The effects of Target Numbers

[Username17]

Consider a dicepool similar to Shadowrun, but let's also consider what would happen if we used a different base target number. Say, the dice are counted as hits if they roll a 3+ or 4+ instead of 5+.

The first and most obvious thing is that what constitutes a 'normal' number of hits would change accordingly. That is: if you get a hit on average on every three dice, then your 'normal' threshold for 6 dice is 2 hits and your 'normal' threshold for 12 dice is 4 hits. That goes to 3 and 6 hits if your TN is 4+, and 4 and 8 hits if your TN s 3+. Let's say an 'easy' task is one where you need to roll at least one less hit than the average, and a 'hard' task is one where you need to roll at least one more than average.

6 Dice

TN	Normal Task	Hard Task	Easy Task
3+	68.0%	35.1%	90.0%
4+	65.6%	34.4%	89.0%
5+	64.9%	32.0%	91.2%

12 Dice

TN	Normal Task	Hard Task	Easy Task
3+	63.2%	39.3%	89.2%
4+	61.3%	38.7%	80.6%
5+	60.7%	36.9%	81.9%

What do we notice?

• As dicepools go up, variance goes up. This means you are less likely to succeed at normal or easy tasks, but more likely to succeed at difficult tasks.
• The higher the base target number, the more likely you are to roll under average, which brings your chance of succeeding on normal rolls down.
• When the target number is above 4 and more dice fail than succeed, there's less room in the RNG for tasks you are likely to succeed at - which means that easy tasks are more likely to succeed.

This makes me strongly favor 4+ or 3+ as a base target number over 5+. One thing that doesn't really come across here is that in the case of the 12 die pool at a TN of 5+, there are four settable success thresholds on the top end that will never come up. That is, it's theoretically possible for a player to roll 9, 10, 11, or 12 - but only the most sadistic MC would ever call for such a thing because there's less than 0.4% of succeeding at even the easiest of them and functionally no chance at all of success at the two top difficulties. Contrasted, at a 4+ difficulty there are only two difficulty thresholds for 12 die pools that have a less than 1% chance of success, and none of them round to 0.00%. At a TN of 3+ you have comparable effects at the low end of threshold that it is virtually impossible to fail at - but I regard that as much less of an issue. Characters succeeding at tasks the player is confident their character can succeed at is totally acceptable.

-Username17

[OgreBattle]

3+ has a nice feel, the pointiness of an odd number adds zest.


What would you pick as an "average human" number of dice to do a strength task that's easy for them, at the edge of impossible?

[Username17]

3+ has a nice feel, the pointiness of an odd number adds zest.


What would you pick as an "average human" number of dice to do a strength task that's easy for them, at the edge of impossible?

At a Target Number of 3+, more rolls give an above average result than give a below average result. The average is maintained by rolls that fall short falling by more than results that beat expectations exceeding the norm. For example, on 6 dice the average is 4 hits, which means that the most it is possible to go above average is 2, and the most it's possible to fall short is 4. And since the average requires an equal number of excess hits above and below the curve, there simply are more high rolls than low rolls.

I regard this as a good thing, because people like rolling well and don't like rolling poorly. So a dice system where people genuinely do roll above average more than they roll below average is a net positive.

But this also means that there simply is less room for extraordinary success. That's more mixed, but ultimately I think it's probably a net positive. Failure is just failure, but extra levels of success should probably mean something - so having lots of theoretically possible levels of success that in practice will probably never happen is probably not great.

However, this also means that without quite significant dice pools there is no possibility of hitting rare levels of success. At dice pools less than 12, there is genuinely no difficulty threshold that would equate to a success rate of less than 1%. At 6 dice, the most difficult possible threshold is 6, which still is achieved on 8.8% of rolls. I would suggest then, that for a TN of 3+ that the 'normal person' have a dice pool of 9 with extremely impressive but normal human stunts having a difficulty of 9 or 10. The normal person succeeds at Difficulty 9 on 2.6% of rolls and can't succeed at Difficulty 10. Total badasses with dicepools of 15 would succeed at Difficulty 9 on 79.7% of rolls, and at Difficulty 10 on 61.8% of rolls. It would take a Dicepool of 17 to succeed at Difficulty 9 on more than 90% of rolls, and a Dicepool of 19 to succeed at Difficulty 10 on more than 90% of rolls

-Username17

[deaddmwalking]

You don't like playing any games like 'a 1 negates a hit' or 'a 6 counts as two hits'?

