Math Behind Opposed Rolls

[Rurik]

Here is some hard math behind the use of opposed rolls.

The Players Update has two major changes that the formulas presented here represent.

1. Opposed Rolls no longer determine a winner on fail/fail checks.

2. Combat, excepting rare ties and criticals, becomes an opposed roll where the results are either a complete success or a complete failure. Armor Points and Minimum Damage Almost never happen (ties and criticals).

This Math DOES NOT include criticals in the calculations. Why is that? Criticals affect the odds very little and complicate the math a lot. If anyone really wants criticals added to these formulas I will be happy to do it. Suffice it to say that criticals will push the odds even further in the favor of the higher skill (as their critical range is greater) - though only by a small amount.

These odds favor the higher skill much more than the old method (where the lower roll wins if both fail) because the lower skill actually gets a larger portion of the Both Fail result (which never happened in the old method).

These results apply to both Combat and Spell Resistance under the new update.

The easiest way to do the math is to set up the following 5 values and then use the provided formulas:

a = Lower Skill
b = Higher Skill
c = 100 minus Lower Skill
d = 100 minus Higher Skill
e = Higher Skill minus Lower Skill

The chance of both participants making their roll and tieing is: a/100
The chance of both participants failing is: (cd)/100
The chance of the lower skill winning is: ((ad)+((aa-a)/2))/100
The chance of the higher skill winning is: ((bc)+(ea)+((aa-a)/2))/100

So, for example lets take skill 70 vs skill 50.

a = 50
b = 70
c = 50
d = 30
e = 20

Tie = 50/100 or .5%
Both Fail = (50x30)/100 or 15%
Lower Wins = ((50x30)+(((50x50)-50)/2))/100 or 27.25%
Higher Wins = ((70x50)+(20x50)+(((50x50)-50)/2))/100 or 57.25%

Some other result sets:

50 vs 30:
Tie = .3%
Both Fail = 35%
Lower Wins = 19.35%
Higher Wins = 45.35%

60 vs 40:
Tie = .4%
Both Fail = 24%
Lower Wins = 23.8%
Higher Wins = 51.8%

95 vs 75:
Tie = .75%
Both Fail = 1.25%
Lower Wins = 31.5%
Higher Wins = 66.5%

In all of the above cases the difference in skills is 20 points, but the higher skill has over twice the chance of winning the contest.

Some other examples:

75 vs 60:
Tie = .6%
Both Fail = 10%
Lower Wins = 32.7%
Higher Wins = 56.7%

80 vs 50:
Tie = .5%
Both Fail = 10%
Lower Wins = 22.25%
Higher Wins = 67.25%

Now what skill advantage means and how much a given skill advantage should affect the odds is subjective. My opinion on this matter is that a relatively small skill advantage leads to a big advantage in real odds of winning the contest - more than I am comfortable with myself. Others opinions will certainly vary. I've tried to give a broad and good representative sampling in my examples.

If anyone has any questions about the math involved or would like to see other comparisons please let me know.

 

[atgxtg]

Sounds like you want a "partial success" state.

 

[Kravell]

Rurik, thanks for doing the math. You've hurt my head.

I'm still using the old rules because I'm new enough that I haven't become annoyed with those charts yet. Could you tell me how the new math fixes the problems with the old tables and maybe what those problems were?

'Cause right now I just have two rolls, two results, check a chart. Seems easier to me than two rolls, two results, compare both, modify result of one, check chart. Also, I like AP and the thought of not using it bums me out a little.

Thanks.

 

[Deleriad]

One of the issues of course is also the effect of adding percentiles if you're over 100. E.g. I would guess that
120 vs 100
plays out significantly differently to 100 vs 80.

As you say, what a skill difference of +20 actually means is contentious.
For example 40 vs 20 is a battle between two incompetents
while 140 vs 120 is a battle between two masters.

The interaction between absolute skill values and relative skill values is quite slippery.

