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The intent of this article is to give the reader a more concrete understanding of damage output and what makes a weapon effective. All the weapon statistics given in this wiki are not entirely useful until they are assembled into a mathematical relationship that allows one to see how a particular statistic affects weapon performance. The analyses given here should reflect personal experience, as any mathematical model should do, but the equations may also reveal things that are not readily seen simply by comparing the numbers in the tables. Even though one weapon can inflict higher total damage than another, how a player uses them under differing circumstances changes their effectiveness greatly. Understanding the math is quite important as misusing the equations for weapon analysis will give erroneous results.

Definition

The Modified Damage Per Second (MDPS) is the amount of damage a weapon can inflict over time, while accounting for the extra damage of random critical hits.

Basic Formula

The following is a generalized derivation of the average damage-per-second (DPS) value used for the Jack.[1] Keep in mind that this is an average DPS and should not be treated as consistent DPS.

\mbox{MDPS} = C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS}

where,
C_{Base} = the player character's base critical chance as a decimal value between 0 and 1.
C_{Mult} = the critical multiplier for a given weapon.
D_{Crit} = the additional critical hit damage for the weapon.
R_{Hits/s} = the number of hits per second, also the rate of fire of the weapon if accuracy is 100%.
B_{DPS} = the base DPS or non-critical hit DPS.

This is a simple formula that allows quick calculation of a weapon's MDPS for statistical purposes assuming 100% accuracy.

Proof

The following is a derivation of the above equation using intuitive logic and unit canceling.

Let,
C_{Hits/s} be the number of critical hits per second.
C_{DPS} be the average critical damage per second.

The weapon's rate of fire is also the rate of hits assuming 100% accuracy. Critical hits make up some portion of those hits, and the player character's final critical chance determines the average number of critical hits in a given number of hits.

\begin{array}{rcl} (C_{Base} \cdot C_{Mult}) \cdot R_{Hits/s} & = & C_{Hits/s} \\
C_{Hits/s} \cdot D_{Crit} & = & C_{DPS} \\ && \\
C_{Hits/s} & = & \dfrac{C_{DPS}}{D_{Crit}} \\ && \\
(C_{Base} \cdot C_{Mult}) \cdot R_{Hits/s} & = & \dfrac{C_{DPS}}{D_{Crit}} \\ && \\
(C_{Base} \cdot C_{Mult}) \cdot R_{Hits/s} \cdot D_{Crit} & = & C_{DPS} \\
(C_{Base} \cdot C_{Mult}) \cdot R_{Hits/s} \cdot D_{Crit} + B_{DPS} & = & C_{DPS} + B_{DPS} \\
C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS} & = & \mbox{MDPS}
\end{array}

QED

MDPS is the sum of the average critical damage per second and the base damage per second; C_{DPS} is what "modifies" the DPS.

Shooting Accuracy and MDPS

If not all of the shots hit, then MDPS will change accordingly.

\mbox{MDPS} = A \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS})

where,
A is the shooting accuracy between 0 and 1.

It is important to note that A is not shooting accuracy as in

\frac{\mbox{number of shots hit}}{\mbox{number of shots fired}}

but rather,

\frac{\mbox{number of shots hit}}{\mbox{number of shots fired} + \mbox{number of shots not fired}}

When taking cover, for example, that time spent could have been used to shoot. This sacrifice of time reduces MDPS in the same way having poor accuracy does.

Time Factor

Time elements can be included in MDPS as follows.

Let,
S_{Hits} be the number of shots hit.
S_{Fired} be the number of shots fired.
t_{Idle} be the time spent idle/not shooting when it is possible to shoot in seconds.

\mbox{MDPS} = \frac{S_{Hits} \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS})}{S_{Fired} + R_{Hits/s} \cdot t_{Idle}}

It becomes difficult to compare different weapons with this because of the difficulty in measuring t_{Idle} since a player will be aiming, shooting, and taking cover. Note that S_{Fired} is simply the rate of fire multiplied by the total time spent firing.

\begin{array}{rcl} \mbox{MDPS} & = & \dfrac{S_{Hits} \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS})}{R_{Hits/s} \cdot t_{Fire} + R_{Hits/s} \cdot t_{Idle}} \\ && \\
& = & \dfrac{S_{Hits} \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS})}{R_{Hits/s} \cdot (t_{Fire} + t_{Idle})}\end{array}

Also note that B_{DPS} = D_{Base} \cdot R_{Hits/s}, where D_{Base} is the base damage of the weapon per hit. It can be observed that t_{Fire} + t_{Idle} is simply the total time elapsed, which shall be called t.

