CHAPTER 9 Mathemati.cal Anthropology - Eclectic Anthropology

78 Pages·2003·58.07 MB·English

From Handbook of S:lclal and Cultural Anthropt)&
logy. John J. Honlgmann, Ed. Copyright 1974
by Rand McNally College Publlshlna Complll11.
All r1aht. re rved.
9
Mathemati.cal Anthropology
CHAPTER
DOUGLAS R. WHITE
INTRODUCTION of the problems currently modeled by
mathematics are of such general rele
This review will introduce the gener
vance to anthropological theory that it
al reader in anthropology to some of
is doubtful that they should be con
the uses of mathematics in the analysis
fined under the title of mathematical
of ethnographic data in order to inten
anthropology. What unites them under
sify the conceptual and explanatory
this rubric is not quantification, which
power of anthropological theory. Many
covers a fraction of mathematics, but
rather the common logical substructure
Early drafts of various sections of this paper were
reviewed and criticized by others, and I have tried to that mathematics shares with science in
incorporate their criticisms and to interpret the ex the use of axiomatic reasoning. The
I
tensive work that has been done in mathematical major thrust of modern mathematical
anthropology in light of various suggestions. I grate~ thought has been to refine the logical
fully acknowledge the assistance and encouragement
of William Geoghegan, Paul Kay, Robert Randall, underpinnings of mathematical systems
fran«ois Lorrain, Michael Agar, Robert Kozelka, of analysis.
Thomas Fararo, Ira Buchler, and Michael Burton,
and the critical and editorial suggestions of G. P. example from his unpublished work on kinship alge
Murdock and Lilyan Brudner. I am also grateful to bra. My professors at the University of Minnesota .. E.
A. Kimball Romney, Roy G. D'Andrade, George Adamson Hoebel, Pertti J. Pelto, and Eldin Johnson,
Collier, Kenneth Morgan, and Bob Scholte for sugges contributed encouragement to the exploration of
tions, contributions, and references, lind to members mathematical anthropology; professors in mathe
of the Summer Seminar in Quantitative Anthropol matical psychology at the University of Michigan,
ogy for (1!scussions that stimulated illY thinking on including Clyde CoombS, Frank Harary, Louis Gutt
the general subject. man, James Lingoes, and Robert Hefner, provided an
G. P. Murdock was particularly helpful in pro important part of my training.
viding information on the Natchez case, Ira Buchler
provided the stimulus for the Kapauku analysis, Wil I In the twentieth century, modern mathemati
liam Geoghegan provided the solution to the Purum cians have shown that all mathematics is based upon
case and generously wen t over his field rna terial on axiomatic reasoning in which four basic concepts are
the Samai address system, and Robert Randall gra involved: (I) a specific.ation of primitive notions; (2)
ciously allowed me to use the American kin-terms a presentation of relevant definitions built up from
369
370 DOUG LAS R. WH ITE
Axiomatic reasoning is central not portant types of general anthropologi
only to mathema tics bu t also to the cal problems that are modeled.
