Why States Exist: The Square Law Theory of The State

There have been many theories concerning the origin and existence of States. I shall show that the existence of the state proceeds from its essential nature, namely, its monopoly on physical violence. As President Woodrow Wilson put it in his 1889 book The State: “ ... in its last analysis, [it] is organised force” (Wilson, 1889, p. 572). We shall see that this explanation for the existence of states explains why states have only existed for some 6,000 years, whereas the human race has existed for much longer, some 200,000 years.

The fundamental reason states exist is that throughout recent human history, states, and states alone, have been able to raise large armies, and that the actual number of soldiers in an army is far more important than the skill of an individual soldier. Thus, a large army of draftees who are forced to join a state army and individually fight incompetently is easily able to overcome far more motivated and skilled individuals fighting to defend their homes and families. Furthermore, the large state army engaged in establishing state authority will suffer far fewer casualties than the better motivated and larger population fighting it as individuals.

To see why size matters far more than skill, let us consider a simple example. Watch the famous John Wayne movie, True Grit. The climax of the movie features the hero, John Wayne, facing four outlaws in a gun battle to the death. As the Hero, John Wayne, of course, wins, but we all know what would happen in real life: the outlaws would win. To see why, let us first simplify the analysis by supposing that John Wayne and the outlaws are individually equal in fighting ability. Later, we will correct for the fact that John Wayne is a Hero, and so a far better fighter than any of the outlaws. But assuming initially that they are equal will allow us to compute just how much better John Wayne would have to be to have a realistic chance of defeating all four of the outlaws.

Since John Wayne and the outlaws are individually equal in fighting ability, this means that all five—John Wayne and the four outlaws—fire their guns at the same rate, and have the same accuracy. This means that for every bullet John Wayne fires, the four outlaws fire four bullets. Furthermore, the fire of the outlaws is four times as concentrated. They are all shooting at John Wayne, whereas John Wayne has to divide his fire between four outlaws. So, on average, for each bullet that John Wayne fires at any given outlaw, John Wayne receives 16 bullets in return fire. This illustrates the Square Law of Combat: the likelihood that a soldier is killed goes as the square of the relative size of the opposing armies.

Now we see just how much more effectively a fighter John Wayne would have to be to have an even chance of defeating the outlaws: John Wayne would have to be sixteen times better. He would have to fire 16 times more often than a given outlaw, or he would have to be 16 times as accurate in aiming his pistol as the average of the outlaws, or a combination of the two. A Hero is certainly better than a Villain, but sixteen times better?

Similarly, individuals defending their homes and families are more motivated to fight than soldiers forced against their wills to join the army of a state. Certainly, individuals protecting their property and loved ones are more motivated to develop fighting skills. But draftees can be forced to fight. Remember Trotsky’s dictum: “men fight best when they face probable death if they advance, but certain death if they retreat.”

As we shall see, another consequence of the Square Law of Combat is that an army far outnumbering another is not only almost certain to win but is likely to suffer far fewer casualties than their opponent. For example, I shall show in the next section that an army of 400 men can expect to completely annihilate an army of 100 men—kill all 100 of the opposing army—while losing only 13 men of their own. This asymmetry in losses is ultimately why states form and continue to exist, despite the obvious disadvantages of states. Size matters and matters in a big way. We can ask, are states forever, or will advances in technology eventually neutralise the Square Law? I shall suggest in Section 3 of this paper that the time of states may indeed be coming to an end, with all the consequences that such an end will have for economies as they are now organised.

Throughout the paper, I shall emphasise that the entire theory I am developing is just a special case of Coase’s Theory of the Firm. As Woodrow Wilson correctly pointed out, a state is merely a firm whose product is physical violence, and thus, all I am doing in this paper is describing exactly how transaction (information) costs arise in such firms.

War was first put on a mathematical basis in 1914 by Frederick W. Lanchester (Lanchester, 1914), and independently in 1915 by M. Osipov (Osipov, 1915); see also (MacKay 2006, 2008) and (Rotte & Schmidt 2002).. Lanchester and Osipov modelled a battle between two armies as a pair of coupled ordinary differential equations: (1) dGdt=−b2B {{dG} \over {dt}} = - {b^2}B (2) dBdt=−g2G {{dB} \over {dt}} = - {g^2}G In this pair of equations, G is the number of Soldiers in the army of Good Guys, B is the number of soldiers in the army of Bad Guys, while g² is a constant which measures the effectiveness of the Good Guys killing the Bad Guys, and b² is a constant which measures the effectiveness of the Bad Guys killing the Good Guys.

The physical justification for the above pair is simple and intuitively obvious. Each equation says that the rate at which an army loses men in combat is proportional to the number of soldiers the enemy has in action on the battlefield. Notice the qualifications in this sentence. Soldiers in uniform but not actually fighting do not count. A solider cannot kill an enemy soldier if he is a 1,000 miles away, or if he is in reserve on the battlefield, or if he is engaged in cooking lunch for his comrades. Or if he is in an inactive theatre of the battle. This latter point is an illustration of the Principle of Concentration, that a battle is won by concentrating an overwhelming superiority of men at a crucial point of the battlefield, the “Schwerpunt,” in German military terminology. “Proportional to” means “equal up to a constant.” This constant will measure the relative average efficiency of one army’s soldiers in killing (or incapacitating) the soldiers of the opposing army. (Lanchester (1914) also considered a “linear theory” of war which is really non-linear and less intuitive. I shall discuss this alternative theory of war in the Appendix.)

The above pair of equations can be integrated by first solving Eq. (1) for dt: (3) dt=−dGb2B dt = - {{dG} \over {{b^2}B}} and plugging this into Eq. (2): (4) dB=−g2Gdt=−g2G[dG(−b2B)] dB = - {g^2}Gdt = - {g^2}G\left[ {{{dG} \over {\left( { - {b^2}B} \right)}}} \right] which can be written (5) b2BdB=g2GdG {b^2}BdB = {g^2}GdG which implies the integral (6) ∫b2BdB=∫g2GdG \int {{b^2}BdB} = \int {{g^2}GdG} Integrating, we obtain (7) b2B2/2=g2G2/2+constant {b^2}{B^2}/2 = {g^2}{G^2}/2 + {\rm{constant}} which can be written simply as (8) g2G2−b2B2=C {g^2}{G^2} - {b^2}{B^2} = C where C is a constant, independent of time, but dependent on the original number of soldiers from each side at the start of the battle. Eq. (8) is the central equation of the Lanchester war theory.

