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.

Building our baseline adventurer

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.

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A better formula for minion damage.

Ever since the monster creation guidelines received errata to cover the new math from Monster Manual 3, I had wondered what had happened with minion damage. The new tables covered other types of monsters, but not minions, so we could only guess what was expected from a modern minion - was the table in Dungeon Master’s Guide 2 still valid, or were new monsters following a different, unknown math? Minions are far less common than normal monsters (particularly at higher levels), so sample monsters in rule books tend to be scarce, complicating the task of verifying the creation guidelines. Still, I found that MM3 had just enough such monsters to prove that the previous formulas were slightly off. Even better, it was possible to come up with a new formula that was a better match for the new monsters, but also for those in Monster Manual 2! Not only that, but it was pretty straightforward, and could be related to that of standard monsters. It was the following:

Minion Damage (normal) = 4+ Level/2

Extending it for brutes and areas

Under Monster Manual 3 rules, brute and limited damage expressions are a 25% higher than normal, whereas low and area damage is 25% lower. This leaves us with the following formulas:

Minion Damage

• Low = 3+ 0.4*Level
• Normal = 4+ 0.5*Level
• High = 5+ 0.6*Level

I have done a bit of rounding for clarity, which shouldn’t add a significant error. As usual in 4E, the formula results should be rounded down. At lower levels, this should be pretty close to the previous table, but for paragon and epic monsters it brings an increase of a couple points of damage.

Interestingly, these formulas are roughly half the damage you’d get from a standard, non-minion monster, which should provide some insight on a minion’s relative value. Compared to a single monster, four minions deal twice as much damage, but get weaker with each hit they take. This suggests that the damage contribution of a group of minions would match that of a normal monster if both took 4-5 turns to kill - though this is rarely the case, as minions’ pitiful survivability means that all are usually dead by the second round of combat.

Some pretty diagrams

I came up with the current formula by analyzing the chart below, which shows the damage values for all non-brute minions in monster manuals 2 and 3, ordered by level. For most levels, there is only a single minion in each book (MM2-1 and MM3-1), but sometimes there were additional ones, which I have displayed as MM2-2 and MM3-2.

As we can see, even though there are a few monsters that show moderate deviations from the norm, most follow a trend of linear increment. Including the formula (4+ 0.5*Level) as blue dots, we get that most values fall over it, or very close to it.

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New monster damage formulas

I have previously reported on how Monster Manual 3 brings a new monster design principle, with several adjustments to monster roles and an overall increase in damage. There was an important detail that was missing, though - the specific damage values under these new principles. This has been addressed in the latest round of rule updates, which introduce a new damage table for monsters and traps, replacing the one in page 185 of Dungeon Master’s Guide.

A nice thing about this new table is that it streamlines the process - instead of two tables with entries for low, medium and high damage each, there is just a single table with values for single-target damage and area damage. For brutes and limited attacks, you just add a 25% extra damage (or up to 50%, for certain limited attacks).

As for the actual numbers, they are much more streamlined - since the tables now have rows for each level, the increases in damage are much more smooth. In fact, I was able to determine the formula for average standard damage by simply looking at the table: it is now just about eight plus monster level. This is how it breaks down, for the different attacks. For a look at the old formulas, you can check this.

- Low (Area): 6 + 0.75*Level

- Medium: 8 + Level

- High (Brute/Limited): 10 +1.25*Level

- Very High (Limited): 12+1.5*Level

It doesn’t look like minion damage is affected, though I’ll try to check the damage for MM3 minions against the formulas from Dungeon Master’s Guide 2, and see if they fit.

So, now that we have these, we can tell how much the damage has really increased. Early reports from D&D editor Greg Bilsland pointed at an increase between 30% and 40%, but actual values deviate a bit from this, and vary with level:

As we can see from the table, the increase varies between 10 and 30% for heroic levels, 40-60% at paragon tier, and 60-70% at epic. We should be able to appreciate a difference at lower levels, but it is at paragon and beyond where the game experience will change the most. In future posts, I will analyze how this affects character survivability, and difficulty in general.

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How much damage is a basic attack worth?

A while ago, we discussed some common methods for calculating the damage dealt by a character on average. However, it is common, particularly for leader classes, to have powers that deal a certain amount of damage and allow one or more allies to make an extra attack. Estimating the exact value of these powers is usually difficult, since it varies with party composition and the specific allies chosen. In today’s article, I’d like to provide some basic guidelines about how much damage to expect from a player character’s basic attack, depending on level.

Although the most immediate application is the evaluation of ‘extra attack’ powers, as mentioned above, knowing about average PC attacks can be useful in other ways. We could, for example, use it to estimate how painful it is for certain monsters to provoke opportunity attacks, or to study encounter duration across levels. I’ll take a look at this kind of issues in future posts.

Character stats

The following stats were chosen for this study. They are intended to include all the options that almost every character, regardless of role, normally takes: good ability scores, expertise feats, and feats to increase critical range at epic. Keep in mind that there is much room for improvement here, through specialized feats, paragon paths, and gear, but this should provide a good baseline. If your character is minimally competent at making basic attacks, he should perform at least as well as the numbers below show - or significantly better, if he focuses on boosting accuracy and damage.

