^8
U4
Dewey
131 |o
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
MANAGERIAL MONITORING OF A SINGLE DATA STREAM
by
Michael E. Treacy
WP 1210-81
April,
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
1981
MANAGERIAL MONITORING OF A SINGLE DATA STREAM
by
Michael
WP 1210-81
E.
Treacy
April,
1981
M.t.T.
Li8P^.RlES
MAY 2 6
t981
RECEIVED
j
.
INTRODUCTION
management
Much of
activity
surveying situations for which there is
purpose
The
action.
potential and thereby
required.
to
indicate
a
and
potential need for corrective
is
when
routinely
to
assess
this
intervention
managerial
payoffs
is never an economic benefit to monitoring, for
only through
scanning
is
is never a need for corrective action, then there
there
If
monitoring
of
monitoring:
involves
actions
which
obtained
are
improve the economic circumstances of the
firm.
Monitoring involves the gathering and assessment of information.
what Simon calls the intelligence phase of
what Pounds
making[1977]
decision
the process of problem finding.
calls
[
action.
and
1969] Monitoring is
activities
differentiated from other managerial information processing
by its
It is
to routinely assess the potential need for corrective
purpose,
Routine implies a repetitive and somewhat structured activity.
Corrective action suggests that the situation is
guided
being
toward
some target or goal
This paper is an inquiry into the
support managerial
design
monitoring functions.
effort is provided by information economics
has observed,
economics
models
affected
systems
information
to
The guiding paradigm in this
theory.
As
Treacy[1981]
of information value concentrate upon
only the choice phase of decision making.
are profoundly
of
Yet, we know that
decisions
by information at the *($lfreriligence and design
phases of the decision making process, because Without
!"
£'*'
information
to
,,<^'^\
\fy
t
i
alternatives, and estimate consequences,
structure
identify problems,
among
present effort
will
alternatives
no choice from
ever
is
Therefore,
made.
this
serve to extend information theory into the
also
first phase of decision making, intelligence.
design
Our primary concern is with the
yet
their valuation,
ability
design without some rudimentary
must
These values
relevant
be
support system design
context in which information
the
Therefore,
contemplated.
is
information
of
use
not
in
values.
relative
assign
to
model must be founded upon
system valuation
the managerial
in
systems,
begin to separate good from bad
hardly
can
we
information
of
information
any
realistic description of
a
even
monitoring,
that
if
description is at odds with prescriptive theory.
The next section
paper
outlines
function
monitoring
the
describing
this
of
of
whether
decision
is
made
The
situation.
or
data
monitored
probability determined
intervening.
from
subsequent
In
personal
and
the
costs
normative
a
determined
intervention,
with
in
is
subjective
the
judgement,
relative
sections,
intervene
to
comparing
by
probability that the situation is in need of
from the
not
for
Monitoring
management.
of
characterized as a decision
framework
general
a
a
threshold
of
and
benefits
cost
investigation
models, which prescribe the determination of the subjective probability
and the
threshold
selecting from
behavior.
This the
probability,
are
reviewed
with
an
aim
toward
them that which is descriptive of managerial monitoring
Unfortunately, not all of the theory is descriptively valid.
researchers
readily
admit
when
they
bemoan
the
minimal
acceptance of their work by managers. [Kaplan 1975;
We shall next move to Pounds work on the process
which provides
probable need
a
intervention.
mathematical
A
developed to complete
a
reconsider
a
finding,
version
this
of
descriptive model of managerial
of an information support system is derived.
simply
problem
Based upon this description, a model of the value
monitoring behavior.
need to
of
description of how managers subjectively evaluate the
for
description is
Magee 1976]
standard
the
This
model
indicates
a
concept of an information system as
collection of information signals.
The developments in this
managerial monitoring
paper
of
one
limited
are
situation.
to
consideration
of
shall not consider the
We
issues of monitoring multiple data streams or of
management attention
a
the
distribution
of
among several situations and how this affects the
design of information support systems.
For one approach to this latter
problem see RockartC 1979].
THE GENERAL FRAMEWORK
Monitoring is performed to assess the need for managerial intervention.
