Author Topic: Dice Pool Crunching  (Read 4835 times)

snakefing

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Dice Pool Crunching
« on: September 11, 2006, 09:25:48 AM »
In this thread, I'll be presenting some statistical data on using d6 dice pools for various purposes. The main reason to do this is to have a good resource to understand what happens to the probabilities as you tweak the parameters.

I'll use the following notations:

DicePool(n,d6) refers to a roll of n dice, each a d6. If for any reason I need to specify the results, it would be in list form, such as:

DicePool(3,d6) gives (2,6,3), meaning three dice were rolled and the results were a 2, a 6, and a 3.

At present, the dice in our pools will always be d6. This might change, but only if for some reason we decide the d6 just aren't working out. For now, that's just a fixed element of the mechanics. Increasing the size of the dice won't change much, but the probability increments would be smaller.

The dice pool mechanic is something like this:

Successes =  #(DicePool(n,d6) + modifiers >= target)

That reads as: Successes is the number of results in the dice pool that meet the target number. If there is a modifier, it is added to each result from the dice pool before comparing. For my first posts, I'll ignore the modifier. Effectively the modifier just reduces the target number that you need to roll.

Depending on the task, the number of successes can be interpreted in different ways. For example, one proposal for combat requires a certain minimum number of successes to hit. Fewer than the required successes means you don't hit.

In this case, we'll be interested in questions like these:

1. How likely is a given combatant (fixed size of dice pool) to hit, given the defender's target number?

2. How does this change if I raise or lower the targt number? Or equivalently, how does this change if the attacker gets a stat increase or spell or something that increases or decreases his modifier?

3. How does this change if I change the number of successes needed to hit?

4. How does this change if the attacker increases his dice pool? Or if some effect decreases the dice pool?

There will be other scenarios, such as damage rolls or skill checks, where the number of successes might be interpreted differently. I'll cover those later, because the questions we'd be asking might be different. In my next post, I'll start laying out some results.
« Last Edit: September 11, 2006, 09:26:14 AM by beejazz »
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snakefing

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« Reply #1 on: September 11, 2006, 10:25:10 AM »
In our dice pool mechanic, there are effectively three inputs:

1. Number of dice in the pool.
2. Target number.
3. Number of successes needed.

Modifiers to the dice pool are another input, but from the probability standpoint they only lower the effective target number. So in all my analysis I'll assume that the modifier is 0.

To plot this fully would be a three dimensional grid - but since holographic displays are not yet a reality, I'll be breaking this down into bit size chewy bits.

The first things to note are the obvious ones. There is no possibility of success if the target number is 7 or higher (after taking into account any modifiers). Also, there is no possibility of success if the number of required successes is greater than the size of your dice pool. These things seem obvious, but they will be serious constraints on the design of the system, because impossible tasks just aren't much fun.

(Shadowrun uses d6 dice pools, but they overcome this particular limit by using open-ended rolls - for each 6 you roll, you can roll another die and add it to the total. This way, there's no theoretical upper limit.)

So, without further ado, here's my first probability table. This one is for rolling DicePool(3,d6) against various target number, requiring only a single success. That's an easy number of successes and a moderate sized dice pool.
[table=Successes >= 1]
[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]
[tr][th]DicePool(3,d6)[/th][td]42%[/td][td]70%[/td][td]88%[/td][td]96%[/td][td]99.5%[/td][/tr]
[/table]

Obviously, with only a single success required, this isn't too hard, even with the hardest difficulty. Reducing the difficulty (or equivalently, increasing modifiers) rapidly increases the probability even more.

Now lets try that with more difficulty. We'll leave the same dice pool, but require 2 successes.
[table=Successes >= 2]
[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]
[tr][th]DicePool(3,d6)[/th][td]7.5%[/td][td]26%[/td][td]50%[/td][td]74%[/td][td]93%[/td][/tr]
[/table]

A big difference! At the high difficulty, it has gone from easy (2 out of every 5) to pretty darn hard (1 out of 12 or so). Even so, if the difficulty drops to 4 (or the attacker has a +2 modifier over the defender), the probability is still fairly high.

