a statistical analysis
ANDREAS FORSTÉN, JOSEFINE LETZNER
Bachelor’s Thesis at CSC Supervisor: Petter Ögren Examiner: Mårten Olsson
June 5, 2015
This paper investigates military formations and tries to give an answer to the question of how geometry affects outcomes of battles. The investigation has primarily been done with the aid of a model based on Markov chains. This method was then complemented with simulations made in Unity 3D. Special focus has been laid on analysing the flanking maneuver and comparisons have been made with recom- mendations from official sources in the military.
The conclusions drawn point toward the importance of ge- ometry during battles.
Om bruket av formationer vid krigföring på land.
Rapporten behandlar militära formationer och försöker ge svar på frågan om hur geometrin hos dessa påverkar ett slag. Undersökningen har i huvudsak gjorts med en modell baserad på Markovkedjor. Denna metod kompletterades se- dan med simuleringar i Unity 3D. Speciell fokus har lagts vid överflyglingsmanövern och jämförelser har gjorts med rekommendationer från officiella källor inom militären.
De erhållna slutsatserna pekar mot att geometrin är av icke försumbar betydelse.
Symbols 1
1 Introduction 2
1.1 Scope and Objectives . . . 2
1.2 Problem Statement . . . 2
2 Background 3 2.1 Formations and Maneuvers . . . 3
2.2 Warfare - a Science or an Art? . . . 4
2.3 Combat Modelling . . . 7
2.4 Alternatives in Combat Modelling . . . 7
2.4.1 Deterministic models . . . 8
2.4.2 Stochastic models . . . 10
2.5 Time and State Discrete Markov Processes . . . 10
3 Method 14 3.1 Model . . . 14
3.1.1 Assumptions and simplifications . . . 14
3.1.2 Model implementation . . . 15
3.2 Unity 3D Simulation . . . 16
4 Results 18 4.1 Investigation of Model Parameters . . . 18
4.2 Comparison of Markov Model and Unity Simulation . . . 25
4.3 Scenarios Tested . . . 28
4.3.1 Model scenarios . . . 28
4.3.2 Unity simulation scenarios . . . 29
5 Discussion 38 5.1 Limitations of Model and Unity Simulation . . . 38
5.2 Sources of Error . . . 38
5.3 Motivation of Standard Setup . . . 39
5.4 Ethics . . . 39
5.5 Applications and Further Work . . . 39
Appendix A 43
Appendix B 44
Γ = Kill rate
dmax = Max range of weapon.
θ= Angle of occlusion.
σ = Distance between centers of armies.
nstat = Number of multiplications of the homogeneous transition matrix.
nnon-stat= Number of multiplications of the non-homogeneous transition matrix.
Introduction
1.1 Scope and Objectives
The usefulness and importance of military formations and maneuvers in warfare has no lack of empirical evidence. Finding the right geometrical deployment of troops for varying situations has been one of the hallmarks of a good commander for thousands of years. But it is not immediately obvious why this is the case. A battlefield will be as chaotic and unpredictable as any situation involving human beings in large groups, and one may wonder if the success of these formations are more due to the psychological effects, or if the geometry itself is actually a major part of it. In this report, we will study some of the commonly used formations and maneuvers from a probabilistic point of view and provide the readers with a brief overview of the problem of military modelling.
1.2 Problem Statement
An investigation of military formations is to be made. The questions to be investi- gated include:
• Can we derive a mathematical model that computes the impact of formations on battle outcomes?
• Do the results of this model agree with empirically drawn conclusions on the use of formations?
• Are the obtained results applicable to different categories of land forces?
Background
In this section, we will first give an explanation of what we mean by the words for- mationand maneuver, and also provide some background to them. What follows is a brief historical overview to introduce the reader to some of the views on mathe- matics in warfare held by prominent military leaders and theorists throughout the last few centuries. A section on combat modelling will then discuss these mathe- matical models and different alternatives one may use when modelling combat. We have chosen time and state discrete Markov processes as the basis of our model, and therefore a longer explanation of these tools is provided in the last section.
2.1 Formations and Maneuvers
A tactical formation may be said to be the geometric arrangement or deployment of some military units. It may include just a few units (often the case with air- crafts, ships or other advanced vehicles) or tens of thousands of units (obviously not as common today as in the past). While scientific advances has rendered many formerly used formations obsolete, such as the square formation which was used against cavalry, other maneuvers and formations have retained their relevance for thousands of years, despite drastically changing circumstances. It should be noted that the usefulness of a formation depends on more factors than just attacking and defending. An example is the maneuverability of the army, which of course depends on the formation chosen. These factors are, however, not investigated in this report.
Two basic formations are the line and column formations that can be seen in 2.1.
The line formation is, as the name suggests, a formation with wide ranks. One of the earliest known military formations, the phalanx, was used by the city states of Ancient Greece. The phalanx is an example of a line formation, and it usually consisted of heavy infantry[1]. A column formation, on the other hand, is a forma- tion in which the files are significantly longer than the width of the ranks in the formation. Variations of these two formations are still used today, both by vehicles and soldiers, even though the reasons for their use have varied with time.
Figure 2.1: The left picture shows a line formation, if it was the Phalanx the soldiers would be equipped with spears, pikes or similar weapons. The right picture shows the column formation.
