aa(2003) a neutral model of stone raw material procurement_brantingham.pdf
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A
NEUTRAL MODEL OF STONE
RAW
MATERIAL PROCUREMENT
P.
Jeffrey Brantingham
Stone tool
assemblage
variability
is
considered a reliable
proxy
measure
of adaptive variability.
Raw material
richness,
transport
distances,
and the character
of
transported
echnologies
are
thought
to
signal
(I)
variation in raw material selec-
tivity
based on
material
quality
and
abundance,
(2)
optimization of
time and
energy
costs associated with
procurementof
stone
rom spatially dispersed
sources,
(3)
planning depth
that
weaves raw material
procurementforays ntoforaging
activ-
ities,
and
(4)
risk
minimization
that sees
materials
transported
in
quantities
and
forms
that are
energetically
economical
and least
likely
to
fail.
This
paper dispenses
with
assumptions
that raw material
type
and abundance
play any
role in the
organization
of
mobility
and raw material
procurement trategies.
Rather,
a
behaviorally
neutral
agent-based
model is devel-
oped involving
a
forager engaged
in a
randomwalk within
a
uniform
environment.
Raw material
procurement
n the
model
is
dependent only upon
randomencounters
with
stone sources and the amount
of
available
space
in the mobile toolkit.
Sim-
ulated
richness-sample
size
relationships,
requencies
of
raw material
transfers
as
a
function of
distance
from
source,
and
both
quantity-distance
and reduction
ntensity-distancerelationships
are
qualitatively
similar to
commonly
observedarchae-
ological patterns.
In some
archaeological
cases it
may
be
difficult
to
reject
the
neutral model.
At
best,
failure
to
reject
the
neutral model
may
mean
that
interveningprocesses
(e.g., depositional time-averaging)
have erased
high-frequency
adap-
tive
signals
in the
data. At
worst,
we
may
have to admit
the
possibility
that
Paleolithic behavioral
adaptations
were some-
times not
responsive
to
differences
between
stone
raw material
types
in the
ways
implied by
current
archaeological theory.
Se considera
la
variabilidad
de las
colecciones
de litica como una medida
confiable
de
la
variabilidadde las
adaptaciones
al
medio ambiente.La diversidad
de materias
primas,
la
distancia
a sus
yacimientosy
la
tecnologia empleada reflejarian
1)
la
selecci6n de la materia
prima
con
base en
su calidad
y
abundancia,
2)
la
optimizacion
de
gastos
de
tiempoy energia
emplea-
dos
en la obtencion
de materia litica de
yacimientos dispersos;
(3)
la
integracion
anticipada
de las visitas a los
yacimientos
con actividades de
caza
y
recoleccion,
y
(4)
una
estrategia
de
reduccion
de
riesgos
que
consiste en
transportar
a
litica
en
las
cantidades
yformas
mas
eficientes
del
punto
de vista
energetico
y
menos
susceptibles
al
desgaste.
En este
trabajoprescindi-
mos
de
suponer que
el
tipo y
la abundancia de materia
prima hayan ugado
un
papel
en la
organizacion
de
estrategias
de
obtencion de
esta
y
de los
desplazamientos
en
general.
En su
lugar,partimos
de un
modelo conductualmenteneutral
basado
en
el
agente
(individuo),
como seria
un
cazador
que
se
desplaza
al
azar
en un medio
ambiente
uniforme.
La
obtencionde
mate-
ria
prima depende
entonces
unicamente
de
hallazgosfortuitos
de
yacimientos
de
litica
y
la
cantidadde
material
que
el
cazador
pueda agregar
a su
equipaje.
La simulaci6n de la relaci6n entre la diversidad
y
el
tamano
de la
muestra,
asi
como de la
fre-
cuencia de
uso,
la cantidad
y
la
reduccion
del volumen
enfuncion
de la distancia
al
yacimiento,
revelan
patrones que
se
ase-
mejan,
de manera
cualitativa,
a los
que
arroja
a menudoel
registroarqueologico.
En
algunos
casos
arqueol6gicos
resultaria
entonces
dificil
descartar al modelo.
En el
mejor
de los casos
la
imposibilidad
de
descartarlo
senialarfa
que
los
procesospost-
deposicion
(p.ej.
la
combinaci6nde
artefactos
de distintas
epocas
en una sola
coleccion)
han
borrado todos los
indicadores
de
la
adaptacion.
En el
peor
de los
casos,
nos
veriamos
obligados
a reconocer
que
las
adaptaciones
del
comportamiento ale-
olftico a veces no obedecian a las
diferencias
entre los
tipos
de materias
primas
de la litica de la manera
que sugieren
las
teorias
vigentes
en
arqueologia.
It is
easy
to
invent a selectionist
explanation
he
richness of stone raw material
types
in
for
almost
any
specific
observation;
proving
it
an
archaeological
assemblage, the
geo-
is another
story.
Such
facile
explanatorye geo-
excesses can be avoided
by being
more
quan-
graphic
distances over which those
materi-
titative.
als were
transported,
nd the
technological
orms
MotooKimura
1983:xiv)
in
which
they
were
transported
have
provided
Motoo
Kimura
1983:xiv)
The Neutral
Theory
of
Molecular Evolution
empirical benchmarks for inferring the organiza-
tion of
Paleolithic
adaptive strategies
(Feblot-
P.
Jeffrey Brantingham
*
Department
of
Anthropology,
University
of
California,
Los
Angeles,
341 Haines
Hall,
Box
951553,
Los
Angeles,
CA
90095-1553.
Email:
American
Antiquity,
68(3),
2003,
pp.
487-509
CopyrightO
2003
by
the
Society
for American
Archaeology
487
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AMERICAN
ANTIQUITY
a
XII
XI
IX
._
C
C
5
E
u
10
b
A
A
A
*
8
Y4
A,r
/
A
2
r2
=
.8397
VII V
III
I
10 20
50
100
200
500 1000
3000
Log Sample
Size
(n)
tratigraphic
evel
0.45
0.40
-
0.35
-
0.30
-
0.25
-
0.20
-
0.15
-
0.10
-
0.05
-
0.00
-
MP2 MP
I
MP3
Quartz
Indet
MP6 MP4
MP7
Raw
Material
Type
Figure
1.
Raw material
richness
in
assemblages
from
the Middle Paleolithic
site of
Grotte
Vaufrey, approximately
204,000-74,000
B.P.