Lastly, try it with three required successes. That means you'd have to succeed on every die:
[table=Successes >= 3]
[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]
[tr][th]DicePool(3,d6)[/th][td]0.5%[/td][td]3.7%[/td][td]12%[/td][td]30%[/td][td]58%[/td][/tr]
[/table]

Here, even if the task is ridiculously easy (target of 2), there's still a good chance to fail. At the high end, this is down to just 1 in 200.

The take home lessons are probably these:

1. Changing the target number has a huge effect on the probabilities.
2. When dice pools are relatively small (3 in this case), changing the number of successes required also has a rather drastic effect.

I rather suspect that things would be rather different if dice pools were larger and the number of successes required were in the area of 1/4 to 1/2 of the number of dice in the pool. So later I will come back and try this again using a larger
dice pool, like 10d6, and successes ranging from 3 to 5 or so.
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beejazz

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Dice Pool Crunching
« Reply #2 on: September 11, 2006, 12:33:06 PM »
Well, as a baseline what about two out of three successes where playres don't really need to hit higher than four?

I mean, all this *is* pre-modifier.

This puts the target number for the average task at seven (on the assumption that most people will have at least +3 mods, where the extraordinary will have seven).
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« Reply #3 on: September 11, 2006, 06:38:23 PM »
Quote from: snakefing

In our dice pool mechanic, there are effectively three inputs:

1. Number of dice in the pool.
2. Target number.
3. Number of successes needed.

Modifiers to the dice pool are another input, but from the probability standpoint they only lower the effective target number. So in all my analysis I'll assume that the modifier is 0.

To plot this fully would be a three dimensional grid - but since holographic displays are not yet a reality, I'll be breaking this down into bit size chewy bits.

The first things to note are the obvious ones. There is no possibility of success if the target number is 7 or higher (after taking into account any modifiers). Also, there is no possibility of success if the number of required successes is greater than the size of your dice pool. These things seem obvious, but they will be serious constraints on the design of the system, because impossible tasks just aren't much fun.

(Shadowrun uses d6 dice pools, but they overcome this particular limit by using open-ended rolls - for each 6 you roll, you can roll another die and add it to the total. This way, there's no theoretical upper limit.)

So, without further ado, here's my first probability table. This one is for rolling DicePool(3,d6) against various target number, requiring only a single success. That's an easy number of successes and a moderate sized dice pool.
[table=Successes >= 1]
[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]
[tr][th]DicePool(3,d6)[/th][td]42%[/td][td]70%[/td][td]88%[/td][td]96%[/td][td]99.5%[/td][/tr]
[/table]

Obviously, with only a single success required, this isn't too hard, even with the hardest difficulty. Reducing the difficulty (or equivalently, increasing modifiers) rapidly increases the probability even more.

Now lets try that with more difficulty. We'll leave the same dice pool, but require 2 successes.
[table=Successes >= 2]
[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]
[tr][th]DicePool(3,d6)[/th][td]7.5%[/td][td]26%[/td][td]50%[/td][td]74%[/td][td]93%[/td][/tr]
[/table]

A big difference! At the high difficulty, it has gone from easy (2 out of every 5) to pretty darn hard (1 out of 12 or so). Even so, if the difficulty drops to 4 (or the attacker has a +2 modifier over the defender), the probability is still fairly high.

Lastly, try it with three required successes. That means you'd have to succeed on every die:
[table=Successes >= 3]
[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]
[tr][th]DicePool(3,d6)[/th][td]0.5%[/td][td]3.7%[/td][td]12%[/td][td]30%[/td][td]58%[/td][/tr]
[/table]

Here, even if the task is ridiculously easy (target of 2), there's still a good chance to fail. At the high end, this is down to just 1 in 200.

The take home lessons are probably these:

1. Changing the target number has a huge effect on the probabilities.
2. When dice pools are relatively small (3 in this case), changing the number of successes required also has a rather drastic effect.

I rather suspect that things would be rather different if dice pools were larger and the number of successes required were in the area of 1/4 to 1/2 of the number of dice in the pool. So later I will come back and try this again using a larger
dice pool, like 10d6, and successes ranging from 3 to 5 or so.




