Dynamics of Homogeneous Fleets in Combat

"No one wins. One side just loses more slowly."
- Roland Pryzbylewski - The Wire

When I published my first set of notes on EVE Online in 2010, I titled the work, "Mathematics of EVE Online." Seeing as my first complete chapter was exclusively on the physics of object motion in the game, it should not have surprised me that one of the criticisms of the work was that I should have named it, "Applied Physics in EVE Online." I had not, at that time, emphasized that there are many other topics to study that have nothing to do with motion physics whatsoever.

The derivation work presented in this note began eight years ago.  It represents a portion of the tactical decision abstracts that were shared internally with Origin. and BPI at that time. It is my hope that the wider community will also find them of interest today, in spite of the significant changes that have taken place in the game.

Now, we all know that fleet battles are extremely complex. It is impossible to determine the outcome of any conflict beforehand. All the same, I want to ask some questions about whether it is possible to identify statistics for outcomes of engagements.  If we make some basic assumptions based on observations, what range of outcomes can we determine for fleet action? What follows in this and subsequent posts is intended to clarify some of the components that go into combat decisions and dynamics, such as:

• Should we engage this hostile force or not?
• At what point should our fleet attempt to flee? 
• Should we prioritize target A or target B?

Readers should be forewarned, however, that the value is not in the instructive nature of the results in these articles, but instead in formalizing the thinking process that is at work in current fleet decision making.  My intention here is more narrow than you might think and I include a note about this below.  

I will add notes to the Executive Summary page to help with readers who are pressed for time. 

Intention

This note presents a simple analysis to help inform decision making for fleets based on understanding the dynamics of combat between two groups of ships. As it stands, the assumptions in this note are very narrowly defined. Instead of seeing this as a negative, see it for what it is — narrowly instructive, but broadly analytical. Any model is a product of a set of assumptions, and by making different assumptions you will get different results.  This model represents a starting point for understanding the statistical dynamics of fleet combat.

These notes are not intended to prove or disprove the balance of strategies, or specific ship-types.  They are not intended as a training guide.  Quite the contrary, all I am trying to do here provide fleet commanders with a tool that can be used to analyze their performance in fleet combat situations and help them clarify which decisions they might change in the future.  If you disagree with the approach or my interpretation of them I encourage you to make an effort to quantify  your decision making and share it with the EVE community.  

With some additional work, some of which I may include in a later post, this approach can take into account averaged locking delay, active tanking and even some electronic countermeasure effects. It can not, however, take into account speed tanking or extensive electronic warfare setups.  Thus it is not at all suitable for analyzing nano-gangs, teams of stealth bombers, electronic warfare-heavy groups, or fleets where capitals are changing state, i.e. siege/bastion/triage-modes, or alpha-maximized fleets whose volley damage is so high that ships are destroyed significantly faster than it takes to lock them.  

Finally, readers who are familiar with dynamic population models might characterize the model I present below as a case of "mutual-predators without replacement."  Not a great deal has been written about this case because, as you will see, one team all dies and the fun is over.  From a strategy perspective, I think there is more to be said about this model, so stay tuned.

On with some assumptions... 

Time And Loss

In fleet combat, as in your life, time is the only thing that you can't go get more of.  If you participate in enough large fleet combat you can observe some of the factors that contribute to the time required to do battle with an enemy fleet.  When the fleet commander gives a command to destroy a ship, a few things have to happen.  First, pilots have to lock the target, then pilots have to activate weapons, then missiles or drones (if they are being used) have to reach the target, and then the damage from attackers has to fully deplete the hit-points of the target.  I will group the hit-point groups (shields, armor and hull) into an effective hit-point number, $HP_{target}$, that also accounts for resist ratios.  In this way, I calculate that time to destroy a ship takes the form,
$\large T_{kill} = T_{lock} + T_{flight time} + \frac{HP_{target}}{DPS - RPS}$

That is, a target ship has $HP_{target}$ hit points and is experiencing a repair rate (either external or internal) of RPS effective hit points repaired per second. This form assumes that you have scaled the repair rate by the resists so that incoming damage can be compared directly to find the total time until the ship is destroyed.  This is not as simple as it looks but assume that we have these numbers. It also assumes that the repair rate is less than the incoming damage rate, which often depends on target because repair effectiveness is critically linked with resist ratio. For this note, assume that $DPS_{eff} = DPS - RPS$ and that hostile $DPS$ is greater than logistics repair rates, or $DPS > RPS$. 

