Original Article
Fight the power: Lanchester's laws of combat in human evolution

https://doi.org/10.1016/j.evolhumbehav.2014.11.001Get rights and content

Abstract

Lanchester's “Laws of Combat” are mathematical principles that have long been used to model military conflict. More recently, they have been applied to conflict among animals, including ants, birds, lions, and chimpanzees. Lanchester's Linear Law states that, where combat between two groups is a series of one-on-one duels, fighting strength is proportional to group size, as one would expect. However, Lanchester's Square Law states that, where combat is all-against-all, fighting strength is proportional to the square of group size. If conflict has been important in our evolutionary history, we might expect humans to have evolved assessment mechanisms that take Lanchester's Laws of Combat into account. Those that did would have reaped great dividends; those that did not might have made a quick exit from the gene pool. We hypothesize that: (1) the dominant and most lethal form of combat in human evolutionary history (as well as among chimpanzees and some social carnivores) has been asymmetric raids in which multiple individuals gang up on a few opponents, approximating Square Law combat; and (2) this would have favored the natural selection of an evolved “Square Law heuristic” that correlated fighting strength not with raw group size but with group size squared. We discuss the implications for primate evolution, human evolution, coalitionary psychology, and contemporary war.

Introduction

“Words are inadequate to describe the emotion aroused by the prolonged movement in unison that drilling involved. A sense of pervasive well-being is what I recall; more specifically, a strange sense of personal enlargement; a sort of swelling out, becoming bigger than life, thanks to participation in a collective ritual.”

William McNeill (1995, p. 2)

“We've got them!”

George Armstrong Custer, at the Battle of the Little Bighorn.

Stephen Ambrose (1975, p. 438)

On 2 August 1867, Crazy Horse led a force of one thousand Sioux warriors in an attack on a US Army outpost near Fort Phil Kearny in northern Wyoming. Captain J. N. Powell gathered 26 soldiers and a handful of armed civilians in a corral of wagons, and they prepared to defend themselves. The Sioux initially circled Powell's position on horseback, firing arrows, intending to exhaust the cavalrymen's ammunition, but to no avail. Powell had stockpiled several thousand rounds, and the soldiers kept up a constant hail of fire. Eventually, Crazy Horse pulled his warriors back into a ravine, where they were partially protected from the gunfire. From here, the Indians attempted to attack on foot. The ravine was narrow which, as Stephen Ambrose describes, meant that “the men in front masked the mass of warriors in the rear, making it impossible for them to fire … Powell only had to deal with a handful of Indians, Crazy Horse and his fellow shirt-wearers [Sioux leaders] at the apex of the charge” (Ambrose, 1975, pp. 294–295). At this point, the outcome of the battle remained far from certain to those present. As one soldier recounted, “It chilled my blood … Hundreds and hundreds of Indians swarming up a ravine about ninety yards [away]… Our fire was accurate, coolly delivered and given with most telling effect, but nevertheless it looked for a minute as though our last moment on earth had come” (Ambrose, 1975, p. 295). Against their volleys of arrows and some astonishingly brave charges, the withering fire from the cavalry's new breech-loading rifles wore the Indians down and, after several hours' fighting, they withdrew to the mountains.

Against the backdrop of the earlier Fetterman massacre of 1866, when Crazy Horse and two thousand Sioux had surrounded Captain William Fetterman's force of 81 cavalrymen and annihilated them to a man, Powell's victory against the odds seemed nothing less than a miracle. But the reason Powell lived to see another day may well have been down to some fundamental mathematical principles of battle. Crazy Horse's congested attack up the ravine meant he was not able to bring his superior numbers and their deadly arrows to bear—even on a tiny enemy force. Meanwhile, Powell's concentrated fire on the lead ranks of Indians meant that, despite Powell's force being outnumbered 25 to 1, any Indian that squeezed onto the frontline fell into the sights of several American soldiers at once. Despite Crazy Horse's numerical supremacy and the advantage of surprise, the deck was stacked against him.

