Statistics

[Tinkgineer]

I enjoy math, but I am very rusty. I am curious about how to model the odds of an opposed roll system like HQ2. Particularly for HQ2 the levels of success I think make it harder for me to understand.

Would you have to make a table for each success level that’s every ability value vs every ability value?

 

[soltakss]

2 hours ago, Tinkgineer said:

Would you have to make a table for each success level that’s every ability value vs every ability value?

That's what I'd do.

[Ian Cooper]

On 4/21/2019 at 5:39 PM, Tinkgineer said:

I enjoy math, but I am very rusty. I am curious about how to model the odds of an opposed roll system like HQ2. Particularly for HQ2 the levels of success I think make it harder for me to understand.

Would you have to make a table for each success level that’s every ability value vs every ability value?

 

Kind of. Opposed D20 resolution maths is covered on a few sites. I think that 'roughly' each point in difference is worth a 3% shift on the odds.

It is why we recommend modifiers in multiples of 3, as they shift by about 10%

[Tinkgineer]

I was asking this because I wanted to try to objectively compare alternative systems. My 8 year old wants to understand what’s going on but she does not really “click” with the success/fail vs a success/fail chart becoming a level of victory. The FATE shifts are a bit more intuitive, where you just subtract the opposed roll to get your shift.

But FATE rolls are purely high number after modifier and easier to math. HQ the roll is first a crit/success/fail/fumble and then afterwards compared. Since the skill modifies the target number and not the roll I am having a hard time turning it into math thingies. 

The fact I say “math thingies” is proof I’m out of my league.

 

[Ian Cooper]

Try looking at https://anydice.com/

 

[Ian Cooper]

Here is something I wrote a while back for someone:

----------------------

So the discussion was about probability in D20 based games. First, its obvious that in an unopposed roll under Target Number (TN)  on a D20, each pip adds a 5% increment. Roll under 10 is 50% (10/20 * 100) and roll under 13 is 65% (13/20 * 100).

 

HeroQuest uses opposed rolls, so the math is a little more complicated. 

 

You can find a number of articles out there which look at the system many D20 games use: Roll D20 and add Dice Modifiers (DMs) and the higher number wins. This is a common approach. The maths is easy, as the probability can derived from the DM, as you are both equally likely to roll any number. Accounting for ties, you can use the difference between the DMs each side has to determine likelihood of victory. This  calculation gives each pip a 4% improvement. For example a +4 DM gives you a 30% chance of losing, 66% chance of winning, and a 4% chance of a tie.

 

However, Heroquest is roll highest but roll under, and the math is more complicated.

 

Let's start with an evenly matched contest of 13 vs. 13.

 

It is easy to understand win vs. lose. It is just 13/20 * 7/20 * 100 = 22.75%.  

 

A situation where both sides succeed is given by 13/20 * 13/20 * 100 = 42.25%.

 

We can also use negation, as well as coincidence.  As either side could win, the chance someone wins is 22.75 + 22.75 + 42.25 = 87.75. which subtracting from 100 means that the chance both  lose is 12.25%

 

Now this is where it gets complex. For now I want to ignore criticals and fumbles as it makes this easier to follow.

 

I want to determine who wins, if both succeed, or both lose, as here the highest roll now wins.

 

If we both win, then I have to roll higher than you. If you roll 1, there are 12 numbers I can roll to beat you, if you roll 2, there are 11 numbers I can roll to beat you. So my chances are 12 + 11 + 10 +... divided by the total number of combinations. Now we already know we are in the 42.25% bracket, so we can treat this as though we had a 13 sided dice, so our possible combinations is 13 ** 2 or 169. This gives us 46% chance that I will win, 46% chance that you will win and 4% ties.

 

Failure is similar, it is the highest roll within the 7 remaining pips. Again my chance of beating you is 6 + 5 + ... divided by the total number of combinations: 49. So 21/49 or 43% with a 14% chance of a tie.

 

Now I can figure out my chance to win. It's (22.75/100) + (46/100 * 42.25/100) + (12.25/100 * 43/100) = 47.45%. Given your chance to win is the same, the chance of a tie is: 5%.

 

And this should meet our expectations: two evenly matched opponents have about a 50% chance of success.