Low Dice, Low Number: Easy Task. Who can do it? ANYONE.

Low Dice, High Number: Intuitive Difficulties. Like art, you either have it or you don't.

High Dice, Low Number: Complex Difficulties. You either know it or you don't.

High Dice, High Number: Extreme Difficulty. You gotta know what you're doing, and even that might fail!
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« Reply #4 on: September 11, 2006, 07:53:20 PM »
So, as a baseline, let's do the 3d6 with starting bonuses ranging from 2 to 7, with three as the norm.

A 1 out of three DC 7 check.

Inept people (2) succeed 70% of the time.
Average people (3) succeed 88% of the time.
(4) succeeds 96% of the time.
(5) succeeds 99.5% of the time.
(6 and 7) succeed automatically.

A 1 out of three DC 8 check.

Inept people (2) succeed 42% of the time.
Average people (3) succeed 70% of the time.
(4) succeeds 88% of the time.
(5) succeeds 96% of the time.
(6) succeeds 99.5% of the time.
(7) succeeds automatically.

A 1 out of three DC 9 check.

Inept people fail.
Average people (3) succeed 42% of the time.
(4) succeeds 70% of the time.
(5) succeeds 88% of the time.
(6) succeeds 96% of the time.
(7) succeeds 99.5% of the time.

A one out of three DC 10 check.

Inept people fail.
Average people fail.
(4) succeeds 42% of the time.
(5) succeeds 70% of the ttime.
(6) succeeds 88% of the time.
(7) succeeds 96% of the time.

A one out of three DC 11 check.

(2, 3, and 4) fail.
(5) succeeds 42% of the time.
(6) succeeds 70% of the time.
(7) succeeds 88% of the time.

In one out of three checks, a range of 7 through 11 looks about right for DCs. A 7 results in something average people can do most of the time, that inept people can do a little less than half the time. An 11 is something that only the better half of society can even do, but once you can do it it's pretty easy.

A 2 out of three DC 7 check.

This means that inept people (2) succeed 26% of the time.
This means that most people (3) succeed 50% of the time.
(4) succeeds 74% of the time.
(5) succeeds 93& of the time.
(6 and 7) succeed 100% of the time.

A 2 out of three DC 8 check.

This means inept people (2) succeed 7.5% of the time.
This means that most people (3) succeed 26% of the time.
(4) succeeds 50% of the time.
(5) succeeds 74% of the time.
(6) succeeds 93% of the time.
(7) succeeds 100% of the time.

A 2 out of three DC 9 check.

Inept people fail.
Most people succeed 7.5% of the time.
(4) succeeds 26% of the time.
(5) succeeds 50% of the time.
(6) succeeds 74% of the time.
(7) succeeds 93% of the time.

A 2 out of three DC 10 check.

Inept people fail.
Most people fail.
(4) succeeds 7.5% of the time.
(5) succeeds 26% of the time.
(6) succeeds 50% of the time.
(7) succeeds 76% of the time.

A 2 out of three DC 11 check.

(1-4) does not succeed.
(5) succeeds 7.5% of the time.
(6) succeeds 26% of the time.
(7) succeeds 50% of the time.

Two out of three checks should probably remain within the range of seven and eleven. Checks of seven leave a 50% chance of success for the ordinary character. Checks of eleven leave a 50% chance of success for the extraordinary character.

A three out of three DC 7 check.

Inept people succeed 3.7% of the time.
Average people succeed 12% of the time.
(4) succeeds 30% of the time.
(5) succeeds 58% of the time.
(6 and 7) succeed 100% of the time.

A three out of three DC 8 check.

Inept people succeed 0.5% of the time.
Average people succeed 3.7% of the time.
(4) succeeds 12% of the time.
(5) succeeds 30% of the time.
(6) succeeds 58% of the time.
(7) succeeds automatically.

A three out of three DC 9 check.

Inept people fail.
Average people succeed 0.5% of the time.
(4) succeeds 3.7% of the time.
(5) succeeds 12% of the time.
(6) succeeds 30% of the time.

A three out of three DC 10 check.

