[Paleopsych] Sigma Xi: Statistics of Deadly Quarrels

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Statistics of Deadly Quarrels

[Click the URL to get the PDF to see the graphics.

[I have a copy of Richardson's book by that title. It's so scarce that 
it's never been reprinted.]

January-February 2002, Volume: 90 Number: 1 Page: 10,
DOI: [28]10.1511/2002.1.10

    [33]Brian Hayes

    Look upon the phenomenon of war with dispassion and detachment, as if
    observing the follies of another species on a distant planet: From
    such an elevated view, war seems a puny enough pastime.
    Demographically, it hardly matters. War deaths amount to something
    like 1 percent of all deaths; in many places, more die by suicide, and
    still more in accidents. If saving human lives is the great
    desideratum, then there is more to be gained by prevention of drowning
    and auto wrecks than by the abolition of war.

    But no one on this planet sees war from such a height of austere
    equanimity. Even the gods on Olympus could not keep from meddling in
    earthly conflicts. Something about the clash of arms has a special
    power to rouse the stronger emotions--pity and love as well as fear
    and hatred--and so our response to battlefield killing and dying is
    out of all proportion to its rank in tables of vital statistics. When
    war comes, it muscles aside the calmer aspects of life; no one is
    unmoved. Most of us choose one side or the other, but even among those
    who merely want to stop the fighting, feelings run high. ("Antiwar
    militant" is no oxymoron.)
    [34]click for full image and caption
    [35]Figure 1. The Great War in La Plata (1865-1870) . . .

    The same inflamed passions that give war its urgent human interest
    also stand in the way of scholarly or scientific understanding.
    Reaching impartial judgment about rights and wrongs seems all but
    impossible. Stepping outside the bounds of one's own culture and
    ideology is also a challenge--not to mention the bounds of one's time
    and place. We tend to see all wars through the lens of the current
    conflict, and we mine history for lessons convenient to the present

    One defense against such distortions is the statistical method of
    gathering data about many wars from many sources, in the hope that at
    least some of the biases will balance out and true patterns will
    emerge. It's a dumb, brute-force approach and not foolproof, but
    nothing else looks more promising. A pioneer of this quantitative
    study of war was Lewis Fry Richardson, the British meteorologist whose
    ambitious but premature foray into numerical weather forecasting I
    described in this space a year ago. Now seems a good time to consider
    the other half of Richardson's lifework, on the mathematics of armed

Wars and Peaces

    Richardson was born in 1881 to a prosperous Quaker family in the north
    of England. He studied physics with J. J. Thomson at Cambridge, where
    he developed expertise in the numerical solution of differential
    equations. Such approximate methods are a major mathematical industry
    today, but at that time they were not a popular subject or a shrewd
    career choice. After a series of short-term appointments--well off the
    tenure track--Richardson found a professional home in weather
    research, making notable contributions to the theory of atmospheric
    turbulence. Then, in 1916, he resigned his post to serve in France as
    a driver with the Friends' Ambulance Unit. Between tours of duty at
    the front, he did most of the calculations for his trial weather
    forecast. (The forecast was not a success, but the basic idea was
    sound, and all modern weather prediction relies on similar methods.)

    After the war, Richardson gradually shifted his attention from
    meteorology to questions of war and international relations. He found
    some of the same mathematical tools still useful. In particular, he
    modeled arms races with differential equations. The death spiral of
    escalation--where one country's arsenal provokes another to increase
    its own armament, whereupon the first nation responds by adding still
    more weapons--has a ready representation in a pair of linked
    differential equations. Richardson showed that an arms race can be
    stabilized only if the "fatigue and expense" of preparing for war are
    greater than the perceived threats from enemies. This result is hardly
    profound or surprising, and yet Richardson's analysis nonetheless
    attracted much comment (mainly skeptical), because the equations
    offered the prospect of a quantitative measure of war risks. If
    Richardson's equations could be trusted, then observers would merely
    need to track expenditures on armaments to produce a war forecast
    analogous to a weather forecast.

