Let's examine the case for new encounter powers which produce half-damage on a miss. (These are purportedly being introduced in 4E Essentials). Let's assume the new encounter powers still follow the same damage of the Heinsoo 4E D&D encounter powers, but with the "half-damage on miss" tacked on.

For non-striker encounter powers with half-damage on a miss and a probability p of a hitting a monster of the same level, the average damage per round scales approximately as (after level 10):

0.5(p+1)*[level*(average[W])/7 + 7*(level-10)/20]

So as the level goes to infinity, "R" approaches:

R -> 2*ROLE/{(p+1)*[(average[W])/7 + 7/20]}

For different [W] weapons attacking a skirmisher monster (ROLE=8) with the player having a p=50% of hitting the monster, we have average number of rounds "R" as the level goes to infinity:

average[d12] = 6.5 --> R = 8.34

average[d10] = 5.5 --> R = 9.39

average[d8] = 4.5 --> R = 10.74

average[d6] = 3.5 --> R = 12.55

average[d4] = 2.5 --> R = 15.08

On average, for a player hypothetically repeatedly spamming a new encounter power (with half-damage on a miss) against a skirmisher monster of the same level, the average number of rounds "R" to kill the monster is approximately shorter by 45% compared to at-will powers, as the level goes to infinity.

For completeness, there's at-will powers with half-damage on a miss. (IIRC, there's a few monsters with such an at-will power from the 4E MM2 and/or MM3).

With half-damage on a miss, the at-will with half-damage on a miss reduces the number of rounds R to kill a monster by a factor = p/(1+p) in comparison to ordinary at-will powers. (p is the probability of hitting a monster).

For different p:

p = 100% --> factor = 50%

p = 75% --> factor = 42.6%.

p = 50% --> factor = 33.3%

p = 25% --> factor = 20%

So with a probability 50% of hitting a monster, an at-will power with half-damage on a miss will reduce the number of rounds to kill the monster by 33.3%, in comparison to the same at-will power without the half-damage on a miss part.

## Wednesday, August 25, 2010

### Scaling of 4E daily powers.

Let's do the same for daily powers, where the the daily powers have half-damage on a miss.

Recall for half-damage on a miss: R = 2N/(1+p).

For a daily power, the average damage per hit scales approximately as (after level 10):

level*(average[W])/5 + 7*(level-10)/20.

For the easy case where the daily powers of non-strikers always miss, "R" approaches

R -> 2*ROLE/{p*[(average[W])/5 + 7/20]} = ROLE/{p*[(average[W])/10 + 7/40]}

as the level goes to infinity.

Comparing this to the expression "R" for at-wills, it means that daily powers always missing and producing half-damage, is slightly worse than generic daily powers.

For non-striker daily powers with half-damage on a miss and a probability p of a hitting a monster of the same level, the average damage per round scales approximately as (after level 10):

0.5(p+1)*[level*(average[W])/5 + 7*(level-10)/20]

So as the level goes to infinity, "R" approaches:

R -> 2*ROLE/{(p+1)*[(average[W])/5 + 7/20]}

For different [W] weapons attacking this skirmisher monster (ROLE=8) with the player having a p=50% of hitting the monster, we have average number of rounds "R" as the level goes to infinity:

average[d12] = 6.5 --> R = 6.46

average[d10] = 5.5 --> R = 7.36

average[d8] = 4.5 --> R = 8.53

average[d6] = 3.5 --> R = 10.16

average[d4] = 2.5 --> R = 12.55

On average, for a player hypothetically repeatedly spamming a daily power (with half-damage on a miss) against a skirmisher monster of the same level, the average number of rounds "R" to kill the monster is approximately shorter by a half compared to at-will powers, as the level goes to infinity.

This means a repeatedly "spammed" daily power (with half damage on a miss) attacking a skirmisher (of the same level), is on average damage-wise approximately equal to two at-wills in general. If the daily power always misses and always produces half damage, it is damage-wise approximately equal to one at-will power.

Recall for half-damage on a miss: R = 2N/(1+p).

For a daily power, the average damage per hit scales approximately as (after level 10):

level*(average[W])/5 + 7*(level-10)/20.

For the easy case where the daily powers of non-strikers always miss, "R" approaches

R -> 2*ROLE/{p*[(average[W])/5 + 7/20]} = ROLE/{p*[(average[W])/10 + 7/40]}

as the level goes to infinity.

Comparing this to the expression "R" for at-wills, it means that daily powers always missing and producing half-damage, is slightly worse than generic daily powers.

For non-striker daily powers with half-damage on a miss and a probability p of a hitting a monster of the same level, the average damage per round scales approximately as (after level 10):

0.5(p+1)*[level*(average[W])/5 + 7*(level-10)/20]

So as the level goes to infinity, "R" approaches:

R -> 2*ROLE/{(p+1)*[(average[W])/5 + 7/20]}

For different [W] weapons attacking this skirmisher monster (ROLE=8) with the player having a p=50% of hitting the monster, we have average number of rounds "R" as the level goes to infinity:

average[d12] = 6.5 --> R = 6.46

average[d10] = 5.5 --> R = 7.36

average[d8] = 4.5 --> R = 8.53

average[d6] = 3.5 --> R = 10.16

average[d4] = 2.5 --> R = 12.55

On average, for a player hypothetically repeatedly spamming a daily power (with half-damage on a miss) against a skirmisher monster of the same level, the average number of rounds "R" to kill the monster is approximately shorter by a half compared to at-will powers, as the level goes to infinity.

This means a repeatedly "spammed" daily power (with half damage on a miss) attacking a skirmisher (of the same level), is on average damage-wise approximately equal to two at-wills in general. If the daily power always misses and always produces half damage, it is damage-wise approximately equal to one at-will power.

### Scaling of 4E encounter powers.

Let's examine the scaling behavior of the encounter and daily powers, in the unrealistic scenario where they can be repeatedly spammed every round. We will calculate the average number of rounds "R" to kill a monster by a player of the same level.