A frequently used maneuver is the flanking maneuver, whereby one attacks the sides, or the rear of the enemy forces, often accompanied by a frontal assault by the rest of the army. One famous example of this being used is the Battle of Cannae, where the outnumbered Carthaginian army led by Hannibal defeated the armies of Rome (Figure 2.2). The Roman army probably included some 90,000 soldiers[2], with about 80,000 infantry units and 6000 cavalry units, while the Carthaginians had about 40,000 infantry units and 10,000 cavalry units[3] at their disposal. The exact numbers have been disputed[4], but the important thing to note is that even though the Romans outnumbered the Carthaginians by a factor of two, the Carthaginians still managed to destroy the entire Roman army by using the flanking maneuver.
2.2 Warfare - a Science or an Art?
Can scientific and mathematical systems accurately predict outcomes in warfare?
Many differing views have been put forth to this question during the last few cen- turies, as military thinkers are of course influenced by the contemporary intellectual views. Some have viewed it as a science to be systematized and understood with objective principles, while others have viewed it more as an art where situations have to be left to the individual good sense of commanders. In the 18th century the view that war is more of a science than an art naturally held sway due to the intellectual currents of that time[5, p.63]. Military theory was dominated by the "ge- ometric" school of thought, which emphasized the importance of maneuvers among other things. Examples of this school includes the Welsh officer Henry E. Lloyd (1719 – 1783), comparing an army to a mechanical device, and writing that war is a branch of Newtonian mechanics[5, p.64]. Another was Adam Heinrich Dietrich von
(a) The Department of History, United States Military Academy. Initial stage of the battle.
Public domain, retrieved from Wikipedia.
(b) The Department of History, United States Military Academy. The surrounding of the Roman army. Public domain, retrieved from Wikipedia.
Figure 2.2
Bülow (1757–1807), an officer in the Prussian army, who tried to obtain a tactical system with mathematical precision[5, p.64]. This view of war as just another branch of science was discredited in the conflicts unfolding after the French revolution, when their ideas failed to explain the outcomes[6, p.25].
It may appear absurd to think that military tactics could be described by mathe- matics alone when we view the complex and irregular way in which modern war is fought, but given the importance of the geometric deployment of the army before the advent of modern weaponry, it is understandable that some theorists were en- thusiastic about the possibility.
Enlightenment theorists such as Lloyd were of course not unaware of the psycho- logical factors of war.[5, p.64] But as German romanticism entered the stage in the 19th century, the view of war as an art became more prevalent.[6, p.26] Clausewitz rejected the idea of mathematical certainty in war, and pointed out that systems fail to account of the infinite complexities of war [6, p.28]. This did not mean that he gave up on the idea of a theory of warfare, but he instead redefined the purpose of such a theory; it was not supposed to be viewed as a manual, but as a guide[6, p.29]. On the opposing side of the Enlightenment theorists we find thinkers such as Hel- muth von Moltke (1800-1891), the German Field Marshal, who said of warfare that[7]: "...everything was uncertain; nothing was without danger, and only with dif- ficulty could one accomplish great results by another route. No calculation of space and time guaranteed victory in this realm of chance, mistakes, and disappointments.
Uncertainty and the danger of failure accompanied every step toward the objective.". Or in more modern times, General S. Patton[8]: War is an art and as such it is not susceptible to explanation by fixed formulae. Yet, from the earliest time there has been an unending effort to subject its complex and emotional structure to dissection, to enunciate rules for its waging, to make tangible its intangibility. One might as well attempt to isolate the soul by the dissection of a cadaver as to seek the essence of war by the analysis of its records."
The difference in opinion amongst these authorities on the subject shows the diffi- culty of the question, and the discussion of the usefulness of mathematics to theorists in warfare goes on until today. It is necessary to clearly delineate what aspects of warfare one is studying when these questions are raised. It is obvious that mathe- matics has been useful in, for example, questions of cryptography, military technol- ogy or fort design in pre-modern warfare, all of which belong to the greater effort of fighting a war. The discussion relevant to this report, which was disputed by the theorists above and which remains an open question to this day, however, is the question of how useful mathematics is in the study of combat outcomes. We will refer to this as combat modelling.
Figure 2.3: Modeling and Simulation of Land Combat, ed L G Callahan, Georgia In- stitute of Technology, Atlanta, GA, 1983. Public domain, retrieved from Wikipedia.
2.3 Combat Modelling
Combat modelling can not be said to be fully scientific, as empiricial testing is difficult if not impossible to perform completely[9, pp.216-217]. It is obviously not moral to start a war merely to test results in a model. One may draw conclusion from historical records or field exercises, but it is not sufficient to call it scientific in the most complete sense. A further problem with combat modelling lies in the fact that conclusions drawn from historical records rely on those records to be true.
War propaganda will often distort the truth about events, and the reliability of the records will therefore often be questionable.
Nevertheless it remains a useful tool, and during the Cold War considerable time was spent on researching the subject. Reliable and available information is sometimes not enough for human beings to make decisions, as the information amount may simply be too large to handle. Combat models are therefore used to aid military decision makers by summarizing the data[9, p. 217].
2.4 Alternatives in Combat Modelling
Many different types of models may be chosen, each of them having its own assump- tions, uses and limitations. One may divide them into these different categories[9, p. 218]:
• Descriptive/Optimization
• Deterministic/Stochastic
• Time continuous/discrete
• State continuous/discrete
In military modelling, descriptive models are used for the most part due to the complexity of trying to apply optimization models to military problems[9, p. 218]. We will therefore limit ourselves to descriptive models in our brief overview of the alternatives.