[OgreBattle]

However, this also means that without quite significant dice pools there is no possibility of hitting rare levels of success.

How about "natural 6 gives you an extra die to roll that can't crit"?

"1= fumble" could do something

[Iduno]

You don't like playing any games like 'a 1 negates a hit' or 'a 6 counts as two hits'?

I think the normal argument is: It's easier to check "die is greater than X" than it is to check "die is greater than X, and also it's a 6, which means this rule applies." It's not a huge increase, but unless you're playing Yatzee, nobody wants to spend all night trying to figure out patterns in dice.

Unless you're using a die roller or other electronic program that can just spit out results. Then go to town.

[echoVanguard]

I would suggest then, that for a TN of 3+ that the 'normal person' have a dice pool of 9 with extremely impressive but normal human stunts having a difficulty of 9 or 10. The normal person succeeds at Difficulty 9 on 2.6% of rolls and can't succeed at Difficulty 10. Total badasses with dicepools of 15 would succeed at Difficulty 9 on 79.7% of rolls, and at Difficulty 10 on 61.8% of rolls. It would take a Dicepool of 17 to succeed at Difficulty 9 on more than 90% of rolls, and a Dicepool of 19 to succeed at Difficulty 10 on more than 90% of rolls.

Setting the baseline at 9-10 for normal and having "badass" be defined at 15 doesn't sound very scalable.

echo

[Username17]

You don't like playing any games like 'a 1 negates a hit' or 'a 6 counts as two hits'?

Such things are fun sometimes, but they make the probability curve dumb. If you make a 1 count as negative and a 6 count double, you haven't actually changes the average at all. But you have increased the variance. By a lot. On 6 dice, you have the increased the range of potential values from 7 numbers to 19 numbers, but the extra six results you've added at the bottom collectively come up on less than two percent of actual die rolls.

From a probability standpoint, there really isn't any justification for special individual results like that.

Setting the baseline at 9-10 for normal and having "badass" be defined at 15 doesn't sound very scalable.

If the TN is 3+, every die matters a lot. In terms of average results, getting 1 bonus die at TN 3+ is the same as getting two bonus dice in Shadowrun (or V:tR). So we're basically talking about a range in SR4 of 12 dice. Like, in SR you might say normal is 6 dice and badass is 18 dice. Same thing.

A dicepool system is functionally logarithmic. The absolute value of the difference in dicepool is the same in terms of its expected bonus to average results regardless of how many dice you start with. You could have a scale of 1-7 or a scale of 20-26 and it would be much the same for a lot of purposes.

I happen to be suggesting an upshift so that normies are rolling 9 dice instead of 5 because dicepools have some weird effects at very low dicepool sizes that go away if you do that.

-Username17

[Foxwarrior]

You don't like playing any games like 'a 1 negates a hit' or 'a 6 counts as two hits'?

What if you had a game where a 1 counted as 1 hit, a 2 counted as 2 hits, a 3 counted as 3 hits, and so on?

This would happen: https://anydice.com/program/1a78c
Sort by "At Least" to get the chance of success.

[deaddmwalking]

You don't like playing any games like 'a 1 negates a hit' or 'a 6 counts as two hits'?

Such things are fun sometimes, but they make the probability curve dumb. If you make a 1 count as negative and a 6 count double, you haven't actually changes the average at all.

I was not making a specific proposal, nor was I trying to suggest that doing both was appropriate. The point was that there are some things you can do that play with those probabilities, and, as you say, they can be fun.

named "rolling above 3 on a d6, counting 6s double"]Here's an Anydice Link to 6d6, count hits (3+), with 6 counting twice using 6d6.

5 is the average result, with at least 5 happening 61% of the time; a 4 happening 81% of the time, and a 6 happening 38% of the time... That's assuming you don't choose the normal 3+ to balance against....

Offhand, that would address:

[quote="FrankTrollman]
there simply are more high rolls than low rolls. I regard this as a good thing, because people like rolling well and don't like rolling poorly. So a dice system where people genuinely do roll above average more than they roll below average is a net positive.[/quote]

It would also allow extra room for extraordinary success if that's something you wanted.

[RadiantPhoenix]

You don't like playing any games like 'a 1 negates a hit' or 'a 6 counts as two hits'?

