Dice and Probabilities

[Torque2100]

I had a thought, and I think that thought might be a good springboard to a game design discussion. It has been many moons since my Intro to Statistics course in college so it might be fun to consider the reasons why certain dice mechanics are used and why.

My thought was this, there many RPG systems. In some systems rolling lower is good, some systems rolling higher numbers is good. Both come with their own set of problems. For instance, if the goal is to roll as low as possible, how do you handle an opposed Static or Skill check in an RPG?

One compromise mechanic that I wish we saw more often is the "roll as high as possible without going over your skill/stat" mechanic or as I like to call it "Blackjack Rolling."

To me it seems like a decent compromise that's simple and straightforward to use. It both means that more adept or skilled characters have a higher range of success and if the system uses "degrees" of success, can get better results. It also makes opposed rolls very simple "Did both parties succeed? If no, the party that rolled a success wins. If yes, then the party that has the highest result on the die wins."

I am surprised I don't see this more often. The only games I know of that really use Blackjack Rolling are Eclipse Phase and the Tabletop Wargame "Infinity."

 

[Baragei]

Mythras does that

And I've yet to get a good answer on the actual probabilities at work with opposed blackjacks.
Character A has a skill of 90, Character B has a skill of 75. They engage in a beer-drinking contest. What are the odds here if I was a betting man?
Explain it like I was 5 (which is probably close to my actual age when it comes to practical applications of mathematical principles).

 

[Raleel]

As the man says, Mythras does that.

And I've yet to get a good answer on the actual probabilities at work with opposed blackjacks.

https://anydice.com/program/1b263 This is a very basic one. I didn’t put in crits on it, but it’s pretty close.

https://anydice.com/program/14337 This one handles differential rolls.

 

[Baragei]

Ah yes. But you're not considering my mathematical handicap here.
To demonstrate: A has a 90/100 chance, B has a 75/100. A has a straight 15/100 of winning outright, and B has a 25/100 of failing outright, with 10/100 of both ending up under the table. If both rolls fall within 1-75, the outcome is basically random.
By themselves all these figures are perfectly fine and obvious and sensible. Together, less so.
I want to eyeball the odds here without resorting to specialized web applications.

 

[Raleel]

Yea, eyeballing the odds is not easy for an opposed roll, and less so for a differential one. Literally why I built the thing.;)

What I can say is that you need a surprisingly large skill difference to feel reasonably assured of success, where reasonably assured we discussed on here a couple years ago and is somewhere between 66% and 75%. For example, a 60/40, if we assume failure always benefits the defender, is really much closer to 50/50 on the attackers chance of succeeding. The attacker needs closer to double the skill of the defender in the middle ranges to get into that psychological sweet spot.

This then turned into be leveraging Easy rolls more in my games.

 

[Lofgeornost]

One compromise mechanic that I wish we saw more often is the "roll as high as possible without going over your skill/stat" mechanic or as I like to call it "Blackjack Rolling."

To me it seems like a decent compromise that's simple and straightforward to use. It both means that more adept or skilled characters have a higher range of success and if the system uses "degrees" of success, can get better results. It also makes opposed rolls very simple "Did both parties succeed? If no, the party that rolled a success wins. If yes, then the party that has the highest result on the die wins."

I am surprised I don't see this more often. The only games I know of that really use Blackjack Rolling are Eclipse Phase and the Tabletop Wargame "Infinity."

Pendragon uses the ‘blackjack’ method for combat rolls, I think from the very first edition back in 1985. I’m not sure if it was the first game to do so.

 

[3rik]

Love blackjack method but every time I tell someone about it, they claim that it doesn't feel intuitive and they prefer the more fiddly roll low and compare margins of succes or success levels.

I think there's more than one d100/BRP variant that does use blackjack opposed rolling.

 

[Fenris-77]

Why not just use the standard deviation from the mean multiplied by Cosign and run through a flahrderiha something-something matrix? Easy-peasy.

 

[Agemegos]

Mythras does that

And I've yet to get a good answer on the actual probabilities at work with opposed blackjacks.
Character A has a skill of 90, Character B has a skill of 75. They engage in a beer-drinking contest. What are the odds here if I was a betting man?
Explain it like I was 5 (which is probably close to my actual age when it comes to practical applications of mathematical principles).

This would be easier if we were considering an 18 and a 15 on d20s, but here goes.

A and B each roll a d100. There are 10,000 pairs of results; each is a different outcome, and all are equally likely if the dice are fair.

• In 75 × 75 = 5 625 outcomes both players have roll 75 or less. 75 of those are ties. Of the rest, A wins half (2 775) and B wins half (2 775)
• In 75 × (90 - 75 = 15) = 1 125 outcomes B rolls 0–75 and succeeds, but A has rolls 76–90 and has a result that B cannot beat whatever they roll. A wins.
• In 75 × (100 - 90 = 10) = 750 outcomes B rolls 01 – 75 and succeeds, and A rolls 91 — 100 and fails. B wins.
• In 25 × 90 = 2 250 outcomes B rolls 76 – 100 and fails, while A rolls 01 – 90 and succeeds. A wins.
• In 25 × 10 = 250 outcomes B rolls 76 – 100 and fails, while A rolls 91–100 and also fails. Tie.

