Most of the current game theory for D&D 4E is focused on measuring and comparing each character's offensive prowess. There are several reasons for this: damage-dealing is the most common role in adventuring parties, and quantifying the result of an attack is relatively straightforward. Also, players just love to destroy monsters (and peers) with absurdly high damage numbers. But even the most well-oiled killing machine needs some degree of defensive capabilities, lest his enemies knock him down in a single turn (as certain players in my group can attest).
But how do you measure survivability? It's trivial to come up with damage as the universal unit for comparing attacks, but which one should be used for toughness? It should factor in Hit Points, Armor Class, and the other three defenses - and also mechanics such as resistance or temporary hit points. After considering the issue for a while, I think I have come up with a good solution: we can (and should) calculate the number of turns that a character can endure monster attacks before becoming unconscious. The basic formula would have the following form:
Survivability = Character HP / Enemy's average attack damage
S = HP / monster DPR
S (survivability), as defined above, provides a good estimate of the effort required to bring a character down. Players can use it to have a better knowledge of the resilience of each member of a party, or of different builds for a certain character. As an example, it would be a useful tool to determine whether it is better to invest in higher defenses, hit points, or resistance, at any given time.
One disclaimer, though: there are aspects of a character's resilience that aren't covered by this stat. Specifically, it doesn't describe how well a character staves off negative conditions - for that, you need to resort to AC and the other defenses. For this reason, when you see two characters with the same survivability, both will be able to endure the same number of attacks, but the one with higher defenses (and, conversely, lower HP and resistance) will suffer less from hindering effects.
This Survivability formula implicitly assumes that the enemy attacks are targeting a specific defense (since we wouldn't be able to calculate average damage otherwise). For this reason, a character will actually have four different S values: S(AC), S(For), S(Ref) and S(Wil). We can define overall survivability (or just S) as the weighted sum of all defense-specific S values:
S= 1/2* S(AC) + 1/6* S(For) + 1/6*S(Ref) + 1/6*S(Wil)
Note that Fortitude, Reflex and Will are all worth the same, and AC counts as much as the three together. This roughly reflects the frequency with which monsters tend to attack each defense - it may not be exact, but I think it's close enough.
The mandatory math section
If you want to play a bit with the idea of survivability, find out actual values for your PCs, and see how long a bunch of theoretical Skirmishers of your level would need to kill you, the quickest way is to go to my online Survivability calculator and start introducing stats. On the other hand, if you are curious as to how it works, this is how the formulas break down:
S(defense) = HP / monster DPR
,where HP depends on the measured character, and monster DPR is (as shown in my monster DPR post, considering standard damage for a skirmisher):
DPR=Hit rate * (8 + 0.5*Level)
I have ignored the monster crit damage because the difference is minor, and it somewhat simplified the code. I'll add it back someday - for completeness' sake, this is how it would look like:
DPR=Hit rate * (8 + 0.5*Level) + 0.225 + 0.01 *lvl
We still need to find out the hit rate. According to the formulas from my post about AC normalization, it would be:
Hit rate (AC) = (26-nAC)/20
Hit rate (Def) = (24-nDef)/20
Or, using plain AC and defenses instead of normalized ones:
Hit rate (AC) = (26 + level -AC)/20
Hit rate (Def) = (24 + level-Def)/20
This leaves us with:
S(AC) = HP*20 / (26 + level -AC)*(8 + 0.5*level)
S(Def) = HP*20 / (24 + level -Def)*(8 + 0.5*level)
Finally, if we want to take a character's resistances into account, we just need to substract any resistance value from the hit damage:
S(AC) = HP*20 / (26 + level -AC)*(8 + 0.5*level - resistance)
S(Def) = HP*20 / (24 + level -Def)*(8 + 0.5*level - resistance)
The above formulas are accurate except for extremey high values of resistance. In particular, any resistance greater than the monster bonus to damage will not be as effective as the formulas show, because monsters never do negative damage. This way, a creature which deals 1d10-5 damage after substracting resist would not average 0.5 damage (5.5 average from the die, -5), but a few points more than that. This is a rare occurrence because such high resistances are very hard to get outside of short term powers, but you should be aware that it exists.
In addition to adding critical damage, there are a few improvements that could be added to this model (and to the calculator!). The most urgent would be the calculation of S values for characters (usually defenders) that can gain temporary hit points each time they attack. This is slightly more complicated than the basic case, as it depends on the character's own hit rate, and varies depending on the number of attacking enemies. I'll write about that in the following days, and add it to the code as soon as I can.
Apart from that, I'd like to hear some reader feedback. Can you think of ways to improve this model? Let me know!