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Demographic variability and preferential marriage patterns.

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Demographic Variability and Preferential
Marriage Patterns’
Princeton University
This is a report of an attempt at computer simulation of a human population.
The immediate purpose of this simulation
is to determine effects of variation in demographic variables such as population
growth, birth rates, death rates, and agespecific marriage rates on the operation
of an ideal pattern of preferred marriage.
In this paper we discuss problems involved
in developing a mathematical model of
a population, some results of application
of the model to a question of a preferential
marriage system, and some suggestions
for other applications of this or similar
The problem
One current fad in social anthropology
is the discussion of cross-cousin marriage
systems. Numerous attempts to explain
their distribution and operation have been
made by Coult (’62), Homans and Schneider (’55), Leach (’51, ’62),L6vi-Strauss
(’49), Needham (’62), and others. Most
of these discussions have been devoted to
logical analyses of the operations of these
patterns as ideal systems with much of the
recent literature being devoted to a distinction between preference and prescription of marriage. Little attention has been
paid, however, to the questions of population dynamics which must be involved in
any such system. Leach’s statement in
this regard is typical:
“The basic confusion is that the ethnographical
literature relating to unilateral forms of crosscousin marriage tends to mix up two entirely different phenomena. (1) A n alleged preference for
a marriage with an actual first cousin as against
a variety of other possible spouses. Such prefer-
ences undoubtedly exist and are seldom difficult
to explain, given the circumstances of the situation. The literature seldom reports the statistical
incidence of the preferred type of union in such
cases, but where figures are given they are usually
low. In some cases the “‘preference” is merely a
verbal formula which does not correspond to the
facts a t all - e.g. Powell’s discovery that out of
85 Trobriand marriages only one was with the
supposedly preferred father’s sister’s daughter.”
(Leach, ’62, p. 153.)
Leach then describes prescriptive marriage, a pattern in which marriage must
be with a certain category of relatives.
The sort of statement Leach has made
about statistics of marriage has led us to
the following questions: What frequency
of the preferred type of marriage can be
expected, given reasonable assumptions
about the demographic parameters of the
population involved? And, what are the
effects of changes in the values of the
demographic variables on the expected
With this sort of question in mind,
a mathematical model was constructed,
within which the following demographic
assumptions were made : Demographic
conditions resemble those found in parts
of the world which have not yet felt the
1This is a revised and expanded version of a
paper presented at the Thirty-Second Annual Meeting
of the American Association of Physical Anthropologists, Boulder, Colorado, May 4, 1963. Preparation
of this paper was made possible by a grant from the
Princeton University Committee on Research i n the
Humanities and Social Sciences. This work made use
of computer facilities supported i n part by National
Science Foundation Grant NSF-GP579. We would also
like to express our appreciation to Elizabeth Kelley
for her able assistance in preparing graphic materials
and supplementary statistical analyses.
full effects of modern medicine. The crude
birth rate was set initially at approximately 41 per thousand and the crude
death rate at 40 per thousand. Age-specific
mortality figures for each sex were derived
from United Nations Model Life Tables
(United Nations, '56) with an expectation
of life at birth of 25 years. Age-specific
marital fertility rates were derived from
average birth rates in rural districts of
India in '51 (Coale and Hoover, '58, p.
352). India was chosen as the source for
these rates because its population is as well
known as that of any area which has not
yet felt the full effects of modern medicine. Although we have used figures from
India, the model should not be thought of
as representing any particular country or
For convenience, we arbitrarily started
with a cohort of 100, 200, or 300 individuals 15 years of age equally divided between males and females. The sex ratio
at birth was set at an average of 105 males
to 100 females. The size of the starting
cohort was varied in order to observe the
effect of population size on the rate of
cross-cousin marriages. For purposes of
comparison, all figures prior to program
year 100 were disregarded. By year 100
of the operation of the program, the age
distribution of the population generated
from the initial cohort has stabilized, and
all living individuals have genealogies of
at least two generations span.
