The Emperor on the Battlefield: Napoleon's Worth as a Military Commander

I Table of Contents

Table of Figures

Thesis
Theory
History
Modelling
Résumé

Bibliography

Appendix

1. Introduction

Theory

2. The Economy of Conflict - The Second Approach

3. Modelling Contest via the Contest Success Functions
a. Contest Success Functions in General
b. The Ratio Contest Success Function
c. The Difference Contest Success Function
d. Choosing the Right Contest Success Function

History

4. The Essentials about Napoleonic Warfare

5. The Data Set

Modelling

6. The Two Ways of Estimating a CSF
a. The OLS Estimator and its Shortcomings
b. The Logit-Model
c. Synopsis of the Estimators

7. Fitting the Ratio CSF via the Linear Probabilistic Model
a. Approach and Parameters
b. Interpretation

8. Fitting the Difference CSF via the Logit Function
a. Approach and Parameters
b. Interpretation

9. Ratio or Difference CSF? - Comparing the Results
a. Decision Time: The Best Model
b. Interpretation of the Best Model
Résumé

10. How much was Napoleon actually worth?
a. Case Study: 40,000 men
b. Case Study: Jena and Auerstedt
c. Case Study: Austerlitz

11. Final Consideration and Summary

III Table of Figures

Figure 1 Ratio Contest Success Function

Figure 2 Difference Contest Success Function

Figure 3 DCSF and RCSF

Figure 4 Frequency of French Forces less Coalition Forces

Figure 5 Ratios of the Effects on the Odds

Figure 6 Relations of Odds and Probabilities

Figure 7 Probability of Winning for French Forces fighting 62 20,000 enemies with and without Napoleon present

I used to say of him that his presence on the field made the difference of forty thousand men.

Arthur Wellesley, 1. Duke of Wellington

1. Introduction

Though centuries have passed since the rise and fall of Napoleon, his military performance continues to fuel discussion among scholars of history and military science. Scientific discourse about him and the battles and campaigns fought during his age are extensive, nevertheless most research on this topic can broadly be divided into two approaches: historical and from a practical perspective employed by military science. Without conflicting the other, these complement one another, as they independently attempt to explore very different aspects of Napoleonic warfare.

The first approach is the historical perspective, practiced - among many others - by the likes of Blanning¹ or Smith². Scholars focus on the broad lines of the military encounters and the grand strategy³ as the central theme, interpreted in light of social and cultural aspects as well. A work of such breadth would usually be written by a historian and would be culminating in a good descriptive impression and an overview of most of the aspects of the Napoleonic times. Due to the nature of historical science, empirical data would normally only be used in a qualitative way without employing in-depth quantitative analysis.⁴ Moreover the broader perspective is normally emphasized at the expense of detailed aspects and so only statistics of the highest importance are included so not to obscure the leitmotif of history.

Military scientists and practitioners using the second approach usually focus on campaigns and describe the course of them and the important battles that took place during them. Their work is highly detailed and covers many different aspects of Napoleonic warfare. Logistics, infantry tactics, cavalry attacks and artillery bombardments are explained in great detail and their effects are scrutinised. These works usually contain high amounts of empirical data about all the different subjects and facets of contemporary warfare. Nevertheless the focus is on the strategy and the tactics and although numbers are taken into account, their exact impact is not worked out in detail through a quantitative and methodical analysis. Their focus on the military factors furthermore prevents these works to give a comprehensive view of the Napoleonic times but makes them dependent on the general approach discussed before.

Although both approaches can be combined for a very detailed qualitative description of Napoleonic warfare, history and times there is an evident lack of thorough empirical analysis of Napoleons military efficiency. Recent research in economic science has seen an increasing number of papers, books and theories addressing the subject of conflict from an economical and rational choice perspective. Starting with the analysis of ‘rent seeking’ by Gordon Tullock, several other important theorists⁵ have ventured out to study the different aspects of conflict that border both social sciences and economics. This thesis attempts to apply their theories of conflict to the battles of the Napoleonic age and to test several assumptions derived from the so called Contest Success Functions that have been put forth as models for the prediction of conflict outcomes. Although these concepts have been around for several years and sparked frequent discussion, there are only some works that actually try to verify these theories by applying them on actual data.