\begin{array}{rcl} \mbox{MDPS} & = & \dfrac{S_{Hits} \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + D_{Base} \cdot R_{Hits/s})}{R_{Hits/s} \cdot t} \\ && \\
& = & \dfrac{S_{Hits} \cdot R_{Hits/s} \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base})}{R_{Hits/s} \cdot t} \\ && \\
& = & \dfrac{S_{Hits} \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base})}{t} \end{array}

This is the most basic definition of MDPS: total damage over time, accounting for average critical damage. There is another observation here: the rate of fire of the weapon is irrelevant as long as it is known how many shots have hit and how much time is spent using the weapon.

Perks and Skills

The following table summarizes what perks and skills affect which variables.

Variable Affected by
C_{Base}
D_{Crit}
D_{Base}

Of course, all of the damage affecting perks cannot be applied at once since different enemies are affected by some of them, but not others.

C_{Base} = (0.01 \times \mbox{Luck}) + (0.05 \times \mbox{Finesse Rank}) + (0.01 \times \mbox{Snide Survival Expert Level}) + (0.15 \times \mbox{Ninja Rank})
D_{Crit} = \mbox{Weapon Critical Damage} \times (1 + 0.50 \times \mbox{Better Criticals Rank})
\begin{array}{ll} D_{Base} = & \mbox{Weapon Base Damage} \times \\
&(1 + 0.05 \times \mbox{Bloody Mess Rank} + 0.20 \times \mbox{Demolition Expert Ranks} + 0.50 \times \mbox{Pyromaniac Rank} + 0.20 \times \mbox{Xenotech Expert Rank} + \\
&0.25 \times \mbox{Auto Axpert Rank} + 0.10 \times \mbox{Black Widow/Lady Killer Rank} + 0.50 \times \mbox{Entomologist Rank} + 0.25 \times \mbox{Robotics Expert Rank}) + \\
&5 \times \mbox{Iron Fist Ranks} + 5 \times \mbox{Ghoul Ecology Rank} + 5 \times \mbox{Superior Defender Rank}\end{array}

Apply only the perks that affect the specific weapon being used and the enemy being faced. Skills affect the base damage as follows.[2]

\mbox{MDPS} = \dfrac{S_{Hits} \cdot [C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base} \cdot (0.5 + 0.5 \cdot \dfrac{P_{Skill}}{100})]}{t}

where,
P_{Skill} is the number of skill points the player has for the weapon skill that corresponds with the weapon being used.

Weapon Decay

The base damage will decrease over time because of decreasing weapon condition as shots are fired. The amount of damage loss varies for different kinds of weapons. The damage at 0% condition is 66% of the base damage rounded to the nearest whole number and it scales linearly.[2] However, it has been found to hold for single-shot firearms only.[3]

The critical chance is also affected by weapon condition.[4] The equation can be modified as follows.

\mbox{MDPS}_{\mbox{Single}} = \dfrac{S_{Hits} \cdot [C_{Base} \cdot C_{Mult} \cdot W_{CND} \cdot D_{Crit} + D_{Base} \cdot (0.66 + 0.34 \cdot W_{CND})]}{t}

where,
W_{CND} is the current weapon condition as a decimal between 0 and 1.

Fully automatic weapons have been found to have 54% of the base damage at 0% condition.[3]

\mbox{MDPS}_{\mbox{Full}} = \dfrac{S_{Hits} \cdot [C_{Base} \cdot C_{Mult} \cdot W_{CND} \cdot D_{Crit} + D_{Base} \cdot (0.54 + 0.46 \cdot W_{CND})]}{t}

Melee weapons have a damage value that is 50% of the base damage at 0% condition.[2]

\mbox{MDPS}_{\mbox{Melee}} = \dfrac{S_{Hits} \cdot [C_{Base} \cdot C_{Mult} \cdot W_{CND} \cdot D_{Crit} + D_{Base} \cdot (0.50 + 0.50 \cdot W_{CND})]}{t}

D_{Base} may already be modified by perks and skills, so this decay modifier is simply multiplied on.

Sneak Attacks & Head Shots

Sneak attack critical hits and head shots double the total damage dealt to the target. [5] These can be factored into the equation as follows.

\begin{array}{rcl} \mbox{MDPS} & = & \dfrac{S_{Hits} \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base}) + N_{HS} \cdot [2(C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base})] + N_{SA} \cdot [2(D_{Crit} + D_{Base})] + N_{SHS} \cdot [2(D_{Crit} + D_{Base})]}{t} \\ && \\
& = & \dfrac{(S_{Hits} + 2N_{HS}) \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base}) + 2(N_{SA} + N_{SHS}) \cdot (D_{Crit} + D_{Base})}{t}\end{array}

where,
N_{HS} is the number of head shots performed while not hidden.
N_{SA} is the number of sneak attack critical hits performed.
N_{SHS} is the number of head shots performed while hidden.