The sum also includes the time it takes to lock the target, or $T_{lock}$.  The locking time term, $T_{lock}$, is a simple function of your scan resolution and the target's signature radius.  Disciplined fleets prelock a secondary target while they shoot at the primary, making this term small or irrelevant. If missiles or drones are your sole damage dealing mechanism, you now wait for them to arrive at the target, or $T_{flight time}$.  Assume turret damage has zero flight time.  

I have included a note about other nonidealities with this rough model in Appendix A at the end of this note, including the 'Dumbo Factor', which can be important to consider. 

The Assumptions

In this article I will consider the case where two fleets, each comprised of homogeneous ships, are in combat.  Imagine two fleets:  The first fleet having $N_A$ total ships at the start of the combat, each ship having effective $HP_A$ effective hit-points and delivering effective damage-per-second of $DPS_A$; The second fleet starts with $N_B$ ships, each with effective $HP_B$ hit-points and delivering effective $DPS_B$ damage-per-second.  I will also assume that neither fleet targets enemy logistics and repair effects are are effectively constant for both fleets.

Now the rate of damage may not be constant, but assume that we chew through the total hit points at a fixed effective target armor, which takes $\frac{HP_{target}}{DPS_{eff}}$ time to destroy the target.  If you are imagining that the damage rate and hit-points are more complicated than that, of course you are correct.  For this article, however, I am using an effective damage rate that takes into account all of the mitigating factors and is effectively computing the time that the target survives while under assault.

In light of the small contributions from the locking time and flight time, I want to make the assumption going forward that $T_{kill}$ is dominated by the amount of time needed for the effective damage per second rate to chew through each target's effective hit-points net of repair activities.  That is,
$\large T_{kill} \simeq \frac{HP_{eff}}{DPS_{eff}}$

This type of fleet combat situation is quite common.  In this fight, the fleet commander has a wide mix of cruiser-class damage dealers.  In spite of this, there is reasonably small variability in the kill-time of targets.  I recorded the kill times for Auguror Navy Issue cruisers starting from the time the target is called to the time it is destroyed in Figure 1, below.  Because of the way I am collecting this data, you will see several outliers, but I include these because it would be misleading not to.  In those outlier cases, however, there are other targets called as priority in the interim and these times also include at least one reload cycle.  If you review the video you'll see that these outliers don't block the progress of the fight.  You can see that under normal conditions kill-time duration is clustered around a statistical mean near seven seconds.

Figure 1: A histogram of the times from target called to target destruction, or kill-time, is shown.  Fight was from GE-8JV example in the text.  Data includes target recalls, damage splitting and reloading cycles in a complex fight scenario.

These data are relevant to a discussion about statistical fleet behavior, but that is not my goal here.  Instead, I want to ask questions given that fleets behave statistically.  That is, what can we conclude about the dynamics and outcomes of these kinds of pitched battles?  If every pilot behaves exactly as the fleet commander orders them, targeting the primary with all their damage and prelocking secondary targets, which fleet will be left standing?  Do the answers to these questions suggest decision points where fleet commanders should choose to disengage, or change positioning to improve effective damage?  

Thinking about time in fights can be powerful.  Before I derive dynamics, a couple more notes:  First, fights where locking time dominates combat duration are usually between groups who are significantly size mismatched, or where target calling is inconsistent. I am not considering this case here.  

Second, in an upcoming article, I will analyze some combat decisions between heterogeneous groups and review some basic results about decision making in that context.  In that post, I will show how having complete information about the properties of a heterogeneous enemy fleet allows target ordering that can optimize for several outcomes.  