growth and development of modern This review examines theoretical
sciences. It provides a logical skeleton contributions of mathematical anthro
upon which systems of explanation and pology in (l) processllaL analysis (2)
veri fica tion are constructed. A mod eI optimization analysis (3) structural
derive·d fl~om an axiomatil- theory con analysis (graph theory) and (4) ethno
tains a logical structure of ,quival nce graphic decomposition. Other areas of
between the et of axioms and the s t mathematization in anthropology, such
of consequences derived from the as data reduction through matrix and
axioms. An interpretation of such a statistical analysis, and inferential gen
model with respect to a body of data eralization through statistical analysis
allows logical explanation> such that if and inference will not be discussed
an axiom set is true, a consequence set her . These om issions are d u partly to
derived from it must also be. true (i.e., limitations of spa e and pm·tly to the
logjcal extension from known facts), fact that these ar as are higllly devel
and a1 0 allows in lire.ct fal ification oped and their anthropological applica
such that if tJle cons quence set con tions are therefore better known than
tains a false stat m nt, the. axiom set the Val·joLls topics I shall be discussing
cannot be true (i.e., indi.rect invalida (but see Burton 1970 and Romney,
tion of first premises). Axiomatic Shepard and Nerlove 1972 on new de
models differ from the conventional velopments in scaling theory).2
and looser use of the teflTl ' lnodel" in The four types of mathematical
social science research. The latter usage modeling that will be discussed) viewed
provides no formal system of explana in conjunction provide a framework
tion and falsification, since here the for explanatory theories of ~)ehavior in
term "lnoder' refers simply to an anal relation to anthropological data. Pro
ogous relationship between one phe cessuaJ analysis and optimization anal-
nomenon and another. ysis focus on the development of theo
In tllis review we shall be concerned lies and models of social process. The
with models in the narrower sense processual models (section 1) dea l with
models derived from mathematizecl
axiomatic theory. Rather than attempt
to trace the developments that encour 2 Previous reviews 3(ld readings on mathematical
anthropology can roughly be divided into six cate
aged the Ll e of .mathematicaJ reason gories with selected refert:nces. as follows:
ing in an thropology the present frarne
1-2. Processua\ and optimization ;;1nalysis (Buchler
of reference is designed to allow the and Selby 1968a, Buchler and Nutini 1969
resultant accomplishments to represent Buchler and Kozelka n.d.).
themselves and be judged largely on 3, StTU tural analysis (graph theory) (l3arnes
1969, Flarncnt 1963).
their own merits. The FraIne of reference
4, Algebraic or ethnographic decomposition
is organized in terms of severa] i111- (White 1963· Hoffmann L970 (gen"ral re
view article]; Kay, ed.. 1971 i also includes
items under 5 and 6)).
the primjtives; (3) a statement of axioms containing 5. Quantification statistics, probability (Mitch
only primitives and auxiliary definitions; and (4) log ell 1963. 1967; Hammel and Fwedman n.eI.;
icaJ derivation of tilcorems from t.he preceding ele Driver 1953,1961, 1965; D'Andrade 1959).
ments, stich th~t if the a~il)ms are tmc, the theorems 6. Data redu tion via matrix analysis and com
necessa.rily follow (for a rigorous anthropological ex pnter analysis (Bu rton 1970; Rornney, Shep
ample see Geoghegan 1971; for an earlier use of the ar{l, and Nerlove t972· Hymes. ed., 1965).
logicodeductive or postulational method, see Mur Only items I through 4 are jncluded in tile present
dock 1949). review.
MATHEMATICAL ANTHROPOLOGY 371
the limits and contingencies of social social forms. The need for a generative
systems, providing a quantitative and approach to social organization and
somewhat mechanistic model of a behav structure was suggested by Murdock
ioral system. The optimization models (1955) and others, and articulated 111
(section 2) add the element of goal greater detail by Barth (1966:2):3
directed behavior, broadening proces
sual analysis to include choice and Patterns are generated through processes of
learning behavior. Structural analysis interaction and in their form reflect the can·
and ethnographic decomposition begin straints and incentives under which people
act. I hold that this transformation from con·
with theories of modeling data by axi
straints and incentives to frequentative pat·
omatic methods. They search for pat
terns of behavior in a population is complex
tern, structure, and regularity in ethno~ but has a structure of its own, and that by an
graphic data, translating insight about
understanding of it we shall be able to explain
structural properties of anthropological numerous features of social form. Indeed ...
data into formal axiomatic structures. the processes which effect that transformation
Structural analysis through graph theo are our main field of study as social anthropol·
ry (section 3) is useful in developing ogists.
