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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
DETERMINISTIC CHAOS IN MODELS OF HUMAN BEHAVIOR:
METHODOLOGICAL ISSUES AND EXPERIMENTAL RESULTS
JOHN D. STERMAN
OCTOBER 1987
WP 1945-87
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
r
'^
DETERMINISTIC CHAOS IN MODELS OF HUMAN BEHAVIOR:
METHODOLOGICAL ISSUES AND EXPERIMENTAL RESULTS
JOHN D. STERMAN
OCTOBER 1987
WP 1945-87
D-3921
Deterministic Chaos in Models of
Human
Behavior:
Methodological Issues and Experimental Results
John D. Sterman
Associate Professor
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
02139
MA
Forthcoming
in the
Special Issue on Instabilities and Chaos
System Dynamics Review
October 1987
D-3921
ABSTRACT
Recent work has shown
literature contain previously
that several
well-known models
model of
the
Two
wave and
of the most
the production-
Beer Distribution Game. The significance of these theoretical
developments hinges on whether the chaotic regimes
space.
system dynamics
unsuspected regimes of deterministic chaos.
extensively analyzed are Sterman's model of the economic long
distribution
in the
lie in
the realistic region of parameter
Further there are major questions regarding the descriptive accuracy of models of
human systems which
exhibit chaos.
experiments mean empirical studies
questions.
An
Data limitations and the
at the
inability to
conduct controlled
aggregate level are not likely to resolve these
alternative approach is based
on laboratory experiments
provide a simulated environment for the study of
human decisionmaking
in
which models
behavior.
Recently
laboratory experiments have been conducted to analyze decisionmaking behavior in the long
wave model and
shows
the
Beer Distribution Game. This paper describes the experiments and
that the behavior
of the subjects
is
explained well with a simple heuristic long used in
system dynamics modeling and well grounded
in behavioral decision theory.
The parameters
of the proposed decision rule are estimated econometrically for each subject.
The parameters
which characterize
a significant minority of the subjects are
direct experimental evidence that chaos can be
real
shown
to
produce chaos. This
produced by the decisionmaking behavior of
people has important implications for the formulation, analysis, and testing of models of
human
behavior.
D-3921
Recent work has shown
literature contain previously
well-known models
that several
in the
system dynamics
unsuspected regimes of deterministic chaos. The work of Day
(1982a, 1982b) provides an early example of chaotic behavior in economic models, while
Mosekilde and others (1987) have developed corporate models which exhibit chaos.
the
most extensively analyzed such models
are Sterman's
Two
of
model of the economic long wave
(Sterman 1985, 1986, Rasmussen, Mosekilde, and Sterman 1985) and the productiondistribution system or
Beer Distribution
Mosekilde and Larsen,
produced
in these
this issue).
systems
results hinges in large
is
Game
While
(Forrester 1961, Jarmain 1963,
Sterman 1984,
the demonstration that chaos can be endogenously
an important theoretical development, the significance of the
measure on whether the chaotic regimes
lie in the realistic
parameter space or whether they are mathematical curiosities never observed
region of
in the real
Further there are major questions regarding the descriptive accuracy of the decision
system.
rules postulated in the
attractors.
The
models of human systems developed
practical significance of chaos
in policy-oriented
modeling remains unclear
to date
which contain strange
and other phenomena such
until
it
as self-organization
can be determined that these phenomena
can occur in models whose decision rules are grounded in empirical study of the actual
decision processes of the agents.
It is
difficult if not
appeal to the aggregate empirical data.
been
at
most
five
reliable results.
wave, for example, there have
industrial revolution, too
few
for statistically
Worse, the data required are simply unavailable, and much of
it is
corrupted
this issue).
alternative approach
is
based on laboratory experiments
simulated environment for the study of
the long
In the case of the long
long-wave cycles since the
by measurement error (Chen,
An
impossible to resolve such issues by
wave model and Beer
in
which models provide
human decisionmaking (Sterman
Distribution
Game
1987a).
Recently,
(traditionally used as a teaching rather
than as a research tool) have been analyzed as experiments in dynamic decisionmaking
a
D-3921
These experiments were conducted primarily
(Sterman 1987b, 1987c).
to study the heuristics
with which people manage a complex dynamic environment. In both experiments the decision
task of the subject
was
manage
to
a stock in the face of losses, delays in acquiring
new
units,
multiple feedbacks and other environmental disturbances.
This paper describes the experiments and shows that decisionmaking behavior
two experiments
is
The departures from optimal behavior
significantly suboptimal.
common
systematic, suggesting subjects use
heuristics for stock
in the
are
managment. The
experiments show that the behavior of the subjects can be modeled well with a simple stock-
management
behavioral decision theory.
The parameters of
econometrically for each subject.
parameters.
shown
to
system dynamics modeling and well grounded
heuristic long used in
the
The systems
The parameters which
in
proposed heuristic are estimated
are then simulated with the estimated
characterize a significant minority of the subjects are
produce chaos. This direct experimental evidence that chaos can be produced by
the decisionmaking behavior of real people has important implications for the formulation,
analysis, and testing of
models of human behavior.
The Stock Management Problem
The regulation of
decisionmaking
tasks.
a stock or
The manager seeks
within an acceptable range.
altering the stock's inflow
as to
system
and outflow
the initiation of a control action and
the stock and the perception of that
may
common dynamic
to maintain a quantity at a target level, or at least
rates.
Typically a manager must set the inflow rate so
compensate for losses from the stock and
vary and
one of the most
Stocks cannot be controlled directly but rather are influenced by
which may push the stock away from
may
state is
its
its
to counteract
desired value.
effect
environmental disturbances
Frequently there are lags between
on the stock, and/or lags between a change
change by
the decisionmaker.
be influenced by the manager's
own
actions.
The duration of
in
these lags
D-3921
Stock management problems occur
macro.
at
many
levels of aggregation
from
the
At the level of a firm, managers must order parts and raw materials so
micro to
the
as to maintain
inventories sufficient for production to proceed at the desired rate, yet prevent costly
inventories from piling up.
They must
adjust for variations in the usage and wastage of these
materials and for changes in their delivery delays. At the level of the individual, people
down
regulate the temperature of the water in their morning shower, guide their cars
highway, and manage
their
checking account balances. At the macroeconomic
Federal Reserve seeks to manage the stock of
economic growth while avoiding
inflation
The generic stock management
control problem
Considering
first
Si=
'
I
may
A
in credit
demand.
be divided into two parts:
the decision rule used
by the manager
the stock and flow structure, the stock of interest S
of the acquisition rate
'
(ii)
level, the
so as to provide sufficient credit for
and compensating for variations
stock and flow structure of the system; and
1).
money
is
^
X
acquisition rate will
(1)
itself,
and may also depend on other
and exogenous variables U:
(2)
depend on the supply
line
SL of units which have been
not yet received, and the average acquisition lag X. In general,
=
The supply
(figure
the accumulation
Lt = fL(St,Xt,Ut).