Richness
changes
through
time
(a)
but is
heavily
dependent upon
sample
size
(b).
Procurement
prob-
abilities
estimated
from the observed
raw material
proportions
(c)
illustrate
the
structure of
the distribution
underlying
procurement
behaviors. The
primary
question
is whether
this
probability
distribution
is derived from
the environmen-
tal densities
of different raw
materials,
or whether
a
biased behavioral
strategy
is
responsible.
Data from
Geneste
(1988)
with
revisions.
Augustins
1993, 1997a, 1997b,
1997c;
Gamble
1986,1999;
Geneste
1988,1989;
Kuhn
1995;
Mel-
lars
1996;
Morala and
Turq
1990;
Potts
1994;
Rensink t al.
1991).
Such nferences
make
appeals
to
optimization
of
mobility
and
technological
strategies,
depth
of
planning
n
landscape
use,
and
risk
minimization.
One
family
of models focuses
on differences
between
generalist
and
specialist
strategies
of raw
materialutilizationas inferred rom the richness
(i.e.,
number
f
types)
of stone
rawmaterials ound
in
archaeological
ssemblages:
generalists xploit
many
different
raw material
ypes,
while
special-
ists
exploit
only
a few
types.
To
complicate
mat-
ters,
observed
assemblage
ichness s
frequently-
if
not
universally-constrained
by
sample
size
(Grayson
1984;
Hayek
and Buzas
1997;
Shott
1989).
Consider,
for
example,
the
Middle
Pale-
olithicsite of
GrotteVaufrey
n
the
Aquitaine
Basin
of
France,
which
spans
the time
interval from
204,000-74,000
B.P.
(Geneste
1988,
1989).
Despite
the
appearance
f
changes
n raw material
richness
through
ime,
a
single sample
size-rich-
ness
relationship
s
apparent
tthis site
(Figure
a,
b) (Geneste 1988).
Changes
n
stone
raw material
richness,
rom
only
two
unique
raw material
ypes
to
as
many
as
nine,
do
not
necessarily
diagnose
switching
between
specialist
and
generalist
trate-
10
.
8
c
6
05
4
ct
2
0
488
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AMERICAN
ANTIQUITY
prise
the s
from
1992
refle
offsi
rials
geog
1993
low
tern,
to be
to
A
20.
?I [I
o :.A
A
, IA
B
1.-1.
ii* I
*
0 50 100 150
200 250
5 35
65
95
125
155
185
215 245
275
DistanceFromSource km)
DistanceFromSource
km)
Figure
3.
Percentage
frequencies
of
specific
stone raw
material
types
in
Middle Paleolithic
assemblages
from
b Central Europe as a function of distance from source. A
[~~~~~A n
=
172
general
contrast between
(A)
high
quantities
of
minimally
reduced
materials
from
nearby
sources and
(B)
low
quan-
tities
of
heavily
reduced materials
from distant sources
is
thought
to
reflect
time-energy
optimization,
depth
of
plan-
ning
in
raw
material
acquisition,
and
formal
technological
strategies
that
minimize
the
risk of
technological
failure.
Data
from
F6blot-Augustins
(1993).
B?
technological
orms-representing
the
end
stages
|||||i
c~
~of
ithic reduction
equences
as
opposed
to
early-
I||L.|
C
stage unmodified blocks or minimally utilized
o
40
8o0
120
160 200
240 280
cores-is
thought
to
signal depth
of
planning
and
0 40
80 120
160 200
240 280
formalrisk
management trategies
Geneste
1989).
DistanceFromSource
km)
In
particular,
echnologies
are
expected
to be
for-
mally designed
from the outset to be
small,
light
e
2.
Frequency istograms
f raw material ransfer
may designed r
to
flure
i he
to
gh
icesfor Middle
Paleolithic
ssemblages
n
(a)
Central
we
and resstant o lure
are to
be
pe
and
(b)
the
Aquitaine
Basin.
High
frequencies
of
transported
over
long
distances
(Beck
et al.
2002;
-distance ransfers
A)
are
interpreted
n
terms of
Kuhn
1994;
Nelson
1991;Torrence
1989).Accord-
lization
of time
and
energy
costs associated
with raw
ing
to
Geneste
1989:80),
these
archaeological at-
rial
procurement.
nternalmodes
B)
are
interpreted
ims
of
logistical
or
seasonal
use of distant
ecological
terns
diagnose
real
behavioral
adaptations
and
les.
Raw material transfers
from
very
distant sources
dynamic
economic
strategies.rethought o mark hemaximum xtentof thegeo-
Here I would like to
introduce thepossibility that
ic
range
of the
populations
in
question.
Data
from
}t-Augustins
1993).
much
of the variation in the
representation
of
stone
raw
materials,
both
within and between lithic
d
of
materials
rom
sources within 15
km
of
assemblages,
ould be
of
no functional r
adaptive
,ite,
while .7
percent
(n
specimens
=
12)
are
significance
whatsoever.
Ultimately,
our
ability
to
i a
source 230
km
away
(Feblot-Augustins
determine
whether
daptive ariability
s
(or
s
not)
):Table
4).
This
general
pattern
s
thought
o measured
y
lithic
assemblagevariability
s
depen-
ct
optimization
of the
time and
energy
trade- dent
upon
having
a
null
modelfor
what
assemblage
inherent
n
the
procurement
f stone
rawmate-
variability
hould ook
like under
ompletely
neu-
from
geographically
adjacent
vs. tral
assumptions.
With
this
goal
in
mind,
I
develop
Yraphically
istant
sources
(Feblot-Augustins
an
agent-based
model
involving
a
forager ngaged
5:220;
Gamble
1999:88;
ee
Metcalfe
and
Bar- in
a
random
walk
in
a
uniform
ood-resource
nvi-
1992).
As
an
extension
of this
decay-likepat-
ronment.