It is first instructive to consider a few examples of the application of Eq. (3). For instance, let us take the John Wayne example. Recall that we assumed that Wayne and his opponents were equally good shots and could fire equally fast. This means that g² = b², so we can divide out these constants, giving (9) G2−B2=C/g2 {G^2} - {B^2} = C/{g^2} Then the initial value of the constant on the left-hand side of Eq. (9) is (10) G2−B2=12−42=−15=C/g2 {G^2} - {B^2} = {1^2} - {4^2} = - 15 = C/{g^2} This constant is independent of time. The fight continues until Wayne is killed, which is to say, the final condition is (11) G(tend)=0 G\left( {{t_{{\rm{end}}}}} \right) = 0 which tells us that the number of Bad Guys remaining at the end of the fight is (12) B(tend)=[−C/g2]1/2 B\left( {{t_{{\rm{end}}}}} \right) = {\left[ { - C/{g^2}} \right]^{1/2}} In the John Wayne case, this is (13) 15=3.87 \sqrt {15} = 3.87

This number is not significantly different from 4, the value of no outlaws being killed at all. That is, the outlaws can expect before the fight that they are likely to all survive, and Wayne to be killed.

Similarly, the result of having one side outnumber the other by four to one scales. To get rid of the minus sign, let us suppose the Good Guys outnumber the Bad Guys by four to one, and that man-to-man, both sides are equally effective. Then, with 40 Good Guys facing 10 Bad Guys, we have (14) G2−B2=(40)2−(10)2=1,500 {G^2} - {B^2} = {\left( {40} \right)^2} - {\left( {10} \right)^2} = 1,500 which gives the number of Good Guys remaining after the battle: (15) 1,500=38.7 \sqrt {1,500} = 38.7 which is to say, the 40 Good guys would expect to lose only one man in the process of annihilating all 10 of the bad guys.

And with 400 Good Guys facing 100 Bad Guys: (16) G2−B2=(400)2−(100)2=150,000 {G^2} - {B^2} = {\left( {400} \right)^2} - {\left( {100} \right)^2} = 150,000 (17) 150,000=387 \sqrt {150,000} = 387 The 400 Good Guys would expect to lose 13 of their men in the process of completely annihilating the company of 100 Bad Guys.

Let us now analyse an example of even larger size, but an example whose numbers scale in an obvious way and whose numbers can be computed without a calculator.

Suppose there are 50,000 good guys and 30,000 bad guys. Five divisions of good guys against three divisions of bad guys. Then the constant is (18) G2−B2=(5×104)2−(3×104)2=(4×104)2 {G^2} - {B^2} = {\left( {5 \times {{10}^4}} \right)^2} - {\left( {3 \times {{10}^4}} \right)^2} = {\left( {4 \times {{10}^4}} \right)^2} If, as above, we assume the battle continues until the smaller army (the bad guys) is wiped out, the good guys will have lost only 10,000 men—one division—in the process of completely annihilating 30,000 bad guys. The bad guys lose three men to the good guys’ one man. The bad guys lose three divisions to the good guys’ one division.

Let us now apply the Lanchester Square Equation to an actual battle, the World War Two battle of Iwo Jima. The United States had 110,000 men, while Japan had only 21,000. The Japanese soldiers fought to the death, so assuming the fight continues until the side with inferior numbers is completely wiped out is a realistic assumption. Assuming as before, equal effectiveness, we compute (19) G2−B2=(110,000)2−(21,000)2 {G^2} - {B^2} = {\left( {110,000} \right)^2} - {\left( {21,000} \right)^2} which tells us that the United States should expect to have 107,977 men after the Japanese had been wiped out, which means that the US, before the battle, should have expected to lose about 2,000 men.

The actual US loss was 7,000 men, >3 times what the Lanchester equation would predict based on equal killing effectiveness. But we should not expect equal killing effectiveness. The Americans were attacking an enemy that was protected by heavy fortifications, and we know now that naval gunfire and dive bombing were relatively ineffective against these fortifications. Putting the actual losses into (8), we obtain (20) (110,000)2−b2g2(21,000)2=(103,000)2 {\left( {110,000} \right)^2} - {{{b^2}} \over {{g^2}}}{\left( {21,000} \right)^2} = {\left( {103,000} \right)^2} which gives the average killing effectiveness of the Japanese soldier relative to the US Marine: (21) b2g2=3.38 {{{b^2}} \over {{g^2}}} = 3.38 This illustrates the central implication of the Lanchester theory, namely that even if the side with inferior numbers is enormously more effective in killing their opponents, this effectiveness is easily overcome with superior numbers.

Eqs (1) and (2) can be solved simultaneously to give G(t) and B(t) explicitly as functions of time, but these solutions were not given in Lanchester’s original paper (though he displayed the solutions in graphical form for special initial conditions), nor are they given in any of the more recent papers on the Lanchester theory, so I shall give these solutions here. Let (22) G(t=0)≡G(0)≡G0 G\left( {t = 0} \right) \equiv G\left( 0 \right) \equiv {G_0} (23) B(t=0)≡B(0)≡B0 B\left( {t = 0} \right) \equiv B\left( 0 \right) \equiv {B_0} be the initial numbers of Good Guys and Bad Guys, respectively. That is, G₀ and B₀ are the numbers of Good Guys and Bad Guys at the beginning of the battle, which is assumed to start at time t = 0.

