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WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
EXPERIMENTAL EVIDENCE OF DETERMINISTIC CHAOS IN
HUMAN DECISION MAKING BEHAVIOR
John Sterman, Erik Mosekilde, Erik Larsen
REVISED June, 1988
WP# 2002-88
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
rr,-:
18 1988
EXPERIMENTAL EVIDENCE OF DETERMINISTIC CHAOS IN
HUMAN DECISION MAKING BEHAVIOR
John Sterman, Erik Mosekilde, Erik Larsen
REVISED June, 1988
WP//
2002-88
D-3957-1
Experimental Evidence of Deterministic Chaos
in Human Decision Making Behavior
John D. Sterman*, Erik Mosekilde**, Erik Larsen"
March 1988
Revised June 1988
Abstract
An
experiment with a simulated microeconomic system demonstrates that the decision-
making processes of human subjects can produce deterministic chaos.
a
commodity production-distribution system
systematically suboptimal.
estimates
show
the
model
A
is
model of the
to
minimize
costs.
Participants
managed
Performance, however, was
subjects' decision rule
is
proposed. Econometric
an excellent representation of the actual decisions.
Simulation
of the estimated rules yields stable, periodic, quasiperiodic, and chaotic solutions.
Analysis
of the parameter space reveals a complex structure including mode-locking, a devil's
case,
and
fractal basin boundaries.
Implications for modeling
human systems
stair-
are explored.
Massachusetts Institute of Technology, Sloan School of Management, Cambridge,
MA
02139
**
Physics Laboratory HI, Technical University of Denmark,
DK-2800 Lyngby, Denmark
2
Sterman, Mosekilde, and Larsen
The prevalence of
deterministic chaos and other nonlinear
phenomena
in physical,
chemical, and biological systems has prompted speculation that these dynamics
may
occur in
Indeed, numerous models have
economic, managerial, and other human systems as well
(1).
shown how economic systems might produce chaos
But the significance of these
(2).
important theoretical developments hinges on whether the chaotic regimes
Further, the decision rules postulated in these models have not
region of parameter space.
been tested experimentally. Despite intriguing
series (3),
it is
the realistic
lie in
difficult to resolve
efforts to identify
chaos
in
such issues by empirical means alone
often unavailable, or too short relative to the frequencies of interest.
economic time
Data
(4).
series are
Measurement
error
and
process noise in social and economic data are significant, further complicating empirical
analysis (5).
A
complementary approach
is
based on laboratory experiments
provide a simulated environment for the study of decision-making
results of
human
The experiment
systems give firms
We report here
common economic
the
flexibility
by establishing a decentralized network of inventories
demand
"Beer Distribution Game,"
at
is
MIT
The experiment
The experiment
is
here, the
a role-playing simulation of a typical production-distribution
to introduce students
of management to economic dynamics and
computer simulation, the game has been widely used for nearly three decades
conducted on a board which portrays
production and distribution of beer (figure
manipulated by the players as the
game
1).
Beer
proceeds.
wholesaler, distributor, and factory (R,
is
to buffer
and production of commodities and
for
business cycles and other fluctuations in industrial economies (7).
Customer demand
Such
Production-distribution systems have long been associated with
manufactured goods.
Developed
context.
creates a simulated industrial production-distribution system.
unanticipated variations between the
retailer,
which models
one such experiment which demonstrates that the decision-making processes of
subjects can produce deterministic chaos in a
system.
(6).
in
is
(8).
in a simplified fashion the
represented by markers which are
Each brewery consists of four
W, D,
represented by a deck of cards.
F).
One
subject
sectors:
manages each
sector.
Each simulated week customers order
Sterman, Mosekilde, and Larsen
beer from the
who
retailer,
3
ships the beer requested out of inventory.
orders from the wholesaler,
who
likewise ships out of inventory.
orders and receives beer from the distributor,
The
factory.
factory brews the beer.
Each week
delays.
(factories decide
the subjects
how much
who
The
Similarly, the wholesaler
and receives beer from the
in turn orders
At each stage there are shipping and order receiving
must decide how much
to order
from
their
immediate supplier
beer to brew).
Subjects seek to minimize cumulative costs incurred during the
weeks).
retailer in turn
game
(36 simulated
Inventory holding costs for each sector are $.50/case/week, and backlogs of unfilled
orders cost $ 1 .00/case/week.