Finally, the concept of effective damage-per-second and effective hit-points are intended to avoid dependence on the obvious challenge that every fleet commander faces in trying to position their fleet for maximum effectiveness, choose ammunition for best effect, choose hardeners to best resist incoming damage and fitting for mobility sufficient to maintain a desirable engagement range.  These are all critically important, but I will assume here that whatever our fleet commanders have done remains constant in this regard, maximizing their total performance given their degrees of freedom.  This is where I think generalization to specific situations will break down, but in return I promise to offer insight into pitched battle dynamics. 

Without further delay, lets dive in to the model. 

Deriving The Model

First, I write the number of ships left alive at time, $t$, by subtracting the number of ships destroyed at time $t$ from the initial number of ships in each fleet. Note that $N_a(t)$ and $N_b(t)$ are functions that represent the number of ships in each fleet at time $t$, and $N_A$ and $N_B$ are the initial number of ships in each fleet.
$\large N_a(t) = N_A - \int_0^t \frac{N_b(t) DPS_B(t)}{HP_A} dt$
$\large N_b(t) = N_B - \int_0^t \frac{N_a(t) DPS_A(t)}{HP_B} dt$     Eq. 2–1

Writing this as a rate of change for the number of ships over time,
$\large \frac{dN_a(t)}{dt} = - \frac{DPS_B}{HP_A} N_b(t)$
$\large \frac{dN_b(t)}{dt} = - \frac{DPS_A}{HP_B} N_a(t)$     Eq.2–2

Note that Eq. 2–2 is just a restatement of Equation 2–1, that the rate of change of ships in a fleet depends on the time that it takes for a ship in the enemy fleet to destroy one ship multiplied by the number of hostiles that are shooting at you. Thus, Eq. 2–2 should make intuitive sense to all readers.   

Now, let's write the equations only in terms of ships in their respective fleets,
$\large \frac{d^2 N_a(t)}{dt^2} = \frac{DPS_A DPS_B}{HP_A HP_B} N_a(t)$
$\large \frac{d^2 N_b(t)}{dt^2} = \frac{DPS_A DPS_B}{HP_A HP_B} N_b(t)$     Eq.2–3

I like these equations because they force us to consider what is really happening in a fleet fight. The first derivative equation (Eq. 2–2) was negative because the number of ships in your own fleet is decreasing and at a rate that is proportional to how many hostiles are shooting at you as well as their damage capability. 

The second derivative (Eq. 2–3) gets to another truth about fleet combat — the rate at which you decrease the harm to your own fleet depends on how quickly you are destroying the enemy's ability to destroy you.  This rate does not depend on the number of hostiles.

Another thing worth noticing about Eq.2–3 is that the coefficients are symmetric between both changes in ship-number rates. These take the form of the time-to-kill for a one-on-one match-up between ships in the respective fleets. That is,
$T_{A \, kills \, B} = \frac{HP_B}{DPS_A}$
$T_{B \, kills \, A} = \frac{HP_A}{DPS_B}$
 

The geometric mean of these times will keep popping up throughout the solution to this system. I will simplify the writing by calling this quantity $T_{mutual}$,
$\large T_{mutual} = \sqrt{ T_{A \, kills \, B} \, T_{B \, kills \, A} } = \sqrt{\frac{HP_B \, HP_A}{DPS_A \, DPS_B}}$.

This time-constant, $T_{mutual}$, is a characteristic duration of dynamics of the combat between these fleets, corresponding to the time when the effect of one fleet's damage overwhelms the damage from the other fleet, meaning that the loss rates of the two groups begin to diverge. If you are in sustained engagement for a duration of $T_{mutual}$ or longer, the fight will be well decided. 

Now that I have written each fleet's ship numbers as a rate in one equation, the method of solution is very easy. I solve for the actual ship numbers versus time, $N_a(t)$ and $N_b(t)$. I've skipped some details here but we know the form of the solution, because differential equations of this type are so common in nature.