Starting main ability score: 18. The character is assumed to make basic attacks with the best (main) ability score. For some classes or builds, this may require taking melee training at level 1. The ability modifier will increase at levels 8, 14, 21 and 28, as usual.

• Attack: Weapon attack vs AC, +2 proficiency bonus, 1d10 damage die, or Implement attack vs For/Ref/Wil, 1d10 damage die. Both are equivalent.


• Expertise: The character gains an appropiate expertise feat at level 5.


• Magic Weapon/Implement: The character gains a +1 weapon/Implement at level 2, and upgrades it at levels 6,11,16,21 and 26. No special enchantments are taken into account, but the crit damage is 1d8 per plus.


• Epic Tier: At level 21, the basic attack deals an extra d10 damage. Also, the critical hit range rises from 5% to 10% - this assumes the character takes a weapon mastery feat or its equivalent for the chosen implement/class.

In addition to these almost mandatory stats, we have considered the extremely common scenario where a character has a magic item granting item bonuses to damage, such as Iron Armbands of Power, Bracers of Archery, Staff of Ruin. We’ll provide damage values for both cases.

• Option - Item bonus to damage: The character gains a +1 item bonus to damage at level 2, and increases it by 1 at levels 6,11,16,21 and 26.

Hit Damage

With the stats described above, we have the following damage progressions. Only average damage on a hit is considered:

Since these tables are a bit cumbersome, and damage increases very linearly, we could approximate the values with simple formulas:

BA Damage (levels 1-20) = 9 + 0.6*Lvl
BA Damage (levels 21-30) = 14 + 0.6*Lvl

BA Damage (Item bonus, levels 1-20) = 9 + 0.6*Lvl
BA Damage (Item bonus, levels 21-30) = 14 + 0.6*Lvl

As we can see in the figures below, the formulas don’t deviate too much from the previous table. As a reference, we have also included monster damage progression, as calculated here.

Since our basic attack represents the lower bound of a PC’s damage, we can see that players will usually deal more damage than monsters of their level, even before taking at-will or encounter powers into account. However, this will not be necessarily true with monsters from Monster Manual 3 and later books, since their damage is higher - though we still haven’t figured out the exact numbers.

Hit Rate

The following figure shows the attack bonuses for our character’s basic attacks throughout levels, and the corresponding hit rates. These have been calculated as explained here. As usual, the character is assumed to be attacking a skirmisher monster of his level. Note that attack bonuses correspond with an implement attack; a weapon attack vs AC would have bonuses higher by 2 points. The chance to hit is the same for weapon and implement attacks, nevertheless.

As you can see, hit rates average 60%, never deviating more than 5% from that, and it tends to be slightly better at heroic levels, and slightly worse at epic.

Damage per Round

The next step, then, is to translate this data into actual damage dealt. We talked in depth about Damage-per-Round calculations in a previous article. The simplified formula that applies here would be:

Average Damage= (Hit rate * Average Hit Damage) + Crit Rate * Extra Crit Damage

We are still missing crit rates and damage. The rates are easy enough - the baseline 1/20 chance for heroic and paragon tiers, which increases to 10% with the appropiate feats at epic. As for the extra damage on a crit, it should amount to 1d8 per plus of the weapon, plus whatever damage you gain from maximizing each d10 (which is worth 4.5 average damage).

The table below shows DPR for characters with and without item bonuses to damage:

Adding striker damage, two-handed weapons

There are a LOT of ways to improve a character’s damage and accuracy beyond the minimum levels provided here. As we explained above, it’s not practical to go into every possible combination, so I chose to go for a straightforward build that can nevertheless be useful as a practical benchmark. That said, there are two hugely frequent character choices that affect attack performance and I wanted to comment: striker damage features, and two-handed weapons.

When talking about striker features, I refer specifically to those that directly add extra damage. Although there is a variety of these, the typical feature grants an extra d6 per tier to the damage roll, or some equivalent amount.

As for two-handed weapons, there is obviously a wide variety of them, including superior weapons, so it’s not easy to settle for a set of stats. Since our one-handed weapon attack assumed a +2 proficiency, 1d10 damage weapon, I’ve gone for the direct damage upgrade without feat investment: a +2 proficiency, 2d6 damage weapon. This would deal an extra 1.5 damage per [W].

The table below shows the increases in hit damage and average damage per attack from these options:

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Changes in Monster Manual 3: Accurate brutes, more damage

UPDATE (12-06-10) - After looking at the monster stats from the official MM3 previews, it looks like the damage increase is even greater than announced. The difference seems to be about 50% more at Paragon Tier, and up to 66% at epic. I've provided a table with the data below.

Monster Manual 3 is coming out soon - and, in fact some stores already have it. Judging from the comments of the lucky owners of early copies, this is going to be an even better book than the previous two in the series. To begin with, there's the revised format for monster stat blocks (presented here and commented here, though both links are sadly for DDI subscribers only), which presents monster information in a much more clear way - this is quite a blessing for the DM, as it greatly reduces the chance of forgetting a crucial ability. But probably the most significant change is one that affects players and DMs alike: an overall increase in monster damage.