Recall that a non-striker repeatedly using at-will powers against a monster of the same level, "R" approaches the limit

R = N/p -> N -> ROLE/{p*[(average[W])/10 + 7/20]}

as the level goes to infinity.

For different [W] weapons attacking this skirmisher monster (ROLE=8) with the player having a p=50% of hitting the monster, we have average number of rounds "R" as the level goes to infinity:

average[d12] = 6.5 --> N = 8, R = 16

average[d10] = 5.5 --> N = 8.89, R = 17.78

average[d8] = 4.5 --> N = 10, R = 20

average[d6] = 3.5 --> N = 11.43, R = 22.86

average[d4] = 2.5 --> N = 13.33, R = 26.66

For non-striker encounter powers, the average damage per hit scales approximately as (after level 10):

level*(average[W])/7 + 7*(level-10)/20.

So for non-striker encounter powers, the average damage per round scales approximately as (after level 10):

p*[level*(average[W])/7 + 7*(level-10)/20].

So as the level goes to infinity, "R" approaches:

R -> N/p = ROLE/{p*[(average[W])/7 + 7/20]}

For different [W] weapons attacking this skirmisher monster (ROLE=8) with the player having a p=50% of hitting the monster, we have average number of rounds "R" as the level goes to infinity:

average[d12] = 6.5 --> R = 12.51

average[d10] = 5.5 --> R = 14.01

average[d8] = 4.5 --> R = 16.12

average[d6] = 3.5 --> R = 18.82

average[d4] = 2.5 --> R = 22.63

On average, for a player hypothetically repeatedly spamming an encounter power against a skirmisher monster of the same level, the average number of rounds "R" to kill the monster is shorter by approximately 4 rounds (compared to at-will powers) as the level goes to infinity. (For a 50% probability of hitting the monster, this would mean that it takes approximately 2 less hits to kill the skirmisher monster).

More generally, a hypothetical spammed encounter power takes approximately 15% to 20% less rounds to kill a monster (of the same level).

Recall that a non-striker repeatedly using at-will powers against a monster of the same level, "R" approaches the limit

R = N/p -> N -> ROLE/{p*[(average[W])/10 + 7/20]}

as the level goes to infinity.

For different [W] weapons attacking this skirmisher monster (ROLE=8) with the player having a p=50% of hitting the monster, we have average number of rounds "R" as the level goes to infinity:

average[d12] = 6.5 --> N = 8, R = 16

average[d10] = 5.5 --> N = 8.89, R = 17.78

average[d8] = 4.5 --> N = 10, R = 20

average[d6] = 3.5 --> N = 11.43, R = 22.86

average[d4] = 2.5 --> N = 13.33, R = 26.66

For non-striker encounter powers, the average damage per hit scales approximately as (after level 10):

level*(average[W])/7 + 7*(level-10)/20.

So for non-striker encounter powers, the average damage per round scales approximately as (after level 10):

p*[level*(average[W])/7 + 7*(level-10)/20].

So as the level goes to infinity, "R" approaches:

R -> N/p = ROLE/{p*[(average[W])/7 + 7/20]}

For different [W] weapons attacking this skirmisher monster (ROLE=8) with the player having a p=50% of hitting the monster, we have average number of rounds "R" as the level goes to infinity:

average[d12] = 6.5 --> R = 12.51

average[d10] = 5.5 --> R = 14.01

average[d8] = 4.5 --> R = 16.12

average[d6] = 3.5 --> R = 18.82

average[d4] = 2.5 --> R = 22.63

On average, for a player hypothetically repeatedly spamming an encounter power against a skirmisher monster of the same level, the average number of rounds "R" to kill the monster is shorter by approximately 4 rounds (compared to at-will powers) as the level goes to infinity. (For a 50% probability of hitting the monster, this would mean that it takes approximately 2 less hits to kill the skirmisher monster).

More generally, a hypothetical spammed encounter power takes approximately 15% to 20% less rounds to kill a monster (of the same level).

### Scaling of standard deviation of N for 4E at-wills.

Let's examine how the standard deviation (stdev) of N changes with level.

Recall the average number of hits to kill a monster by a non-striker player character (of the same level) using an at-will power with weapon damage dice [w] = level*[W],

N = [ROLE*(level+1) + CON]/[(average[w])/10 + 7*(level-10)/20]

where one of the egregious assumption made, was that the damage of at-will powers [W] increases each ten levels after level 10, in the pattern of:

level 21-30 --> 2[W]

level 31-40 --> 3[W]

level 41-50 --> 4[W]

level 51-60 --> 5[W]

etc ...

For weapon damage dice 2[W], it means that the number of weapon dice is doubled. For example if the weapon dice [W] is a d6, then 2[W] means 2d6, 3[W] means 3d6, etc ...

The average of damage dice "level*[W]", turns out to be equal to level*average[W].

But the standard deviation of damage dice "level*[W]", turns out to be [sqrt(level)]*stdev[W]. ("sqrt" is the square root). This can be found in any college statistics/probability textbook.

(For example, (stdev[3d6])^2 = (stdev[d6])^2 + (stdev[d6])^2 + (stdev[d6])^2 which gives stdev[3d6] = sqrt(3) * stdev[d6]).

Recall that for individual dice like d4, d6, d8, d10, d12, etc ... the standard deviation is:

stdev[dN] = sqrt[(N^2 -1)/12]

where sqrt is the square root. (A single die follows a discrete uniform distribution).

For example,

stdev[d4] = 1.118

stdev[d6] = 1.708

stdev[d8] = 2.291

stdev[d10] = 2.872

stdev[d12] = 3.452

To get the standard deviation of N, there's the formula:

stdev[N] = |dN/dw| * stdev[w]

where w = level*[W]. The term in between the | | absolute value brackets is the first derivative of N with respect to [W].

Doing the calculation of the standard deviation of N and scaling the level to infinity, we get:

stdev[N] = N^2 *(stdev[W])/[ROLE*sqrt(level/10)]

when the level becomes larger and larger.