It may conveniently be modelled as a process in which a
data for
the
situation.
purpose
of
views
manager
choosing whether or not to intervene into a
This is compatible with
Simon's
characterization
of
the
intelligence phase of decision making as finding an occasion for making
a
decision
before
possible
considered. [Simon 1977, p.
courses
40] It
is
of
action
decision
making
have
with
been
minimal
structure.
Let us assume that a situation is being monitored using a single stream
of reported data, one datum arriving
denoted by
period
one of two states, in control (s
control
or out of
)
(s
is associated a payoff u.
He can
For each state-action pair there
or not intervene (a^,)-
)
each
In
).
the manager has two possible actions open to him.
time period,
each
and
Also let us assume that the situation always exists in
y.
intervene (a
time
each
in
These payoffs are shown in Figure
in control
out of control
a
:
don't investigate
u(s
,a
)
u(s ,a
)
a
:
investigate
u(s
,a
)
u(s ,a
)
Figure
1.
1
By definition, we have that:
u(s^,a^)
>
u(s„,aj
(1)
1
u(s
and
,a
)
<
u(s
,a
(2)
)
That is, in the out of control state it is better to intervene
the in
control
state it is better not to intervene.
intervene
want to
intervention exceeds
let p(s
)
represent
situation is
a^, if:
out
the
in
the
the
of
situation
if
the
and
The manager will
expected
payoff
expected benefits of nonintervention.
manager's
subjective
in
probability
that
of
If we
the
control, then he should intervene, choose action
[1
- p(s )].u(s ,a
[1
=>
p(s^)
)
+p(s
).u(s
,a
>
)
- p(s^)].u(sQ,aQ) + p(s^).u(s^,aQ)
(3)
= p^
>
(4)
[uCSq.Sq) - u(sQ,a^)] + [u(s^,a^) - uCs^.a^)]
T
p
represents
of being
out
want to act.
threshold probability.
a
subjective
If the
probability
of control exceeds this threshold, then the manager will
Data is observed to determine p(s
),
the
probability
of
the situation being out of control.
Notice that the general framework has made two restrictive assumptions,
that the situation can assume only one
manager monitors
only
one
stream
behavior.
states
two
of data.
first assumption is sufficient for an
managerial monitoring
of
and
that
the
We shall argue that this
descriptive
initial
theory
The simplification is made to reduce
both the complexity and the information requirements of the model as
parallel to
simplifications
processing requirements.
subsequent paper.
The
made
by
managers
limit of two actions is not
before
the
the
a
just that, notation.
in
a
restriction, but
earliest
phase
of
problem has more than minimal structure.
The rest of the notation, including the derivation of the
rule is
a
to reduce information
The second assumption shall be relaxed
by definition of the intelligence activity as
decision making,
of
It
intervention
provides no restriction upon possible
managerial behavior other than consistency.
DETERMINATION OF THE THRESHOLD PROBABILITY FOR INTERVENTION
How does
a
T
manager determine
p
the
,
probability
of
the
One approach is to directly estimate
intervention?
situation needing
threshold
the four payoffs, u, and to compute p
T
This appears
from equation (4).
to be an unlikely description of managerial behavior.
Dyckman[1969], in
a
paper on the prescriptive theory of
cost
variance
investigation, suggested that the manager should subjectively determine
two values:
the cost of investigation and L, the present value of
C,
the gross savings obtained from an investigation when the situation
out of
control.
cost of investigation is independent of the
the
If
is
state, then these values correspond to payoffs as follows:
C
=
uCs^.a^) - u(sQ,a^)
L - C = u(s
,a
)
- u(s
,a
(5)
(6)
)
Equation (5) reflects the benefits of choosing the correct action
the situation is in control.
out of
a
when
Equation (6) reflects those benefits when
The threshold probability of intervening now takes on
control.
simple form:
p'^
=
C/L
(7)
Dyckman concedes that "the precise determination
each future
period
is not an easy matter."
of
the
savings
for
He observes that "where a
corrective action is not forever binding, the calculation of L needs to
be adjusted to
periods" and
reflect
that
"data
the
on
possibility
of
future
out
of
control
the average number of periods before the
process is discovered to be out of
1969, p.
218]
These
are
control
may
helpful ."[Dyckman
be
types of tradeoffs that managers must
the
implicitly make in their decision to intervene.
