Sunday, 11 June 2017

Fairness of Dice Modifiers and Advantage / Disadvantage

In discussions on Delta's post on Advantage and Disadvantage the question arose about the merits of percentage increases versus percentage point increases in probabilities and which is the more important consideration. I'm going to compare a few systems to see the effect of modifiers on the results and explain in the process why I place such importance on the proportional increase of probabilities. At the end I'll cover Advantage / Disadvantage and I'm surprised by the findings.

Comparing Linear and Bell Curves

Firstly I'll compare a linear system versus a bell-curve system and analyse what the differences are - so I'll choose D% for the linear system, and 3D6 for the bell-curve system.

If you need 51+ on the D% system, or 11+ on the 3D6 system, then in both cases you have a precisely 50% chance on success. It makes no difference to the outcome which system you used. In fact, for any target number you need on the 3D6 there is an almost exact equivalent target number on the D% system. Hence in this first analysis it makes no difference to the outcome - the only difference is:

The chance of making a target number is clear in a linear system, and obscure in a bell-curve system.

In this first respect the bell-curve system has no advantage. However, people maintain that a bell-curve system is better because it more accurately models the real world, where things typically have normal distributions. In many things, such as the heights of humans, this is obviously true. But we're not actually considering the distribution of 100 arrows shot at a target, we're considering what proportion of them hit the target. Hence what matters is how we determine the target number for hitting the target. In both systems (linear and bell-curve) it is traditional to have a standard target number which is modified by bonuses or penalties for skill level and conditions. Hence the question becomes, when we give a bonus in the two different systems, is there a different effect?

Most obviously:

Any bonus/penalty in a linear system has a clear effect on the outcome in any given situation, it can be obscure in a bell-curve system.

That is, if you need 41+ on a D% to hit something, and due to a +10 bonus it is now 31+ you can clearly see that you had a 60% chance of hitting, and now you have a 70% chance of hitting. In comparison say you needed 13+ on 3D6, you get a +1 bonus and you now only need a 12+. That's a 25.9% chance of success changed into a 37.5% chance of success - hardly clear.

What is a fair bonus / penalty?

The next observation is that bonuses in the linear system always change the chance of success by a fixed number of percentage points. Is this an advantage in itself? The main consequence is the clarity of the system, which we have already covered - but I don't see any other inherent advantage to it. That may seem to be a contrary position, so I'll explain myself.

For example, in a game mixing skill and luck I offer both players a 5% chance to win the game outright before they play. I roll a D20 and on a 1 player A wins outright, on a 20 player B wins outright, else they play the game. In this case they'd both get the same chance of winning from the die roll. But if one player is great at the game, the other a novice, then the great player will not accept the offer as it reduces his chances of winning. It is an equal 5% for both sides, but that statement does not make it a fair proposition.

Just as the bell-curve's normal distribution doesn't inherently make it better, to see advantages / disadvantages of the systems we should examine what effect they have on the results in the game (and I see no other way of determining it).

Let us consider two opponents in combat. Some situational modifier comes into play which either gives both sides an advantage, or both sides a disadvantage. The opponents would consider it "fair" if it affected both sides equally. What do I mean by that? Do I mean it increases their chances of hitting by the same percentage points, or do I mean it improves them proportionally the same amount? What I mean is that the effect can be considered "fair" if it has no effect on the outcome of the contest. A "fair" effect is one which both sides could agree to before the contest, an unfair effect is one which would give one side an unfair advantage.

Given this definition, a fair effect is one which does not alter the ratio of the average damage per round for the two combatants. That is, an effect which doubles the average damage caused by one combatant should also double the average damage caused by the other combatant for it to be considered fair. This is the same as saying a fair effect on the chance to hit is one which does not alter the ratio of the chances to hit for the two combatants.

Are modifiers for D% or 3D6 fair?