Hence the standard deviation of the average number of hits "N" to kill a monster by a player repeatedly using at-will powers against a monster (of equivalent level), scales as:

stdev[N] ~ 1/sqrt(level)

This means the standard deviation of N moderately gets smaller and smaller as the level gets bigger and bigger. The number of hits to kill a monster (of the same level) deviates less and less from the average, as the level becomes bigger and bigger.

Basically while one is "always fighting orcs", the "orcs" are becoming more and more "predictable" as the level gets higher and higher.

Recall the average number of hits to kill a monster by a non-striker player character (of the same level) using an at-will power with weapon damage dice [w] = level*[W],

N = [ROLE*(level+1) + CON]/[(average[w])/10 + 7*(level-10)/20]

where one of the egregious assumption made, was that the damage of at-will powers [W] increases each ten levels after level 10, in the pattern of:

level 21-30 --> 2[W]

level 31-40 --> 3[W]

level 41-50 --> 4[W]

level 51-60 --> 5[W]

etc ...

For weapon damage dice 2[W], it means that the number of weapon dice is doubled. For example if the weapon dice [W] is a d6, then 2[W] means 2d6, 3[W] means 3d6, etc ...

The average of damage dice "level*[W]", turns out to be equal to level*average[W].

But the standard deviation of damage dice "level*[W]", turns out to be [sqrt(level)]*stdev[W]. ("sqrt" is the square root). This can be found in any college statistics/probability textbook.

(For example, (stdev[3d6])^2 = (stdev[d6])^2 + (stdev[d6])^2 + (stdev[d6])^2 which gives stdev[3d6] = sqrt(3) * stdev[d6]).

Recall that for individual dice like d4, d6, d8, d10, d12, etc ... the standard deviation is:

stdev[dN] = sqrt[(N^2 -1)/12]

where sqrt is the square root. (A single die follows a discrete uniform distribution).

For example,

stdev[d4] = 1.118

stdev[d6] = 1.708

stdev[d8] = 2.291

stdev[d10] = 2.872

stdev[d12] = 3.452

To get the standard deviation of N, there's the formula:

stdev[N] = |dN/dw| * stdev[w]

where w = level*[W]. The term in between the | | absolute value brackets is the first derivative of N with respect to [W].

Doing the calculation of the standard deviation of N and scaling the level to infinity, we get:

stdev[N] = N^2 *(stdev[W])/[ROLE*sqrt(level/10)]

when the level becomes larger and larger.

Hence the standard deviation of the average number of hits "N" to kill a monster by a player repeatedly using at-will powers against a monster (of equivalent level), scales as:

stdev[N] ~ 1/sqrt(level)

This means the standard deviation of N moderately gets smaller and smaller as the level gets bigger and bigger. The number of hits to kill a monster (of the same level) deviates less and less from the average, as the level becomes bigger and bigger.

Basically while one is "always fighting orcs", the "orcs" are becoming more and more "predictable" as the level gets higher and higher.

### Scaling for different levels between player and monster.

One can look at the scaling behavior of the average number of hits to kill a monster being attacked by a player, but for slightly different levels for the player character and monster.

For the level of the monster and player not being equal, we'll indicate their levels by level_m and level_p respectively. So from a previous post, the average number of hits "N" to kill a monster (of level_m) being attacked by a player character (of level_p) is:

N = [ROLE*(level_m+1) + CON]/[level_p*(average[W])/10 + 7*(level_p-10)/20]

Taking the limit where both level_m and level_p are brought to infinity but keeping the ratio (level_m/level_p) constant, we get:

N -> (level_m/level_p) * ROLE/[(average[W])/10 + 7/20].

From this result, we can examine what happens to "N" when (level_m/level_p) is changed.

The easiest case is when the level of the monsters is double the level of the players. In this case (level_m/level_p) = 2, which implies the average number of hits to kill a monster double in level from the players, will take on average two times as many hits to kill in comparison to a monster of the same level as the players.

The second easiest case is when the level of the monster is half of the level of the players. In this case (level_m/level_p) = 0.5, which implies the average number of hits to kill a monster half in level from the players, will take on average half as many hits to kill in comparison to a monster of the same level as the players.

For the level of the monster and player not being equal, we'll indicate their levels by level_m and level_p respectively. So from a previous post, the average number of hits "N" to kill a monster (of level_m) being attacked by a player character (of level_p) is:

N = [ROLE*(level_m+1) + CON]/[level_p*(average[W])/10 + 7*(level_p-10)/20]

Taking the limit where both level_m and level_p are brought to infinity but keeping the ratio (level_m/level_p) constant, we get:

N -> (level_m/level_p) * ROLE/[(average[W])/10 + 7/20].

From this result, we can examine what happens to "N" when (level_m/level_p) is changed.

The easiest case is when the level of the monsters is double the level of the players. In this case (level_m/level_p) = 2, which implies the average number of hits to kill a monster double in level from the players, will take on average two times as many hits to kill in comparison to a monster of the same level as the players.

The second easiest case is when the level of the monster is half of the level of the players. In this case (level_m/level_p) = 0.5, which implies the average number of hits to kill a monster half in level from the players, will take on average half as many hits to kill in comparison to a monster of the same level as the players.

### Scaling of 4E monsters fighting against players.

Let's look at things from the monster's point of view. We'll calculate the average number of hits "N" to kill a player character (of the same level) and see how it scales when the level goes to infinity.

For this analysis, we'll ignore healing surges and stat increases to CON.

For the player characters, the hit points H scale approximately as:

H = *constant* + CON + CLASS*level

where *constant* is a number between 6 to 10 (depending on class) and:

CLASS = 4 --> wizard, invoker

CLASS = 5 --> cleric, druid, ranger, rogue, bard, sorcerer, warlock, warlord, shaman

CLASS = 6 --> fighter, paladin, barbarian, avenger

CLASS = 7 --> warden.