I can't give first hand information as I still don't have Monster Manual 3, so the details will have to wait for a future post. What we know, for know, is that damage from all monsters (except maybe those in heroic tier?) is going up by 30-40%, and the balance between monster roles will be shifting, with Brutes (typically seen as the weakest type of monsters) now making attacks at normal accuracy (rather than at a -2, compared to other roles).

The source is as official as it gets: Greg Bilsland, D&D game editor and one of the authors of MM3, has commented the issue on an interview at Critical Hits, as well as his own blog. Here are some of the highlights:

"...we increased monster damage output by about 30-40%."
"...most older monsters up until about level 10 are performing just fine. It’s only around paragon tier that the damage really needs some adjustments."
"We reexamined the various roles and ended up adjusting brute accuracy back to baseline"

Without having seen the changes in play, I think they made a good call. Our campaing is still finishing heroic tier, but we're already starting to feel a decrease in monster threat. And I recently posted an analysis showing that the time it takes for a monster to kill a PC more than doubles between level 1 and 30. The greater monster damage will be a change for the good, particularly if they make the increase proportional to level, that is, you add more damage percentage-wise at paragon and epic tier. And anything that makes brutes more balanced with other roles (and not downright unplayable against higher level parties) is very welcome.

Some Tentative Formulas.

A while ago, I worked out a few simple formulas to describe the average monster damage across levels. I thought it would be an interesting exercise to see how these might look like, for post-MM3 monsters. I have applied a linear increase to them, so that level 1 monsters deal the same damage, whereas level 30 ones get the full 40% boost, and we have the following:

Attack damage, normal attacks

• Low: 6 + 0.6*Level
• Medium: 7.5 + 0.85*Level
• High: 8.5 + 1*Level

A quick patch for old monsters

Following the formulas above, we can patch a current monster to MM3 levels by adding a fixed amount of damage to its attacks. The following values can work:

Monster damage increase

• Low: 1/5 * Level
• Medium: 1/3 * Level
• High: 2/5 * Level

You should modify these accordingly if a monster's damage is split among multiple attacks (i.e. divide the bonus by the number of attacks), unless it's an elite or solo, in which case you should probably add the full bonus.

Update: Sampling actual monsters

Out of curiosity, I checked these numbers against the monsters from the official previews. Sample size is small (just 7), but the results are consistently higher than expected. The damage increase seems to be about 50% for paragon monster, and 65% for epics.

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Character Survivability Calculator (advanced)

UPDATE (16/05/10): Fixed script formulas for sustained damage)
UPDATE (01/12): Updated for MM3 monster stats

I've been making some improvements to my calculator of character survivability, and now we can incorporate PCs with attacks that grant temporary hit points, resistances, or defense bonuses on a hit, that depend on the PC's chance to hit. Defender characters (who are typically the most interested in measuring their endurance) should find this particularly useful, since most make regular use of these mechanics.

I've chosen to focus exclusively on effects that a character can grant himself, rather than external bonuses and healing. That stuff is left for future posts. If your character isn't interested in gaining THPs and the like, you'll probably prefer the basic version of the calculator, which is still available here.

An explanation of the new formulas used will be added soon.

Character Stats:

HP:Resistance: Regeneration:

AC:For:Ref:Wil:

Conditional damage mitigation on a hit:

Bonus to AC:, to For:, to Ref:, to Wil:

Resistance/THP: Attack Bonus: vs AC For/Ref/Wil

Chance to Mitigate:Sustained THP?: No Sustained: 2 turns Sustained: 3 turns Sustained: inf. turns

Chance to mitigate damage:

Chance to sustain THP:

Monster Stats:

Level: Type: Skirmisher Soldier Brute

Marked Combat Advantage

Average monster DPR:

Number of turns it takes a Skirmisher monster of level 1 to kill you:

Attacking vs AC: vs For: vs Ref: vs Wil:

Overall survivability:turns.

Chance for a Skirmisher monster of level 1 to hit you:

Attacking vs AC: vs For: vs Ref: vs Wil:

Overall Hit %:

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Character survivability across levels

After defining the concept of character survivability (and providing a tool for calculating it), I think it's time to explore its applications. Today I'll study the S (survivability) values of three different characters, representative of each party role: a defender, a controller, and a striker/leader (since both roles share similar amounts of HP and defenses). I'll compare their progression across the 30 levels of the game, examining survivability for individual defenses as well as overall survivability.

I'd like to find out whether the relative resilience of each role remains stable throughout tiers, or it presents large variability due to scaling differences. Also, I want to analyze absolute survivability values, and their implications on combat length - having defined the metric as the number of turns until a PC is killed by a monster of the same level, these numbers should provide a very rough estimation (i.e. before accounting for healing and stunned monsters) of the maximum duration of an encounter of the PC's level. The minimum duration, of course, would be dependant on the party DPR... but that is a story for another day. Let's proceed with the experiment.

The sample characters.