Mixing in a boundary condition that the initial number of ships and the initial rate of ship destruction, I find the time domain equation for the number of ships during the battle,
$\large N_a(t) = N_A cosh(t/T_{mutual}) - N_B \sqrt{ \frac{DPS_B HP_B}{ DPS_A HP_A} } sinh(t/T_{mutual})$
$\large N_b(t) = N_B cosh(t/T_{mutual}) - N_A \sqrt{ \frac{DPS_A HP_A}{ DPS_B HP_B} } sinh(t/T_{mutual})$         Eq. 2–4

I've written this as a number of ships remaining alive at time $t$.  Equations Eq.2–4 is a competition between hyperbolic functions representing the two forces. There is something deeper going on here, but it will take a bit more work to tease apart what is happening. In the meanwhile, lets look at some example dynamics of this model by plotting the number of ships over time.

Note that the solution in Eq.2–4 is valid for $t > 0$ and $N_a(t) > 0$ and $N_b(t) > 0$ as well as the initial conditions where the fleet properties are not identical, an infinite solution artifact arising from modeling the number of ships as a continuous variable.

Example Dynamics

I show a few examples of this fleet combat dynamic model below.  These examples are chosen to illustrate how these solutions behave all the way to when one of the fleets is completely eliminated even though most fleet commanders will make an effort to leave the field long before that time.  I've also constructed the example to show some basic trade offs related to number of ships versus quality of ships.  In Figure 2 below, I show the model dynamics for two fleets with equal numbers but where Fleet A has only 10% higher DPS per ship than Fleet B.  This would be a close fight, and Fleet A wins the fight, eliminating hostiles but with only 30% of their ships surviving.  

Figure 2: Example of two fleets with identical starting numbers but differing ship quality is shown.  Fleet properties: $N_A = N_B = 100$, $HP_A = HP_B = 30000$, $DPS_A = 110$ $DPS_B = 100$, $T_{mutual} = 286 \, seconds$.

The second example in Figure 3, below, changes the conditions slightly, showing the model dynamics for a case where Fleet A this time brings 10% more pilots than Fleet B, but the ship performance in both fleets, DPS and HP, are now identical.   Note that the surviving number of Fleet A ships is now 42% of the starting number.  I hope to explain the intuition behind this difference in the Discussion section below.  

Figure 3: Example of two fleets with identical ship quality but differing initial numbers is shown.  Fleet properties: $N_A = 110$, $N_B = 100$, $HP_A = HP_B = 30000$, $DPS_A = DPS_B = 100$, $T_{mutual} = 300 \, seconds$.

In both of these examples I have shown 'Time' on the x-axis, per the model, which does not include locking time or time-of-flight for missile or drone damage so actual engagement duration may be different. 


I'll share one more example where far superior ships can defeat superior numbers in Figure 4.  I constructed this case as a hypothetical situation where heavy assault cruisers might best some Tech I cruisers but the exact numbers are not critical to demonstrating the qualitative nature of dynamics for this model. 

Figure 4: Example of two fleets with differing ship numbers and quality is shown.  Fleet properties: $N_A = 50$, $N_B = 100$, $HP_A = 70000$, $HP_B = 30000$, $DPS_A = 200$, $DPS_B = 100$, $T_{mutual} = 324 \, seconds$.

I have shown engagements examples above that result in complete destruction of one of the two fleets. Note that in all of these examples, as the losing fleet's numbers diminish to zero, the rate of loss for the winning fleet levels off, reflecting the withering incoming damage from the vanquished.

With these pictures in mind, I'd like to derive a few other results that arise from this viewpoint.  The most obvious question we want to be able to answer being, "given the numbers of ships and their properties, which group will be eliminated first?"  Of course, shrewd fleet commanders will make an effort to escape long before complete destruction but we can use the answer to this question to guide that decision as well.  After all, holding the field after the fight advantages the survivors with the ability to loot, so holding the field by complete destruction of the opponent, or forcing them to flee will both be treated as desirable outcomes. 