Solve (8) for b²B: (24) b2B=b(gG)2−C {b^2}B = b\sqrt {{{\left( {gG} \right)}^2} - C} and insert this into Eq. (1): (25) dGdt=−b(gG)2−C {{dG} \over {dt}} = - b\sqrt {{{\left( {gG} \right)}^2} - C} which is an ordinary differential equation involving the single function G(t). This equation can be written in integral form: (26) −gb∫dt=∫d(gG)(gG)2−C - gb\int {dt} = \int {{{d\left( {gG} \right)} \over {\sqrt {{{(gG)}^2} - C} }}} and both indefinite integrals can be easily evaluated: (27) −gbt+D=ln[(gG)+(gG)2−C] - gbt + D = ln\left[ {\left( {gG} \right) + \sqrt {{{(gG)}^2} - C} } \right] where D is a constant of integration. We can exponentiate both sides of (27) and obtain: (28) Ae−gbt=(gG)+(gG)2−C A{e^{ - gbt}} = \left( {gG} \right) + \sqrt {{{\left( {gG} \right)}^2} - C} Since C is a constant, we can express it by evaluating (8) at t = 0: (29) C=(gG0)2−(bB0)2 C = {({gG_0})^2} - {({bB_0})^2} Inserting this expression for C into (28), also evaluated at t = 0 gives us the value of the constant A: (30) A=gG0+(gG0)2−[(gG0)2−(bB0)2] A = {gG_0} + \sqrt {{{\left( {{gG_0}} \right)}^2} - \left[ {{{\left( {{gG_0}} \right)}^2} - {{\left( {{bB_0}} \right)}^2}} \right]} or (31) A=gG0+bB0 A = {gG_0} + {bB_0} Rearranging Eq. (28) to have nothing but the square root on one side of the equation, and inserting Eq. (28), yields: (32) (gG)2−C=(gG0+bB0)e−gbt−gG \sqrt {{{\left( {gG} \right)}^2} - C} = \left( {{gG_0} + {bB_0}} \right){e^{ - gbt}} - gG Squaring both sides of Eq. (32) gives (33) (gG)2−C=[(gG0+bB0)e−gbt−gG]2 {\left( {gG} \right)^2} - C = {\left[ {\left( {{gG_0} + {bB_0}} \right){e^{ - gbt}} - gG} \right]^2} which is a linear algebraic equation for gG(t) (since the quadratic factor (gG)² cancels from both sides of Eq. (33). Solving this linear equation for G(t) gives: (34) G(t)=[(gG0)2−(bB0)2]egbt+[gG0+bB0]2e−gbt2g(gG0+bB0) G\left( t \right) = {{\left[ {{{\left( {{gG_0}} \right)}^2} - {{\left( {{bB_0}} \right)}^2}} \right]{e^{gbt}} + {{\left[ {{gG_0} + {bB_0}} \right]}^2}{e^{ - gbt}}} \over {2g\left( {{gG_0} + {bB_0}} \right)}} We see from Eq. (1) that we can obtain B(t) by simply differentiating Eq. (34) with respect to t, and dividing by—b². The result is (35) B(t)=[(gB0)2−(bG0)2]egbt+[gG0+bB0]2e−gbt2b(gG0+bB0) B\left( t \right) = {{\left[ {{{\left( {{gB_0}} \right)}^2} - {{\left( {b{G_0}} \right)}^2}} \right]{e^{gbt}} + {{\left[ {{gG_0} + {bB_0}} \right]}^2}{e^{ - gbt}}} \over {2b\left( {{gG_0} + {bB_0}} \right)}} Both Eqs (34) and (35) can be simplified by using the identity (36) (gG0)2−(bB0)2=[gG0+bB0][gG0−bB0] {\left( {{gG_0}} \right)^2} - {\left( {{bB_0}} \right)^2} = \left[ {{gG_0} + {bB_0}} \right]\left[ {{gG_0} - {bB_0}} \right] This yields the final solution for the functions G(t) and B(t): (37) G(t)=(gG0−bB0)egbt+(gG0+bB0)e−gbt2g G\left( t \right) = {{\left( {{gG_0} - {bB_0}} \right){e^{gbt}} + \left( {{gG_0} + {bB_0}} \right){e^{ - gbt}}} \over {2g}} (38) B(t)=(gB0−bG0)egbt+(gG0+bB0)e−gbt2b B\left( t \right) = {{\left( {g{B_0} - b{G_0}} \right){e^{gbt}} + \left( {{gG_0} + {bB_0}} \right){e^{ - gbt}}} \over {2b}} Assuming that gG₀ > bB₀, so that the Good Guys win, the end of the battle comes at a time t_end that satisfies the condition (39) B(tend)=0 B\left( {{t_{{\rm{end}}}}} \right) = 0 Putting this into Eq. (38) and solving for t_end gives (40) tend=1gbln[gG0+bB0gG0−bB0] {t_{{\rm{end}}}} = {1 \over {gb}}ln\left[ {{{{gG_0} + {bB_0}} \over {{gG_0} - {bB_0}}}} \right] This calculation of t_end assumes that gG₀ > bB₀. If we have gG₀ = bB₀, then C = 0, and so gG(t) = bB(t), which implies in turn that (41) dGdt=−gbG {{dG} \over {dt}} = - gbG (42) dBdt=−gbB {{dB} \over {dt}} = - gbB both of which have the solutions (43) G(t)=G0e−gbt G\left( t \right) = {G_0}{e^{ - gbt}} (44) B(t)=B0e−gbt B\left( t \right) = {B_0}{e^{ - gbt}} So in this special case, strictly speaking, the battle never ends, since at all times both G(t) and B(t) are greater than zero, but in practice we can consider the battle to be over when one side or the other is reduced to less than one man. A number of people come in integer multiples. Thus, we can assume the battle is really over when either G(t_over) = 1 or B(t_over) = 1 whichever gives the smallest value for t_over.

Eqs (43) and (44) give (45) toverG=1gblnG0 t_{{\rm{over}}}^G = {1 \over {gb}}ln{G_0} (46) toverB=1gblnB0 t_{{\rm{over}}}^B = {1 \over {gb}}ln{B_0} So t_over equals the smaller of the two numbers (45) and (46).

Colonel T. N. Dupuy of the US Army has written two books (Dupuy, 1977, 1987) wherein he has attempted to include many other factors, such as individual morale and relative concentration, into the Lanchester theory developed in this section. However, Dupuy only appears to be mathematically rigorous. He never attempts to integrate the generalised differential equations he presents in his books, and indeed the “equations” have terms whose definitions are too vague to be actual mathematics. I suspect that Dupuy exhibits his “equations” only to overawe non-mathematicians.

States are very rarely destroyed by invasions of foreigners, who, unless they are themselves organised into states, will have numbers far inferior to the numbers in the armies of the states. Instead, what typically happens is that states, being based on nothing but military force, apply this force to their own citizens, increasing the taxes and regulations to such an extent that the ability of the citizenry cannot respond to new economic challenges or environmental changes.

