Monday, 4 February 2013

Inspirations: the Sands of Time, part five

A while ago I started looking at whether you could take anything interesting from Prince of Persia: The Sands of Time for use in RPGs. I've had a quick overview of general ideas and issues, and then considered how you might handle things in Pathfinder or BRP. I've also looked at the combat system in detail, and considered how it might be converted for tabletop.


For the sake of interest, I'm going to consider an entirely new mechanism. We want PCs to successfully fight against multiple opponents; in POPSOT, the PC attacks faster and more accurately than the monsters, and is better at defending.

The Necromunda and old Warhammer 40,000 combat system appeals to me for this. For each opponent, you both roll d6s equal to your Attack score, add your Weapon Skill and any modifiers (such as terrain or multiple attackers), and the difference in scores is the number of hits taken by the loser. A more skillful fighter will typically win a fight without a scratch, as their high Weapon Skill gives a higher final figure. A fast-striking fighter can overwhelm an opponent, because they roll more attack dice and have more chance of getting an optimal roll. A single fighter can fight multiple opponents, rather than having to pick one to fight while the others attack freely. A skillful fighter can defeat multiple opponents without automatically being overwhelmed, because each fight is largely independent; but statistically, fighting several opponents is still more dangerous, because there are more chances for a poor die roll to scupper you.

As I'm aiming for a fairly streamlined system, I'm looking at something inspired by this, but reducing its complexity somewhat. Here, you'd simply roll opposed dice, but different creatures would have different die sizes. A PC might have a d8 compared to a normal enemy's d4, for example. Weak enemies have small dice, and powerful enemies have large dice. What would that look like?

Crunching dice

With a PC d10 versus an NPC d4, the PC will win 30/40 combats. With a PC d8 versus an NPC d4, the PC will win 22/32 combats. With a PC d6 versus an NPC d4, the PC will win 14/24 combats.

Matrix of possible results
Die size PC die d3 d4 d6 d8 d10 d12
Enemy die 1 2 3 4 5 6 7 8 9 10 11 12
1 ----- 1 2 3 4 5 6 7 8 9 10 11
2 LOSE ----- 1 2 3 4 5 6 7 8 9 10
d3 3 LOSE LOSE ----- 1 2 3 4 5 6 7 8 9
d4 4 LOSE LOSE LOSE ----- 1 2 3 4 5 6 7 8
5 LOSE LOSE LOSE LOSE ----- 1 2 3 4 5 6 7
d6 6 LOSE LOSE LOSE LOSE LOSE ----- 1 2 3 4 5 6

Proportion of PC wins
Die size d3 d4 d6 d8 d10 d12
d3 1/3 1/2 1/3 3/16 2/15 5/48
d4 1/4 3/8 7/12 11/16 3/4 19/24
d6 1/6 1/4 5/12 9/16 13/20 17/24
d8 1/8 3/16 5/16 7/16 11/20 5/8
d10 1/10 3/20 1/4 7/20 9/20 13/24
d12 1/12 1/8 5/24 7/24 3/8 11/24

For multiple opponents, each combat could be rolled separately, or PCs could roll one die per opponent and then allocate them against the enemy results. The second would give them a significant advantage, since the combat system already favours the PC, allocation reduces randomness, and randomness favours the underdog. I think this might be too strong as the standard mechanic, and I also suspect it would make combat against multiple opponents actually easier than against one.

Sequential versus allocated rolling in mass combat

Logically, let's say an enemy can only beat you on opposed d6s if it rolls a 6 and you roll a 1. This will happen, on average, one out of 36 times. If you fight two opponents in succession, by rolling separately for each opponent, the chance of you losing at least one fight is...

At this point, I realise I've forgotten nearly all the statistical knowledge that I painstakingly absorbed at school. I spend some time researching probability distributions.

Okay. The other version is easier. If you win unless you roll a 1 while they roll a 6, your probability of victory is 35/36. If you fight two opponents and roll separately for each, the probability of you winning both is 35/36 x 35/36. This makes 1225/1296, or 0.95. The probability of losing one or more fights is 0.05, or about 5%. For three opponents, the probability of flawless victory is 42875/46656, or 92%, while there's an 8% chance of you losing at least one.

Next, you fight two enemies using the roll allocation mechanic. If the enemies roll 5 & 6, and you roll 1 & 2, you simply allocate your 1 against the 5, and your 2 against the 6, countering both hits. The only way for you to either combat is if either both roll a 6 and you roll at least one 1, or if you roll two 1s and they roll at least one 6.

Wow, my maths is somewhat rusty, but that actually looks like a much nastier calculation than I initially thought... I'm sure there probably is a nice, statistical way to manage that, probably using Poisson distributions and factorials, but I'll stick to basics.