Hence this thesis seeks to explore the following two working hypotheses: Firstly, that Napoleon’s alleged military superiority in terms of skill and battlefield competence over his peers can be empirically quantified and proven. Secondly, that the results of Napoleonic warfare can be predicted by applying the theory of Contest Success Functions to these battles.⁶

To address these claims this paper is organized into this introduction and four different sections, with eleven chapters in total as follows:

Theory

The first of the conceptual sections summarizes the theoretical underpinning behind the economical understanding of conflict. This so called ‘second approach’⁷ and its merits are outlined and the history of these theoretical concepts is explained. Chapter three introduces the Ratio Contest Success Function (RCSF) put forth by Tullock and the Difference Contest Success Function (DCSF) employed by Hirshleifer, the concepts for predicting probabilities of success in conflict theory.

History

The fourth and fifth chapters are used to outline the actual conditions during the Napoleonic wars and the data used for this study. The focus of this part is especially what we actually do now about these battles and how it may be used. The fourth chapter gives a brief report on warfare during the Napoleonic ages. A special emphasis lies on an analysis that evaluates if the key parameters have been homogenous over the time and what kind of technology was employed during these battles. The results are then compared with the demands of conflict theory. The fifth chapter then explicates the data set. The different variables that could be obtained are introduced and at last the scope of the further analysis is specified. This is done by picking the variables that actually can be used for an in-depth quantitative.

Modelling

The third part of the thesis is of especial importance, as the focus of this work is to answer the two hypotheses by empirical work. In the four chapters that deal with the actual modelling, the theory is applied on the historical data to yield the results we need to verify the working hypotheses. After the two different estimators used have been introduced in chapter six, the chapters seven and eight deal with utilising each of the estimators to answer these questions. The results from the estimates are interpreted and are compared in chapter nine. In addition, chapter nine attempts to weigh the explanatory value of the two approaches and places them in the historical perspective.

Résumé?

The last section of my thesis contains two chapters. Chapter ten answers comments on Napoleon’s personal worth on the battlefield and applies the findings of the empirical work on three short case studies.⁸ The subsequent summary then merges the results of the whole study and concludes with follow-up questions for future research.

I Theory

There are but two powers in the world, the sword and the mind. In the long run the sword is always beaten by the mind.

Napoleon Bonaparate

2. The Economy of Conflict - The Second Approach

To understand how economist approach conflict, we first have to master one very important concept of modern economics, that forms the centre of economic science as we do know it. This is the fictive homo oeconomicus. This conceptual person takes everything into account and then tries to maximise its utility⁹ by deciding what to do. Possibilities can include such trivial decision like the one between going to work or staying at home¹⁰ or between buying a car or saving the money. In the later case the homo oeconomicus would buy a car if he values owning and using it more than the money he has to pay for it. We always assume that this individual would make a rational choice, based on its assumptions about the different possibilities. Conventional economics do know only one method for this person to make a living: producing useful goods or services and trade.¹¹ Although this constitutes by far the biggest part of economic transactions it nevertheless does not catch these in total and omits the other side of human nature and behaviour.

Interestingly, this strict focus on producing goods evolved only over time. The works that founded economic science did refer quite often to a very different aspect of economic behaviour:

“ The efforts of men are utilized in two different ways: they are directed to the production or transformation of economic goods, or else to the appropriation of goods produced by others . ” - Vilfredo Pareto Adam Smith¹² and Vilfredo Pareto both made many references to conflicts and how these shaped human interactions and especially decisions. The merit for reintroducing this dualism to economic acting goes to Thomas Schelling, who in his book The Strategy of Conflict¹³ outlined many concepts¹⁴ that nowadays are part of the standard economic curriculum, after they for some time had been of no importance¹⁵. Especially the game theory¹⁶ profited from his works, which mainly deal with the underlying concepts of behaviour. Several later scientists then started to sketch out a theory of conflict interactions. During the 1970es Tullock¹⁷ “was […] the first to employ standard analytical building blocks […] for dealing with conflict interactions”.¹⁸ By this inventive and new approach it was for the first time possible to conduct a thorough analysis of conflicts from an economic perspective and to foster a better understanding of how conflicts actually do work.

