Friday, May 14, 2010

+1, -1, +1, -1 ≠ 0

Yet another post on the interesting dice pools of Warhammer Fantasy RPG 3rd Edition. This time, we'll be taking a look at Fortune and Misfortune dice (the white and black d6's that are half blank, and which represent situational modifiers).

GM Tip: If your players do something crazy in character, that might just work but could also backfire, the proper reward for that is to add 2 white and 1 black die. It's mostly a benefit, but has a small chance of making things fail spectacularly.

We're going to start this one off with a graph that's not even Warhammer. Here's your standard graphs of normal rolls of ordinary 2d6, 3d6, or 1d20, the cornerstones of many a gaming system.

Now, let's say you were rolling one of these systems, and there were a lot things going on. The GM applies a bunch of modifiers, and they just happen to be an equal number of positive and negative modifiers. Maybe it's an archery roll, and you're shooting a top-quality bow (+1) with a magic arrow (+1) at a stationary target (+4), but it's kinda windy out (-2) and the sun is low on the horizon (-2) and it's beyond your first range increment (-2). What would that do to the graph of your odds to hit a particular target in one of these kinds of systems?

The answer, of course, is nothing. The graphs would look exactly the same, because the equal modifiers balanced out. Despite all these crazy details being in play, it ends up with you having the exact same odds of making the shot that you would normally if none of the modifiers were in effect. In most systems, conflicting modifiers just cancel each other out (sometimes literally, sometimes just effectively, but the result is the same). If a truly complicated situation comes up, you can end up doing a fairly large amount of math only to discover that the end result is the same, or very similar to, the unmodified default roll. I don't know about you, but I find that a little frustrating.

In contrast to that, Warhammer fantasy role-play involves less math, and makes sure the conflicting modifiers actually have an impact. Nearly all modifiers in Warhammer are abstracted out to either one Fortune Die, or one Misfortune Die. It's easy to remember, and the GM is free to tweak it on the fly. If something crosses your mind as a possible modifier, you throw another die into the pool. As you can probably guess from the tone of my description, I'm inclined to like this. I'm a sucker for elegant flexible systems, so I'll acknowledge that bias before going on.

One area I was worried about though, was whether or not these dice would effectively just cancel each other out. I know that in general with normal numbered dice, the more you put into a pool, the steeper the bell curve gets. I was a concerned and suspicious that a big handful of fortune and misfortune dice would just make the average result happen a lot more often. Much to my surprise, the exact opposite occurs. Additional dice leads to greater variance in the dice pools.

On the right is a simple graph of a very small dice pool (3 Characteristics dice vs 1 Challenge Die). The blue line shows what happens when we add an equal number of Fortune and Misfortune dice to the pool. Odds of basic success and failure barely changed at all. Success dropped 2 points from 59% to 57%, a variation that's hardly going to be noticed. At the same time, however, the chance of scoring the important triple hammer uber-success line of most action cards nearly doubled, shooting up from 6% to over 12%.

How exactly that happened is better illustrated by the next graph, which also shows the same die pool, but breaks out the possible results a little differently. Instead of showing the odds of scoring a particular result or higher, it gives the percentages for every discrete number of successes possible. Here we can see where those extra results are coming from. The odds of getting exactly one success have actually dropped rather far, but the improved odds of scoring the bigger hits have compensated for most of it.

Extrapolating from the 2 charts, we see that with the Fortune and Misfortune dice added, you're only 25% likely to score exactly 1 success, but you're 32% likely to score 2 or more successes. Without the extra modifier dice, the numbers are essentially reversed, and you're more likely to score 1 success than 2 or more.

Of course, as I noted in last week's articles about the Challenge Dice, looking at successes is only half the equation in Warhammer 3rd. To get the full picture, we have to look at boons. Here's a pair of graphs of a somewhat larger pool of dice. I've graphed what it looks like as an unmodified pool, as well as what it looks like with 1, 4, and 7 sets of matched Fortune and Misfortune dice. Not that I have any desire to roll a 20+ dice pool that's been loaded down with 14 situational modifier dice (on top of the realistic pool of Stance, Characteristics, Skill and Challenge dice) but since the Warhammer probability tool made it easy for me to calculate those odds, I thought I'd share them. They do make for some interesting graphs.

