Dumpshock Forums > Opposed test statistical study

ZorbaTHut

Oct 29 2004, 12:56 PM

I remember seeing somewhere (possibly on this board) that opposed tests are massively weighted towards the side with even slightly higher stats. The person posting had a house rule that all opposed tests are made with a target number of 4, so the only difference is the number of dice. I was curious how true this was. So I decided to run a simulation.

I've got two orthogonal rule variations that I used - various target number calculations and a modified rule-of-six where every 6 above the target number counts as an extra success. I went up to 12 because I was interested, and didn't go above that because I wasn't. This isn't any fancy mathematical proof, it's just the result of doing 100,000 tries and checking the results. (Ties don't count, it just retries it.) That's why none of the identical-stat results are precisely 50%.

The target number options include standard opposed tests, target number of 4, and target number of 2 + opponent/2, which I added to see how well it worked. (Badly.)

CODE

Modified rule-of-six off
Standard opposed target

1 2 3 4 5 6 7 8 9 10 11 12
01 49.84 97.00 99.67 99.96 99.99 100.00 99.99 100.00 100.00 100.00 100.00 100.00
02 3.08 49.86 95.79 99.56 99.96 99.99 99.99 100.00 100.00 100.00 100.00 100.00
03 0.30 4.29 49.74 87.96 97.85 99.74 99.91 99.97 99.99 99.99 100.00 100.00
04 0.03 0.49 12.11 49.91 84.21 97.41 98.50 99.39 99.77 99.92 99.98 99.99
05 0.00 0.04 2.17 15.89 49.93 85.38 89.59 94.41 97.18 98.75 99.51 99.87
06 0.00 0.00 0.20 2.65 14.60 50.04 56.42 68.75 79.37 87.70 93.77 97.71
07 0.00 0.00 0.10 1.51 10.27 43.62 49.93 63.68 75.45 85.32 92.52 97.16
08 0.00 0.00 0.01 0.59 5.59 31.01 36.57 50.21 63.98 76.43 87.09 94.91
09 0.00 0.00 0.00 0.21 2.89 20.69 24.38 36.08 50.02 64.60 79.41 91.29
10 0.00 0.00 0.00 0.07 1.27 12.32 14.81 23.41 34.65 49.91 67.26 84.83
11 0.00 0.00 0.00 0.02 0.52 6.24 7.44 12.65 20.48 32.41 49.94 73.05
12 0.00 0.00 0.00 0.00 0.14 2.40 2.80 4.93 8.69 15.26 27.13 49.76

Modified rule-of-six on
Standard opposed target

1 2 3 4 5 6 7 8 9 10 11 12
01 50.22 91.11 98.37 99.63 99.93 99.99 99.99 99.99 100.00 100.00 100.00 100.00
02 8.67 50.14 89.47 97.95 99.63 99.96 99.98 99.99 100.00 100.00 100.00 100.00
03 1.66 10.52 50.00 83.17 95.86 99.35 99.69 99.89 99.95 99.99 100.00 100.00
04 0.34 2.14 17.02 50.03 80.79 95.74 97.30 98.74 99.41 99.78 99.94 99.98
05 0.06 0.35 4.12 19.22 49.75 82.41 87.09 92.42 95.76 97.86 99.09 99.71
06 0.00 0.03 0.63 4.17 17.36 49.83 55.98 67.38 77.35 85.70 92.05 96.91
07 0.00 0.01 0.28 2.70 12.92 43.89 50.10 61.92 73.47 82.90 90.71 96.28
08 0.00 0.00 0.11 1.21 7.71 32.63 37.81 49.92 62.54 74.93 85.19 93.76
09 0.00 0.00 0.03 0.56 4.18 22.87 26.64 37.33 50.05 63.69 77.42 89.94
10 0.00 0.00 0.01 0.20 2.11 14.49 16.87 25.28 35.62 49.94 65.99 83.45
11 0.00 0.00 0.00 0.06 0.93 7.80 9.28 14.78 22.62 33.96 49.77 71.99
12 0.00 0.00 0.00 0.01 0.31 3.12 3.66 6.22 10.02 16.65 28.03 50.15

Modified rule-of-six off
Target number 4

1 2 3 4 5 6 7 8 9 10 11 12
01 50.07 79.95 91.70 96.28 98.26 99.13 99.60 99.82 99.89 99.95 99.97 99.99
02 19.95 50.04 72.68 85.66 92.25 96.08 97.86 98.88 99.38 99.66 99.82 99.88
03 8.36 27.28 49.93 68.72 81.43 89.26 93.79 96.42 97.98 98.84 99.37 99.62
04 3.70 14.36 31.33 49.91 66.33 78.34 86.45 91.59 94.99 96.95 98.19 98.87
05 1.69 7.38 18.35 33.64 50.07 64.51 76.06 84.21 89.61 93.44 95.81 97.42
06 0.85 3.87 10.66 21.73 35.26 49.76 63.15 73.92 82.29 88.08 92.20 94.90
07 0.39 2.15 6.29 13.59 24.08 36.84 50.04 62.08 72.06 80.33 86.52 90.75
08 0.19 1.18 3.54 8.31 15.75 25.98 37.73 50.29 61.21 71.02 79.01 85.02
09 0.09 0.58 2.01 5.10 10.15 17.64 27.28 38.71 50.18 60.49 70.04 77.61
10 0.05 0.34 1.11 3.07 6.57 11.85 19.70 28.81 39.08 49.83 59.80 69.13
11 0.02 0.16 0.69 1.84 4.16 7.85 13.51 20.94 29.81 39.90 49.99 59.48
12 0.01 0.08 0.38 1.11 2.59 5.15 9.24 14.66 22.40 30.94 40.39 50.07