We made the following assumptions
about the ideal pattern of preferred marriage:'
All women should get married. The
initial age-specific probabilities of marriage were derived from the proportion of
females never married, reported for Assam
province, India, in 1931.3 According to
this schedule, approximately 80% of all
women will have been married at least
once by age 20. In subsequent trials the
age-at-marriage schedule for females was
varied covering a range from a median
age of 11 years to approximately 15 years
of age.
Women can marry starting at age 5, but
they do not start bearing children until
age 11.4
Widows or widowers can remarry, but
there is no plural marriage.
The ideal pattern of marriage for a man
is with his mother's brother's daughter;
for a woman, with her father's sister's son
(i.e., matrilateral cross-cousin marriage).
The search for an ideal mate starts early
in childhood (age 2 for girls, age 5 for
boys). If no ideal mate (matrilateral crosscousin) is available at the time of marriage, a mate eventually will be selected
at random from unmarried individuals
with whom marriage is not restricted by
the incest taboo, nor by exogamy within
the patrilineage.
We assume that no births occur except
to married females.
These assumptions were made in order
to attempt to maximize the proportion of
marriages of the ideal type, within the
limits set by specified demographic processes. The mathematical model is intended to incorporate the variation due to
chance in births, deaths and marriages
within natural populations. It is, in fact,
a model of a stochastic process. The agespecific birth and death probabilities and
the probability of a birth being male or
female are constant over time, but the
probabilities of marriage, that is locating
a n eligible spouse (ideally the preferred
cousin) vary with the current state of the
system resulting from the operation of
these probabilities in previous years. Analysis of the mathematical properties of the
system is deferred in order to explore
quickly the behavior of some more or less
typical cases. To what extent Monte Carlo
procedures will be required for the further
development of this research remains to
be seen.
2 Again it should be noted that this .is a model
which does not represent any real population although
it is demographicall typical of a certain class of
populations. A seam% of the ethnographic literature
reveals that there are insufficient data f o perm+. the
accurate approxlmation of demographic conditions
and marriage customs of real popuJations to construct
a n accurate and fully representative model.
3 Figures given by S. N. Agarwala ('62,.p. 54).
These figures are for women never married; the
probability of rst marriage at a iven age can be
derived from tge complement of tiese figures. The
probability of first marriage is not a n accurate representation of proportion of women Farried at any
given age. In order to allow for remart.iae;e of widows,
so that marital fertility rates will be in balance with
death rates, we ad'usted upwards the figures
Aganvala. If we i a d failed to do so, the proportion
of married females in the population would be too
low, due to death of their spouses. Then the population would die out because our model depends only
on kgitimate births, and death rates were chosen
to be in balance with a given rate of marital fertilitv.
4The marital fertility schedule in use starts with
a probability of 0.05 at age 11, increases to a maximum of 0.281 at age 23, and declines to 0 at age 45.
The computer program
Using these assumptions, a model was
con~tructed.~Important features of this
model include the following:
1. Length of the cycle of operation.
For convenience we used a one-year cycle.
Length of the cycle is an important consideration in the model, because, although
in a real population events such as birth,
marriage and death are taking place continuously, in a computer simulation it is
necessary to consider each of these eventualities in some kind of a systematic, discrete sequence. Length of cycle is also
important because of its direct influence
on amount of computer time required per
2. Initial state of the population. The
results presented here were produced by
machine runs in which we varied the
initial population size from 100 to 300,
with an initially equal number of males
and females, all age 15, all unrelated to
one another. Our results suggest that the
initial size of the population has little
effect on proportion of ideal marriages
after a period of yearly cycles.
3. Order of the operations-program flow
chart. The initial population is read into
the machine, and each individual is identified by age, sex, and marital status (for
details see footnote 5). Part I of the program involves the search for “ideal” engagements. At the beginning of the yearly
cycle, an attempt is made to find suitable
marriage partners. This is done by examining each individual’s record, looking
for unmarried, unengaged women of engageable age (2-50). For each of these
women a search is made for all of her
father’s sisters’ sons who are alive, unmarried, unengaged, and of marriageable
age (that is, age 5-50). If any of these
are first cousins, the girl is “engaged to
the eldest. We can refer to engagements
of this type as “ideal.” If there is none of
this type for the first woman, the machine
repeats the operation for the next person,
and so on for the entire population.