The equation shows that normal head shots count as two normal hits, as would be expected since it doubles the damage dealt per head shot. It must be emphasized that S_{Hits} is the number of regular hits, not head shots or sneak attacks, otherwise there would be double-counting. The N variables work independently of S_{Hits} and they do not increase together. Head shots while hidden are implied to be sneak attacks, and the damages of regular sneak attacks and head shots while hidden stack, so N_{SA} increases with N_{SHS}. Since sneak attack criticals are guaranteed, they are not subject to the player's base critical chance or weapon critical multipliers.

Enemy Damage Resistance

Enemy damage resistance (DR) can also be included into the formula as a percentage of the damage dealt.

\mbox{MDPS} = \dfrac{(S_{Hits} + 2N_{HS}) \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} + (1 - D_R) \cdot D_{Base}) + 2(N_{SA} + N_{SHS}) \cdot (D_{Crit} + (1 - D_R) \cdot D_{Base})}{t}

where,
D_R is the enemy damage resistance as a decimal number between 0 and 0.85, since 85% is the maximum usable DR.

DR is applied after any modifiers (e.g. Weapon Decay, Perks, Skills) to D_{Base} is applied. Critical damage actually ignores DR, so the modifying term affects D_{Base} only.[2] Including DR in MDPS calculations is useful when determining whether the final damage is enough to kill within reasonable time. When comparing weapons, including DR can be quite important since a weapon that has a low chance for critical hits will be hindered. Weapons that ignore DR will not be hindered and are treated as though D_R = 0.

Implications

The MDPS equation can be used to quantitatively measure a player's performance with various weapons. This equation can also be used to calculate DPS by setting the critical chance, critical multiplier or critical damage to 0. Since critical hits are an integral part of Fallout 3 combat, MDPS is the preferred terminology and is more useful than DPS. MDPS is constantly varying because of the nature of S_{Hits}, D_{Base}, and t. Thus, MDPS, in the broadest sense, is the aggregate of a player's ability to do damage over time with particular weapons under certain conditions.

Time

Short-run and long-run analysis on weapons can be performed using the MDPS formula. The difference between short-run and long-run is that analyzing in the short run gives a weapon's effectiveness in combat, while analyzing in the long-run gives a weapon's reliability (i.e. how often it will be used). Since t_{Idle} is part of t, changing t_{Idle} also changes t.

Combat

In the short run, t_{Idle} increases when the player is not firing, which includes reloading and taking cover. Tactical reloading (i.e. reloading when the magazine is not empty during pauses in combat) increases overall idle time (which decreases MDPS in the long-run) in order to increase short-run MDPS by allowing the player to fire more shots before reloading. Because of reloading, DPS and MDPS are not simply the weapon's rate of fire multiplied by the damage. Such a definition works only on a per magazine basis and does not account for the loss in time because of reloading.

The MDPS is always changing even when firing at the maximum rate of fire. Between individual shots, the time will still count, which lowers the MDPS. The moment the next shot is fired, the MDPS jumps back up to a peak value and starts decreasing again until the next shot is fired. When reloading a weapon, time will increase longer than in between shots and it will reduce the average MDPS. The steady-state MDPS value for a weapon uses it's magazine capacity for S_{Hits}, and the minimum time to fire all the shots plus the reload time for t.

In the event that a weapon breaks, it is treated as though the player has stopped shooting, so t_{Idle} increases while S_{Hits} remains the same; weapon breakage influences MDPS in this manner.

Weapon Reliability

When analyzing weapons in the long-run, weapons that use scarce resources increases t_{Idle} substantially because the time will be spent finding additional ammo and weapons for repair. If the player stops using a weapon indefinitely due to unreliability, t \rightarrow \infty and \mbox{MDPS} \rightarrow 0. An MDPS that approaches 0 means the weapon is impractical and ultimately not very effective regardless of high total damage (e.g. Experimental MIRV uses rare Mini nukes, yet it fires 8 of them at a time).