    Mathematical models of arms races have been further refined since
    Richardson's era, and they had a place in policy deliberations during
    the "mutually assured destruction" phase of the Cold War. But
    Richardson's own investigations turned in a somewhat different
    direction. A focus on armaments presupposes that the accumulation of
    weaponry is a major cause of war, or at least has a strong correlation
    with it. Other theories of the origin of war would emphasize different
    factors--the economic status of nations, say, or differences of
    culture and language, or the effectiveness of diplomacy and mediation.
    There is no shortage of such theories; the problem is choosing among
    them. Richardson argued that theories of war could and should be
    evaluated on a scientific basis, by testing them against data on
    actual wars. So he set out to collect such data.

    Others had the same idea at roughly the same time. The Russian-born
    sociologist Pitirim A. Sorokin published a long list of wars in 1937,
    and Quincy Wright of the University of Chicago issued another
    compilation in 1942. Richardson began his own collection in about 1940
    and continued work on it until his death in 1953. Of the three
    contemporaneous lists, Richardson's covers the narrowest interval of
    time but seems to be best adapted to the needs of statistical

    Richardson published some of his writings on war in journal articles
    and pamphlets, but his ideas became widely known only after two
    posthumous volumes appeared in 1960. The work on arms races is
    collected in Arms and Insecurity; the statistical studies are in
    Statistics of Deadly Quarrels. In addition, a two-volume Collected
    Papers was published in 1993. Most of what follows in this article
    comes from Statistics of Deadly Quarrels. I have also leaned heavily
    on a 1980 study by David Wilkinson of the University of California,
    Los Angeles, which presents Richardson's data in a rationalized and
    more readable format.

"Thinginess Fails"

    The catalogue of conflicts in Statistics of Deadly Quarrels covers the
    period from about 1820 until 1950. Richardson's aim was to count all
    deaths during this interval caused by a deliberate act of another
    person. Thus he includes individual murders and other lesser episodes
    of violence in addition to warfare, but he excludes accidents and
    negligence and natural disasters. He also decided not to count deaths
    from famine and disease associated with war, on the grounds that
    multiple causes are too hard to disentangle. (Did World War I "cause"
    the influenza epidemic of 1918-1919?)

                                      [36]click for full image and caption
                                   [37]Figure 2. Magnitude of a war . . .

    The decision to lump together murder and war was meant to be
    provocative. To those who hold that "murder is an abominable selfish
    crime, but war is a heroic and patriotic adventure," Richardson
    replies: "One can find cases of homicide which one large group of
    people condemned as murder, while another large group condoned or
    praised them as legitimate war. Such things went on in Ireland in 1921
    and are going on now in Palestine." (It's depressing that his
    examples, 50 years later, remain so apt.) But if Richardson dismissed
    moral distinctions between various kinds of killing, he acknowledged
    methodological difficulties. Wars are the province of historians,
    whereas murders belong to criminologists; statistics from the two
    groups are hard to reconcile. And the range of deadly quarrels lying
    between murder and war is even more problematic. Riots, raids and
    insurrections have been too small and too frequent to attract the
    notice of historians, but they are too political for criminologists.

    For larger wars, Richardson compiled his list by reading histories,
    starting with the Encyclopaedia Britannica and going on to more
    diverse and specialized sources. Murder data came from national crime
    reports. To fill in the gap between wars and murders he tried
    interpolating and extrapolating and other means of estimating, but he
    acknowledged that his results in this area were weak and incomplete.
    He mixed together civil and international wars in a single list,
    arguing that the distinction is often unclear.

                                      [38]click for full image and caption
                        [39]Figure 3. Frequency of outbreaks of war . . .

    An interesting lesson of Richardson's exercise is just how difficult
    it can be to extract consistent and reliable quantitative information
    from the historical record. It seems easier to count inaccessible
    galaxies or invisible neutrinos than to count wars that swept through
    whole nations just a century ago. Of course some aspects of military
    history are always contentious; you can't expect all historians to
    agree on who started a war, or who won it. But it turns out that even
    more basic facts--Who were the combatants? When did the fighting begin
    and end? How many died?--can be remarkably hard to pin down. Lots of
    wars merge and split, or have no clear beginning or end. As Richardson
    remarks, "Thinginess fails."