I do personally prefer a "partial success" setting because now that the lower skilled character is losing more often you want to mitigate that. Of course that means when the lower skilled character does win, odds are that the higher skilled will manage a partial success.

Anyway, thanks for the maths.

 

[Rurik]

Sounds like you want a "partial success" state.

I like the math better. I'm working on the formulas for that but might not get it up today. It basically takes part of the attacker success range and makes it a partial success for the defender. I should have constructed things Attacker vs. Defender rather than Higher Vs. Lower for consistency - it matters with partial successes. Doh.

 

[Rurik]

Rurik, thanks for doing the math. You've hurt my head.

I'm still using the old rules because I'm new enough that I haven't become annoyed with those charts yet. Could you tell me how the new math fixes the problems with the old tables and maybe what those problems were?

'Cause right now I just have two rolls, two results, check a chart. Seems easier to me than two rolls, two results, compare both, modify result of one, check chart. Also, I like AP and the thought of not using it bums me out a little.

Thanks.

Well, I'm not sold on the new chart. The above math is part of the reason why. I also agree it adds complexity while reducing the likely results (to an all hit or miss system).

Grab some asprin as there is more math coming when I get to it. I want to show comparative systems as well (mainly the 'old' opposed formula, which I only need to dig up from old posts, and partial success odds).

 

[Rurik]

One of the issues of course is also the effect of adding percentiles if you're over 100. E.g. I would guess that
120 vs 100
plays out significantly differently to 100 vs 80.

As you say, what a skill difference of +20 actually means is contentious.
For example 40 vs 20 is a battle between two incompetents
while 140 vs 120 is a battle between two masters.

The interaction between absolute skill values and relative skill values is quite slippery.

I do personally prefer a "partial success" setting because now that the lower skilled character is losing more often you want to mitigate that. Of course that means when the lower skilled character does win, odds are that the higher skilled will manage a partial success.

Anyway, thanks for the maths.

I think that 140 vs 120 is pretty much the same as 100 vs 80. Remember that you are still rolling 1d100 vs 1d100, so you have a grid of 10,000 possible results, even if the skills are 500 vs. 480. You are just shifting results with "add amount over 100" and "subtract amount over 100" adjustments. I will work it out though and see. It may take me a bit. I am no math wiz - I actually graph out the results on a grid then figure out the formulas behind them which takes me a bit.

 

[Kravell]

One confusion I have with all these changes is that my rulebook seems to be written differently (and in my opinion better) than the SRD. For instance, the SRD states: if both Both Characters Fail
Whoever rolled the lowest in their skill test wins the opposed test.

While my actual rulebook states: Both Characters Fail
Re-roll until one or both characters succeed their skill test.

The rulebook version makes more sense to me. It appears you continue to struggle if you fail your test.

 

[atgxtg]

Sounds like you want a "partial success" state.

I like the math better. I'm working on the formulas for that but might not get it up today. It basically takes part of the attacker success range and makes it a partial success for the defender. I should have constructed things Attacker vs. Defender rather than Higher Vs. Lower for consistency - it matters with partial successes. Doh.

It might be easier just to go with- if the defender7s roll is successful but the attacker won, it is a partial success for the defender.

That way you could keep the hit but get Parry APs, 2xAP stuff.

 

[atgxtg]

I think that 140 vs 120 is pretty much the same as 100 vs 80. Remember that you are still rolling 1d100 vs 1d100, so you have a grid of 10,000 possible results, even if the skills are 500 vs. 480. You are just shifting results with "add amount over 100" and "subtract amount over 100" adjustments. I will work it out though and see. It may take me a bit. I am no math wiz - I actually graph out the results on a grid then figure out the formulas behind them which takes me a bit.

No, it isn't quite the same. Going over 100% actually causes some funky things to happen. Since there is almost no failure range per say, the both fail chance drops to 0.01%,and things seem to slide ever more towards the higher skilled character. Also crticals start to become more of a factor and can't be discounted (the chance of someone critically goes from 17.2% for a 100 vs 80 confict to 23.9% in a 140 vs 120 conflict).