Li
[
1970] has criticized the theoretical validity of Dyckman 's
T
for setting
p
for another
reported
because it does not anticipate the benefits of waiting
subjective probability
datum
of
which
with
being
sequential
nature
of
periods increases.
attendant on
solving
large
manager's
the
out of control, p(s
He advocates
).
which
[1969]f
explicitly
the monitoring process and computes
threshold probabilities which converge
number of
sharpen
to
using Kaplan's dynamic programming approach
models the
approach
to
a
constant
value
Dyckman responded that "the difficulties
complex
and
programming
dynamic
real
problems can limit the succesful application of this technique."
supported by
Magee[1976],
whose
comparison
of
approach for not
'3
considering future actions, while valid theoretically, may have
on
the
incremental
savings."
cost
Jacobs[1978] using an experimental
significant differences
between
setting,
of
little
comparison
Another
failed
results
the
He is
the approaches using
simulation concluded that "Li's criticism of Dyckman
effect
the
as
to
by
indicate
using Dyckman
any
and
's
Kaplan's approach.
Our interest is in using parts of these
variance investigation
dynamic programming
decisions
approach
would
description of managerial behavior.
although theoretically
which
prescriptive
of
models
cost
have descriptive validity.
hardly
Dyckman
appear
's
to
be
a
A
valid
approach, alternatively,
flawed, has a certain face validity which makes
We shall incorporate it into
it appealing.
model
our
managerial
of
monitoring behavior.
NORMATIVE DETERMINATION OF THE POTENTIAL NEED FOR INTERVENTION
We have assumed, to this point, that the manager uses a
of data
to
monitor
The data is observed to assess the
situation.
a
p(s
probability of need for intervention,
these circumstances,
there
can
manager's determination of p(s
only
be
):
stream
single
),
any
in
upon the
influences
three
Under
period.
his initial judgement, the stream of
data, and the passage of time.
We shall present the normative approach to the determination of
as developed
the
in
cost variance investigation literature.
between
approach, there is little disagreement
other theoreticians,
although
found in Dyckman [ 1969] and Kaplan [ 1975].
descriptive validity
of
the
we
and
Details may be
shall
assess
the
the theoretical treatment and, unfortunately,
be forced to reject the approach as a
determination of
Next
On this
Kaplan,
Dyckman,
formulations do differ.
p(s,)
description
of
potential need for intervention.
the
managerial
A new approach
shall be presented in a subsequent section.
Let us define p'(s
)
of intervention
at
passage of
more
one
as the probability that the situation is
need
in
the start of a period, before the new datum or the
time
period
have
been
considered.
p'(s
)
information since the last intervention;
the
manager's judgement of the need for intervention immediately after
the
summarizes all
relevant
intervention and
prior
all
observations
and
time
since
the
last
intervention.
The new datum, y, is prescribed to be incorporated into the probability
estimate using Bayesian revision.
f(y|s )p'(s
i
=
P(s !y)
)
i
(8)
f(ylsQ)p'(sQ) + f(y!s^)p'(s^)
For this
calculation,
distributions,
manager
the
f(y|s
and
)
requires
f(yls.).
These
knowledge
are
of
two
conditional
the
probabilities of the observed datum under each state.
Incorporation of the time information requires
nature of the process underlying the situation.
from state
s
probability g,
to state s
assumptions
about
If the situation moves
with probability g^. and from
to s^ with
s
during any period without intervention, then:
p(s^)
=
p(sQ|y)gQ, + p(s^|y)[1 - g^g]
=
[1
(9)
- p(s^|y)]gQ^ + p(s^|y)[1 - g^Q]
If it is assumed that the situation never moves from s
^
to
(10)
Sq
without
intervention (g,Q =0), then the process is one of geometric decay.
longer the
the
situation
is
left unattended, the greater the probability
that it is in need of intervention.
p(s^)
=
The
pCs^ly) +
>p(s^|y)
[1
In such a case:
- p(s^ly)]gQ^
(11)
(12)
10
Geometric decay
usually
is
literature, probably
quality control
much
because
theory,
correcting ability.
assumed
where
p(s
)
>
effect
of
time on p(s
investigation
was borrowed from
machine
no
has
self
not generally true in a management
is
situation, where one is monitoring the
Here the
theory
the
misad justed
a
same
The
of
cost
the
in
outcomes
of
is ambiguous.