Part I1 of the program involves a search
for “other” engagements. When the search
for ideal mates has been completed, the
machine again searches the population for
unmarried, unengaged women of marriageable age (5-50). For each of these
females, potential marriage partners are
selected by searching for unmarried, unengaged males of marriageable age. When
such a man is found, a comparison is made
between the woman’s father, father’s
father, and father’s father’s father, and
her potential fiance’s father, father’s
father, and father’s father’s father. If
any of these patrilineal relatives are shared
(i.e., if any of them are the same person)
no marriage is allowed. Such a search
eliminates all primary relatives and all
patrilineal relatives up to three generations
removed from the concerned parties. If
none of the patrilineal relatives is shared,
a marriage is possible. From among the
possible mates, one male is selected at
random, and engaged to the woman. This
process is repeated for each member of
the population.
Part 111 of the program results in marriage or the breaking of the non-ideal engagements. The machine searches the
population for engaged women, and applies the age-specific probability of marriage to them. A random number is then
5The program was written for the I.B.M. 7090
Computer. It has two short subroutines written i n
FAP used to pack and unpack addresses. the rest is
i n F’ORTRAN. The way in which data (i.e., the population) are represented in the storage of the computer has gone through several versions, movin in
the direction of maintaining more informatlon a%out
each person. As of this writing three words are used
for each person. The populatiAn can be thought of
as an 8,000 by three matrix, using 24,000 words of
storage for data out of the 32 768 words available.
The rest is occupied by.the pro’gram itself.
The information camed for each person includes
seven zero- or one-type indicators; alive or not, crosscousin engaged or not, engaged in any way or not,
married or not male or female widow (or widower)
or not, and an“‘initial” person br not. By “initial” is
meant a person who was generated at the beginning
of the run, and who therefore, has no lineage. Each
oerson’s aee is carri’ed.
references: father mother, mate (person engaged or
married to), first ’child, younger brother or sister of
same mother, and younger brother or sister of same
father. The last three allow rapid scanning for all
potential cross-cousin mates.
A girl who is to be engaged is defined as a row i n
the matrix that is alive, unmarried, unengaged, not
male. For each such person her father’s father’s
oldest child is found. T is mGy be her father or an
older brother or sister (or half-brother or half-sister)
of her father. If this oldest .child is female: a search
is made for her oldest child, and for his or her
brothers and sisters (each having the address of the
next younger) until an address of zero is found
sto ping the c‘hain. In this way a chain of aunt;
an$ uncles is searched, and for each aunt, the chain
of eliglble cousins is examined. It is erhaps interesting to note that the least used axdress in the
system 1s that of a person’s mother.
A listing of this program can be obtained by writing
Computer Center Princeton University Princeton,
New Jersey. Attendon: MI. Roald Buhle;
generated. If this number is less than or
equal to the probability of marriage, the
marriage takes place. If the random number is greater than the probability of marriage, the marriage does not take place.
(A similar procedure is followed each time
a probability is to be applied in the program.) If the couple is married, this fact
is recorded on each of their individual
records. If they do not get married, and
if the engagement is of the ideal type, the
engagement is continued. Otherwise, the
engagement is broken. Thus jilted, the
girls go back into the pool of prospective
brides for the following year.
Part IV results in the birth of children.
A search is made of the population for
married women, and the probability of
their having a chiId (as determined from
an age-specific birthrate) is applied to
them. If a child is born, it is given an
identification number, the identifications
of its parents are recorded, and a sex is
assigned to the child at random on the
basis of a sex ratio for live births.
The way this program has been written,
we have storage capacity for only 8,000
individuals (living and dead). For this
reason, if a newborn child would be assigned number 8001, we would be unable
to store his record. In order to solve this
problem, when number 8,000 is reached,
the first 500 individuals are dropped from
the machine memory (or placed in dead
storage), and everyone else’s identification number or address is decreased by
Part V results in the application of mortality to the population. The machine
searches the entire living population and
applies the age- and sex-specific probabilities of dying to each individual. The
fact of death is recorded for those who die.
If they are married, the marriage is broken
(but the fact that they were married is
retained), and the widow or widower becomes eligible for re-marriage in the next
yearly cycle. If the deceased person was
engaged, the engagement is broken, and
the fiancd re-enters the marriage pool.
