Wednesday, October 15, 2008

Diminishing Returns in Savage Worlds

While righting that last post, I realized something that almost was a bullet point, but deserves it's own attention as a separate topic. I've long wondered about exploding d4s.

Not some clandestine CIA plot to eliminate Castro's D&D group, though that could make for an interesting tale, I'm sure. Hmm... there's a one-shot concept in there somewhere.

Instead, by "exploding", I'm referencing dice that you get to reroll and add to the total if they score their highest face. I think Savage Worlds calls it Acing the dice, but I first encountered it in 7th Sea, where it's said they explode. If you rolled a "10" you'd roll again and add 10 to the total. If you rolled a second ten, you'd roll a third time and add 20 to that roll.

Exploding dice have strange interactions with systems that use multiple die types. Barring explosion, the average roll of each standard die type is 1 higher than the die type preceding it (except the jump from d12 to d20, of course), so a d4 roll averages 2.5, a d6 roll averages 3.5, etc. However, that flat progression falls away when you start messing with exploding dice. The average roll of an exploding d4 is about 3.56, and the average roll of an exploding d6 is about 4.19. Those numbers aren't quite accurate - I'd have to take the math out past several more explosions to figure out the actual repeating digits. But, it should be clear from these aproximations, that there's diminishing returns. Each consecutive upgrade in die size gives you a smaller boost to your average roll.

It's worth noting that since dice explode, there's technically no upper limit to rolls. Sure, you're probably not ever going to get "6" two dozen times in a row and score around 150 on a roll. But getting a d4 to roll "13" will happen from time to time, and I've yet to see (only 3 sessions of actual play) a difficulty above 10. If that holds true, then die size doesn't end up mattering much. An exploding d6 has a 1 in 12 chance of hitting difficulty 10, and a d10 only has a 1 in 10 chance. The d10s a little better, but not a lot.
The most common difficult I've experienced as a player is "4". A d4 has a 25% chance of hitting that, a d6 has 50%, a d8 has 62.5% chance, and a d10 has 70%. Again diminshing returns.
All this does for me is reinforce my deep-rooted desire to buy my d4s up to d6s.

So, with diminishing returns, and the possibility (though infrequent) of the lowliest die scoring any number the best die can score, you'd expect the system to make high die-code skills fairly cheap.

Nope. To get a high die-code in a skill, you typically first buy a high rating in the Attribute it's linked to. That costs points from a very limited pool, so some other attribute will "suffer" for it. (Again, having the low die in something ain't that horrible, but Attributes also generate some static numbers that are based on non-exploding averages or maximums, so it can hurt if you chump stat the wrong the Attribute.)

If instead you choose to buy the skill above the corresponding Attribute, then the higher levels cost double. That double cost hardly seems worth it, since you're at the point of diminishing returns. Especially since those same value in points will buy you another Edge. Edges are special powers (like Feats in D&D) - they make your character unique, and they are almost always of greater benefit than just rolling a slightly larger die.

Damnit. Now I've worked myself into a weird place. I've pretty much convinced myself that there's illogical flaws in the Savage Worlds system. However, I have friends playing said system, and I've very much enjoyed meeting them, gaming with them, etc. I'm quite certain I'll continue playing with them. I'm tempted to steer them to Fate (or more Wushu), but doubt they'll make that switch based on my recommendation. After all, I haven't even read the Savage Worlds book, and haven't given it a fair try, and I'm still the new guy of the group. I've done an analysis that tells me that taking a skill above a d8 is dubious, but I see d10 skills on PC sheets at game night quite often. Either there's something I'm missing, or they just haven't analyzed the math. My personal code of conduct won't let me sit quiet on this discovery. Munchkins game the system and keep the secrets to themselves - not me. But you can't put the genie back in the bottle. Clearly, any future PCs I make will be from an informed viewpoint. So I'm going to have to point it out to the group, which will lead to spirited conversation. Natural human reaction is to say "no, I'm not in error" and debate, so they'll want to. Since I've never read the book, I'm inadequately prepared for that debate - there could be balancing factors I'm missing. So that means I have to go buy a book for a game I would not choose to run. I'll be buying it for the express purpose of talking my friends out of running it, and/or figuring out the optimal character builds, should talking them out of it be impossible. That whole situation stinks. I kinda wish I'd never run the math that lead to this conclusion.