One requirement of this study is to have a set of sample characters whose survivability values could be representative of their role. Since defensive statistics can vary considerably, even for a given class/race combination, depending on the choice of feats and items, this is no trivial task. In order to keep complexity at a manageable level, we will make the following assumptions:

- Each character starts at level 1 with average defensive stats for their role.

- It is assumed that the character takes the feats and ability increases necesary to keep AC stable as they level up. Other than that, no survivability-boosting feats or items are considered. Any actual PC who invests in survivability should show values above these, particularly at higher levels.

- The leader role and striker role are considered equivalent from a survivability point of view. From now on, when I mention the "striker" character, it applies to either role.

This is how we assigned values for each statistic:

Hit Points

- Defender: 30 + 6 per level above the first. This assumes a starting Con of 15

- Striker: 24 + 5 per level above the first. This assumes a starting Con of 12

- Controller: 22 + 4 per level above the first. This assumes a starting Con of 12.

For simplicity, any posible increases in Constitution haven't been taken into account. The impact should be minimal, relative to total amounts of HP.

AC

- Defender: Normalized AC of 18, equivalent to AC 19 (scale + shield) at level 1.

- Striker: Normalized AC of 15, equivalent to AC of 16 (chain, or leather and +4 starting ability modifier) at level 1.

- Controller: Normalized AC of 15, equivalent to AC of 16 (chain, or leather and +4 starting ability modifier) at level 1.

It is impossible to keep normalized AC completely stable across levels, but the deviations are usually minor. I have opted for using constant values because these variations depend more on the choice of heavy or light armor than on actual role, and are usually compensated over a character's whole career.

Non-AC Defenses

For Fortitude, Reflex and Will, I decided to make a small abstraction, and consider only "good defenses" and "bad defenses", regardless of how they were allocated. I have also considered them role-independent, so all three characters show the same progression. The progression itself was taken from the values generated for the character PC2 in this article. PC 2 is a character with 2 "good" defenses, and we defined it as follows:

"The second sample character, PC2, has well-balanced NADs, with only one bad defense and a starting ability array of 16, 16, 12, 12 (pre-racials). This character would have racial bonuses to both primary and secondary abilities, and a +2 class bonus to the best defense, for starting NAD scores of 16, 14, and 11 (from best to worst defense). This character could be a Whirling Barbarian or a Chaos Sorcerer, among others."

Only the best and worst defense for PC2 have been used (as "good" and "bad" values, respectively), discarding the middle one. For the overall survivability of our three characters, we will make two calculations: one assuming they have two good defenses, and another assuming they have two bad ones. This introduces a small error, since the best defense of a character tends to be slightly higher if it is the only good one but, again, the gain in simplicity outweights the loss in accuracy.

The results

This graph shows the Survivability progression for attacks against AC:

This is where the greatest differences among roles show up. As we can see, the values for defenders roughly double those for controllers. As for strikers they start out very close to controllers, but slowly overtake them as the difference in HP grows more important over levels, capping at around 20% more at level 30.

Also interesting is the fact that absolute survivability values experiment considerable growth, more than doubling between level 1 and 30. This is in spite of the fact that the monster hit rates remain constant - the relation between PC hit points and monster damage becoming more and more favorable to the PCs.

The following figures show S for "good" and "bad" non-AC defenses:

Since non-AC defenses don't vary depending on role, the relative S values here are much closer, reflecting only the HP diferences of each role. Keep in mind that normalized defenses are actually diminishing as the character levels up (specially for the bad defense), but overall survivability still grows because of the unbalance between PC hit points and monster damage. Nevertheless, this growth is not as pronounced as AC survivability, and values at maximum level end up slightly below double those of level 1. Also of note is the fact that defenders are much more vulnerable against NAD attacks than against AC attacks, whereas strikers and controllers find their AC roughly equivalent to their good NADs. Bad defenses, as expected, are terrible for all characters, though it is interesting to see that their S values remain more or less constant, rather than getting worse as the characters level up.

Finally, we can average the values shown above to figure out the overall survivability of each character. As we can see, the differences between the builds with 2 good NADs and 1 good NAD aren't actually very significant, with regards to overall survivability. This can be explained by the fact that having an additional good NAD will only come up on 1/6 of attacks. Nevertheless, these figures hide other negative consequences of bad NADs, such as the higher variability among encounters (increasing the chance that enemies auto-hit you), as well as a huge vulnerability to attacks that inflict status effects (as opposed to raw damage, which is what S measures).

It is also possible that the coefficients used to average overall S (1/3 for AC, 1/6 for other defenses) need to be adjusted at paragon and epic tiers, since there seems to be a greater amount of NAD attacks for monsters of these tiers. However, since I lack solid data on that matter, I'll stick with my original approximation until I find some statistical analysis of the different monster manuals.

Conclusions

The relative resilience of the different roles is more or less as could be expected, with defenders being twice as tough, and controllers falling slightly behind strikers. However, we must remember that these are just baseline values - for real builds, non-defenders tend to have a lot more potential for improvement, specially for AC, so the gap with a defender should be much narrower if a player decides to invest in resilience. On the other hand, the temporary hit point gaining of some defenders are more rare among other roles, and would grant them an edge. The same could be said of damage resistance. I'll try to talk about this in a future post.