models of structural properties, and in
examining the fit between these theo In this section we shall examine the
retical principles and complex bodies of uses of mathematical models in the in
data about empirical structures. Eth vestigation of social processes, focusing
nographic decomposition (section 4) ap chiefly on the problems of relating the
plies primarily to the qualitative and principles of social behavior at the indi
sY111bolic areas of rule systems in cul vidual level (residence, marriage align
ture, including rules of appropriateness, ment, recruitment to social groups, ac
ordering rules, decision rules, transfor ceptance of innovation, etc.) to social
mation rules, and the like. Structural forms and frequencies of behavior at
analysis and ethnographic decomposi the group level. The social processes are
tion, as part of an integrated approach mediated by a large number of contin
to data analysis, provide a basis on gencies. Numbers of people in the sys-
which theoretical statements about
human behavior can be constructed, se 3 Murdock (J 955: 361), for example, stated:
"However useful, and indeed indispensable, is ihis
cure in both empirical adequacy and a
lask of claSSification, it is by no means the ultimate
seIt-consistent logical structure. Proces
goal of science. Any typological system is, by its
sual and optimization analysis provide a very nature, siatic in character. It takes on full
more general and theoretical account of meaning only when scientists are able to demon·
sociocultural and behavioral systems. strate the dynamic processes which give rio e to the
phenomena thus classified. The Linnaean system, for
example, came alive only afler Darwin had discerned
rhe processes of variation and natural selection, and
I. PROCESSUAL ANALYSIS
especially after the geneticists had laid bare the dy·
Social anthropology is concerned namic mechanisms of heredity."
with explaining similarity and diversity Simila.rly, Barth (1966:2) stated: "Explanation
(of social formsl is not achieved by a description
of forms of social life. The focus on
of patterns of regularity, no matter how meticulous
explanation in anthropology, which and adequate, liar by replacing this description by
builds lipan and supersedes description other abstractions congruent with it, but by exhibit·
and classification, represents a shift ing what makes the pat tern, i.e., certain prOcesses.
To study social forms, it is certainly necessa.ry but
from the study of forms of soeiallife to
hardly sufficient to be able to describe them. To
th investigation of the generative rela
give an explanation of social forms, it is suffiCient to
tionship between social processes and describe the prOcesses that generate the form."'
372 DOlJGLAS R. WHITE
tern (i.e., delnographic factors), distri difference equa tions). Compu tel' simula
bution of resources (e.g.) ecological and tions provide a mathernatical model
economic factors), and other contin through mllnerical computation when a
gencies mitigate the effect of cultural generat mathematical so1ution to a prob
rules or indiv id ual lnotives and stra te len) is not already worked out. Simula-
gies. They constitute the fnHnework or tions may be either probabilistic or de
ground rutes within which a social sys terministic. A decision tree may also
tem operates (Buchler and Selby be used as a processual model if a pop
1968a). The study of social process can ulation (defi.ned in tern1S of variable
be viewed from the perspective of sta situations of individuals within it) is
tistical factors in demography) resource "mapped" by the tree into a set of out
distribution, situations, etc.> or [raIn conles. Tills model Inay again be either
the perspective of the interaction be cletenninlstic or pro ba bi.listic, depend
tween statistical factors and cultural or ing on whether the outcomes are certain
jural rules. or probable.
Until recently there was no foun In cOIllparison to a body of ethno
dation in anthropology for a systematic graphic data, axiomatic mathematical
investigation of vaTious types of social theories and derived models provide a
process. Earlier processual accounts much greater number of theoretical
for example, of marriage systems-"had possibilities than are contailled in a
few theoretically determined base lines finite bocly of data. The data used to
from whiCh to draw conclusions regard test various models may be statistical
ing the differential effect of ecolog distributions or observations over time.
icat~ demographic, and cultural varia The nlodels may be tested at those
tions on social organization~' (Gilbert points where data are available, but
and Han1mel 1966:72). Recently, they may also help clarify other areas
Inathematical Inodels, including com." of the theoretical problem for future
pl1ter SDnulatlol1, have provided the research. They may serve as a guide to
needed framework for the analysis of the collect jon of new data needed to
interactions between c01l1plex sets of test and refine the theory or model.