Firstly lets consider 3D6:
Target Needed
% Chance Hit
% Chance with +1 bonus
Multiplier on average damage / rnd
% Chance with -1 penalty
Multiplier on average damage / rnd
3
100.0%
100.0%
1.00
99.5%
1.00
4
99.5%
100.0%
1.00
98.1%
0.99
5
98.1%
99.5%
1.01
95.4%
0.97
6
95.4%
98.1%
1.03
90.7%
0.95
7
90.7%
95.4%
1.05
83.8%
0.92
8
83.8%
90.7%
1.08
74.1%
0.88
9
74.1%
83.8%
1.13
62.5%
0.84
10
62.5%
74.1%
1.19
50.0%
0.80
11
50.0%
62.5%
1.25
37.5%
0.75
12
37.5%
50.0%
1.33
25.9%
0.69
13
25.9%
37.5%
1.45
16.2%
0.63
14
16.2%
25.9%
1.60
9.3%
0.57
15
9.3%
16.2%
1.75
4.6%
0.50
16
4.6%
9.3%
2.00
1.9%
0.40
17
1.9%
4.6%
2.50
0.5%
0.25
18
0.5%
1.9%
4.00
0.0%
0.00

So +1/-1 can either have little or no effect up to quadrupling / quartering the damage, and at the extremes the penalty means a hit becomes impossible (without special natural 18 = a hit rules).

In contrast we'll consider D% (with the targets chosen to match the previous table as closely as possible):
Target Needed
% Chance Hit
% Chance with +5 bonus
Multiplier on average damage / rnd
% Chance with -5 penalty
Multiplier on average damage / rnd
1
100.0%
100.0%
1.00
95.0%
0.95
2
99.0%
100.0%
1.01
94.0%
0.95
3
98.0%
100.0%
1.02
93.0%
0.95
6
95.0%
100.0%
1.05
90.0%
0.95
10
91.0%
96.0%
1.05
86.0%
0.95
17
84.0%
89.0%
1.06
79.0%
0.94
27
74.0%
79.0%
1.07
69.0%
0.93
39
62.0%
67.0%
1.08
57.0%
0.92
51
50.0%
55.0%
1.10
45.0%
0.90
64
37.0%
42.0%
1.14
32.0%
0.86
75
26.0%
31.0%
1.19
21.0%
0.81
85
16.0%
21.0%
1.31
11.0%
0.69
92
9.0%
14.0%
1.56
4.0%
0.44
96
5.0%
10.0%
2.00
0.0%
0.00
99
2.0%
7.0%
3.50
0.0%
0.00
100
1.0%
6.0%
6.00
0.0%
0.00

In both cases the values are reasonably consistent in the top half of the table, but the bottom half of the table is anomalous - towards the bottom end bonuses and penalties can have a disproportionate effect. The 3D6 system is not a clear winner with this measure of fairness. Thus we have seen:

Bonuses / penalties in a bell-curve system are not necessarily much "fairer" than those in a linear system.

We could choose a system on purpose so the bonuses / penalties are "fair", but clearly any closed system is going to have anomalies at the ends of the distribution where a bonus/penalty makes a result a certainty/impossibility OR ceases to have an effect. Hence:

Only an open-ended system can have "fair" bonuses/penalties throughout the range.

That doesn't mean all open-ended systems are "fair" - in fact many of them are quite wacky. (There can also be different non-modifier based systems that are "fair"). What would an open-ended fair system look like?

A Fair Open Dice System

The fairest system would be one where +1/-1 always modified your chance by a fixed proportion. You can do this easily, however there are other disadvantages of that as I always like to include a chance of failure. As a compromise I chose one where a +3 bonus halved your chance of failure (for failure<50%), or doubled your chance of success (for failure>50%). I approximated this with my open-dice system. (Note this is a bell-curve, but not a normal distribution). Here's the fairness test repeated for that system:

Target Needed
% Chance Hit
% Chance with +1 bonus
Multiplier on average damage / rnd
% Chance with -1 penalty
Multiplier on average damage / rnd
2
100%
100.0%
1.00
99.0%
0.99
3
99.0%
100%
1.01
97.0%
0.98
4
97.0%
99%
1.02
93.9%
0.97
5
94%
97%
1.03
89.8%
0.96
6
90%
94%
1.05
84.6%
0.94
7
85%
90%
1.06
78.3%
0.92
8
78%
85%
1.08
70.8%
0.90
9
71%
78%
1.11
62.3%
0.88
10
62%
71%
1.14
52.5%
0.84
11
52%
62%
1.19
43.5%
0.83
12
44%
52%
1.20
35.5%
0.82
13
35%
44%
1.23
28.2%
0.79
14
28%
35%
1.26
21.8%
0.77
15
22%
28%
1.29
16.3%
0.75
16
16%
22%
1.33
12.1%
0.74
17
12%
16%
1.35
8.6%
0.71
18
9%
12%
1.40
6.4%
0.74
19
6%
9%
1.35
5.2%
0.81
20
5%
6%
1.23
4.1%
0.80
21
4%
5%
1.25
3.3%
0.79
22
3.3%
4%
1.27
2.6%
0.79
23
2.6%
3%
1.27
2.0%
0.78
24
2.0%
3%
1.28
1.5%
0.76
25
1.5%
2%
1.32
1.2%
0.79
26
1.2%
2%
1.27
0.9%
0.77
27
0.9%
1%
1.29
0.7%
0.78
28
0.7%
1%
1.28
0.6%
0.85
29
0.6%
1%
1.17
0.5%
0.77
30
0.5%
1%
1.29
0.4%
0.79

Thus this system isn't completely "fair" but is a reasonable compromise. A +1 bonus can at most make you 40% better (and is generally between 20% and 40% better) and a -1 penalty can at most make you 29% worse (generally at least 20% worse). You could also come up with a different resolution system that better approximates my stated goal distribution.

Is this fairness an advantage that outweighs the loss of clarity of the linear system? That's entirely subjective - but there are other advantages of this approach.

For example I've previously noted that if you double the distance to a target, then it presents one quarter the size target to the archer, hence it is reasonable beyond a certain range for 2* distance to equate to 1/4 the probability of hitting or a -6 modifier.

Another question is whether you want the modifiers to be fair or not!

Fairness of Advantage / Disadvantage Mechanic

Now modifiers are not the only way of giving people bonuses - one currently popular method is the Advantage / Disadvantage system of 5th edition. How "fair" is this?

Target Needed
% Chance Hit
% Chance with advantage
Multiplier on average damage / rnd
% Chance with disadvantage
Multiplier on average damage / rnd
1
100.0%
100.0%
1.00
100.0%
1.00
2
99.0%
100.0%
1.01
98.0%
0.99
3
98.0%
100.0%
1.02
96.0%
0.98
6
95.0%
99.8%
1.05
90.3%
0.95
10
91.0%
99.2%
1.09
82.8%
0.91
17
84.0%
97.4%
1.16
70.6%
0.84
27
74.0%
93.2%
1.26
54.8%
0.74
39
62.0%
85.6%
1.38
38.4%
0.62
51
50.0%
75.0%
1.50
25.0%
0.50
64
37.0%
60.3%
1.63
13.7%
0.37
75
26.0%
45.2%
1.74
6.8%
0.26
85
16.0%
29.4%
1.84
2.6%
0.16
92
9.0%
17.2%
1.91
0.8%
0.09
96
5.0%
9.8%
1.95
0.3%
0.05
99
2.0%
4.0%
1.98
0.0%
0.02
100
1.0%
2.0%
1.99
0.0%
0.01

We can see that at the bottom end the advantage system roughly doubles the chance of success. As you get to the top the effect switches to halving your chance of failure, but it rapidly reduces that towards zero. Apart from the top end it equates quite closely to a +3 in my open dice system, and is similarly fair. Hence, rather surprisingly, neither side in a combat would have much to complain at if both sides got advantage on all rolls - those with a low chance to hit might have doubled their chance to hit, but those with a high chance to hit would have almost eliminated their chance of missing.

In contrast the disadvantage system roughly doubles the chance of failure in the top half, but in the bottom half the chance of success dwindles almost to nothing. So although there is no cliff to fall off at the bottom (it never reaches zero) it is far from "fair". Disadvantage is a slight issue for people who are mostly successful, but is dire for people that are unlikely to succeed.