At
itself
X may be
ordered but
a function of the
and the other endogenous and exogenous variables:
fA(SLat).
line
is
(3)
simply the accumulation of the orders which have been placed
O
less those
which have been delivered:
SL,=
the
(A,-L^)dx+S,
K
endogenous variables
supply line
(i)
less the loss rate L:
Losses from the stock must depend on the stock
The
the
I
•^0
(O,- A,)dx+SL,
°
(4)
D-3921
The
structure represented by figure
that the functions
arbitrarily
and
1
eq. (1-4)
is
There
quite general.
governing losses and the acquisition lag are
complex feedbacks among the endogenous
no presumption
There may be
linear.
variables,
is
and the system may be
influenced by numerous exogenous forces including noise and nonstationarity of the
underlying equilibrium.
Consistent with the behavioral foundations of system dynamics modeling (Morecroft
1983, 1985, Sterman 1987a), behavioral decision theory (Hogarth 1987, Tversky and
Kahneman
1974), and the theory of
bounded
rationality
(Simon 1979, Cyert and March 1963),
the proposed decision rule utilizes information locally available to the decisionmaker
not presume the manager has global knowledge of the structure of the system.
decision rule recognizes three motives for ordering which any stock
and does
The generic
management
heuristic
must include:
Order enough
to (1) replace
expected losses from the stock, (2) reduce the
discrepancy between the desired and actual stock, and (3) maintain an adequate
.supply line of unfilled orders.
1.
when
Replacement of losses. The replacement motive
the desired
is
straightforward.
and actual stock are equal, the manager must continue
replace ongoing losses.
Losses
may
arise
from usage
(e.g.
would cause
2.
demand
a
the stock to fall
Stock adjustment.
mechanism
below the desired
to order
enough
to
shipments from an inventory of
finished goods) or decay (e.g. the depreciation of plant and equipment).
losses
In equilibrium,
Failure to replace
level, creating steady-state error.
Errors in forecasting losses or changes in the desired stock
to adjust orders
above or below replacement. Orders
to reduce the
discrepancy between the desired and actual stock form a negative feedback loop which
regulates the stock
(shown
in the
bottom part of figure
1).
Any
rule
compensate for discrepancies between the desired and actual stock
at all.
Such
rules could not respond to a
change
which
fails to
fails to control the
in the desired stock,
stock
nor restore the stock to
^
D-3921
The stock would follow
the desired value if displaced.
a
random walk
as the
system
is
bombarded by shocks.
Supply
3.
give stock
Delays between the
line adjustment.
management systems
significant inertia
to ensure a stable response to shocks.
initiation
and impact of control actions
and should be accounted for by managers
Failure to account for the supply line results in
overcorrection and instability. Consider cooking dinner on an electric range. If one turns the
heat
down
comes
just as the pot
continue to heat the pot, boiling
to a boil the supply line of heat in the coils of the range will
it
over and ruining dinner.
The following equations formalize
in
most
real life situations
the ordering heuristic proposed above.
First, orders
must be nonnegative,
Ot = MAX(0,IOt)
where 10
is
The
(5)
the indicated order rate, the rate indicated by other pressures.
indicated order rate
is
based on the anchoring and adjustment heuristic (Tversky
and Kahneman 1974). Anchoring and adjustment
an
unknown
quantity
estimated by
is
first
recalling a
then adjusting for the effects of other factors which
obscure.
a
is
common
known
may
Anchoring and adjustment has been shown
judgmental strategy
in
which
reference point (the anchor) and
be less salient or whose effects are
to apply to a
wide variety of
decisionmaking tasks (Einhom and Hogarth 1985, Davis, Hoch, and Ragsdale 1986, Johnson
and Schkade 1987, Lopes 1981, Hines 1987, Sterman 1987dX Here the anchor
expected loss rate L^.
Adjustments are then made
desired and actual stock
AS
in
is
the
response to discrepancies between the
and between the desired and actual supply
line
ASL:
lOt = L^t + ASt + ASLt.
The expected
loss rate
may
(6)
be formed in various ways.
economics and management science include
static
Common
assumptions
expectations L^t = L* (a constant or
in
D-3921
equilibrium value), regressive expectations L^t= yLt-l + (l-Y)L*, 0<y<l, adaptive
expectations L«t= 6Lt-i + (l-6)L^t-l, 0<9<1, and extrapolative expectations, AL^t =
lcoi*ALt-i,
where
A
is
The adjustment
regulates the stock.
the first-difference operator and coi>0.
for the stock
The proposed
linear in the discrepancy
AS
creates the chief negative feedback loop
heuristic
assumes for simplicity
which
adjustment
that the
is
between the desired stock S* and the actual stock:
ASt = as(S*t-St)
(7)
where the stock adjustment parameter as
is
the fraction of the discrepancy ordered each
is
formulated analogously as
period.
The adjustment
for the supply line
ASLt = asL(SL*twhere SL*
line.
O*
is
SLt)
(8)
the desired supply line and
The desired supply
asL
is
the fractional adjustment rate for the supply
line in general is not constant but
depends on the desired throughput
and the expected lag between ordering and acquisition of goods:
SL*t = Xh*0*t.
The adjustment
(9)
overordering and also compensates for changes in the acquisition lag X.
example,
ASL
which avoids
for the supply line creates a second negative feedback loop
induces sufficient additional orders to restore
possible representations for X^ and
If
A=0*. There
X
rises, for
are a varier>' of
O*, ranging from constants through sophisticated
forecasts.
Experiment
The experimental
The Economic Long Wave
protocol for the long
Meadows 1985 and Sterman
economics are discussed
I:
wave model
is
described in Sterman and
1987a, 1987b). The methodological foundations of experimental
in the
seminal work of Smith (1982) and Plott (1986). The model
represents the aggregate capital-producing sector of the
economy
(figure 2).
The dynamic
D-3921
hypothesis behind the long
the process by
wave can be
stated in
about 15-25 years
in
First,
Second,
in the
orders and acquires capital from
damped
due
to construction lags,
demand
is
inherently
fluctuations with a period of
aggregate the capital-producing
itself.
This multiplier effect or
which destabilizes the
ordering' creates a positive feedback loop
individual firms, changing the
to
would produce damped
response to a shock.
economy
sector of the
parts.
which individual firms adjust production capacity
In isolation an individual firm
oscillatory.
two
'self-
oscillatory tendencies of
oscillation to a limit cycle with a
40-60 year period.
Simulation and formal analysis confirm the dynamic hypothesis (Sterman 1985; Rasmussen,
Mosekilde and Sterman 1985). As the self-ordering loop becomes stronger, damping drops
The system goes through
rapidly.
a
Hopf
bifurcation and produces a limit cycle.
Further
increases in the strength of self-ordering proceed through period doublings and ultimately to
chaos.
model
In the experiment the
is
transformed into a
capital-producing sector of a simple economy.
demand
for capital
for capital arise
orders.
by adjusting
its
in
which subjects manage
meet desired production. Orders
from the exogenous consumer goods sector and from
essential structure
the
game
reflect a simplified
and dynamics.
the
Subjects must balance the supply of and
their production capacity to
The equations underlying
preserve
game
In the
game
the subject's
own
form of the original model but
decisions are
made
in discrete
time intervals representing two years, and the model becomes a third-order difference
equation system.