 

So what happens if the numbers are 17 and 13 instead, in other words I use one of my key abilities against one of your also rans.

 

Win vs. Lose. For me 17/20 * 7/20 * 100 = 29.75% 

Lose vs. Win. For me (or you win) 3/20 * 13/20 = 9.75%

Both Lose: 3/20 * 7/20 = 5.25%

Both Win 13/20 * 17/20 = 55.25%

 

Now we need to look at again at outcomes where both win or lose. If you roll 1 there are 16 numbers I can beat you with and so on. So my chances are 16 + 15 + … divided by the total number of combinations and again we can treat this as though we have a 17-sided dice and a 13-sided dice or 221 possible combinations. That gives us 136/221 or a 61% chance that I win. You win on 12+11+.. divided by 221 occasions, or 35% of the time, with a 4% chance of a tie.

 

Failure is a little trickier. The contestant with the higher skill has a small set of numbers, 3, whilst the one with the lower skill has, 7 possible numbers. Of those 21 combinations, on a 20 I can beat  6 numbers, on a 19, 5 and on an 18, 4  so I have 15 of the 21 combinations or 71%, you have 3 of the 21 or 14%, and there is 15% chance of a draw.

 

So I win on (29.75/100) + (55.25/100 *61/100) + (5.25/100 * 71/100) = 67.18% 

 

So to compare the two: 13 vs. 13 is a 47.45% chance of victory, but 17 vs 13 is a 67.18% chance of victory. It is of the order of 5% per pip, which is to be expected, but its certainly a sizeable difference.

 

So it does not differ significantly from opposed rolls with a DM of +4 above.

 

 

[Ian Cooper]

15 minutes ago, Ian Cooper said:

You can find a number of articles out there which look at the system many D20 games use: Roll D20 and add Dice Modifiers (DMs) and the higher number wins. This is a common approach. The maths is easy, as the probability can derived from the DM, as you are both equally likely to roll any number. Accounting for ties, you can use the difference between the DMs each side has to determine likelihood of victory. This  calculation gives each pip a 4% improvement. For example a +4 DM gives you a 30% chance of losing, 66% chance of winning, and a 4% chance of a tie.

...

 

So to compare the two: 13 vs. 13 is a 47.45% chance of victory, but 17 vs 13 is a 67.18% chance of victory. It is of the order of 5% per pip, which is to be expected, but its certainly a sizeable difference.

 

So it does not differ significantly from opposed rolls with a DM of +4 above.

 

 

For those who want the headline. Take the difference between the higher and lower abilities, multiply by 5%, add to 50% and that is a good rule of thumb for the higher abilities chance of victory.

So, simplifying grossly

13 vs 13 is roughly 50-50

14 v 13 is roughly a 55 % chance of victory

17 v. 13 is roughly as 70 % chance of victory

20 vs. 13 has a difference of 7, which is roughly an 85% chance of victory.

[boradicus]

13 hours ago, Ian Cooper said:

Here is something I wrote a while back for someone:

----------------------

So the discussion was about probability in D20 based games. First, its obvious that in an unopposed roll under Target Number (TN)  on a D20, each pip adds a 5% increment. Roll under 10 is 50% (10/20 * 100) and roll under 13 is 65% (13/20 * 100).

 

HeroQuest uses opposed rolls, so the math is a little more complicated. 

 

You can find a number of articles out there which look at the system many D20 games use: Roll D20 and add Dice Modifiers (DMs) and the higher number wins. This is a common approach. The maths is easy, as the probability can derived from the DM, as you are both equally likely to roll any number. Accounting for ties, you can use the difference between the DMs each side has to determine likelihood of victory. This  calculation gives each pip a 4% improvement. For example a +4 DM gives you a 30% chance of losing, 66% chance of winning, and a 4% chance of a tie.

 

However, Heroquest is roll highest but roll under, and the math is more complicated.

 

Let's start with an evenly matched contest of 13 vs. 13.

 

It is easy to understand win vs. lose. It is just 13/20 * 7/20 * 100 = 22.75%.  

 

A situation where both sides succeed is given by 13/20 * 13/20 * 100 = 42.25%.

