Math on low-roll wins vs blackjack...

[overgeeked]

This is a thread for hashing out the math on low-roll wins vs blackjack-style highest roll without going over wins.

ENWorlders with better maths knowledge than me, please explain this to the rest of us.

 

[overgeeked]

Here's one post on the topic from the Dragonbane thread. To get us started.

did a bit of programmatic analysis...
with "a" having skill 7, and "b" having 17...
HW a= 42 b= 312 t= 7 both f= 39
LW a= 112 b= 242 t= 7 both f= 39
Note that high wins vs low wins is a significant change.
Low wins means low will win roughly 1/4 of the time, instead of 1/10th, given the 7 vs 17.
lets see an 8 vs 13
HW a= 84 b= 224 t= 8 f= 84
LW a= 124 b= 184 t= 8 f= 84
That's still pretty profound.
While the game doesn't permit ties (the active wins ties), by checking them separately, the program becomes active agnostic.

## low = [1,2,3,4,5, 6, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
## high = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,0,0]
low = [1,2,3,4,5,6,7,8,0,0,0,0,0,0,0,0,0,0,0,0]
high = [1,2,3,4,5,6,7,8,9,10,11,12,13,0,0,0,0,0,0,0]
na = 0 ## high wins
nb = 0
nt = 0
oa = 0 ## low wins
ob = 0
nf = 0
for a in low:
    for b in high:
        if (a > 0) and (b > 0):
            if a > b:
                ob += 1
                na += 1
            elif b > a:
                oa += 1
                nb += 1
            else:
                nt += 1
        elif a > 0:
            na += 1
            oa += 1
        elif b > 0:
            nb += 1
            ob += 1
        else:
            nf += 1
print ("HW\ta=",na,"\tb=",nb,"\tt=",nt,"\tf=",nf)
print ("LW\ta=",oa,"\tb=",ob,"\tt=",nt,"\tf=",nf)

 

[TwoSix]

I was a little confused at first, because the results didn't match my gut intuition. But if you plug it into a spreadsheet, and draw a little picture, it makes more sense.

Assuming A is success on a 15- (75% chance), and B is success on a 5- (25% chance).

A and B both fail 75 out of 400 combinations, or 3/16. Importantly, I'm assuming that "high wins" and "low wins" DOESN'T MATTER in this case, since both sides failed. Since "both fail" is the converse of "both succeed", but high/low doesn't matter, this means the portion of the odds where both succeed has a larger influence on the overall ratio of win vs losses than might initially be intuited.

A passes, B doesn't: 225/400, or 9/16
B passes, A doesn't: 25/400, or 1/16

This leaves the block from [1,1] to [15,5], 75 combinations, where both succeed.

5 are ties (the diagonal from [1,1] to [5,5]).

The remaining 20 combinations in the square from [1,1] to [5,5] are going to be divided evenly, 10 and 10, between A and B no matter which way the high/low rule is applied.

Where the high/low rule matters are the last 50 combination, the block from [6,1] to [15,5]. Since that block ONLY has values where A>B, then the high/low rule completely determines whether or not A or B wins the block. And those 50/400 combinations are what tilt the odds so much depending on whether or not you use "low roll wins" or "blackjack style".

 

[Bill Zebub]

When I get a chance I'm going to make some color block diagrams, but on a 4x4 grid to keep things simple.

 

[Committed Hero]

In a low roll wins scenario, there will be more instances where the more skilled individual succeeds while the less skilled one fails altogether. But in the cases where the latter succeeds, it negates the former's higher skill level.

 

[Bill Zebub]

In a low roll wins scenario, there will be more instances where the more skilled individual succeeds while the less skilled one fails altogether. But in the cases where the latter succeeds, it negates the former's higher skill level.

That's a good way of expressing it, and of explaining the 1-in-10 vs. 1-in-4 result.

 

[Wolfpack48]

Think of it like this: A skill 4 character, when he succeeds, forces the 17-skill character to roll a master class to win when using the low roll approach. The skill 4 character is completely inept but is also occasionally a kung-fu master.

 

[overgeeked]

The skill 4 character is completely inept but is also occasionally a kung-fu master.

Right. Which is why I really dislike low-roll wins and prefer blackjack. It gives the higher-skilled character the advantage rather than the lower-skilled character.

 

[overgeeked]

Where the high/low rule matters are the last 50 combination, the block from [6,1] to [15,5]. Since that block ONLY has values where A>B, then the high/low rule completely determines whether or not A or B wins the block. And those 50/400 combinations are what tilt the odds so much depending on whether or not you use "low roll wins" or "blackjack style".

Exactly. That's where the discussion has been mostly focused. That light purple block where it makes the difference.

Low-roll wins gives the lower-skilled character an advantage.

Blackjack gives the higher-skilled character an advantage.

Which is "right" or "better" is the question.

 

[Wolfpack48]

I was a little confused at first, because the results didn't match my gut intuition. But if you plug it into a spreadsheet, and draw a little picture, it makes more sense.

Assuming A is success on a 15- (75% chance), and B is success on a 5- (25% chance).

A and B both fail 75 out of 400 combinations, or 3/16. Importantly, I'm assuming that "high wins" and "low wins" DOESN'T MATTER in this case, since both sides failed. Since "both fail" is the converse of "both succeed", but high/low doesn't matter, this means the portion of the odds where both succeed has a larger influence on the overall ratio of win vs losses than might initially be intuited.

A passes, B doesn't: 225/400, or 9/16
B passes, A doesn't: 25/400, or 1/16

This leaves the block from [1,1] to [15,5], 75 combinations, where both succeed.

5 are ties (the diagonal from [1,1] to [5,5]).

The remaining 20 combinations in the square from [1,1] to [5,5] are going to be divided evenly, 10 and 10, between A and B no matter which way the high/low rule is applied.

Where the high/low rule matters are the last 50 combination, the block from [6,1] to [15,5]. Since that block ONLY has values where A>B, then the high/low rule completely determines whether or not A or B wins the block. And those 50/400 combinations are what tilt the odds so much depending on whether or not you use "low roll wins" or "blackjack style".View attachment 391859

Roll low: Favors the underdog
Roll high: Makes sure the little guy really is an underdog!