As for the implications of survivability on encounter length, we find that a group of same-level monsters will need more than twice the amount of time to kill a level 30 party, compared to one of level 1. That is, if the level 1 PCs can expect the monsters to defeat them after 6 or 7 turns of combat, the level 30 ones would live for more than a dozen turns. That is, of course, assuming that the PCs just stand around doing nothing - in an actual encounter, PCs will get healed, and monsters will gradually die, so the real time before the defeat should be significantly higher.

We can also apply these results to analyze the effect of focused monster attacks on a single character. Using the standard of 5 characters per party, the monster group would be able to knock out a level 1 non-defender in a single round, whereas the defender would survive for another round. At epic tier, the non-defender would endure 2 whole rounds of punishment, and the defender more than 3.

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Character Math: Measuring Survivability in Turns to Die.

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

or, abbreviated:

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.

Future refinements

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!

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Character Survivability Calculator

Update: A more advanced version of the calculator has been released, featuring compatibility with temporary hit points, regeneration, and defense-boosting attacks. You can find it here.

Here is a tool I've been working on lately. Just type in your PC's stats, and it will calculate the time a skirmisher monster of a given level needs to bring you to 0 HP. It's still a bit rough, as it doesn't take into account crits, temporary HP, or high resistance values. Still, I have found it both useful and entertaining.
Expect discussion of the underlying math, the approximations involved, and the (brand new!) concept of character survivability in an upcoming post. Until then... enjoy!

Note: I have tested this with Firefox and Chrome. Unfortunately, it doesn't seem to work yet with Explorer, but I'm working on it.

Character Stats:

HP:Resistance:

AC:For:Ref:Wil:

Monster Stats:

Level: Type: Skirmisher Soldier Brute

Marked Combat Advantage

Average monster DPR:

Number of turns it takes a Skirmisher monster of level 1 to kill you:

Attacking vs AC: vs For: vs Ref: vs Wil:

Overall survivability:turns.

Chance for a Skirmisher monster of level 1 to hit you:

Attacking vs AC: vs For: vs Ref: vs Wil:

Overall Hit %:

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Monster Math: Damage Formulas

The most important statistics of a monster in D&D 4E can be easily derived from simple formulas depending on level and role: Attack bonuses, defenses, hit points... with one remarkable exception: attack damage. In order to determine a monster's damage, you need to consult a couple of tables in the Dungeon Master's Guide (p.184, to be precise). Because these tables only show you an expression with damage dice plus a bonus, which is not immediate to evaluate, a while ago I generated new set of tables with the average damage values for each level. Today, I'd like to go one step further, and reduce these tables to short, approximate formulas.

These are the formulas for normal attacks; I'll leave limited attacks for a future post. I provide some comments and discussion on their deviations from the original values below:

Attack damage, normal attacks

• Low: 6.1 + 0.4*Level
• Medium: 8 + 0.5*Level
• High: 9.4 + 0.6*Level

Minion Damage

• Low: 3 + (Level -1)/3
• Medium: 3 + (Level -1)/3 + (1* Tier)
• High: 3 + (Level -1)/3 + (2* Tier)

Explanation

The formulas for regular monsters came up naturally. Once you see the averages, the progression is almost linear, so I just drew a line from the level 1 values to the level 30 ones, and tweaked the numbers a bit so that they ended up as rounded as possible. I was pleased to see that damage actually increases by 1 per two levels, as per the monster customization guidelines in DMG, p.174. Anyway, this is how the formulas look like, compared to the tables:

If we substract the estimated (formula) values and the original ones, the absolute value of the result is the estimation error, in damage points. I divided these absolute errors by the original damage values, to come up with relative error values, as a percentage of these original values. This is shown below:

As you can see, these relative errors peak at 15%, staying at 10% or below for most levels. Though not perfect, I find this quite acceptable. In absolute terms, the aproximation is off by a maximum of 2 points of damage for medium attacks, or 2.6 points for strong attacks - but for most levels, it will be much closer.

As for the minion formula, it comes from the tables in DMG2, p.133. In this case, damage increased by 1 per 3 levels, with an additional jump of 1 extra point at certain levels. Since these jumps came roughly - but not exactly - with the change of tier, I chose to tweak it into an easier to express, more logical progression, by moving these extra increases to levels 11 and 21. Damage for most levels remains unchanged, it's just the 1-2 levels around 11th and 21st that have varied slightly. Also, although there was no 'low damage' entry for minions, they suggested reducing normal damage by 25% for weak attacks. My expression is just a bit above that, but with such numbers, some rounding was unavoidable.

Critical damage

If we want to find the Damage per Round of a monster, we should take into account the contribution from critical hits. I have made formulas for that, too. They are the following:

Crit extra damage, normal attacks

• Low: 2.5 + 0.2*Level
• Medium: 4.5 + 0.2 * Level
• High: 5 + 0.3*Level

I got these the same way as the previous formulas, except I used the maximum damage values from the DMG table rather than the averages. Minions don't get an entry because they lack any extra critical damage.