Different
tone
raw
material
ypes
occur
the
tendency
or
materials
rom
distant
ources at
equal
densities,
but
are distributed
andomly
n
introduced
o
sites
as
small,
heavily
reduced the
environment. imulated
raw
material
procure-
50
'A
0
oD
D
o
A
-I
E
0
=3
E
z
c)
o
CZ
q
E
Figur
distai
Eurol
short
optim
mater
in
ter
patch
(C) al
graph
Feblo
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AMERICAN ANTIQUITY
the
foraging
environment
s uniformwith
respect
to food
resources
i.e.,
it is a
single-resourcemega-
patch);
(2)
stone
raw materials occur as
point
sources
distributed
andomly
within the environ-
mentandeachpointsource s arbitrarilyabeledas
a
unique
raw
material
ype;
(3)
foragers
ollow a
"random
walk"
oragingpath;
4)
each
forager
has
a "mobile
oolkit"of fixed size
(i.e.,
a
forager
can
carry
a maximum
amountof stone
material); 5)
if
a material ource
is
encountered,
aw material s
collected
contingent upon
the amount of
empty
space
there s
in the
toolkit;
and
(6)
if the mobile
toolkit
containsraw
material,
hen a fixed amount
is selected
at randomwith
respect
o raw material
type, consumedand discardedfrom the mobile
toolkit.
Assumptions
1 and
3
ensure
mobility-neutral
foragingpatterns.
A
random
walk
through
a uni-
formenvironment
s neither
ogistical
norresiden-
tial
(sensu
Binford
1980).
More
technically,
a
random
walk does not seek
to
optimize
any spe-
cific
currency
ssociatedwith
movement,
nvolves
no
depth
of
planning
n thatall move
directions re
equally
likely,
and is risk insensitive
in that the
resultsof
previous
move decisions
have no
impact
on the
probability
of
the next move direction.
Assumption
2
ensures
ource-neutrality
n
thatall
raw
material ources
occur at
equal
densities
and
are
randomly
distributed
n the environment.
n
other
words,
here
areno abundance iases
or non-
random
patial
clustersof
individual aw material
types.
Moreover,
he random-walk
oraging
strat-
egy
employed
by
the
forager
nsures
hateach raw
material ource
has,
n the
imit,
a
probability qual
to one
of
being
encountered
nd thateach
unique
source
ultimately
will
be
encounteredan
equal
number f times.
Assumptions
and6
are
unit
raw
material-neutral.
ssumption
ensures
hatall raw
materials avean
equal
probability
f
procurement
whenencountered. ince
all raw
materials re
alike,
save
for an
arbitrary
abel,
they
are
necessarily
ol-
lected
if there s
empty space
in the mobile toolkit
(assumption
).
Once
rawmaterials avebeen
pro-
cured from
a
source,
assumption
6
ensures that
each unit
of material
n the toolkit has a
probabil-
ity
of
being
consumed and discarded
dependent
only
upon
ts
relative
requency
n the toolkit.
Technical
Meanderings
The simulation
s basedon
the RePast
agent-based
modelingplatform http://repast.sourceforge.net).
The simulated"world" onsists of a
two-dimen-
sional
grid
(500
x
500
cells)
that
holds
all
of the
rawmaterial ourcesand a
single foragerengaged
in a randomwalkthrough he environment.Each
grid
cell is assumed o containa
uniform,
nfinite
food
supply, nsuring
hat hereareno
patch
choice
decisions to be made
in
forager
movement
(see
below).
The world can containa variablenumber
of raw material ources
(Figure
4).
In
most simu-
lations,
heworld s seededwith
5,000
point
sources
of stone raw materialand each of these sources s
arbitrarily
ssigned
a
unique ype
label
i
=
1, 2,..
.5,000.
The
coordinate ocationof each raw mate-
rial sourceon thegridis chosen at random rom a
uniform distribution
(x,
y)
=
[1,
500],
without
replacement.
There are
250,000
cells that could
potentially
hold a
unique
raw material
ource.
The
probability
hat
any
one
cell contains stone raw
material s
approximately
02
(i.e.,
5,000
sources
250,000
cells)
and,
since each
point
source
repre-
sents an
arbitrarily nique
raw material
ype,
the
probability
hat
any
one cell containsa
specific
raw
material
ype
is
approximately
.0
x
106
(i.e.,
1
type
/
250,000 cells).Alternatively,
hese numbers
may
be
thought
of as the environmental
ensities
of all stone raw materialsand
specific
raw mater-
ial
types, respectively.
100
* *
**6
-
***.?
* f0:
;:* :0
:
. :: f : ::-
: : :*
:
:
:
90 o-
*
80
6
*80
.
*:
*
*-
::
:0
**
66
60
50
,0 :0
10 *
011
0
,0
0
0?
?
?
0
0 0 00
0
80
90 100
3lation wod
(l
se00
x
00
cells)
*
*showi the rand
10
.
.
;
mximm
=
85
0
:*
:
* :: 0X:: -
::
:
: : *: *:
?:: *; *
0 1
6
3046
50 60
76
80
90 1001
Figure
4.
A
snapshot
of a
100-x-100
cell area of the simu-
lation world (totalsize 500 x 500 cells) showing the random
distribution of raw material sources in the environment.
The mean
distance
between nearest
neighboring
raw
material sources is
3.72
grid
cells
(standard
deviation
=
1.85
cells;
minimum
= 1.0
cells;
maximum
= 8.25
cells).
The entire simulated world contained
5,000
raw material
sources each
arbitrarily assigned
a
unique type
label.
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AMERICAN
ANTIQUITY
6000
.
n
=
22,830
5000
,
4000
.E
t
3000
o
Z
2000
1000
0 1
2
3
4 5
6
7
8
9 10
11
Raw Material
Richness
(n
Types)
Figure
5. Structural and
dynamic
components
of
a
neutral
model of stone raw material procurement. Building the
simulation
environment entails
defining
the
size
of
the
sim-
ulation
world and
adding
a fixed number of
unique
raw
material
sources.
A
single
forager
with a mobile
toolkit of
fixed
maximum
capacity
is
then added to the environment.
At
each
time
step,
the
forager
moves
to one
of
the
eight
nearest
neighbors,
or
remains
in
its current
location,
with
equal probability,
and a fixed
amount of raw
material is
consumed
dependent only upon
its
frequency
in the mobile
toolkit.
If
a raw material
source
is
encountered,
the toolkit
is
reprovisioned
up
to its
maximum
capacity
before
mov-
ing again
at
random.
If
no
raw material
source
is
encoun-
tered,
the
forager
moves
immediately
at
random.
Simulations are run until 200 unique raw material sources
are
encountered,
or the
edge
of
the simulation
world
is
reached.
are
generated
on raw
material
ichness,
quantities
of
unique
raw
material
ypes,
and
transport
dis-
tances
for
unique
raw
material
ypes
contained
n
the mobile
toolkit.These
datacombine
to
provide
additional
xpectations
or
the
relationship
etween
the
quantity
of
a
material
n
the toolkitand
the dis-
tancefrom
ts sourceand
he
ntensity
of
rawmate-
rial
reduction
s
a
function
of
distance rom
source.