But conflict theory is not so very one sided to explain only how conflict evolves - it even does not predict that all the time clashes have to occur. Nash-equilibriums not only include the best amount of input for a conflict, but can explain cooperation as well. To show how the parts of conflict theory interact, we have to distinguish four aspects that together explain what happens before, during and after:¹⁹

Sources of Conflict

These can range from different preference sets up to totally irrational reasons for conflicts, e.g. hate, distrust or love. They are the starting point for the whole affair. Depending on their shapes they can make conflicts more intense or make it easy to bridge the gap and cooperate.²⁰ Because this thesis analyses the battles and their outcomes, the sources of conflict during the Napoleonic ages are not to be discussed here in depth. Suffice to say that there are several brilliant books written by historians that cover these conflicts and how they evolved.²¹ For our analysis it is only necessary to assume that there is conflict and that during the battles it can not be solved by any kind of cooperation or agreement short of a surrender, so that the battles have to be fought out.

Technology of Conflict

This category includes the analysis of the technology actually used to fight out the conflict - either in a metaphorical way where leaflets or television campaigns could be the technology, or when these measures are actually used to have fights and wars. The importance of the tools used to wage a conflict lies within the differences of them. Without distinguishing between them, a rational choice is not possible. Therefore it is not only necessary to differentiate among these instruments, but for every choice it has to be considered how to get this kind of technology, how to use it, what the costs are and what the impact upon the conflict will be.

In contrast to conflicts with diverging sources, it is not

always possible to compare conflicts them when the technologies are not the same among them. Each set of technology offers a unique set of choices and possibilities to the actors. Hence different conflicts can only be compared if some of consistency holds true for the situations that are to be analysed. It will be of importance to have a closer look at the technology used during the Napoleonic wars to make sure that the battles can be compared without blurring the results.

Modelling of Conflict

The actual modelling of the conflict tries to put the interactions during it into a theory, which most of the time is expressed at least partially through mathematics. This is generally a problem of “optimization on the decision-making level”,²² where we are interested in how probable it is to win a certain conflict or how big a share of the booty we can expect.

During the modelling several aspects have to be taken into account. Technology, resources, the intensity of the conflict - all these are variables that might influence the result and therefore have to be checked for their impact. There are umpteen different approaches to actually modelling a conflict, from black-box-models, where no assumptions are made about what actually happens, to the highly complex war games of the military, where thousands of variables are taken into account. Small level, multi party conflicts with homogenous technologies among the parties are of a special interest. These approaches include the Contest Success Functions (CSF), which model the probability of winning a certain contest or conflict subject to the inputs by the parties involved and which shall be used later on.

Consequences of Conflict

Consequences can range from getting no or less money to very grave ones like getting killed or defeated in a military struggle - even cooperation and peace are possible. Military battles often not only yielded partial results in the style of a won battle, but could lead to crushing defeats of whole countries. These consequences have to be taken into account when the actual modelling takes place, although some interesting results can be obtained about them during modelling as well.

It would be beyond the scope of this work to include all the different consequences into the analysis of the battles. For this thesis the consequences of a certain battle are either to hold one’s ground or the retreat from the battlefield and hence the loss of terrain, initiative and cohesion among the forces.

Conclusion

But in how far is this of reference for the two working hypotheses? The first hypothesis states, that the presence of Napoleon had an impact on the French probability of winning a battle. This is actually both an amendment to the technology of conflict and the modelling. By making the presence of Napoleon possible, the technology set is augmented with one more option and this addition has to be integrated into the modelling as well.

The second hypothesis postulates that the battles of that era can be modelled via the Contest Success Functions. These, like we shall see in the next chapter, say that the input of the competitors - the committed forces in military terms - can be used to predict the outcome of such a conflict situation.

