Damage per Heat (DPH)
Balancing two weapons is understood as equalizing the efficiency of both weapons where the efficiency is defined as gain/cost.
Eq.(1):
E = gain/cost
Taking a look at laser weapons the damage/heat ratio (DPH) is probably the first important factor that comes to mind to describe the efficiency of theese weapons as damage is what you want to maximize (gain) and heat is the main ressource at hand during the battle (cost).
Eq.(2):
DPH = damage/heat
Understanding DPH as the value to be balanced neglects range, tonnage, number of slots, damage spread in time (pinpoint damage) and alpha strike potential of the weapon (so far), but you gotta start somewhere i guess.
The difference in DPH between S,M and L could be set to some scale as desired from a gameplay perspective. I.e. (just taking natural numbers as an arbitrary example for easier reading.):
SL_DPH = 4
ML_DPH = 2
LL_DPH = 1
where the scale of this example is [4,2,1]. The same lasers in clan version could be constructed by introducing clan DPH increase (C_DPH_INC) as a factor to that scale.I.e.
CSL_DPH = SL_DPH*C_DPH_INC
CML_DPH = ML_DPH*C_DPH_INC
CLL_DPH = LL_DPH*C_DPH_INC
Doing so reducedes the number of variables to balance as the clan weapons do not have to be balanced individualy but are just the IS weapons times some factor. You could then i.e. set
C_DPH_INC = 1.1
if you want the clan weapons to be 10% more heat efficient then the IS coutnerpart. This one value, and the scale itself is what is to be balanced, not the single values of the individual weapons themselfs as seen over the last few years
Weapon range
The same principle can be applied to range. (again the numbers are just arbitrary choosen natural numbers/placehodlers):
SL_range = 100
ML_ range = 300
LL_ range = 1000
Introducing the clan range increase factor (C_R_INC)
CSL_range = SL_range* C_R_INC
CML_ range = ML_range* C_R_INC
CLL_ range = LL_range* C_R_INC
which might be set to
C_R_INC = 1.2
if you want clan lasers to have 20% increased range. Same principle as above, the number of variables is reduced a lot.
DPH over weapon range
The above was meant as an easy example where range and DPH of the lasers where thought of as beeing independed variables which they are not from a balancing perspective. You could i.e. say that a CERML has approximately the range of an IS LL and should as such have more the LL_DPH as its base instead of the ML_DPH. In fact lets just come up with one function that assigns one DPH value to every range of the laser weapon and take that as a basis for all laser weapons (further reducing the number of variables)
Such function could look like the following blue line in the picture.:

Figure 1
I made it quadratic, as the area of effect you can cover with your laser in a 2D enviroment scales quadratic with its range (compare area of a circle). By adjusting the steepness of the function you can shift from brawling to sniping gameplay. This "steepness" is a single variable shifting from brawling to sniping gameplay for all lasers.
The plot can i.e be read as: "A IS laser with a range of 400m has a DPH of 1.6. A clan laser at the same range has a DPH increased by a factor of 1.1 (the C_DPH_INC factor we defined earlier). No matter the name of the laser its DPH is just defined by its range. You can now place an arbitrary numbers of different laser weapons on that function i.e. IS S,M,L, ERS,ERM,ERL). This arbitrary number is probably discretized to only a few by the round number of tonnage an slots required for each of them.
Tonnage and required slots
Heat is not the only "cost" for a weapon, there is also its tonnage and the ammount of slots required. The question is how to add up heat, tonnage and slots where they have different units to find the "cost" for the efficiency of the weapon. This are 3 variables so we lack 2 equations to define cost as a function of heat, tonnage and slots.
The first one is given by the double heatsink (DHS). Lets take the Clan DHS as example. It sinks 1.4 heat/s for 1 ton and 2 slots.
Eq.(3):
1.4 Heat = 1t + 2 slots
Now we need one more equation. I choose to find out how many tons and slots the average mech has available for construction. if you have i.e. 10 times more tons available then slots, then the value of a slot is 10 times higher.