I think it's quite surprising that advantage and disadvantage have such different effects.

To clarify this: as a simple example, is it better for you to be given advantage - or your opponent to be given disadvantage? Consider A hits 1/4 of the time, B hits 3/4 of the time:

Combatant
Standard chance to hit
With Advantage
With Disadvantage
A
1/4
7/16
1/16
B
3/4
15/16
9/16

Note there's not much difference between the two choices for A. Initially B is hitting 3 times as often, and their choice is to change that to 1.71 times as often (A gets adv) or 2.25 times as often (B gets disadv). It's slightly better for them to get advantage.

In contrast for A they are initially hitting B 3 times as often and their choice is to change that to 3.75 times as often (B gets adv), or to 12 times as often (A gets disadv).

It's not intuitive to me that one choice is so much better than the other for B. In fact it's always better to place advantage/disadvantage on the person whose least likely to hit - A puts advantage on themselves, B puts disadvantage on A.

Of course, this may be the effect that you're looking for!

12 comments:

  1. This is more the sort of thing I was looking for on Delta's post. I personally don't think that fair is necessarily what we're after.

    Example: Let's say two warriors face off in combat, with a slight skill difference between them. One warrior has the odds in their favour, but it's not a certain win. Now a high wind appears, whipping dust into the faces of both our warriors. Is the wind fair, and the outcome the same as before? Or does it more harshly affect the lower skill warrior, making the fight a more certain win for their better?

    By the way, Delta's post on Normalising Resolutions makes an interesting read for why things like doubling the distance does not just 1/4 the probability of hitting. In that he came up with a better resolution mechanic, but ultimately rejected it as overly complex in play. With that approach, I can see why he favours +/-2 over the Dis/Adv mechanic.

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    Replies
    1. That’s a very good point - complete fairness isn’t what we’re after. Firstly it is impossible as you cannot increase your chance of success above 100%, so the fairest distribution (pure logarithmic decay) isn’t fair at the top end anyway. I went further than that by tailing off towards 100% success so you can never get there, and I rather glossed over why I do that in the post (as I ran out of time and the post was over complex as it was). Thus in my open dice system, -3 (for the wind) will halve the poor warrior’s chance to hit but double the great warrior’s chance to miss. Thus the great warrior drops from 90% to 80%, and the poor warrior drops from 40% to 20%. So it does exactly what you suggest – the poor warrior is put at a far greater disadvantage. My system is truly "fair" only really when you’re talking about <50% chance of success.

      In contrast "disadvantage" knocks the 90% to 81%, and the 40% to 16% - so quite similar so far – but it knocks a 20% chance down to 4%, and 10% down to 1%. So this is also like what you suggest but far more severe for the poor warrior. Personally I think that’s too unfair – but it may, of course, be your desired effect.

      I don’t see what you’re saying in the Normalising Resolutions post (though I only skimmed it) – he covered archery accuracy in a subsequent post where he used bivariate normal distributions. I modelled that and at long distances found that it tends towards a logarithmic fall I use, which is why I proposed it!

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    2. Yes you're right, I cited the wrong post, but it looks like you found the right one!

      With regards to fairness, I think unfair is what I'm after, but I have no idea how unfair. Also I'm not certain if the same reasoning applies to advantages (e.g. who does the laser-guided gyro-stabilised crossbow help more?)

      Incidentally, I've read through your posts on your Explore system. As a mathematician (moving between Glasgow and Edinburgh), I really appreciate the logarithmic approach you've taken. It's a great way to handle huge differences in scale (I found it unfortunate that Gurps 4e chose to fix 3e's linear scale with a quadratic one -- so near yet so far).

      I've only used the open dice mechanic once before, when playing Maelstrom, and it definitely worked for rolling damage (pass/fail is d%), but I'm not sure if it wouldn't be a bit heavy for all resolutions. Mass combat might quickly get bogged down, whereas rolling a handful of d20s is quick (but then the dis/ad mechanic becomes awkward).

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    3. How fair/unfair it should be in your game is the tricky judgement call - the proof is in the play, but it can take a while for these issues to become clear.