Therefore this hypothesis augments the technology set as well by introducing the input of forces and makes a statement about the kind of modelling actually to be used.

The most important aspects of conflict for testing these hypotheses therefore are the actual modelling of the conflict and - to a much smaller degree - the technology of the conflict.

3. Modelling Contest via the Contest Success Functions

It is important to understand some basic concepts about Contest Success Functions in general before moving on to the special Contest Success Functions by Tullock and Hirshleifer. Hence this chapter will first introduce basic aspects of Contest Success Functions, before the two special ones are introduced and analysed. At last it will be discussed when which function might be applicable.

a. Contest Success Functions in General

The concept of economic contest evolved, when Tullock introduced a new approach to measuring the welfare costs of monopolies.²³ In his paper he argued that the traditional approach to calculate the costs of a monopoly for society were flawed, because they omitted the costs that are connected with the struggle about the monopoly. In principle the patrons are overreached in a monopoly case, as the producer can charge them prices that are higher then they would be in the case of competition. Therefore the monopolist can make a higher profit then he could have by selling the same amount on a competitive market - which is highly desirable for him.

Additionally to these burden to the consumers, there is another burden to society: the costs of the struggle for this desirable case of being the monopolist. These costs can be varied and can include lobbying costs, marketing, new facilities or even bribes - to name only a few. Tullock showed that these costs also have to be taken into account when calculating the costs a monopoly had for society. Starting with this problem of the so-called ‘rent-seeking’²⁴, many more conflicts were recognised.

Hirshleifer then put forth the argumentation that these

thoughts should be broadened and that the models that had been designed to describe rent-seeking compromised only a small part of a much bigger family of Contest Success Functions. He argued that these Contest Success Functions could be applied to every conflict situation to provide it with a theoretical background. Furthermore his impression was, that on the border of social sciences and economics these Contest Success Functions would offer new opportunities to shed light on how conflicts evolve and especially on who will be the winning party.²⁵ During further research it emerged, that there are some characteristics that all Contest Success Functions have to share and which can be stated with an axiomatic character.²⁶

As these are best expressed in mathematical terms, we first need to define the variables we are going to use:

illustration not visible in this excerpt

Then we can express the three absolutely needed axioms about Contest Success Functions this way:

Abbildung in dieser Leseprobe nicht enthalten²⁷

Axiom 1 states that all the probabilities sum up to one in the end, the probability of winning of every contestant is at least zero and that if you put any kind of effort into the contest, your probability of succeeding will not be zero. Axiom 2 dilates this with the presumption that if you put more effort into the contest your probability of succeeding will rise and that it will fall if any other contestants decides to invest more. The anonymity property statement of Axiom 3 then predicates, that the probability of winning for every contestant does only rely on his effort, not on who he his.

Any kind of function that tries to model contest has to fulfil these requirements to give predictions that are at least conceivable. Nevertheless, these axioms are only the lowest bar every kind of economic modelling has to take. In no way do they differentiate good models from bad ones - but only models that could describe reality from those models that never could do so. During research two families of Contest Success Functions evolved, which dominate conflict economics nowadays. Both have unique features and fulfil all the three axioms introduced above, so that they can be used for modelling. Named after their most striking characteristics these are the Ratio Contest Success Function and the Difference Contest Success Function, which will be discussed and explained now.

b. The Ratio Contest Success Function

In Tullock’s basic model of ‘rent-seeking’ the probability of winning for a contestant²⁸ was determined by the ratio of the own effort and all efforts together. For the case of only two players this can be expressed as:

illustration not visible in this excerpt

This fairly basic model was in turn amended by several papers²⁹ and evolved by adding a factor m, the so called ‘masseffect-parameter’, and a factor k i which measures how efficient the effort of the ith player is.³⁰ The amended and generalized Tullock function then evolves into the generally applicable Ratio Contest Success Function:

illustration not visible in this excerpt

The Ratio Contest Success Function obviously fulfils all the three axioms we established during the general approach to Contest Success Functions and therefore does not conflict with reality per se.³¹ Furthermore, it has a special attribute that characterises it:[Abbildung in dieser Leseprobe nicht enthalten]. This implies that

if all the inputs are multiplied by a fixed amount λ , no results

and probabilities will change.³² Using the Ratio Contest Success Function the probability of winning a conflict does only depend on the ratio of the efforts involved, the ‘mass- effect-parameter’ (m) and the efficiency of the individual players (k i). If all players increase their efforts with the same ratio, no effects will occur. The contrary happens when the players increase their efforts by the same amount - the probabilities will change then.³³