Calculating DPR

Since monsters don't usually deal damage on a miss, the average DPR can be simplified to the following expression:

Average Damage= (Hit rate * Average Hit Damage) + Crit Rate * Extra Crit Damage

Replacing the known values (Crit Rate= 0.05) we have:

Average Damage (low) = (Hit rate * (6.1 + 0.4*Level)) + 0.125 + 0.01 *lvl
Average Damage (medium)= (Hit rate * (8 + 0.5*Level)) + 0.225 + 0.01 *lvl
Average Damage (high) = (Hit rate * (9.4+ 0.6*Level)) + 0.25 + 0.015 *lvl

In order to simplify the crit contribution, we can make another aproximation and use fixed values for each tier:

Crit damage contribution to DPR, per tier (heroic/paragon/epic):

• Low: 0.2/0.3/0.4
• Medium: 0.3/0.4/0.5
• High: 0.35/0.5/0.65

Average Damage (low) = (Hit rate * (6.1 + 0.4*Level)) + 0.2/0.3/0.4
Average Damage (medium)= (Hit rate * (8 + 0.5*Level)) + 0.3/0.4/0.5
Average Damage (high) = (Hit rate * (9.4+ 0.6*Level)) + 0.35/0.5/0.65 Read More......

Character Math: Damage per round

Today I'm starting yet another article series, this time devoted to the calculation of a player character's statistics. The idea is to provide ways to measure and compare different aspects of a PC's performance, much like the post on attack and defense normalization, a while ago. I have a few cool formulas lying around to quantify stuff that hasn't really been explored in depth... but I should start from the beginning. So this post will focus on damage dealing, which is pretty much all that is looked at, nowadays. You will find some basic definitions and formulas, as well as guidelines for calculating damage in some common special scenarios.

Most message boards have adopted the term "Damage Per Round" (DPR) as a measure of the amount of damage that can be dealt during a game turn. Although I can't tell for sure where this expression originated, it does look awfully similar to the "Damage Per Second" (DPS) commonly used in MMORPGS.

Definitions

• Damage per round (DPR): Average damage inflicted by a character to a single enemy in a round. It takes into account hit rates and extra damage from critical hits. DPR typically measures the abilities of a single character, so bonuses from other party members are ignored.

We usually clasify DPR in two types, depending on whether it involves a character using spendable resources or not: At-will DPR, and Nova DPR.

• At-Will DPR: Damage per round that can be achieved by using exclusively at-will powers. Bonuses from daily or encounter effects do not apply, and spending of action points or other consumable resurces is not considered. Exceptionally, an encounter power granting encounter-lengh effects can be considered a valid modifier because of being permanently active.
  
  At-will DPR is the most commonly used metric in optimization boards. It isn't the most realistic estimation of actual PC damage, since the contribution of encounter and Daily powers is usually very significant. On the other hand, it's relatively straightforward to calculate, and a reliable indicator of damage in the worst case scenario for a damage dealing PC.

• Nova DPR: Damage that can be dealt in a round with all the tools that a single character has at his disposal, including action points and daily powers. Used to measure a character's maximum offensive potential.

In addition, any DPR value can be against a single target (the default assumption) or against multiples (Area DPR).

• Area DPR: When attacking multiple enemies, damage that can be dealt to each of these enemies in a round. Area DPR measurements should specify what kind of area or how many enemies can be covered: "Area DPR (area burst 1)" or "Area DPR (2 targets)".

Calculating DPR

The DPR of a character making a single attack each round is equal to the average damage of that attack. For scenarios with more than one attack per round, see below.

Average Damage of an attack.

The following calculations assume that a character is attacking a standard skirmisher monster of his own level. If you want to find out DPR values against different types of enemies, you will need to adjust Hit Rates accordingly.

Average Damage= (Hit rate * Average Hit Damage) + (1 - Hit Rate) * Average Miss Damage) + Crit Rate * Extra Crit Damage

Or, abbreviated:

AD= HR*Dh + (1-HR)*Dm + CR * Dc

• Hit rate (HR), chance to hit a Skirmisher Monster of the same level. Can be calculated from here. If an attack has an intrinsic bonus to the attack roll, add it to the HR (+0,05 per extra point).


• Crit Rate (CR), chance to score a critical hit.
  
  Roll to crit - Crit rate
  Crit on a 20 - 0.05
  Crit on 19-20 - 0.1
  Crit on 18-20 - 0.15


• Average Hit Damage (Dh). To calculate, add the average value of all rolled dice, and any damage modifiers that apply.
  
  Die - Average value
  d4 - 2.5
  d6 - 3.5
  d8 - 4.5
  d10 -5.5
  d12 -6.5
  
  The Brutal property adds 0.5 to the average value of a die per point of brutal.


• Average Miss Damage (Dm). Usually 0, but can be half the hit damage (Dh/2) for certain Daily attacks, or a fixed falue in certain cases.


• Extra Crit Damage (Dc), or how much more damage is dealt by a critical hit, on average, in excess of the average damage of a normal hit.
  
  Dc = Dh - Average Crit Damage.
  
  (Average Crit Damage is the sum of all maximized damage dice, plus damage modifiers, plus extra crit dice).