Modified rule-of-six on
Target number 4

1 2 3 4 5 6 7 8 9 10 11 12
01 49.96 75.92 88.12 93.78 96.71 98.17 98.92 99.43 99.67 99.80 99.89 99.93
02 23.71 49.91 69.50 81.64 88.93 93.33 96.02 97.47 98.57 99.07 99.44 99.66
03 11.73 30.36 49.91 65.70 77.46 85.38 90.55 93.88 96.02 97.39 98.41 98.93
04 6.22 18.30 34.00 49.99 63.85 74.48 82.42 87.92 91.88 94.59 96.39 97.65
05 3.23 10.95 22.34 36.19 50.10 62.27 72.29 80.07 85.98 90.07 93.16 95.20
06 1.78 6.57 14.69 25.59 37.47 50.01 61.26 70.47 77.92 84.01 88.33 91.63
07 1.03 4.03 9.39 17.67 27.70 39.09 50.29 60.12 69.22 76.47 82.34 86.92
08 0.60 2.45 6.19 12.08 19.91 29.37 39.23 50.08 59.67 67.89 75.06 80.83
09 0.36 1.48 3.90 8.18 14.13 22.06 30.86 40.46 50.02 59.27 66.84 73.97
10 0.18 0.89 2.51 5.48 10.05 15.96 23.47 31.99 41.08 49.94 58.66 66.25
11 0.09 0.51 1.53 3.53 6.97 11.54 17.63 25.03 33.05 41.41 49.94 58.13
12 0.04 0.33 1.02 2.39 4.77 8.26 13.16 18.91 26.11 33.75 42.05 50.08

Modified rule-of-six off
Target number 2 + n/2

1 2 3 4 5 6 7 8 9 10 11 12
01 49.84 97.70 99.68 99.96 99.99 100.00 100.00 100.00 100.00 100.00 100.00 100.00
02 2.18 49.99 81.07 96.15 98.58 99.74 99.89 99.98 99.99 99.99 99.99 100.00
03 0.33 18.73 50.26 87.67 95.06 99.10 99.63 99.97 99.99 99.99 99.99 100.00
04 0.04 3.97 12.04 49.97 66.56 90.48 94.39 99.21 99.56 99.71 99.84 99.94
05 0.00 1.39 5.01 33.78 49.99 84.22 90.24 98.59 99.21 99.52 99.75 99.89
06 0.00 0.23 0.82 9.43 15.50 49.95 59.59 89.94 92.56 94.57 96.10 97.99
07 0.00 0.09 0.36 5.59 9.67 40.49 50.20 86.85 90.15 92.68 94.61 97.23
08 0.00 0.01 0.02 0.83 1.39 10.13 13.30 49.96 55.84 60.70 65.04 75.56
09 0.00 0.00 0.01 0.45 0.77 7.37 9.84 44.60 50.01 55.25 59.94 71.52
10 0.00 0.00 0.00 0.27 0.44 5.37 7.34 39.43 44.80 49.98 54.89 67.35
11 0.00 0.00 0.00 0.16 0.25 3.89 5.36 35.04 40.11 45.32 50.04 63.30
12 0.00 0.00 0.00 0.06 0.09 2.04 2.72 24.44 28.19 32.33 36.57 49.94

Modified rule-of-six on
Target number 2 + n/2

1 2 3 4 5 6 7 8 9 10 11 12
01 50.13 93.19 98.31 99.65 99.90 99.98 99.99 99.99 100.00 100.00 100.00 100.00
02 6.70 50.21 74.99 92.86 96.68 99.19 99.61 99.93 99.97 99.98 99.99 99.99
03 1.66 24.60 49.86 83.00 91.21 97.92 99.00 99.83 99.91 99.96 99.98 99.99
04 0.37 7.15 16.99 49.86 63.45 87.22 91.53 98.35 98.97 99.32 99.61 99.79
05 0.08 3.39 8.80 36.54 49.97 81.06 86.93 97.44 98.43 98.97 99.28 99.71
06 0.02 0.82 2.09 12.84 19.09 49.85 58.41 87.46 90.26 92.38 94.14 96.67
07 0.01 0.36 1.01 8.40 13.05 41.86 49.70 83.97 87.78 90.23 92.48 95.64
08 0.00 0.07 0.12 1.69 2.48 12.57 15.81 50.32 54.78 59.62 63.47 73.53
09 0.00 0.02 0.07 1.05 1.59 9.86 12.51 45.17 49.97 54.43 58.83 69.44
10 0.00 0.00 0.03 0.62 1.02 7.58 9.55 40.71 45.31 50.10 54.26 65.60
11 0.00 0.00 0.01 0.38 0.67 5.79 7.34 36.47 41.12 45.36 49.88 62.03
12 0.00 0.00 0.00 0.17 0.29 3.31 4.27 26.75 30.14 34.39 37.94 49.97