    In organizing his data, Richardson borrowed a crucial idea from
    astronomy: He classified wars and other quarrels according to their
    magnitude, the base-10 logarithm of the total number of deaths. Thus a
    terror campaign that kills 100 has a magnitude of 2, and a war with a
    million casualties is a magnitude-6 conflict. A murder with a single
    victim is magnitude 0 (since 10^0=1). The logarithmic scale was chosen
    in large part to cope with shortcomings of available data; although
    casualty totals are seldom known precisely, it is usually possible to
    estimate the logarithm within ±0.5. (A war of magnitude 6±0.5 could
    have anywhere from 316,228 to 3,162,278 deaths.) But the use of
    logarithmic magnitudes has a psychological benefit as well: One can
    survey the entire spectrum of human violence on a single scale.

Random Violence

    Richardson's war list (as refined by Wilkinson) includes 315 conflicts
    of magnitude 2.5 or greater (or in other words with at least about 300
    deaths). It's no surprise that the two World Wars of the 20th century
    are at the top of this list; they are the only magnitude-7 conflicts
    in human history. What is surprising is the extent to which the World
    Wars dominate the overall death toll. Together they account for some
    36 million deaths, which is about 60 percent of all the quarrel deaths
    in the 130-year period. The next largest category is at the other end
    of the spectrum: The magnitude-0 events (quarrels in which one to
    three people died) were responsible for 9.7 million deaths. Thus the
    remainder of the 315 recorded wars, along with all the thousands of
    quarrels of intermediate size, produced less than a fourth of all the

    The list of magnitude-6 wars also yields surprises, although of a
    different kind. Richardson identified seven of these conflicts, the
    smallest causing half a million deaths and the largest about 2
    million. Clearly these are major upheavals in world history; you might
    think that every educated person could name most of them. Try it
    before you read on. The seven megadeath conflicts listed by Richardson
    are, in chronological order, and using the names he adopted: the
    Taiping Rebellion (1851-1864), the North American Civil War
    (1861-1865), the Great War in La Plata (1865-1870), the sequel to the
    Bolshevik Revolution (1918-1920), the first Chinese-Communist War
    (1927-1936), the Spanish Civil War (1936-1939) and the communal riots
    in the Indian Peninsula (1946-1948).

    Looking at the list of 315 wars as a time series, Richardson asked
    what patterns or regularities could be discerned. Is war becoming more
    frequent, or less? Is the typical magnitude increasing? Are there any
    periodicities in the record, or other tendencies for the events to
    form clusters?

    A null hypothesis useful in addressing these questions suggests that
    wars are independent, random events, and on any given day there is
    always the same probability that war will break out. This hypothesis
    implies that the average number of new wars per year ought to obey a
    Poisson distribution, which describes how events tend to arrange
    themselves when each occurrence of an event is unlikely but there are
    many opportunities for an event to occur. The Poisson distribution is
    the law suitable for tabulating radioactive decays, cancer clusters,
    tornado touchdowns, Web-server hits and, in a famous early example,
    deaths of cavalrymen by horse kicks. As applied to the statistics of
    deadly quarrels, the Poisson law says that if p is the probability of
    a war starting in the course of a year, then the probability of seeing
    n wars begin in any one year is e ^-p p ^n/n!. Plugging some numbers
    into the formula shows that when p is small, years with no onsets of
    war are the most likely, followed by years in which a single war
    begins; as n grows, the likelihood of seeing a year with n wars
    declines steeply.

    Figure 3 compares the Poisson distribution with Richardson's data for
    a group of magnitude- 4 wars. The match is very close. Richardson
    performed a similar analysis of the dates on which wars ended--the
    "outbreaks of peace"--with the same result. He checked the wars on
    Quincy Wright's list in the same way and again found good agreement.
    Thus the data offer no reason to believe that wars are anything other
    than randomly distributed accidents.

    Richardson also examined his data set for evidence of long-term trends
    in the incidence of war. Although certain patterns catch the eye when
    the data are plotted chronologically, Richardson concluded that the
    trends are not clear enough to rule out random fluctuations. "The
    collection as a whole does not indicate any trend towards more, nor
    towards fewer, fatal quarrels." He did find some slight hint of
    "contagion": The presence of an ongoing war may to some extent
    increase the probability of a new war starting.
    [40]click for full image and caption
    [41]Figure 4. Distribution of wars in time . . .