Multiple attacks

When your character can make several attacks during a single round, calculate separately the average damage of each one and add them together.


Conditional attacks

When you have an attack that only triggers after hitting with a previous power, multiply that attack's Average Damage by the Hit Rate of the previous attack.

Attacks that trigger on conditions not completely under your control, such as a Fighter's Combat Challenge should not be considered on DPR calculations.

Automatic damage

If an attack deals automatic damage in addition to, or instead of a conventional attack roll, that damage is directly added to the total Average Damage.

Multiple rolls

If an attack allows you to roll twice and use the highest value, the effective Hit Rate and Crit Rate are increased. You can calculate these new values, HR' and CR' from the original HR and CR as follows:

HR' = HR * (1-HR)
CR' = CR * (1-CR)

Once-per-turn damage bonuses

Most striker classes have features that deal extra damage once per round. When calculating DPR, you can add this extra damage to Average Hit Damage only if you are making a single attack per round. When making multiple attacks, calculate this extra damage separately, as follows.

Striker damage contribution = Chance to hit once * average extra damage

The chance to hit once (H1) depends on the number of attacks and the Hit Rate HR:

2 attacks: H1= HR * (1-HR)
3 attacks: H1= 1 - (1-HR)*(1-HR)*(1-HR)
N attacks: H1 = 1 - ((1-HR) ^ N) Read More......

Fixing the math: Fortitude, Reflex, Will (II)

After the lengthy explanation of current issues with Fortitude, Reflex, and Will progression, I think we are ready to look at the crunchy bits. The house rule I propose for the non-AC defenses is not, in fact, a single rule, but a set of three different modifications: the introduction of masterwork neck slot items, the creation of a new type of magic item that ensures a minimum ability modifier to defenses without costing any additional item slot, and the ban of a series of feats.

1- The masterwork necks

The first step of the patch is to allow the use of masterwork neck slot items. Following the same principle as masterwork armor (see PHB, p.212), a masterwork neck grants a higher bonus to Fortitude, Reflex and Will than a non-masterwork neck. The type and level of a magic neck item determines if it can be masterwork, as detailed in the table below. The cost of a masterwork neck is included in the cost of the magic item, so higher-level characters should seek out masterwork necks instead of normal magic necks.

MAGIC NECK SLOT ITEMS

Masterwork necks are a functional replacement for defense-enhancing feats, providing equivalent bonuses without any cost for the PCs. As such, they fix, by themselves the problems associated with good defenses (as defined in the previous article). However, an additional measure is needed in order to patch bad defenses.

2- The Lucky Badges

A player's bad defenses suffer from lack of progression in ability modifiers. In order to compensate for that, you can allow players the use of the Badge of the Lucky Hero magic item.

If you are using this house rule, a level-appropiate Badge of the Lucky Hero is an item that every Player Character needs and is assumed to have, much like PCs in the basic rules need and are assumed to have level-appropiate magic Weapons/Implements, Armor, and Neck items. DMs will have to assign additional treasure parcels to account for that: for a normal 5-player party, an additional magic item of the PCs' level for each level would be enough.

Alternately, you can have all characters in a party earn a Badge of an appropiate level upon completion of a major quest, instead of granting additional parcels.

Lucky Badges ensure that a character's defenses don't fall below a minimum threshold regardless of having low ability scores. This threshold is set so that monsters of the same level will hit these defenses roughly 70% of the times - a value noticeably worse than that of any defense affected by primary or secondary abilities, but high enough for enemies to occasionally miss.

3- The ban

While using this rules patch, the following feats should be disallowed, since their bonus is effectively given for free:

• Paragon defenses
• Robust defenses
• Epic Reflexes
• Epic Fortitude
• Epic Will

The following feats are still permited, since they allow players to customize their defenses through modest bonuses, and are neither dangerously strong nor required for most builds:

• Great Fortitude
• Iron Will
• Lightning Reflexes
• Opportune reflexes
• Unyelding Fortitude
• Indomitable will

4- Changing your defense values in Character Builder

Unlike attack bonuses, your defenses can be modified in a Character Builder sheet. In order to do that, you only need to double click on the defense score in the character sheet, and type the desired value.

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Fixing the math: Fortitude, Reflex, Will

They start out all right, neither too easy nor too difficult to hit. But as the levels increase, they lag behind and never recover. Fortitude, Reflex and Will, also known in certain circles as NADs (non-AC defenses), are broken. Heroic characters barely notice it, Paragons start to suffer, and Epic heroes just have to assume that certain attacks will always hit them. Today I will talk about why this happens, just how bad it is, and how the official "feat patch" fails to solve the problem - and in the following post I will suggest a way to fix it!