The average mech with an XL engine of 250 or above, no endo steel, no ferro and maximum Armor has ~17 tons of free payload for equipment. You can get to that value when you average over equipping all engine sizes above and including 250 to all mech chassis tonnages. Its not only the average but also the median (most common value).
Every mech with XL, no endo, no ferro and full arm activators has 45 free slots.
Every XL engine of 250 or above has 10 built in heatsinks (which is why we restricted to engine sizes 250+) of 2.0 heat/s which come "for free". You can see that they come for free if you plot mech speed over engine tonnage for all engine sizes. Using Eq.(3) this 20 heat correspond to
20 Heat = 14.29t + 28.57 slots
Adding the average payload tonnage of 17t and the 14.29t which you get via the 10 DHS inside the engine you have on average of 17t + 14.29t = 31.29t avaiable.
Adding the 45 free slots of the mech and those 28.57 you get indirectly by the 10 DHS in the engine results in 45 + 28.57 = 73.57 free slots on a mech.
So on the average mech the ratio between tonnage and slots is
Eq.(4):
31.29 t = 73.57 slots
which can be used as the 2nd eqation required to combine heat,tonnage and slot requirements of a weapon to a single cost value.
The efficiency of a weapon was earlier (Eq.1) discussed as.:
E = Gain/Cost
Eeshaping Eq.(3) and Eq.(4) we find that
1 Heat = 0.3236 slots
1 Heat = 0.7528 t
and can thus express the weapon efficiency as a function of damage, heat, tonnage and slots.:
Eq,(5)
DPHTS = Gain/Cost = Damage/(Heat + 0.7528*t + 0.3236*slots)
where DPHTS is the damage per heat,tonnage and slots.
Now we can directly compare lasers based on theire DPHTS efficiency.
=> for a max efficient mech you should use all its tons (no brainer)
=> for a max efficient mech you should "run hot or die" (constantly have your heatsinks sink the maximum ammount they can during a match)
=> for a max efficient mech you should use all its slots (not that obvious)
==> removing the hand activators potentialy increases the mechs damage output (funny but true :þ)
Comparison of IS and Clan none pulse laser efficiencies
Now that we have a model for the DPHTS weapon efficiencies lets plot them for the four groups IS and Clan normal and pulse lasers over their respective range.:

Figure 2
The little greenish squares represent current MWO weapon data. The thick lines just connect weapons of the same group (like IS normal lasers). IS weapons are blue, Clan weapons are read. Normal lasers are solid lines. Pulse lasers are dashed lines. The y-axis shows the efficiency of the weapon. The x-axis is the squareroot of the range of the weapon (compare the area of a circle model above).Under that scale all weapons of one group (like clan pulse lasers) need to be locaed on a straight line in order to be balanced inside theire group. I did fit a 1st order polynomial (a straight line) through each of the four groups (the thiner lines). Those lines can be understoof as the average efficiency over range behavior of that weapon group.
Some information that we can get from the above plot
Clan lasers are located higher as they are more efficient then IS lasers.
Pulse lasers are less efficient then the corresponding normal lasers. They have the advantage of decreased burn duration which is not modeled here at all (ill come to that later) so without taking that benefit into account they must be less efficient then the normal lasers to be balanced.
Overall the deviation from the fitted thin line is not that large suggesting that:
- the choosen model is quite accurate
- the balance inside each weapon group is quite good but still improoveable.
Deriving other weapon groups from existing ones based on Fig.2
The first thing to do now in terms of balancing is to bring all lasers inside one group directly onto theire groups line. Every weapon above its groups thin line is performing above average and every weapon below the thin line is performing below average.