Love Thy Neighbor

    If the temporal dimension fails to explain much about war, what about
    spatial relations? Are neighboring countries less likely than average
    to wind up fighting one another, or more likely? Either hypothesis
    seems defensible. Close neighbors often have interests in common and
    so might be expected to become allies rather than enemies. On the
    other hand, neighbors could also be rivals contending for a share of
    the same resources--or maybe the people next door are just plain
    annoying. The existence of civil wars argues that living together is
    no guarantee of amity. (And at the low end of the magnitude scale,
    people often murder their own kin.)

    Richardson's approach to these questions had a topological flavor.
    Instead of measuring the distance between countries, he merely asked
    whether or not they share a boundary. Then, in later studies, he
    refined this notion by trying to measure the length of the common
    boundary--which led to a fascinating digression. Working with maps at
    various scales, Richardson paced off the lengths of boundaries and
    coastlines with dividers, and realized that the result depends on the
    setting of the dividers, or in other words on the unit of measurement.
    A coastline that measures 100 steps of 10 millimeters each will not
    necessarily measure 1,000 steps of 1 millimeter each; it is likely to
    be more, because the smaller units more closely follow the zigzag path
    of the coast. This result appeared in a somewhat out-of-the-way
    publication; when Benoit Mandelbrot came across it by chance,
    Richardson's observation became one of the ideas that inspired
    Mandelbrot's theory of fractals.

    During the period covered by Richardson's study there were about 60
    stable nations and empires (the empires being counted for this purpose
    as single entities). The mean number of neighbors for these states was
    about six (and Richardson offered an elegant geometric argument, based
    on Euler's relation among the vertices, edges and faces of a
    polyhedron, that the number must be approximately six, for any
    plausible arrangement of nations). Hence if warring nations were to
    choose their foes entirely at random, there would be about a 10
    percent chance that any pair of belligerents would turn out to be
    neighbors. The actual proportion of warring neighbors is far higher.
    Of 94 international wars with just two participants, Richardson found
    only 12 cases in which the two combatants had no shared boundary,
    suggesting that war is mostly a neighborhood affair.

    But extending this conclusion to larger and wider wars proved
    difficult, mainly because the "great powers" are effectively
    everyone's neighbor. Richardson was best able to fit the data with a
    rather complex model assigning different probabilities to conflicts
    between two great powers, between a great power and a smaller state,
    and between two lesser nations. But rigging up a model with three
    parameters for such a small data set is not very satisfying.
    Furthermore, Richardson concluded that "chaos" was still the
    predominant factor in explaining the world's larger wars: The same
    element of randomness seen in the time-series analysis is at work
    here, though "restricted by geography and modified by infectiousness."
    [42]click for full image and caption
    [43]Figure 5. Web of wars is constructed from Richardson's data . . .

    What about other causative factors--social, economic, cultural? While
    compiling his war list, Richardson noted the various items that
    historians mentioned as possible irritants or pacifying influences,
    and then he looked for correlations between these factors and
    belligerence. The results were almost uniformly disappointing.
    Richardson's own suppositions about the importance of arms races were
    not confirmed; he found evidence of a preparatory arms race in only 13
    out of 315 cases. Richardson was also a proponent of Esperanto, but
    his hope that a common language would reduce the chance of conflict
    failed to find support in the data. Economic indicators were equally
    unhelpful: The statistics ratify neither the idea that war is mainly a
    struggle between the rich and the poor nor the view that commerce
    between nations creates bonds that prevent war.

    The one social factor that does have some detectable correlation with
    war is religion. In the Richardson data set, nations that differ in
    religion are more likely to fight than those that share the same
    religion. Moreover, some sects seem generally to be more bellicose
    (Christian nations participated in a disproportionate number of
    conflicts). But these effects are not large.

Mere Anarchy Loosed upon the World

    The residuum of all these noncauses of war is mere randomness--the
    notion that warring nations bang against one another with no more plan
    or principle than molecules in an overheated gas. In this respect,
    Richardson's data suggest that wars are like hurricanes or
    earthquakes: We can't know in advance when or where a specific event
    will strike, but we do know how many to expect in the long run. We can
    compute the number of victims; we just can't say who they'll be.