The equations of
production
DP
the
game
are given below.
or Production Capacity PC. Capacity
capital/output ratio
k
is
Production
is
PR
is
the lesser of desired
proportional to the capital stock.
The
one period (two years):
PRt = MIN(DPt,PCt)
(10)
PCt =
(11)
Kt/K.
D-3921
The
capital stock of the capital sector
depreciation
CD. Depreciation
capital stock x
The
is
is
is
augmented by acquisitions
proportional to the stock.
AK
and diminished by
The average
lifetime of the
10 periods (20 years):
Kt+i = Kt + (AKt-CDt)
(12)
CDt =
(13)
Kt/t.
acquisition of capital by both the capital and goods sectors
(AK and AG) depends on
supply line of unfilled orders each has accumulated (the backlogs
BK
and
BG) and
the fraction
of demand satisfied FDS. Each period both the capital and goods sectors acquire the
supply line of unfilled orders unless the capital sector
amount. In
this
acquisition lag
case acquisition by each sector
X
is
thus 1/FDS, and
is
AK+AG=PR
is
the
full
unable to produce the required
reduced
in proportion to the shortfall.
at all times,
ensuring that output
The
is
conserved:
AKt = BKt*FDSt
(14)
AGt = BGt*FDSt
(15)
FDSt = PRt/DPt
(16)
DPt = BGt + BKt.
(17)
The supply
lines of unfilled orders for
BG
and
BK
are
augmented by orders
for
by each sector and emptied when those orders are delivered:
capital placed
BKt+i = BKt + (OKt
New
each sector
AKt)
(18)
BGt+i = BGt + (OGt-AGt).
(19)
-
orders placed by the goods sector are the exogenous input to the system to which the
subject of the experiment
for their
own
must respond by choosing an appropriate amount of
capital to order
use:
OGi = exogenous
(20)
OKt =
(21)
determined by subject.
Subjects are responsible for only one decision
- how much
capital to order.
The goal
D-3921
9
of the subjects in making these decisions
score
minimize
to
is
their total score for the trial.
defined as the average absolute deviation between desired production
is
production capacity
PC
over the
T
DP
The
and
periods of the experiment:
T
S = (1/T)
S
I
DPt
-
PCt
(22)
|.
t=0
The score
indicates
how
equally for both excess
The experiment
well subjects balance
demand and excess
is
implemented on
demand and
supply.
Subjects are penalized
supply.
IBM
PC-type microcomputers.
A
'game board'
is
displayed on the screen providing subjects with perfect information. Color graphics and
animation highlight the flows of orders, production, and shipments to increase the
transparency of the structure. The conversion of the original model to a form suitable for
experimental testing
is
described in Sterman 1987a.
The subject population (N=49) consisted of
students in
MIT
undergraduate, master's and doctoral
management and engineering, many with extensive exposure
contro^ theory; scientists and economists
from various
institutions in the
to
economics and
US, Europe, and
the
Soviet Union; and business executives experienced in capital investment decisions including
several
company
presidents and
CEOs.
Typical experimental results are shown in figure
period 3 there
is
a one-time step input of
10%
3.
All trials begin in equilibrium.
in the orders of the
goods
sector.
In
The optimal
response returns the system to equilibrium within 6 periods, producing a score of
19.
contrast the vast majority of subjects produced significant oscillations (table
The average
1).
In
score for the sample was 591; the lowest was 77.
Next
the
proposed stock management heuristic (equations 5-9) was tested against
the ordering behavior of the subjects.
straightforward.
The stock
to be
Adapting the heuristic to the experimental context
managed
is
the capital stock K.
The
desired stock
is
is
10
D-3921
The
proportional to desired production.
stock
CD. The
DSL
desired supply line
loss rate
L
is
simply the depreciation of the capital
was specified according
to equation 9, with the
0*= CD.
expected acquisition lag X^
=X=
additive disturbance term
the proposed ordering rule for capital investment
OKt
=
E,
1/FDS and desired throughput
MAX(0,CDt + ASt + ASLt +
Allowing an
becomes:
(23)
Et)
ASt = as(DKt-Kt)
ASLt = asUDSLt
-
(24)
BGt)
(25)
'
DSLt = ?.e*CDt= (l/FDSt)*CDt.
To
test the rule
(26)
only the two adjustment parameters as and asL need be estimated.
All other
Maximum
likelihood
data required to determine orders are presented directly to the subjects.
estimates of the parameters for each
trial
Assuming
subject to the constraints as,
asL
normally distributed then the
maximum
^0.
were found by grid search of the parameter space,
the errors £ are independent, identical, and
likelihood estimates of such nonlinear functions are
given by the parameters which minimize the
sum of squared
Such estimates
errors.
are
consistent and asymptotically efficient, and the usual measures of significance such as the
test are
t-
asymptotically valid.
Estimates for 49
trials
together with t-statistics are given in table
ability to explain the ordering decisions of the subjects is excellent.
R2
2.
The model's
varies between
33%
and 99+%, with an overall R^ for the pooled sample of 85%.2 All but two of the estimated
capital stock adjustment parameters are highly significant.
parameter
is
The supply
line adjustment
significant in 22 trials.
Sterman 1987b analyzes the estimation
results
and identifies several 'misperceptions
of feedback' which are responsible for the subjects' poor performance.
tendency for subjects to give insufficient attention to the supply
line,
One of
these
is
the
causing subjects to
D-3921
11
continue ordering even after the construction pipeline contains sufficient units to correct any
stock discrepancy.
The present concern, however,
is
the relationship
The estimated parameters
parameters and the regimes of behavior in the model.
making behavior of
the decision
When
actual people.
between the estimated
characterize
simulated in the model are the
estimated decisoin rules of the subjects inherently stable, or do they produce limit cycles,
period multiples, or chaos?
Table 2 also indicates the mode of behavior produced by simulation of the decision rule
The parameters estimated
with the estimated parameters.
stable.
Most of
these produce
overdamped behavior of
for thirty subjects
Seven parameter
multiples.
The parameters which
sets
are
the capital stock in response to the
produce limit cycles of period
step input.
(61%)
characterize ten subjects
1,
and 2 produce period
(20%) produce chaos. Inspection
of table 2 shows that the subjects whose parameters are stable performed best in the task
while those whose parameters produce limit cycles and chaos generally had the highest
scores.
Figure 4 shows the phase portrait for several of the more interesting parameter
Note the period 5 cycle produced by (as, asL) =
produced by
(.42, 0) is,
upon closer inspection, seen
Figure 5 locates the modes of behavior
intuition
and formal
values of asL.
(.46, .33).
in
stability analysis, stability is
More
The apparent period
sets.
2 cycle
to be chaotic.
parameter space.