Totals:

1. A wins 2 775 + 1 125 + 2 250 = 6 150 out of 10,000 outcomes, which is 61.5%
2. B wins 2 775 + 750 = 3 525 out of 10,000 outcomes, which is 35.25%
3. They tie on 75 + 250 = 325 outcomes out of 10,000, which is 3.25%

Here's a diagram to try to make things clearer. I have constructed this using a blackjack roll on d20s, to make the cells bigger.



Observe that B wins in two conditions: if B succeeds and A fails (pink rectangle on the lower left) or if both succeed and B's roll is higher (pink triangle in the upper right of the rectangle on the top left). Outcomes where both succeed and A wins include a wedge of green that mirrors B's wins, plus a rectangle below the dotted line where A succeeds on a roll higher than B could succeed on.

 

[Agemegos]

Still no? Jeez! Tough audience!

When the "blackjack" system is used for an opposed roll on d100, each contestant can roll any number from 1 to 100; if the roll is higher than their target number their score is 0, otherwise their score is equal to the number they rolled. The contestant with the higher score wins; if the two scores are the same the contest is a tie.

To make the explanation simple, I'm going to call the contestant with the higher target number "A" and their target number "a" and the one with the lower target number "B" and their target "b". The formulas that we work out will work when both are the same, but not when B's target is higher than A's.

Here's a big chart showing all the possible outcomes. A's roll is the in first column; their corresponding score is in the second column. B's roll is in the top row, their corresponding score is in the second column. In the 100-row by 100-column block that makes up the rest of the chart, a spreadsheet "if" statement and conditional formatting colour the cell blue if A's score is higher, yellow if B's score is higher, and red if the two scores are the same. Formulas in the cells with violet text count all the cells of the appropriate colour and divide by the total number of cells (ten thousand of them) to calculate a probability: the result is expressed as a percentage.



You can see outcomes in which B wins get plotted in two parts. There is a rectangle at the bottom left that is (100 - a) rows deep and b rows wide, containing (100 - a)b cells. And there is a triangle in the middle of the top that is (b - 1) cells wide and high, containing b(b-1)/2 cells (that's the formula for the b-1th triangular number).

So the probability that B wins is

P(B) = b × {(b-1)/2 + 100 - a)} / 10 000​

You can see that outcomes in which A wins get plotted as a big rectangle a columns deep and 100 columns wide, that has had a triangle b cells wide deep cut out of it. The number of cells is 100 a minus b(b+1)/2 (that's the formula for the bth triangular number).

P(A) = {100a - b(b+1)/2} / 10 000​

And you can see that tied outcomes get plotted in two areas: b cells in a diagonal where both contestants succeed and their scores are the same, plus a rectangle (100-a) by (100-b) in which both contestants have a score of zero.

P(tie) = {b + (100-a)(100-b)} / 10 000​

That's probably still not intuitive, but at least now you have formulas in closed form.

 

[Baragei]

This is great. Now to condense this down to eyeballing..

 

[Agemegos]

This is great. Now to condense this down to eyeballing..

Well. if you consider the diagonal of ties as a neglible thin line, the chance of B winning looks like the chance that they will succeed while A fails (75% of 10%) plus half the square of the chance they'll succeed ( ½ (75% × 75%)).

 

[Agemegos]

Okay. Last try.

Disregarding the diagonal line of ties (which are 1% of B's 75%), B's chance of winning is

≈ Chance of succeeding × (chance A fails + ½ × chance A rolls under B's target)​

​

≈ 75% × (10% + ½ × 75%)​

​

≈ 0.75 × 37.5%​

​

≈ 35.625%​

​

The rounding error in that approximation is +0.375%, i.e half of the ties on the diagonal.

 

[Baragei]

And the formula for A winning is..?

Ok, sorry

As an aside, the odds in this particular example turned out pretty much the same as if we'd used the ol' (and scorned) resistance table, only slightly skewed in favour of B. The RR took me all of ..12 seconds to grok when I first encountered it.

 

[3rik]

The Resistance table only works with two opponents, and only the active party gets to roll. You can apply blackjack to multiple characters competing at any stat.

 

[Dr Jack]

I am surprised I don't see this more often. The only games I know of that really use Blackjack Rolling are Eclipse Phase and the Tabletop Wargame "Infinity."

I know this system from Pendragon. I ran a long campaign for getting on for two years. It is the only system I've ever played where my players consistently got it wrong after such a timescale. IMO, the main reason that it is not often used it that it is simply an extremely unintuitive system.

 

[xanther]

I had a thought, and I think that thought might be a good springboard to a game design discussion. It has been many moons since my Intro to Statistics course in college so it might be fun to consider the reasons why certain dice mechanics are used and why.

You had me at statistics. One of my favorite courses of all time was Statistical Mechanics, kind of different from die statistics but statistics none the less.