variables. These models have been This provides the research with a para
applied to problems such as residential digmatic quality (Kuhn 1962), in that
patterns, marriage choice, recruitment past, present, and future data can be
patterns, and diffusion of infofl11ation related to a. formal model or theory,
as wen as material artefacts. Formal and findings can feed back to mod ify
theories pertinen t to these phenolnena the general theoretica1 framework.
aJlow models to be derived which can In the choice of a model for a theo
be compared with actual ethnographic ry, the distinction between pro babilis
data. The criterion of a well-formed tic and deterministic models is iJnpor
theory in such cases is the extent to tant and should be understood. Deter
wltich a logical stnh"ture of explanation ministic systems, exemplified by New
of the process is provided, and how ton's laws, aSSUlne that the entire
well the mode} of the theory can ma tch future of the system can be predicted
actual cia tao given sufficient infonnation about the
Mathematical models that may be past. Probahilistlc systems, such as sta
utilized in pJocessual analysis include tistical mechanics, can predict only the
the probabilistic (e.g., stochastic or probabilities of certain future occur
time-dependent statistical processes) rences (Kemeny and Snell 1962: 5),
and the detenninistic (e.g., linear or usually becoming tess accurate as the
MATHFMATICAL ANTHROPOLOGY 373
time span lengthens. Probabilistic mod abilities. Finite state probability mod
els arc suited to a host of contingency els have only recently begun to be used
factors that play important parts in so in anthrop.ology but th y havc proved
cial processes. De term inistic models useful in relation to questions of change
come into play either as approximations and stability, and in the analysis of un
of the probability model (e.g., exact cal intended consequences of social pro
culation of the most likely or expected cesses.
outcome) or else as an approximate Buchler and Selby (l968a:58-68)
representation of a set of cultural rules. have suggcsted that studies of devel
It is highly unlikely that the social sci opmental cycles and change in residen
ences will ever approach deterministic tial patterns can benefit from stochastic
laws in the Newtonian sense of strict modeling. Residential patterns were
prediction. As we shall see, deter one of the first ethnographiL: phenom
ministic models that do take social con ena to be modeled processually (see
tingencies into accollnt and may pre Fortes 1945, 1949). Fortes' work stim
dict behavior very accurately at a given ulated others (e.g., Mitchell 1956,
point in time have been developed; Smith 1956) to study resid ntial pat
these models will not, however, explain terns developmentally in terms of
change in the system- that is, they do ho useho Id compositio n. The emphasis
not predict well over time. The closer on process also led to new conceptual
we come to an exact model of the interests in decision pro cesses under
structure of social behavior, the less we lying residential choice (Goodenough
m'e able to ac 'ollnt for change, and the 1956b).