Inept and average people fail.
(4) succeeds 0.5% of the time.
(5) succeeds 3.7% of the time.
(6) succeeds 12% of the time.
(7) succeeds 30% of the time.

A three out of three DC 11 check.

(2, 3, and 4) fail.
(5) succeeds 0.5% of the time.
(6) succeeds 3.7% of the time.
(7) succeeds 12% of the time.

Three in three checks should remain in the 7-10 range, except for truly extraodinary things.

Now, with this as a baseline, what happens when we add one more die?

What happens when we up the required number of successes?
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« Reply #5 on: September 12, 2006, 12:51:37 AM »
Thus far, I'm seeing stat generation that starts with two points per score and distributes three.
Then a three dice pool with a minimum DC of 7 and a maximum DC of 11.
Success, or degrees thereof, might rely on number of successes as well as DCs.
Thoughts?
Yes?
No?
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snakefing

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« Reply #6 on: September 12, 2006, 07:51:17 PM »
[blockquote=beejazz]
Low Dice, Low Number: Easy Task. Who can do it? ANYONE.

Low Dice, High Number: Intuitive Difficulties. Like art, you either have it or you don't.

High Dice, Low Number: Complex Difficulties. You either know it or you don't.

High Dice, High Number: Extreme Difficulty. You gotta know what you're doing, and even that might fail!
[/blockquote]
I think I might analyze this a bit differently:

1 Success needed: This is never very hard, even at the highest target number and lowest dice. Appropriate to tasks that are fairly easy if you've got the prerequisites. For example, crafting a basic item - if you've got the skills and tools, you can do it pretty well. Might also be appropriate for fairly easy things if they are done in a hurry. For example, jumping from a small height, there is a small chance if you are clumsy that you might twist an ankle.

2 Successes needed: This can be fairly hard at low dice pools, even if the target number isn't that high. As your dice pool increases, this won't be that bad. For example, two successes with (net) target number 6 and dice pool 4d6 sounds pretty hard (two 6's out of four dice) but in fact it succeeds about 1 in 7 - which is uncommon but not outrageous. This is probably appropriate for things that are contested or considered challenging but not too hard. For example, crafting a quality weapon or cutting a fine gem. Or things like Observe checks.

3 Successes or more: This will generally be pretty hard unless the dice pool is pretty high or the target number is quite low. I'll have to look at this in more detail. I suspect that in such cases you might valuably look at this in terms of Successes as a fraction of your dice pool. (That is, what fraction of your dice have to be successes.)

When I have time, I'll start looking at stats as you vary successes required and dice pools. But right now, I've got to make the kids some dinner.
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beejazz

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« Reply #7 on: September 12, 2006, 08:18:24 PM »
I was just thinking, in each instance, keeping the numbers needed and successes needed identical, what happens for each die you add. Because progression and training go on number of dice (or such is my understanding thus far).

Here's my understanding thus far:

SO FAR

ABILITIES:
Strength
Coordination
Reflex
Toughness
Luck
Characters start with two points in each and have three points to distribute.

ROLLING:
We roll on dicepools of 3d6. Difficulties run from 7 to 11 and from one to three required successes. In combat or anything else, dicerolls are always on the offensive side of the check.

ADVANCEMENT:
Characters advance via character points. They can spend points to add extra dice to their pools or gain new special abilities. Also, they can spend character points on powerful magic or technological items (as far as money goes, adventurers live hand-to-mouth, often having to fight just to eat... even if they did have money, you don't just go out and buy the Lance of Longinus).

ADVENTURE POINTS:
On a critical success (a success with four dice) a character gets an adventure point. A character can have a miximum number of adventure points equal to his luck score. To use adventure points, a character must declare that he is using them and what he intends to do. He automatically succeeds. This costs the diferrence between the number of actual successes and the number of required successes.

COMBAT:
Combat runs like everything else, rolls of the d6.

Attack= Ability+Pool vs. Reflex+Armor
Damage= Ability+Pool vs. Toughness+Armor

The number of successes required to hit would depend on armor (I suppose) and critical hits happen only on four successes.
Damage done is equal to the number of damage successes, where the damage dice pool is determined by weapon (between one and five, I guess)
Hit points themselves run on something like the vitality and wound points in Star Wars. They'd be some fixed number each, with fewer wounds than vitality.
Critical successes (for AP, anyway) don't happen on either of these two checks.