The “Wagon Box Fight” of 1867 reflects the mathematical patterns of Lanchester's Laws of Combat (Lanchester, 1916). These “laws” are mathematical equations that model the dynamics of conflict and its outcomes, and were originally developed with modern human warfare in mind. Although they have long been used in military operational research (for reviews, see MacKay, 2006, Wrigge et al., 1995), they have only recently been applied to explain variation in the patterns of conflict in animals such as ants, birds, lions, and chimpanzees (Franks and Partridge, 1993, Mosser and Packer, 2009, Plowes and Adams, 2005, Shelley et al., 2004, Whitehouse and Jaffe, 1996), including manipulation experiments showing variation in fighting behavior as parameters were changed (McGlynn, 2000, Wilson et al., 2002).

Much of the literature on Lanchester's Laws looks at models and data with regard to combat outcomes. In this paper we make a rather different kind of argument. First, we argue that Lanchester's Square Law, under which imbalances in numbers are disproportionately advantageous to the larger side, is especially applicable to pre-military human conflict, and is likely to have influenced its dynamics for several million years. This provides substantive support to theories about the importance of human groups and coalitions in early warfare (Alexander, 1987, Bingham, 2000, Pitman, 2011, Wrangham, 1999a).1

Second, the question then naturally arises: Have we evolved corresponding assessment strategies that influence when (and how) we choose to fight? Violent conflict is argued to have played a major role in our ancestral past (Buss and Shackelford, 1997, Ferguson, 2012, Gat, 2006, Guilaine and Zammit, 2004, Keeley, 1996, LeBlanc and Register, 2003, Potts and Hayden, 2008, Wrangham and Peterson, 1996, though for an earlier, contrasting view see Knauft 1991). Empirical studies suggest that warfare accounted for around 15% of male deaths among archeological and ethnographic data (and much more in some societies Bowles, 2006, Keeley, 1996, Otterbein, 1989), implying strong selection pressure on adaptations for fighting—and winning. We therefore hypothesize that natural selection should have favored assessment mechanisms that take the Square Law into account, leading to an evolved “Square Law heuristic” in the context of coalitionary conflict. Thus the Square Law becomes more than a post hoc model of conflict outcomes: rather it may be an evolved heuristic that influences decisions about whether or not to fight in the first place, continuing to affect decisions about conflict today. If so, this carries major implications for understanding human conflict in our past, present, and future.

Section snippets

Lanchester's Laws of Combat

Although there are variations in how the models are set up, and in real life there are many complicating factors (Adams and Mesterton-Gibbons, 2003, Johnson and MacKay, 2011, MacKay, 2011), the underlying logic of Lanchester's Laws capture the essence of conflict processes irrespective of species or setting—“elementary principles”, as Lanchester called them, “which underlie the whole science and practice of warfare in all its branches” (Lanchester, 1916, p. 39). The key insight is the

Which law applied in human evolution?

The dramatic difference between the Linear and Square Laws suggests that humans may have evolved rather different adaptations to conflict, depending on what type of conflict was dominant in human evolutionary history. We discriminate two distinct possibilities in particular:

Hypothesis 1 (H1)

If lethal conflict in human evolutionary history was primarily either duel-like (only one individual tended to fight one other at a time, so there was no opportunity for multiple attacks) or in the form of formal pitched

Implications for modern war

While a Square Law heuristic may have been adaptive in the small-scale inter-group conflicts of our evolutionary past (for which we argue it was designed), it may be maladaptive in conflict today, and especially in modern warfare. An evolved Square Law heuristic may prompt aggression in modern settings where there is numerical asymmetry but where the underlying dynamics, owing to weapons or circumstances, are no longer Square Law. Of course an unarmed crowd is unlikely to attack a small number

Possible empirical tests

The possibility that humans have an evolved Square Law heuristic leads to a variety of predictions that future empirical and experimental studies could test (summarized in Table 3). The core prediction is that human assessments of fighting strength increase in proportion to group size squared (relative to opponent group size). An auxiliary prediction is that we should observe a Square Law heuristic among men but not (necessarily) among women, since they were less likely to have participated in

Conclusions

The predominant form of deadly conflict in human evolutionary history, as well as among chimpanzees and some social carnivores, appears to have been asymmetric raiding, in which an attacking group has an overwhelming numerical advantage and uses it to ambush and kill members of rival groups at little cost to themselves (Manson and Wrangham, 1991, Wrangham, 1999a). Attack is concentrated while defense is divided, leading to Lanchester's Square Law, under which the fighting strength of a group is

Acknowledgments

We are grateful for comments and criticisms from Dan Blumstein, Oliver Curry, Jaimie Krems, Robert Kurzban, Steven LeBlanc, Anthony Lopez, Sarah Mathew, Raphael Sagarin, John Orbell, Michael Wilson, Richard Wrangham, and the participants at the Institute of Cognitive and Decision Sciences' 2008 meeting “Evolutionary Perspectives on War: An Interdisciplinary Conference”, in Eugene, Oregon.

