On the Use of Formations in Land Warfare a statistical analysis ANDREAS FORSTÉN, JOSEFINE LETZNER Bachelor’s Thesis at CSC Supervisor: Petter Ögren Examiner: Mårten Olsson June 5, 2015 Abstract 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. Referat 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. Contents Symbols1 1 Introduction2 1.1 Scope and Objectives........................... 2 1.2 Problem Statement............................ 2 2 Background3 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 5.6 Conclusion ................................ 40 References 42 Appendix A 43 Appendix B 44 Notation 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. 1 Chapter 1 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? 2 Chapter 2 Background In this section, we will first give an explanation of what we mean by the words for- mation and 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. 3 CHAPTER 2. BACKGROUND 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 4 CHAPTER 2. BACKGROUND (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 5 CHAPTER 2. BACKGROUND 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.