What if you had a game where a 1 counted as 1 hit, a 2 counted as 2 hits, a 3 counted as 3 hits, and so on?

This would happen: https://anydice.com/program/1a78c
Sort by "At Least" to get the chance of success.

Change the dicepool to a fixed 1 die, use a d20, and add automatic positive/negative hits for modfiers and then you're just talking D&D.

[OgreBattle]

Two step rolls, opposed rolls

So in anydice how do I make dice explode on a 6?

[pragma]

Some of these premises seem crazy to me. The choice of threshold doesn't seriously affect your probability of succeeding in easy medium or difficult tasks as defined above: Frank's numbers don't change by more than 6% (a 1.2 in 20 chance) in any column. You can see also see that the easy, medium and hard points have the same y value for each curve in this anydice link (I find it clearest to use the "at least" and "graph" views).

That said, I like Frank's argument that that the total number of dice needed for a beginner or expert's pool to score changes with the required d6 roll for the same number of required hits. Having a smaller maximum pool size appeals to me because my players get mad at the higher variance when they throw around 18 dice, even when they still succeed ("dang, only 4 hits this time!"). I also like being able to put the number of hits required for easy tasks on the flat, guaranteed success portion of the cumulative distribution (the "at least" curve), which is easier when players usually get more successes.

And now for pedantic disambiguations!

The higher the base target number, the more likely you are to roll under average, which brings your chance of succeeding on normal rolls down.

The average is at the center of a symmetric probability distribution. You have an equal chance of rolling above or below average regardless of your threshold. Maybe you're referring to the chance for half the dice to be hits?

A dicepool system is functionally logarithmic. The absolute value of the difference in dicepool is the same in terms of its expected bonus to average results regardless of how many dice you start with.

This is describing a linear behavior, not a logarithmic one. And, indeed, both the expected value of a dice pool (size*probability of success) and the variance of a dice pool ( size*p_success(1-p_success) ) are linear in its size.

[pragma]

Two step rolls, opposed rolls

For opposed dice pool rolls, like Shadowrun "output 10d{0,0,1}-5d{0,0,1}" to get net successes on a 10 die pool vs. 5 die.

So in anydice how do I make dice explode on a 6?

"output [explode d6]"

[Username17]

The average is at the center of a symmetric probability distribution. You have an equal chance of rolling above or below average regardless of your threshold. Maybe you're referring to the chance for half the dice to be hits?

No. The average is the sum of all possible rolls divided by the number of different rolls needed to get there. If the results are left shifted or right shifted, it's entirely possible for there to be more above average results than below average results or vice versa. There being an equal chance of being above or below average is a special case of symmetrical data - if the TN is anything other than 4+ that will not be true.

The simple example is the Doubling Die from Backgammon. It's a d6 with faces 2, 4, 8, 16, 32, 64. The average result is 21, which means that two thirds of the time it rolls below average and one third of the time it rolls above average.

For dice pools, a high target number means that the average is much lower than the maximum and only slightly higher than the minimum. That means small numbers of outlier high results are balanced with much larger numbers of slightly low results and people roll below average more often than they roll above average. For example: at TN 5+ the average is 2 hits, and you roll 3 or more hits 32.0% of the time and you roll 1 or less hits 35.1% of the time. For a TN of 5+ you get exactly the opposite - the average is 4 hits and you get 5 or more 35.1% of the time and 3 or less 32.0% of the time.

5 is the average result, with at least 5 happening 61% of the time; a 4 happening 81% of the time, and a 6 happening 38% of the time... That's assuming you don't choose the normal 3+ to balance against....

Offhand, that would address:

That's not addressing anything. That's the opposite of addressing things. If you have 6s count for 2 hits, you get a maximum that's very high compared to the average. And that means that people fail more because they roll under average more often. It drives the chance of rolling under average up to 39.0%, which is even worse than what you had with TN 5+.

Rolling above average more than you roll below average is a feature, and die fuckery that adds variance and makes the opposite true are bad.

-Username17

[OgreBattle]

If my goal is to reduce dice pool size, is exploding dice or ‘6= 2 hits’ a solution?

Example: I hit on 3+ and a 6 = explodes. I have a pool of 1, a 2hit task is still achievable which is risky but possible.

Per d6 rolled its a 77% chance to get 1-2 successes
11% chance to get 2 successes

[pragma]

There being an equal chance of being above or below average is a special case of symmetrical data - if the TN is anything other than 4+ that will not be true.

