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!

Inspirations from Castle Menzies

Last week we visited Castle Menizes - a sixteenth century Scottish Castle located near Aberfeldy on the edge of the south-eastern highlands.

On the drive back to where we staying in Killin I was struck by how many game-able ideas there were in just one castle:

1. A strong room within the thick walls of the keep, only accessible by a stone trap door in the floor above.

2. A lifelike model of a ship in a glass case (magic?)

3. A cellar full of barrels (there's one in nearly every adventure but it's great to wander into one)

4. A table made from barrels.

5. Doors reinforced with iron gateways

6. Spiral Staircases with wall sconces (once again, it's just great to wander past something like this)

7. Each tower has one more floor than in the main building, so each room is joined to the main building by confusing stairs - some up, some down.

8. A secret compartment hidden in the thick walls behind secret door in the wooden panelling.

9. A room in the roof-space accessible only through a narrow opening high up in the wall with no ladder.

10. One wing of the castle is uninhabitable with collapsed roof and rotten floors

11. Due to later extensions, some rooms are now inaccessible, stairways lead to blank walls, and fireplaces are boarded up.

This vein of ideas lead me to ponder the rest of the holiday:

12. A wooden house built on pillars over the lake. The inhabitants avoid the woodland around as it is inhabited by bears and wolves, and they travel up and down the lake by boat, and trading. For some inexplicable reason, no one eats the fish in the lake. (Truly - in the archaeological digs no fish remains have been found)

13. Strange unexplained changes in the weather in moments from blazing hot sunshine to near darkness, freezing hail, then torrential rain.

14. The level of the lake is unusually low, revealing remains of a ruin poking out of the water. (Actually this holiday it was a sandy beach, but previously it's been ruins).

15. A small island in a lake accessible with care from the shoreline across a narrow spit of land. It turns out to be a treacherously boggy jungle marshland. The island shows signs of ancient civilisation. The smoldering remains of a fire indicate someone has been here only recently but there is no sign of them.

16. A well built ancient paved way leads over a pass from one valley to the next. Its purpose is unclear, and it is now used only by the brave. Locals try to prevent you from treading its path with dire warnings. In reality it was made by an ancient people, now seldom seen, to transport heavy machinery when creating underground water pipelines.

17. The river is crossed by a rickety old wooden bridge with broken planks - the only alternative is hoping from one rock to another across the raging torrent.

With this mindset, even mundane happenings on holiday turned into game-able ideas:

17. A magical vanishing box: when items are placed into it and the lid is closed they vanish and reappear in a secret compartment elsewhere. (I placed my sunglasses in the compartment below the steering wheel in the car and they vanished. I thought I was mistaken so bought new sunglasses. The second pair also vanished... I found them both down a hole at the top at the back!)

18. The painting looks normal at first glance - but then you notice that one of the people appears twice. You recognise their face as someone you know, and they are not a twin...

19. The door is held open for you by a small fox. It stares at you, and there is clearly deep intelligence behind the eyes, but it makes not a sound.

Charging Elephants

There are a few apparently unrelated areas of Explore I've long wanted to address - grappling, animal attacks, and spell area attack saves - but I've come to understand how they're strongly connected through a rather circuitous path. Starting off considering climbing, you'll see how answering that comes round to the simpler issue of charging elephants, and how answering that has wider applications.

Climbing Power

Back in January I half finished a blog post on climbing - it isn't strength that's important but your power-to-weight ratio - but in Explore your size is derived from your strength and constitution. This lead to some odd consequences (your con gave you a negative bonus on climbing as high con implied being well built and in climbing being slimmer is better), so I determined that it should be the reverse: your height and build should influence your strength and constitution. This changes around how stats work during character creation, but not how the game works - whichever system I decide upon, when you need a climbing stat bonus you sum it up and write it on the character sheet and it doesn't alter.

This meant I started revising what the stats were. Instead of getting up to +3 from Strength, you get up to a +1 from each of Height, Build, and Muscle. Con similarly gets a bonus of up to +3 from Height, Build, and Toughness. (Note that this means that Str and Con have the same range as before and still cannot differ by more than 2). It is then only the Muscle that gives you a bonus on Climbing, and that's your power-to-weight bonus. The bonuses also tie into the sizing system. A height of 0 means you're between 5'6'' and 5'11''.  As per my revised size categories, height of +1 would put you in the 6' to 6'5'' range, and a height of -1 would put you in the 5' to 5'5'' range. If you also had +1 build that would push you up two weight categories from say 160lb to 192lb (assuming the character is not overweight!) or -1 build and -1 height would knock you down to 96lb.

Do you Dodge or Parry a Giant?

From this I wondered what other advantages / disadvantages you might get from being big / small. To work this through I considered large differences - with a human versus a giant is the giant slower to swing and thus it's harder to hit the human? Or is the sword a big attack which is tricky to dodge and impossible to parry? If you give bonuses how would this affect giant versus giant combat? Should it be the same as two humans, or would it be qualitatively different? I started to think about the different situations and what the desired outcome might be, and to temporarily stop considering mechanics.

When examining a big problem with lots of variables it's easier to break it down and consider one variable at a time, so I concentrated on dodging. Looking for extreme situations where dodging was clearly the only option, I considered an Elephant charging.

The Charging Elephant

When an Elephant charges you, you're not going to be able to outrun it.

I've already assumed that size doesn't affect your speed - so by default a huge animal like an Elephant would not get any bonuses or penalties due to its size - but you can get many bonuses due to other factors than size. In fact according to this speed of animals resource, an Elephant's top speed is 24.9mph . Usain Bolt's top speed is about 30mph, so Elephants are indeed very close in speed to a well-trained, naturally fast human, so aren't getting much in the way of bonuses. However, even if you are that fast, an Elephant will tire much less quickly than you - you'll soon tire and slow down and be caught and be squashed.

Alternatively you could try and face it down, like this man did (who doesn't even seem that fazed by the experience), but for today I'm going to assume you wait until the last moment and then try and dive out of the way. I observed that as the Elephant is 7' wide you have to move a fair amount to dodge it.

Alternatively consider a bee attacking you - it is hard to hit the bee because it is small, but it is hard to dodge because you are large.

Hence a charge attack is something along the lines of a bonus for speed and agility and skill plus a bonus for the size of the larger individual (attacker or defender), versus a dodge based upon speed and agility and skill.

Dodging Fireballs

This lead to a realisation that the Elephant misses because you have to move out of the way! So when grappling a giant, it will win easily in a contest of strength, but it has to catch you first - and avoiding it means moving around (not just staying stationary and rolling a dice and saying the giant missed).

If that's the case, then the reason why the fireball missed is because you jumped out of the blast zone. (I'm thinking 5' radius blast here incidentally, not D&D's mass effect fireballs from Chainmail and war-gaming). Similarly for a rock from a trebuchet, or thrown by a giant. Treating dodging an arrow in the same fashion would mean that you got a bonus for the larger of the arrow and the target, so you'd get a bonus for the size of the target, which is exactly what I already have.

In conclusion, I don't know whether I'll ever differentiate mechanically between "I dodge" and "I parry", but I can see combat effects becoming far more dynamic and size bonuses being applied more consistently. I can see different combat tactics being effective against different size adversaries. Giants are slow but deadly and are best taken on with missile attacks, smaller creatures have to be taken on hand-hand, but area effect attacks become more useful.