## Wednesday, January 11, 2012

### Game Math: Attack of the average adventurers

How hard does an adventurer hit? The underlying math behind monster stats in D&D 4E is well known by this point, but working out the numbers for player characters is a much trickier proposition, due to the insane amount of customizability that the game offers. Indeed, if one looks to the most extreme build options and loopholes available out there, we come up with characters that can one-shot standard monsters of their level, and it’s even possible to engineer wizards capable of dealing hundreds of damage in a single turn... while charging with a melee weapon.

Since the optimized scenarios present so much variability, I want to focus on the most basic builds. What kind of numbers should you expect from characters of a certain level, provided they have the essential options and gear, but nothing else to boost their attacks? The answer is in the table below:

Note that this builds on some previous articles, like my study on basic attacks, and on character survivability. Damage Per Round, or DPR, is defined here.

I will devote the rest of the article to analyze these numbers, and explain how I came up with them. For now, keep in mind that although it is possible to have characters with attack stats slightly below these, it won’t be a common scenario - in most games, you can expect PC attacks to deal at least as much damage as shown in the table, if not considerably more.

My philosophy for building the reference character was to give it all the offensive resources that can be considered essential - and nothing more. Any character built without particular attention to damage dealing should have attack stats very similar to those of the reference character, whereas damage-focused PCs (even if not particularly optimized) should easily outperform it. For reference, I also added damage numbers for a baseline striker (adding just the striker extra damage class feature), though in practice you will very rarely see a striker character which doesn’t devote feats, powers and equipment to improve his attacks.

Following these ideas, the character was build under these assumptions:

• For simplicity, we only examine the character at certain critical levels: 1, 6, 11, 16, 21, 26, 30.
• Starting 18 on primary ability score, with the usual boosts for levelling up. At epic, the PC gains a +2 to his ability score from an epic destiny .
• At-will attacks are implement vs For/Ref/Will or weapon (with +2 proficiency) vs AC (same hit rate), and on a hit deal 2d4 + primary ability modifier damage (4d4+mod at level 21).
• Encounter attacks are like at-will, but on a hit they deal 4d4+mod damage (levels 1,3,7), 6d4+mod (levels 11, 13,17) or 8d4+mod (levels 23,27).
• Magic weapons/implements by level: Level 6 (+2), Level 11 (+3), Level 16 (+4), Level 21 (+5), Level 26 (+6). Extra crit damage is 4 per plus (rounded for convenience).
• Only two feats are considered: Weapon/Implement expertise, and Weapon/Implement focus. Both are gain at level 6.
• Magic items granting item bonus to damage (like Iron Armbands of Power or Rod of Ruin) are assumed. Item bonus to damage by level is: Level 6 (+2), Level 16 (+4), Level 26 (+6).
• For the striker damage numbers, a class damage bonus of +4/tier is added.
• Character themes are not considered.

Most of these points represent very common choices. The use of 2d4 for attack damage is unusual, but I chose it because the most common damage dice are d8s and d10s, so this is an intermediate point between those, with the advantage of averaging an integer value ( 5), allowing for much cleaner results. The starting 18 ability score is more or less standard (though 20s are also common). Likewise, the magic weapon/implement progression and use of expertise feats are pretty much universal. The most controversial points are probably the addition of weapon/implement focus and item bonuses to damage. Focus feats are often ignored at lower levels (though usually because players take superior weapons instead, wich are roughly equivalent), but tend to become too tempting to pass on by paragon tier. As for item bonuses to damage, virtually every character who can take them does so, though certain builds (i.e. implement PCs not using staves) have a hard time acquiring them. I have come to accept that game math works better with them, to the point of giving them for free as a house rule.

Note that attack powers (both at-will and encounter) have been greatly simplified. Non-damaging effects of the attacks are ignored, and we assume that the attacks themselves don’t provide extra damage, above that of a basic attack (for at-wills) or a basic attack plus extra damage dice (for encounters). Power damage shows great variance, though a very common implementation for attacks with extra damage consists on adding a secondary ability modifier to the damage roll, which can be roughly approximated as an extra 3 damage per tier (or about 25% more damage than the attacks in the table).