2.4.1 Deterministic models
In 1916[10], Frederick Lanchester devised a set of differential equations (eq. 2.1, 2.2) that models attrition in warfare. A and B are the number of units in the respective armies, which we refer to as force levels. The coefficients Ka and Kb describe the respective effectiveness of the armies. By effectiveness we mean the rate of fire times the probability each shot has of killing[11, p.94]. It is important to note that they are not meant to model battles as a whole, but only attrition. The two sets of equations are[12]:
dA(t)
dt = −KbB(t), dB(t)
dt = −KaA(t) (2.1)
Which is the equation used for a certain set of conditions which will lead to the so-called "square law". The one which leads to the "linear law" is:
dA(t)
dt = −A(t)KbB(t), dB(t)
dt = −B(t)KaA(t) (2.2) From (2.1) and (2.2) one may obtain Lanchester’s laws, which he presented in the paper Mathematics in Warfare[10].
Lanchester’s linear law
Lanchester’s linear law is the less known of the two laws. Lanchester thought that it was mainly useful in modelling ancient combat and other cases where so-called indirect fire occurs[11, p.100]. By indirect fire we mean that the units are not firing against a specific target, which we call concentration of fire, but that the shots they fire are being spread out evenly across a given area. The units that fire do not have information on the impact of their fire, and they do not shift to another target once a target has been killed[11, p.100]. This lack of consideration of the effects of concentration of fire is the reason for the relative neglect of it[11, p.103]. An obvious example of where this law is relevant is artillery duels or other such long-range duels. The fire will be spread out, and knowledge of whether an enemy has been hit or not is limited.
From (2.2) one may obtain that (to see how this is done, see Appendix B):
KaA(t) = KbB(t) + C (2.3)
Where C = KaA(0) − KbB(0). C = 0 if the initial strengths (effectiveness times force levels) of the armies are equal. For this situation we obtain:
KaA(t) = KbB(t) (2.4)
Neither effectiveness nor force levels are given special importance here, whence the name "linear law" comes. The equal importance of effectiveness and force levels is not true for Lanchester’s square law though, as we shall see.
Lanchester’s square law
Lanchester’s square law, on the other hand, allows for concentration of fire[11, p.93]. It is therefore more relevant when studying modern weapons, as there is not the obvious limit of how many units can attack a given enemy at the same time as there was in the case of pre-modern combat where close-range weapons were used. Fire is now directed, by which we mean aimed, and it is distributed evenly over targets.
We assume that each army has full tactical knowledge, that is to say, that it can detect every enemy unit on the battlefield and immediately know if an enemy has been neutralized.
The equation for the square law is (2.1), and we will now obtain (a complete expo- sition of the math behind the square law may be found in Appendix B):
KaA2(t) = KbB2(t) + C (2.5) Where C = KaA2(0) − KbB2(0). With initally equal strengths we will now get:
KaA2(t) = KbB2(t) (2.6)
As is seen, numbers are now postulated to be more important than effectiveness;
the total strength of an army depends on the force level squared. This means that doubling the force level on one team has to be matched by an increase in the effectiveness of a factor four by the other team in order to obtain equal inital strength. The advantage of directed and concentrated fire is well known in military thought[11, p.96], an example being the principle of not dividing one’s army. One sees that if the square law is correct, adding more numbers to an army would decrease its losses as it can destroy the enemy quicker. Note that movement is not a factor in the Lanchester equations[11, p.97].
The Guerilla model
Another example of a Lanchester type model is the so-called Guerrilla model. The equations have the form:
dA(t)
dt = −δbB(t)A(t), dB(t)
dt = −δaA(t)B(t) (2.7) The equations are similar to (2.2), but the constants have been changed so as to model Guerrilla warfare. Detection is now the limiting factor instead of effectiveness.
The model basically says "if you have detected an enemy unit, you have killed it".
The equations are non-linear, but do not exhibit any chaotic behaviour.
Figure 2.4: A two state Markov chain
2.4.2 Stochastic models
A stochastic model on the other hand will involve probabilites, and therefore yield different outcomes even though the initial conditions are the same. Stochastic ver- sions of the models above have been made, but the mean outcome of these differ from the deterministic ones, and the difference increases when the number of units gradually decrease during the simulated battle. The deterministic model therefore becomes more unreliable when the number of units decrease. For this and other rea- sons not mentioned here, stochastic models are preferrable[9, p. 227]. The drawbacks of stochastic models are mostly performance related, as the number of differential equations increase drastically compared to the deterministic models.
For our model we chose to use a Markov chain since this is a mathematically and operationally simplistic method[13]. Another advantage with Markov chains is that one only has to keep data for the current time, not for the future or the past.
2.5 Time and State Discrete Markov Processes
A Markov chain is a certain sort of random process that can be used to describe the statistical behaviour of a phenomena over time. What characterizes a Markov chain is that one can make predictions for the future solely based on the current state of a system, in other words, the history of a process has no influence on the future states[14]. The transitions of a Markov chain can be described by a type of matrix called a transition matrix (also known as stochastic matrix, probability matrix or Markov matrix). Every entry of the matrix is called a transition probability, which describes the probability to go from one state to another.