I think the final odds are a little over 60/40 favoring the 140%er. The increasing crticals are the biggest difference, and important since the chance of someone crticalliny is higher than the difference in success chance. In other words, the crticals will probably be the deciding factor in a fight between people of great skill.

 

[Rurik]

One confusion I have with all these changes is that my rulebook seems to be written differently (and in my opinion better) than the SRD. For instance, the SRD states: if both Both Characters Fail
Whoever rolled the lowest in their skill test wins the opposed test.

While my actual rulebook states: Both Characters Fail
Re-roll until one or both characters succeed their skill test.

The rulebook version makes more sense to me. It appears you continue to struggle if you fail your test.

Well the First Printings in matched the SRD. You have a later rulebook that probably has a slightly different combat table than the first printings and the first SRD as well.

The 'Old' opposed rolls (where lowest roll wins if both fail) was designed to determine a winner in one roll (an opposed roll between skills of say 20-25% can go on for some times otherwise).

The odds are better using the old system with lower skills - 60 vs. 40 actually works out to about 60 vs 40 in odds as well. The reason for this is that a portion of the 'both players fail' chunk of the odds goes to the lower skill, kind of equalising the odds. As the skills go higher there is a smaller "both players fail" range and the odds start to look like the odds I posted above for 95 vs. 75.

Edit: Here is a link to a BlueJay's MRQ Calculator for the odds of 'old' opposed rolls. Be sure to uncheck the Test Combat box.

 

[Rurik]

That way you could keep the hit but get Parry APs, 2xAP stuff.

That is pretty much what some of us were working on. There is a thread from a few weeks back and a version on the wiki that includes partials successes and removes the table completely. I've revised mine (and may again based on some of the recent threads) and will post again.

I think that 140 vs 120 is pretty much the same as 100 vs 80. Remember that you are still rolling 1d100 vs 1d100, so you have a grid of 10,000 possible results, even if the skills are 500 vs. 480. You are just shifting results with "add amount over 100" and "subtract amount over 100" adjustments. I will work it out though and see. It may take me a bit. I am no math wiz - I actually graph out the results on a grid then figure out the formulas behind them which takes me a bit.

No, it isn't quite the same. Going over 100% actually causes some funky things to happen. Since there is almost no failure range per say, the both fail chance drops to 0.01%,and things seem to slide ever more towards the higher skilled character. Also crticals start to become more of a factor and can't be discounted (the chance of someone critically goes from 17.2% for a 100 vs 80 confict to 23.9% in a 140 vs 120 conflict).

I think the final odds are a little over 60/40 favoring the 140%er. The increasing crticals are the biggest difference, and important since the chance of someone crticalliny is higher than the difference in success chance. In other words, the crticals will probably be the deciding factor in a fight between people of great skill.

Remember you are still rolling 1d100 vs 1d100 - there are only 10000 possible results, even if the skills involved are well over 100. And a 96+ is still an automatic failure. I'll figure it out for sure, but a 140 vs a 120 may well in fact work out to be the same as a 95 to 75 contest. But I'll post on this again when I know for sure.

 

[Halfbat]

Thanks for this Rurik.

I calculated the results using criticals and it does seem to throw things even _more_ in favour of the higher skill, and very quickly. For me, as you are aware, it means that determining the "ECL"* of a combat is considerably more difficult as even a small difference weights results significantly.

I agree with atgxtg - the partial success system works well.

* Sorry to use the d20 term, but i think everyone knows what I mean.

 

[Space Coyote]

However, the (few) combats I've run all ended up pretty quickly. This means that the number of rolls wasn't nowhere near a statistical significant one (that is, the odds didn't even out). What I found was that given the large range of the d00 distribution and the all or nothing nature of the results, it all felt a little bit too random.

What would be more interesting, in my opinion, is to estimate (maybe via a Montecarlo or something alike) after how many opposed rolls the higher skill really dominates.