Raw Material Richness
The simulated
mobiletoolkit
spends
variable
peri-
ods of
time
in
different
ichnessstates
(Figure
6).
Figure
6.
Frequency histogram showing
mobile toolkit
richness for one complete simulation run of 22,830 time
steps.
These simulation results
provide
empirical probabil-
ities of
observing
a
given
raw material
richness
within the
mobile
toolkit.
In
this
representative
simulation
run the
mobile
toolkit contained no
raw
material
approximately
25
percent
of the simulation
time,
between one and
four
unique
raw material
types
61.1
percent
of
the
time,
and
five or
more
types only
13.7
percent
of
the time. On aver-
age,
one would
expect
to see different richness states
in
roughly
those
proportions.
Model
parameters:
world size
=
500
x
500;
number of
unique
sources
=
5,000;
mobile
toolkit
size
=
100;
consumption
rate
=
1.
Most often the toolkit contains
few raw material
types,
25
percent
of the time
containing
no mater-
ial at
all.
Occasionally
he toolkitsustains ichness
levels
of more than
10
uniquetypes.
Ten
separate
simulation
runs
using
baseline
parameters
stab-
lish
a median raw material
richness within
the
mobile toolkit
of
two
unique
material
ypes
and
maximum
awmaterial ichness
of
11
unique
ypes.
Thesevalues
are
not
significant
n
any
global
sense,
though
the
generalshape
of the
distribution
may
be
(see
Hayek
and
Buzas
1997;
Hubbell
2001).
What s
interesting,
however,
s
that richness
ev-
els are
always very
low relativeto the
theoretical
maximum,
determined
by
the size
of the mobile
toolkit
(equation
1] above).
In
Figure
6,
the
sim-
ulated
maximum
ichness
s
only
11
percent
of the
theoretical
maximumof
100
unique ypes.
The
proportion
f time
spent
n
various
ichness
states
and the observedmaximumrichnessof the
mobiletoolkit
are
dependent
n
the
density
of
raw
materials
n
the
environment,
he maximum
apac-
ity
of
the mobile
toolkit,
and
the raw
material on-
sumption
ate.
A
few
analytical
teps
are
required
to
explicate
hese constraints.
Consider
irst henumber f time
steps
Nit
takes
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94
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P.
Jeffrey
Brantingham]
NEUTRAL
MODEL
OF
STONE
RAW
MATERIALROCUREMENT
Table
1.
Variablesand Baseline Parameter
ettings.
Variable
description
Variable Units
Baseline Parameter
Setting/Range
Simulated world
size
in X
dimension
X
grid
cells
500
Simulated world size
in Y
dimension
Y
grid
cells
500
x-coordinatepositionof rawmaterial/forager x gridcells random rom uniform= [1, 500]
y-coordinate
position
of raw
material/forager
y
gridcells random
rom uniform
=
[1,
500]
Number
of
unique
raw materialsources n
sources
5,000
Raw
material
ype
label
i
1, 2,
...
5,000
Quantity
of material rom source i
in
mobile
toolkit
vi
arbitrary
nits minimum
=
0;
maximum
=
100
Total
material
of
all
types
in
mobile
toolkit;
maximum
5vi
arbitrary
nits minimum
=
0;
maximum
=
100
toolkit
capacity
Amount of materialcollected from
source
i
ai
arbitrary
nits minimum
=
1;
maximum
=
100
Probability
of
consuming
materialof
type
i
in
mobile toolkit
ci
[0.0,
1.0]
An
observed numberof simulation ime
steps
N
time
steps
Estimateddistance
traveled n N
time
steps;
effective d
grid cells minimum
=
0;
maximum
=
707
foraging
radius
Maximumforagermove lengthat each time step I gridcells 1
Raw
material
consumption
rate r
arbitrary
nits / time
step
1
Raw
materialrichness in
mobile toolkit k
number
of
types
minimum
=
0;
maximum
=
100
Quantity
of materialdiscarded
n
making
room for
newly
qi
arbitrary
nits minimum
=
1;
maximum
=
100
procured
material
Most abundantmaterial
n
the
mobile
toolkit
at a
max[v,]
arbitrary
nits
minimum
=
1;
maximum
=
100
given
time
step
to
get
from
one
raw
material ource o the nextran-
domly
encountered
raw
materialsource. This is
approximately:
N
-
d
)
2
(4)
where
d is
the distancebetween the two sources
and
1
is the move
length
at each time
step
(see
Denny
and
Gaines
2002:110-114).'
When 1 is
unity,
he numberof time
steps
it
takes to travela
given
distance
d
is
approximately
he
square
of
the
distance.
Remembering
that
a
fixed amount of
material s consumedat each time
step,
it is
possi-
ble
also
to
establish
he
relationship:
xVi
N=
i
r
,
r>0
where he sum
of v.
s
thetotalamount
of raw
mate-
rial in
the mobile toolkit at a
given point
in time
and
r
is the
consumption
ate.
Equation
5)
defines
the toolkit
"clearing
ate,"
he number f
time
steps
it
takes o consumeall
of
the rawmaterial
resently
in
the mobile toolkit at a
consumption
ater.
Substituting
nto
equation
4)
and
rearranging
yields:
d=l
vi(6)
U
1
(6)
r
Equation
6)
states hat
he
distance
hatcan
be
traveled
before
all
materials n the
mobile toolkit
are
consumed s
approximately
he
square
oot
of
the
present
oolkitsize divided
by
the
consumption
rate,
assumingagain
that he
step length
1
s
unity.
Witha
consumption
ateof one unitof
material
er
time
step,
the distance hatcan be
traveled
before
the
toolkit s "cleared"s
simply
the
square
oot
of
the amount
of material
n
thetoolkit:a
toolkit
illed
to
a
maximum
capacity
of
100 units would
be
"cleared" f all
materials
by
the time the
forager
had
traveled
approximately
=
10 cells from
the
current
osition,provided
o
othermaterial
ources
were encountered
long
he
foragingpath.
The
for-
aging
area arounda raw
material ource that
can
be
effectively
exploited
with
material
from
that
source s
thus defined
by
radius
d.
Additional
material ourcesencountered
efore
thetoolkit s
"cleared"ncreasemobile
toolkitrich-
ness.