Part VI, the final portion of the program, represents the end of the year. This
is the end of the annual cycle - all marriages, births, deaths and engagements
have been recorded, and the population
is now updated by adding one year to each
individual’s age. The results of the year’s
operation can now be printed, and the
program can resume for the next year.
Information which we have obtained routinely for each year includes number of
persons alive in the population, number
of males, marital fertility, crude death
rate, number of ideal engagements, number and sex of newborn children, number
of deaths, number of ideal marriages, ideal
marriages expressed,as a percentage of the
total number of extant marriages, number
of year of machine operation. A more
complete census can be taken periodically
when desired (usually at 10 or 20 year
intervals) giving distribution of age, sex
and marital status of the population.
A variation of this program allows marriage to take place within the population
at random (except for the previously
stated restrictions on age at marriage, and
incest and exogamy proscriptions). This
allows a comparison of the number of
cross-cousin marriages which take place
by “accident” with the number which will
take place given attempts at maximization
of cross-cousin marriages.
Realism of the model and
possible modifications
Several objections may be raised concerning the realism of this model. The
demographic conditions (vital rates), of
course, are subject to question and are
easily modified by substitution of other
rates in the tables of probabilities within
the program, Objections may also be raised
concerning the operation of the preferential marriage rules. In its present form,
the program is general enough so that it
could be modified to allow for a change
to other systems of preference or prescription (e.g., patrilateral cross-cousins, or
second cross-cousins) and likewise the incest and exogamy rules are subject to
A more serious question may be raised
with respect to the possibility of population
growth or decline. Real populations have
homeostatic devices available to them
which tend to control population growth
(e.g., increased disease rates, famine, female infanticide, migration), or control
decline (e.g., earlier age of marriage, re-
laxation of incest taboos, bettter care for
children, larger desired size of family).
Our model has no control built in to stabilize the population size.
Population expansion, with its resultant
increase in machine running time presents
a practical problem. Given the close balance of birth and death rates, expansion
or decline is probable in this program
because of the several random operations.
For example, purely by chance a large
number of females may be born and survive to childbearing age. This will result in population increase. Likewise, by
chance, for a series of years a small number of females may be born, and the population will tend to decline.
Several solutions have been considered
for this problem, but none has yet been
implemented. These include : alteration of
fertility and mortality schedules dependent on specified limits of gross size of
population; fertility and marriage rates
which insure population growth, in combination with “epidemics” (killing at random a significant proportion of the population in a given year over and above usual
mortality) when the population reaches a
certain size; “migration” - insuring population growth through marriage and fertility rates, and then removing a portion
of the population periodically to reduce
the remaining population’s size to some
specified limit with those removed possibly
selected by certain demographic characteristics, depending on the type of migration being simulated.
We now have a number of results of
machine runs on populations continued
up to as long as 740 years, and we are
able to indicate some of the effects of
demographic variation on rates of matrilateral cross-cousin marriage. It should be
noted that all results discussed here refer
to the structure and changes in populations after the first 100 years of operation,
so that age distributions have had time to
stabilize and genealogies have been generated for all members of the population.
It is important to emphasize that the
results presented below are quite tentative
in respect to relationships suggested between the proportion of cross-cousin mar-
riages and population size and rate of
growth. Most of our evidence results
from experimenting rather unsystematically with the age schedule of marriage
probabilities for females. The probabilities of dying and the probability of a
married female having a birth remained
unchanged in all trials; only marriage
probabilities by age were altered which
thus produce changes in total fertility and
the rate of population growth. Despite
the early stages of this research, however,
we are reasonably convinced that the estimates of cross-cousin marriage rates assembled below are sufficiently reliable to
permit theoretical inferences about the
viability of prescriptive marriage norms
in a social system with normal demographic variability. One further caution
- we have employed a simple test of
statistical significance (chi square) in
spite of several questions about its appropriateness only to facilitate inferring suggestive generalizations.