There have been a huge number of books published describing how civilisations and states are destroyed by internal mechanisms. Gibbon’s Decline and Fall of the Roman Empire (Gibbon, 1946), Oswald Spengler’s Decline of the West (Spengler, 1991), and Arnold Toynbee’s A Study of History (Toynbee, 1972) are only the best known. But a careful reader will note that the ultimate reason for state and civilization collapse described in these three books and as described in more recent books on the subject, such as Herman (2007), Kennedy (1987) and Ferguson (2011) (the best most recent is [Acemoglu & Robinson, 2012])—is the application of force against their own citizens. Once a firm whose product is physical violence obtains a monopoly in a region—that is, once it becomes a state—other firms pay (via bribes, lobbying, etc.) the monopoly producer of physical violence to regulate any potential competitor out of existence. And the monopoly producer eventually learns that it can use physical violence to extract so much tax revenue that the non-state sector is paralysed. Then the state collapses. The reader should not be fooled into thinking that the fact that the state rarely has to kill or injure anyone to regulate or tax. No one, including the reader, would pay any tax unless the threat of force were not present.

Luxembourg was invaded by Germany in both 1914 and 1940. In the first invasion, they put up no resistance, and very little in the second. This did not mean that the German invasions were not acts of violence. It only meant that the Luxembourgers were not stupid. In the second invasion, the Germans had three panzer divisions, about 50,000 men and 600 tanks. Luxembourg had only a few hundred 100 men and no tanks. The only reason Luxembourg put up any resistance at all in the second invasion (resistance in which they had no men killed in the fight) was due to the fact that they hoped for support from the French army.

States have existed for between (6 × 10³)/(20 × 10⁵) × 100% = 3% and (10 × 10³)/(5 × 10⁵) × 100% = 20% of human history. (Anatomical modernity arose in the human line about 200,000 years ago, but there is evidence that behavioural modernity did not appear until about 50,000 years ago. Depending on what level of organisation one uses to define “state”’ states have existed for between 6,000 years and 10,000 years.) The reason states have existed for such a small fraction of human history is that before 6,000–10,000 years ago, a combination of climate and technological limitations did not permit the concentration of large numbers of soldiers. Before the invention of farming, human food was supplied by hunting and gathering, and this technology did not allow for significant storage that would support armies of any size. Farming was difficult in ice age conditions. Farming allowed the storage of grains, which in the form of flour could be supplied by military seizure (taxation) and a supply sufficient for a week or more could be carried by an individual soldier.

But it is possible that there will come a time when the technological environment will no longer permit the existence of states. The theory of states developed above shows that there are only two ways in which technological development will preclude the existence of states.

First, technology gives an increase in the military prowess of individuals that is so great that it cancels the square law of the advantage of numbers. Recent history shows that this may be imminent.

For example, in the Second Iraq War, the United States easily defeated the numerically superior Iraqi army with its overwhelmingly superior military technology, particularly air supremacy.

The second way whereby an individual may overcome the square law advantage of states is the possibility that technology (think cell-phone-enabled social media, and the recent phenomena of flash crowds) may allow the individual to form a temporary coalition with other individuals against the state. Alexander Hamilton and James Madison proposed this as the reason for allowing the American people to have arms and the ability to form in state militias. Hamilton and Madison argued in the Federalist Papers that states could only support a standing army of at most 1% of the state’s population, whereas a militia of free armed individuals would be made up of nearly the entire male population. thereby giving the individual the advantage of the square law. The history of the United States suggests, however, that Hamilton and Madison were mistaken. It is all too easy for the state to convert the militia into a reserve for the army of the state.

The efficacy of a militia was tested in the Third Iraq War, where the United States initially had difficulty with irregular militias, but this was due to the fact that the US army ignored the Principle of Concentration. The Surge Strategy showed that the state still has the upper hand against the militia. The Israeli Swarming Strategy is another innovation that allows states to take advantage of the square law.

Plato proposed in The Republic that small states could overcome the size advantage of large state armies—he knew the power of the Persian Empire, which had nearly conquered Greece in the generation before his—by subversion. That is, by inducing the larger states to split into warring factions. We all know how well this worked out. In the generation following Plato, all of Greece was conquered by Alexander the Great, leading the army of the unsubverted, unified state of Macedonia.

So, judging by history, if states are to be abolished, it will be due to advances in military technology that allow individuals to cancel out the square law advantage of states.

Improvements in technology allow weapons to become cheaper with time, and this allows more people to own them. The Third World is awash with Kalashnikov AK-47s, which, before they were banned in the United States in the early 1990’s, were sold for under $100. This Russian automatic rifle is superior to the Thompson submachine gun, a weapon that sold for $200 in the 1920’s, a sum roughly equivalent to $4,000 today. If we project this decrease in price into the future, we can envisage extensive private ownership of nuclear weapons within 400 years. Nuclear weapon physicists know more than they are willing to discuss in public. They know, for example, that it is technically much easier to manufacture nuclear weapons than is generally realised, even by would-be nuclear-armed states. As I shall show in a moment, it is not essential to have fissile materials such as thorium, uranium, or plutonium to make a thermonuclear bomb; such a weapon can be made using only chemical explosives.

Private possession of nuclear bombs would cancel the Principle of Mass. The case of a single nuclear-armed individual’s ability to annihilate an entire army can be handled by Lanchester equations. Such destructive capability just means that the constants b² and g² are very large. Consider the expression for the constant in Eq. (9). It is sufficient to consider the case b² = g². The limit as the killing power of both sides approaches infinity is (47) limg2→∞(Cg2)=0 \mathop {\lim }\limits_{{g^2} \to \infty } \left( {{C \over {{g^2}}}} \right) = 0 which means that in the limit of infinite killing power, both sides are wiped out, whatever the initial relative sizes. An Evil Tyrant with a huge army and a nuclear arsenal attacks a Heroic Individualist, who also has a nuclear arsenal. The Heroic Individualist is killed—but so is the Evil Tyrant, along with his entire huge army. Everybody is killed. (The opposite limit, as b² = g² goes to zero—this means that both sides are becoming increasingly inept at killing one another—results in the constant C/g² going to infinity, implying that the side with the largest numbers annihilates the smaller side, but with essentially zero losses -— at least at first. Since both sides are becoming increasingly inept at killing, eventually the constant will be so large that for any given ratio of initial army sizes, the larger army will be unable to completely annihilate the smaller army. When the limit b² = g² = 0 is attained, both sides are completely inept at killing each other, and no one at all is killed. This is clear from Eqs (1) and (2): both time derivatives are zero.)