A Figure of Merit

I intentionally cropped the example graphs above at the point where the losing fleet's ship numbers dropped to zero. When does this happen? If I write the solution for $N_*(t) = 0$, solving for the time when the opposing force has zero ships, I find,
$t_{death,B} = T_{mutual} \, tanh^{-1} \left ( \frac{N_B}{N_A} \sqrt{ \frac{DPS_B HP_B}{DPS_A HP_A} } \right )$
$t_{death,A} = T_{mutual} \, tanh^{-1} \left ( \frac{N_A}{N_B} \sqrt{ \frac{DPS_A HP_A}{DPS_B HP_B} } \right )$         Eq 2–5

The defeated party in the engagement will be the one who has the lesser of these two death times.  Just remember that $tanh^{-1}(x)$ is only real for $|x|$ less than 1.  So, the fleet that is defeated will be the fleet having the lower $N \times \sqrt{DPS \times HP}$ product.

This is an important result because the benefit of ship numbers and ship quality are not the same. I hinted at this in my dynamics examples above. Because the $N$ term has squared ratio to the $DPS \times HP$ product, bringing 10% more ships is approximately as effective as bringing 20% more damage per ship.  Or, bringing 10% more ships is approximately as effective as bringing 20% more hit points per ship. 

This makes sense intuitively because each additional pilot that you bring is also bringing more damage capability as well as more hit points that opponents have to chew through to remove that damage capability from the field.  Everyone who is experienced with EVE combat knows that it is good to bring more pilots if you can, but we now have a quantitative way (albeit under nominal conditions) to compare the trade-offs between the quality of pilots/ships, and the quantity of pilots/ships.

Let's call this number the Fleet Figure of Merit,
$\Large FOM = N \times \sqrt{DPS \times HP}$  Eq. 2–6

So, you get more mileage toward victory from ship quantity than you do from ship quality. I don't need to provide examples of this to any reader familiar with the history of conflict in EVE, although being able to show it with clarity is new to the EVE discussion. Fleet commanders who are risk averse stubbornly avoid fights where they are outnumbered precisely because number imbalance results in rapid losses, as this simple dynamic model confirms formally.
 

Returning to the dynamics equation Eq. 2–4, which I wrote in terms of the numbers of ships in both fleets, I can now write this with more insight into the factors that determines the outcome of the fight. Rewriting the dynamics equations in terms of the Fleet Figures of Merit and how these change with time by simply rearranging the constants,
$\large N_a(t) \sqrt{DPS_A HP_A} = FOM_A cosh(t/T_{mutual}) - FOM_B sinh(t/T_{mutual})$
$\large N_b(t) \sqrt{DPS_B HP_B} = FOM_B cosh(t / T_{mutual}) - FOM_A sinh(t / T_{mutual})$  Eq. 2–7

This viewpoint suggests that we can think of an engagement as a competition between two dynamically updating figures of merit, which in an aggregate statistical sense, is a useful guide for thinking about large engagements.  

Discussion and a word of caution

In the derivation above, I used two state variables, the number of ships in each fleet, and two sets of constants, damage per second and hit points, to model dynamics of two fleets in combat.  This model for the dynamics of fleet combat assumes that everyone is doing their job and that fleet compositions are homogeneous.  The model suggests that engagement  outcomes under these circumstances are statistically determined by a figure of merit that has squared benefit for pilot numbers in comparison with ship quality or pilot skill points.  

I have made no effort to show that this model, based on assuming all pilots take commands in the same way, will be equal to the experimental average of fleet combat in game.  So this is a good place to caution the reader on the limitations of seeing fleets as only Figures of Merit.  As with all of the conclusions based on abstract models, the effective damage rate, and even the effective hit points depend on what you are fighting because of target resists, range, speed and signature radii.  As I will illustrate in an example below and in a following post, the $FOM$ number hides many limitations. Successful fleet commanders have to balance the qualities of the ships that are reflected in this number with those properties that are not, such as mobility, damage projection, tackle and electronic warfare. Using $FOM$ to analyze, say, Alliance Tournament teams, might be misguided.

Next, I want to elaborate on a metric that fleet commanders can use (and already do use) during the fight to assess the status of the engagement.  