Example:
On the planet Xzorg 552 only two possible weather conditions are possible: Rain or sun. Let us call the rainy state A and the sunny state B. A Xzorgian meteorologist has determined the exact probabilites for having rain or sun the next day given that it rains or is sunny today. The transitions between the states are vizualised with arrows in Figure 2.4. The black arrow that points back to the current state
A would in this case be "the probability that it will rain tomorrow given that it is raining today". And the arrow that points from state A to B is "given that it is raining today, what is the probability of sun tomorrow". So the meteorologist has determined the probabilities to be:
A → A: PAA = 0.6 A → B: PAB = 0.4 B → B: PBB = 0.3 B → A: PBA = 0.7 which corresponds to the following transition matrix:
T =
0.6 0.4 0.7 0.3
It is worth noting that there are different types of Markov chains, the one used in our model is an absorbing Markov chain. This is a chain which can reach an absorbing state, in other words a state that once reached can never be left. In our case, a state is the status (dead or alive) of every unit of the battlefield. Therefore, given a finite number of units, a finite number of states are possible.
The states possible in our model can be seen in the matrix of Figure 2.5, where • and ◦ represent a living unit on each team. Note that in order for the model to be realistic the states have to depend not only on how many living units there are on one side, but also on the units being viewed as unique (because each unit has a fixed position on the battlefield). The corresponding transition matrix will have 16 ∗ 16 = 256 entries. The first entry represents probability to continue the state where every unit is alive. The second entry is the probability to go from "every unit is alive" to "one unit in the black team is dead, all the other units are alive", and so on. Many of the entries will be zero; this is a feature of absorbing states. The absorbing states are therefore the states where either all of the units on one team are dead, or where every unit on the battlefield is dead. Multiplying the transition matrix with itself is the equivalent of taking a discrete time step, in this way the long term behaviour of the situation can be studied.
Transition matrices which change with time are called non-homogeneous. The pro- cedure for finding the long-term behaviour of these is the same, which may be proved by using the Chapman-Kolmogorov equation. According to this, two non-stationary transition matrices between t0 and tn satisfy:
P(t0, tn) = P (t0, tn−s) · P (tn−s, tn), ∀s= 1, 2, . . . , n − 1 (2.8) It may then be proven that[15]:
P(t0, tn) = P (t0, t1) · P (t1, t2) . . . P (tn−1, tn) (2.9)
It is readily seen that the problem of Markov processes lies in finding the transition probabilites. If the probabilites assigned are far from the correct ones, any results derived will be faulty.
P ossible states=
(◦ ◦ ••) (◦ ◦ • ) (◦ ◦ •) (◦◦ )
(◦ • •) (◦ • ) (◦ •) (◦ )
( ◦ • •) ( ◦• ) ( ◦ •) ( ◦ )
( ••) ( • ) ( •) ( )
Figure 2.5
Method
3.1 Model
This chapter will provide details for the model setup and Unity3D simulation, and how the Markov property applies to the model. The model was implemented in Python with the aid of the packages specified in Appendix A.
A comparison between model and simulation, simulation and Lanchester’s law was then made, the outcome of these are found in the chapter "Results". The first comparison was made in order to verify that the outcomes later obtained would not be platform dependent properties but instead be consequences of the geometry and other parameters that we were investigating. The analysis of the simulation and Lanchester’s law were performed to see how our model and simulation (that proved to be quite eqvivalent) agreed with already acknowledged methods.
3.1.1 Assumptions and simplifications
As mentioned in the introduction the aim of this project was not to simulate a completely realistic battlefield with all of its aspects, but to provide evidence for the geometrical advantage or disadvantage of certain formations. Therefore, when setting up the model some assumptions and simplifications were made. The moti- vation for these are found under chapters "Results" and "Discussion". In particular the units (represented as dots in the Figure below) were given two properties and one major simplification was made:
• The units had a given chance of hitting each other, called the kill rate or Γ.
The kill rate was specified by some function of distance f(d).
• They could not shoot through team members (Figure 3.1). How much space a unit occupied was specified by the angle of occlusion θ.
• Only a few units were used in the model, due to the performance issues in- volved in large matrices. The question of whether this affects the results is
brought up in the discussion.
Figure 3.1: In the white team it is only the first unit that is able to shoot all enemies (left picture). The right picture shows the angle, θ, blocked due to a team member standing in the line of fire. In this report, θ is referred to as the angle of occlusion
3.1.2 Model implementation
After assigning all units a position according to the formation that was to be used, the main part of the code was set up in order to obtain the transition matrix. The overall procedure for this is shown in the example below:
Example:
Calculate the entry P16 of the transition matrix, that is, from the state
(◦1 ◦2•3•4) to (◦1 •3 ). Γ is set to a constant 0.5, we call it Hij here for read- ability purposes. In other words, Hij is the probability that unit i is killed by unit j.
First the chance of surviving for each unit is calculated. The probability to miss another unit is denoted Mij, where i is the unit to survive an attack from j. For ◦1
the M:s will be:
M11= 1 M12= 1 M13= 1 − H13= 0.5 M14= 1 − H14= 0.5.
Note that the probability for a unit to hit itself or to be hit by a team member is set to zero. Therefore, M11 and M12 are set to one, since the chance Hij of hitting is given by Hij = 1 − Mij =⇒ M11 = M12 = 1. So the probability for ◦1 to stay alive is:
P(◦1 | ◦1) = M11· M12· M13· M14 (3.1) The same calculations are done for the other units, but for units that die during a step the probability for the complementary event is calculated instead:
P( 4| •4) = 1 − (M41· M42· M43· M44) (3.2)
The transition between the two states in the example will be given by the joint probability:
P[(◦1 •3 ) | (◦1◦2•3•4)] = P (◦1 | ◦1) · P ( 2 | ◦2) · P (•3| •3) · P ( 4 | •4) (3.3) These calculations are repeated for every entry in the matrix. Simplifications can be done for some cases; for instance if a unit goes from being dead to being alive.