Maximum oolkit richnesscan be
estimated
empirically
rom the maximumnumber
of
unique
sources hat
might
be found
within
a
foraging
area
of radius
d
=
10. Table
2
lists the
minimum
near-
est
neighbor
distances
between raw
material
sources for two
simulatedworlds with
different
global
raw material
densities.2
Minimum
nearest
neighbor
distances
are calculated
by
measuring,
for each
source,
the distance to all n
-
1
other
sources
n
the environment
nd
then
selecting
the
495
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AMERICAN
ANTIQUITY
Table 2.
MinimumNearest
Neighbor
Distances
between
Raw Material
Sources and PredictedMaximum
Toolkit Richness
for
Two SimulatedEnvironments.
5,000
Unique
Sources
minimum distance
to source
(grid
cells)
3000
Unique
Sources
minimum distance to source
(grid
cells)
Nearestsourcerank1 1.00 1.00
Nearest
source
rank
2
1.00
3.00
Nearest
source
rank 3
2.24
4.24
Nearest source rank
4 3.61
6.08
Nearest source rank 5
4.47
6.40
Nearest
source
rank 6 6.40
8.49
Nearest source rank
7 7.07
9.22
Nearest source rank 8
7.07
11.18
Nearest
source
rank 9 7.81
12.04
Nearest source rank 10
8.00
12.73
Nearest source rank
11 9.22
14.21
Nearest source rank
12
10.20
14.56
Nearestsource rank 13 10.82 16.03
Nearest source rank
14
11.18 17.49
Nearest
source
rank 15 11.31
18.25
Nearest source rank 16 12.53
19.03
Nearest source rank
17 13.00
19.31
Nearest source rank 18 13.34
19.92
Nearest source rank 19
13.60
21.02
Nearest source rank 20
14.14
21.21
Nearest
source
rank 21 14.32
22.02
Nearest source rank
22
15.52
22.67
Predictedmaximum tool
kit richness
11 7
Observed
maximum
toolkit richness IIa
6b
Average
distance to
a
neighbor
within 5.69
5.49
the
foraging
area
Note: Toolkit size is 100
and d
=
10
in both
cases;
boldface
numbersmark
the minimum
nearest
neighbors
that fall within
the
effective
foraging
radiusd.
aTen
separate
simulation
runs.
bFive
separate
simulation
runs.
observed minima for
the
population
of
sources
overall.
Two
sources
chosen at random
rom
the
environment,
or
example,
achhave
nearest
neigh-
bors of
rank
1, 2,
3 ...
5,000.
The
corresponding
distances
o
neighbors
of
each rank
may
be,
hypo-
thetically,
1.0, 1.25,4.3
...
698,
and
1.2,
1.3,
2.6.
. 687
grid
cells
for
each
source,
respectively.
n
this
hypothetical
ase,
the first- and second-order
minimumnearest
neighbors
redefinedon
thebasis
of
source
1,
while the third and
5,000th
nearest
neighbors
are
defined
on
the basis of source
2.
This
process yields
the
set of minimum
nearest
neigh-
bors
1.0, 1.25,
2.6
...
687,
which
subsequently
defines hemaximum
ossible
packing
of
resources
into
a local
foraging
area. When
the simulated
world contains
5,000
unique
sources-a
global
density
of .02
sources/grid
ell-the
set
of
mini-
mum nearest
neighbors
will on
average
nclude
11
sources
within
he
boundaries f a
foraging
areaof
radiusd
=
10. With
a total of
3,000
sources-.012
sources/grid
cell-on
average
there
are seven
unique
sources
within a
foraging
area
of radius
d
=
10
(Table2).
These sets
provide
reasonable sti-
matesof maximum
ttainableawmaterial ichness
underdifferent
aw
material
density
conditions.
In
general, ncreasing
he
global density
of raw
materials
n
the
environment,
hich ncreases
rob-
abilistically
the
maximum
number
of
materials
found in a
foraging
area
of radius
d,
will also
increase he
maximum
attainable ichness
or
the
mobile toolkit.
Similarly, increasing
maximum
toolkit
capacity,
leaving
both the environmental
density
of
resources
andraw
material
onsumption
rate
unchanged,
ffectively
ncreases
he
foraging
radiusd. For
example,
rom
equation
6),
doubling
the maximum
apacity
of
the
mobile
toolkit
o 200
unitsof material ncreases
he
foraging
radiusd to
14.1
grid
cells.
This
expands
he areaover which
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Jeffrey Brantingham]
NEUTRAL
MODELOF STONE
RAWMATERIAL
ROCUREMENT
Table
3.
Procured
Raw Material
Package
Sizes
and
Expected Consumption
Rates.
Package
Size Procured
(Arbitrary
Units)
Toolkit Richness Number of Occurrences
Mean Std. Deviation Minimum Maximum
1 160 63.6
45.1
2
100
2 104 30.3 29.0 2 95
3
93 29.7
30.0
2
95
4
49
26.6 25.8
2
95
5
54 14.9 18.6
2
70
6
18 20.8 19.2
3
66
7 3
6.7
4.7 3 12
Total 481
38.7
38.5 2
100
Expected
Amount
Remaining
At Distance da
d=10 d=20 d=30 d=40
20.7 9.0 3.9 1.7
Note: Data
are
for one
simulationrun
using
baseline
parameters.
"Expected
amount
(arbitrary
nits)
calculatedfrom the
equation
y
=
47.726e-0'0834X
iven
in
Figure
11.
rawmaterial ources
may
be encountered ndcon-
comitantly
increases the
maximum attainable
toolkit
richness.
Using
Table
2,
maximumattain-
able richness ncreases o km
=
19
in the
case
of
max
5,000
unique
sources
n the environment nd
kma
=
10
in the case of
3,000
unique
sources.
Finally,
increasing
he
consumption
ate
r
reduces heeffec-
tive
size
of the mobiletoolkit
andtherefore owers
the
maximum
attainable oolkitrichness.
Note that here
s a tradeoff etween oolkit
rich-
ness and he amount
f time
spent
ata richness tate.
From
equation
5),
N
=
max[vi]/r
s
the amountof
time it takesto "clear" he
most abundantmaterial
type
from the mobile toolkit.
The
dynamic
behav-
ior
of the mobile
toolkit is similarto
a
zero-sum
game
(see
Hubbell
2001;
MacArthur
960).