Interestingly the behaviour of the Ratio Contest Success Function depends only on the ‘mass-effect-parameter’ m when only one player changes his efforts. Regardless of the level of m it always holds true that [Abbildung in dieser Leseprobe nicht enthalten]. For m ≤1 the player will face diminishing returns to his efforts all the time. When m > 1 increasing returns to effort are possible at the start, with diminishing returns later on. This is illustrated by Figure 1.

Figure 1. Ratio Contest Success Function

illustration not visible in this excerpt

c. The Difference Contest Success Function

Although Tullock’s model obviously has its merits - for instance it is easy to understand and work with - there are nevertheless several flaws that can render it less usable for some applications. Hirshleifer criticises, that “there is an enormous gain when your side’s forces increase from just a little smaller than the enemy’s to just a little larger”.³⁴ Especially when the efforts of both sides are high the Ratio Contest Success Function will not award further input with a considerable increase in the probability of winning³⁵ but does predict only a very small effect on the probabilities. Instead of amending the Ratio Contest Success Function to compensate for these flaws, he proposed another sort of model that is based not on the ratio of the efforts but on the differences between them.³⁶

Keeping in mind the axioms A1 to A3 and further postulating that success in contest depends upon the difference between the resources committed, he offered the following model, which is now known as the Difference Contest Success Function:

illustration not visible in this excerpt

This formula is a specific case of the family of the logistic functions, where γ is the ‘mass-effect-parameter’ for the difference form. This family of functions has several special features that make them very attractive for describing probabilities. The most important of these attributes in the case of the Difference Contest Success Function is the returned value, which always is between 0 and 1. Similarly to the case of the Ratio Contest Success Function, this simple case with two contestants can be generalized by introducing an efficiency parameter k i for the individual players and by raising the amount from 2 to n. The result is the general form of the Difference Contest Success Function:

illustration not visible in this excerpt

Where the Ratio Contest Success Function was not affected by proportional increases of all efforts, the Difference Contest Success Function is not affected when all players increase their efforts by the same absolute amount.³⁸ Contrary to the Ratio Contest Success Function, in this case the ‘mass- effect-parameter’ γ does not change the general shape of the returns to efforts curve but does influence how steep these effects are.

Figure 2. Difference Contest Success Function

A visual comparison of the two Contest Success Functions shows us, that under some circumstances both can look remarkably alike. This is the case when the players commit nearly the same amount of effort. For every Ratio CSF there exists then a Difference CSF that has exactly the same increase in probability for a further unit of effort.³⁹

illustration not visible in this excerpt

Figure 3. DCSF ( γ = 0.04) and RCSF (m = 4 )

Hence it can be complicated to differentiate between the two kinds of Contest Success Functions practically, especially when the measured efforts of the contestants do not vary much among each other and among the contests. In cases where either the efforts among the contestants or among the contests do vary more it gets easier to differentiate and to decide upon the function with the greater explanatory value.

Theoretically we would expect the RCSF to be of higher value if the contest is conducted under optimal circumstances - where total information for all the players and perfect homogeneity of the efforts can be assumed, the RCSF is normally chosen. When we expect frictions to happen and decisions problems for the players to arise, we would assume that the DCSF would prevail.

This is a result of the way the functions are modelled and is much easier to understand, if one assumes the analogy of a battlefield. The RCSF would describe the battle pretty accurately if the battlefield were flat, had no fog, offered no places to hide and we could use all our units simultaneously and wherever we wanted them to be right at this moment. We then would expect the larger force to use its superior numbers for a concentrated assault. It would be logical for the defending force to concentrate as well, as smaller units would have even smaller chances of winning. Therefore we expect an all-out battle of attrition that concentrates in one spot with all the forces.