The transition probability for this case is of course zero. The same happens if we want to calculate the kill rate for an enemy that is dead. The resulting matrix is a 2N x 2N matrix, where N is the total number of units on the battlefield.
In order to make the model more developed, other factors were introduced in the calculations of the transition probabilites. Γ was set to a be function of distance, instead of a constant. An angle of occlusion θ was also introduced to make units unable to fire through their own team members, see Figure 3.1. The kill rate was set to zero if a team member occluded the unit firing.
To calculate the outcome of the battle, the final absorbing state of the matrix first had to be reached, which was done as specified in the background. The only inter- esting case in this model was when all units start off alive, which is state one. All the transition probabilites for state one are found in the first row of the transition matrix, which was therefore the one to be studied.
Both the model and Unity simulation were implemented to show either the station- ary case (the units remain at starting position) and the moving case where the two armies approached each other, which we refer to as the non-stationary case. The movement was set to re-position one army a specified, constant distance closer to the other every time step. The difference in calculations for the separate cases was that the non-stationary case will have a transition matrix that changes at every time step. The matrices changed because Γ was set to be a function of distance. One also sees in Figure 3.1 that the black units were able to fire upon more white units as the armies came closer to each other. This also contributed to the time-variation in the matrices.
3.2 Unity 3D Simulation
To enable easier investigation of larger battles we implemented a simulation in Unity3D. The simulation was therefore made to be equivalent to the model, but be- cause of the difference in software it was implemented in a slightly different manner.
The main differences in simulation and model were:
• Shooting targets - In the model the units did not concentrate their fire to a single unit; at every time step they had a chance of killing every enemy unit (that was not occluded). In the simulation on the other hand, one unit could
only target a single enemy at each time step. The unit to be attacked was randomized. The kill rate was calculated using the same function as in the model.
• Angle of occlusion - To implement the angle of occlusion trigger colliders were used. A trigger collider is the green sector as seen in Figure 3.2. If a fired enemy shot hit the trigger collider, the unit was killed. The size of the trigger collider corresponded to the angle of occlusion, this lead to the angle of occlusion being determined less exact than in the model where it was specified in numbers directly.
Figure 3.2: A unit and it’s trigger collider.
Results
First an investigation of the parameters used in the model is presented. Changing these parameters will partially provide an answer to the question of whether the theory used for this model is applicable to different types of land forces or not.
A comparison between the outcomes of the model and the simulation can then be found in section 4.2. Section 4.3 provides the results obtained from both the model and the simulation and will show the effect of geometry and also how the obtained results agree with empirically drawn conclusions. Note that outcome "win" was defined as one team killing all other units while still having at least one unit alive.
A tie result corresponds to both armies being extinguished. The notation used is:
Γ = Kill rate
dmax = Max range of weapon.
θ= Angle of occlusion.
σ = Distance between centers of armies.
nstat = Number of multiplications of the homogeneous transition matrix.
nnon-stat= Number of multiplications of the non-homogeneous transition matrix.
4.1 Investigation of Model Parameters
To ensure that the model was set up in a reasonable manner, in other words, that the result it provided would be credible, some tests were performed. This was done by varying model parameters while studying the outcome and evaluating if the re- sults were plausible or not. While running the tests the default setup used for the parameters were:
Γ(d) = 1+e0.5(d−5)1
dmax= 15 (beyond this point, Γ = 0) θ= 7 °
σ = 8.5 (σ defined as in figure 4.1) nstat= 50 nnon−stat = 0
Number of units in an army: 4
Note: In the tests some of these parameters were altered. If a change was made it is specified.
How varying number of units affects outcome Setup:
Formations: Flanking (White) vs. Column (Black).
Unit deployment is found in Figures 4.1 - 4.3.
Number of units in an army: varied from three to five
Figure 4.1: Three soldiers. Flanking
Figure 4.2: Four soldiers. Flanking
Figure 4.3: Five soldiers. Flanking
The outcome of this test is seen in the diagram of Figure 4.4. A reason for the variations of outcome is that the formation geometries are not identical for different army sizes. We decided to view the differences as being small enough to be disre- garded. This implied that we did not have to reject the assumption that, as long as symmetry is kept, the number of units does have a dramatic impact on results.
Figure 4.4: Outcome of the battle for different number of units.
Angle of occlusion Setup:
Formations: Flanking (White) vs. Column (Black).
Unit deployment is found in Figure 4.2.
0 °≤ θ ≤ 24 °
From the graph (Figure 4.5) it is clear that the angle has a significant effect on battle outcome. An error is apparent when viewing the percentage for low angles. The white team has a lower win percentage of ≈ 1%, even though no advantage is given to any team for low θ, as no occlusion occurs and no movement is incorporated. The win percentages should for this reason be equal. Tests done on the code showed:
• In the stationary case, the error only occured when different formations were set up against each other.
• Entries in the transition matrix that should be equal displayed an error (less than 10−3 %).
• The error was inversely proportional to the number of units.
• The error decreased if Γ was set to a constant as opposed to using a distance dependent function. For some constants no error could be seen in the printed results, but we suspect that this was due to the error being less than the printed number of decimals.
• In the non-stationary case errors also occured when equal formations were set up against each other.
No faults in the code was found. We therefore drew the conclusion that the error was due to the fact that the code executed many operations involving arithmetics.