Due
avoiding backlogs.
unfilled orders
may
Subjects must therefore keep their inventory low while
and shipping
to the order receiving
build up.
The
may
lag in receiving beer
lags, a substantial supply line of
vary:
if
the wholesaler can cover
the retailer's orders, the retailer will receive the beer desired after four weeks.
But
if
the
wholesaler has run out, the retailer must wait until the wholesaler receives additional beer.
Only the
factory, the primary producer, faces a constant delay in acquiring inventory (for
simplicity there
is
highly simplified,
no
it
limit to production capacity).
nevertheless shares
many
negativity constraints on orders and shipments:
if
inventory
is
is
features with real economies, including
The
multiple feedbacks, nonlinearities, and time lags.
but
Although the experimental system
nonlinearities arise
from non-
shipments normally equal incoming orders,
depleted, shipments must equal zero, and the unfilled orders are held in the
backlog for future delivery.
The experiment follows standard protocols
initialized in equilibrium.
cases per week.
demand
Each inventory contains 12
The system
to 8 cases per
Forty-eight
week
trials
(6, 9)
is
in
and
cases.
is
detailed in (10).
Customer demand
week
is initially
5.
(192 subjects) were analyzed.
lower costs than the
trial is
disturbed by an unannounced step increase in customer
Trials in
accounting errors were discarded, leaving eleven (44 subjects).
slightly
Each
full
which subjects made
This subset generated
sample, indicating that these subjects understood and
4
4
Sterman, Mosekilde, and Larsen
performed best
MIT
Participants
in the experiment.
were graduate and undergraduate students
and senior executives from a number of U.S. firms.
Results are summarized in table
behavior
is
minimize
regularities, suggesting subjects
1;
figure 2
shows a
typical
More
costs.
used a
common
initial
The
participants'
it is
not surprising
interesting, the results exhibit strong
heuristic in ordering.
and inventory are dominated by a large amplitude fluctuation.
required to recover
trial.
Given the complexity of the system
significantly suboptimal.
that subjects are unable to
inventory levels. 2. Amplification:
orders rises steadily from customer to retailer to factory.
8
at
On
1.
Oscillation: Orders
average 21 weeks are
The amplitude and variance of
Customer orders increase from 4
to
cases/week; responding to this disturbance, factory orders rise from 4 to a peak averaging
32 cases.
3.
Phase
Orders tend to peak
lag:
later as
one moves from
retailer to factory (not
surprising since orders propagate upstream through decision-making and processing delays).
We
next estimate a model of the subjects' decision rule (see (10) for details).
We
hypothesize that subjects utilize the anchoring and adjustment heuristic to determine orders
(11).
Anchoring and adjustment
quantity
is
estimated by
adjusting for factors
person's weight
first
is
common
recalling a
which may be
may be
a
judgmental strategy
known
which an unknown
reference point (the anchor) and then
less salient or
whose
effects are obscure.
estimated by recalling the subject's
differences in height, girth, etc.
in
own weight and
For example, a
adjusting for
Anchoring and adjustment has been documented
in a
wide
variety of decision-making tasks (11, 12).
Here the anchor
is
expected demand D^, the subject's forecast of incoming orders.
Replacement of incoming orders,
ceteris paribus, maintains inventory at
The asymmetric cost function means
nonnegative value
S'.
the desired stock S'
line of unfilled orders.
received.
The supply
Finally, orders
O
current value.
subjects should maintain their inventory at a
Thus an adjustment
and actual stock
its
S.
line
to the
Subjects
SL
is
anchor arises from the discrepancy between
may
also consider a fraction P of the supply
the accumulation of orders placed but not yet
must be nonnegative (order cancellations are not allowed). Thus,
Sterman, Mosekilde, and Larsen
Ot= MAX(0,
5
D^t + a(S'
Sf
-
where the stock adjustment parameter
pSLj))
(1)
a represents
the fraction of any discrepancy corrected
each time period.
The adjustment term a(S'
regulate the stock.
Sf
-
pSLj) creates two negative feedback loops which
Discrepancies between the desired and actual stock induce additional
orders until inventory reaches the desired value.
on order induces additional orders
Likewise an insufficient quantity of goods
- depending on
If
p =
orders placed are forgotten until they arrive, causing overordering and instability.