(And no, I can't just ask to borrow theirs. We've known each other for a short time, or rather, a small number of events over several months. Long enough that someone might spontaneously offer, but still new enough you'd seem impolite to ask. Besides, if I'm going to turn a whole group off of a company, the least I can do is throw one final sale in that companies direction. Luckily, I think there's a $10 or $12 pocket guide to the game, which will assuage my guilt without hurting my budget.)


jamused said...

I think you should check your math. By my calculations, the amount the expected value of each die goes up is increasing, albeit only slightly. The real oddity of the system isn't diminishing returns, it's that at the bumps (where the target = exactly the size of the die) you're actually about 1% more likely to hit it with a die one size smaller.

r_b_bergstrom said...

My average roll numbers came out very similar to yours on d6+ – close enough to be a rounding error. But I’m consistently coming out higher when I calculate the d4’s average roll. What’s the equation or method you’re using to generate the d4 average rolls? Is it taking into account how much more likely a d4 is to ace than a d12? I only ran the math on d4s, d6s and d8s initially, so I missed part of the pattern. However, I keep getting higher numbers than yours for d4s.

To allow the conversation to continue, let’s assume your average rolls data is correct. Your raise columns also puzzle me, as it’s not clear what they mean, but for the sake of the discussion we’ll just assume the first two columns from the left are accurate.

D4-2 rolls 1.30
D4 rolls 3.30
D6 rolls 4.19
D8 roll 5.14
D10 rolls 6.11
D12 roll 7.09

So, for 1 die increase, I raised my average roll by +2.
For a second die increase, I raised it by +0.89.
The third die increase raised it by +0.95
Fourth die increase by +0.97
Fifth die increase by +0.98

So, yes, that is an increase (after the weird d4 to d6 dip, which is really a disproportionate d4-2 to d4 jump), if you only take into count one factor: the average roll.

But there’s more than one factor that matters.

If, however, you compare the odds of hitting the typical difficulty of “4”.
A d4 hits it 25% of the time.
A d6 hits it 50% of the time. (+25 percentage points, and also doubling the overall likelihood you’ll succeed)
A d8 hits it 63% of the time. (+13 percentage points, a far step down from doubling your chance of succeeding since the last increase.)
A d10 hits 70% (+7 percentage points)
A d12 hits 75% of the time. (+5 percentage points)
source of percentage data

Using “chance to hit a 4” as the yardstick, each die type gives you less benefit than the previous time. And since that’s the to hit number of most gunshots and archery attacks, and the majority of Trait tests, it’s a pretty important yardstick to consider.

Also relevant to the discussion at hand is the cost of getting a given die type.
If your attribute is a typical d6, the skill costs 1 point for d4, 2 points for d6, 4 for d8, 6 for d10, 8 for d12. The cost-to-benefit ratio just took a punch in the nose for a d8 or better skill.
While you can certainly take a higher attribute to reduce that skill point cost, that adds another cost. You only get 5 Attribute points, so taking one at a d8 means having to carry a d4 weakness in another attribute. D10 means two at d4, d12 means three at d4. That’s a pretty hefty cost.

So, if I raise my skill from d8 to d10, I do indeed get +0.97 to my average roll. But my chance of scoring a success on most skill tests only went up by 7%, and it cost me two skill points or two attributes at d4 to get that minimal bonus. If you ask me, that’s diminishing returns.

r_b_bergstrom said...

Pardon my poor manners - I should be saying "thank you for sharing your opinion," not trying to prove you wrong. How rude of me.

jamused said...

No problemmo.

The spreadsheet I used did the math in as naive a way as possible, by doing the recursion out by hand. (The columns labeled Raises should really be called explosions, it just happens that each explosion on a d4 is also a raise. I fixed it in the latest version) So for d4 the expected value is

f(0) =(.25 * 1) + (.25 * 2) + (.25 * 3) + (.25 * f(1))

f(n) = 4 + (.25 * 1) + (.25 * 2) + (.25 * 3) + (.25 * f(n+1)), for n > 0

It's absolutely true that if what you care about is beating 4, which is the most frequent TN in the game, there's diminishing utility in spending more and more character build resources on it. I thought you were complaining that for each successive die type the fact that it can explode adds less and less to the expected value, which doesn't seem to be the case unless I've overlooked something.