For the monster damage, the average damage from the "normal damage expression" table on page 185 of the 4E DMG1, follows approximately:

low damage: 6 + level*5/12

medium damage: 8 + level*7/15

high damage: 10 + level*3/5.

(These were found by doing a simple linear fit to the average values).

Calculating "N" for the monster attacking players and scaling the level to infinity, we get:

low damage: N -> 12*CLASS/5 = 2.4*CLASS

medium damage: N -> 15*CLASS/7 = 2.14*CLASS

high damage: N -> 5*CLASS/3 = 1.67*CLASS

For the new monster damage table from the 4E DMG1 errata for page 185, the average damage follow approximately:

single target: 8 + level

two or more targets: 6 + level*3/4.

Calculating "N" for the monster attacking players and scaling the the level to infinity, we get:

single target: N -> CLASS

two or more targets: N -> 4*CLASS/3 = 1.33*CLASS.

Compare the old damage scheme to the errata update one. For the average number of hits "N" of a monster killing a player character, it is suggestive the errata updated monster damage is a lot more formidable.

Using the new errata monster damage, for example a monster fighting a typical defender player character, it takes around 6 to 8 hits to kill the defender without any healing surges. Using the old monster damage table from page 185 of the 4E DMG1, it would take the same monster around 12 hits to kill the defender (without any healing surges).

In contrast from previous posts which calculated the scaling for non-striker and striker player characters, we calculated that a non-striker player character would take around 8 to 10 hits to kill a skirmisher monster using weapons with damage dice d8, d10 or d12. A striker player character would take around 6 to 8 hits to kill a skirmisher monster using weapons with damage dice d6, d8 or d10.

Indeed, the old 4E monsters using the old damage scheme from page 185 of the 4E DMG1, don't appear to be as much of a threat to the players (even without any healing surges). A reverse "always fighting orcs" where the "orcs" are consistently losing to the player characters.

In contrast with the new damage scheme from the 4E DMG1 errata, the "orcs" can actually win sometimes against the players.

For this analysis, we'll ignore healing surges and stat increases to CON.

For the player characters, the hit points H scale approximately as:

H = *constant* + CON + CLASS*level

where *constant* is a number between 6 to 10 (depending on class) and:

CLASS = 4 --> wizard, invoker

CLASS = 5 --> cleric, druid, ranger, rogue, bard, sorcerer, warlock, warlord, shaman

CLASS = 6 --> fighter, paladin, barbarian, avenger

CLASS = 7 --> warden.

For the monster damage, the average damage from the "normal damage expression" table on page 185 of the 4E DMG1, follows approximately:

low damage: 6 + level*5/12

medium damage: 8 + level*7/15

high damage: 10 + level*3/5.

(These were found by doing a simple linear fit to the average values).

Calculating "N" for the monster attacking players and scaling the level to infinity, we get:

low damage: N -> 12*CLASS/5 = 2.4*CLASS

medium damage: N -> 15*CLASS/7 = 2.14*CLASS

high damage: N -> 5*CLASS/3 = 1.67*CLASS

For the new monster damage table from the 4E DMG1 errata for page 185, the average damage follow approximately:

single target: 8 + level

two or more targets: 6 + level*3/4.

Calculating "N" for the monster attacking players and scaling the the level to infinity, we get:

single target: N -> CLASS

two or more targets: N -> 4*CLASS/3 = 1.33*CLASS.

Compare the old damage scheme to the errata update one. For the average number of hits "N" of a monster killing a player character, it is suggestive the errata updated monster damage is a lot more formidable.

Using the new errata monster damage, for example a monster fighting a typical defender player character, it takes around 6 to 8 hits to kill the defender without any healing surges. Using the old monster damage table from page 185 of the 4E DMG1, it would take the same monster around 12 hits to kill the defender (without any healing surges).

In contrast from previous posts which calculated the scaling for non-striker and striker player characters, we calculated that a non-striker player character would take around 8 to 10 hits to kill a skirmisher monster using weapons with damage dice d8, d10 or d12. A striker player character would take around 6 to 8 hits to kill a skirmisher monster using weapons with damage dice d6, d8 or d10.

Indeed, the old 4E monsters using the old damage scheme from page 185 of the 4E DMG1, don't appear to be as much of a threat to the players (even without any healing surges). A reverse "always fighting orcs" where the "orcs" are consistently losing to the player characters.

In contrast with the new damage scheme from the 4E DMG1 errata, the "orcs" can actually win sometimes against the players.

### Scaling of 4E ranger's twin-striker power.

This is an analysis of the ranger's "twin strike" power. (We'll ignore the additional complication of hunter's quarry).

We will calculate the average number of rounds "R" it takes to kill a monster (of the same level as the ranger) by being repeatedly twin striked.

We'll assume twin strike is two attacks in one round, where each attack is separately rolled for attack and damage. To make the problem general, we will use the notation:

p1 = to-hit probability of first attack

p1 = to-hit probability of second attack

d1 = damage of first attack

d2 = damage of second attack

The expectation value ED of the twin strike damage in one round is:

ED = p1 p2 (average[d1] + average[d2]) + p1 (1-p2) average[d1] + (1-p1) p2 average [d2] + (1-p1)(1-p2)*0

(EDIT: The first term is when both attacks hit. The second and third terms are when one attack hits and the other misses. The fourth term is when both attacks miss).

Doing the algebra, we get:

ED = p1 average[d1] + p2 average[d2]

(More generally this result can be generalized to any number of attacks. For example, three attacks in one round: ED = p1 average[d1] + p2 average[d2] + p3 average[d3]).

If we assume p1 = p2 = p, and d1 = d2 = d, we have ED = 2p*average[d].

For twin strike, the damage is d = [W] + magic enhancement for each attack.

The magic enhancement scales approximately as level/5, assuming the progression on page 225 of the 4E PHB1.

For the sake of argument, we will look at the case where the weapon damage dice per attack remains [W] for all levels to infinity. (ie. Weapon damage does not change to 2[W] per attack at level 21).