Possession of nuclear weapons by individuals would also cancel civilisation, in the most literal sense that living in cities of concentrated humanity would be too dangerous, because cities would be tempting targets for terrorists, or even individual citizens who, when they get mad at city hall, vaporise it, and the city around it. Hopefully, the internet will eventually make cities obsolete before they are vaporised. Non-physicists do not generally appreciate that the construction of nuclear bombs by an ordinary person in his garage is a real possibility. Sam Cohen, the inventor of the neutron bomb, pointed out (Cohen, 1996; see also McPhee, 1994) three decades ago that a pure fusion nuclear bomb of roughly 10 tons TNT equivalent can be set off by only 100 pounds of high explosives, no fissionable uranium or plutonium required. Such a weapon would be a neutron bomb, capable of killing with radiation everyone within half a mile. The cost of this weapon would be only $250,000 in1996. The details of how to construct such a bomb are fortunately top secret, but no secret is forever, and even if the details of the government calculations are never revealed, advances in computer power will inevitably enable huge numbers of individuals to repeat the calculations made on government supercomputers. A mini-fusion bomb set off by a lightning bolt was recently observed (Bowers, 2017). This spontaneously formed nuclear bomb by lightning bolt used the pure fusion mechanism described by Cohen—an extremely intense electric pulse.

Information technology may make states obsolete before privately owned nuclear weapons become unavoidable. We are all familiar with how advances in information technology have allowed the development of precision bombs and missiles. Imagine a similar development occurs in bullet technology, and return to the John Wayne example with which I began this paper. Suppose that John Wayne’s revolver is loaded with bullets which can alter their trajectory in flight so that they are virtually certain to hit their targets. Smart bombs and missiles have this capability today. Suppose further that if there were several targets—in the John Wayne case, there were four targets—the bullet would divide into several independently guided bullets. Multiple Independently targetable Reentry Vehicles (MIRVs) have this capability today. With this bullet technology, the Hero John Wayne would fire once, and all four of his opponents would drop dead. Of course, if the Hero has this weapon technology, we must assume the Villains have it also. Whoever fires first wins. If all fires simultaneously, everyone is killed, and we once again have the Eq. (47) case.

Cancellation of the Principle of Mass is probably inevitable. It is well-known to particle physicists (e.g., the Nobel-prize-winning physicist Gerard ‘t Hofft (‘t Hofft, 1976); Tipler, (2024) has recently shown experimentally that the ‘t Hofft mechanism works) that if indeed there is a mechanism in the early universe that generated the observed excess of matter over anti-matter, this process can be utilised to construct anti-matter bombs. Such bombs would be to nuclear bombs as nuclear bombs are to spit balls. Dan Brown’s Angels and Demons (2003) science fiction story is based on this fact. In a recently published science fiction story about a war between a state and an anarcho-capitalist society, such a weapon was used by the anarchists against the state, with the planet of the statists being blown up (Bieser, 2025; p. 50, p. 80, p. 92). States, in other words, are just a passing phase in human history, due to last only a few 1,000 years.

In Coasean terminology, this means that firms whose product is physical violence will one day be as obsolete as firms that produced buggy whips or slide rules.

The battle of Salamis (Hanson, 2014), fought in September 480 B.C., where a Greek fleet under the Athenian admiral Themistocles annihilated a superior Persian fleet, is interesting because it one of the most decisive battles of history—Persian advance into Europe was stopped forever—but also because it is an excellent example of how a great commander can apply the Principle of Concentration against a numerically superior force, and destroy it. Themistocles first tricked the Persian king Xerxes into sending a substantial part of his fleet around the island of Salamis to “block the escape of the Greek fleet,” but in reality, this splitting of the Persian fleet reduced its numbers in the crucial battle zone. Themistocles then drew the remaining Persian ships into the narrows between Salamis and the Attic mainland, a location where the Persian superior numbers could not be deployed, and where the Greek ships could concentrate against the Persians in locally superior numbers. The Persian fleet was annihilated, with the estimated (Hanson 2014) [] Persian losses of 80,000 sailors. Without the fleet, Xerxes could not keep his huge army in Greece supplied, so he withdrew most of it, and the remainder of the army was defeated by the Greeks at the battle of Plataea. The Persians never returned.

The Battle of Leuctra, fought between the Greek cities of Thebes and Sparta in 371 B.C., is another example of the decisive application of the Principle of Concentration. The Theban commander Epaminondas, with a total army of 6,500 hoplites (heavy infantry) opposing a superior Spartan army of 8,500 hoplites, formed a phalanx 48 men deep on his left wing, and threw them against the Spartan right phalanx, which had the standard depth of 12 men. The rest of the Theban army was deployed in echelon formation to conceal the thinness of the rest of the Theban line. Outnumbered four to one, the Spartan right was thrown back, with the loss (according to Xenophon) of 1,000 men, the cream of the Spartan army, as against the Theban loss of either 300 (according to Diodorus) or 47 (according to Pausanias). Assuming the entire Spartan right wing was wiped out, so that the initial numbers were 4,000 Thebans fighting 1,000 Spartans on the Theban left, we would expect the Thebans to have lost 130 men, about midway between these two reports of the battle.

One might wonder why phalanx formation would also be subject to the Concentration Principle. After all, in a phalanx, most of the men would be too far from the front line to actively engage the enemy. Actually, the depth of the phalanx is indeed a way to make use of the Concentration Principle. Victor D. Hanson (Hanson, 1990) has studied Greek phalanx battles at length, and shows (chapters 13, 14, 15, entitled “A Collision of Men,” “Tears and Gaps,” and “The Push and Collapse,” respectively) that the added weight of more men in the deeper rows would allow the phalanx to push holes in the enemy phalanx, thereby allowing a larger area of combat between the opposing sides. This would allow the Concentration Principle to be applied.