Cheers, Alex.

 

[Mongoose Pete]

For me, as you are aware, it means that determining the "ECL"* of a combat is considerably more difficult as even a small difference weights results significantly.

Personally I find that using Opposed Rolls actually makes it easier for me to predict who the winner should be.

From mid skill levels upwards an advantage of 20% gives the superior skill between a 2/3 and 3/4 chance of winning. So if I want to give players a contest with a decent chance of them winning, then this is a good skill level difference. If I wanted to give them a hard time then I'd match the skill values. And an impossible task/fight then give the opponents a 10-20% higher skill.

I think it gives skill percentages much more significance, without needing to boost skills way into the 100+ ranges to gain a decisive effect.

 

[Halfbat]

What I found was that given the large range of the d00 distribution and the all or nothing nature of the results, it all felt a little bit too random.

And this makes the combat less predictable, too. It's the combination that really counts.

 

[atgxtg]

That way you could keep the hit but get Parry APs, 2xAP stuff.

That is pretty much what some of us were working on. There is a thread from a few weeks back and a version on the wiki that includes partials successes and removes the table completely. I've revised mine (and may again based on some of the recent threads) and will post again.

YOu could even do a degree of success sort of thing. For instance, if the parial success for the parry means a block for APs, you could just give then parry an extra +1 AP per 10% difference. OR 2xAP for a 50% difference, 3x for a 75% and so on.

Remember you are still rolling 1d100 vs 1d100 - there are only 10000 possible results, even if the skills involved are well over 100. And a 96+ is still an automatic failure. I'll figure it out for sure, but a 140 vs a 120 may well in fact work out to be the same as a 95 to 75 contest. But I'll post on this again when I know for sure.

Yeah, but as the skills increase more and more of the results get "eaten" up by the criticals. For example, a 540% vs. 520% is still a 20% difference in skills, but there is over a 75% chance that somone will critical.

Yeah, I went to an unrealsticly high extreme, but to point out how the odds do shift as you go over 100%. A 140 vs 120 fight favors the 140 character more than, say a 40 vs 20 fight. For higher skilled guy gets a few more perceililes for his win chance, plus the increased critical chance will make those wins more signficant.

 

[Mongoose Pete]

Yeah, but as the skills increase more and more of the results get "eaten" up by the criticals. For example, a 540% vs. 520% is still a 20% difference in skills, but there is over a 75% chance that somone will critical.

That is precisely why I prefer the 'Take the best skill, subtract 100 and use the remainder as a penalty to the skill of everyone participating in the contest'.

It doesn't warp/break at higher percentages, can be summed up in a single sentence, and doesn't require fiddling about with exceptions (i.e. only add when a normal success, you fumble on the original roll - not the modified one, etc).

 

[atgxtg]

Yeah, but as the skills increase more and more of the results get "eaten" up by the criticals. For example, a 540% vs. 520% is still a 20% difference in skills, but there is over a 75% chance that somone will critical.

That is precisely why I prefer the 'Take the best skill, subtract 100 and use the remainder as a penalty to the skill of everyone participating in the contest'.

It doesn't warp/break at higher percentages, can be summed up in a single sentence, and doesn't require fiddling about with exceptions (i.e. only add when a normal success, you fumble on the original roll - not the modified one, etc).

Oh, the RQ2 method
Fine with me, but most people don't seem to want any old RQ rules.




Another way to handle partial successes would be to eliminate separate attack and defense rolls. Instead each combatant rolls, and the winner gets to roll damage and the loser gets parry APs for a partial success.



The thing is, D100 is good for a lot of things, especially "degrees of success", but they are lousy for opposed rolls. For opposed rolls it would work out much easier to used smaller dice. Especially if you want to cut down on the math.

It might make more sense to take the amount over 100 and just roll it separately and take the better of the two results. SO a guy with 140 could roll a 100% skill and a 40$ skill and pick the best roll.