 

Clues in the Rates

The challenge with applying the work I have presented above is that the effective damage rates and hit points are almost impossible to judge prior to a fight. Even if you had perfect information about the numbers and fittings of a hostile force, their skill at focusing fire, applying logistics and electronic warfare, would be unknown. Instead of trying to use only scouting information, how can they help us during a fight?

Gathering information takes time, something which fleet commanders have little of. Is there an easy way to determine the fleet advantage during combat? If it is too time consuming for a fleet commander to determine this information without assistance from fleet mates, what is the right digest of information that he or she can request from lieutenants to help make decisions about whether to fight or GTFO (i.e. get the fleet out)? 

Some fleet commanders will ask that pilots of a certain ship type put an 'x' in fleet chat after they have lost their ship so that everyone can see how quickly the fleet is losing ships. If you watched the Zarvox T1 fleet that I linked earlier in the article, you probably also noticed that he requests, with repeated urgency, that pilots who lost their ships report it in fleet chat.  This method is trying to gauge the situation by looking at a rate of ship loss.

Other fleet commanders will ask that one pilot report to them at the time that they have lost a certain number of ships of a certain type.  In the 'Engaging & Disengaging' edition of Jin'Taan's 'Fundamentals of FCing' series, he discusses several factors related to when to decide to leave an engagement.  Among many other considerations in the 'Situational Awareness' section, he uses a comparative loss rate between fleets to make that decision.




The insight from these very experienced FCs is consistent with my simple dynamical model for homogeneous fleet engagement.  I claim that their intuitive approach of comparing rates of loss is, in fact, the same as comparing the dynamically changing figures of merit. That is, comparing the fraction of ships lost in a given time by each fleet is the same as comparing the effective fleet figures of merit.  This is true for homogeneous fleets of equal or unequal size.  

Intuitively, you are computing $\Delta N_a / N_A$ and comparing it with $\Delta N_b/N_B$ for a given unit of time.  If Fleet A has 30 ships, and Fleet B has 10 ships, the loss rate of ships in A must be less than three times those of Fleet B for Fleet B to hold the field. As with most of the conclusions in this note, it should come as no surprise that the emergent behavior of successful fleet commanders is already achieving the same objective function for time analysis of combat dynamics as I derive from the dynamic model.

Looking at the proof of this approach, by writing the rates from our Eq.2-4, at a time $T_s$,
$\large \left ( \frac{1}{N_a(T_s)} \right ) \left. \frac{dN_a(t)}{dt} \right |_{T_s} = \frac{DPS_B N_b(T_s)}{HP_A N_a(T_s)}$
$\large \left (\frac{1}{N_b(T_s)} \right ) \left. \frac{dN_b(t)}{dt} \right |_{T_s} = \frac{DPS_A N_a(T_s)}{HP_B N_b(T_s)}$

When we compare these as a ratio, the result is the ratio of the fleet figures of merit sampled at time $T_s$, or $\frac{N_a^2(T_s) HP_A DPS_A}{N_b^2(T_s) HP_B DPS_B}$.  This is a dynamic update of how the actual damage rate and number of ships are having as an effect on the potential outcome.  What adds to the challenge of keeping running totals of the rates of ship-loss in your fleet and hostile fleets is that in most complex engagements, hostiles can enter and leave the field constantly. Fleet commanders update their internal models for the status of hostile and friendly forces based on the rates of ship destruction.  Specialized overview setups can help with small engagements, however, large engagements would need a more dedicated approach, which is usually not practical. Perhaps some better tools could be written along these lines? 

When you have a logistics wing in your fleet, you have the advantage that a dedicated group of pilots are constantly examining survival timing for ships in your own fleet. Some fleet commanders try to gather information from this group by asking whether, "reps are holding?" Some fleet commanders rely on this type of information as well to get a sense of the dynamics of the engagement. 

An Applied Example

I want to share an example of how features of these dynamics manifest in real fleet combat situations. This is the most dangerous part of this blog post because readers may misinterpret my intent here.  My objective in sharing this example is just to show that; (1) There can be qualitative (and in some cases quantitative) agreement between the simple dynamical model, and; (2) That rates of ship loss are diagnostic of the fleet figures of merit.  