Thoughts: Yes, it's pretty bad. 4-vs-5 is an 85% win for 5. Adding the modified rule-of-6 brings it down to 80%. That's still not great. With a target number of 4, it's a 66% win for 5 (and a 63% win if you add the modifed rule-of-6.)

It's much better when you get above 6, but if you often have people with skills of 6 doing opposed checks, you need to rethink your game balance IMHO.

The weird 2+n/2 target number didn't work out as well as I was hoping, combining the bad parts of both. 4-vs-5 ends up equivalent to target-number-of-4, 5-vs-6 is nasty, and 3-vs-4 is nasty (though not as bad as the original opposed number tests). Stupid stairsteps. If we were using d12s this would work much better, I think.

Hmmm . . . well, maybe I'll try that later.

Whether you prefer the standard exaggerated differences or the target-of-4 downplayed differences probably depends on what kind of a game you're running. It's fun to be overpowered and rampage, and it's also fun (in a different way) to defeat enemies who are incredibly powerful compared to you. It's probably more realistic to have it be more random.

I think personally I'll stick to target-of-4 except when I need to override it for dramatic purposes.

If anyone has other statistical breakdowns they'd like me to run, let me know. I find these things kind of fun.

Botch

Oct 29 2004, 04:12 PM

Average speed increases due to successes at athletic tests for quickness 1 through 10 would be nice.

ZorbaTHut

Oct 29 2004, 05:11 PM

That one's easy. I don't even need a simulation for it, it's just linear.

runningspeed = ( quickness + dice / 2 ) * 3

Since the target number is 4, which leaves you with precisely 50% chance.

With the modified rule-of-six it gets more interesting - you end up with an average of 1/2 + sum( 1/(6^n), n=1..inf ) successes. Which (I think) comes out to 1/2+1/5, or 7/10. (I didn't realize it was quite that many!)

So with the modified rule-of-six, you get

runningpseed = ( quickness + dice * 7/10 ) * 3

which can be easily calculated for any numbers you want. But basically, each quickness point will give you 3 m/s, and each Athletics dice will give you - on average - 1.5m/turn or 2.1m/turn depending on your rolling style.

Or, if you like modified-rule-of-six but don't like the added speed, you could push the target number up to 5 which would make it 1.6m/turn.

Note that dwarves run at 2/3 of this speed.

Deadeye

Oct 29 2004, 05:15 PM

QUOTE

It's much better when you get above 6, but if you often have people with skills of 6 doing opposed checks, you need to rethink your game balance IMHO.

Interesting opinion, but do you consider two (starting) characters with HtH or Edged Weapons, ect, that specialized to have a 6 or 7 in their respective skill unbalanced? What about a Face who is negotiating with an experienced Fence or charasmatic leader where each character might be very good at what they do? What I mean is--and I'm not being critical, I'm honestly curious to see where you and others stand on this--are there certain skills that strike you as being overpowered at 6 and others that aren't? I realize that not everything is going to be a resisted test, but for those that could be, where do you stand?

ZorbaTHut

Oct 29 2004, 05:33 PM

Coincidentally I'm actually reading the base melee rules right now, and they never bring up the subject of opposed tests

Although I did miswrite my original statement - I meant to write "if you often have people with skills far above 6 etc etc etc". So that's my bad. 6 itself isn't so bad, and IMHO 7 and 8 are okay if the character's experienced (or is a metahuman - a cyclops with a strength of 7 would be kinda pathetic.)

I personally, though, dislike overpowered games. In fact, house rules on my games is that you may only start with one stat at 6, one skill at 6, and two skills at 5. And I'm considering changing the first part to "one stat at 6 or two stats at 5, everything else 4 or below". (Before racial modifiers.)

(Which is kind of amusing when you consider that I also believe in "anything is possible, if you can roleplay it, pay for it, and are lucky enough" - I don't mind power levels reaching absolutely absurd levels as long as the players have worked for it.)

Back on subject - most of the opposed tests I've seen are ones where failure is really really important, like negotiation or holding a door closed, and in those cases the massive penalty you get for being one point too low (or bonus for one point too high) makes players prioritize those skills. Which means the GM has to raise the opponent's skills for them to not be pushovers, and it turns into a massive arms race where every skill point is an 80% bonus. My feeling is that players shouldn't *have* to raise their skills above 6 in order to be effective, and over-six skills and stats should really only be considered if the player wants to be a world-class expert, or kill them.