For dice pools, a high target number means that the average is much lower than the maximum and only slightly higher than the minimum. That means small numbers of outlier high results are balanced with much larger numbers of slightly low results and people roll below average more often than they roll above average. For example: at TN 5+ the average is 2 hits, and you roll 3 or more hits 32.0% of the time and you roll 1 or less hits 35.1% of the time. For a TN of 5+ you get exactly the opposite - the average is 4 hits and you get 5 or more 35.1% of the time and 3 or less 32.0% of the time.

Got it, and I agree with what you're saying. Backgammon is a nice example of a very non-symmetric distribution.

I think our only point of contention is that I am approximating 3+, 4+ and 5+ distributions as all essentially symmetric. The relatively small difference you cite in your example -- a 3% higher likelihood of rolling above or below average -- suggests the approximation isn't bad. However, I'll acknowledge that my approximation breaks down for very small pools.

In summary: you're technically correct, the best kind of correct.

If my goal is to reduce dice pool size, is exploding dice or ‘6= 2 hits’ a solution?

Depends on your pool size and your thresholds. That change does increase the expected hits per die, but not by much, and it increases the variance quite a bit in small pools. On average, you would expect to get one additional hit for every six die in your pool. If you want to play around in anydice you can use "output Nd{0,0,1,1,1,2}" where you substitute your desired pool size for N.

[Orca]

One advantage of TN 4 on a d6 is that I can work out exact chances for smallish dice pools getting some number of hits in my head pretty quickly. This breaks down beyond ~5 dice, true.

[Username17]

I think our only point of contention is that I am approximating 3+, 4+ and 5+ distributions as all essentially symmetric. The relatively small difference you cite in your example -- a 3% higher likelihood of rolling above or below average -- suggests the approximation isn't bad. However, I'll acknowledge that my approximation breaks down for very small pools.

I don't think it holds at any sized dice pools. With TN 5+, it is always shifted so that rolling above average is more likely than rolling below average. With TN 3+, it is always shifted so that rolling below average is more likely than rolling above average.

The curve flattens as dice pools increase in size and variance increases. But the discrepancy always remains. With an 18 die pool, rolling actually average is something that happens less than twenty percent of the time. With a TN of 3+ the over-average portion is 41.2% and the under average portion is 39.1%. With a TN of 5+, that's precisely reversed.

It may not sound like a big difference, but we're actually talking about a difference in what the single most likely result of every single die roll is. At TN 5+, the most likely result is that you fall short (for all dice pools larger than five dice). At TN 3+, the most likely result is that you roll above average (for all dice pools larger than 5 dice). At TN 4+, rolling above or below average is precisely equally more likely than rolling dead average (for all dice pools larger than 4 dice).

We aren't talking about a once-per-session roll, we're talking about literally every roll. When failure is the single most likely outcome, that definitely is something that you end up noticing.

If my goal is to reduce dice pool size, is exploding dice or ‘6= 2 hits’ a solution?

TN=4+ gives you the most 'space' on the RNG above and below average. If you left shift or right shift your data by having the TN be higher or lower than 4+, the numbers crunch up on one side or the other.

A good way to look at this is by looking at 6 dice because the number of numbers at or above the TN is also the expected number of hits. TN 5+ averages 2 hits, TN 4+ averages 3 hits, and TN 3+ averages 4 hits. There are factually 7 possible results regardless. Three results are above average and three results are below average. At TN 5+ you only have two below average results and at TN 3+ you only have two above average results. So both are squeezed more by having smaller dice pools.

Having botches or crits on the dice increases variance and gives you more 'dice space' to work with. But it's not very efficient space because most of the numbers don't come up practically speaking. So for example, we could get the same average results as TN 4+ by having 5s count 1 hit, 6s count for 2 hits, and missing on a 1-4. The average is the same, but there are now thirteen potential results instead of only seven. But all of the extra possible results combined only come up 4% of the time - less than the chances of rolling a natural 20 on an icosahedron. So in practice you end up having to make rules or guidelines that cover the situation where you get 10 or more hits on six dice, but that only comes up on one roll in a thousand so it's a bad use of game space.

-Username17

[Roog]

Why not set the standard DC based on the median (or whatever cdf percentage works) instead of the average?
That makes the shape of the tail independent of the standard success chance, and allows the game effect of the distribution's tail shape to be considered separately.