more \VC are able to deal with change A major problem in the devel
(e.g., through probabilities) the less opmental studies of residence and of
exact our knowledge of the structure. household affiliation and composition
There is thus no possibility (or human (e.g., Goocly,ed., 1958,HammeI1961,
istic "danger"") of reducing human be Fortes 1962, and Miller 1964 for the
havior to a set of mathematical pre former; Fisch r 1958, Barnes 1960, and
dictions, although mathematics may Romney n.d. for the latter) was in
help to make both process and struc ga uging the eff 'cts of vital ra tes, repro
ture understandable as aspects of ductive ages, segmentation processes,
human behavior. A review of anthro and so forth on household composi
pological uSeS of proc ssual mathe tion. A stochastic model can, in effect,
matical models may show the potential ha ndle all of these factors sirl'l ul
complementarity of probabilistic and taneously by setting appropriate transi
deterministic models. tion probabilities for household compo
sition types over time. Buchler and
Selby (1968a :63-66) show how such a
A. PROBABILISTIC MODELS
model could be constructed using the
Stocha~tic 01' probabilistic approxi example of Freeman's (1958) data on
mation models represent one of the the Iban bilek famjly system. If the
modern developments in statistics, ill various demographic factors are rela
the statistical modeling of empirical tively constant over time, it is possible
phenomena. In a processual model, or to assume that the transition probabili
stochastic process, a phenomenon is ties representing the normal frequencies
represented as a set of tates that are of maintenance and change in house
maintained over time or altered in ac hold types are also constan t. A stochas
cordance with a set of transition protr tic process with stable transition prob-
374 DOUGLAS R. WHITE
abilities, called a Markov process, has and has disrupted this cycle: young
an important Inathen1atical property: it men and women move off the island
can be proved that the expected fre for wage labor, al1d often send their
quencies in successive states of the sys children back to be cared for by grand
tem, as cornputed by the transition parents. In comparing two sllccessive
probabilities, approach a limiting set of censuses (1961 and 1968), Otterbein
values or an equilibrium vector. Regard found that the actual patterns of main
less of the initial set of frequencies of tenance or change in household com
the systen1, the same limiting vector position differed considerably from the
will be reached for a given matrix of normative developmental cycle in the
transition probabilities. I shall attempt past. The types and nun1bers of changes
to illustrate a Inathematical model of can be indicated in a finite-state dia
such a system using Otterbein's (1970) gram (Figure 2). The numbers to the
data on J"esidence changes on Andros rjght of the diagram indicate formation
Island. of new families and dissolution of old
Otterbein (1970: 1414) has classified families (new/old).
Andros Island households into nine In spite of the small numbers in this
types. These have been relabeled in sample, the frequencies of stability and
Figure 1 to show basic family types and change give a fairly accurate indication
their composition in tenns of principal of the relative frequency or probability
adult menlbers. The anows on the dia of the transition processes during the
gram indicate the traditional develop seven-year period. The transit jon prob
mental cycle from a married couple ability matrix can be computed for
(Z2) to a nuclear fan1i1y (X2) to an these data to show transitions for the
extended family (Y2), followed by a three types of families (nuclear, extend-
decline of the faInily; typically the ed, adult) as well as for the category
grandfather dies first (leaving type Y D, which stands for the demographic
1 ),
and finally the children and grand processes of formation of new house
children move away> leaving the grand holds from the delnographic pool of
mother alone (Zl ). Migration from the Unn1alTied adults, etc. (D!), and dissolu
island, however, has been increasing, tion of households through death or
Basic Family Types
Principal Nuclear Extended
Adult (with (with Adults Demogl'aphic
f11embers Children) Grandchildren) Only Factors
Dissolution
Female XI YI > 2\ )- by death or
1 migration
Couple X2 ) Y2 ~Z2( Formation of
~ new famiHes
Male X3 Y Z3
3
Figure 1. Normal developmental cycle among household types in traditional Andros society.
MATHEMATICAL ANTHROPOLOGY 375
Frequency data: Numbers of households of To: later time period
each type. X Y Z Dd
XI X2 X3 YI YZ Y3 ZI Z2 Z3 Ftirmoem :p eeraiorldie r XY 00..50 00..36 00..20 00..22
1961 4 25 I 9 15 2 II 8 10 Z 0.0 0.1 0.4 0.5
1968 4 17 a II 20 I 7 5 8 Df 0.5 0.5 0.0 0.0
Diagram: Transitions for the nine household We sec from the matrix, for example,
types, including demographic factors of for
that there is an 0.5 probability that a
ma tion and dissolution of households.
type X household in 1961 will remain
so in 1968, an 0.3 probability that it
DIIDd
x will change to type Y, 0.0 to type Z,
Y Z (formation/dissolution)
and 0.2 that it will disappear. If these
G -2-0
transition probabilities are stable, the
_2_Y 2 3/1 4/3 1/7
1 I 3/; I ·2- I same will be true for any seven-year
"0/,q:::l;). period in the future.