Consistent with both
enhanced by small values of as and large
aggressive attempts to correct the discrepancy between the desired and
actual capital stock are destabilizing: by ordering
more aggressively
the self-ordering loop
causes a larger increase in total demand, thus exacerbating disequilibrium and encouraging
still
is
larger orders in future periods.
stabilizing
Conversely, more aggressive response to the supply line
by constraining orders as the supply
line fills.
1^
D-3921
the periodic and chaotic attractors tend to occur for large values of
Thus
values of asL-
Yet the route to chaos
is
apparently
way
Stability gives
chaos as as increases for asL=0.
2 at (.42, 0), and ultimately to chaos for as ^.55.
slightly chaotic period 2 are
period
lies
1
limit cycles.
results
performance
in a
to period
one
the route to
limit cycles, then to period
Yet the parameters which produce the
by parameters which produce
surrounded on the ray asL =
Similarly, the period 5 attractor produced by the parameters (.46, .33)
between the region of
The
more complex. Consider
as and small
show
stability
and the region of period
that the parameters
common and
1
behavior.
which characterize actual decisionmaking
important macroeconomic situation can produce a wide range of
behavior, including chaos.
Experiment
The "Beer
Distribution
II:
The Production-Distribution System
Game"
and distribution system developed
at
is
a role-playing simulation of an industrial production
MIT
to introduce students
of management to the
concepts of economic dynamics and computer simulation. In use for neariy three decades, the
game
has been played
all
over the worid by thousands of people ranging from high school
students to chief executive officers and government officials.
The game
is
played on a board which portrays in a simplified fashion the production
and distribution of beer (figure
6).
Orders for and cases of beer are represented by markers
and pennies which are physically manipulated by the players as the game proceeds. Each
brewery consists of four sectors:
One person manages each
sector.
Customers demand beer from the
retailer in turn orders beer
wholesaler's inventory.
retailer,
wholesaler, distributor, and factory (R,
Customer demand
retailer,
who
is
W,
D,
F).
represented on a deck of cards.
ships the beer requested out of inventory.
from the wholesaler, who ships the beer requested out of the
Likewise the wholesaler orders and receives beer from the
The
D-3921
13
who
distributor,
and receives beer from the
in turn orders
factor^'.
The
factory produces the
At each stage there are shipping delays and order receiving delays. These represent
beer.
the time required to receive, process, ship, and deliver orders, and as will be seen play a
crucial role in the dynamics.
The
subjects' objective is to
minimize cumulative team costs over the length of the
game. Inventory holding costs for each sector are $.50 per case per week, and stockout costs
(costs for having a backlog of unfilled orders) are $1.00 per case per
The
problem.
decision task of each subject
a clear
example of the stock management
Subjects must keep their inventory as low as possible while avoiding backlogs and
satisfying customer
The
is
week.
demand. Inventory cannot be controlled
lag in receiving beer
directly but
must be ordered.
potentially variable: if the wholesaler has beer sufficient to cover
is
the retailer's orders the retailer will receive the beer desired after three weeks.
But
if
the
wholesaler has run out, the retailer must wait until the wholesaler can receive additional beer
from
the distributor.
Only the
acquiring inventory (there
The protocol
in equilibrium.
week.
To
is
no
limit to the production capacity of the factory).
for the experiment
The
is
described in Sterman 1987c.
Each inventory contains 12
disturb the system customer
and remains
cases.
demand
Customer demand
The game
is initially
increases to eight cases per
is
initialized
four cases per
week
in
week
5
at that level thereafter.
results reported here
A
over a period of four years.
consistency.
for
factory, the primary producer, faces a constant delay in
more than
Trials in
a
were drawn from four dozen games (192 subjects) collected
computer model of the game was used
to test the records for
which there were accounting errors of more than a few cases per week
few weeks
in
any of the four sectors were discarded from further analysis.
Eleven games were retained, providing 44 subjects.^ That sample consists of undergraduate,
MBA,
and Ph.D. students
at
MIT's Sloan School of Management, executives from
a variety of
D-3921
14
firms participating in short courses on computer simulation, and senior executives of a major
computer firm.
Typical experimental results are shown in figure 7 and summarized in table
results
show
3.
The
from optimal. The average team cost
the behavior of the subjects to be far
is
Like the macroeconomic experiment, the results exhibit
ten times greater than optimal.
several strong regularities.
1.
Oscillation:
The
trials are all
pattern of orders and of inventory
is
characterized by instability and oscillation.
The
dominated by a large amplitude fluctuation with an
average period of 21 weeks.
2.
Amplification: The amplitude and variance of orders increases steadily as one
moves from customer
to retailer to factory.
The peak order
average more than double the peak order rate generated
increase from 4 to 8 cases per week.
factor)', the
3.
By
rate at the factory level is
at the retail level.
on
Customer orders
the time the disturbance has propagated to the
order rate averages a peak of 32 cases.
Phase
to the factory.
lag:
The peak order
The phase
lag
is
rate tends to occur later as
one moves from the
retailer
not surprising since the disturbance in customer orders must
propagate through decisionmaking and order delays from retailer to wholesaler and so on.
Next the proposed decision
game and
cast in a
form
rule
must be adapted
to the particular situation in the beer
suitable for estimation of the parameters.
to the inventory of the subject
and the supply
line
SL
to the
sum
The stock S corresponds
of orders in the mail delays,
the backlog of the subject's supplier (if any), and the beer in the shipping delays.
rate is the rate at
which each subject receives orders. To
test the rule
it is
The
loss
necessary to
specify expected losses L^, the desired stock S*, and the desired supply line SL*.
Expected losses from the stock are the
immediate customer
to place orders, that
is,
rate at
which each subject expects
the retailer's forecast of the
their
customer order
rate,
D-3921
15
the factory's forecast of the distributor's order rate, etc.
postulated.
Adaptive expectations are widely used
economic systems, they
are often a
in
Adaptive expectations are
simulation modeling of corporate and
good model of the evolution of expectations
in the
aggregate (Sterman 1987d, Frankel and Froot 1987), and they are one of the simplest
formulations for expectations flexible enough to adapt to a nonstationary process.
the subjects lack the information
the desired stock
and time
S* and desired supply
Because
to determine optimal inventory or supply line levels
line
SL*
are both
assumed
to
be constants which
must be estimated.
The generic decision
rule of eq. (5-9) thus
becomes:
Ot = MAX(0,IOt)
(27)
IOt=Let + ASt + ASLt
(28)
Let = eLt-l + (l-e)Lei.i.
ASt = as(S*
-
0<e<l
(29)
St)
ASLt = asL(SL*
-
(30)
SLt).
(31)
Defining P = asUct-s and S' = S* + PSL* yields
lOt =Let + as(S'-St-pSLt).
Note
that since S*,
place
more emphasis on
SL*, asL and as are
(32)
all
>0, S'>0.
Further,
the supply line than on the inventor)'
it is
unlikely that subjects will
itself:
the supply line does not
directly enter the cost function nor
is it
asLS
can be interpreted as the fraction of the supply line taken
as, meaning 0<(3<1. Thus
into account
by the subjects.