Once the weapons of one group themself are "in line" we can then shift and tilt that line in relation to the other weapon groups in order to create balance between the different weapon groups. The line (and therefor the whole weapon group) can then be expressed by just two variables (recall math stuff about straight lines).:
y = mx+b
Efficiency = m*sqrt(weapon_range) + b
IS lasers -> Clan lasers
I would recomment to define that clan normal lasers are on every range more effective then IS normal lasers by a ratio of i.e. 80%. 80% beeing the one number to be balanced. At least thats the most simple approach. One could also assign a ramp with which the efficiency of clan vs. IS weapons increases over range if we want the clans to have a bigger sniping then brawling advantage - a gameplay decision. Apart from that all clan normal laser weapon values can just be derived from the IS normal laser weapons further decreasing the number of variables to be balanced by a lot.
normal lasers -> pulse lasers
We then derive IS pulse lasers from IS normal lasers by giving them i.e. 85% efficiency of the normal IS laser but at the benefit of 75% burn duration of the IS normal lasers. 75% and 85% beeing the values to be balanced, all other IS pulse laser stats are just derived from IS normal lasers.
Clan pulse lasers are normal clan lasers dirived using the same 75% respectively 85% found for the IS normal to pulse laser relation. So this weapon group can be derived without any new variable or balancing efford required.
AC and UACs
All UAC weapons have been calculated for "fired while constantly doubletapping". I recalculated the damage and heat for the weapon in DT mode as if it would be single firing with increased damage and heat. It does in principle double the damage and heat while DT, but the cooldown is also increased. So the effective increase in cooldown while DT is related to normal cooldown, jamming chance and jamming duration.:
Effective_DT_Cooldown = Cooldown + Jam_Chance * Jam_Cooldown
The relative increase in shots is then.:
Shots_increase = (2/1) / (Effective_DT_Cooldown / Cooldown)
Effective Dmg and Heat has then be scaled by that.:
Effective_Dmg = Dmg*Shots_increase;
Effective_Heat = Heat*Shots_increase;
Ammo:
For weapons carriing ammo we can add the average ammo required per match in the form of 1t and 1slot per ammo using Eq.(5). In order to estimate the ammount of required ammo for all ACs and UACs i just defined that the required ammo for a CUAC5 = 2t. This value is based on ingame experience and maybe a bit low. Actualy i usualy bring 2.5t of ammo, but as you sometimes die early i guess that on average it would be effective to bring less. I then calculated how many seconds i can fire with that gun until the ammo is spended. I then assumed that every other AC needs to be able to fire for the same ammount of time for it to have enough ammo. Taking different ammounts of ammo per ton and other ACs firing rates into account you can then calculate how much ammo all other ACs would need to fire for the same time as the initial assumption of the CUAC5. By doing that i found the required ammount of ammo for each AC which adds to the tonns and slots cost of each AC.
So heres the plot. Same thing as the L4z0rs plot.

Figure 3
=> All ACs of size 2 underperform.
While interpreting the fitted thin lines you also have to realize, that the weak AC2 on the right side make them drop quite fast which might i.e. let the AC5 appear stronger then it is inside the group.
And some more colorful pictures. Lasers and ACs together in one plot for comparison. The thin lines of the polynomial fit are not displays as the plot is already overcrowded.

Figure 4
=> the damage fallof over range is worse for ACs then for lasers. If i remember correctly theire ranges have been nerved here and there. well here you see what you get.
=> Clan ACs and UACs of size 20 and 10 are kinda in line with theire lasers.
=> Clan AC5 and UAC5s are not such a good choice
=> All ACs of size 2 are just plain bad choices. We always knew they are to hot i guess. The range nerfs they got also factors into that.
Burn duration
There are more variables involved in the cost of a weapon then just heat, tons and slots (although that describes it pretty well if i.e. the burn duration is similar among compared weapons). The burn duration is not considered here at all. This variable in the extreme case touches the pin point accurate weapons like gauss, PPC, IS ACs and such. It is difficult to evaluate the exact impact of that factor as it is difficult to find an equation which links it to eigther heat, tonnage or slots or damage.
All of the above is just meant to outline the general idea based on examples. Balancing could be made a lot easier if you reduce the complexity of the problem before you start turning on all the single nobs available without seeing the bigger dependencies.
Link to Matlab code:
https://www.dropbox....l%20V4.rar?dl=0
Older patch stats:
26.08.2014
http://i.imgur.com/8oOzuTs.jpg
http://i.imgur.com/PaTO6HI.jpg
http://i.imgur.com/rpc8I4J.jpg
Edited by MadTulip, 05 September 2014 - 11:18 AM.