If
until the pipeline is filled
p.
then subjects fully recognize the supply line and do not double order. While p=l
prior experiment
"
showed
P«l
for
many
is
0, then
P =
1,
optimal, a
subjects in a similar task (13).
Expected demand EK represents each participant's forecast of incoming orders.
Adaptive expectations are assumed:
D^
= eDt.i + (i-e)D^t-i,
o<e<i.
(2)
Consistent with behavioral decision theory (11, 12, 14), the rule utilizes information
locally available to the decision
maker and does not presume managers have global knowl-
edge of the system structure or the cognitive capability
The parameters
(0,
44
the proposed rule
subjects.
A
optimum performance.
a, P, and S') were estimated for each participant by nonlinear
least squares subject to the constraints
power of
to solve for
is
excellent:
0<6<1 and a,
the
mean R^
P,
is
S'
>0
(table 2).
71%; R2
is
The explanatory
less than
50%
for only 6 of
large majority of the estimated parameters are significant, and the estimated
parameters are systematically related to performance (10).
Though
the subjects' disequilibrium response
is
suboptimal,
we
expected that their
decision rules would be stable and swiftly return the system to a low-cost equilibrium.
To
Sterman, Mosekilde, and Larsen
test this
hypothesis
we
6
simulated the experimental system using each set of estimated
Thirty parameter sets (68%) do indeed produce stable behavior.
parameters (15).
However,
one periodic and three quasiperiodic solutions appear. Ten (23%) yield aperiodic (chaotic)
behavior.
Compare
Figure 3 shows several characteristic simulations.
=
strange attractor produced by the parameters of subject 4 (0, a, P, S'
S' = 0, 0.3, 0.15, 17).
subject 36 (0, a,
(3,
produced by the
(stable)
To
parameters for subject 27
values of .25 and 17, respectively, and varied
The space
chaotic solutions.
The unstable
(0,
a, p, S' = 0.2, 0.3, 0.05,
complex, including
solutions arise
when
there
each sector's immediate supplier,
saler's inventory.
Low
is
is
and S'
e.g.,
when
frequencies arise
when
at representative
stable, periodic, quasiperiodic,
from the nonlinear coupling of several
in the distribution chain.
and
oscil-
These
in the system.
High
sufficient inventory so that orders can be filled
to wait for the factory to receive orders,
Periodic solutions arise
set
8).
and P over the managerially meaningful
coupled through the availability of inventory
frequencies are produced
frequencies
a
we
by the multiple inventories and time delays
latory feedbacks created
oscillators are
(figure 4) is
1.0, 0.65, 0.40, 15) to
Note the long and apparently chaotic transient
explore the structure of the parameter space,
interval [0,1].
the shape of the
by
the retailer's orders aie filled out of the whole-
when
inventories are inadequate, forcing customers
produce the beer, and finally ship
it
downstream.
the parameters of the heuristic are such that the ratio of these
rational; quasiperiodic solutions arise
when
this
winding number
is irrational,
and chaotic solutions arise when the individual oscillators are sufficiently unstable (16).
In general, larger values of
a
and smaller values of p are destabilizing (figure
4).
To
the extent a subject ignores the supply line of unfilled orders (P<1), a stock shortfall causes
orders to be placed each period even after sufficient orders are in the supply line, leading to
excess inventories and oscillation.
stock adjustment (a>0.4) since
However,
the transitions
Such overordering
more
is
exacerbated by more aggressive
will be ordered in response to a
given stock
shortfall.
between modes are not monotonic. Note particularly the two
narrow fjords of stable solutions which snake between the periodic and chaotic solutions.
Sterman, Mosekilde, and Larsen
7
Figure 5 magnifies a region of parameter space
unstable solutions.
The
at the transition
oscillators
staircase is complicated; however, by the fingers of stable solutions
These regions of
from
for a=0.90.
chaos.
The
structure of the
which cut across the
appear to have fractal boundaries as well.
chaos also exists and also appears to be
As P decreases from .444
to .434 the
fractal.
A
transition
Figure 6 varies P
system moves irregularly from
stability to
Successive magnification reveals the self-similarity of the fractal boundary.