MOVEMENT:
A character gets 10 action points a round, and can do anything in a round that adds up to less than 10. A character can move one square for every point he spends, and can attack for seven points (though this figure could probably be reduced). A turn lasts five seconds.

We may include the option to act out of turn by spending five points, but that's a maybe.

How we determine who goes first, I don't know yet.

[close]


And as for advancement, extra dice costing x^2*10, where x is the number of extra dice?
So 1=10, 2=40, 3=90, 4=160.
And only teh spend of 1/3 character points on any one thing?
Just a thought.
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« Reply #8 on: September 12, 2006, 10:42:56 PM »
Some quick results, organized by # of successes:

[table=1 Success]
[tr][th]Net Target[/th][th]6[/th][th]5[/th][th]4[/th][th]3[/th][th]2[/th][/tr]
[tr][th]DicePool(3,d6)[/th][td]742%[/td][td]70%[/td][td]88%[/td][td]96%[/td][td]99.5%[/td][/tr]
[tr][th]DicePool(4,d6)[/th][td]52%[/td][td]80%[/td][td]94%[/td][td]99%[/td][td]99.9%[/td][/tr]
[tr][th]DicePool(5,d6)[/th][td]60%[/td][td]87%[/td][td]97%[/td][td]99.5%[/td][td]100%[/td][/tr]
[tr][th]DicePool(6,d6)[/th][td]67%[/td][td]91%[/td][td]98%[/td][td]99.9%[/td][td]100%[/td][/tr]
[/table]

[table=2 Successes]
[tr][th]Net Target[/th][th]6[/th][th]5[/th][th]4[/th][th]3[/th][th]2[/th][/tr]
[tr][th]DicePool(3,d6)[/th][td]7.5%[/td][td]26%[/td][td]50%[/td][td]74%[/td][td]93%[/td][/tr]
[tr][th]DicePool(4,d6)[/th][td]13%[/td][td]41%[/td][td]69%[/td][td]89%[/td][td]98%[/td][/tr]
[tr][th]DicePool(5,d6)[/th][td]20%[/td][td]54%[/td][td]81%[/td][td]95%[/td][td]99.5%[/td][/tr]
[tr][th]DicePool(6,d6)[/th][td]26%[/td][td]65%[/td][td]89%[/td][td]98%[/td][td]99.9%[/td][/tr]
[/table]

[table=3 Successes]
[tr][th]Net Target[/th][th]6[/th][th]5[/th][th]4[/th][th]3[/th][th]2[/th][/tr]
[tr][th]DicePool(3,d6)[/th][td]0.5%[/td][td]4%[/td][td]12%[/td][td]30%[/td][td]58%[/td][/tr]
[tr][th]DicePool(4,d6)[/th][td]1.5%[/td][td]11%[/td][td]31%[/td][td]59%[/td][td]87%[/td][/tr]
[tr][th]DicePool(5,d6)[/th][td]3.5%[/td][td]21%[/td][td]50%[/td][td]79%[/td][td]96%[/td][/tr]
[tr][th]DicePool(6,d6)[/th][td]6%[/td][td]32%[/td][td]66%[/td][td]90%[/td][td]99%[/td][/tr]
[/table]

[table=4 Successes]
[tr][th]Net Target[/th][th]6[/th][th]5[/th][th]4[/th][th]3[/th][th]2[/th][/tr]
[tr][th]DicePool(4,d6)[/th][td]0.1%[/td][td]1.2%[/td][td]6%[/td][td]20%[/td][td]48%[/td][/tr]
[tr][th]DicePool(5,d6)[/th][td]0.3%[/td][td]4.5%[/td][td]19%[/td][td]46%[/td][td]80%[/td][/tr]
[tr][th]DicePool(6,d6)[/th][td]0.9%[/td][td]10%[/td][td]34%[/td][td]68%[/td][td]93%[/td][/tr]
[tr][th]DicePool(7,d6)[/th][td]1.8%[/td][td]17%[/td][td]50%[/td][td]82%[/td][td]98%[/td][/tr]
[/table]

As a sort of rule of thumb, it looks like adding a die more or less doubles the probability of success, at least for the harder tasks. On the other hand, a +1 modifier (reducing the net target number) seems to triple the likelihood (or more).

