Monday, July 26, 2010

Color-Blinded

Here I am reflecting on the Warhammer 3rd dice yet again. This time it's the red (reckless) and green (conservative) dice. In previous posts, I've looked into how the red dice are a bit of a mixed bag. They generate some big hits, but run a lot more risks in the process. Largely, this is balanced by the way that most action cards have a much-better red side. Or do they? As it turns out, many of the red actions only look much better than their green counterparts. I think Vegas casinos would love those red dice and red sides of the cards, as they look much sexier to the players than they really are.


Let's examine the action Troll-Feller Strike. I chose it partly at random (it was in the first PC ziplock I opened), partly because the math on it was reasonably easy for me to figure out, and partly because the other special attack in the PC bag I pulled that one out of is a bit of an under-performer. Troll-Feller Strike, though, while not one of the absolute best cards, but it's pretty darned good. It's a solid tier-2 card, and the tier-1 above it is comprised of just 2 or 3 broken cards, 2 of which have seen major errata from the publisher. I imagine you could easily argue that Troll-Feller Strike is in a 12-way tie for 4th best non-magic attack card. It certainly doesn't suck, and is the type of action you'd be happy to have in your arsenal.

Looking at the Troll-Feller Strike card, one's initial instincts are that this card is much better on the red side than the green side. The red side has that juicy double-boon line and the very respectable comet line, both of which are absent from the green side. What's more, the green side adds an extra black die of difficulty to your roll. First impressions are that this card is much better on red than green. In theory, you could roll a hit for + 7 damage, +2 criticals, and a number of bonus wounds based on the severity of one those criticals. While that's not gonna happen very often, it's still clear that the red side does tons more damage than the green!

Except it doesn't. After running the math, I've concluded that on average it does 0.2 damage more per attack. Average damage for the red side is 12.8725 minus target's toughness. Average damage for the green side is 12.66 minus the target's toughness.

Here's how I arrived at those figures.

I used an attack pool of 1 purple, 3 blue, 2 stance, 1 yellow. In my experience, most attack pools also have at least one white and at least one black die, but since they kinda come close to cancelling each other out (and the math is much simpler without them), I figured we'd leave the white and black out and just stick with this. (Also, I started on this line of thought in a reply to a post on the FFG forum, and the original poster had sited that pool when they asked their question.)

I ran the numbers through the online probability tool at http://www.jaj22.org.uk/wfrp/diceprob.html, as that saved me a lot of time, though it meant I don't have numbers for odds of rolling 4 or more successes, or similarly large numbers of boons and banes.

I rounded those results to the nearest percentage point and made little results charts.

Given those numbers, the red side has the following Hit or Miss percentages:
10%: Miss
35%: Hit +1 damage
55%: Hit +3 damage

The red side has the following boon or bane odds:
1%: 2 Fatigue from Banes*
16%: 1 Fatigue from Banes*
21%: 0 boons or banes
25%: If it hits, gets +1 damage, ignore armour soak
19%: If it hits, gets +3 damage, +1 critical.
18%: If it hits, gets +4 damage, +1 critical, ignore armour soak

To determine the overall damage odds, I multiplied those two percentages. These are rough numbers, for several reasons. (The accurate math is actually more complicated because some results are less likely to occur concurrently. With the red dice, you get slightly more banes on rolls that have fewer successes, and some sides have more than one symbol so you get more extreme rolls. I played around with that for a little bit, and decided the numbers weren't different enough to justify the extra effort. This is part of why I rounded them to the nearest percentage point for the charts here, as the long flowing digits aren't any more accurate than the shorter numbers that are easier to read.)

This multiplication came up with the following percentages for the damage results of any given roll.
10%: Miss
13%: Damage N+1
21% Damage N+3
9%: Damage N+2 (+ignore soak)
7%: Damage N+4 (+1 crit)
6%: Damage N+5 (+1 crit, ignore soak)
14%: Damage N+4 (+ignore soak)
10%: Damage N+6 (+1 crit)
10%: Damage N+7 (+1 crit, ignore soak)

If we assume that N=10 (Strength 5 + Hand Weapon), and that the ability to ignore armour soak adds on average 2 points of damage to the total**, then this generates an average damage per attack roll of 12.86.