I have chosen to consider a battle between Imperium and FAS dreadnoughts in July 2017.  Choosing a capital fight might have simplified the discussion in this section because siege-mode combat means that remote logistics will not interfere in effective HP assumption and it also means that ease of tackle means that we can watch the fight until the bitter end with one side being completely destroyed.  

The fight in question is occured in MDD-79 on July 11th, 2017 from 18:15 to 18:29.  There is also a video record of this engagement.  There is also a subcapital fight happening at the same time, and as you will see in the analysis there is cross-over between these that affects the outcome, but I will focus on the dreadnought numbers only.  The damage dealers present on grid for both fleets at the start of the fight are:

	Fleet A	Fleet B
Mixed Dreadnoughts	19	30

The mixture of capital types may cause readers to question the assumption that both fleets are homogeneous, however, I argue that dreadnoughts are intentionally chosen to have relatively similar hit-point and and damage rates.  They are similar enough that it should not affect our conclusions.  From the battle report data, I determine the total remaining ships at 60 second intervals and plot these in Figure 5.  These curves exhibit qualitative characteristics of the mutual-predation model, which only goes to show that our assumption about constant loss rates is true within a margin of error.  I would restate this as large fleets behave statistically.  If we look even deeper, there are clues about the performance of both fleets.  

Figure 5: Number of dreadnoughts surviving for both fleets is shown versus time.  Also shown is the model dynamics based on average parameters for both fleets determined from kill mail records.

For starters, lets plot the results of what the dynamic model shows as the total number of remaining ships as time moves forward.  These dashed lines show that Fleet B (Imperium) lost rather more ships than would have been expected to accomplish complete elimination of the FAS dreadnoughts.  

Looking at who was shooting who during this period resolves this question.  Reviewing the number of involved ships on all kill mails reveals that the FAS battleship fleet was also targeting dreadnoughts during the 150s to 400s period, trimming Imperium numbers.  Such a choice of course comes with other trade-offs considering there was also an Imperium battleship fleet present on grid as well.  

This model-data playback does not achieve quantitative agreement and this is because this fight violates some of the initial assumptions.  From the outset, the mutual-predation model will be helpful when all damage sources are accounted for.  This example does serve to show that fleet numbers behave statistically when fleets are even moderate in size.  Second, it shows that when the assumptions of homogeneous fleet combat are met that mutual-predation dynamics dominate the time-evolution of ship losses and overall combat outcome.  Stated in terms of the model parameters, numbers of ships, hit points and damage per second, the rates of ship loss correspond to the ratios of fleet figures of merit. 

The outcome of this engagement was largely inevitable given the numbers and timing, and I chose this example for simplicity.   Thus the model does not suggest any change in tactics for either fleet (apart from not getting dropped by a large number of Imperium dreadnoughts), a dynamical model should enable us to identify possible choices where alternative uses of auxilliary or subcapital fleets might make other outcomes and opportunities possible.  This is a much larger topic, and beyond the scope of this note. 

Now that we have seen an example, what other uses can we imagine for a model for homogeneous fleet engagements?

Fleet Capability: Can We Quantify Risk Taking?

Can we use these results to determine the number and composition of enemy fleets that I can engage safely?  Let's imagine that I am in command of a fleet and I am willing to lose some fraction, $S$, of my fleet.  What are the properties of hostile fleets that I can safely engage without exceeding this loss ratio?  

Before you discard this questioning as being inconsistent with how you command fleets, consider that if you know how your opponent is answering this question, you might use the result to determine what fraction of your numbers to hide to entice a fight that you can win!  Of course, don't forget the amount of time it takes for your reinforcements to come to your aid. 