They roll two 6s and you roll one 1: 1/36 x 10/36
They roll one 6 and you roll two 1s: 10/6 x 1/36
They roll two 6s and you roll two 1s: 1/36 x 1/36

10/1296 + 10/1296 + 1/1296 = 21/1296
= 7/432
= 0.016

If my maths is accurate (a dubious thing), the chance of you taking at least one hit in a fight against two opponents is only 1.6%. What about three opponents?

They roll three 6s. You roll a 1. (1/216 x 25/216) - three arrangements
They roll three 6s. You roll two 1s. (1/216 x 5/216) - three arrangements
They roll three 6s. You roll three 1s. (1/216 x 1/216) - one arrangement
They roll two 6s. You roll two 1s. (5/216 x 5/216) - nine arrangements
They roll two 6s. You roll three 1s. (5/216 x 1/216) - three arrangements
They roll one 6. You roll three 1s. (25/216 x 1/216) - three arrangements

Multiplying all the possible arrangements of all the possible combinations gives a probability of 406/46656, or 0.009ish. This means that in combat against three opponents, the chance of you taking one or more hits is a mere 0.9%.

Under sequential rolling, you have a 3% chance of injury when facing one opponent, a 5% when facing two, and an 8% when facing three. These figures are, however, exactly the same as if you fight three enemies one after the other.

Under allocated rolling, you still have a 3% chance of injury when facing one opponent. Against two, that drops to only 1.6%. Against three, it's down again to 0.9%. Combat gets easier the more enemies you're facing! While this might be a nice mechanic in some situations - for example, if you're going for a comedy tone where the clever PCs can make opponents get in each other's way - it's not really appropriate for this one. Too much Law of Ninja for my taste.

Sorry about that. Probably my longest tangent ever, but I'm not going back to check.

So yeah, as I thought, sequential rolling for the win. I'd probably give a small modifier for each subsequent opponent, just so there's some advantage in numbers.

Combat Options

Defensive fighting might let shrink your die size. For every size you drop, you can parry two hits you would otherwise take from each opponent you face. Is this worth it?

If you're fighting an opponent with d8, and you have a d8, on mean you'll take zero hits. However, you can take anywhere up to 7 hits if you roll badly. One to four hits is relatively likely (mode 1, standard deviation 3.5). A PC in bad shape, or who has a lot of opponents to fend off without much support, may not be able to take that. By dropping a die size to d6, you slightly increase the number of hits you're likely to take (and significantly reduce your chance of scoring any hits yourself), but can ignore two of them: the likely pool of damage increases from about 3 to about 4, but two are ignored, giving a slight defensive bonus. Dropping even further, to a d4, means you could readily take up to 5 hits, but can parry four of them. A d3 allows up to six parries, at which point it's very difficult to damage you at all. While I don't claim to be mathematical expert, it looks to me like this mechanism would actually do what's intended.

versus d12 d8 d6 d4 d3
MEAN -2.18182 -3.27273 -4.36364 -4.90909
STD 4.305958 3.940111 3.615798 3.476381
Parry 0 2 4 6
RESULT -6.48778 -5.21284 -3.97943 -2.38547

versus d8 d8 d6 d4 d3
MEAN 0 -1.14286 -2.28571 -2.85714
STD 3.495452 3.064919 2.650746 2.455315
Parry 0 2 4 6
RESULT -3.49545 -2.20778 -0.93646 0.687542

versus d6 d8 d6 d4 d3
MEAN 0.571429 0 -1.2 -1.8
STD 3.064919 2.690981 2.23842 2.00713
Parry 0 2 4 6
RESULT -2.49349 -0.69098 0.56158 2.19287

There could also be a reckless fighting stance, which gives you more hits but makes you more likely to take damage yourself. I'm not quite sure how to do this, as simply increasing the die size just makes you more likely to win the combat, without risk. A slightly complex option would be to grant an increased die size, but to calculate enemy hits with a penalty, perhaps half the number rolled. For example, you might upgrade to a d10 and roll a 7, which compared to the enemy 5 grants two hits. However, for defensive purposes it only counts as a 4, and so you also take a hit.

This system doesn't immediately lend itself to any kind of active defence, which is a slight downside. It ought to allow fairly fast and simple combat, though. But if we're looking for something more tactical, I'm not sure this would be great - it might actually be better to have a more complex system if combat is going to be a major aspect of the game, just as D&D fleshes out things significantly. While I could go with a Necromunda-style system of simple modifiers to die rolls, it's just not very interesting for an RPG, however well it works in a tactical miniatures game.

This system also has the downside that it's a homebrew. Which means I'd still have to establish systems for handling everything other than combat, and it's probably riddled with flaws. So to be honest I'm probably better off sticking with something established. Still, I had fun.

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