Overkill

Overkill affects MDPS by bottlenecking the total damage to the hitpoints of the target(s). Although a weapon may have extremely high total damage per shot, most of the damage may become wasted if the target has few hitpoints. Area of effect weapons affect multiple targets and the total damage of such weapons is multiplied by the number of targets hit. However, the total damage is still affected by the bottleneck on a per target basis. This forces the MDPS to decrease because the total damage is reduced while the number of shots fired and the amount of time elapsed while using the weapon has not changed. Overkill holds the greatest implications for slow-firing, high damage weapons such as the Gauss Rifle. For a weapon that has low damage per shot and high rate of fire, this bottleneck is negligible since the killing shot will exceed the target's hitpoints by only a little and the high rate of fire allows the player to change targets and continue firing. This kind of distribution of damage among multiple targets allows a weapon to maintain high MDPS.

Examples

Marginal Change in Critical Chance

What happens when the base critical chance is increased by 1%? In other words, what if Luck is raised by 1?

If we raise C_{Base} by 0.01, we have,

\begin{array}{rcl}\mbox{MDPS}_{C_{Base} + 0.01} & = & (C_{Base} + 0.01) \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS} \\
& = & C_{Base} \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + 0.01 \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s} + B_{DPS} \\
& = & \mbox{MDPS} + 0.01 \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s}\end{array}

which means the MDPS value increases by 0.01 \cdot C_{Mult} \cdot D_{Crit} \cdot R_{Hits/s}. For a weapon like the Precision Gatling laser, this means MDPS increases by 0.24 for each 1% increase in critical chance. Whereas, for the Sniper rifle, MDPS increases by 2.1428. Therefore, a character with high base critical chance will benefit greatly from weapons that have high multipliers and critical damage. Since fully automatic weapons have their multipliers divided out by the rate of fire, high rate of fire barely adds to the MDPS, with the exception of the Jack, where this is not the case. For players that prefer to use fully automatic weapons, having a high critical chance adds only marginal benefit unless you sustain fire for several seconds while making every shot hit.

Gauss Rifle vs. Lincoln's Repeater

In one experiment, the Gauss Rifle could fire 10 shots within 25 seconds, while the Lincoln's repeater could fire 21 shots in 25 seconds, both of which take reloading into account. The 25 seconds is arbitrary; however, in the experiment, neither weapon was close to firing an extra shot at the 25 second mark. Having an 18% base critical chance, Better Criticals, and Bloody Mess is assumed. Real-time (non-V.A.T.S.) combat is also assumed.

\begin{array}{lcl} \mbox{MDPS}_{\mbox{Gauss}} & = & \dfrac{10 \cdot (0.18 \cdot 5 \cdot 75 + 105)}{25} \\
& = & 69 \\
\mbox{MDPS}_{\mbox{Lincoln}} & = & \dfrac{21 \cdot (0.18 \cdot 2 \cdot 75 + 52.5)}{25} \\
& = & 66.78\end{array}

There is one important attribute of the Gauss Rifle: knockdown. In the heat of combat, it is likely the player will be spending time taking cover (i.e. increasing t_{Idle} for something other than reloading). However, knockdown disables the enemy and removes any player incentive to sacrifice shooting time to take cover. Suppose we add 5 seconds of total take-cover time to Lincoln's Repeater.

\begin{array}{lcl} \mbox{MDPS}_{\mbox{Lincoln}} & = & \dfrac{21 \cdot (0.18 \cdot 2 \cdot 75 + 52.5)}{30} \\
& = & 55.65\end{array}

The Lincoln's Repeater loses to the Gauss Rifle under this circumstance. The downfall of the Lincoln's Repeater in this case was due to its reloading. Suppose the weapons are fired for 10 seconds, when the Repeater doesn't need to reload. The Gauss Rifle was observed to fire 4 shots, while the Lincoln's Repeater could fire 9.

\begin{array}{lcl} \mbox{MDPS}_{\mbox{Gauss}} & = & \dfrac{4 \cdot (0.18 \cdot 5 \cdot 75 + 105)}{10} \\
& = & 69 \\
\mbox{MDPS}_{\mbox{Lincoln}} & = & \dfrac{9 \cdot (0.18 \cdot 2 \cdot 75 + 52.5)}{10} \\
& = & 71.55\end{array}

The Lincoln's Repeater wins by a margin. However, if the player takes cover for even 1 second while using the Lincoln's Repeater,

\begin{array}{lcl} \mbox{MDPS}_{\mbox{Lincoln}} & = & \dfrac{9 \cdot (0.18 \cdot 2 \cdot 75 + 52.5)}{11} \\
& = & 65.05\end{array}

