Friday, 16 June 2017

Struggling with Attributes

Moving Away From The Big 6*3d6

In Explore I’ve gone through several iterations of attributes, moving further and further away from the traditional big six with 3d6.

Originally I started with the big six but wanted more balanced attributes without resorting to the usual solutions of escalating scores / point buy acrobatics / cookie-cutter stat arrays – I used this method to randomly generate six values 3-18 via 3d6 which always add up to 63 whilst keeping the same distribution as 3D6.

The next iteration was to replace the mental attributes (INT, WIS and CHR) with Memory and Intuition, so as to remove Logic and Personality attributes to make those issues purely down to roleplaying – and to split DEX into Agility and Reflexes.

Next I wanted some correlation between related stats. That’s tricky to resolve, but dropping the 3-18 and instead working only with the bonuses -3 to +3 made it possible. (It was a shame to lose the balanced 3D6 system though). I arranged the attributes in pairs – Strength/Constitution, Agility/Reflexes, Memory/Intuition – where the total sum of bonuses is 0, and each pair differs by no more than 2. Thus if you have STR 1 you cannot have a CON of -2 or -3. This was generated via rolling two D% and using this lookup table.

Using this system I then derived Height and Weight from the stats, which also gave me the consistency between the physical character and their stats.

This is the version I’ve used for the last two years. It works reasonably well, and creates varied, balanced and consistent characters. However when considering Climbing, which is about power-to-weight ratio, I encountered difficulties - when I put the numbers together something wasn't making sense. At this point I realised that if in the real world Height and Weight influence Strength, it would be simpler if it worked that way round in the game.

Taking a step back I bit the bullet and stopped blogging the revised rules, and decided to allow myself to follow any train of thought without worrying about the consequences, to see what I could come up with. Not for the first time I was glad I don't have a Kickstarter to complete!

Sacred Cows

Freed up to consider anything this quickly lead to the germination of a new system, but finalising it to something I'm happy with has been something I’ve been wrestling with on and off for the last five months. As you’ll see, mechanically it is only a slight evolution of the previous system, but the approach is very different. In particular it keeps the key benefit of the previous system - closely related skills still have bonuses which can differ by no more than 2 - and doesn't require any alterations to the rest of the rules (aside from a couple of +3s added to target numbers). In fact, apart from making character creation simpler and more transparent, it doesn't really impact on gameplay.

In the process I’ve ended up with a lot more attributes – something which I’ve previously been quite opposed to – but the reasons behind my opposition to that have evaporated. In addition I’ve removed randomness from character generation – something I’ve had in since day one. Both of these were a bit of a sacred cow, but it was making character creation difficult.

My goals are:
  • Attributes are broad brush bonuses to represent aspects of your character to give consistent characters.
  • Aspects of your character which you role play (problem solving and conversations) are not covered by attributes.
  • I want clarity of what an attribute bonus means – it should tell me something specific about the character.
  • I would like you to be able to choose attribute bonuses based upon what you would like your character to be like - that is without having to make dissociated decisions* during character creation.
  • There should be as small an overlap between attributes as possible.
  • It should be reasonably clear which attributes should give bonuses to skills.
  • I want to keep the same size range for bonuses to skills from attributes – seven steps from worst to best – as that has worked well so far.
  • I want only bonuses not penalties – my group was particularly enthusiastic about this change, despite it being only psychological.
  • It should be possible to excel at activities by having bonuses in different attributes. For example, you can be strong by being tall, or by being well built, or muscled.
  • The skills which have similar bonuses due to taking bonuses from similar attributes should be plausible. For example, acrobatics should be strongly linked to dancing, but not to lock picking.
In some respects it now resembles Rolemaster – which was a surprise as I'm a refugee from Rolemaster – and Size is of course an attribute in Runequest. Nothing is new under the sun – but every journey is different. On the other hand it still strongly resembles the prior system - strongly related skills (those sharing two out of three attributes) can only defer by at most 2 - exactly as before - but this "related skills" concept is now quite wide-ranging.

In the next post I'll unveil the attributes and what skills they apply to, and illustrate with a nifty diagram how I found problems and refined them away (well, I like it anyway!).

*It is often claimed that all character creation is dissociated - what I mean by dissociated character creation is the common situation where you make a choice about an aspect of the character purely based upon the mechanical effectiveness of the choice in the game rules, rather than your character concept.

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!