The DCSF describes a totally different battlefield, on which movement, dispersion and knowledge about the enemy are all limited. We then can interpret the difference between the two parties involved as reserves that the superior force still can use, after the whole enemy force has been engaged. This force then could be used to fight only at certain spots and not against all of the enemy force. Hence we expect this kind of battle to be divided into smaller encounters where different units fight each other, with one side having reserves in the backhand. This model hence should provide to be a better fit if we try to analyse battles that only lasted a certain time and so could not evolve into battles of attrition.

II History

History is the version of past events that people have decided to agree upon.

Napoleon Bonaparate

4. The Essentials about Napoleonic Warfare

During the discussion of economic conflict theory it emerged, that there are four aspects to every analysis of conflict on which light has to be shed. We established before, that the sources of conflict and the consequences do not belong to the topics of this thesis. In the last chapter we examined a way of modelling the conflict via the Contest Success Functions. Nevertheless there still does remain the question of the technology of conflict. Necessarily this question could not be answered in an abstract way, as everything can belong to it. We therefore need to discuss it directly along the lines of the conflict explored. The following chapter is based upon several historical works that are listed in the references in detail. Two of these were of special importance: The Art of Warfare on Land by David Chandler⁴⁰ and Tactics and the Experience of Battle in the Age of Napoleon by Rory Muir⁴¹.

As a battle only forms a small cut-out of a war, war itself constitutes only one way of conflict. During a war everything can be an instrument to be turned against the enemy - together they make the technology of this war. From the more subtle ones to the brutish attacks all have to be taken into account, when deciding what constitutes the technology. Recurring on the working hypotheses, we do not need to deal with the technology of these wars in total, but only with what could be applied during the battles that formed the culminations of warfare in these times. We now will have to examine the technology of Napoleonic era battles and will start with the armament of these times. The soldiers that could be fielded, the organisation of the armies and their tactics are examined afterwards.

Armament

Although Napoleon is known to be one of the greatest generals of all time, he never was very interested in new technologies and how these could be applied to warfare.⁴² Since the times of Frederick the Great only small inventions had been made and generally the armies were still equipped with the same weapons.

The infantry was armed with muskets and, in the case of the light battalions, with rifled guns. Special importance was laid upon the use of the bayonet, as the firearms were still clumsy and took long times to reload. Cavalry was equipped with sabres and pistols, the heavy ones⁴³ still wearing armour and the light cavalry only protected by the normal uniform. Only the artillery equipment had changed noteworthy after lighter guns, the so-called field guns, had been introduced into service. The Royal French Army had been the first to utilize these advanced guns and soon the other countries had to improve as well. Nevertheless no big inventions were made during the Napoleonic era that had a notable impact on the armament of the troops fielded. Hence armament can be treated as having been homogenous in nature.⁴⁴

[...]

---

¹ Blanning (1996).

² Smith (2005).

³ In this thesis I will use the definitions of grand strategy, strategy, grand tactics and tactics as proposed by Chandler (2001).

⁴ E. g. statements along the lines of ‚he fielded more troops’ or ‚he had a higher amount of artillery at his disposal’. These statements, although backed by numbers, do not give insight into the size of the effect or the nature of it and therefore can not qualify as quantitative analysis.

⁵ I especially owe much to the work of Hirshleifer, who studied conflicts for years and always encouraged other economists to apply the economic theories to other field of scientific work.

⁶ Interestingly, a similar approach was chosen by two research teams before. In 1962 the Research Analysis Corporation conducted a study on the Lancaster Equations for the United States Department of Defence. [The Lancaster Equations being early developments of Contest Success Functions there are some similarities in the approach, especially in the use of regressions. Willard concluded that the Lancaster Equations only had a poor predictive value for his data. Compare Willard (1962) for further information. The second research work is the so-called Quantitative Judgment Method Analysis developed by Colonel Dupuy. [Dupuy(1985)] This analysis started from a historical perspective by manually fitting curves until the conduct of a battle could be predicted. Although this method has high value for predicting the outcome of battles, this is only accomplished by using dozens of variables to increase the predictive value. Although some of the curves are variants of the Logit-Function this thesis relies on as well, the method used and the sheer magnitude of explaining variables makes comparison only possible for small aspects.