In the initial transition matrix the error was of a magnitude of 10−3 %, but since the transition matrix was multiplied several times this error increased as it propagated.
We concluded that the error was too small to nullify any results, since the purpose of this paper was to see the general advantage of formations, and not to find perfectly accurate percentages.
Figure 4.5: Win percentage for different angles.
Angle of occlusion for the non-stationary case Setup:
Formations: Flanking (White) vs. Column (Black). Unit deployment is found in Figure 4.2.
nnon−stat = 25 nstat = 0 Starting σ= 16 Ending σ = 6 0 °≤ θ ≤ 24 °
From Figure 4.6 it is clear that angle of occlusion has a great impact on result. The error discussed above is also present here but it is not possible to spot because the effect of occlusion is too large in comparison with the order of magnitude of the error.
Figure 4.6: Win percentage for different angles in the non-stationary case.
Distance function and kill rate
The kill rate, Γ, is a function of distance, which means that we have Γ = f(d).
When choosing a function we could not rely upon empirical data, as this would involve the individual skill of the soldier, weather conditions and other factors. But three criteria had to be fulfilled for the probability distribution:
• The function should go towards one (or at least, be close to one) as the distance goes towards zero.
• The function should drop to zero beyond a certain distance, simulating the range of the weapon. We call this dmax.
• A decrease in kill rate should occur as the distance increases, Df (d)Dd < 0,
∀d ∈[0, dmax).
As there are an infinite number of functions which satisfy these requirements, we had to limit ourselves to a few test cases. The following were chosen (Figure 4.7):
• f(d) = d+11
• f(d) = d2+11
• f(d) = 1
1+e0.5(x-5), a logistic curve with sigmoid mipoint 5, and steepness -0.5.
• f(d) = 1 − dmaxd , a linear curve.
• f(d) = C, a constant. This function does not fulfill the criteria, but is included for testing purposes.
(a) f(d) = d+11 (b) f(d) = d2+11
(c) f(d) = 1
1+e0.5(x-5) (d) f(d) = 1 −dmaxd
Figure 4.7: Functions tested.
A test was performed for the different functions and their outcomes compared as seen in table 4.1. Only the function was varied; the other parameters were the same as in the standard setup. The formations used were Flanking and Column. Missing percentages are ties.
Function White team winning Black team winning
1
d+1 0.6726 0.3087
1
d2+1 0.6526 0.3456
1 −dmaxd 0.6381 0.2589
1
1+e0.5(x−5) 0.6015 0.3876
Table 4.1: Outcome of battle when using different functions
4.2 Comparison of Markov Model and Unity Simulation
A comparison between outcomes of the model and the simulation was performed.
This was done for two reasons. The first was to test the results of the model with the aid of another platform. The second was to see if we could use the simulation as an extension of the model, as more intricate computations could then be performed and larger battles be analysed.
The mean outcomes of the simulations closely matched the results obtained from the model, as is seen in Figures 4.9 - 4.11. This was expected as the assumptions made in them were similar. We therefore saw it as justified to use the simulation for tests where the model could not be used due to performance issues.
First stationary case Setup:
Formations: Flanking (White) vs. Column (Black). Figure 4.2 σ = 9.
Results in Figure 4.9 Second stationary sase Setup:
Scenario: Encirclement as seen in Figure 4.8.
θ= 10°.
Results in Figure 4.10
Figure 4.8: Encirclement
Non-stationary case Setup:
Formations: Flanking (White) vs. Column (Black). Figure 4.2 Start: σ = 16
End: σ = 6 nstat= 0 nnon−stat= 30
Results in Figure 4.11
Figure 4.9: Comparison of win percentages for model and simulation. First station- ary case.
Figure 4.10: Comparison between model and simulation. Second stationary case.
Figure 4.11: Comparison between model and simulation. Non-stationary case.
4.3 Scenarios Tested
4.3.1 Model scenarios Setup:
Formations: Oblique order (White) vs. Line (Black). See Figure 4.12.
nstat= 0 nnon−stat= 25
Here the efficiency of a flanking maneuver, often referred to as the oblique order, is put to test. The oblique order also incorporates the concept of concentration of force. Outcome of the battle:
White army winning ≈ 59.23%
Black army winning ≈ 39.97%
Tie ≈ 0.79%
Figure 4.12: Oblique order, a flanking maneuver.
Setup:
Γ = 0.2
Simplified version of the Battle of Cannae. Five Carthaginian infantry units have surrounded the Roman army. Deployment may be seen in Figure 4.13.
Outcome of battle:
White army winning: ≈ 71.90%
Black army winning ≈ 27.45%
Tie ≈ 0.65%
Figure 4.13: Battle of Cannae
4.3.2 Unity simulation scenarios
Lanchester’s square law and Force concentration
For more information about Lanchester’s laws, see section 2.4. A comparison of the deterministic model in equation (2.1) with a simulation in Unity3D is shown here.
The simulation is well suited for comparison with the Lanchester model if one sets Γ to a constant, as the assumptions in the respective models will be the same.
The tests in Unity3D were done by varying the number of units in a team, while keeping the number of units in the other team a constant ten. No advantage in firepower or fire rate was given to any team, so the effectiveness K may be set to one. By comparing the initial values in equation (2.10) with the final values one gets what we will call the Lanchester outcome. Example:
A0 = 10 units, B0 = 20 units. Kb and Ka are set to one. For the initial values we get:
202−102 = constant = 300. (4.1)
We may now see what the Lanchester outcome of B is as A is set to zero:
B2−02 = 300 =⇒ B2= 300 =⇒ B ≈ 17.3 (4.2) Here we see the drastic effect of the square law; a force of double size will only lose
∼13.5% of it’s forces on average while the enemy team is eradicated. By running a Unity3D simulation with all factors equal except for numbers, we obtained results as seen in Figure 4.14.