 

We can also use negation, as well as coincidence.  As either side could win, the chance someone wins is 22.75 + 22.75 + 42.25 = 87.75. which subtracting from 100 means that the chance both  lose is 12.25%

 

Now this is where it gets complex. For now I want to ignore criticals and fumbles as it makes this easier to follow.

 

I want to determine who wins, if both succeed, or both lose, as here the highest roll now wins.

 

If we both win, then I have to roll higher than you. If you roll 1, there are 12 numbers I can roll to beat you, if you roll 2, there are 11 numbers I can roll to beat you. So my chances are 12 + 11 + 10 +... divided by the total number of combinations. Now we already know we are in the 42.25% bracket, so we can treat this as though we had a 13 sided dice, so our possible combinations is 13 ** 2 or 169. This gives us 46% chance that I will win, 46% chance that you will win and 4% ties.

 

Failure is similar, it is the highest roll within the 7 remaining pips. Again my chance of beating you is 6 + 5 + ... divided by the total number of combinations: 49. So 21/49 or 43% with a 14% chance of a tie.

 

Now I can figure out my chance to win. It's (22.75/100) + (46/100 * 42.25/100) + (12.25/100 * 43/100) = 47.45%. Given your chance to win is the same, the chance of a tie is: 5%.

 

And this should meet our expectations: two evenly matched opponents have about a 50% chance of success.

 

So what happens if the numbers are 17 and 13 instead, in other words I use one of my key abilities against one of your also rans.

 

Win vs. Lose. For me 17/20 * 7/20 * 100 = 29.75% 

Lose vs. Win. For me (or you win) 3/20 * 13/20 = 9.75%

Both Lose: 3/20 * 7/20 = 5.25%

Both Win 13/20 * 17/20 = 55.25%

 

Now we need to look at again at outcomes where both win or lose. If you roll 1 there are 16 numbers I can beat you with and so on. So my chances are 16 + 15 + … divided by the total number of combinations and again we can treat this as though we have a 17-sided dice and a 13-sided dice or 221 possible combinations. That gives us 136/221 or a 61% chance that I win. You win on 12+11+.. divided by 221 occasions, or 35% of the time, with a 4% chance of a tie.

 

Failure is a little trickier. The contestant with the higher skill has a small set of numbers, 3, whilst the one with the lower skill has, 7 possible numbers. Of those 21 combinations, on a 20 I can beat  6 numbers, on a 19, 5 and on an 18, 4  so I have 15 of the 21 combinations or 71%, you have 3 of the 21 or 14%, and there is 15% chance of a draw.

 

So I win on (29.75/100) + (55.25/100 *61/100) + (5.25/100 * 71/100) = 67.18% 

 

So to compare the two: 13 vs. 13 is a 47.45% chance of victory, but 17 vs 13 is a 67.18% chance of victory. It is of the order of 5% per pip, which is to be expected, but its certainly a sizeable difference.

 

So it does not differ significantly from opposed rolls with a DM of +4 above.

 

 

I suppose that in order to see how each system really stacks up against the other it would be helpful to graph the probability for each column (opposed ability) for each row (ability) a row at a time; or you could probably generate a 3d graph as well.  Then you could see how the curves for each method look side by side.

 

[Joerg]

14 hours ago, Ian Cooper said:

20 vs. 13 has a difference of 7, which is roughly an 85% chance of victory

20 vs. 13 has an effective difference of 6.

[JonL]

Behold, calculated odds and result distributions, masteries excluded.

[boradicus]

On 4/29/2019 at 9:18 AM, JonL said:

Behold, calculated odds and result distributions, masteries excluded.

This is awesome!  I am a little confused as to what the rows vs. the columns are, however.  Are the columns the Target Number?

[el_octogono]

Rows are the PCs target number and columns the opposition's TN. Inside the chart you'll find the results distribuition.

 

[JonL]

1 hour ago, boradicus said:

This is awesome!  I am a little confused as to what the rows vs. the columns are, however.  Are the columns the Target Number?

The columns are the Resistance rating the GM is rolling with. The rows are the rating of the PC's ability. 

[JonL]

To be more clear, the rows and columns are what the actual rolls are, you enter the ability rating and resistance down below.

The colors and labels in the result matrix adjust  based on what ability and resistance values you set.

[boradicus]

6 hours ago, JonL said:

To be more clear, the rows and columns are what the actual rolls are, you enter the ability rating and resistance down below.