Hence the damage scales approximately as:

average[d] = average[W] + level/5

and ED = 2p*(average[W] + level/5)

Now the average number of rounds "R" to kill a monster by a player of the same level is:

R = [ROLE*(level+1) + CON]/(2p*[average[W]+ level/5])

As the level goes to infinity, R approaches

R -> 5*ROLE/2p = 2.5*ROLE/p.

Recall from the previous post that a similar result for generic at-will powers having weapon dice damage [W] for all levels to infinity was:

R -> N/p -> 20*ROLE/7p = 2.86*ROLE/p.

So even without the stat mod added to the damage to twin strike, the scaling limit for R is slightly better than for generic at-will powers as the level goes to infinity.

If the same analysis is done again for twin-strike, but adding in an egregious assumption where the weapon dice damage per attack increases with level in the form:

level 21 - 30 --> 2[W] per attack

level 31 - 40 --> 3[W] per attack

level 41 - 50 --> 4[W] per attack

etc ...

We will assume the magic enhancement scales approximately as level/5.

Hence the damage per attack scales approximately as:

average[damage] = (level/10)*average[W] + level/5

The expectation value of the damage per round for twin strike with these egregious assumptions is:

ED = 2p*[(level/10)*average[W] + level/5]

Now the average number of rounds "R" to kill a monster by a player of the same level is:

R = [ROLE*(level+1) + CON]/(2p*[(level/10)*(average[W]) + level/5])

As the level goes to infinity, R approaches

R -> ROLE/[2p*( (average[W])/10 + 1/5)].

Plugging in some numbers for a ranger twin striking a skirmisher (ROLE=8), the weapon dice [W] for ranger weapons could be d6, d8 or d10.

average[d6] = 3.5 --> N = 7.27

average[d8] = 4.5 --> N = 6.15

average[d10] = 5.5 --> N = 5.33

(We used R = N/p to get N = average number of hits to kill monster).

In the end twin-strike is also "always fighting orcs" as the level scales to infinity.

We will calculate the average number of rounds "R" it takes to kill a monster (of the same level as the ranger) by being repeatedly twin striked.

We'll assume twin strike is two attacks in one round, where each attack is separately rolled for attack and damage. To make the problem general, we will use the notation:

p1 = to-hit probability of first attack

p1 = to-hit probability of second attack

d1 = damage of first attack

d2 = damage of second attack

The expectation value ED of the twin strike damage in one round is:

ED = p1 p2 (average[d1] + average[d2]) + p1 (1-p2) average[d1] + (1-p1) p2 average [d2] + (1-p1)(1-p2)*0

(EDIT: The first term is when both attacks hit. The second and third terms are when one attack hits and the other misses. The fourth term is when both attacks miss).

Doing the algebra, we get:

ED = p1 average[d1] + p2 average[d2]

(More generally this result can be generalized to any number of attacks. For example, three attacks in one round: ED = p1 average[d1] + p2 average[d2] + p3 average[d3]).

If we assume p1 = p2 = p, and d1 = d2 = d, we have ED = 2p*average[d].

For twin strike, the damage is d = [W] + magic enhancement for each attack.

The magic enhancement scales approximately as level/5, assuming the progression on page 225 of the 4E PHB1.

For the sake of argument, we will look at the case where the weapon damage dice per attack remains [W] for all levels to infinity. (ie. Weapon damage does not change to 2[W] per attack at level 21).

Hence the damage scales approximately as:

average[d] = average[W] + level/5

and ED = 2p*(average[W] + level/5)

Now the average number of rounds "R" to kill a monster by a player of the same level is:

R = [ROLE*(level+1) + CON]/(2p*[average[W]+ level/5])

As the level goes to infinity, R approaches

R -> 5*ROLE/2p = 2.5*ROLE/p.

Recall from the previous post that a similar result for generic at-will powers having weapon dice damage [W] for all levels to infinity was:

R -> N/p -> 20*ROLE/7p = 2.86*ROLE/p.

So even without the stat mod added to the damage to twin strike, the scaling limit for R is slightly better than for generic at-will powers as the level goes to infinity.

If the same analysis is done again for twin-strike, but adding in an egregious assumption where the weapon dice damage per attack increases with level in the form:

level 21 - 30 --> 2[W] per attack

level 31 - 40 --> 3[W] per attack

level 41 - 50 --> 4[W] per attack

etc ...

We will assume the magic enhancement scales approximately as level/5.

Hence the damage per attack scales approximately as:

average[damage] = (level/10)*average[W] + level/5

The expectation value of the damage per round for twin strike with these egregious assumptions is:

ED = 2p*[(level/10)*average[W] + level/5]

Now the average number of rounds "R" to kill a monster by a player of the same level is:

R = [ROLE*(level+1) + CON]/(2p*[(level/10)*(average[W]) + level/5])

As the level goes to infinity, R approaches

R -> ROLE/[2p*( (average[W])/10 + 1/5)].

Plugging in some numbers for a ranger twin striking a skirmisher (ROLE=8), the weapon dice [W] for ranger weapons could be d6, d8 or d10.

average[d6] = 3.5 --> N = 7.27

average[d8] = 4.5 --> N = 6.15

average[d10] = 5.5 --> N = 5.33

(We used R = N/p to get N = average number of hits to kill monster).

In the end twin-strike is also "always fighting orcs" as the level scales to infinity.

### Average number of rounds R vs. average number of hits N.

To get the average number of rounds to kill a monster (including the attacks which missed) by a player of the same level as the monster, it turns out it is related to "N". Let's call this average number of rounds to kill a monster, as "R".

We won't include stuff like critical hits, attacks with half-damage on a miss, and other stuff which I haven't thought of yet. With that being said, it turns out the average number of rounds "R" to kill a monster for a player with probability p of hitting the monster, is:

R = N/p

To get this result, this is from the geometric probability distribution. A geometric probability distribution model includes the hits and misses when a player attacks a monster.