That doesn't take the comet line into account. The comet line turns one point of damage into a crit, and then adds bonus wounds equal to the severity of that crit. Crit severity will vary wildly depending on what cards have been handed out already as wounds, and which expansions you have. I did a number crunch on my deck and found the average is 2.25 damage. This assumes a fresh deck and no pre-existing “flesh wound” crits on the foe.

The next step was to calculate what percentage of hits can actually find the comet effect useful. Obviously, you don't want to use the comet line if you didn't net any other successes, or if it could be used as a boon to do +1 damage AND cancel a couple points of armour soak. So, this is only going to be an additional boost in the situations where your roll shows 2 or more potential successes, AND you also already have 1 or more banes or 3 or more eagles as your final boon/bane result from all the other dice but the yellow one, AND you roll a comet. You have a roughly 19% chance of rolling a comet (17% from your initial roll of a yellow die, plus 2% from rolls that get one or more Righteous Successes before rolling the comet), a 76% chance of scoring 2+ successes, and 35% (17% + 18%) chance of scoring banes or 3+ boons. So at best, the comet line is a smart move on 5% of all rolls (as .19 * .76 * .35 = .05054). So, on 5% of all rolls, we add .26 damage, meaning we add an average of 0.125 damage per attack roll.

The end result being that we do an average damage on the red side of 12.8725 minus the target's toughness.

Now let's look at the green side. Same process (except we can skip the obnoxious part about the comet, since there's no comet line on the green side).

Green Side:
Hit vs Miss Odds:
11%: Miss
40%: Hit +1 damage
49%: Hit +3 damage

Boon and Bane Odds:
0%*: 2 Fatigue
10%: 1 Fatigue
19%: 0 boons or banes
71%: If it hits, gets +1 damage, ignore armour soak

Overall damage odds:
11%: Miss
12%: Damage N+1
14% Damage N+3
28%: Damage N+2 (+ignore armour soak)
35%: Damage N+4 (+ignore armour soak)

Assuming the same value for N (10) and for armour soak (2) as we did for the red side, this results in an average damage of 12.66 minus the target's toughness.

About 33% of all red rolls will do more damage than the best green roll, which seems pretty good. However, many of the high red rolls are just 1 point higher than green. What's more, the red dice have higher chances (than the green) of rolling the two lowest possible damage results as well. It all works out to the red side of the card having considerably more variance, but just a fractionally better performance in the long run.

The red side does an average damage of 12.8725, just 0.2125 damage per hit more than the green. As I indicated, most of the above numbers were rough, and there could be rounding errors that I've missed. But the margin of error on my numbers is likely to be smaller than the margin of damage bonus that the red side of the card has. Whenever there was doubt (such as the comet effect), I chose for the numerical result that gives the larger boost to the red side, yet it still only got ahead of the green by 2 tenths of a point of damage.



The upshot of all this is that the two sides of the cards are deceiving. The red side often looks much better, when it's actually just minimally better or just breaking even. This all has to do with the extra banes on the red dice, which make you much less likely to score beneficial boon lines. For a card like Accurate Shot, where the green actually looks better than the red, I say with confidence that the red side is actually quite horrible and the gap between them is huge.

I'm continually impressed with the extents to which the designers of WFRP 3rd went to to make a well-balanced game despite it's highly opaque and unique mechanics. Somebody over at FFG must really like math.


CAVEAT: This took some significant time, so I doubt I'll do this level of analysis with any other cards. Which means I can't be certain I didn't just luck into the Troll-Feller Strike as my first card chosen at random. Confirmation Bias may be influencing my conclusions. I expect the other cards to hold true to the same patterns this card did, but I can't be certain without doing a lot more work.