If
(3
P=
1,
as salient as inventory.
Therefore
it is
probable that
then the subjects fully recognize the supply line and do
not double order. If P = 0, then orders placed are forgotten until they arrive, encouraging
overordering and
instabilitj.'^
In contrast to the
macroeconomic experiment,
the stock
management decision
rule as
D-3921
16
implemented
in the
beer
game
contains four parameters to be estimated (6, as, S', and p).
The parameters were estimated by
An
additive disturbance term
is
the
same procedure used
in the long
wave experiment.
assumed, and the disturbances are assumed to be
normal. Estimates for each sector of each
trial
i.i.d.
were found by grid search of the parameter
space subject to the constraints 0<6<1 and as, S', p >0.5
The estimated parameters
than
50%
for only 6 of
significant.
As
44
subjects.^
in the long
A large
wave experiment
decisionmaking behavior of the subjects.
the decision rule
Sterman 1987c
It is
feedback structure which cause oscillation
in the
Next
given far too
the beer
trial
is less
is
an excellent model of the
relates the estimated parameters to
macroeconomic experiment
are responsible
game. Specifically, the supply
line
of
weight by most subjects.
game was simulated using
estimated parameters. Each
parameters.
little
71%; R^
noteworthy that the same misperceptions of
for the poor performance of the subjects in the beer
is
is
majority of the estimated parameters are
the dysfunctional performance in the task.
unfilled orders
The mean R2
are displayed in table 4.
the stock adjustment rule with the
involves four subjects, each of
In the simulations, however, the
whom
could have different
same parameters were assumed
for each
sector, vastly simplifying the analysis.*^
Table 4 indicates the
one parameter
1,
sets
mode
(70%) produce
of behavior produced by each set of parameters. Thirtystable behavior.
period 5, and period 12. Ten yield chaos.
behavior produced by the system
projections of phase space.
is
One each produces
Since the beer
game
is
limit cycles of period
a 19th order
exceedingly complex and there are
many
system the
possible
Figure 8 shows several of the more interesting phase portraits.
Note, for example, the different shape of the attractor produced by simulation with the
17
D-3921
parameters of the Bassbeer Factory
HeinekenS Factory
Factory
(9, as, p,
as, P, S' = 0,
(9,
S' = .25,
1
The
.
as, P, S' =
1,
.65, .40, 15)
The
retail sector
difference in
mode
is
compared
The parameters estimated
.3, .15, 17).
18) reveal a cycle of period 12
.3, 0,
plotted against factory inventory.
limit cycle of period
(9,
when
to those of the
for the
Coors
distributor inventory
is
of the same team, however, produces a
probably due to the integer constraint on
orders.
The
Figure 9 shows the distribution of the modes in parameter space.
between the adjustment parameters and
macroeconomic experiment, and
for similar reasons.
smaller values of P are destabilizing.
line of unfilled orders
even
stability are similar to those
That
is,
apparent in the
In general, larger values of
since
more
stock.
(P=0) a stock discrepancy will cause orders to be placed each period
Such overordering
is
and
exacerbated by more aggressive stock adjustment (as>>0)
will be ordered in response to a given discrepancy
between the desired and actual
Figure 9 also indicates that in general the smaller the desired inventory S' the less
stable the system.
Again, intuition readily explains
the retailer to an unanticipated increase in customer
lags in receiving beer.
The
the stock adjustment term.
(S'»0) incoming
The delay
why
this is so.
If the
orders can
demand. Retail inventory
all
wholesaler has adequate inventory to
be filled and the
still
orders swell the supply line in a vain attempt to replenish inventory.
retailer finds inventory
fill
retailer finds orders
drops
falls
due
to the
wholesaler through
retailer's acquisition lag
in receiving beer lengthens, the retailer's stock
does arrive, the
Consider the response of
retailer will place additional orders with the
however, the wholesaler runs out of inventory the
finally
as and
to the extent the subjects ignore the supply
after sufficient orders are in the supply line, leading to excess inventories
oscillation.
If,
relationships
incoming orders
remains constant.
placed
further,
When
and
fail to arrive.
still
more
sufficient beer
overshooting the desired level by huge margins
18
D-3921
(figure 7).
S' plays such a crucial role in determining the stability of the system because
determines
how
it
close the system operates to the fundamental nonlinear contraint that
shipments can only be made
if
inventor)' exists.
Small values of S' make
it
more
likely the
system will run out of inventory thus entering the unstable region in which the acquisition lag
X
is
thus
long and variable.
more
stable
Those subjects who maintained
on average than those who attempted
larger target inventory levels
to cut costs
by reducing
were
their buffer
stock.
There appears
the system.
to
be no simple relationship between the parameter 6 and the modes of
6 controls the speed with which the adaptive forecast of incoming orders
Faster adjustment of the
adjusted and has two opposing effects on stability.
(9=1) increases the gain between orders placed and incoming orders.
demand
is
forecast
Shocks are transmitted
up the distribution chain more readily, destabilizing the system by causing larger inventory
excursions upstream in response to a given change in orders downstream.
On
the other
hand, sluggish adjustment (0=0) means variations in incoming orders are not quickly
countered by the replacement term of the ordering rule, causing inventory to decline further
than
it
would
if
replacement of incoming orders was swift. As a
result, inventories
may be
exhausted, moving the system into the less stable zone in which backlogs build up and the
acquisition lag lengthens.
As
behavior
is
in the
macroeconomic experiment
the results
show an unexpectedly
rich array of
generated by the interaction of the decision processes of real people with the
feedback structure of the environment.
Discussion
The discovery of nonlinear phenomena such
as deterministic chaos in the physical
world naturally motivates the search for similar behavior
the social scientist faces difficulties in that search
in the
world of human behavior. Yet
which do not plague the
physicist, at least
D-3921
19
not to the
exist for
same degree. Aggregate data
many of
most important
the
sufficient for strong empirical tests simply
social systems.
from the environment. The huge temporal and
Social systems are not easily isolated
spatial scales
of these systems, vast number
of individual actors, considerations of cost and ethical concerns
on the systems themselves
difficult at best.
do not
make
Finally, the laws of
controlled experiments
human behavior
are not as
stable as the laws of physics.
The
social scientist is thus left with
relevance of nonlinear
systems
may be
phenomena
in the
modes
in a
in the quest to
understand the
domain of human behavior. Formal models of
constructed and analyzed.
existence of chaotic
two main options
social
Previous work of this type has demonstrated the
wide array of
social
and economic models. However, analysis
of these models frequently showed that the chaotic regime lay far outside the plausible region
of parameter space.
the
Worse, without empirical
models was open
phenomena remained
to doubt.
In the absence of empirical data the relevance of nonlinear
is
to
develop laboratory experiments with simulated social
Since experiments on actual firms and national economies are infeasible, simulation
models of these systems must be used
people.
of the decision rules the verisimilitude of
questionable.