When
all of
the most
abundantmaterial s
consumed,
he
richness
state
of the toolkit
will decrease
by
1. If
the toolkit is
completely
ull
(i.e.,
Xvi
=
100),
any
increase
n richnessof thetoolkit-from
k to
k+
-
must be
accompaniedby
a decrease
qi
in
a
quan-
tity
of material
already
n the toolkit. The
most
abundantmaterial
ype
in the
toolkit
s
usually
the
one that
must
yield space
since the
probability
f
consuming
a material
ype
is
dependent nly upon
its
frequency
n the oolkit
equation
3] above).
The
estimated ime
spent
at
the
new richnessstateN
is
necessarily
owersince
max[vi
qi]
40
km
from
the
site,
but
may
fail to
explain
the low
expectedpro-
curement
probability
or
MP4,
which is substan-
tially closer (Geneste 1988, 1989). As a first
approximation,
he differential
representation
f
these materials
may
simply
reflect
different
prob-
abilities
of
finding
viable
(i.e.,
"pre-clearing")
an-
dom walk
paths
betweensourcesand
sites,
and
not
necessarily
raw material
electivity.6
Raw
material ransferswithinthe MiddlePale-
olithic of Central
Europe
and the
Aquitaine
Basin
of France
display
distributions
haracterized
by
high
frequencies
of transfers rom
nearby
ources,
withinthe "local"portionof a foragingrange,and
a dramaticdecline in the
occurrenceof transfers
from
greater
istances.
n
theformer
egion,
but
ess
so
in
the
latter,
herearealso "internalmodes"
rep-
resentingrelativelyhigh
frequencies
of transfers
from sources
beyond
the "local"
portion
of the
range
(for
individualMiddle
Paleolithiccases see
Geneste
[1989];
for individual
Upper
Paleolithic
cases see
Feblot-Augustins
1997a,
1997b,
1997c]
andMoralaand
Turq
1990]).
The
longest
distance
transfers
230-300 km)
within the Middle
Pale-
olithic
in
Central
Europe
are between
4.6 and 6
times distance
represented y
the
primary
nternal
mode
(50 km).
The
ongest
distance ransferswithin
the
Aquitaine
Basinwouldbe
only
twice
thismodal
distance.
The neutralmodel
generates
requency
distrib-
utions
displaying
similar
characteristics
compare
Figures
2
and
9).
In
particular,
when observations
of the simulatedmobile
toolkit
are
spatially
biased
toward
procurementpoints,
there is a
frequency
peak
at
very
low distances from
source. There
remains,however,
an
nternal
mode
corresponding
to raw materialtransfers rom intermediatedis-
tances.The modeledmaximumstone
transferdis-
tances n this
case
are
between hreeand our imes
the distance
represented y
the
internalmode.
The
neutral
nterpretation
s that he
spike
in
short-dis-
tancetransfers s
primarily
resultof
a
spatial
bias
in
the
sample
of sitestoward hose ocatedator
near
raw material ources.The
"internal
modes"
define
the effective
foraging
radiusd around aw mater-
ial
sources:
generally,
how far a
forager
can
travel
in a random
walk with
material
from a source.
These
"internalmodes"arecontrolled
by
the clear-
ing
rate or the toolkitandneed not
imply special-
ized
logistical
or
seasonal
exploitation
of
patches.
Finally,
the
archaeologically
observed
maximum
transport
istances n these
cases are
qualitatively
similar o those
predicted
by
the
neutral
model.
It
remains o be seenwhether hequantitative iffer-
ences
between observedand
predicted
maximum
transport
istancesarerobust
under urther
esting.
As a first
approximation,
owever,
the
similarity
between the
Middle Paleolithic
data and the
neu-
tralmodel
suggests
that
raw-material
ransferdis-
tances of this order
need not reflect
optimization
of stone
procurement
and
transport.
Moreover,
there is no
necessary
reason to invoke
social
exchange
o
explain
maximum
ransport
istances
in theseinstances.7Moregenerally, t is not clear
that
maximum raw
material
transport
distances
translate
n
any
direct
way
into
geographic
erri-
tory
size.
Lastly,
he
Central
European
Middle
Paleolithic
also illustrates
hat materials rom
nearby
ources
are
transferred
n
relatively arge quantities,
while
materials rom
distant sources are
transferredn
small
quantities
Feblot-Augustins
993;
Geneste
1988,
1989;
but
see
F6blot-Augustins
997c).
Nor-
mally
this
pattern
s
paralleled yincreasing
educ-
tion
intensity
(and/or
design
formality)
with
increasing
istance rom
source.The
neutralmodel
similarly
anticipates
othof these
empirical
bser-
vations
compare
Figures
3
and
11).
Materials
rom
nearby
sources are
usually
transferred
n
large
quantities,
utwith
a
high degree
of
variability,
nd
these
materialsare
usually minimally
consumed
relativeto the initial
quantitiesprocured.
n
con-
trast,
materials rom distant
ourcesare
nvariably
found in
small
quantities
and have been
heavily
consumedrelative
o initialraw
material
package
sizes.
The
modeled
relationship
is
accurately
described
by
an
exponential
unctionof
the form
y
=
b
*e-ax.
The
neutral
nterpretation
s that he
quan-
tity
of
material ransferred
s constrained
y
a fixed
toolkit
size,
precedence
given
to
materials
lready
in
the toolkit and a
fixed
consumption
ate.8
Simi-
larly,
reduction
ntensitymay
simply
be a
function
of the
length
of time
a materialhas
spent
in
the
mobile oolkit
and hus
degree
of
exposure
o "con-
sumption
risk"
(see
Dibble
1995).
Quantities
of
materials ransferred nd
reduction
ntensity
n
this
case are
ndependent
f
specific
rawmaterial
ypes
and,
because he
mechanism
riving
oolkit
dynam-
ics is
entirely
stochastic,
hese
patterns
need not
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AMERICAN
ANTIQUITY
imply any depth
of
planning,optimization,
r risk-
management trategies.
Discussion
Manyanethnography-andcommonsense-tells
us
that
oragersemploy complex
optimal oraging
strategies
Binford
2001;
Kelly
1995;
Smith
1991).
It
is assumed
hat
optimal oraging
strategies
must
influence
and, herefore,
e
diagnosed
by
stone
raw
material
procurement atterns.
