Demographic Variability and Preferential Marriage Patterns’ PETER KUNSTADTER, ROALD BUHLER, FREDERICK F. STEPHAN AND CHARLES F. WESTOFF 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 models. 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 frequency? Assumptions 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. 511 512 P. KUNSTADTER, R. BUHLER, F. F. STEPHAN AND C. F. WESTOFF 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 area. 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 given.by 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. DEMOGRAPHY A N D PREFERENTIAL MARRIAGE 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 generation. 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 513 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 to: Computer Center Princeton University Princeton, New Jersey. Attendon: MI. Roald Buhle; 514 P . KUNSTADTER, R. BUHLER, F. F. STEPHAN AND C . F. WESTOFF 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 500. 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 modification. 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- DEMOGRAPHY A N D PREFERENTIAL MARRIAGE 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. RESULTS 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- 515 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 516 P . KUNSTADTER, R. BUHLER, F. F. STEPHAN AND C . F. WESTOFF DE M OGR APH I C VA R IA B I LITY AN D PREFER ENTI AL MA RR IAGE percentage of cross*cous i n morr i o g 8 s so r I I 100 150 I 200 2 5 0 I 1 I 350 400 1 300 I I 1 450 500 Time in I 550 600 .--.- Years 650 700 750 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. TABLE 1 Population size and proportion of matrilateral cross-cousin marriages Population size at 50 year intervals after year 100 Proportion of. cross-cousin marriages Under 24% 2631% Over 31% Total observations ~~ 100-299 300 and over 20 6 47 27 11 10 16 11 Chi square = 3.3; d.f. = 2; p between 0.20 and 0.10 TABLE 2 Population growth and proportion of matrilateral crosscousin marriages Population growth Proportion of cross-cousin marriages 1 Under 24% Declining Stationary Growing 24-31% 7 8 5 10 6 6 Chi square = 8.0; d.f. = 4; Over 31% 5 1 9 Tota! observations 17 19 21 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. 1 TABLE 3 Age at marriage and proportion of matrilateral cross-cousin marriages Age a t marriage 1 High Low Proportion of cross-cousin marriages 2 Under 24% 20 13 24-31% 6 16 Over 31% 6 17 Tota! Observations 32 46 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. DEMOGRAPHY AND PREFERENTIAL MARRIAGE 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 marriages. 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- DEMOGRAPHIC VARIABILITY AND PREFERENTIAL MARRIAGE Population A with Marriage Preferences Population A without Marriage Preference6 Population B with Marriage Preferences Population B without Marriage Preferences Percentage of Cross-Cousin Marriage6 35 517 .... 30 25 20 15 10 5 Fig. 2 Percentage of cross-cousin marriages in populations with and without marriage preferences. 518 P . KUNSTADTER, R. BUHLER, F. F. STEPHAN AND C . F . WESTOFF gram, and one was run without marriage preferences. 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 approximated. 3. The model, with little modifkation or amplification, can be applied directly to studies of the effects of various forms CONCLUSIONS AND IMPLICATIONS 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. DEMOGRAPHY AND PREFERENTIAL MARRIAGE LITERATURE CITED 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. 519 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.