The second approach
systems.
tests
to explore the
Such experiments create 'microworlds'
institutional structures, information,
decisionmaking heuristics of real
which the subjects face physical and
in
and incentives which mimic
(albeit in a simplified
fashion) those of the real world.
The experiments described here presented
subjects with a straightforward stock-
adjustment task. Results show that the subjects' behavior can be modeled with a high
degree of accuracy by a decision rule long used
is
in
system dynamics models. The decision rule
consistent with a vast body of empirical knowledge developed in behavioral decision
theory, organizational studies, and psychology.
The stock-adjustment
heuristic appears to
20
D-3921
be robust: the feedback structure, information available, loss function, time available, "cover
story"
and many other dimensions of the experimental environment differed markedly.
Performance
in
both settings was decidedly suboptimal.
feedback structure of the system are apparent
in
The same misperceptions of
the
both cases.
The experiments reported here demonstrate
that
chaos can in fact be produced by the
decisionmaking processes of real people. The results are significant in several respects. The
demonstration that chaos can be produced by the decisionmaking heuristics people actually
Chaos thus appears
use strengthens the argument for the universality of these phenomena.
to
be a
common mode
of behavior not only in physical systems but in social and economic
systems as well. These results do not show
States has in fact been chaotic.
be
settled.
peril.
But the
Further empirical work
results suggest
Models of economic and
evolution of
that, say, the
GNP
required before such questions can
is
modelers can ignore nonlinear dynamics only
social
all
times or that adjustment processes are stable.
formulated so that they are robust in extreme conditions, since
it
is
These principles have long been central
Forrester and Senge 1980, Richardson and
to
economy
is in
or
Models should be
the nonlinearities
necessarily introduced by robust formulations that crucially determine the
system.
at their
dynamics should portray the processes by which
disequilibrium conditions are created and dissipated, not assume that the
near equilibrium at
United
in the
modes of
the
system dynamics (Forrester 1961,
Pugh 1981) and behavioral economics
(Da)' 1984)
and are important even for systems which do not contain limit cycles or strange attractors
(Sterman 1985b).
At
the
same time
a
number of questions regarding
and other nonlinear phenomena
in social
the practical significance of chaos
system modeling remain unresolved. Real social
systems are bombarded by broadband noise, and
it
is
well
known
that
such random shocks
severely degrade the point-predictability of most systems (Forrester 1961, App. K).
Does
21
D-3921
the existence of stochastic shocks
How
swamp
does the existence of chaotic regimes
the predictability of that response?
the uncertainty in trajectories caused by chaos?
model influence
in a
Particularly troubling here
the simulated beer
response to policies, and
the apparent nonuniformity
Mosekilde and Larsen's
of the distribution of modes in parameter space.
bombing experiments with
is
its
game show
(this issue) carpet
that there are islands of chaos in
parameter space, and that the boundary between the periodic and chaotic solutions
a finding consistent with analysis of such classic systems as Buffing's equation
and Stewart 1986). The response of
stability to
may
parameter variations
Similarly, chaos
frames, but
all
many
et al.
a steady-state
such systems
is
a major area for future research.
phenomenon which manifests over very long time
much
1986, Weidlich et
al.
learning and evolutionary pressures.
this issue).
How
Over such extended time horizons
static
decision functions remain constant throughout.
here, for example,
Yet there
that subjects begin to learn within just a
is
the behavior of
The parameters of
the
few cycles, modifying the parameters
at least,
the chaotic region into the region of stability.
in policy-oriented
show
evidence (Sterman 1987b,
of their ordering function. In the macroeconomic experiment,
chaos and other nonlinear phenomena
the
but will themselves evolve with
systems simulated for thousands of years and weeks, respectively.
away from
(e.g.
does learning influence chaotic systems? The
macroeconomic and corporate experiments described
subjects
dynamics
shorter than those used in the analysis of chaotic
parameters of the system cannot be considered
Bakken 1987)
The dev-
policy-oriented models are concerned with transient dynamics and nearly
with time horizons
Mosekilde
is
in
(Thompson
not be monotonic
or even predictable in such systems, raising major questions for policy analysis.
elopment of principles for policy design
is fractal,
The
learning
moves
the
practical significance of
models of social and economic
behavior remains clouded while these questions are unanswered.
More
important, the results demonstrate the feasibility of subjecting theories of
22
D-3921
human behavior
to
experimental
Many models
test.
of
human systems may be
built
which
produce an astounding variety of behaviors. Unfortunately many of these models are
formulated without regard to the vast body of empirical knowledge documenting the decision
processes of individuals.
For example, rational expectations models
economics presume
in
agents have perfect knowledge of and the cognitive capability to solve the system of
equations which characterize the
economy (Lucas 1976, Shaw
strongly contradicted by evidence at
all
levels of aggregation
1984).
These assumptions
are
(Simon 1984, Hogarth and
Reder 1986) and form a poor foundation for descriptive models of economic dynamics.
Similarly, physical scientists frequentiy specify decision processes in
behavior by analogy with natural systems (Beylich 1986,
Zeeman
1977,
models of human
Zimmerman
1986),
or simply assert that the processes studied in natural systems extend by analogy to social
dynamics (Ruelle 1984,
May
1976).
The
natural sciences can indeed be a fruitful source of
metaphor for theories of human behavior, and there
of nonlinear science to social systems
empirical grounding in
tiie
(e.g.
is
encouraging progress
Prigogine and Sanglier, 1987).
psychology of choice behavioral
empirical
test.
as accepted
But without
scientists will continue to
such models as ad hoc and continue to question their relevance.
assess the validity of such
in the application
It
a firm
view
has been difficult to
models because the assumed decision rules were not subject
In the future experimental tests of models of
and commonplace as they are
to
human behavior should become
in the natural sciences.
Over time
the application
of these methods should identify the processes of judgment people actually use in dynamic
decisionmaking tasks and which are acceptable as the basis for accurate modeling of human
affairs.
Models formulated on
the basis of
with the tools of simulation and
offer the best
systems.
hope
to
modem
knowledge of individual decisionmaking, analyzed
nonlinear science, and subjected to experimental
improve our understanding of the dynamics and evolution of
social
test
D-3921
23
Notes
1.
Order cancellations are sometimes possible and may exceed new orders
conditions (e.g. the U.S. nuclear
power
likely to be subject to different costs
industry in the 1970s).
in
extreme
Since cancellations are
and administrative procedures than
new
orders
they should be represented separately as a distinct outflow from the supply line of
unfilled orders rather than as negative orders.
2.
Note
that the function
OK=f(-) does not contain an estimated regression constant. Thus
the correspondence of the estimated and actual capital orders, not just their variation
around mean values, provides an important measure of the model's explanatory power.
Since the residuals e need not satisfy Zet =
The
alternative
R2 =
1
-
Set^
/
lOKt^
is
the conventional
used (Judge
et al.
R^
1980).
is
not appropriate.
This
R2 can be
interpreted as the fraction of the variation in capital orders around zero explained by the
model.