⁷ Hirshleifer (1994).

⁸ The case studies then should answer if Napoleon really had the impact of 40,000 soldiers, like Wellington attributed it to him.

⁹ The concept of utility will not be detailed here, as it is common economic knowledge. For our uses it should be sufficient to understand utility as the amount one does value a specific situation.

¹⁰ But even these decisions can be quite hard when the structural conditions are a bit more complicated - e. g. when welfare could be paid out.

¹¹ Compare the definition of economics of most standard textbooks, e. g. Marshall (1977).

¹² Adam Smiths „The Wealth of Nations“ cites war and conflict regularly.

¹³ Schelling (1960).

¹⁴ The most widely known of these is the concept of commitment, where one side constrains the options of its adversary by binding itself.

¹⁵ Nevertheless these mostly ‚conflict-less’ times saw the development of the Lancaster Laws, which were early special cases of the Ratio Contest Success Function which will be discussed later.

¹⁶ The game theory attempts to capture behaviour in strategic situations by mathematical terms.

¹⁷ Tullock (1974).

¹⁸ Hirshleifer (2001), p. 4.

¹⁹ These four aspects are presented in Hirshleifer (2001), p. 13.

²⁰ Amazingly it can even happen, that intense sources of conflict can ease cooperation, e. g. when both parties involved are willing to give everything up for the fight and both parties know this and do not want to take the chance.

²¹ Although all the books in the references should be able to shed some light on the sources of the Napoleonic wars, especially „The Napoleonic Wars: Rise and Fall of an Empire“ by Barnes and Fisher (2004) gives a good first impression of them.

²² See Hirshleifer(2001), p. 18.

²³ Tullock (1967).

²⁴ This term was actually coined by Krueger (1974).

²⁵ Compare Hirshleifer (1989) for more information.

²⁶ For a formal analysis check Skaperdas (1994). For the sake of simplicity only the three most important axioms will be discussed here. It will become evident, that only these will be needed to create econometric models by which to estimate the exact CSF later.

²⁷ This approach owes greatly to Skaperdas (1994), who was among the first to point out these fundamentals in such a clear manner.

²⁸ Contestant and player will from now on be used as synonyms.

²⁹ Hirshleifer (1994) credits among others Hillmann and Katz (1984), Corcoran and Karels (1985) and Hillmann and Samet (1987) with the improvement of the Tullock model.

³⁰ This factor k i can be the same for all players - a special case when it looses it meaning - but we can not assume this without proof.

³¹ Proof for this is trivial and can be found in Skaperdas (1994).

- 16 -

³² Proof: [Abbildung in dieser Leseprobe nicht enthalten]

³³ Except for the special case when all the players invest the same effort and are equally effective in applying it.

³⁴ Hirshleifer (1994), p. 93.

³⁵ This is because of the diminishing returns to effort.

³⁶ It can be argued that amending the Ratio Contest Success Function would not have been possible at all.

³⁷ The proof that the Ratio Contest Success Function does comply with A1 to A3 is trivial. See Skaperdas (1994) for further information.

³⁸ Mathematically this can be expressed as[Abbildung in dieser Leseprobe nicht enthalten]. d. Choosing the Right Contest Success Function

³⁹ To get the same slope in this point γ = m c has to be fulfilled. For proof of this skip to Appendix A.

⁴⁰ Chandler (2001).

⁴¹ Muir (1998).

⁴² Chandler (2001) argues that he even disbanded the experimental balloon

companies.

⁴³ The cuirassiers.

⁴⁴ It is important to differentiate between the arms in general having been homogenous and all units being homogenously armed. The former was the case, as every country knew about guns, sabres and rifles. The latter we can not assume, as the countries armed their troops differently.