The issue with NADs is similar to the one with attack bonuses and expertise, only worse. Essentially, attacks and defenses in 4E are assumed to increase at a rate of 1 point per level, but they don't quite. Over the course of 30 levels, a character's attacks fall behind by 4 points, compared to their expected values. (hence expertise). In the case of NADs, they also fall behind, but at different rates, depending on whether they are 'good' or 'bad' defenses:

• A good defense is affected by a primary or secondary ability modifier. Since this ability is increased whenever possible (for a maximum of an additional +4 modifier at maximum level), the defense follows the same progression as attack bonuses, sharing the same 4 point gap . This is annoying, but not catastrophic.
• A bad defense, on the other hand, depends on tertiary abilities, which only get increases at levels 11 and 21. As a consequence, the gap for these defenses reaches a whopping 7 points at level 30. This means that, at Epic Tier, a monster making an attack against a bad defense can hit a player character with a roll of 2 or more. Hit rates at Paragon aren't that extreme, but can easily be higher than 80%, which is also problematic. Every character in the game suffers from at least one bad defense (or two, if both primary and secondary abilities affect the same defense), so this is a critical problem.

Example 1: Basic scenario

I have built two sample characters to illustrate defense progression across levels. The first one, PC1 is an extreme (but still relatively common) worst-case scenario of a character with two bad defenses. Assuming that the character uses an ability score array of 18, 14, 11 (pre-racial modifiers), that he has a racial bonus to his primary ability and a +2 class bonus to his best defense, his starting NAD scores would be 17 for the best defense, and 11 and 10, respectively, for the second and third best. 1 point in the second-best defense comes either from a racial bonus to a tertiary ability, or from a racial bonus to a defense. This character could be a Battlerager Fighter, a Wand Wizard, or a Devoted Cleric, among others.

The second sample character, PC2, has well-balanced NADs, with only one bad defense and a starting ability array of 16, 16, 12, 12 (pre-racials). This character would have racial bonuses to both primary and secondary abilities, and a +2 class bonus to the best defense, for starting NAD scores of 16, 14, and 11 (from best to worst defense). This character could be a Whirling Barbarian or a Chaos Sorcerer, among others.

The following graphs show the NAD progression for PC1 and PC2 in the basic scenario (no defense-boosting feats). These values have been normalized (i.e. they are substracted the PC's level) for clarity. Two additional lines, "cap" and "auto-hit", are shown, representing the defense values where a skirmisher monster of equal level would hit on a 20+, or on a 2+, respectively. For more information on normalization, you can read this article. Enhancenment bonus from Neck slot items are added at levels 2, 7, 12, 17, 22 and 27. The detailed calculations can be followed on this spreadsheet.

We can calculate the average hit rates for monsters of equal level against these defenses, using the method described in the normalization article. They are the following:

As we can see, for PC 1 the worst defense moves into auto-hit territory at level 15. PC1's second defense and PC2's worst one, on the other hand, don't get that bad until level 23. Nevertheless, their values are worryingly low even before reaching this cap, with hit rates of 90% starting at levels 13 and 17, respectively.

As for the good defenses, they are acceptable but noticeably lower at epic tier, compared to their starting values.

Example 2 - NAD-boosting feats

Since the examples above assumed that no NAD-boosting feats were taken, now we are going to analyze how these feats mitigate the problem. There are eight feats in the game whose sole purpose is to provide a bonus to Fortitude, Reflex, and Will. Half of them are Paragon-Tier feats: Paragon Defenses, Great Fortitude, Lightning Reflexes, and Iron Will. The rest are Epic clones of the former, but with better bonuses: Robust Defenses, Epic Fortitude, Epic Will, and Epic Reflexes. It is unfortunate that they chose to duplicate feats, rather than having less of them, but with scaling bonuses, a la Weapon Focus.

A player who wants to close the NAD gap by taking these feats is presented two options: the cheap solution, and the expensive one. The cheap solution consist in taking Paragon Defenses at Paragon tier, and retraining to Robust defenses at Epic. This costs the player a single feat, but has a reduced effect - good defenses become a bit lower than they should, whereas the auto-hit cap in bad defenses is delayed a few levels but eventually appears again. For this reason, we will focus on the expensive solution: taking the full set of 3 feats for fortitude, reflex and will at paragon, and retraining them at epic.

For that hefty cost, one would assume that you could forget about NAD problems, but it turns out that you can't. The figures below show defenses for PC1 and PC2, assuming that they somehow take the three appropiate feats at level 11, and retrain at 21. (Actual PCs would have to wait until levels 11 and 22 for the third of these feats, but the graphs ignore this for simplicity).

And the corresponding hit rates:

This progression is definitely better, and the dreaded bugbear of auto-hit is gone. However, upon close examination, we can see that bad defenses are still terrible at Epic tier, with hit rates of 80% or worse. At such low values, the feat bonuses are underutilized: 80% is only a 15% improvement above the cap, instead of the theoretical 20% that a +4 bonus should net you. And this diminished improvement practically disappears when facing a higher-level monster or a soldier. For this reason, players might be better off giving up on bad defenses altogether. Clearly, this expensive solution is unsatisfactory.

Example 3 - The house rule.

Since this is already quite long, I will leave my suggestion for house rules for the following post. However, I'd like to show the graphs for these revised rules here, so that they can easily be compared with the current ones. Here they are:

Although some oscillations are inevitable, this is a very regular progression across tiers. A mechanism to ensure that bad defenses don't fall below a certain threshold has been added, so that they remain weak but can actually be missed at epic levels.

Next: The Fix!

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