Focused Fire, part 2

Today I’m going to take a closer look at how varying levels of focused fire affect combat. This is a bit math heavy, but it does give use a good theoretical base to work from.

Formulas

Note:In the following formulas, I’m using square brackets to set up the abbreviations for each of my terms.

First off, we’re going to be looking for events that change the course of the battle. The most notable of these is the loss of a combatant. We can calculate the timing of these events as follows.
time to defeat [TtD] = initial health [Hi] / health loss rate [HLR]

Usually health loss is the sum of all damage sources directed at the target. For scattered fire, this work out as follows:
HLR from scattered fire [HLRs] = damage rate for enemy team [DRet] / number of enemy targets [NTe]
HLRs = (number of enemies [Ne] * average enemy damage rate [DRea]) / number of allies [Na]
HLRs = Ne * DRea / Na

For focused fire, we get:
HLR from focused fire [HLRf] = damage rate for enemy team [DRet]
HRLf = Ne * DRea

If the group can only manage focused fire a certain percentage of the time, we can calculate the results as follows:
mark’s HLR [HLRm] = chance of focused fire [Cf] * HLRf + chance of scattered fire [Cs] * HLRs
Since we have only 2 tactics: Cf + Cs must add up to 1. This gives us:
HLRm = Cf * HLRf + (1 – Cf) * HLRs
HLRm = Cf * Ne * DRea + (1 – Cf) * Ne * DRea / Na
HLRm = (Ne * Drea) * (Cf + (1 – Cf) / Na)

In contrast, those who aren’t being focused on take no damage from focused fire, so we get:
total HLR for non-marks [HRLn] = Cf * HLRf + (1 – Cf) * HLRs
HRLn = Cf * 0 + (1 – Cf) * Ne * DRea / Na
HRLn = (1 – Cf) * Ne * DRea / Na

Finally, the gradual loss of health can be modeled as follows:
health at end of interval [Ht] = initial health [Hi] – time [t] * average health loss rate for the interval [HLRa]
Ht = Hi – t * HLRa

Testing

Let’s start by assuming each unit has a damage rate of 1. This gives us:
HRLm = Ne * (Cf + (1 – Cf) / Na)
HRLn = (1 – Cf) * Ne / Na
HLRs = Ne / Na

I used a spreadsheet to crunch though these number for varying team sizes and degrees of focused fire. For focused fire, I left one team at 0 focus so we’d see how different levels of focused fire stand up against scattered fire. Here are the results.

Health Remaining after Battle

Focus Chance 2 on 2 3 on 3 4 on 4 5 on 5
100% 75% 67% 63% 60%
75% 79% 71% 68% 66%
50% 83% 78% 75% 73%
25% 90% 87% 85% 84%

Conclusions

We’ve already noted that full focus fire can be handle with the equation: Health Loss [HL] = (team size [TS] + 1) / (2 * teams size [TS]). It’s interesting to note the health loss values I got seem to follow a similar pattern for all levels of focus fire. For example, take a look at some of the health loss numbers when they’re written as fractions.

Fractional Health Loss

Focus Chance 2 on 2 3 on 3 4 on 4 5 on 5
100% 3/4 4/6 5/8 6/10
50% 5/6 7/9 9/12 11/15
33% 7/8 10/12 13/16 16/20
25% 9/10 13/15 17/20 21/25

This follows a very definite pattern. In each case, the numerator increase by a set amount each time the team size increases. This rate seems to equal the inverse of the focus chance. In other words the increase equals 1/Cf. This gives us a numerator of (1/Cf)*TS + 1. The denomination shows similiar, but slightly faster scaling. This works out to being 1 step faster than the numerator, which works out to (1 + 1/Cf)*TS. Combing these gives us:
health loss [HL] = ((1/Cf) * TS + 1) / ((1 + 1/Cf) * TS)

We’ve already seen we can invert health loss to get the allied offense multiplier which will match the tactic. The interesting thing is an allied defense modifier would have a similar effect. As such, we’ll just refer to this general increase in effectiveness as a force multiplier. This gives us:
force multiplier [FM] = (1 + 1/Cf) * TS / ((1/Cf) * TS + 1)

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Published in: on May 26, 2011 at 7:44 am  Leave a Comment  
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