The colors and labels in the result matrix adjust  based on what ability and resistance values you set.

So, the table is a roll versus roll table rather than a roll versus a Target Number table?

[JonL]

12 hours ago, boradicus said:

So, the table is a roll versus roll table rather than a roll versus a Target Number table?

That's right. You enter the Player and GM TNs below the grid where it says, "EDIT THESE."  The grid then changes to show you the outcomes of all 400 possible combinations of Player and GM die rolls. Player rolls are numbered down the left hand side, while GM rolls are numbered across the top.

The counts and percentages on the lower left show the frequency of each outcome within the result space along with a few useful aggregations, while the graph on the lower right shows the distribution curve (with ties split evenly between Marginal Defeat and Marginal Victory for graphing purposes).

I like in particular to show people the graph when they complain that rolling 1D20 is "too swingy." The player's individual roll may have a flat distrubution, sure, but when you oppose that roll with another and matrix the results you get very nice bell-ish curves with about 2/3 of the results being Marginal Victory or Marginal Defeat when Ability rating and Resistance rating are equal.  

Plug in various Ability and Resistance values, and watch how the distribution responds. In particular, enter common starting ability ratings, 13, 15, and 17, and compare them to Low(8), Moderate(14), and High(20) Resistances. Note in particular how the Any Victory/Any Defeat,  Marginal or Tie, and Minor+ Victory/Defeat aggregate percentages vary with respect to the matchups. 

[boradicus]

10 hours ago, JonL said:

That's right. You enter the Player and GM TNs below the grid where it says, "EDIT THESE."  The grid then changes to show you the outcomes of all 400 possible combinations of Player and GM die rolls. Player rolls are numbered down the left hand side, while GM rolls are numbered across the top.

The counts and percentages on the lower left show the frequency of each outcome within the result space along with a few useful aggregations, while the graph on the lower right shows the distribution curve (with ties split evenly between Marginal Defeat and Marginal Victory for graphing purposes).

I like in particular to show people the graph when they complain that rolling 1D20 is "too swingy." The player's individual roll may have a flat distrubution, sure, but when you oppose that roll with another and matrix the results you get very nice bell-ish curves with about 2/3 of the results being Marginal Victory or Marginal Defeat when Ability rating and Resistance rating are equal.  

Plug in various Ability and Resistance values, and watch how the distribution responds. In particular, enter common starting ability ratings, 13, 15, and 17, and compare them to Low(8), Moderate(14), and High(20) Resistances. Note in particular how the Any Victory/Any Defeat,  Marginal or Tie, and Minor+ Victory/Defeat aggregate percentages vary with respect to the matchups. 

Thanks!  I've got it now.  This is actually very nice, indeed.  It is quite bell-ish, and even though there are some odd dips here and there, overall, they don't really affect the aggregates too much - in fact, you could say that they add "character."  Is there a way to add modifiers?  When adding modifiers, what happens to the curves?  I like the fact, that as the basic d20 vs d20 roll statistics stand - without modifiers - that the curve still encompasses the full gamut of possible outcomes; whereas, I would be concerned that after adding in modifiers that the curve would shift so that some outcomes become either partially or fully truncated.  Now, I suppose, that if the outcomes are only partially truncated that it really would not make that much of a difference, because despite the fact that the curve might shift beyond a certain set of die roll outcomes, that the success/failure descriptions would not be entirely occluded. 

Edited May 2, 2019 by boradicus
Grammar

[JonL]

For modifiers, just change the TN  values accordingly. A 15 with a +3 augment is no different than an 18. You never modify the die roll itself in HQ, or most* roll-under systems. 

(* Pendragon's handling of skills > 20 is a noteworthy exception. and an interesting counterpoint to HQ's Mastery scaling.)

Edited May 2, 2019 by JonL

[boradicus]

7 hours ago, JonL said:

For modifiers, just change the TN  values accordingly. A 15 with a +3 augment is no different than an 18. You never modify the die roll itself in HQ, or most* roll-under systems. 

(* Pendragon's handling of skills > 20 is a noteworthy exception. and an interesting counterpoint to HQ's Mastery scaling.)

I don't have Pendragon.  How does it handle skills > 20?  And how does HQ and BRP handle skills > 20?  