The
present
model,
however,
s
based
on
assumptions
hat
oragers
do
not
optimizeany specificcurrency
ssociatedwith
movement,
do not
depend
on
any
formof
planning
depth,
and are
risk
insensitive
n
all
of theirmove-
ment and procurementdecisions. Surprisingly,
these
extreme
assumptions
ead to
patterns
n raw
material richness and
transport
hat
are
qualita-
tively
similar o
commonly
observed
archaeologi-
cal
patterns.
What shouldwe conclude
from these
results?
There
are at least
two answers o
this
question.
First,
it is
possible
that the
processes
of
archaeo-
logical
site
formation-especially
time-averag-
ing-may
eradicate the fine details
of discrete
procurementvents eavingapalimpsestof behav-
ioral
traces
indistinguishable
from
the neutral
model.
In
general,
time-averaging
occurs
when-
everevents
hat
happened
t
different
oints
n
time
appear
synchronous
in
the
geological
record
(Kowalewski
1996).
It influences
our understand-
ing
of
behavioral,
ecological,
or
evolutionary
processes
whenever he sedimentation
ate s slower
than
he
time scale of
the
process
n
question
Bush
et al.
2002;
Kowalewskiet al.
1998;
Stem
1994).
Stone
procurement trategies
were
implemented
on time
scales of minutes o
perhaps
months,
f
sea-
sonal
planning
was
in
play,
while sedimentation
s
generally
a
much
slower
process.
Moreover,
geochronological
controls
rarely
offer
such
fine-
scale
resolution.
There
s thus little
question
that
time
averaging
will be
a
concern
in
interpreting
most
cases of stone raw material
procurement
nd
transport
but
see
Close
2000).
It
may
be the case
that
discrete tone
procurement
vents
were
highly
adaptive
n
all of the
ways suggested
by
current
archaeological heory,
but n some cases
it
may
not
be
possible
to
distinguish
he
aggregate
patterns
f
procurement
ehavior rom he neutral lternative.
Second,
it is
conceivable-though perhaps
unlikely
n
many
environments
but
see
Branting-
ham et
al.
2003;
Brantingham
t al.
2001;
Turchin
1998)-that
random-walk
foraging
and a
raw-
material
procurement
trategy
hat s indifferent o
stone
types
could
be
optimal.
The
conclusion
here
would be thatthe absence of bothplanningdepth
and
explicit
risk
management
trategies
would
yield
higher
itness
han heir
more
complex
alternatives.
It s
perhaps
more
ikely
that
other
aspects
of
for-
aging
behavior
were
subject
o
optimization, epth
of
planning,
and
risk
management
n the
ways
sug-
gestedby
contemporary
rchaeologicalheory,
but
that stone raw
material
procurement
nd
transport
were
not,
at least as reflected n the
empirical
mea-
suresdiscussed
here.9
Foraging trategies ptimized
withrespectto mobileprey,or seasonallydistrib-
uted
plant
esources,
may
be
entirely
tochasticwith
respect
o raw
material
ourceencounters nd raw
material
procurement.
Moreover,
whetheror
not
stone
procurement
ehaviors
were
optimized
(or
sensitiveto
risk)
may
have had little or no
impact
on the
optimality
of
food
procurement
trategies.
The conclusion
here would
be thata
random-walk
stone
procurement
trategy
nd
ndifferenceo stone
raw material
ype
may
not
entailthe fitness
penal-
ties commonlyassumedand,importantly,hatthe
more
complex
alternatives
ay
not
necessarily ield
higherpayoffs.
This is a
logical
extensionof
argu-
ments
suggesting
hat
stone raw material
procure-
ment
was
completely
embedded
within other
foraging
ctivities
Binford1979).
Absent
any
direct
fitness
consequences,
tone
procurement
nd rans-
port
behaviors
would be
neutralwith
respect
to
selection andwould
be free
to
vary
n
a
stochastic
manner.
The
presentmodel,
ike
many
neutral
models n
community ecology
(Gotelli
and
Graves
1996:5-6),
raises
questions
about
how we infer
adaptation
rom
empiricalpatterns.
This
said,
here
will be
temptation
orcritics o
reject
hemodelout-
right by
claiming
that
foragers
"would" never
engage
in
a random
walk
and"would" evermake
raw-material
rocurement
ecisions
withoutcon-
sidering
raw material
ype.
Similarly,
motivated
objections
have
been
voiced over neutral
models
discounting
he
importance
of
interspecific
om-
petition
n
structuringcological
communities
see
Connell
1980;
Conner and
Simberloff
1979;
Roughgarden
1983;
Simberloff
1983).
In
this
lat-
ter
case,
conventional
wisdom
suggests
that com-
petition
is the critical determinant of
whether
504
[Vol.
68,
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2003
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P.
Jeffrey Brantingham]
NEUTRAL
MODEL
OF
STONE
RAWMATERIAL
ROCUREMENT
species
can coexist on
given
a limited resource
(MacArthur
nd Levins
1967).
Many
researchers
have
rejected competition-free
neutralmodels
in
communityecology only
because
they
seemed
to
deny the unique behavioral and morphological
charactersonsidered
daptivelymportant
o,
if not
definitive
of,
biological
species.
That a neutral
model contradicts rthodox
heory,
however,
s
not
sufficient
grounds
or
rejecting
he model
(Gotelli
and Graves
1996:13).
A
relatedcriticism
might
concede
that
he
neu-
tral model works for the
Middle Paleolithic
and
behaviorally
archaic
hominids,
but that behav-
iorally
modern humans "would" never have
employedsuchstrategies.The choiceof examples
discussed here was not
motivated
by
a
desire
to
paint
archaichominidsas
"acultural" utomatons
engaged
in
random
behavioral
trategies.
Rather,
the
Western
and Central
European
Middle
Pale-
olithic
archaeological
records offer some of the
best
(and
most
explicit)
treatments
f
long-term,
regional patterns
of stone raw
material
procure-
ment and
transport.
The
models and behavioral
interpretations
made
explicit
in
these studies are
also
commonly
nvoked
n LowerPaleolithic
e.g.,
Feblot-Augustins
1997b,
1997c;
Kimura
2002;
Martinez
1998;
Potts
1994),
Upper
Paleolithic
e.g.,
Blades
1999;
Feblot-Augustins
1997a, 1997b,
1997c)
and
even Holocenecontexts
see
Bamforth
2002).
The neutral
model
provides
a
baseline
for
comparison
in all of
these
contexts. It
may
be
rejected
n
some,
or
perhaps
even
most
empirical
tests
(see
Feblot-Augustins
1997c:Tables62
and
64).