3.
Analysis showed a slight tendency for the
trials
with the most extreme amplitude and
highest costs to be most prone to accounting errors.
trials is
The
biased slightly towards those
effect
is
who
Thus
the final sample of eleven
understood and performed best in the game.
modest, however, and reinforces the conclusions drawn below regarding
misperceptions of the feedback structure by the subjects.
experiment and
4.
In the
5.
The parameters
respectively.
in the
6, as, p,
and S' were estimated to the nearest
The search was
capturing the global
simulations orders are restricted to the positive integers.
minimum
.1, .05, .05,
and
1
carried out over a sufficiently large range to ensure
of let^.
units,
i^- ^ J ^
6.
1
Note
that S' functions
nonlinearity
much
Ot>0 means
like a regression constant in eq. 32.
However
the residuals will not, in general, satisfy Let =
and actual orders need not have a
common
an appropriate measure of goodness of
fit.
mean). The conventional R^
The
alternative
R^ =
(estimated
is
r^ is used,
the coefficient of correlation between estimated and actual orders (Judge et
7.
Subjects were assigned positions randomly, and one-way
ANOVA
the
not therefore
where
al.
r is
1980).
of the estimated
parameters showed no strong relationship between the values of the estimated
parameters and the position one plays
significant at the
4.3%
in the
game
(with the exception of
as which was
level).
N.B. The long wave game runs on any
IBM PC
or compatible computer; disks are available
from the author. The estimation and simulations for both experiments were carried out
on Macintosh computers using the TrueBASIC language. The data and computer
programs are available from the author upon request. To avoid
plots
shown
in figure
discarding the
first
1500 weeks. The
the first 1000
4 were generated by simulateing the model
transients, the
phase
for 10,000 periods
and
9000. Similarly, the phase plots shown in figure 8 were simulated for
first
500 were discarded
were discarded.
for all except the Bassbeer factory, for
which
D-3921
25
References
Bakken, B. 1987. Learning System Structure by Exploring Computer Games, unpublished
manuscript, System Dynamics Group, Sloan School of Management,
MIT, Cambridge
MA
02139.
Beylich, A. 1986.
On
the Modelling of Vehicular Traffic Flow. In C. Kilmister (ed.)
Disequilibrium and Self-Organization. Dordrecht: D. Reidel.
Cyert, R. and
J.
March. 1963. A Behavioral Theory of the Firm. Englewood
Cliffs N.J.:
Prentice Hall.
Davis, H. L., S.
J.
Hoch, and E. K. Easton Ragsdale. 1986. An Anchoring and Adjustment
Model of Spousal
Predictions. Journal of
Consumer Research. 13:25-37.
Day, R. 1984. Disequilibrium Economic Dynamics:
Journal of Economic Behavior and Organization.
A Post-Schumpeterian
Contribution.
5, 57-76.
Day, R. 1982a. Irregular Growth Cycles. American Ecoru>mic Review. 72, 406-414.
Day, R. 1982b. Complex Behavior
Einhom, H.
J.,
in
System Dynamics Models. Dynamica.
and R. M. Hogarth. 1985. Ambiguity and Uncertainty
8,
82-89.
in Probabalistic Inference.
Psychological Review. 92:433-461.
Forrester,
J.
W.
Forrester,
J.
W. and
1961. Industrial Dynamics.
P.
M. Senge.
Models. TIMS Studies
Frankel,
J.
in the
Cambridge,
MA: MIT
Press.
1980. Tests for Building Confidence in System
Management
Sciences.
Dynamics
14:201-228.
A. and K. A. Froot. 1987. Using Survey Data to Test Standard Propositions
Regarding Exchange Rate Expectations American Economic Review 77:133-153.
Hines,
J.
1986
A
Behavioral Theory of Interest Rate Formation. Working paper
Sloan School of Management, MIT.
Hogarth, R.
M.
1987. Judgement
and Choice. 2nd
ed.
New
York: Wiley.
WP- 177 1-86,
26
D-3921
M. and M. W. Reder
Hogarth, R.
(eds). 1986.
The Behavioral Foundations of Economic
Theory Journal of Business. 59:S181-S505.
Jarmain,
W.
Johnson, E.
E. (ed.) 1963.
Problems
Dynamics. Cambridge,
in Industrial
MA: MIT
Press.
and D. A. Schkade. 1987. Heuristics and Bias in Utility Assessment.
J.
Unpublished manuscript, Wharton School, University of Pennsylvania, Philadelphia.
Judge
et al. 1980.
The Theory and Practice of Econometrics.
New
York: Wiley.
Lopes, L. L. 1981. Averaging Rules and Adjustment Processes: The Role of Averaging in
Inference. Report 13, Wisconsin
Human
Information Processing Program, University of
Wisconsin, Madison.
Lucas, R.E. 1976. Econometric Policy Evaluation:
(eds).
A
Critique.
The Phillips Curve and Labor Markets. Supplement
In K.
Brunner and A. Meltzer
to the Journal
of Monetary
Economics.
May, R. M. 1976. Simple Mathematical Models with Very Complicated Dynamics. Nature.
261(5560):459-467.
Bounded
Omega. 11:131-142.
Morecroft,
J.
1983. System Dynamics: Portraying
Morecroft,
J.
1985. Rationality in the Analysis of Behavioral Simulation Models.
Rationalit>'.
Management
Science 31:900-916.
Mosekilde, E., D. Rasmussen, H. Jensen,
in a Generic
Sturis,
and
J.
Jespersen. 1987. Chaotic Behavior
Management Model. European Journal of Operations Research. Forthcoming.
Mosekilde, E. and E. Larsen.
Dynamics Review.
Nicolis, G.
J.
and
I.
1987.
Chaotic Behavior
in the
Beer Distribution Game. System
4(1).
Prigogine. 1977. Self-Organization in Nonequilibrium Systems.
New
York:
Wiley.
Plott, C. 1986.
Laboratory Experiments
in
Economics: The Implications of Posted Price
D-3921
27
Science. 232 (9 May), 732-738.
Institutions.
Prigogine,
and M. Sanglier
I.
(eds.) 1987.
Laws of Nature and Human Conduct.
Brussels:
Task Force of Research Information and Study on Science.
Rasmussen,
a Simple
S.,
E. Mosekilde, and
Model of
the
J.
D. Sterman. 1985. Bifurcations and Chaotic Behavior
Economic Long Wave. System Dynamics Review. 1(1):92-1
in
10.
Richardson, G.P.and A.L. Pugh. 1981. Introduction to System Dynamic Modeling with
DYNAMO.
Cambridge
MA: The MIT
Ruelle, D. 1984. Strange Attractors.
Adam
Press.
In P. Cvitanovic (ed.)Universality in Chaos. Bristol:
Hilger, Ltd.
Shaw, G.K. 1984. Rational Expectations.
New
Simon, H. A. 1979. Rational Decisionmaking
York:
in
St.
Martins Press.
Business Organizations. American Economic
69:493-513.
