Recent conflicts in Iron Kingdoms (which culminated in my character’s necessary death) have introduced me to the fascinating problem of feat point budgets, and methods for estimating the optimal use of feat points. Basically in Iron Kingdoms every PC has three feat points (in Warhammer 3rd Edition these would be fortune points; I think many games have this system). Feat points can be used to boost attacks or damage (or for various other tasks), and in the case of trollkin for regeneration. They are regained through rolling criticals or killing enemies or through GM fiat. Thus expending a feat point to kill someone can be cost free. But you only have three, so expending them too early or in an inefficient way can be catastrophic (as my party discovered, to Carlass’s great cost!) So it’s important to decide where to spend them.
The combat system in Iron Kingdoms is very simple:
- Attack: roll 2d6 + attack value, you hit if you beat the target’s defense
- Damage: roll 2d6 + weapon power, all points greater than the target’s armour do damage
That is, you have a threshold for success followed by a threshold for damage, with results above the latter threshold being more important if they are higher. Typically an enemy will have between 5 and 15 hps you can knock down, so a good result on the damage roll can be fatal. However, the attack roll is 2d6 so small improvements in bonuses are very important when attacking high-defense enemies.
Feat points can be spent to add 1d6 to either of these rolls. Adding a feat point to the attack roll increases the chance of hitting, but can be wasted if your target has high armour; adding a feat point to the damage roll can do a lot of extra damage but only works if you actually hit.
This scenario has an equivalent in epidemiology: it’s called a double-hurdle model, and is commonly used for estimating models of health-care expenditure in situations without health insurance. The first step (the first “hurdle”) is the decision to spend money on healthcare – this is often voluntary and poor people won’t always make it. The second step is the amount spent, which is inherently random. Amounts spent above a threshold lead to financial catastrophe (this threshold is defined by various means depending on how you spend income) and the intensity of expenditure is determined by the threshold. In the double hurdle model the decision to spend may be assigned a distribution, and the amount spent is often Gamma-distributed with a high probability of low cost and a small probability of extremely high cost.
In both cases (Iron Kingdoms or out-of-pocket expenditure analysis) the problem is made more complex by the fact that we don’t usually know the thresholds. Usually in the double hurdle model we’re interested in identify risk factors for exceeding the threshold. Typically in Iron Kingdoms we want to know which decision to boost to get over the second threshold – should we boost the consumption (attack) or expenditure (damage) decisions? We’re also often interested in guessing the threshold values – the GM knows them but we don’t, and we may for example roll a 9 and fail to hit, or hit on an 8 but do no damage on a 9, and then someone else boosts and hits on a 9 but does damage on a roll of 15, so the question is – what is the armour threshold?
In my last Iron Kingdoms session this came up in a beautiful way: our opponent was going to finish off the entire group if it lasted another round, and Alyvia had one feat point left. Unboosted, she was guaranteed to achieve nothing. We knew our enemy was hard to hit and hard to damage, but we didn’t know the exact values. What should she spend her last feat point on? Naturally, since I’m a statistician in my day job, all eyes turned to me. What to do? This sparked a new interest for me: I think there are methods that can be used to answer these questions. So, over the next few weeks I aim to do a few analyses to present some answers to the following questions:
- Under assumed thresholds and attack/damage values, what are the best ways to spend your feat point budget?
- Are there guidelines for these decisions when you don’t know the thresholds but have a rough idea of what they might be?
- If you don’t know the thresholds, are there simple formulas you can use to guess what they are, or to assign probabilities to given thresholds, given that you know the results of other players’ rolls?
- Can these ideas be extended beyond Iron Kingdoms to other games?
The first question can be answered easily using basic probability theory. The second and third problems are actually a slightly challenging problem in estimating boundary values of a distribution using Bayesian statistical analysis, and I’m going to have a crack at it. The fourth question is related to the third, and is most easily explored through d20/Pathfinder: in this case my naive guess is that you can set a uniform distribution on the prior probability of any threshold value, and because the observed values (the likelihood) are also uniform, get a uniformly distributed posterior distribution for the threshold given the observed data (other players’ rolls). I think I will work from this example back to the Iron Kingdoms example (which may require simulation). If the fourth question has an analytical solution it will lead to a formula I can post on the Pathfinder forums that will allow players to second-guess their GMs’ monsters, and my guess is that a party of 3+ PCs can work out the most likely threshold required to hit within a round of combat. That’s a convenient little trick right there!
Finally, it’s possible that this information may be actually informative for the out-of-pocket spending problem, which I occasionally study at work. I doubt it, but wouldn’t it be great if random ponderings on gaming helped to improve our understanding of health insurance issues in Bangladesh?!
Stay tuned for some Bayesian nastiness, if I can find the time over the next few weeks …