    This view of wars as random catastrophes is not a comforting thought.
    It seems to leave us no control over our own destiny, nor any room for
    individual virtue or villainy. If wars just happen, who's to blame?
    But this is a misreading of Richardson's findings. Statistical "laws"
    are not rules that govern the behavior either of nations or of
    individuals; they merely describe that behavior in the aggregate. A
    murderer might offer the defense that the crime rate is a known
    quantity, and so someone has to keep it up, but that plea is not
    likely to earn the sympathy of a jury. Conscience and personal
    responsibility are in no way diminished by taking a statistical view
    of war.

                                      [44]click for full image and caption
              [45]Figure 6. Long-term catalogue of global conflicts . . .

    What is depressing is that the data suggest no clear plan of action
    for those who want to reduce the prevalence of violence. Richardson
    himself was disappointed that his studies pointed to no obvious
    remedy. Perhaps he was expecting too much. A retired physicist reading
    the Encyclopaedia Britannica can do just so much toward securing world
    peace. But with larger and more detailed data sets, and more powerful
    statistical machinery, some useful lessons might emerge.

    There is now a whole community of people working to gather war data,
    many of whom trace their intellectual heritage back to Richardson and
    Quincy Wright. The largest such undertaking is the Correlates of War
    project, begun in the 1960s by J. David Singer of the University of
    Michigan. The COW catalogues, like Richardson's, begin in the
    post-Napoleonic period, but they have been brought up close to the
    present day and now list thousands of militarized disputes. Offshoots
    and continuations of the project are being maintained by Russell J.
    Leng of Middlebury College and by Stuart A. Bremer of Pennsylvania
    State University.

    Peter Brecke of the Georgia Institute of Technology has begun another
    data collection. His catalogue extends down to magnitude 1.5 (about 30
    deaths) and covers a much longer span of time, back as far as a.d.
    1400. The catalogue is approaching completion for 5 of 12 global
    regions and includes more than 3,000 conflicts. The most intriguing
    finding so far is a dramatic, century-long lull in the 1700s.

    Even if Richardson's limited data were all we had to go on, one clear
    policy imperative emerges: At all costs avoid the clash of the titans.
    However painful a series of brushfire wars may seem to the
    participants, it is the great global conflagrations that threaten us
    most. As noted above, the two magnitude-7 wars of the 20th century
    were responsible for three-fifths of all the deaths that Richardson
    recorded. We now have it in our power to have a magnitude-8 or -9 war.
    In the aftermath of such an event, no one would say that war is
    demographically irrelevant. After a war of magnitude 9.8, no one would
    say anything at all.

    © Brian Hayes


      * Ashford, Oliver M. 1985. Prophetor Professor?: The Life and Work
        of Lewis Fry Richardson. Bristol, Boston: Adam Hilger.
      * Cioffi-Revilla, Claudio A. 1990. The Scientific Measurement of
        International Conflict: Handbook of Datasets on Crises and Wars
        1945?1988. Boulder and London: Lynne Reinner Publishers.
      * Richardson, Lewis F. 1960. Statistics of Deadly Quarrels. Edited
        by Quincy Wright and C. C. Lienau. Pittsburgh: Boxwood Press.
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        Study of the Causes and Origins of War. Edited by Nicolas
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      * Richardson, Lewis F. 1961. The problem of contiguity: An appendix
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      * Richardson, Lewis Fry. 1993. Collected Papers of Lewis Fry
        Richardson. Edited by Oliver M. Ashford, et al. New York:
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      * Richardson, Stephen A. 1957. Lewis Fry Richardson (1881-1953): A
        personal biography. Journal of Conflict Resolution 1:300-304.
      * Singer, J. David, and Melvin Small. 1972. The Wages of War,
        1816-1965: A Statistical Handbook. New York: John Wiley.
      * Sorokin, Pitirim A. 1937. Social and Cultural Dynamics Vol. 3:
        Fluctuations of Social Relationships, War, and Revolution. New
        York: American Book Company.
      * Wilkinson, David. 1980. Deadly Quarrels: Lewis F. Richardson and
        the Statistical Study of War. Berkeley: University of California
      * Wright, Quincy. 1965. A Study of War, with a Commentary on War
        Since 1942. Second edition. Chicago, Ill.: University of Chicago


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