## Friday, June 28, 2019

### Fleet Combat Dynamics II

"All models are wrong, but some are useful."
- George Box

In my post in March 2018, I presented a continuous dynamical model that enabled analysis of homogeneous fleet combat in aggregate terms.  By modeling only the damage rates, effective hit points and number of ships in each fleet, I showed that a pair of separable differential equations could model dynamics of ship losses with time.

Using this model as a starting point, I showed a couple basic things.  First, I showed that the number of ships in your fleet has a squared effect on fight outcomes.  Stated simply, bringing just 10% more people is roughly equivalent to fitting every ship with 20% more hit points per ship.  Second, I showed that at any given time during a fight, comparing the fractional rate of ship loss for two fleets is equivalent to comparing the figure of merit for each fleet.

In this short post I want to show a good example of how the dynamic model behaves for a real fight, and suggest extending the interpretation of this dynamical model for fleet combat to help us diagnose the outcome of another engagement.

I'm including footage of the fight in the link below.  Thanks to BigChols and the Chols Media Group for providing the community with videos of important PVP engagements!

I highlight this fight because it has many of the properties that I relied upon in the fleet combat dynamics model that I shared in March 2018.  That is, this fight involves two fleets that are largely homogeneous and have similar numbers, when counting their damage dealing ships.

Even more intriguing is that the damage dealing ships are of the same type in both fleets which is a good case to study with our dynamic model.  I will first make the assumption that both fleets are fitted and trained identically and that hit-points and damage rate are the same for both ship types, i.e. $HP_A = HP_B$, as well as, $DPS_A = DPS_B$.  This makes the assumption that both fleets have the same fitting, use the same ammo type, and on average, have the same level of skill point training associated with the fittings of their ships.  These assumptions are not far fetched, and as you will see below, are mostly correct.

The battle report for the fight is here (ZMV9-A battle report) and I summarized the initial numbers involved below:

 Ship Type \ Fleet Fleet A(Black Legion) Fleet B (Test Alliance) DPS Muninns 104 94 Logistics Loki 5 2 Shield Logistics 6 0

Figure 1 below shows the time evolution of the number of damage dealing ships surviving with time.  The plots in this post show time zero aligned with 15.**.  Also, note that I only plot the solution to the dynamic model until the 480 second point, or eight minutes.  From the footage, my understanding is that Test Alliance Munnins had disengaged at around 400 seconds and were leaving grid after that point, so they were not applying damage as effectively.  Futher model calculation wouldn't be relevant to understanding the outcome.

 Figure 1: Damage dealers (Munnins) remaining alive over time.

Clearly Fleet A had superior damage-dealer numbers as well as logistics.  The results aren't surprising in the number of ships destroyed when Fleet B decides to leave the field.

If I ignore the effect of logistics, and whatever fitting and ammo differences there are, can our simple statistical dynamical model capture the overall outcome of the engagement?  If I only modeled the damage dealer behavior and ignored everything else, how well would our simple model predict the outcome of this fight?

Based on the fleet composition information and basic information from ship fitting I use the following parameters in the dynamic model:

 Ship Type \ Fleet Fleet A(Black Legion) Fleet B   (Test Alliance) EHP 75000 75000 Effective DPS 120 120

Note these are just estimated from the usual fittings, resists and ammo used by artillery Munnins.  $T_{mutual}$ is estimated at ~610 sec.

So, I want to compare the model assuming that both fleets are equally coordinated and using the same ammunition.  Recall that the closed form of the dynamical model estimates the time evolution of the surviving numbers, as the number $N_a(t)$ and $N_b(t)$,
$N_a(t) = N_A cosh(t/T_{mutual}) - N_B \sqrt{ \frac{DPS_B HP_B}{ DPS_A HP_A} } sinh(t/T_{mutual})$
$N_b(t) = N_B cosh(t/T_{mutual}) - N_A \sqrt{ \frac{DPS_A HP_A}{ DPS_B HP_B} } sinh(t/T_{mutual})$

You can review the derivation of the model here.

Plotting these equations in Figure 2, I compare the remaining number of damage dealers during the eight minutes of the engagement.  There is surprising agreement between the model and the actual loss rates for both fleets.  Looking a bit more closely, the loss rates are almost constant, indicating that the fight lasted less than a single engagement time-constant, $T_{mutual}$.  This is quite different than the example I included in my previous post, because in this case there are no external damage dealers confusing the assumptions of the model.

Note also that the loss rate of the BL fleet appears to drop to zero after the six minute mark (360 seconds), while Test fleet losses continue to mount for another two minutes.

 Figure 2: Dynamical model playback with $HP_A = HP_B = 75000$.  Qualitative agreement is surprisingly good considering the complexity of this fight and the presence of a small number of logistics.

Looking more closely at Figure 2, there are a couple things to mention.  First, you'll notice that I put the zero start time after some of the Fleet B ships had already been lost.  This is because in the first 60 seconds, Fleet B was shooting logistics ships and Lokis.  Considering the zero-time I'm using in this plot is helping me to ignore logistics, because some of these have been eliminated.  Second, there are some notes that might be worth adding on damage application and command links [bunch more to be added later].

Having mentioned these factors, another thing that can contribute to difference between the model and the fight outcome is that Test Muninns appear to be loading artillery with Titanium Sabot, while Black Legion is loaded with Fusion.  Seeing as the fleets are at the same aggregate distance from one another, there does not appear to be any reason to use longer range ammunition.

#### Executive Summary

• The strength of your doctrine fleet scales with the square of the number of damage dealing ships, i.e. $O(N^2)$.  Another way to think of this is that once you have lost 30% of your combat ships (or 70% of damage dealers are remaining) you have lost more than half of your combat capability.
• Ammunition choice, resists and target order have an effect on fight outcomes.  Don't let that part of the fleet doctrine and command go unexamined or unpracticed.