3/4 1/1 0/1 Given the matrix, the operation of
11 '2,,!
the stochastic (Markov) process on a
I -1-'E)4
X3 l3 0/1 J/O 0/6 population for successive time intervals
can be illustrated_ We will start with a
hypothetical population of eighty
household s in 1961, plus a pool of
Reduc_e_d_ _st_a t1e- -d..i.a.g,.r.a m: potential householders (category D,
which also in this hypothetical case
1s Ct-8--cY:;::~~-?' 6/6 5/4 1/14 represents the number of out-migrants),
16 12
distributed to correspond to the census
data figures fo l' 1961 for the household
types X, Y, and Z. We shall assume that
Reduced state matrix
the numbers for the various types of
(corresponding to reduced stale diagram:
families in 1961 are X = 28, Y = 25, Z '"
27, and that in category D there are 20
x
Y Z Dd potential households. The expected val
ues for household distributions for
X 15 8 I 6
a 1968 can then be computed, for exam
Y 16 6 4
Z a 3 12 14 ple:
a
Df 6 5 1
X(l968) = O.5X( 19(1) + 0.OY(l961)
Figure 2. Construction of transition matrix + O.OZ( 1961) + O.SD( 1961)
= 0.5 x 28 + D.5 x 20
from Andros household censuses for 1961
= 14 + 10" 24
and 1968.
The equilibrium vector can be com
migration (Dd)_ The approximate tran puted by solving a set of linear equa
sition probabilities, computed from the tions where X, Y, Z, and D have the
reduced state matrix in Figure 2, are as same value before and after transitional
follows: probabilities are computed. The values
376 DOUGLAS R. WHITE
at successive time intervals can then be same cycle or set of grades as their
compared with the final equilibrium fathers (Hoffmann 1965, 1971). In the
values: Shoa Galla system, for example, in
which the age-grade cycle is forty
1961 1968 1975 1982 ... Equilibrium years long (five grades at eight-year
x 28 24 24 22 20 intervals), stability would represent a
y 25 36 42 44 45 problem if the age span between father
Z 27 16 14 14 ]5 and son were much less or much greater
D 20 24 20 20 20 than forty years.4 Hoffmann's (1971)
assumption that a Markov pro cess
model is applicable to this case is q ues
The equilibriu111 point (20, 45) 15)
tionable, because the transition prob
20) will be the same regardless of the
abilities cannot be assumed to approx
initial frequencies. In this case, the
imate stability. On the contrary, the
initial frequencies, after two intervals, likelihood of a son's entering his fa
lead to expected values that approach
ther's grade at the same age as his father
within 4 percent of the equilibrium
did obviously depends on the age of the
vector and are in three intervals within
father when the son was born.
2 per~ent of equilibrium values. Thus,
Unless the conditions for the use of
whereas the past domestic cycle on
stochastic or Markov models are ful
Andros Island may have involved a low
filled (i.e., satisfaction of the assump
percentage of extended families (e.g.,
tions of the axioms of the model and
about 20 percent), the new conditions
nonarbitrary detennination of transi
would stabilize within fifteen years,
tion probabilities), the use of these
when the frequency of extended falni
models simply replicates initial faulty
lies approaches 45 percent. By the use
assumptions without providing better
of the stochastic model, then, aspects
understanding of the mechanisms in
of processes of stability and change can
volved. The decision to use a stochastic
be evaluated more precisely. As Buchler
or Markov process model should be
and Selby (19680.:67) state: based upon the plausibility of assunip
tions that can be made about the phe
This perspective opens up a variety of nomenon being modeled, and, as far as
approaches and poses a series of questions its predictive adequacy is concerned,
relevant to pro cess and change. Consider one on the extent to which (1) the actual
example: residence rules. We may sa~ that .a
given society is «matrilocal," typo!ogtze~esl­
dential processes (Fischer 1958), or conSlder
the decision processes underlying residential 4 This is an example of the more widespread
choice (Goodenough 1956b). But . .. it is of Gada, or cycling, system i.n North Africa (Murdock
considerable importance to construct and 1959: 326). Entrance into the system is crucial for
examine a transition rna trix in order to deter~ adult status, so stability of the system is a rea1
mll1e the stability of the . . . tendency, to question.