It is
the
stability to
stability
from
and form a devil's staircase (16). Further
magnification reveals the characteristic fractal steps of the staircase.
directly
stable to
alternating bands of periodic and quasiperiodic solutions arise
mode-locking between the various
staircase.
from
common
in the social sciences to
dynamics of human systems
that formal rules
are, if
assume
that
not optimal, then
decision-making behavior and thus
at least stable.
results
show
which characterize actual managerial decision making can produce an
extraordinary range of disequilibrium dynamics, including chaos.
number of
These
issues for social scientists.
Such complexity
raises a
Policy interventions often imply changes in the
parameters of a decision rule or model.
How
space" contains fractal basin boundaries?
In such systems changes
produce unpredictable qualitative changes
circumstances which differ only slightly.
can policy analysis be conducted
in behavior.
Do robust
Experience
if
the "policy
on the margin may
may
not transfer to
principles of policy design exist in such
systems?
Even though
deterministic cause-effect relations exist in chaotic systems,
impossible to predict the effects of small changes in
initial
it is
conditions or parameters.
Does
such 'randomness' slow the discovery of cause and effect by agents in the economy and thus
hinder learning or evolution towards efficiency? Indeed, does learning alter the parameters of
decision rules so that systems evolve towards or
further
away from
development of theory and experiment are required
assess the significance of chaos in
human
systems.
the chaotic regime?
to
Much
answer these questions and
..
8
Sterman, Mosekilde, and Larsen
C. Grebogi, E. Ott,
1
J.
A. Yorke, Science, 238, 632 (1987), and the references therein; R.
May, Nature (London) 261, 459 (1976);
Bristol, 1984);
J.
Thompson and H.
P. Cvitanovic,
M.
Ed. Universality in Chaos (Hilger,
Stewart, Eds. Nonlinear Dynamics
and Chaos (Wiley,
Chichester, 1986).
R. Day,
2.
Am. Econ.
3 (1986);
J.
M. Grandmont and
J.
M. Grandmont, Econometrica,
8,
397 (1987). E. Mosekilde
M.
J.
Stutzer, /. Econ.
3.
W.
4.
A. Wolf,
5
O. Morgenstem,
A. Brock,
J.
J.
Malgrange,
P.
53, 995 (1985); H.
and Organization,
35, 80 (1988);
.
Rev., 72, 406;
et al.,
W.
J.
Lorenz,
Econ. Theory, 40,
/.
Econ. Behavior
European J. of Operational Research,
Dynamics and Control,
2,
353 (1980).
Econ. Theory, 40, 168 (1986); P. Chen, System Dynamics Review, 4, 1988.
Swift, H. Swinney,
On
the
J.
Vastano, Physica 16D, 285 (1985).
Accuracy of Economic Observations, 2nd
ed. (Princeton, Princeton,
1963).
6.
J.
D. Sterman, Management Science, 33, 1572 (1987).
7.
J.
W.
Forrester, Industrial
Dynamics (Cambridge, MIT, 1961); V. Zamowitz, Orders,
Production, and Investment
-
Cyclical
and Structural Analysis (New York, NBER,
Dynamics, (Cambridge, MIT, 1963).
8
W.
9.
C. Plott, Science, 232, 732 (1986); V. Smith,
10.
J.
E. Jarmain, Ed.,
Problems
in Industrial
D. Sterman, System Dynamics Review,
1973).
American Economic Review, 72, 923 (1982).
148 (1988);
4,
J.
D. Sterman, Management Science,
forthcoming.
1
1.
A. Tversky, and D. Kahneman, Science, 185, 1124 (1974).
12. R. Hogarth,
13.
J.
Judgement and Choice, 2nd
ed.
1987).
D. Sterman, Organizational Behavior and Human Decision Processes, forthcoming 1988.
14. H. A.
Simon, American Economic Review, 69, 493 (1979); R. Cyert and
Behavioral Theory of the Firm (Englewood
Management Science,
15.
(New York, Wiley,
31,
Cliffs, Prentice Hall, 1963);
J.
J.
March,
A
Morecroft,
900 (1985).
Analysis of variance showed no strong relations between a subject's position in the distribution
chain and the estimated parameters.
We therefore assume identical parameters for each sector.
Sterman, Mosekilde, and Larsen
reducing the dimension of the parameter space from 16 to
9
4.
