Two By Three

In yesterday's analysis of the dice from Warhammer Fantasy Role-Play 3rd Edition, I came to the conclusion that 1 Challenge die was worth 2 Misfortune dice. That was, as it turns out, an over-simplification.

Let's look again at the chart that shows the relative impact of challenge dice and misfortune dice. Here it is, a very typical starting character dice pool, 4 trait dice, 1 skill die, and 1 fortune die. The graph shows how that pool performs against various difficulty ratings.
As you'll recall from yesterday's post and can see today, 1 challenge and 2 misfortune is almost identical to 2 challenge. The math of it gets a little complicated as you try different dice pools, but this chart sums it up nicely, being represented by the way the yellow line and brown line overlap, and the way the green line overlaps with the light blue line.
The impact on the success or failure of the action is essentially the same whether you add 1 challenge or 2 misfortune.

But this Warhammer 3rd we're talking about, and success is only half the picture. The game also has boons and banes, which represent silver linings and nasty side-effects. A roll can "fail" but still generate some sort of minor beneficial effect. An attack can hit, but fatigue the attacker. You can save the day, but still make things worse while you're at it. That's part of the charm of the system.

Let us now examine the boon and bane odds of those same die rolls.

For ease of comparison, I kept the same color coding of the lines, and kept the chart oriented where the better rolls are on the right. But since we're looking at Boons and Banes now, the chart doesn't run from 0 to 3. It runs from -3 to +3, essentially. The middle column of the chart represents rolls where either no boons or banes come up, or where equal numbers come up and they cancel each other out.

Two things become obvious about this.

• One, the brown and yellow line no longer overlap, which tells us that 2 challenge dice is worse than 1 challenge and 2 misfortune, afterall.
• Two, the green line is still hidden. Except now, the green line is hidden behind the brown one, instead of the light blue. That suggests a challenge die has a similar impact on boon/bane odds as three misfortune dice.

So I ran the numbers quite a ways up (as well as down to just misfortune with no challenge), and discovered that yes, the boon and bane numbers for 1 challenge die are about equal to the boon and bane numbers on three misfortune dice. I'm not posting the full chart here as it became an eyesore of overlapping rainbow colored lines, and to get any useful information out of it you have to keep turning various columns and dice pools on and off so they don't obscure the data layered beneath them. The edited version above should convey the general point clearly enough.

Does 1 challenge die equal 2 or 3 misfortune dice? The answer is "yes".

• For purposes of affecting whether your action succeeds or fails, it's worth two misfortune.
• For purposes of determining whether the side effects that go with your action are positive or negative, it's worth three misfortune.

That has an interesting effect on those Improved Active Defenses we were talking about yesterday. Taking "Improved Parry" does not improve your odds of completely avoiding an attack, at least not significantly. But it does reduce the quality of that attack, and make it more likely the enemy will suffer some sort of drawback or side-effect. How much of a boost that is, and how dire the side effects will be, depends entirely upon the action card you're being attacked by. It's beyond the player's control, and even beyond their ability to know in most situations.

It will definitely impact how I design monsters and NPCs for my own campaigns, though. I'll make certain I include plenty of juicy bane lines on monster actions to benefit the PCs who took the improved defenses.

One more interesting observation: As you add more and more negative dice to the roll, the odds of getting boons goes down, but after a certain point the odds of getting a net-zero roll stops advancing as well. If you roll the die pool above against 3 challenge dice, for example, you end up with a 45% chance of getting boons, a 30% chance of getting banes, and only a 25% chance of getting neither. You can see it starting in the second graph in this post, where the light blue line shows equal odds of getting 1 bane or zero. That plateau turns into a peak (or a capital "M" shape), and then grows ever more extreme as you throw more dice at it. In a future post, I'll revisit this interesting aspect of variance in the warhammer dice, and how it differs from what you'll find in most other RPG mechanics.

Credit: In my previous post, I did all the math myself, but for this one, I saved myself some trouble and lots of drudge work by using a wonderful internet resource, John Jordan's WHFRP 3rd Probability tool. It saved me hours, and allowed me to test graphs up into dice pools you'd never be able to fit in your hand.