I actually think that diminishing returns on build-points for specialization is a feature, not a bug. Systems where getting your best attribute or talent as high as possible is obviously optimal seem broken to me; at the very least they really cut down on character diversity. I like that with SW there's no one right way to build a competent character, or if there is I haven't found it yet.

r_b_bergstrom said...

I don’t think it’s a bug, either. I think it’s an intentional feature, just one I could have done without.

Better yet, they could have included it, but intentionally mentioned it in the character creation or advancement sections. As it is, a player is inclined to think that, since high dice cost twice as much, they must be better. The way the dice work is not transparent in this game. I feel that, as much as possible, RPG designs should avoid “hidden” benefits, hidden costs, and non-obvious optimal builds. Including such things rewards the munchkins and point-weasels. Those are not the behavior patterns that I think should be rewarded.

Systems where getting your best attribute or talent as high as possible is obviously optimal seem broken to me; at the very least they really cut down on character diversity.

Though it may not be as obvious, character diversity is also threatened by a system that rewards generalization over specialization.

I just wrapped up a Scion campaign that ran over a year. That game had 9 “magic powers” that were so cheap and so incredibly useful, every PC bought at least the first two levels of all nine before the campaign was over, and only one PC in the group didn’t have the first 3 levels of all 9 (he did on 8). To me, that was a real downer – the characters uniqueness dwindled session by session. Prior to this experience, I saw things more like you do.

I still don’t want a single optimal build to exist in a game, I want several different solid and competitive builds to exist. But I also want the concept of several of them briefly sketched out, so that players don’t have to devote lots of time analyzing the system and trying to figure out what does and doesn’t work well. I want the first PC you make under the system to be one you’re not afraid to play for 4 to 6 months.

Gotta say that the “roles” are my favorite part of 4th Ed. I think they do an incredible job of giving players a simple focus to hang their character on. Savage Worlds has nothing like that, which is fine if your game is pure role-play, but (between that and the open-ended point build methodology) it’s really hard to balance PCs or ensure everyone has their own spotlight during the action scenes.

jamused said...

I haven't played any Savage Worlds campaigns out to Legendary status, but I remain unconvinced that experience will wash away character differences or indeed that refusing to raise attributes from a d6 is truly optimal. There are too many attractive Edges that require a d8 or d10 in an attr or skill, the payoff from an advance to a skill past its governing attr is too small, there are several really important figured stats, and there are too many situations where you have minuses for me to just eyeball it and say, well obviously nobody is going to want to do that.

I agree with you completely about non-obvious benefits and hidden costs (see my blog entry on desirable generic RPG qualities), but so far that's not my impression of SW at all.

r_b_bergstrom said...

I double-checked my post, and I see that I wrote:

I've done an analysis that tells me that taking a skill above a d8 is dubious,

Not the same as saying “I’ll never take an attribute above d6” which is what you seem to be arguing against. I don’t think I ever said “I’ll never take an attribute above d6,” I said taking a skill to d10 or d12 is of dubious value. In other words, I’d need a good reason to go investing those points – if I had my eye on a sweet Edge that required a d10, that’d be a good enough reason.

However, looking at the Edge chart, there’s several that require Fighting or Shooting at d10, and one that requires your Arcane Skill at d10, but that’s it. (Technically, there’s the professional/expert/master tree, but all those do is boost your roll, so if I’m not interested in boosting my roll above a d8 in the first place, I’m not going to take them.)

If my character concept is just “I’m really good at Investigation”, there’s very little benefit to taking investigation above a d8.

jamused said...
This comment has been removed by the author.
jamused said...

Little reason to take a skill above d8... until you face modifiers or are engaged in a contest of skills. Since that's mostly up to the GM, I think if the GM regards it as a problem there's a solution ready-to-hand.

Sorry if I misread what you were saying, btw.

r_b_bergstrom said...

No troubles.

Thanks for engaging in this conversation with me, btw, as it's definitely resulted in me understanding (and liking) the Savage Worlds rules better than I had.