)
others'
actions.
In the general case,
p(s |y), if and only if:
p(s
P(s !y)
g-,
_yi
1
>
g^Q
1
- p(s^|y)
!y)
1
=
(13)
pCSply)
Since the ratio on the right varies from period to period,
of time
may
one
in
decrease it.
Even this ambiguous
upon assumptions
situation.
be to increase p(s
period
about
the
result
nature
of
and in another to
|y)
fundamentally
is
process
the
effect
the
dependent
underlying
the
Different assumptions lead to different theory.
DESCRIPTIVE VALIDITY OF THE NORMATIVE APPROACH
Having described normative theory for the determination
must assess
behavior.
its
relevance
a
stream
assess these three
of
aspects
data,
of
p(s
),
we
description of managerial monitoring
The theory considers three
judgement, the
evidence.
to
of
and
the
determinants
the
theory
of
p(s.):
passage of time.
against
other
prior
We shall
available
11
Normative theory
directs
determination of p(s
)
that
prior
judgements
through Bayesian revision.
affect
In that fashion:
1
1,
=
p(sQ!y)
(14)
f(ylsQ) p'Cs^)
In words, managers should form odds of being in s, versus S-
the product
s„.
of
ignored.
Kahneman and Tversky [1972,
the most compelling evidence of
prior information.
£t a]^[1976],
who
equal
to
the likelihood ratio of y and the prior odds on s. and
There is significant evidence that in the assessment,
are largely
the
f(y|s.) p'(s,)
P(s.|y)
1—
should
systematic
this
prior
odds
19733 have produced
underutilization
of
Their basic conclusions are supported by Swieringa,
conclude that the degree to which the prior odds are
ignored are directly related to the strength of association between the
data stream and the situation. [p.
182]
The evidence suggests that Bayesian revision is a poor
how managers
utilize prior judgements.
description
We posit that there is just as
little support for Bayesian revision as a description of how
are
utilized.
Our
claim
rests
of
upon
the
conditional probability functions f(y|s„) and
observation
f(y|s.)
are
new
data
that
the
enormously
difficult to generate, especially in a monitoring context.
Consider, for
conditions.
a
The
weather
moment,
forecast
is
forecasts
and
information;
it
tomorrow's
corresponds
weather
to
Tomorrow's weather condition (rain or sun) is the uncertain state.
think of your favorite weather forecaster and estimate the
that he
will
forecast
rain
given
that
it
y.
Now
probability
will be sunny tomorrow.
12
f(y|s).
An important difficulty immediately arises.
backwards to
states.
normal
the
fashion
The probability of sun tomorrow, given
p(s|y), is
more
a
and
measures
it
information.
This
revision as
a
is
about information and
forecast
a
of
rain,
assessment, because it is chronologically
natural
ordered (first an information
state)
thinking
of
problem
The
signal,
descriptive
illustrates
the
where
theory
an
notion
natural
the
example
then
inference
of
about
the
reliability
of
inadequacy
prior
of
Bayesian
judgements
are
not
revised, but largely ignored.
must
Although this paper is in the information economics tradition, we
reject the
Bayesian revision formulation of the manager's intervention
decision, for we wish to base our information value theory upon
The evidence
description of managerial behavior.
with Kahneman
and
his evaluation of
Bayesian:
man
apparently
is
better
suited
to
a
to
agree
not
conservative
a
We maintain that a non-Bayesian
he is not Bayesian as all."
formulation is
us
valid
450] when they conclude that "in
Tversky [1972, p.
evidence,
leads
a
descriptive theory of probability
assessment in the managerial monitoring context.
Finally there
is
assessment of
the
the
matter
potential
of
need
the
for
influence
time
of
intervention.
upon
The normative
theory is fundamentally dependent upon assumptions about the nature
the process
underlying
different theory.
consistent result:
the
situation.
Different
Any realistic assumptions, though,
the
of
assumptions yield
will
yield
one
the influence of the passage of time will be small
compared to the influence of the data stream.