[JonL]

48 minutes ago, boradicus said:

I don't have Pendragon.  How does it handle skills > 20?  And how does HQ and BRP handle skills > 20?  

In Pendragon, ratings above 20 get added to your die roll. If your total is over 20, you score a critical success. (Normally, you score a critical by rolling your rating exactly).

HQ breaks ratings down into 20pt brackets, called Masteries. When a rating hits 21, you have reached the first Mastery level, notated as 1M, with the 'M" representing 20. a total of 25 would be written as 5M, while 47 would be 7M2 ( i.e. 7+(20 * 2)). Your TN is the number before the M, but your result gets bumped up one grade (Fumble -> Fail, Fail -> Success, Success -> Critical) for every M you have over your opposition.

BRP games vary a bit from one implementation to the next, but speaking broadly your special or critical success thresholds are based on a fraction of your skill percentage, your chances of attaining them rise even when you hit the maximum overall success chance. Percentages over 100 can also soak up penalties, help with splitting for multiple actions, penalize your opponents in opposed rolls, etc. - depending on the specific game.

All three approaches work reasonably well. The Pendragon and BRP approaches start to fray when you approach double the base range, though as a practical matter that isn't really a problem in any actual game I've heard of. (I think Lancelot has a ~30 in his Lance skill in Pendragon. Highest any character I played ever got was a 24.)

The HQ approach , while perhaps harder to grasp at first and having a little oddness right at the breakpoints, has the benefit of continuing to gracefully and  meaningfully scale up and up. While the spreadsheet I linked to doesn't encompass Mastery, the odds and result distributions for 17M3 vs 15M3 would be exactly the same as 17 vs 15.

Edited May 2, 2019 by JonL

[boradicus]

5 hours ago, JonL said:

In Pendragon, ratings above 20 get added to your die roll. If your total is over 20, you score a critical success. (Normally, you score a critical by rolling your rating exactly).

HQ breaks ratings down into 20pt brackets, called Masteries. When a rating hits 21, you have reached the first Mastery level, notated as 1M, with the 'M" representing 20. a total of 25 would be written as 5M, while 47 would be 7M2 ( i.e. 7+(20 * 2)). Your TN is the number before the M, but your result gets bumped up one grade (Fumble -> Fail, Fail -> Success, Success -> Critical) for every M you have over your opposition.

BRP games vary a bit from one implementation to the next, but speaking broadly your special or critical success thresholds are based on a fraction of your skill percentage, your chances of attaining them rise even when you hit the maximum overall success chance. Percentages over 100 can also soak up penalties, help with splitting for multiple actions, penalize your opponents in opposed rolls, etc. - depending on the specific game.

All three approaches work reasonably well. The Pendragon and BRP approaches start to fray when you approach double the base range, though as a practical matter that isn't really a problem in any actual game I've heard of. (I think Lancelot has a ~30 in his Lance skill in Pendragon. Highest any character I played ever got was a 24.)

The HQ approach , while perhaps harder to grasp at first and having a little oddness right at the breakpoints, has the benefit of continuing to gracefully and  meaningfully scale up and up. While the spreadsheet I linked to doesn't encompass Mastery, the odds and result distributions for 17M3 vs 15M3 would be exactly the same as 17 vs 15.

That is cool.  I like the idea of Mastery in HQ.  What I would like to do is to design a bell-ish system where the ability/skill progression is virtually open-ended at the top, yet has an ease of playability that scales well as the "bell" simultaneously stretches and becomes shaped  to reflect a weightedness toward higher levels of mastery.  Being that I am not a statistics genius, I think that the logical approach to doing this would be to incorporate the repetition of a simple two-stage process until a result with good balance has been obtained.  The first stage would be determining ways in which the "bell" could be "extended" (obviously it will start to look more like a wave than a bell at some point), and the second stage would be looking at various ways to make the system easy and fun to play (e.g. charts, roll comparisons, electronic aids/apps - such as electronic dice which could include complex calculations not seen by the player/GM, etc). 