From a pure damage standpoint, both weapons are roughly similar. But since the Gauss Rifle has knockdown, the player can stand out in the open and shoot as fast as possible at his/her downed opponent rather than alternating between shooting and taking cover. One other important factor that needs mention is overkill per shot. When fighting multiple weak enemies, the total critical modified damage per shot may not exceed each enemy's hitpoints. If shooting a Raider with 85 hitpoints,

\begin{array}{lcl} \mbox{MDPS}_{\mbox{Gauss}} & = & \dfrac{4 \cdot (85)}{10} \\
& = & 34 \\
\mbox{MDPS}_{\mbox{Lincoln}} & = & \dfrac{5 \cdot (79.5) + 4 \cdot (5.5)}{10} \\
& = & 41.95\end{array}

When up against these Raiders, the lower damage per shot of the Lincoln's Repeater allows the player to distribute the damage over all enemies present better than the Gauss Rifle. Therefore, there is a tradeoff between the weapons. When fighting Super mutant overlords, the Gauss Rifle fares better because of its knockdown capability and high damage per shot. When fighting a gang of Raiders, the Lincoln's Repeater excels because it can distribute damage better. The knockdown advantage of the Gauss Rifle is negated when it takes only one shot to kill, and reloading after each shot becomes it's downfall.

For the Lincoln's Repeater, the total damage is divided into two parts. The first half of the shots does the full 79.5 damage, while the other half only does 5.5 damage. Since one normal shot does not kill, the second shot does only the remaining hit points of the Raiders in damage. It's not possible to shoot a fraction of a shot and it does not make sense to fire more "second shots" than "first shots" when dealing with a group of enemies that have full health, so the shots hit is rounded in favour of higher MDPS.

The Gauss Rifle is also more dependent on player skill to use. Calculus may be applied by differentiating with respect to S_{Hits}, which gives the rate of change of MDPS as S_{Hits} is changed by 1.

\begin{array}{ccl} \dfrac{\partial \mbox{MDPS}}{\partial S_{Hits}} & = & \dfrac{C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base}}{t} \\ && \\
\dfrac{\partial \mbox{MDPS}_{\mbox{Gauss}}}{\partial S_{Hits}} & = & \dfrac{0.18 \cdot 5 \cdot 75 + 105}{25} \\
& = & 6.9 \\ && \\
\dfrac{\partial \mbox{MDPS}_{\mbox{Lincoln}}}{\partial S_{Hits}} & = & \dfrac{0.18 \cdot 2 \cdot 75 + 52.5}{25} \\
& = & 3.18
\end{array}

If one shot from the Gauss Rifle is missed, it's MDPS drops by more than what Lincoln's Repeater drops by. This means the Gauss Rifle punishes the user much more for missed shots. From the opposite perspective, if the player was originally poor at aiming, but practiced to become better, he/she will notice that increasing the number of shots hit for the Gauss Rifle is more rewarding than for the Lincoln's Repeater.

Shotgun Weapons

Weapons that fire many shots in one attack have the unique property of having critical damage applied to each shot outside of V.A.T.S.[5] The MDPS equation need not be modified to show this, since each shot can be treated as adding to the number of shots hit while the base damage is the damage of an individual shot. For the Combat Shotgun, S_{Hits} simply increases by 9 at a time while D_{Base} is 6.111, the damage of one pellet. In most cases, only some of the pellets will hit, some will hit the head, and the rest will miss. The head shot, sneak attack damage of a single attack with The Terrible Shotgun can be calculated by using the numerator of the MDPS equation only.

\begin{array}{lcl} \mbox{DMG}_{\mbox{Terrible Shotgun}} & = & (S_{Hits} + 2N_{HS}) \cdot (C_{Base} \cdot C_{Mult} \cdot D_{Crit} + D_{Base}) + 2(N_{SA} + N_{SHS}) \cdot (D_{Crit} + D_{Base}) \\
& = & (0 + 2 \cdot 0) \cdot (0.18 \cdot 1 \cdot 60 + 8.889) + 2(9 + 9) \cdot (60 + 8.889) \\
& \approx & 2480\end{array}

This is the ideal case, where all the pellets hit the head on a sneak attack.

References

  1. By 68.192.9.90: Talk:Jack_(weapon)#Ridiculous_critical_damage
  2. 2.0 2.1 2.2 2.3 Weapon Damage Formula - GECK Wiki
  3. 3.0 3.1 Talk:Fallout_3_armor_and_clothing#VALUES
  4. Critical Hit Chance Formula - GECK Wiki
  5. 5.0 5.1 Fallout 3 combat