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!
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