A look at the numbers

A spreadsheet with the calculations used for the stat table can be found here.

Some quick facts that can be derived from the table and spreadsheet:

• Hit rate against same level monsters averages 65%.
• Base damage of strikers is about 30% more than that of non-strikers.
• Average base damage actually decreases a bit between level levels 26 and 30.
• Interestingly, crit damage is almost exactly twice the normal damage for most levels.

One very interesting parameter that can be calculated from these attack stats is the average time it takes for a PC to kill a monster, which allows us to estimate how many turns combat encounters usually last:

And for striker PCs:

These tables show the number of turns that the reference character would need, on average, to kill a skirmisher monster. The PC first uses all his encounter powers, and then attacks with his at-wills. Daily attacks were not considered because they are hard to characterize and not always available. Action point usage is also ignored - for the purposes of this table, using an action point is equivalent to taking an extra turn.

In the tables and spreadsheet, we see that:

• Non-strikers take 4-6 turns to kill a monster at heroic, 7-8 at paragon, and 8-10 at epic.
• Strikers take 25% less turns.
• Very challenging encounters can take almost twice as long.
• The contribution of encounter attacks amounts to about 1 turn of saved time.

An important point about these numbers is that they suggest that combat at paragon and epic tiers takes too long. In my opinion, easy encounters (between level and level +1) should last about 4 turns at heroic and 5-6 turns at paragon and epic, to allow characters to use all their attacks without requiring them to spend a long time spamming at-wills. However, it should be noted that the deviation between the stats of our base character and actual damage-focused PCs increases with level - in my experience, it is perfectly possible to build a party of characters that end fights in reasonable times with little or no optimization effort.

1. Interesting article. Also notable is the effect of multi-attacks and out of turn attacks, which are commonly employed by strikers but may be available to PCs of any role. This is very important for reducing the amount of rounds it takes to kill things (obviously the numbers in your tables are higher than ideal, leading to overly long combats). Furthermore, the higher your level the more likely you are to gain extra attacks, which is why in reality combat at higher levels doesn't take all that much longer (factoring out option paralysis, which is of course more of a player issue).

2. The thing about multi-attacks and off-turn attacks is that they have been applied very inconsistently - The original ranger was plagued with them, but other than that, they tend to appear rather randomly across strikers and non-strikers alike. And, of course, their power level is off the charts, compared to your standard, "single attack plus a bunch of extra dice" attacks that make up 90% of encounter powers in the game. So each class has a random number of god-tier powers for the purpose of damage dealing - hardly the ideal model for handling striker balance.

If I could rewrite the game today, I would:
- Change a few hundred powers (multiattacks, minor action attacks, reactions, granted basic attacks) to prevent PCs from making more than 1 damage roll per turn (barring action points).
- Set a goal for encounter length (say, 5 rounds at heroic, 6 at paragon/epic), and use it to calculate an ideal PC damage per round number for each level. Adjust damage bonuses to match that number.
- Adjust area attacks to deal a bit less damage than single-target ones (say, 75%)

Unfortunately, I don't think the 4E designers bothered to do that -or, if they did, something went horribly wrong along the way. With unknown amounts of multiattacks for each class, damage scaling is all over the place, so it's not possible to just adjust damage bonuses and expect it to work fine for all classes...

Anyway, I'm deviating from the topic at hand. Multiattacks, along with lots of damage bonuses that haven't been considered here, make high level combat much shorter than the numbers above indicate. However, this is build-dependent and requires a bit of system mastery. A group of inexperienced players could conceivably generate a high level party with damage numbers as low as these, and get bored to death as a result. The game system shouldn't allow for that.

3. I've never had a problem with my current epic game's combats taking too long. The slayer isn't terribly optimized, though the executioner/artificer takes advantage of both a couple of minor-action attack powers and off-turn attacks (synergizing his striker feature).

4. Thanks for the great article as always. I think you've left out the link to the spreadsheet by the way. I'd quite like to see it.