compute a limiting vector that may well indi~ Hoffmann (1965, 1971) shows that stability of
the system can be expressed in an equation, where
cate a drift .... Here, then, is a new way of
Dn is defined as the difference in age at entry of a
coming at the problem of process.
descendant "11" generations hence, the length of the
cycle is denoted by "k" (in this case, forty years),
and the average reproductive age by "P." Then Dn =
Equilibrium conditions in a Markov
n· k- ~ p .. If P is greater them 40, D n is negative; if
process model have also been used to i I .
analyze age-grade systems of the type P js less than 40, Dn is positive. The system Will be
stable only when Dr! = 0, which occurs when P is
in which sons must go through the equal to 40.
MATHEMATICAL ANTHROPOLOGY 377
transition probabilities can be logically mel developed a formal mathematical
derived or estimated from known data model of the probability of cousin mar
and (2) the model can be tested against riages as a function of the probability
time-sequence data for goodness of fit. of endogamous marriage, and they were
Marriage choice is another area of able to compare the results of the two
traditional anthropological interest that models.5 The close fit between the
has only recently been studied proces formal mathematical model and the
sually. Models of marriage choice phe computer simulation tended to confirm
nomena within cultural, social, and the conceptual validity of each.
demographic constraints are so com Models of the type used by Gilbert
plex as to require mathematical treat and Hammel can be adapted to similar
ment. This represents the first major studies in other societies. Variables
area in anthropology in which com such as territorial marriage preferences,
puter simulation has been used (Kun residence rules, and demographic fac
stadter et al. 1963; Coult and Hammel tors can be changed without affecting
1963; Randolph and CouIt 1968; Gil the basic structure of the problem, so
bert and Hammel 1965, 1966), al that their effects upon expected mar
though later refinements have made use riage frequencies may be observed.
of stochastic processes as well (Morgan Through the use of experimental mod
1969, 1970). In general, the operation els (probabilistic equations and simu
and consequences of a cultural rule or lations) it is possible to examine rela
marriage preference is studied by con tionships between variables, and to
structing a model of a population of a evaluate the kinds of empirical data
given size, with specified vital rates and that would be needed to test or falsify
probabilities of choice within a range of the hypotheses involved.
social categories. Computer simulation was used by
For example, in exploring Ayoub's Kunstadter et al. (1963) for analysis of
(1959) view that patrilateral parallel the effect of demographic constraints
cousin marriage was an "epiphenom on an ideal pattern of marriage: pref
enon of a general tendency toward erential matrilateral cross-cousin mar
kin-group endogamy," Gilbert and riage. In construction of the demo
Hammel (1966:72) took advantage of graphic model, the factors allowed to
the experimental potential of the com~ vary were population size and marriage
puter by simulating a territorially sub probability by age (which in turn affect
divided population with specific mar ed average age at marriage and at
riage rules (e.g., incest prohibition, ter
ritorial preference for mates), residence
rules (patri-virilocal), and demographic that5 TEhgeo mwatihlle mmataicrrayl eqFuBaDti onis :f orP rth(FeB pDro) ba=bi l(iPtyr
factors (average reproductive age, age
(ENDOG) X Pr(FBDIENDOG)) + (Pr(EXOG) X
matching in . marriage, etc.). They Pr(FBDIEXOG)). [n ordinary language, the probabil·
hypothesized that the resultant pattern ity that a man will marry his father's brother's
of patriJateral parallel cousin marriage daughter is equal to the sum of the probability of his
choosing her from among the marriageable girls
could be accounted for "as an epiphe
within the village (the probability of endogamy
nomenon of territorial endogamy
times the probability of his marrying his father's
alone, without necessarily specifying brother's daughter if he marries within the village)
kinship-phrased preferen ce." plus the probability of his choosing her from among
After logical clarification of the kin the marriageable girls outside the village (the proba·
bility of exogamy times the probability of his mar·
ship preference problem was obtained
rying his fa ther's brother's daughter if he marries
through simulation, Gilbert and Ham- outside the village) (Gilbert and Hammel 1966:80).