The proportion of matrilateral crosscousin marriages occurring under maximal conditions (see above description of
Part I of the program) for each of nine
populations at 50-year intervals is plotted
in figure 1. These populations vary either
in initial size and/or in the schedule of
age - at-marriage probabilities. Although
there is some variation reflecting sampling
variability and fluctuating supply and demand of eligible cousins, the range is
fairly restricted with over three quarters
of all observations falling in the 20 to 35%
range and over nine out of ten in the 15
to 40% range. The average proportion of
matrilateral cross-cousin marriages is calculated to lie between 27 and 28%.
There seems to be a direct but not statistically significant relationship between
population size ( all populations observed
at 50-year intervals) and the proportion
of matrilateral cross-cousin marriages. Results are presented in table 1.
The direction of population change that is whether the population is growing,
remaining stationary or declining - appears to be positively associated with the
proportion of ideal marriages (see table 2).
And, viewing the same phenomenon from
the perspective of age at marriage (dichotomized for presentation in table 3 ) we see
percentage of
cross*cous i n
morr i o g 8 s
so r
200 2 5 0
Time in
700 750
Fig. 1 Proportion of matrilateral cross-cousin marriages in simulated populations. Note: The
proportion existing a t each year was averaged for every ten year interval beginning with the year 95.
Population size and proportion of matrilateral cross-cousin marriages
Population size
at 50 year
after year 100
Proportion of.
cross-cousin marriages
Under 24%
Over 31%
300 and over
Chi square = 3.3; d.f. = 2; p between 0.20 and 0.10
Population growth and proportion of matrilateral crosscousin marriages
Proportion of cross-cousin marriages 1
Under 24%
Chi square = 8.0;
d.f. = 4;
Over 31%
p between 0.10 and 0.05
Ten-year averages of percentages of cross-cousin marriages observed at 5 0 ear intervals during
extended periods of population decline, stability or growth in simulated popurations.
Age at marriage and proportion of matrilateral cross-cousin marriages
Age a t
marriage 1
Proportion of cross-cousin marriages 2
Under 24%
Over 31%
Chi square = 9.0; d.f. = 2; p = 0.01
1 The
designations “high” and “low” refer to the entire female age-at-marriage schedule. Roughly
the “low” values include schedules below a median age at first marriage of 13.
2 Ten-year averages of percentages of cross-cousin marriages observed at 50-year intervals after
the year 100.
a similar relationship with the proportion
of cross-cousin marriages.
The factor underlying this pattern of
positive association between population,
population change or age at marriage and
the proportion of ideal marriages would
seem to be changes in family size, that is
changes in the average number of children
per couple. As noted earlier the only two
parameters that were varied in these simulation trials were either the initial size of
the population or its age at marriage
schedule. The age-specific mortality and
marital fertility probabilities were held
constant throughout the series. The factor
responsible for population increase is of
course the reduction in age at marriage
which, with a constant schedule of marital
fertility, implies an increase in family size.
And, an increase in family size means an
increase in the number of cousins thereby
increasing the probability of finding ideal
Fortuitous cross-cousin marriage (that
is, marriage between cross-cousins when
the program does not attempt to maximize
such marriages) appears to be relatively
infrequent in the populations generated
by use of this model. Again, variability in
frequency is inversely related to the size
of the population. In relatively large (over
200) and stationary populations the proportion of marriages which conform to
the ideal merely by accident appears to
be between 1 and 2%. Figure 2 shows
the percentage of cross-cousin marriages
which resulted from machine runs of two
pairs of populations. Each pair started
with the same conditions of initial size,
birth rates, death rates and marriage rates.
One member of each pair was run with
marriage preferences in the machine pro-
Population A with Marriage Preferences
Population A without Marriage Preference6
Population B with Marriage Preferences
Population B without Marriage Preferences
Percentage of
Fig. 2 Percentage of cross-cousin marriages in populations with and without
marriage preferences.
gram, and one was run without marriage
Thus, it would appear that a small proportion of cross-cousin marriages can be
expected to occur in a population merely
by chance. It is also evident from the
preceding illustrations that the members
of a population could raise the frequency
of cross-cousin marriages to about 25-30%
by consistently following preferential rules
used in the model. It then becomes a
problem to explain the relatively low frequency of observed cross-cousin marriages
in real societies whose members profess a
marriage preference, in addition to the
problem of explaining the maintenance
and functions of such an ideal (cf. Leach,
'62, quoted above).