In 60 A.D., Boudica (traditionally, her name was spelt Boadicea), Queen of a Celtic tribe in Britain, rose in revolt against the Romans. According to Tacitus (Agricola, 14–16; Annals, 14: 29–39), Boudica had raised an army of 230,000 men, and in the first encounter, enveloped the single Roman Legion (the 9th legion) of 10,000 men, and annihilated it. Suetonius (also known as Paulinus), the Roman governor of Britain, tried again with the 14th Legion and some detachments of the XX Legion. Meeting Boudica at the Battle of Watling Street, Suetonius concentrated his men in a narrow defile with a wood behind him, thereby neutralising Boudica’s numerical superiority: she could send against the Roman line only a number equal to theirs, with the result that the battle was decided by the relative skill of the soldiers of the two opposing armies. Needless to say, the Romans won. Once the morale of the Britons had been broken, and they turned to flee, the Roman soldiers were able to concentrate detachments against individual opponents, and the British were slaughtered. Tacitus reported that 80,000 Britons perished, as against 400 Romans.

The Romans did not fare as well when the Principle of Concentration was used against them, rather than by them. In the Battle of Teutoburg Forest in 9 A.D., three Roman legions were tricked by the Germanic tribal commander (later known to history as Herman the German) into extending their force into a long, thin line nearly 20 kilometres long. Herman’s army, hidden in the surrounding forest, concentrated and attacked parts of the Roman line in sequence: after one group of the Roman line was wiped out, the Germans would march through the forest and then concentrate on the next part. On one side of the long Roman line was a swamp, and on the other side were German soldiers behind a previously prepared wall, which allowed the Germans to hold the thin Roman ranks while marching most of their troops to concentrate against other parts of the Roman line. The Romans were completely annihilated. German losses are unknown, but given what we have learned above, German losses would be expected to be considerably less than the Roman losses of some 20,000 men, about 10% of the total military strength of the entire Roman Empire. Confirming the relatively light German losses, archaeologists have discovered at the battle site 6,000 Roman relics, but only 1 German relic. Roman advance into Germany was stopped permanently, giving the Battle of Teutoburg Forest the status of one of the most decisive battles in history.

In 1346 A.D., the French army under Philip VI engaged the English army under Edward III at the village of Crecy. Edward, as Herman and Suetonius did a 1,000 years earlier, met the French with a forest on one side, and Philip did not deploy his superior numbers, but instead decided to attack the English as each of his separate detachments came around the forest from his encampment (Costain, 1958, pp. 299–300). The English army numbered about 10,000, the French about 35,000–100,000. The English won, losing about 300 men to the French 14,000.

Ten years later, in 1356 A.D., the French army under Philip VI’s son and successor, John II, engaged the English army under Edward III’s son, the Black Prince, at Poitiers. Once again, the French failed to deploy, advancing against the English up a narrow path that allowed only four men abreast (Costain, 1958, p. 336). The English army was about 7,000 strong, the French about 21,000. The English won with minimal casualties, whereas the French lost about 2,500 killed and wounded, and about 2,000 captured.

The great Chinese military writer Tzu Sun was well aware of the principle of concentration. His book The Art of War is filled with maxims based on this principle. To give only three examples:

(1) Maxims 12–17 of Chapter III (“Offensive Strategy”) are: “Consequently, the art of using troops is this: When 10 to the enemy’s one, surround him [this makes maximum use of the principle of concentration, since only then will all of your troops be in contact with the enemy]. When five times his strength, attack him; If double his strength, divide him. If equally matched, you may engage him. If weaker numerically, be capable of withdrawing. And if in all respects unequal, be capable of eluding him, for a small force is but booty for one more powerful (Tzu Sun [500 B.C.], 1963, pp. 79–80).”

(2) Maxim 52 of Chapter XI (“The Nine Varieties of Ground”) is “Now when a Hegemonic King attacks a powerful state he makes it impossible for the enemy to concentrate. He overawes the enemy and prevents his allies from joining him.” An ancient Chinese commentator (Ts’ao Ts’ao) elucidates: “In attacking a great state, if you can divide your enemy’s forces your strength will be more than sufficient (Tzu Sun [500 B.C.], 1963, p. 138).”

(3) Maxim 13 of Chapter VI (“Weaknesses and Strengths”), is: “If I am able to determine the enemy’s dispositions while at the same time I conceal my own, then I can concentrate and he must divide. And if I concentrate while he divides, I can use my entire strength to attack a fraction of his. There, I will be numerically superior. Then, if I am able to use many to strike few at the selected point, those I deal with will be in dire straits (Tzu Sun [500 B.C.], 1963, p. 98).”

So, the first book on military strategy, written roughly in 500 B.C., contained the essence of the principle of concentration. For if the superiority of numbers did not enable the superior force to inflict far more casualties on the inferior force than the superior force suffered, there would be no reason to concentrate.

The Confederate general Nathan Bedford Forest also understood the importance of the superiority of numbers. He is famous for saying that his strategic aim was to get in the decisive position “first with the most men (Hurst, 1993, p. 5), a statement that is usually misquoted to make Forrest appear ignorant (Catton, 1982). Unschooled Forrest certainly was; he had only 6 months of formal schooling (Hurst, 1993, p. 6). But Lee, Sherman and other Civil War military leaders considered him to be the most “remarkable soldier the war produced” (Hurst, 1993, p. 8). He had contempt for the standard West Point rule that a third of an army should be kept in reserve. Forrest, in the battle of Sand Mountain, kept no reserve at all, sending in even the men he normally used to guard his horses (he was a cavalry general, but one who used his troops as mounted infantry), saying “If we are whipped, we will not need any horses” (Hurst, 1993, p. 5). Forrest is thought to have taught himself military strategy by experience on the battlefield.

Carl von Clausewitz, the author of the classic and enormously influential On War, (von Clausewitz 1943) certainly had battlefield experience. He played a crucial role in the Battle of Waterloo, one of the most decisive battles in history (Rogers 2002). Clausewitz was Chief of Staff of Johann von Thielmann’s Third Corps of the Prussian Army. The Prussian Third Corps, with 17,000 men, held up an important part of the French army, 33,000 men under Marshall Grouchy, for sufficiently long to prevent this force from joining Napoleon, and further. permitting the main part of the Prussian army to attack the main French force facing the Duke of Wellington. It should be pointed out that the officer “chief of staff,” in the later German army, was the one responsible for drawing up the battle plan, which the army commander then approved or disapproved, generally the former.