References (137)

  • S.E. Ambrose

    Crazy Horse and Custer: The Epic clash of two great warriors at the little bighorn

    (1975)
  • I. Arreguin-Toft

    How the weak win wars: A theory of asymmetric conflict

    (2005)
  • A. Baudry

    The naval battle: Studies of tactical factors

    (1910)
  • P.M. Bingham

    Human evolution and human history: A complete theory

    Evolutionary Anthropology

    (2000)
  • P.M. Bingham et al.

    Death from a distance and the birth of a humane universe

    (2009)
  • G.A. Blainey

    The causes of war

    (1973)
  • C. Boehm

    Hierarchy in the forest: The evolution of egalitarian behavior

    (2001)
  • M. Boot

    The evolution of irregular war: Insurgents and guerrillas from Akkadia to Afghanistan

    Foreign Affairs

    (2013)
  • R.M. Borges

    Warring ants: Lessons from Lanchester's theory of combat?

    Journal of Biosciences

    (2002)
  • S. Bowles

    Group competition, reproductive leveling, and the evolution of human altruism

    Science

    (2006)
  • S. Bowles

    Did warfare among ancestral hunter-gatherers affect the evolution of human social behaviors?

    Science

    (2009)
  • R. Boyd et al.

    Coordinated punishment of defectors sustains cooperation and can proliferate when rare

    Science

    (2010)
  • N.A. Chagnon

    Yanomamo

    (1997)
  • C.A. Chapman et al.

    Party size in chimpanzees and bonobos: A reevaluation of theory based on two similarly forested sites

  • S.E. Churchill et al.

    The evolution of the human capacity for “killing at a distance”: The human fossil evidence for the evolution of projectile weaponry

  • R. Collins

    Violence: A micro-sociological theory

    (2008)
  • A.W. Crosby

    Throwing fire: Projectile technology through history

    (2002)
  • S. Cubaynes et al.

    Density-dependent intraspecific aggression regulates survival in northern Yellowstone wolves (Canis lupus)

    Journal of Animal Ecology

    (2014)
  • P. de Souza

    The Greek and Persian wars 499–386 BC

    (2003)
  • F.B. de Waal

    Peacemaking among primates

    (1989)
  • F.B.M. de Waal

    Chimpanzee politics: Power and sex among apes

    (1998)
  • F.B.M. de Waal et al.

    Bonobo: The forgotten ape

    (1997)
  • S.J. Deitchman

    A Lanchester model of guerrilla warfare

    Operations Research

    (1962)
  • J.H. Engel

    A verification of Lanchester's law

    Journal of the Operational Research Society

    (1954)
  • J.M. Epstein

    The calculus of conventional war: Dynamic analysis without Lanchester theory

    (1986)
  • J.M. Epstein

    Why model?

    Journal of Artificial Societies and Social Simulation

    (2007)
  • B.R. Ferguson

    Tribal warfare

  • C.M. Fleming

    New or old wars? Debating a Clausewitzian future

    Journal of Strategic Studies

    (2009)
  • R.D. Fricker

    Attrition models of the Ardennes campaign

    Naval Research Logistics

    (1998)
  • J.F.C. Fuller

    The foundations of the science of war

    (1926)
  • A. Gat

    Social organization, group conflict and the demise of neanderthals

    Mankind Quarterly

    (1999)
  • A. Gat

    War in human civilization

    (2006)
  • L. Glowacki et al.

    The role of rewards in motivating participation in simple warfare

    Human Nature

    (2013)
  • D.J. Goleman

    What is negative about positive illusions? When benefits for the individual harm the collective

    Journal of Social and Clinical Psychology

    (1989)
  • J. Goodall

    The chimpanzees of Gombe

    (1986)
  • J. Guilaine et al.

    The origins of war: Violence in prehistory

    (2004)
  • J.C. Halfpenny

    Yellowstone wolves in the wild

    (2003)
  • S. Hearne

    A journey from Prince of Wales's fort in Hudson's Bay to the Northern Ocean, 1769, 1770, 1771, 1772

    (1958)
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