of the ideal form. This leads to a question
of what could in reality be meant by marriage prescription as contrasted with marriage preference. A model similar to the
one discussed here can be used to see
whether in fact a population could perpetuate itself if marriage had to be within
a certain prescribed class of people. It
seems likely from the findings of this study
that there will be a considerable divergence from the ideal pattern, merely as a
result of the operation of chance. This
raises a question as to how such an ideal
is maintained in the face of the impossibility that it can be achieved or even
3. The model, with little modifkation
or amplification, can be applied directly
to studies of the effects of various forms
of marriage, or various sets of incest and
Although this paper does not represent exogamy proscriptions on distributions of
the final summary of the project which genes within a gene pool. Such a model
we have described, several conclusions and is more realistic in the study of human
populations than an assumption of ranimplications seem justified.
1. It is clear that demographic processes dom mating (cf. Brues, ' 6 3 ) .
4. The model, with relatively little modiaffect and set limits on the operation of
ideal marriage patterns. These effects can fication, can be applied to questions of the
be predicted quantitatively. The results operation of various kinds of inhertitance
presented here suggest that the proportion systems under differing demographic conof ideal marriages is directly related to ditions. There are clear and immediate
population growth, and to marriage rates, applications of such results to theoretical
and that variability in proportion of ideal studies of the origin and perpetuation of
marriages is inversely related to population social stratification. For example, assumsize. This suggests that no description of ing a certain pattern of inheritance and
a marriage system, even in ideal terms, is property transmission at marriage, will
complete without a statement of demo- property tend to become concentrated in
graphic conditions within which that sys- the hands of a relatively small group, or
tem operates. Without information of this will i t tend to be dispersed throughout
type we cannot state the maximum ex- the population? What are the effects of
pected number of ideal marriages, nor can changes in population parameters (e.g,
we say whether an observed frequency of growth rate, population size, age-specific
marriage of a certain type is higher or marriage rates) on the concentration of
lower than would be expected under the property?
5. Finally, work on this model has sugoperation of the ideal system, or under the
gested a number of specific areas where
operation of chance alone.
2. Given a model such as the one de- more knowledge is needed. These include
scribed in this paper, it is possible to state demographic information for populations
quantitatively how close actual behavior which have yet to be exposed to modern
can be expected to resemble ideal patterns medicine. We especially need to know
under given ideal forms of marriage and more accurately such things as their agegiven demographic conditions. The data specific marriage rates. We also need to
presented in this paper suggest that even know, with greater precision and detail,
if the ideals are rigidly adhered to, under the strategies followed in various societies
reasonable demographic conditions only a with the intended purpose of promoting
minority of all marriages can possibly be such customs as cross-cousin marriage.
Agarwala, S. N. 1962 Age at Marriage i n India.
Kitab Mahal Pvt., Allahabad.
Brues, Alice M. Further contributions to the
problem of ABO blood group polymorphism.
Paper delivered at the meeting of the American
Association of Physical Anthropologists, Boulder, Colorado, May 4, 1963.
Coale, A. J., and E. M. Hoover 1958 Population
Growth and Economic Development in LowIncome Countries. Princeton University Press.
Coult, Allan D. 1962 A n analysis of Needham’s critique of the Homans and Schneider
theory. Southwestern Journal of Anthropology, 18(4): 317-335.
Homans, George Casper, and David M. Schneider
1955 Marriage, Authority and Final Causes.
Free Press, Glencoe.
Leach, Edmund R. 1951 The structural implications of matrilateral cross-cousin marriage.
Journal of the Royal Anthropological Institute
LXXXI parts 1 and 2, pp. 23-56.
1962 The determinants of differential
cross-cousin marriage. Man, Oct. 1962 vol.
LXII, no. 238.
LCvi-Straws, Claude 1949 Les Structures E1Cmentaires de la Parente. Paris.
Needham, Rodney 1962 Structure and Sentiment. University of Chicago Press.
United Nations 1956 Methods for Population
Projections by Sex and Age. Manual 111, No. 25.
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