Clausewitz thus knew by personal experience at Waterloo about the importance of numerical superiority, and in fact was personally responsible for making sure that the Allies had numerical superiority over Napoleon at Waterloo. Clausewitz devoted an entire chapter of On War, namely Chapter 8 of Book III, to “Superiority of Numbers.”

His opening sentence of the chapter was “THIS is in tactics, as well as in strategy, the most general principle of victory, and shall be examined by us first in its generality ...”

Clausewitz continues:

But this superiority has degrees, it may be imagined, twofold, threefold or four times as many, etc., etc., and everyone sees that by increasing in this way, it must (at last) overpower everything else. ...

The direct result of this is that the greatest possible number of troops should be brought into action at the decisive point. ...

Frederick the Great beat 80,000 Austrians at Leuthen with about 30,000 men, and at Rosbach with 25,000 against some 50,000 allies; these are, however, the only known instances of victories gained against an enemy double, or more than double in numbers. ... Buonaparte had at Dresden 120,000 against 220,000, therefore not the double. At Collin, Frederick the Great did not succeed, with 30,000 against 50,000 Austrians, nor Buonaparte in the desperate battle of Leipsic [Clausewitz’s spelling], where he was 160,000 strong, against 280,000, the superiority therefore considerably less than double.

From this, we may infer that it is very difficult in the present state of Europe for the most talented general to gain a victory over an enemy double his strength. Now, if we see double numbers, such as a weight in the scale against the greatest generals, we may be sure that, in ordinary cases, in small as well as great combats, an important superiority of numbers, but which need not be over two to one, will be sufficient to ensure the victory, however disadvantageous other circumstances may be. Certainly, we may imagine a defile which even tenfold would not suffice to force, but in such a case, it can be no question of a battle at all ...

The first rule is therefore to enter the field with an army as strong as possible. This sounds very like a common place, but still really not so.

In order to show that for a long time the strength of forces was by no means regarded as a chief point, we need only observe, that in most, and even in the most detailed histories of the wars, in the eighteenth century, the strength of the armies is either not given at all, or only incidentally, and in no case is any special value laid upon it. ...

Another proof lies in a wonderful notion which haunted the heads of many critical historians, according to which there was a certain size of an army which was the best, a normal strength, beyond which the forces in excess were burdensome rather than serviceable.

Lastly, there are a number of instances to be found, in which all the available forces were not really brought into the battle, or into the war, because the superiority of numbers was not considered to have that importance which in the nature of things belongs to it.

There remains nothing, therefore, where an absolute superiority is not attainable, but to produce a relative one at the decisive point, by making skilful use of what we have ...

Much more frequently the relative superiority—that is, the skilful assemblage of superior forces at the decisive point—has its foundation in the right appreciation of those points, in the judicious direction which by that means has been given to the forces from the very first, and in the resolution required to sacrifice the unimportant to the advantage of the important—that is, to keep the forces concentrated in an overpowering mass. In this, Frederick the Great and Buonaparte are particularly characteristic.

We think we have now allotted to the superiority in numbers the importance which belongs to it; it is to be regarded as the fundamental idea, always to be aimed at before all and as far as possible.

Which is the central idea of the paper you are now reading?

The Confederate general Thomas “Stonewall” Jackson agreed with Clauswitz about the basic point of numerical superiority, as shown in Jackson’s advice to his subordinate General John D. Imboden: “Always mystify, mislead, and surprise the enemy, if possible; and when you strike and overcome him, never let up in the pursuit so long as your men have strength to follow; for an army routed, if hotly pursued, becomes panic-stricken, and can then be destroyed by half their number. The other rule is, never fight against heavy odds, if by any possible manoeuvring you can hurl your own force on only a part, and that the weakest part, of your enemy, and crush it. Such tactics will win every time, and a small army may thus destroy a large one in detail, and repeated victory will make it invincible (Johnson & Buel, 1884–1888, p. 297 of volume 2).”

The importance of numerical superiority was also recognized by naval commanders. The seventeenth-century English admiral Cloudsley Shovell (who served under William and Mary), said “Without a miracle ‘tis numbers that gain the victory (Herman, 2004, p. 192).” Shovell defeated Louis XIV’s main fleet precisely because in the crucial battle, he outnumbered the French two to one (Herman, 2004, p. 223).

It is interesting that James Madison, in one of his contributions to The Federalist Papers, specifically number 46, “The Influence of the State and Federal Governments Compared,” which first appeared in the New York Packet on Tuesday, January 29, 1788. Madison provided a calculation, based on the same idea of superiority of numbers, to show that a Federal Army could never destroy the freedom of the American People, even if the Federal Government wished to use the US Army for this purpose:

Extravagant as the supposition is, let it, however, be made. Let a regular army, fully equal to the resources of the country, be formed; and let it be entirely at the devotion of the federal government; still, it would not be going too far to say that the State governments, with the people on their side, would be able to repel the danger. The highest number to which, according to the best computation, a standing army can be carried in any country, does not exceed one hundredth part of the whole number of souls; or one twenty-fifth part of the number able to bear arms. This proportion would not yield, in the United States, an army of more than twenty-five or thirty thousand men. To these would be opposed a militia amounting to nearly half a million citizens with arms in their hands, officered by men chosen from among themselves, fighting for their common liberties, and united and conducted by governments possessing their affections and confidence. It may well be doubted whether a militia thus circumstanced could ever be conquered by such a proportion of regular troops. Those who are best acquainted with the last successful resistance of this country against the British arms will be most inclined to deny the possibility of it. Besides the advantage of being armed, which the Americans possess over the people of almost every other nation, the existence of subordinate governments, to which the people are attached, and by which the militia officers are appointed, forms a barrier against the enterprises of ambition, more insurmountable than any which a simple government of any form can admit of. Notwithstanding the military establishments in the several kingdoms of Europe, which are carried as far as the public resources will bear, the governments are afraid to trust the people with arms. And it is not certain that with this aid alone they would not be able to shake off their yokes. But were the people to possess the additional advantages of local governments chosen by themselves, who could collect the national will and direct the national force, and of officers appointed out of the militia, by these governments, and attached both to them and to the militia, it may be affirmed with the greatest assurance, that the throne of every tyranny in Europe would be speedily overturned in spite of the legions which surround it.