Notes:
*: In addition, there's a 36% chance of getting a fatigue from the red dice's exertion symbols, which happens almost independently of the boon-bane status of the rest of the roll. Overall, the roll has a 47% chance of generating 1 or more fatigue, with a maximum fatigue gain of 3 points per roll.

By contrast, the green die only has a 10% chance of getting fatigue at all, and a less-than two-tenths of a percent chance of getting 2 fatigue (and no chance at all of getting a third). Given that every roll of the green pool has a 48% chance of qualifying for recovering a fatigue, they can pretty much ignore their exertion status.

36% of green rolls will end up adding 2 extra recharge tokens to an action or dropping the parties best initiative token down by two. Just how bad that is compared to the red sides fatigue has a lot to do with the situation and character build, not to mention just how sadistic the GM is feeling at the moment.

**: I chose to equate “ignore your target's armour soak value for this attack” with +2 damage as 2 is approximately the average soak of foes in the Tome of Adventure. One could argue that this is below average for the sorts of foes you'd use Troll-Feller Strike against. However, since the green dice has very reliable odds of getting the single-boon trigger to ignore the soak, if we assume soak is higher it just makes things worse for the red side. If armour soak is 3, then the red side does an average damage of 13.42 and the green side does 13.29, closing the gap by nearly by half.

Conversely, the red side fairs a tiny bit better against low-armour foes. The red side would do 12.47 damage vs the green sides 12.03 if the armour soak were just 1 instead of 2. The higher the foes armour soak, the better the green side of the card does.

2 comments:

r_b_bergstrom said...

UPDATE: Since making this post, I've discovered that my numbers were actually a little too kind to the red side of the card (the one I was trying to make the case against). I was using the probability tool, with a yellow die in the pool. Turns out the yellow die is calculated wrong in that pool. As a result, those numbers were inflating the results of the red side, because it was getting more successes and boons to spend than it should have.

For example, those numbers implied that 16% of all the red attacks were going to do more damage than the max the green attacks could do. In reality, it's only about 8% that will.

___________________________


Here's the test of the yellow die:

I went to the probability tool and entered 1 yellow die, nothing else in the pool.


A yellow die has 6 sides, being 2 w/ boons, 1 w/ success, 1 w/ righteous success, 1 w/ comet and 1 blank.


So, if the probability tool is counting comets as a success, then the chance of getting 1 or more successes should be exactly 50% (because 3 sides are giving 1 or more success). If it's not counting comets as successes, then the chance should to about 33%.


Likewise, if it is counting comets as boons, the numbers would be a little more than 50% (since three sides would be a boon, and there's also the possibility of getting 1 or more righteous successes followed by a boon). It's not counting comets as boons, then we should expect to see a little more than 33% chance of boon odds


Put in 1 yellow die, and it says:
"Chance of 1 or more successes: 0.5000"
"Chance of 1 or more boons: 0.5833"


Clearly, every time you get a comet, it counts them as both a success AND a boon.

This suggests that the probability generator will frequently over-estimate the power of a yellow die, and over-estimate the success and boon rate of any roll that uses a yellow die.


I also noticed another weird mathematical anomaly to the probability tool and yellow dice: It over-estimates the ability of a single yellow die to score 3 or more successes. To do so, the expertise die would need to roll a righteous success (1 in 6) followed by a righteous success (1 in 6) followed by either a success, a comet, or another righteous success (3 in 6). 1/6 * 1/6 * 3/6 = 3/216, about a 1.38% chance. The probability tool instead lists it at 2.78%, about double what it should be. All the other numbers for a single die of any type appear to be correct, not sure why it's got this one wrong.

From these two errors with the yellow die, we can conclude that the odds of scoring the really high damage results on troll-feller strike are actually much lower than what my initial analysis suggested.

Erik said...

Let's see if it works.
Doing exact calculations is definitely time consuming. That's why I usually take the lazy route and do a Monte Carlo approximation.