Figure 4.14
We see that the shape of the functions are equal, but that the randomized outcome in the simulation gives a slightly higher percentage of the dominant force being killed. This is most likely due to either the fact that Lanchester gun fire is modelled as continuous, while gun fire in the model is discrete, or that the units used are too few. The second point about using a few number of units was brought up in the background (section 2.4). The solution to the system of equations has a tendency to vary compared to stochastic models when the number of units decreases. It is seen that the percentage difference decreases as the number of units increases in Figure 4.15.
Figure 4.15
The data is not sufficient to determine whether or not it will converge to zero, but the small differences between the deterministic model and the stochastic for large numbers show us why Lanchester’s equations have retained their relevance to modellers of military tactics to this day, as they accurately give equal outcomes to stochastic models. It also shows why force concentration has been an important factor when setting up formations.
Tank platoon movement
The white platoon is trying to move through a relatively flat territory, while the black platoon has set up an ambush. The units were given a rudimentary "AI"
which made them fire upon the closest enemy target within the span of surveillance (as defined in figures 4.17 and 4.19).
Figure 4.16: The black platoon has set up an ambush in the rocky area The battlefield is covered in dust, which makes the respective platoons unable to see each other until the white tanks have come sufficiently close. When the enemy situation is unclear or when a commander suspects that contact may occur, the wedge formation is often used[16, p. 7](figure 4.17).
Figure 4.17: Wedge formation, lines indicate the span which units surveille. Mecha- nized Infantry Platoon and Squad (Bradley). Department of the U.S Army. p. 3-7.
2010. Public domain.
A comparison was made between the mean outcome of the battle when the white platoon used the recommended wedge formation, and when it used a column for- mation (figure 4.19).
Figure 4.18: Wedge formation. Surveillance span is the same as in 4.17
Figure 4.19: Column formation, lines indicate the span which units surveille.
A significant difference between the mean outcomes occured, as seen in figures 4.20 and 4.21.
Figure 4.20: Mean outcome for the wedge formation.
Figure 4.21: Mean outcome for the column formation.
Tank platoons and Artillery
The line formation (figure 4.22) is recommended for traversing open areas, maxi- mizing fire power towards the front, and for situations when the platoon is under artillery fire and therefore needs quick movement[16, p. 7].
Figure 4.22: Line formation, lines indicate the span which units surveille. Mecha- nized Infantry Platoon and Squad (Bradley). Department of the U.S Army. p. 3-7.
2010. Public domain.
The black team consists of a tank platoon with Γ = 0.2 and an artillery piece. The artillery has Γ = 1, but the shots fired have an uncertainty. The white team consists of a tank platoon with Γ = 0.3. The starting distance between the armies was
approximately 250 (dimensionless number) in the y-direction, while the uncertainty of the artillery shots is ±20 in the x-direction (dotted white arrows, figure 4.23).
The fire rate of the artillery was set to be three times slower than the tank fire rate.
Tanks were only vulnerable to the artillery fire if it struck within a distance of ±6.
Figure 4.23: Battle deployment
The distance between the tanks (white arrows in 4.23) was varied, and the mean outcomes after sixty simulations were recorded. A decrease in the amount of wins for the white team occured, as the artillery fire will hit more often when the vulnerable areas of the tanks come closer to each other. A "missed" artillery shot may then hit another unit in the white team instead.
Discussion
5.1 Limitations of Model and Unity Simulation
An apparent limitation of the model was the number of units it could handle. This made it impossible to set up certain formations, and made the model more of an abstraction than it could have been otherwise. On the other hand, it was shown in section 4.1 that varying the number of units did not make a significant difference, as long as the symmetry of the unit deployment was kept. This may be an indicator that an increased amount of units would yield results of an equivalent character.
However, a side effect that was prominent compared to reality was that ties occured more frequently, both in the model and the simulation. If the number of units had been greater this would not have been the case, because a tie only occurs if every re- maining unit on the battlefield is killed during the same time step. The probability of every unit being killed obviously drops as the number of units increases. We drew the conclusion that the abstraction from many to few units is not in and of itself a serious restriction, as the focus of this paper was on the qualitative behaviour of the results, which seemed to be kept despite variations (Figure 4.4). In addition to this the simulation provided an outcome close to the model, and it could thus be used to obtain reliable results for more units.
The limitations of the simulation are mostly the same as in the model, since it was implemented in such a way as to be similar to it. The main difference lies in the way units fire, as has been mentioned above.
5.2 Sources of Error
A potential source of error in the model is the rate of convergence towards the stationary distribution. This was especially seen in the non-stationary case, where transition matrices become non-homogeneous. As these matrices changed with each time step the rate of convergence was markedly slower than in the stationary case.
To handle the slower convergence rate it was necessary to increase the number of
matrix multiplications. For some cases where the rate of convergence was especially slow, we decided that the added time consumption for this was not worth the small increase in accuracy. When this occured we settled for a solution where no non- absorbing state had a transition probability greater than 10−2 %. Most of these non-absorbing values were at that point less than 10−5 %. As the percentages of the absorbing states were an order of magnitude two to three larger than the largest percentages for the non-absorbing states, no qualitative effect on the outcomes ensue from this.