My thought was this, there many RPG systems. In some systems rolling lower is good, some systems rolling higher numbers is good. Both come with their own set of problems. For instance, if the goal is to roll as low as possible, how do you handle an opposed Static or Skill check in an RPG?

That's an easy one. Simply keep in mind roll over or under is mathematically equivalent. I call such systems "target number" systems.

In a roll under system. Say my attack is 10, and your defense is 4. Then my target number is 10 - 4 = 6, and I need to roll under 6 to succeed. It makes intuitive sense that a good defense subtracts from you attack, or in this case your target number. If I need to roll over, then your defense subtracts from my roll instead of my target number.

In all cases where one is roll over or under, etc. You can get equivalence by simply moving the modifier from ones roll to ones target number.

When there are no "skills" and just rolls, it is simply a matter of comparing which person succeeds the most, that is succeeds by the greatest margin. Be it under the most or over the most. You can even have one impact the other. However, it's easier to implement such things in skill systems (less likely to need cascading rolls) than a roll only system.

For example, my attack and defense above is a "skill" like system. All you do is compare and then use that modify your target number and the success is determined in 1 roll. In contrast, say you only a roll to hit and roll to defend system. If I roll to hit, and hit, then you roll to defend (parry, block etc.) and after that 2nd roll we know if I succeeded. So 2 rolls instead of 1 to get to the same result.

One compromise mechanic that I wish we saw more often is the "roll as high as possible without going over your skill/stat" mechanic or as I like to call it "Blackjack Rolling."

To me it seems like a decent compromise that's simple and straightforward to use. It both means that more adept or skilled characters have a higher range of success and if the system uses "degrees" of success, can get better results. It also makes opposed rolls very simple "Did both parties succeed? If no, the party that rolled a success wins. If yes, then the party that has the highest result on the die wins."

I'll assume any modifiers get added to the target number here, that is what you need to roll under and not added to your roll, otherwise one is penalized for good gear if modifiers add to the roll.

I like your version, it does give the underdog a chance to beat the one who is more skilled. I believe the part at the end I highlighted is key.

It does seem to have more things that can be done with it and a better way to do "degree of success" than systems where it is flat (I roll a 20), or how much over or under I am (which requires subtraction).

I am a huge proponent for knowing where one stands by looking at the dice on the table (the only math being known what your target number was).

I am surprised I don't see this more often. The only games I know of that really use Blackjack Rolling are Eclipse Phase and the Tabletop Wargame "Infinity."

Good point. I suspect it has nothing to do with the merits of the mechanic and everything to do with knee-jerk reactions or it could also be the games that use it are not popular for other reasons so people don't know.

Also d20 as done in most systems doesn't have an active opposition aspect, i.e. defense is all static in AC and HP. So this kind of opposed roll is not part of the design. Now I think it could be...and a better way to do any defense than a separate cascaded roll IMHO.

 

[xanther]

Love blackjack method but every time I tell someone about it, they claim that it doesn't feel intuitive and they prefer the more fiddly roll low and compare margins of success or success levels.
...

I chalk that up to hobgoblins...the of the mind kind.

Calculating margins by subtraction is much slower, and now will read on to see if see if calculating a margin vs the under but highest number wins are statistically comparable. My gut says they are very close statistically (so much so it is a matter of taste how much the help the weaker and harm the stronger).

Logistically, so much easier to look at the die than to subtract...players only have 1 set of dice to deal with, the Referee? Got 24 goblins and 6 bugbears...in that case at a glance mechanics for the win

 

[xanther]

Still no? Jeez! Tough audience!

.....

That's probably still not intuitive, but at least now you have formulas in closed form.

You had me convicted in the first post. I'm just late to the party.

Love math and statistics, to persuade would say stick with a d20...it's completely applicable to d100.

On intuition, for me would explain along these lines, using a d20.

Looking for an easy to implement degree of success in opposed rolls.

Without such things in bog standard d20, say I need to roll 10 or under and you need to roll 6 or under. We both roll a 5, what then?

One way is to say well 6-5 = 1 and 10-5 = 5 so the person who rolled 5 under wins (or is it the other way...if the other way it's wonky but I doubt most will intuit that)...but that requires subtraction for each die on the table, as players you have 1 set, as Referee I have many, many more.

Another way, is to just look at the die and not doing any subtraction. The odds won't be exactly the same but the intended effect (sometimes the weaker opponent wins because they succeeded more) is still there and to essentially the same degree, and the speed of play make up for it. In what proposing we both know we succeeded and we look and see 5's, tie breaker to higher skill no need to do subtraction.

Now you could give them the odds, in a example. I can say the odds for the "do subtraction" are actually far less intuitive (i.e. people are likely way off on what they really are) versus your mechanic. Simply based on the fact the subtraction statistics are harder to set up and do right.

The thing is, your way is just as good at verisimilitude as the subtraction way...there is no objective data or math behind the subtraction way, it's just a way of flattening the power curves and providing for comparative degrees of success.