More generally, "R" is the ratio:

(hit points of monster)/(expectation value of damage per round).

Let's call

H = hit points of monster

ED = expected value of damage per round.

So R = H/ED.

Let's assume a player has a probability p of hitting a monster.

For an at-will power with weapon damage dice [W] at lower levels, the expectation value of the damage for one round is:

p*average[W + stat mod + magic enhancement] + (1-p)*0

which is simply: ED = p*average[W + stat mod + magic enhancement]

For a hypothetical at-will which does half damage on a miss, the expectation value of the damage for one round is:

p*average[W + stat mod + magic enhancement] + 0.5*(1-p)*average[W + stat mod + magic enhancement]

which is simply: ED = 0.5 (1+p)*average[W + stat mod + magic enhancement].

If every at-will power had "half damage on a miss", then "R" would be:

R = 2N/(1+p)

If one includes critical hits to an at-will power with weapon damage dice [W] at lower levels, the expectation value of the damage for one round is:

ED = (p-0.05)*average[W + stat mod + magic enhancement] + (0.05)*(maximum[W + stat mod + magic enhancement] + average[criticalextra]).

("criticalextra" is the extra damage dice from magic weapons on a critical).

As one can see, incorporating "half-damage on a miss" or "critical hits", can make the equations look a lot messier. They don't fit into a nice form like R = N/p.

We won't include stuff like critical hits, attacks with half-damage on a miss, and other stuff which I haven't thought of yet. With that being said, it turns out the average number of rounds "R" to kill a monster for a player with probability p of hitting the monster, is:

R = N/p

To get this result, this is from the geometric probability distribution. A geometric probability distribution model includes the hits and misses when a player attacks a monster.

More generally, "R" is the ratio:

(hit points of monster)/(expectation value of damage per round).

Let's call

H = hit points of monster

ED = expected value of damage per round.

So R = H/ED.

Let's assume a player has a probability p of hitting a monster.

For an at-will power with weapon damage dice [W] at lower levels, the expectation value of the damage for one round is:

p*average[W + stat mod + magic enhancement] + (1-p)*0

which is simply: ED = p*average[W + stat mod + magic enhancement]

For a hypothetical at-will which does half damage on a miss, the expectation value of the damage for one round is:

p*average[W + stat mod + magic enhancement] + 0.5*(1-p)*average[W + stat mod + magic enhancement]

which is simply: ED = 0.5 (1+p)*average[W + stat mod + magic enhancement].

If every at-will power had "half damage on a miss", then "R" would be:

R = 2N/(1+p)

If one includes critical hits to an at-will power with weapon damage dice [W] at lower levels, the expectation value of the damage for one round is:

ED = (p-0.05)*average[W + stat mod + magic enhancement] + (0.05)*(maximum[W + stat mod + magic enhancement] + average[criticalextra]).

("criticalextra" is the extra damage dice from magic weapons on a critical).

As one can see, incorporating "half-damage on a miss" or "critical hits", can make the equations look a lot messier. They don't fit into a nice form like R = N/p.

### Scaling of 4E at-will powers for strikers.

Let's examine how "N" changes with extra damage from striker classes. (The previous post was for non-strikers).

For the ranger's hunter's quarry and warlock's curse, the extra damage is:

levels 1-10 --> 1d6

levels 11-20 --> 2d6

levels 21-30 --> 3d6

while for a rogue, the extra damage for sneak attacking is:

levels 1-10 --> 2d6

levels 11-20 --> 3d6

levels 21-30 --> 5d6.

(This is not very realistic for a rogue to repeatedly sneak attack a monster every round to infinity levels, but in principle it could be done if the monster is marked and tied up by a defender).

Let's make an egregious assumption and extrapolate this extra striker damage as:

- ranger or warlock

levels 31-40 --> 4d6

levels 41-50 --> 5d6

levels 51-60 --> 6d6

etc ...

- rogue

levels 31-40 --> 7d6

levels 41-50 --> 9d6

levels 51-60 --> 11d6

etc ...

So for the ranger or warlock, the average extra damage approximately scales as:

1 + [average(d6)]*level/10

while the rogue's average extra damage scales approximately (at high levels) as:

1 + [average(d6)]*2*level/10

So for the average number of hits to kill a monster (N) by the above striker players of the same level, as the level approaches infinity becomes:

- rangers or warlocks

N -> ROLE/[(average[W])/10 + (average[d6])/10+ 7/20]

- rogues

N -> ROLE/[(average[W])/10 + (average[d6])/5 +7/20]

To plug in some numbers, a rogue repeatedly sneak attacking a skirmisher (ROLE = 8). The weapons the rogue is proficient in, typically have d4 or d6 [W] damage dice.

[W] = d4 --> N = 6.15

[W] = d6 --> N = 5.71

For ranger or warlock at-will powers (excluding twin strike) repeatedly attacking a skirmisher, the damage dice can be d6, d8, or d10.

[W] = d6 --> N = 7.62

[W] = d8 --> N = 6.96

[W] = d10 --> N = 6.4

In contrast, a non-striker attacking a skirmisher with weapons dice [W] of d8, d10, d12:

average[d12] = 6.5 --> N = 8

average[d10] = 5.5 --> N = 8.89

average[d8] = 4.5 --> N = 10

On average, the extra striker damage reduces the number of hits to kill a monster by around 1 or 2 hits as the level goes to infinity.

For the ranger's hunter's quarry and warlock's curse, the extra damage is:

levels 1-10 --> 1d6

levels 11-20 --> 2d6

levels 21-30 --> 3d6

while for a rogue, the extra damage for sneak attacking is:

levels 1-10 --> 2d6

levels 11-20 --> 3d6

levels 21-30 --> 5d6.

(This is not very realistic for a rogue to repeatedly sneak attack a monster every round to infinity levels, but in principle it could be done if the monster is marked and tied up by a defender).