In the simulations orders are real-
valued while in the experiment orders are integer-valued. The experiment
is
a difference
equation system; a continuous-time version also produces chaos with plausible parameter values
[E.
16.
Mosekilde and E. Larsen, System Dynamics Review,
M. H. Jensen,
P.
4, 131, (1988)].
Bak, T. Bohr, Phys. Rev. A, 30, 1960 (1984).
Sterman, Mosekilde, and Larsen
Figure
"Beer Distribution
1.
Game"
10
board.
Subjects carry out five steps each simulated
week:
1.
Receive inventory and advance shipping delays.
2.
Fill orders.
3.
Record inventory or backlog.
4.
Advance
5.
Place orders.
make
Subjects
Figure
the order slips.
2.
their decisions in step 5; steps 1-4
handle bookkeeping and other mechanics.
Typical experimental results. Top: customer orders increase from 4 to 8
cases/week
week
in
5.
Middle: The resulting orders placed by subjects (from bottom to top,
Retailer, Wholesaler, Distributor, Factory).
inventory=Inventory-Backlog).
Bottom: Effective inventory levels (Effective
Tick marks on y-axes denote 10
amplification, and phase lag as the disturbance propagates
Figure
3.
showing
Note the
from customer
oscillation,
to factory.
Simulation of the decision rule with estimated parameters. Top: phase portraits
distributor inventory vs. factory inventory.
transient;
units.
flow
is
Weeks 1000-1600 shown
to
remove
clockwise. Bottom: The parameters for subject 27 produce a long chaotic
transient (Retailer orders shown).
Figure
Parameter space of the simulated system. The stock and supply
4.
parameters
S'=17.
a
and P are varied over the interval
[0,1] in
line adjustment
increments of .025, holding 0=.25 and
Green=stable; red=periodic; blue=quasiperiodic; orange=chaotic. Note the two fjords
of stable solutions separating regions of unstable behavior.
Figure
5.
Magnification of parameter space by a factor of 100 compared to figure
4.
Green=stable; red=periodic; blue=quasiperiodic; yellow=period-doubling. The periodic
H
Sterman, Mosekilde, and Larsen
solutions have rational winding number, indicating the high frequency oscillator locks into a
rational multiple of the
low frequency
cycle.
low frequency cycles completed before
Figure
6.
The numbers
3, 4, 5, etc. indicate the
the system repeats.
Transition from stability to chaos as P varies, holding a=.90.
orange=chaotic.
number of
Green=stable;
Successive magnification reveals the irregularity characteristic of a fractal
boundary. Rightmost column of figure reflects magnification of 10^ compared to figure
4.
12
Sterman, Mosekilde, and Larsen
Table
1.
Summary of
experimental results.
Data are means of 11
trials.
Actual costs exceed
optimal levels by a factor of 10, a highly significant margin.
Total
COSTS
$2028
Mean
Benchmark*
Ratio
t-statistic:
Hq:
Mean
cost =
Benchmark
Retailer
Wholesaler
Distributor
Factory
Sterman, Mosekilde, and Larsen
Table
2.
Summary of
estimation results for 44 subjects.
The ordering
determines the speed of adjustment of the demand forecast;
1-2.
response to inventory shortfalls;
subjects; S'
is
the desired stock.
Parameter:
Mean
13
(std. dev.)
Median:
Minimum:
Maximum:
6
(3
is
heuristic is given
a
determines the
the fraction of the supply line accounted for
See (10) for
by eqs.
by the
details.
a
P
0.36(0.35)
0.26(0.18)
0.34(0.31)
0.25
0.00
1.00
0.28
0.00
0.80
0.30
0.00
1.05
R^
S'
17
15
38
(9)
0.71
0.76
0.10
0.98
(0.22)
Sterman, Mosekilde, and Larsen
f
14
Sterman, Mosekilde, and Larsen
40
15
Customer Orders
10
30
Weeks
Trial 5
-^
10
UJ
20
Weeks
30
U
Sterman, Mosekilde, and Larsen
30
'1
1 SutJiect
4
'O
a
-30
10
-10
-80
n
Factory Inventory (cases
-10
Factory Inventory (cases)
-80
SuDiect 27
i
15
5:
10
o
5
6G0
Sterman, Mosekilde, and Larsen
17
a«
Sterman, MDsekilde, and Larsen
8
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as
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Sterman, Mosekilde, and Larsen
20
Appendix: Equations for Production-Distribution System
The equations
simulations (figure
backlogs =
are presented in the sequence of calculation used in the experiment and
equilibrium conditions are:
Initial
1).
