[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,
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.)
click for full image and caption
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.
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?)
click for full image and caption
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.
click for full image and caption
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.
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
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
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.
click for full image and caption
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."
click for full image and caption
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
click for full image and caption
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
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.
* Richardson, Lewis F. 1960. Arms and Insecurity: A Mathematical
Study of the Causes and Origins of War. Edited by Nicolas
Rashevsky and Ernesto Trucco. Pittsburgh: Boxwood Press.
* Richardson, Lewis F. 1961. The problem of contiguity: An appendix
to Statistics of Deadly Quarrels. Yearbook of the Society for
General Systems Research, Ann Arbor, Mich., Vol. VI, pp. 140?187.
* Richardson, Lewis Fry. 1993. Collected Papers of Lewis Fry
Richardson. Edited by Oliver M. Ashford, et al. New York:
Cambridge University Press.
* 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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