Let's make an egregious assumption and extrapolate this extra striker damage as:

- ranger or warlock

levels 31-40 --> 4d6

levels 41-50 --> 5d6

levels 51-60 --> 6d6

etc ...

- rogue

levels 31-40 --> 7d6

levels 41-50 --> 9d6

levels 51-60 --> 11d6

etc ...

So for the ranger or warlock, the average extra damage approximately scales as:

1 + [average(d6)]*level/10

while the rogue's average extra damage scales approximately (at high levels) as:

1 + [average(d6)]*2*level/10

So for the average number of hits to kill a monster (N) by the above striker players of the same level, as the level approaches infinity becomes:

- rangers or warlocks

N -> ROLE/[(average[W])/10 + (average[d6])/10+ 7/20]

- rogues

N -> ROLE/[(average[W])/10 + (average[d6])/5 +7/20]

To plug in some numbers, a rogue repeatedly sneak attacking a skirmisher (ROLE = 8). The weapons the rogue is proficient in, typically have d4 or d6 [W] damage dice.

[W] = d4 --> N = 6.15

[W] = d6 --> N = 5.71

For ranger or warlock at-will powers (excluding twin strike) repeatedly attacking a skirmisher, the damage dice can be d6, d8, or d10.

[W] = d6 --> N = 7.62

[W] = d8 --> N = 6.96

[W] = d10 --> N = 6.4

In contrast, a non-striker attacking a skirmisher with weapons dice [W] of d8, d10, d12:

average[d12] = 6.5 --> N = 8

average[d10] = 5.5 --> N = 8.89

average[d8] = 4.5 --> N = 10

On average, the extra striker damage reduces the number of hits to kill a monster by around 1 or 2 hits as the level goes to infinity.

### Scaling of 4E at-will powers.

Let's analyze the scaling behavior of the average number of hits to kill a monster in 4E D&D itself, with some very egregious extrapolations.

Recall that the average number of hits to kill a monster (N) is the ratio:

N = (monster hit points)/(average amount of damage per successful attack).

From the 4E PHB1, the damage done by at-will powers is typically:

[W] + stat mod + magic enhancement.

Stat increases happen at levels 4 and 8, where +1 is added to two stats of choice. We assume one of the stats goes into the primary stat.

At the paragon and epic tiers, the stat mod increases happen at levels 11, 14, 18 (paragon) and levels 21, 24, 28 (epic). At levels 14, 18, 24, 28, the stat mod increases are similar to the ones at levels 4 and 8. At levels 11 and 21, the stat mod increases are +1 to every stat. (This is on page 29 of the 4E PHB1).

The magic enhancement for different levels assumes the table from page 225 of 4E PHB1.

1 -5 -> +1

6 - 10 -> +2

11 - 15 -> +3

16 - 20 -> +4

21 - 25 -> +5

26 - 30 -> +6

(ie. Magic enhancement increases by +1 every five levels).

To make things simple, we will examine the levels 11-30 of paragon and epic tiers as one entity. Over levels 11-30, the total additional damage contributed by the stat increases and magic enhancement is +7, due to +3 from stat increases and +4 from magic enhancement, by the time one reaches level 30. (Heroic tier by level 10, typically already has a +6 to +8 contributed to the damage, where: +3 to +5 is from the stat mod, +1 from the two stat increases, and +2 from magic enhancement).

One egregious assumption we will make, is that this pattern of stat increases and magic enhancement remains the same every 20 levels as one goes to higher levels beyond level 30. For example, stat increases at levels 31, 34, 38, 41, 44, 48, etc ... and magic enhancement increases of +1 every five levels. In effect at level 50, the total additional damage contributed by stat increases and magic enhancement is +7.

So above level 10, the increase to damage from stat increases and magic enhancement scales approximately as: 7*(level-10)/20

Another egregious assumption we will make, is that the damage of at-will powers [W] increases each ten levels after level 10, in the pattern of:

level 21-30 --> 2[W]

level 31-40 --> 3[W]

level 41-50 --> 4[W]

level 51-60 --> 5[W]

etc ...

So above level 10, the average damage from a successful hit scales approximately as:

level*(average[W])/10 + 7*(level-10)/20

(We will ignore the heroic tier stuff, since it will just drop out as a constant when things scale with level. But for reference, the heroic tier will contribute a +6 to +8 to the damage by the time one reaches level 10).

On the monster side, the hit points of various monsters from page 184 of the 4E DMG1 are:

ROLE*(level +1) + CON

where ROLE is:

Artillery, Lurker --> ROLE = 6

Skirmisher, Soldier, Controller --> ROLE = 8

Brute --> ROLE = 10

(Elites double the hit points, while Solos quadruple the hit points).

Now the average number of hits to kill a monster being attacked by a player (of the same level as the monster) is approximately the ratio (for a high level):

N = [ROLE*(level+1) + CON]/[level*(average[W])/10 + 7*(level-10)/20]

Taking the level to infinity, this ratio approaches the limit of

N -> ROLE/[(average[W])/10 + 7/20].

(If one actually ignores the egregious assumption of weapon damage [W] increasing every ten levels, then the average number of hits to kill a monster is just:

N -> ROLE*20/7

as the level increases to infinity. In this scenario, the at-will power [W] remains the same for all levels. At level 21, [W] remains [W] and does not increase to 2[W]).

Let's plug in some numbers. For a skirmisher monster, ROLE = 8. For a weapon with d8 damage dice, average[W] = 4.5. For these numbers, the average number of successful hits to kill a skirmisher being attacked by a [W] = d8 weapon is N = 10.

For different [W] weapons attacking this skirmisher monster, we have average number of attacks N as the level goes to infinity:

average[d12] = 6.5 --> N = 8

average[d10] = 5.5 --> N = 8.89

average[d8] = 4.5 --> N = 10

average[d6] = 3.5 --> N = 11.43

average[d4] = 2.5 --> N = 13.33

More generally for different monster types with ROLE:

Artillery, Lurker --> ROLE = 6

Skirmisher, Soldier, Controller --> ROLE = 8

Brute --> ROLE = 10

(Elites double the hit points, while Solos quadruple the hit points).

