". So we will never see a damage score of N, 2N, 3N, 4N, etc .... The damage possibilities are:

- 1, ..., (N-1) -> each damage case individually with a probability 1/N
- N+1, ..., N + (N-1) -> each damage case individually with a probability 1/N^2
- 2N+1, ..., 2N + (N-1) -> each damage case individually with a probability 1/N^3
- 3N+1, ..., 3N + (N-1) -> each damage case individually with a probability 1/N^4
- etc ...

(An exercise for the reader: show that the dN "exploding dice" probabilities all sum up to 1).

To get the average damage of this dN exploding dice system, we just take the weighted sum of the probabilities with each damage case value:

[1 + ... + (N-1)]/N + { [N+1] + ... + [N+(N-1)] }/N^2
+ { [2N+1] + ... + [2N+(N-1)] }/N^3
+ { [3N+1] + ... + [3N+(N-1)] }/N^4
+ ...

Though not entirely mathematically rigorous, we will rewrite this sum (handwaved in a cavalier manner) by grouping the terms in a more suggestive form:

[1 + ... + (N-1)]*[1/N + 1/N^2 + 1/N^3 + 1/N^4 + ...]
+ (N-1)*[1/N + 2/N^2 + 3/N^3 + ...]

As an exercise for the reader, this infinite sum becomes quite simple:

N(N+1)/[2(N-1)]

for the average damage of this dN "exploding dice" system.


For various dN die, we have the average "exploding dice" damage N(N+1)/[2(N-1)]:

d4 -> 3.33
d6 -> 4.2
d8 -> 5.14
d10 -> 6.11
d12 -> 7.09
d20 -> 11.05

in contrast to the average damage of a single dN die (N+1)/2:

d4 -> 2.5
d6 -> 3.5
d8 -> 4.5
d10 -> 5.5
d12 -> 6.5
d20 -> 10.5

 

[Octangula]

The system I have now has a damage roll, where, depending on your power, you roll d4 to d12. Part of the issue with this one is that any system that relies on a "random chance" of happening, that can happen on the d4, is LESS likely to happen on the d12. If I say "Okay, if you roll maximum, it's a crit!" the d4 will crit 25% of the time, but the d12 only about 8% of the time. And I still want to involve the d4 in it. Also, any number that's going to come up on a d4 is going to be a LOT more common than any current crit is.

Still mulling this over...hmm....

Maybe you need to turn it on its head. What if the dice roll is not the amount of damage you are doing, but the amount of damage you're not doing?

I don't know how well this assumption would work for your system, but let's assume that the ideal amount of damage a hit does is 12 (or in the region of that number; this would require testing), and that the roll is how far away from the ideal you fall. A smaller dice is much more consistent than a larger dice. It still lets you have crits on a 1, if you feel you need them. It also lets you have complete misses, if you are too inconsistent (and roll too high), but once you have a smaller dice, you can never do so badly that you miss.

The only drawback to this is that it's very counter-intuitive. But then, so was THAC0...

 

[mmadsen]

How might you keep these kinds of elements, in a system where (almost) every hit is automatic?

Why do you want hits to be (almost) automatic?

 

[I'm A Banana]

Why do you want hits to be (almost) automatic?

Mostly, because of pyschology. Having a game that is action-reaction runs a lot smoother when those actions are fairly well-assured, and they give the player a sense of empowerment when they're sure their attempt is going to actually happen (though it might not happen as well as they hope).

It also helps with a cinematic feel in that there is no "wasted time." Every time a character gets a turn, something is going to change.

 

[mmadsen]

Mostly, because of pyschology.

But people prefer gambling to grinding. That's why I'm confused.

Having a game that is action-reaction runs a lot smoother when those actions are fairly well-assured, and they give the player a sense of empowerment when they're sure their attempt is going to actually happen (though it might not happen as well as they hope).

I can see why the players don't want to fail -- or to succeed and then fail, due to some defense roll -- but I'm not sure that's at all symmetrical. I think they're perfectly happy to have their opponents fail or to use some "you missed me!" power.

It also helps with a cinematic feel in that there is no "wasted time." Every time a character gets a turn, something is going to change.

I agree that making every turn count is a good thing, but I wouldn't make every swing a hit to achieve that. It's certainly not realistic, and it doesn't feel right in action fiction, either.

 

[Celebrim]

Mostly, because of pyschology. Having a game that is action-reaction runs a lot smoother when those actions are fairly well-assured, and they give the player a sense of empowerment when they're sure their attempt is going to actually happen (though it might not happen as well as they hope).

It also helps with a cinematic feel in that there is no "wasted time." Every time a character gets a turn, something is going to change.

Personally, I don't find this a very strong answer and it wasn't the one I was expecting you to give.

In fact, it resurrects the old specter of the logic that runs, "Since failure is not fun, we'll just remove failure from the game.", in a way that I wish I had as a firm example of the slippery slope back when that was one of the dominent arguments on the forums.

I think that it is reasonable to want to speed up play by throwing one dice instead of two, in which case I would encourage you to think of the 'to hit' dice as also being the damage dice.

But I don't find it very reasonable to suggest that the possibility of failure isn't fun, or that the assurance of success is fun. I think not only is that wrong, but it is a fool's errand and the entire mindset behind it is bad for the game and indicative to appealing to 'ego gaming'. Unless this is a game for 5 year olds who can't emotionally handle delayed gratification, I'm not seeing the point.