I think that such a system would be useful for a number of purposes.  Monsters, animals, machines, etc, could be more proportionally designed with respect to PC characters (which usually humanoid); skill progression could be asymmetrical (for instance, certain classes of skills might have an easier or harder learning curve; also, certain skills could have different ranges or "caps" where after a certain point, further progression would become trivial with respect to how good someone could become at something); simulation could be more realistic; critical successes and critical failures could be dealt with entirely within the scope of the curve rather than via the addition of an ad hoc rule-set such as rolling one's score or a natural 1 or a natural 20.

While I am partial to actual dice, I don't at all think that it is unreasonable to assume that "electronic dice" will continue to gain in popular appeal, and in fact, gaming platforms such a roll20 already make use of "electronic dice."  Therefore, it would naturally be the next step in the evolution of advancing the use of chance in gaming to go beyond the mere representation of physical dice.  Of course, there will always be people who will want a more portable (and reasonably affordable) system when they are not using platforms such as roll20 to play.  Such devices and apps can be built - and of course one of the keys to their distribution will be good marketing - and, of course, good design: just think like Steve Jobs !!

I'm not stuck on migrating to "electronic dice," by any means, but, I thought that I should include this brief argument in their favor rather than to leave my inclusion of their consideration unsupported.

I know that various dice systems have been experimented over the past few decades, including the use of dice pools.  Dice pools are interesting, and they do offer a way to extend the curve by adding dice (and, if so desired, one can subtract 1 for each die added to normalize the lowest result to still being a 1).  I'm not necessarily sold on dice pools, however, because, for one, polyhedra do not scale in their number of faces linearly.  Additionally, adding more dice does tend to rather quickly decrease the potential for outlying results.  This makes me more partial toward using charts and tables.  But, then the question becomes: how does create a chart or a table that is virtually/indefinitely extendable (without going to a Rolemaster like system - which has a chart for everything and an exploding dice system that I think leaves quite a lot to be desired)?

I would be interested to have your thoughts on these ideas!  I am currently inching (millimeter-ing, if we are talking progress via page thickness!) my way through a book on statistics in order to catch up on the topic, being that I never took a such a course when in school.

Edited May 3, 2019 by boradicus
Grammar & Style

[JonL]

A lot of the design concepts that seem to interest you have been done before, though not all at once. 

In addition to HQ, I suggest you study the following games, look at what they did well, and look at the interconnected bits that support that...

GURPS - variable skill progressions and bell curve task resolution. 3d6 for classic bell distribution. 

DC Heroes/Underground/Blood of Heroes - logarithmic attribute ratings for arbitrary power levels that never break the mechanic as they scale up. 2d10 added for ramp distribution.

TORG - log scale mechanics similar to the above, but with a different feel. 1d20

Rolemaster/Spacemaster - Bridges class  based and skill based character models. Later editions suffer from skill bloat, but are also very complete. MERP and Cyberspace are their lighter siblings. D100 open ended roll high. Best example of a game that could use an  assistant app.

Fate/Fudge - Fudge dice actually roll deviations from the mean result represented by your stat.

I would look at what these games get right and wrong, and learn from that.

[Aprewett]

I always liked the Harn system for skills, development and rolls.

[boradicus]

Thanks to you both :D.  How exactly does the Harn system work and differ from the ones that JonL mentioned above?  Thanks again.

[boradicus]

On 5/2/2019 at 8:26 PM, JonL said:

A lot of the design concepts that seem to interest you have been done before, though not all at once. 

In addition to HQ, I suggest you study the following games, look at what they did well, and look at the interconnected bits that support that...

GURPS - variable skill progressions and bell curve task resolution. 3d6 for classic bell distribution. 

DC Heroes/Underground/Blood of Heroes - logarithmic attribute ratings for arbitrary power levels that never break the mechanic as they scale up. 2d10 added for ramp distribution.

TORG - log scale mechanics similar to the above, but with a different feel. 1d20

Rolemaster/Spacemaster - Bridges class  based and skill based character models. Later editions suffer from skill bloat, but are also very complete. MERP and Cyberspace are their lighter siblings. D100 open ended roll high. Best example of a game that could use an  assistant app.

Fate/Fudge - Fudge dice actually roll deviations from the mean result represented by your stat.

I would look at what these games get right and wrong, and learn from that.

D.C. Heroes looks a bit hard to find in an inexpensive format.  I have some Rolemaster already, and I have wishlisted the others.