inventories = 12 units,
Customer demand
is initially
4 and
The computer programs
estimation are available from the
Step
The contents of
1:
sector's inventory (I) are
right (Dl) are
same
moved
added
into
first
for the simulations
W,
and parameter
author.
the shipping delay (Dl)
immediately to the right of each
The contents of
to inventory.
the shipping delay
on the
Dl. Factories advance the production delays Dip and D2p
far
in the
fashion.
Ij(t)
= Ij(t-l) + Dlj(t-l)
(Al)
Dlj(t) = D2j(t-l)
Step
2:
Retailers
examine Incoming Orders
new
week
rises to 8 in
Subscripts denote the sector of the distribution chain (Retailer R, Wholesaler
Distributor D, and Factory F).
all
= 4 units (orders placed, incoming orders, production delays,
0, all other variables
shipping delays, and expected orders).
5.
all
(A2)
examine
(10).
All sectors
B from
orders plus any Backlog
retailer's
the top card
on the Customer Order deck (CO);
fill
orders.
The Shipment
rate
all
S must equal the
the prior period, to the extent inventory permits.
shipments go directly to the customer and leave the system.
others are placed in the shipping delay
D2
Shipments of
of the downstream sector (A4).
others
The
all
Shipments also
reduce inventory (A5).
SR(t) =
Si(t)
MIN(CO(t)
-f
= MIN(IOi.i(t-l) + B|(t-l),Ii(t)),
D2k(t) = Sk^i(t),
Ij(t)
= Ij(t)-Sj(t)
Step
3:
(A3)
BR(t-l), lR(t))
k=
1
=
W,D,F
(A3')
R,W,D
All sectors record inventory or backlog.
(A4)
(A5)
•
The net change
ence between incoming orders and shipments. Effective Inventory EI
is
in
backlog
is
the differ-
inventory less backlog.
Sterman, Mosekilde, and Larsen
BrCD =
BR(t-
1 )
+ CO(t)
21
-
SrCD
(A6)
Bi(t) = B,(t-l) + IO,.i(t-l)- S,(t),
EIj(t)
=
The
=
W,D,F
(A6')
(A7)
Ij(t)- Bj(t)
Step 4: Each sector advances the
orders.
1
week's Order
slip containing last
O
to
incoming
factory puts last week's order in the top production delay D2p.
I0,,(t)
= 0j,(t-1),
R,W,D
k =
(A8)
D2F(t) = OF(t-l)
Step
supply line
5:
is
(A8')
Each sector places
the
sum
orders.
First the
Supply Lines
are calculated.
of units in the two shipping delays, the backlog of the supplier
and orders placed the previous week. Since the factory
line is
SL
is
the primary producer,
(if
any)
supply
simply the contents of the production delays.
SLk(t) = Dli.(t) + D2i,(t) + Bk^i(t) +
I0j,(t),
k=
R,W,D
(A9)
SLF(t) = DlF{t) + D2F(t)
Each
its
The
sector's forecast of
(A9')
incoming orders (expected Demand) D^
is
formed by adaptive
expectations, with smoothing parameter 6.
D-(t) = ejlOj(t-
1 )
+
Finally, orders are the given
between the desired Stock
line.
S'
(
l-ej)D.(t-
(AlO)
1
by demand forecast adjusted by a fraction
a
of the difference
and the effective inventory, including a fraction p of the supply
Orders must be nonnegative.
Oj(t) =
MAX(0,D-(t) +
aj(S'j- Elj(t)- PjSL/t)))
(All)
Sterman, Mosekilde, and Larsen
22
Appendix. Estimated parameters, Beer Distribution
Trial
& Position
R
W
D
F
R
W
D
F
R
W
D
F
R
W
D
F
R
W
D
F
R
W
D
F
R
W
D
F
R
W
D
F
R
W
D
F
10
R
W
D
F
11
R
W
D
F
e
0.90
a
P
S'
Game
r2
Mode
u
u
u56
V^
^b
Date Due
JUL
JAN.
3
2
1
1
19a
199*
3
TQfiD
DOS 37b ISM