      Thanks for the comments about the Logarthmic approach - I think it takes a bit of effort getting the mechanics right but it really pays for itself.

      On average you roll 2.22 dice each time, which with the adding up is obviously slower than rolling a single dice. I think this is mitigated by the lack of hitpoints, which means that combats at high level run quicker, and these cancel out. I've run reasonably large combats quite quickly - on Sunday one of the party was following a group of 12 Ogres and watched them get into a fight with 3 Giants. I could have resolved it via DM fiat, but I gave them the option of letting the combat play out with the players controlling the Giants. They went with that, and it didn't take up much time, and it was quite satisfying for him to go back and report what *really* happened. When I ran Rolemaster (with open dice) it was slow - but that was the double digit arithmetic and the table look up and the tracking of wounds, not the open dice - so you definitely have to keep an eye on it.

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  2. Thanks for the work on this thoughtful post! I'm not entirely sure I can personally sign off on the definition of "fair" being used here. Nor do I think this was the motivation for 5E advantage/disadvantage (frankly it just looks like lazy, sloppy design to me). Interesting to see your conclusion that even if that was the motivation, it wouldn't solve the problem.

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    Replies
    1. Thanks. I'm looking forward to the new release "Chess 2nd edition" where I hear there are some really cool *fun* new mechanics ;-)

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  3. Completely aside to the point, there is another method which attempts to reach some kind of fairness by tying the effect of a modifier to the skill to which the modifier is applied, such as in BTRC's games TimeLords, SpaceTime, and WarpWorld. In this system, the effect of a modifier is taken as a percentage of the skill, so that a +2 (+10%) modifier would give a +2 value to a skill of 20, or a +1 value to a skill of 10 - or for that matter a +3 value to a skill of 30, with intermediate values rounded. This becomes cumbersome enough in play that those games center on a modifier chart (which is also used as a shortcut for figuring out percentage values, as when damage is done as a percentage of total "body points"), and is probably why those games have been abandoned by that company in favor of progressively simpler mechanics (first CORPS, now EABA).

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    1. Marvel Super Heroes (FASERIP) also had a table, which to me seemed the perfect opportunity to have a custom distribution - but oddly it was just almost exactly the same as d20 resolution, which seemed to waste the point of having a look up table. I tried replacing this with a custom lookup table, similar to the one you're talking about, but found that even a single table was too cumbersome in play.

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    2. In the right circumstances, a table (we called them "black box systems") can be the best design choice. FASERIP is, of course, the most successful at this, being so successful that it took over pretty much all of TSR's design philosophy for a time. A few other examples where it worked to some degree include Fantasy Wargaming, James Bond 007, and perhaps RuneQuest/Basic Roleplaying, if one considers the "Special", "Critical", and "Fumble" results as a sort of proto-table. None of those really runs with the "BTRC System" (for lack of a better name) method of varying the effect of a modifier by the value it is modifying, though.

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    3. I've read bits of Zebulon's Guide (the awful update to Star Frontiers) which had a colour chart like FASERIP, but looking at it again I'm surprised to discover that it's not like FASERIP but almost exactly an exponential system (i.e. what I described as the "fairest" type of system). I've looked at James Bond and found that's also an entirely different system where success = < skill * "ease factor", with benefits for better than 10%, 20% or 50% of the target. Tables do allow you to have whatever distribution you want, and they're taking advantage of that!

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    4. Yes, indeed. What I've been trying to get at, though, is that modifiers in the BTRC System were relative to the skill level. So, for example, if you have a skill of 10 and a modifier of +4, then you would roll hoping to achieve 12 or less. That is, skill 10 + ([+4 modifier × 5% =] 20% of skill 10). The modifier chart in this case existed to facilitate the math, rather than being a table of results as in these other examples.

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    5. Ah, I get your point. Amusingly the James Bond example also has a table of results of a mathematical operation - it has a multiplication table at the back. If your skill is 17 and the ease level is 3 then you need to get <= 3*17 = err... err.. I'll just look it up on the multiplication table!

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