For various weapons [W] dice, this limit is:

average[d12] = 6.5 --> N = ROLE

average[d10] = 5.5 --> N = ROLE/0.9 = 1.11*ROLE

average[d8] = 4.5 --> N = ROLE/0.8 = 1.25*ROLE

average[d6] = 3.5 --> N = ROLE/0.7 = 1.43*ROLE

average[d4] = 2.5 --> N = ROLE/0.6 = 1.67*ROLE

average[d2] = 1.5 --> N = ROLE/0.5 = 2*ROLE

(d2 is flipping a coin, where one side is 2 and the other side is 1).

Indeed with these egregious extrapolations, 4E D&D is "always fighting orcs" as the levels go to infinity.

Recall that the average number of hits to kill a monster (N) is the ratio:

N = (monster hit points)/(average amount of damage per successful attack).

From the 4E PHB1, the damage done by at-will powers is typically:

[W] + stat mod + magic enhancement.

Stat increases happen at levels 4 and 8, where +1 is added to two stats of choice. We assume one of the stats goes into the primary stat.

At the paragon and epic tiers, the stat mod increases happen at levels 11, 14, 18 (paragon) and levels 21, 24, 28 (epic). At levels 14, 18, 24, 28, the stat mod increases are similar to the ones at levels 4 and 8. At levels 11 and 21, the stat mod increases are +1 to every stat. (This is on page 29 of the 4E PHB1).

The magic enhancement for different levels assumes the table from page 225 of 4E PHB1.

1 -5 -> +1

6 - 10 -> +2

11 - 15 -> +3

16 - 20 -> +4

21 - 25 -> +5

26 - 30 -> +6

(ie. Magic enhancement increases by +1 every five levels).

To make things simple, we will examine the levels 11-30 of paragon and epic tiers as one entity. Over levels 11-30, the total additional damage contributed by the stat increases and magic enhancement is +7, due to +3 from stat increases and +4 from magic enhancement, by the time one reaches level 30. (Heroic tier by level 10, typically already has a +6 to +8 contributed to the damage, where: +3 to +5 is from the stat mod, +1 from the two stat increases, and +2 from magic enhancement).

One egregious assumption we will make, is that this pattern of stat increases and magic enhancement remains the same every 20 levels as one goes to higher levels beyond level 30. For example, stat increases at levels 31, 34, 38, 41, 44, 48, etc ... and magic enhancement increases of +1 every five levels. In effect at level 50, the total additional damage contributed by stat increases and magic enhancement is +7.

So above level 10, the increase to damage from stat increases and magic enhancement scales approximately as: 7*(level-10)/20

Another egregious assumption we will make, is that the damage of at-will powers [W] increases each ten levels after level 10, in the pattern of:

level 21-30 --> 2[W]

level 31-40 --> 3[W]

level 41-50 --> 4[W]

level 51-60 --> 5[W]

etc ...

So above level 10, the average damage from a successful hit scales approximately as:

level*(average[W])/10 + 7*(level-10)/20

(We will ignore the heroic tier stuff, since it will just drop out as a constant when things scale with level. But for reference, the heroic tier will contribute a +6 to +8 to the damage by the time one reaches level 10).

On the monster side, the hit points of various monsters from page 184 of the 4E DMG1 are:

ROLE*(level +1) + CON

where ROLE is:

Artillery, Lurker --> ROLE = 6

Skirmisher, Soldier, Controller --> ROLE = 8

Brute --> ROLE = 10

(Elites double the hit points, while Solos quadruple the hit points).

Now the average number of hits to kill a monster being attacked by a player (of the same level as the monster) is approximately the ratio (for a high level):

N = [ROLE*(level+1) + CON]/[level*(average[W])/10 + 7*(level-10)/20]

Taking the level to infinity, this ratio approaches the limit of

N -> ROLE/[(average[W])/10 + 7/20].

(If one actually ignores the egregious assumption of weapon damage [W] increasing every ten levels, then the average number of hits to kill a monster is just:

N -> ROLE*20/7

as the level increases to infinity. In this scenario, the at-will power [W] remains the same for all levels. At level 21, [W] remains [W] and does not increase to 2[W]).

Let's plug in some numbers. For a skirmisher monster, ROLE = 8. For a weapon with d8 damage dice, average[W] = 4.5. For these numbers, the average number of successful hits to kill a skirmisher being attacked by a [W] = d8 weapon is N = 10.

For different [W] weapons attacking this skirmisher monster, we have average number of attacks N as the level goes to infinity:

average[d12] = 6.5 --> N = 8

average[d10] = 5.5 --> N = 8.89

average[d8] = 4.5 --> N = 10

average[d6] = 3.5 --> N = 11.43

average[d4] = 2.5 --> N = 13.33

More generally for different monster types with ROLE:

Artillery, Lurker --> ROLE = 6

Skirmisher, Soldier, Controller --> ROLE = 8

Brute --> ROLE = 10

(Elites double the hit points, while Solos quadruple the hit points).

For various weapons [W] dice, this limit is:

average[d12] = 6.5 --> N = ROLE

average[d10] = 5.5 --> N = ROLE/0.9 = 1.11*ROLE

average[d8] = 4.5 --> N = ROLE/0.8 = 1.25*ROLE

average[d6] = 3.5 --> N = ROLE/0.7 = 1.43*ROLE

average[d4] = 2.5 --> N = ROLE/0.6 = 1.67*ROLE

average[d2] = 1.5 --> N = ROLE/0.5 = 2*ROLE

(d2 is flipping a coin, where one side is 2 and the other side is 1).

Indeed with these egregious extrapolations, 4E D&D is "always fighting orcs" as the levels go to infinity.

### 4E D&D scaling analysis articles.

We'll be reposting the series 4E D&D scaling analysis articles from therpgsite.

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