In other words, the army of any government could never be >1% of any country’s population, and thus it would always be outnumbered by an army composed of the freedom-loving citizens of a Republic. The citizen’s army, Madison estimated, would be made up of one sixth of the entire population and would thus outnumber the central government by at least 10 to 1. As Clausewitz pointed out, a superiority of 10 to 1 would indeed be decisive.

Provided, of course, that the citizens could be organised into an army of this size, and their force could be applied at the decisive point. If the Federal Army could arrange to face the citizen piecemeal, then the Federal Army would win.

Alexander Hamilton, one of the writers of The Federalist Papers, had military experience in the American Revolutionary War. argued, in Federalist Paper Number 25, that a free society nevertheless needed a professional army to hold off an invading army until the militia could be mobilised: “Here I expect we shall be told that the militia of the country is its natural bulwark and would be at all times equal to the national defence. This doctrine, in substance, had like to have lost us our independence. It cost millions to the United States that might have been saved. The facts which, from our own experience, forbid a reliance of this kind are too recent to permit us to be the dupes of such a suggestion. The steady operations of war against a regular and disciplined army can only be successfully conducted by a force of the same kind ...”

War, like most other things, is a science to be acquired and perfected by diligence, by perseverance, by time, and by practice.

Whether Madison and Hamilton are correct depends on the ease with which the People can organise against the central government. That is to say, it depends ultimately on information (transaction) costs, which brings us, finally, to the work of Ronald Coase.

The difficulty which individuals have in forming coalitions against states is a manifestation of information cost (often called “transaction cost”), specifically, the cost of forming contracts to organise against the state. Thus, in the absence of information cost, we would expect states not to exist. States, in other words, are just firms whose product is physical violence. Because of the Square Law and the problem individuals have in organising against it, there is an economy of scale for military force. Coase’s classic paper “The Nature of the Firm” thus applies to the state as a special case.

But in “The Nature of the Firm,” Coase himself reversed the order. His paper could have been titled “Why Firms Exist” because that was what Coase was really trying to explain. Coase pointed out that a firm was actually a “mini-state” because it ignored the price mechanism within itself. With the price mechanism, a factor A moves from X to Y “... until the difference between the prices in X and Y ... disappears. ... If a workman moves from department Y to department X, he does not go because of a change in relative prices, but because he is ordered to do so” (Coase, 1937, 1990, p. 35). Action imposed by orders, not the price mechanism, is what a state does. For Coase, a firm is a “state,” and thus, since it does not use the price mechanism is less efficient that an anarchy—unless some other factor, in the case of the firm, the “cost of using the price mechanism” which is the way Coase himself viewed transaction cost, was significant (Coase, 1990, p. 6). What I have done in this paper is identify the reason for the cost of using the price mechanism in the case of the state.

All firms aim to become a monopoly, the sole producer of a product. Attaining a monopoly would enable a firm to enjoy monopoly profits (in the sense of Peter Thiel [Thiel, 2014], see especially Chapter 3, “All Happy Companies Are Different”). Firms that produce violence are no different, except in one crucial way: they can back up their monopoly with force, because what they produce is force. So, we should not be surprised to see violence-producing firms—states—become monopolies in definite geographical areas and divide the entire Earth into such monopoly areas. They grant themselves “legal” monopolies in these areas. For states, legal monopolies and natural monopolies are the same.

The traditional counter to an opponent with a much larger army (Gladwell 2013; Arregu’in-Toft. I. 2001) is the Fabian Strategy, named after Quintus Fabius Maximus Verrucosus, the Roman consul, who applied it against the Carthaginian general Hannibal during the Second Punic War (218–202 BC). In the Fabian Strategy, the inferior army hits the superior army’s supply lines and other small detachments rather than the main army. Since only small detachments are attacked—by surprise, so that reinforcements cannot be sent—it is possible for the smaller army to attain numerical superiority in the small-scale attacks on the supply line and small detachments. It is obvious that a Fabian Strategy can only work if the larger army is unaware of the location of the smaller army’s plan of attack, and indeed, of the location of the smaller army. Only if the Coasean transaction cost of getting this information is high, in other words. A Fabian Strategy can only be applied if the army applying it can act over a long period of time, since it requires a considerable period to wear down the larger army. In Coasean terms, a Fabian Strategy can be applied only if the smaller violence-producing firm can outlast the monopoly reserves of the larger violence-producing firm. The Romans lost patience with Fabius, dismissed him, and assembled an army which met Hannibal at Cannae, which the Carthaginians won due to Hannibal’s clever use of the Principle of Concentration. The Romans then returned to the Fabian Strategy and won.

George Washington (who has been called the American Fabius) applied a Fabian Strategy against the British after the Battle of Long Island, a serious American defeat in which the numerically superior British army applied the Principle of Concentration. John Adams grew impatient with Washington, saying, “I am sick of Fabian systems in all quarters!” Eventually, the British were tired of the war.

Again, a Fabian Strategy can be applied only if the information cost that must be paid by the larger army to learn the location of the detachments of the smaller army, and its planned attacks, is too great. Often, the soldiers of the smaller army hide among a population friendly to them. The standard counter to such a technique is to enforce a Law of Collective Responsibility, in which every member of a population near the location of the smaller army’s attack is held responsible for the attack. (Collective Responsibility is considered a “war crime,” since it necessarily involves killing innocent civilians, but it is an often-successful counter to the Fabian Strategy.) An alternative to Collective Responsibility is the internment of the entire suspect population in concentration camps, as was done by the British, under General Herbert Kitchener, to the Boers in the Boer War. (The term “concentration camp” was first used to describe the camps into which the Boer population was interred.)

The Square Law theory of the state, which I have advanced in this paper, explains why states did not exist in the first 140,000 years of human existence, when tribes were the only social groupings of human beings, and only came into existence when cities did. The same technology that allowed cities to exist, farming, also allowed the storage of food to feed standing armies. I argued that technology will eventually cancel the Square Law. In Coasean terminology, this means that firms whose product is physical violence will one day be as obsolete as firms that produced buggy whips or slide rules.