[infinitederpgeneration]

What would be a good "human" range of dice for a TN 4+ system? Say you wanted to do World of Darkness with real big boy math and playtesting. What would be your range of sickly child to Olympic athlete, and what would be the range of superhuman to superman?

[erik]

For a d6 Dice pool with TN4+* I went with something like this:

Dice	Ability Description
0	Utterly unable to perform
1	Deficient
2	Average potential, but not developed
3	Average and competent
4	Professional expertise
5	Top tier professional expertise
6	Legendary, at the limit of human ability
7	Absurdly proficient, super human

The notion being that you can trade in dice so 4 dice can automatically succeed at DC needing 2 hits. 6 dice automatically can succeed at DC needing 3 hits.

Hits needed	Difficulty
0	Absurdly simple task, can only fail with extraneous pressure
1	Moderate amateur task or simple professional task
2	Difficult amateur task or moderate professional task
3	Difficult professional task
4	Extremely difficult professional task
5	Legendary task

* (the TN can vary a little bit in my game with some skills they may be TN5+ or TN6+)
Oh, and I updated my Nexus thread with a most recent DL link for the game.

[Foxwarrior]

How can you be Legendary with 6 dice if you need 5 hits to do a Legendary task? That'd barely work with TN 2+.

[erik]

I probably needed a better thesaurus when writing it. A difficulty 5 task is something that you'd need items to boost you into being able to expect to accomplish it, or be legendary to even have a hope of succeeding (it's about 10.94%).

[pragma]

I think our only point of contention is that I am approximating 3+, 4+ and 5+ distributions as all essentially symmetric. The relatively small difference you cite in your example -- a 3% higher likelihood of rolling above or below average -- suggests the approximation isn't bad. However, I'll acknowledge that my approximation breaks down for very small pools.

I don't think it holds at any sized dice pools. With TN 5+, it is always shifted so that rolling above average is more likely than rolling below average. With TN 3+, it is always shifted so that rolling below average is more likely than rolling above average.

The curve flattens as dice pools increase in size and variance increases. But the discrepancy always remains. With an 18 die pool, rolling actually average is something that happens less than twenty percent of the time. With a TN of 3+ the over-average portion is 41.2% and the under average portion is 39.1%. With a TN of 5+, that's precisely reversed.

It may not sound like a big difference, but we're actually talking about a difference in what the single most likely result of every single die roll is. At TN 5+, the most likely result is that you fall short (for all dice pools larger than five dice). At TN 3+, the most likely result is that you roll above average (for all dice pools larger than 5 dice). At TN 4+, rolling above or below average is precisely equally more likely than rolling dead average (for all dice pools larger than 4 dice).

We aren't talking about a once-per-session roll, we're talking about literally every roll. When failure is the single most likely outcome, that definitely is something that you end up noticing.

I disagree. In your example, you are 5% (.412/.391-1) more likely to roll above average than below average. The "extra" below average rolls are as common as rolling a 20 in a D&D session. You're correct that this difference is always present and that it is present in every roll. But I am not convinced that the slight difference in over/under-average ratio matters to players (especially compared to other hard to control factors like "are there snacks nearby"), and if I'm wrong then I think players attitudes will be influenced subtly enough that you can't measure it without concerted effort. But, to hedge, I don't have the psychology training to comment on how this probability discrepancy would play out in an average player's psyche. I think there's room for disagreement here given that it would take a determined grad student a few months to get to the fact of this matter.

To summarize my views: I think the small probability differences you're harping on are a bad reason to change the hit number for a die, but I think that bounding the upper size of a pool (to reduce variance) justifies a change in hit number from 5+ to 3+.

However, One aspect of this change deserves more comment: you'll need to be stingier with bonus dice if you set the hit number to 3+ because each die is more likely provide an extra hit. That seems to cause a problem similar to D&D 5e, where you can't give players too many nice things without them falling off the difficulty scale. I could imagine a solution where player toys mostly mitigate penalties, but I'm curious about others.

Why not set the standard DC based on the median (or whatever cdf percentage works) instead of the average?
That makes the shape of the tail independent of the standard success chance, and allows the game effect of the distribution's tail shape to be considered separately.

You may be on to something, and I'd be interested to see a specific proposal about how you'd implement this -- what type of dice, what rolls count as hits and what a set of success thresholds looks like -- but I suspect that you'd be hamstrung by the strong quantization effects inherent in using only integer numbers of dice.