All the graphs I compiled for all the pool sizes shared a few traits in common. In general, more black and white dice meant more extreme results. They increased the chance of triple-hammer successes, but also the chance of failure. They increased the odds of scoring banes, but also increased the odds of scoring 2 or more boons. Reckless dice (the red d10's) are a little more complicated, and probably deserve a blog post of their own, but the charts shown here are very representative of a wide array of pools composed using Conservative Dice or Neutral Dice. In many of the pools, adding more black and white dice (at least in the quantities likely to be seen in-game) would have almost no impact on the odds of scoring at least one boon, but would pump up the more extreme boon and bane results. It would accomplish this by reducing the odds of scoring a "boon neutral" roll where the boons and banes didn't come up, or came up in equal numbers and cancelled each other out. The odds of a single boon goes down at the same rate that the odds of multiple boons goes up.

Ultimately, the amount of variance provided by the Fortune and Misfortune dice wasn't huge. The green lines on these last too charts look impressive, but that represents 14 dice of modifiers being added to the pool. You're just not likely to use enough modifiers or dice to get to the point that the numbers really get interesting.

I find that I'm happy that the dice don't just cancel each other out, and that they lead to increased chance of the more extreme rolls occurring. At the same time, I'm a little let down that it takes so many dice to achieve these effects.

Warhammer 3rds die rolls get interesting because of the Boons and Banes, they provide the subtle nuance of the system, the extra oomph that other rules sets lack. Boons and banes are also more sensitive to large variance. There's very few cases in the default rules where getting 5 successes is functionally better than getting 3 successes (mostly just for first aid and recovery rolls). But, there's often something useful to do with a 4th or even 7th boon, and frequently some horrible side-effect waiting to be triggered by yet another bane. Looking at it from that perspective, there's plenty of reason to want boons and banes to be the most likely result of the Fortune and Misfortune dice. Sadly, they're not. Each such die has 2 Success or Failure symbols, and only 1 Boon or Bane symbol. They're twice as likely to impact success as boons. I see them as a step in the right direction (since a ton of modifiers actually does something, instead of just canceling out), but they strike me as being a little bit of a missed opportunity. If they had it all to do over again, I would encourage FFG (the publisher of Warhammer 3rd) to swap the Boons and Successes ratios on these dice. That way more modifiers on a roll would result in a more dramatically increased chance of side effects stemming from the roll.

One last observation (and this is where I stole the GM tip from for the start of the article): Let's say a PC at your table does something really risky, but which might just pay off. Swinging across a chandelier and kicking an enemy, for example. Throwing themselves against a foe on or near a clifftop. Muckraking and name-calling during a social event. Having seen these numbers, I'd feel pretty comfortable "rewarding" such behavior with both a Fortune and a Misfortune die - the net result will be about a percentage point or two against them, but with a big impact on the odds of making the likely result more extreme either way.

If you want to genuinely reward such behavior, and encourage it in the future, then I'd consider giving 2 Fortune and 1 Misfortune die to the action. That will probably help them out a bit, but also open up a small chance of it backfiring. I think this is a lot better than the example in the rulebook where a PC dives out of a tree at a beastman. In the book, they just give him two Misfortune dice (and no Fortune) because it's tricky and dangerous, which is just going to discourage your players from taking such colorful actions again in the future.

2 Fortune and 1 Misfortune is also fun because it mirrors the effects of being drunk. The "intoxicated" condition card gives 1 Fortune and 2 Misfortune. The subtle difference of these pools shades the distinction between bravery with a side of foolishness, or foolish with a side of bravery. Plus, if a PC is ever drunk, and the player does a good job of hamming it up with outrageous actions, they'd stand to roll 3 Fortune and 3 Misfortune total, which as we've seen does a great job of making something unusual (for good or ill) come out of the roll.

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