378 DOUGLAS R. WHITE
chlldbirth, average number of children, generating process is known as Monte
and the growth and size of the popula Carlo simulation. In simulating small
tion). The results showed the impossibil populations, this process introduces a
ity of prescriptive marriage of this type realistic fluctuation in population size
in relatively small populations. The over time, and successive runs of the
maximum frequency of the marriage program can be used to establish gener
type in populations of about a hundred al trends as well as the probabilities of
persons is generally between 15 and 24 a society's being reduced to extinction.
percent for the simulation model, and By the use of this baseline rnodel,
generally between 31 and 45 percent overlay modifications in tenns of differ
for initial popUlations of two to three ent marriage rules can be introduced
hundred which are ex panding. The au into the program to determine their
thors note that in societies with larger effect upon population growth or de
popUlations and with a nlatrilateral cline. By this method, Morgan (1970)
cross-cousin marriage preference, it is found that for a model where popUla
often a pro bleJn to explain why the tion tends to slight natural increase
actual frequency of this marriage type without specified marriage rules (ran
is lower than might be expected from d01TI mating), the introduction of incest
their model. prohibitions (without consideration of
Baseline models and overlay models deleterious in breed ing effects) tended
are effectively utilized to study in ter slightly to auglnent population growth,
acting effects in complex systems while the effects of clan endogamy
(Coleman 1964:519-22). The two tended to lead to dying out of a popu
examples of simulation of marriage lation in about four hundred years,
choice illustrate a means by which com because of restrictions in the availabil
plex phenomena can be broken down ity of mates.6
into simpler subsystems by the use of The use of a stochastic process in
baseline (i.e., given demographic condi Monte Carlo simulation to estimate the
tions) and overlay (i.e., tnarriage prefer chances that a society (or household
ence) models. More sophjsticated base type or age grade) will die out under
line models can be constructed in certain conditions is one of the impor
which variance jn the baseline D10del is tant contributions of stochastic models
siInulatecl to help understand the effect (Coleman 1964:527-28). Gilbert and
upon the overlay phenomenon (e.g., Hammel (1966:87), in their model,
marriage ru les). noted that Monte Carlo sin1ulation "a 1-
Morgan (1969) uses a model of dem lows one to speak of the dispersion of
ographic constraints, including differ the results." The suitability of sto
ent marriage rules (Morgan 1970), to chastic models for analysis of "survival
study the survival possi bilities of small, chances" of the components of social
closed populations. He uses a demo
graphic baseline model where the vital 6 The conditions for the baseline population
rules (e.g., birth, death) are set by a mode} were found to be stable in the earlier simUla
stochastic process. A computer simula tions (Morgan 1969), and correspond to low-fertil·
ity, low-mortality demographic rates. Thus the re
tion utilizes the sto chastic pro cess in
sults of this simulation are not necessarily general
such a way that the population under izable to other conditions. High-fertility, high-mor
goes successive demographic transitions tality rates in smalJ popUlations were found to
governed by random fluctuation in the exhibit such a degree of random fluctuation that they
tended to become extjnct in the absence of any
vital rates as detennined by the transi
marriage rules after an average span of about four
tion probabilities. The use of a random hundred years.