In
those cases we
will
be
more confident hat
observed
archaeologicalpatterning
does reflect
some
formof
adaptation.
However,
ejection
of
the
neutralmodel s not
assured
priori
simply
because
of
an
assumption
of
behavioral
modernity.
A more
appropriate
riticism of the
present
model would
suggest
thata
forager
"could"never
engage
in a
random-walk
oraging strategy
and
"could" ever
gnore
he differencesbetweenstone
raw material
ypes.
In this
case,
the model would
be
behaviorally
rrelevant.
contend,
however,
hat
the neutralmodel of
stone raw
material
procure-
ment
developed
here
is both
behaviorally xplicit
and
behaviorally
ealistic.
Regarding
he
first
point,
there
are
no hidden
variablesand there should
be
no confusion about
the behavioral mechanics
underlying
he
model.
Regarding
he
second
point,
the
model
offers
a
simplest-case
scenario
or the
form and
sequence
of
behaviors nvolved in raw
material
procurement.
Use of the word"realistic"s
likely
to
raisesome
hackles.Nonetheless, t is critical orecognize hat
the simulationdoes
not
intend to
capture
all fea-
turesof Paleolithic
oraging
adaptations,
ut
rather
only
thosebehaviors
directly
nvolved
n
rawmate-
rial
procurement.
f
anthropologists
ccept
as rea-
sonable
and realistic hat
Paleolithic
foragers
had
the
adaptivecapacity
to
implement
patch
choice
strategies
and even
long-term
seasonal
foraging
plans,
should
t not
also be
withintheir
capacity
o
not
mplement
hese
behaviors?
imilarly,
f
anthro-
pologists accept thatforagersare able to distin-
guish
betweenrawmaterial
ypes
and
make urther
choices about which
materials
o
transport
nd
in
which
forms,
should
we not
also attributeo them
the
ability
not
to
make
these distinctions
and deci-
sions?
If
the
answersto
these
questions
are affir-
mative,
s I
believe
hey
are,
hena neutral
modeling
strategy
s
appropriate
nd
both
a random-walkor-
aging strategy
and
the
absence of raw material
selectivity
offer
simplest-case
tartingpoints.
One
may
also
object
to
the
appropriateness
f
parsimony
as
a
principal
for
choosing
between
alternative
ypotheses
hat
generate
imilar,
r even
equivalent,
esults.
In
truth,
here is no
guarantee
that
he
simplestexplanation
s the correct
one.
In
defense of
parsimony,
I
would
suggest
that
the
probability
f
producing
a
Type
II error
i.e.,
false
positive)
is
minimized
by
favoring
the
simplest
model available
and
also that the
opportunity
or
identifying
and
rectifying
such
an error s
greater.
Finally,
t
is clear
that
a
general
neutralmodel
of stone raw material
procurement
s
only
a first
step.
Rigorous,
quantitative
development
of the
observations
resented
herein
requires
alibration
of
the
agent-based
model to run
in
simulated
"worlds" uilt around
he known
geographic
dis-
tributionsof actual
raw
material sources.
Such
integrated
GIS
agent-based
models arethe
subject
of
current ndeavors.
Subsequent
o
this,
it will be
important
o
begin
systematically
modifying
the
neutral
assumptions
of
the model to
explore
how
different
ssumptions
nfluence
results.Forexam-
ple,
it will be
important
o
explore
the
impact
of
partial-
r
fully
non-random
walk-foraging
trate-
gies
on mobile
toolkit
dynamics,
holding
the
other
assumptions
about unit raw
materialneu-
505
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21/24
AMERICAN
ANTIQUITY
trality
constant.
Similarly,
t will be
instructive
o
modify
the
assumptions oncerning
raw material
selectivity
by,
for
example,
rank
ordering
stone
materials
n the environment
ccording
o
quality.
Ultimately,
he
goals
of
these futuremodels will
be to establish whether
changes
in
assumptions
lead
to
quantitatively
or
qualitatively
different
results
from the neutral model.
Perhaps
more
importantly,
he
goal
will
be to establish whether
these extended results find
empirical
support
n
the
archaeological
ecord.
Conclusions
The
richnessof raw
materials,
ransport
istances,
and characterof
the
transported
materials
ound
in
archaeological assemblages
are often inter-
preted
n
termsof
adaptiveoptimization,
depth
of
planning,
and risk minimization. In
many
instances,however,
hese
patternsmay
be
qualita-
tively indistinguishable
from a
non-adaptive
model
of
Paleolithic
foragers engaged
in
a
ran-
dom-walk
foraging strategy
and
procuring
stone
raw materialswithout
any
regard
or raw
material
type.
The neutral model
of
stone raw
material
procurementdeveloped
here
provides
the
sim-
plest-case
behavioral
nterpretations
f (1) rich-
ness-sample
size
relationships;
2)
a
"decay-like"
pattern
n
the
frequencies
of
raw
material
rans-
fers
from sources at differentdistances
from
site;
and
(3)
the
tendency
for
materialsfrom distant
sources
to
be
imported
n
low
quantities
and
as
heavily
reduced
technological
forms.
Though
provocative
n
its
challenge
of conventional
he-
ory,
the model is nonetheless
explicit
and,
more
importantly,
behaviorally
realistic. The
primary
implicationof the neutralmodel is that our infer-
ences
about
adaptivevariability
ased on
patterns
of
raw material
ichness
and
transportmay
be
dif-
ficult
to
prove.
The neutral model
provides
an
alternative et of
expectations
where we can be
sure that
adaptation
s
not
in
evidence. In
princi-
ple,
patterns
n raw materialrichness and
trans-
port
that
deviate
from
the
neutral
baseline
expectations may
indicate where
optimization,
depth
of
planning,
and risk
management
are
potentiallyelementsof raw materialprocurement
adaptations.
Acknowledgments.
This researchwas
supported
n
part
by
a
postdoctoral
fellowship
from
the Santa Fe
Institute.
Many
thanks
o John
Pepper,
Van
Savage,
Cosma
Shalizi,
andTodd
Surovell
for
guidance
along
the
way.
I am
grateful
to
David
Madsen,
David
Rhode,
J.
Feblot-Augustins,
and
two
anony-
mous
reviewersfor comments on earlierdraftsof this
paper.
I
thank
Alex
Borejsza
for
translating
he
abstract.
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