Another source of error was the suspected round-off mentioned in section 4.1. The extent of this has already been treated.
5.3 Motivation of Standard Setup
In section 4.1. four different functions for Γ were proposed. We decided to use the logistic function and a specified set of parameters as our standard setup. The choice may have seemed arbitrary and careless, but it becomes more credible if one recognizes that f(d), dmax, θ and σ are to be chosen with respect to each other . By the relationship between them, a Γ which one deems reasonable for the situation to be modelled can be obtained. It may be seen in Chapter 3, that Γ is in fact a function of d, dmax and θ, which means that both the parameters and the function can be chosen almost arbitrarily as long as the output, Γ, is reasonable for the purpose one has.
5.4 Ethics
One may wonder if it would have been better to spend an equal amount of time on solving problems leading to war, as time spent studying them. This is a legimtimate question to ask oneself when studying formations and other factors for optimizing succees in war. The problem is that the underlying factors for this question stretches far beyond the question itself. In society today many might say that war has been, and still is, necessary to defend the rights of democracy. If this is the case or not will not be claimed here, but at least it is the authors belief that war is, and until an uncertain date will be, an important tool to keep the world in order.
5.5 Applications and Further Work
Many improvements could be done to the simulation or the model. A more devel- oped movement pattern for the units and the addition of terrain should probably be amongst the first. To improve the modelling of vehicles the addition of correctly placed armor should be implemented, as vehicles will have varying weak points which must be taken into account if the model is to be realistic.
In the background (section 2.1) it was mentioned that formations have not only been used for the extra fire power that they add. Maneuverability is an important factor, especially with modern land vehicles in narrow city environments. The built- in features of Unity3D are well accommodated for such a problem. Another factor to study would be reconnaissance and visibility, as different formations will provide different visibility to friendly units, especially when one considers large vehicles.
The angle of occlusion used in this project should be applicable for that problem as well. Unity3D may be used here too. Optimization of the code for the model could certainly be done, but it is not certain if any enhancement would be noticeable as the size of the matrices is fixed for a given army size.
5.6 Conclusion
Two geometric factors were identified in the results as the main reason for why some formations have an advantage compared to others. One of them is shown in Figure 5.1 where the white army is moving towards the black. The horizontal lines in the picture represent the range of the units. In the initial stage this means that the white team has a numerical advantage of four to one. The rest of the black army can not fire until the white team has moved within range. The second advantage was the so-called angle of occlusion. This factor was especially present in pre-modern warfare, where tightly packed soldiers marched together and were unable to attack until the friendly soldiers in front of them had been killed. It should be noted that the results for the angle of occlusion are not necessarily applicable to real battles.
The model does not take area covering weapons into account, and it will therefore give an unrealistic advantage to the team that cluster their units in a small area.
Variations were made in the values of the two factors, which generated different outcomes, but the results in chapter four show that a significant advantage is given to the army which has the geometrically favourable position and formation. This may indicate that the results are applicable to different types of units. And so the answer to the question if the obtained results are applicable to different land forces is most likely yes. Thus, it may with confidence be said that the advantages won in battles have been partly due to the geometrical deployment, and not only the other factors as dicussed in the introduction.
The two reasons alone suffice to show us why commanders have spent so much effort on the deployment of their units, and the search for ideal maneuvers and formations will continue in the future as the circumstances of war changes.
Figure 5.1: Effects of movement
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NumPy: http://www.numpy.org SciPy : http://www.scipy.org Matplotlib: http://matplotlib.org
Derivation of Lanchester’s linear law The system of equations from 2.2 is:
dA(t)
dt = −A(t)KbB(t), A(0) = A0 (5.1) dB(t)
dt = −B(t)KaA(t) B(0) = B0 (5.2) To study A as a function of B, one may use the chain rule to obtain:
dA dB = Kb
Ka (5.3)
Separation of variables (with integration limits 0 and t) then gives us:
Kb(B(t) − B(0)) = Ka(A(t) − A(0)) (5.4) Derivation of Lanchester’s square law
The system of equations from 2.1 is:
dA(t)
dt = −KbB(t), A(0) = A0 (5.5)
dB(t)
dt = −KaA(t) B(0) = B0 (5.6)
To study A as a function of B, one may use the chain rule to obtain:
dA dB = Kb
Ka
B
A (5.7)
Separation of variables gives us:
KbB2 = KaA2+ C (5.8)
Where C is some constant to be determined by initial conditions (see section 2.4).
It may be more illuminating to express it as:
KbB2− KaA2= constant (5.9)
If one is interested in the behaviour of the equations as a function of time, they may be solved using Laplace transforms:
sL[A(t)](s) − A0 = −KbL[B(t)](s) (5.10) sL[B(t)](s) − B0 = −KaL[A(t)](s) (5.11) We set X(s) = L[A(t)](s) and Y (s) = L[B(t)](s). This gives us:
sX(s) − A0 = −KbY(s) (5.12)
sY(s) − B0 = −KaX(s) (5.13)
Or:
X(s) = sA0− KbB0
s2− KaKb (5.14)
Y(s) = sB0− KaA0
s2− KaKb (5.15)
Or in the time-domain:
A(t) = A0cosh(pKaKbt) − KbB0
√KbKasinh(pKbKat) (5.16)
B(t) = B0cosh(pKaKbt) − KaA0
√KbKa
sinh(pKbKat) (5.17)