If I calculate the number of Fleet A's ships alive at the time when the Fleet B's numbers reach zero, at $t_{death,B}$, the fraction of ships surviving is,

$\large S = \frac{N_a(t_{death,B})}{N_A} = \sqrt{ 1 - \frac{N_B^2 DPS_B HP_B}{N_A^2 DPS_A HP_A} }$

This is valid for $FOM_A > FOM_B$ but the goal here is to make Fleet B think that $FOM_B \gt FOM_A$.  If you believe that a hostile fleet $B$ with $FOM_B$ will accept a loss of $S$ fraction of their fleet, you would choose to hide a fraction of your numbers to achieve,
$\large FOM_A \leq \frac{FOM_B}{1 - S^2}$

So, in this context we can think of $S$ as the risk aversion of a hostile fleet commander.  Ranking fleet commanders in this way could produce an interesting survey across New Eden.  Given the information from scouts, what target FOM are they willing to engage compared to their own fleet?  Of course, most of the time this will be overshadowed by logistics and electronic warfare considerations, so a practical survey is not likely using this simple model but it is interesting to consider. 

Executive Summary

• In EVE, as in life, time is the only thing you can't go get more of.  When fighting opponents in same-size or smaller-size targets, therefore, prelocking secondaries can have a dramatic effect on combat effectiveness.
• For homogeneous fleets in combat, the rates of target destruction are determined by the ratio of each fleet's figures of merit, or FOM: $FOM = N \times \sqrt{ DPS \times EHP}$.  Some notes on the FOM concept:
• The number of ships in the fleet has squared benefit over ship quality, i.e. $DPS \times HP$ product.  This makes obvious intuitive sense considering that bringing one more ship means you have also brought more $EHP$ and more $DPS$, both. 
• Ship quality factors, i.e. effective hit points and effective damage per second, are interchangeable in this dynamic model.  In reality, of course, they are not because our definition for $T_{kill}$ in this model assumes that the logistics repair rate is constant. 
• It is usually impractical to calculate a fleet figure of merit during combat.  Instead, by comparing the fractional rate of loss for your own fleet and your opponent's fleet for any time period, you are comparing the effective fleet figures of merit. Methods along these lines are already widely used. 
• With a model for fleet advantage, it is possible to quantify the risk aversion of fleet commanders by comparing their fleet FOM against hostile fleets FOM where they choose whether or not to engage. 

Acknowledgements

I want to thank Jin'Taan and Elo Knight for assisting me as a sounding board for ideas over the past months and years.

Appendix A: The Dumbo Factor

Can we account for the fact that not all pilots in damage-dealing ships will be doing what they are told?  Some of them may be eating a burrito.  This can affect how the lock time for the target, as well as the fraction of pilots applying damage to the intended target.  Famed fleet commander and fluent francophone, Elo Knight, refers to this as the "Dumbo Factor".  

We know that our pilots aren't perfect, so we can try to lump their imperfections into some of the functions we have used in the derivation above.  If we were to encode this inefficiency in our DPS model for the $T_{kill}$ expression, it would include both a locking-time delay factor, $T_{dumbo}$, and account for a number of pilots in the fleet not applying any DPS to the target, $N_{dumbo}$.  Our new expression would be, 

$\large T_{kill} = T_{lock} + T_{dumbo} + T_{flight time} + \frac{HP_{target}}{ \left ( 1 -  \frac{N_{dumbo}}{N} \right ) DPS - RPS}$ 
In this way, we can factor pilot imperfection into our effective DPS consideration.  In Figure 1, I illustrate a hypothetical time-evolution of the damage for a target-calling event, assuming the target is broadcast at $t = 0$.  We can easily group ineffective pilots into the effective DPS and locking time of the fleet.   Of course, the dynamical model presented in this note does not include finite delay in locking, so some extensions may be needed going forward.  

Figure A-1: Example DPS plot versus time with one-second intervals.  In this example, I show $N=100$ ships each doing one unit of DPS, and $T_{lock}=4s$.  I have also illustrated how the 'Dumbo Factor' could influence this profile by including $N_{dumbo}=5$, and $T_{dumbo} = 2s$. A Poisson arrival process approximates the lock-time of imperfect pilots although a model with added delay of $T_{dumbo}$ would likely be sufficient to capture the performance of the fleet.

Thank you for reading.