I've got data for 5, 6, 7 successes for up to 7d6 pools as well. We may need that when it comes to damage dice.
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« Reply #9 on: September 12, 2006, 11:14:21 PM »
Doubles?!
*Jawdrop*
This will take some thinking through, no doubt!

And yes, good thoughts on the damage dice!
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« Reply #10 on: September 12, 2006, 11:36:45 PM »
Well, keep in mind that the rule of thumb only applies to fairly hard tasks. For example, 3 successes using 4 dice, rolling against a 5 is hard (11%, like rolling 19 or 20 on d20). Making that 5 dice raises the probability to 21% (nearly double, more like rolling a 17 or higher on d20), but then, doubling 11% isn't that big an increase.

For tasks where the probability is less hard (like 20% or more), the doubling rule doesn't quite apply. But it is still true that this is not a system that allows for small adjustments. That's pretty much inevitable when you are using a die like d6, where even the hardest roll (needing a 6) is still 17% likely.

I don't think this is necessarily fatal to the system. But you have to accept that you aren't going to be giving out modifiers for small things like higher ground or morale bonuses. There's just no scope in the system for simulating such things. It will tend to lend a somewhat more heroic/cinematic feel, I think, because you can't get as bogged down in tactical details.

I wonder whether that makes the idea of detailed action points a little incongruous. Anyway, time enough to look at that when we get to it.
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« Reply #11 on: September 12, 2006, 11:46:54 PM »
When it comes to tactics, there is the melee/ranged combat divide to explore.

Also, I suppose that for five points, a character can do shiz "out of turn."

And *some* tactical advantages can offer movement penalties and bonuses... Just some thoughts.

Any better ideas?
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beejazz

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Dice Pool Crunching
« Reply #12 on: September 13, 2006, 02:23:04 PM »
So, I've been thinking. Price extra dice at 10(x^2) and limit people to spending more than 1/5 their xp on one ability. So dice cost as follows:
1=10
2=40
3=90
4=160

And "the highest dice I can buy", which is bears an uncanny resemblance to "levels" runs as follows:

1=50
2=200
3=450
4=800

Sound about right?
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I don't believe in it anyway.
What?
England.
Just a conspiracy of cartographers, then?

snakefing

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« Reply #13 on: September 13, 2006, 08:19:08 PM »
Regarding tactics: You are right. What I should have said is that you won't be able to grub around after small tactical advantages that in d20 might give you a +1 bonus. There's no way in this system to represent such small advantages.

There's still plenty of scope for tactics on the larger scale - how and when to use magic vs. arms, missile fire vs. hand to hand, an armored screen to protect more vulnerable combatants, etc. There may be some scope for tactical advantages based on initiative and so on too.

My main thinking was that if the base mechanic isn't capable of representing small advantages in combat, it might be a little inconsistent to employ tactical mechanics that try to capture equally small advantages, but in a different way.

For example, some tactical advantages could offer movement penalties or bonuses, for sure. That's not directly affected by dice pools. But if the blow-by-blow of melee is highly abstracted, and movement is highly detailed, it might create a kind of Frankenstein's monster - made of different parts that just don't seem to fit together. So we need to think about how to create movement, initiative, and other tactical rules that support the same kind of feel that dice pools are going to give to melee.
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CYMRO

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« Reply #14 on: September 13, 2006, 08:26:33 PM »
KISS.

And this doesn't seem to do it.

The more I look at this, the less I like dice pools.  Not when there is a simple opposed roll system in place with a simple and convenient 5% increment built in.


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And "the highest dice I can buy", which is bears an uncanny resemblance to "levels" runs as follows:


What is the point of getting rid of levels if you just rename them and hide them in a more complicated mechanic.

After some discreet questioning around several different kinds of boards, reduction of complications in rules is the biggest objective for those looking beyond d20.
This does not do it.