Revie^^.
Simon., H. A. 1984. The Behavioral and Rational Foundations of Economic Dynamics. Journal
of Economic Behavior and Organization. 5:35-55.
Smith, V. 1982. Microeconomic Systems as an Experimental Science. American Economic
Review. 72, 923-955.
Sterman,
J.
D. 1984. Instructions for Running the Beer Distribution Game. System Dynamics
Group working paper D-3679, Sloan School of Management, MIT.
Sterman,
J.
D. 1985a.
A
Behavioral Model of the Economic Long Wave. Journal of Economic
Behavior and Organization. 6:17-53.
Sterman,
J.
D. 1985b.
A
Skeptic's
Guide
to
Computer Models. Working Paper D-3665,
System Dynamics Group, Sloan Scholl of Management, MIT, Cambridge
Sterman,
J.
MA 02139.
D. 1986. The Economic Long Wave: Theory and Evidence. System Dynamics
Review. 2(2):87-125.
28
D-3921
Sterman,
J.
D. 1987a. Testing Behavioral Simulation Models by Direct Experiment.
Management
Sterman,
J.
Science.
D. 1987b. Misperceptions of Feedback
WP- 1899-87.
paper
Sterman,
J.
33(12).
in
Dynamic Decisionmaking. Working
Sloan School of Management, MIT, Cambridge
MA.
D. 1987c. Managerial Behavior in Dynamic Decisionmaking: Further
Misperceptions of Feedback.
System Dynamics Group working paper D-3919, Sloan
School of Management, MIT.
Sterman,
J.
D. 1987d. Expectation Formation in Behavioral Simulation Models. Behavioral
Science. 32:190-211.
Sterman,
Game
J.
D. and D. L. Meadows. 1985.
STRATEGEM-2: A Microcomputer
Simulation
of the Kondratiev Cycle. Simulation and Games. 16:174-202.
Thompson,
J.
and H. Stewart. 1986. Nonlinear Dynamics and Chaos.
Tversky, A. and D.
Science
.
Kahneman
1974. Judgment
New
York: Wiley.
Under Uncertainty: Heuristics and Biases.
185 (27 September): 1124-1 131.
Zeeman, E. C. 1977. Catastrophe Theory: Selected Papers, 1972-1977. Reading,
MA:
Addision Wesley Publishing Co.
Zimmerman, R. 1986. The
Century. In C. Kilmister
Transition from
(ed.).
Town
to City: Metropolitan
Behavior
in the 19th
Disequilibrium and Self-Organization. Dordrecht: D. Reidel.
29
D-3921
Figure
1.
The generic stock-management system.
u
Exogenous
Variables
Stock Acquisition System
Expected
Loss Rate
Ordering Heuristic
D-3921
Figure
2.
30
The stock management system applied
to aggregate capital investment.
portrays a firm which produces capital plant and equipment.
is
exogenous, as
multiplier
is
is
the delivery delay for acquiring
introduced, the firm's
the delivery delay for capital
new
The demand
capital stock.
The model
for the firm's product
When
the self-ordering
orders for capital add to the exogenous demand, and
the firm's own delivery delay.
own
becomes
CAPITAL
N
^ DISCARDS
o
CARTAL
STOCK
SUPPLY
UNE
CAPITAL
ACQUISITIONS
/
S^
ill
(
DELIVERY
L
DELAY
7
fpRooucnoNj
\
J
BACKLOG
K-not
_
-/OTHER
\ORDERS
D-3921
31
Figure
4
3.
Typical results of the macroeconoinic experiment.
iitKaMaMMMMMa
<
<
•
It
II
a
14
a
32
D-3921
Table
1.
Macroeconomic experiment: Summary of results.
35
D-3921
Table
Trial
1
Score
2.
Macroeconomic experiment: Estimated parameters.
as
Std. Error*
asL
Std. Error*
R2
Mode
D-3921
34
Figure
4.
Simulation of the decision rule with estimated parameters.
HOOn Game 36
1200n Gome 18
1200
10001000
600
600
1200
800
1200
eoo
100
Production CapecUij
Proaucllon Capacity
*
eoo
Game
12
750
c
TOO
o
880
o
a.
^
600
•>
o
550
500
1230
150
700
600
500
eoo
Production Capacity
1200-1
Game
10
100
800
Production Capacity
1200
1250
Production Capacity
1270
35
D-3921
5. Modes of the system in parameter space.
Lower graph zooms in to area between (0,1).
Figure
b-
a
U- J3 L.
1
:
:
1?
37
D-3921
Figure
7a:
7.
T^'pical experimental results of the
Compare
Beer distribution Game.
week 5 customer
orders rise from 4 to 8 cases per week.
against the oscillations in the subjects' orders (7c).
Customer Orders.
In
3025
-
ro15 '
J
-
5
-
-
Figure 7b.
Key
a
to experimental results (figure 7c).
Factory
^
Distributor
_ Wholesaler
Retailer
5
10
U
20
25
30
Orders placed by sector.
From bottom to top,
R. W. D, F, each offset by 1 5 cases/week.
Major tick-marks=l5 cases/week.
Minor tick-marks=5 cases.'week
Initial orders=4 cases/week in all sectors.
:5
Weeks
Factory
Effective Inventory
o
c
by seaor.
Effective lnventory=lnventory-Backlog.
01
>
c
Distnbutor
From bottom to top,
R, W. D, F, each offset by 40 cases.
o
o
Wholesaler
Maior tick-marks=40 cases.
Minor tick-marks=lO cases.
Initial inventorysl2 cases in
Retailer
3
10
13
20
23
Weeks
30
33
all
sectors.
38
D-3921
Figure 7c. Typical experimental results.
Effective Inventory
Orders
»-
D-3921
39
Table
3.
Summary
of experimental results, Beer Distribution
Data are averages of
Total
COSTS
Mean
Benchmark*
Ratio
1 1
Game.
trials.
Retailer
Wholesaler
Distributor
Factory
D-3921
40
Table
Trial
&
Position
Bassbeer
6
4.
Estimated parameters, Beer Distribution
as
p
S'
Game
r2
Mode
41
D-3921
Figure
8.
Simulation of decision rule with estimated paraemters, Beer Distribution Game.
50
60
HemekenJ Retailer
Coors Factory
10
50
20
20
-20
10
-10
-60
-10
-eo
-20
20
Factory Inventory
Factory Inventory
80
1
—
I
42
D-3921
Beer Distribution Game. Modes of
Figure 9.
the system in parameter space.
1.0
Stable
0.8
•
Period
•
Period 5
•
1
Period 12
Chaos
0.6
0.4-
0.2
D
B
o
f
0.0
0.0
0.2
»
*
—
n
I
—D-
o
1
0.4
0.6
a
0.8
1.0
_
40
Stable
•
Period
•
Period 5
•
Period 12
35
1
30
Cliaos
25S'
20-
•
B
I
.
15
10
5
—
0.0
0.2
0.4
a
g
I
0.6
0.8
Date Due
iOV 151990
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