For a
demonstration
of
13
this, consider
equation
geometric decay.
g
is
equation
is
the process is assumed to be in
A reasonable assumption about this
quite small;
than the rule.
where
(11),
process
that
is
the need for intervention is the exception rather
Therefore the second term on
negligible
its
in
right
the
influence
upon
side
p(s.),
of
the
which
is
approximately equal to p(s.!y), determined from the influence of y.
In realistic managerial monitoring situations, time
has
influence
only
upon
mechanism that
the
intervention
distributes
competing situations.
paper.
a
decision,
manager's
a
but
attention
some
through
among
factor in
our
Therefore, we shall
initial
the
several
will not consider this issue in the present
We
In the assessment of a particular situation, though,
negligible role.
minimal
descriptive
choose
model
not
to
include
managerial
of
time plays
this
monitoring
behavior.
A DESCRIPTIVE MODEL OF THE DETERMINATION OF p(s^)
We require a model of the
assessment
of
p(s
)
that
utilizes
prior
judgements in inverse proportion to the association of observed data to
the situation
f(yiS-).
and
that
does
not
rely
upon knowledge of f(y|s_) or
For guidance, we turn to Pounds' seminal work which describes
the process of problem finding.
of problem finding activity.
[
1969] Monitoring is an important
class
14
Pounds defined a
problem
present condition
of
as
significant
a
difference
between
situation and how it ought to be.
a
the
He observed
that "the manager defines differences by comparing what he perceives to
the output of a model which predicts the same variables. "[p. 5] A model
need not be any formal piece of logic.
simply what
manager
a
historical results,
ought
judges
from
plans,
More
to
from
often
than
not
is
It may be derived from
be.
comparison
a
it
similar
of
situations, or by mandate.
The logic of this approach is simple and clear.
the
condition,
present
difference, d, and that
(state s
)
and
k,
a
the
model
manager decides
or not (state s
)
Pounds
observed
norm,
or
that
define
n,
a
a
problem
on the basis of this difference.
Because
whether
he
has
1
the norm may be uncertain, the
because the
association
difference
may
be
uncertain.
Also,
between the data stream and the situation may
not be exact, any particular difference may not indicate with certainty
Thus, a manager's uncertainty as to whether there
either Sq or s..
a need
for
intervention
is
uncertainty as to
derives from two sources:
how the data stream ought to be and the inadequacy of the
stream
data
as an indicator of the present situation.
Differences are defined so that greater
indicative of
problem situation.
a
differences
observed value is more indicative, then d
d
=
In
- kl.
a
more
If a larger observed value is more
indicative of an in control situation, then d
deviation from
always
are
=
=
n - k.
k - n.
If
a
smaller
Finally, if smaller
standard in either direction is more indicative, then
For the rest of this paper,
we
shall
assume
that
the
15
manager is
dealing
with data of the second type and define d
=
k - n.
Parallel results for the other two cases are left to the reader so that
we may avoid unnecessary repetition.
because we
The second case has
been
chosen
wish to compare our results with those of the cost variance
investigation literature and cost data is of this type.
Let f(d|y) represent
assessment of
the
the
distribution
of
indicated
difference
represent the manager's assessment
of
control, for
Then:
a
given difference d.
=
=
jQiy)
P(3^ly)
We may mathematically
the
by
manager's
datum
y and let p(s
probability
the
subjective
of
being
|p(s„!d)f(d|y)dd
jp(so
represent
Pounds'
'd)
in
(15)
descriptive
theory
of
the
managerial assessment of p(s^) as:
p(s^)
=
p(s^ly)
=
1
- p(s_!y)
(17)
=
1
- rp(sQ|d)f(d!y)dd
(18)
We have chosen in equations (17) and (18) to
residual of
p(s_!y).
(16)
represent
p(s.)
as
the
This obviates the need to clarify the nature of
s., which may represent an amalgam of out of control possibilities.
The difference, d, equals k - n, the subjectively
between the
present
condition and the norm.
Only
difference
The present condition is
represented by the data stream, which is subject
error.
assessed
to
bias
and
random
know bias will affect a subjective assessment and we will
define our data stream as already adjusted for known bias.
The
effect
.
16
of random
error
reliable.
We may represent the assessment
upon receipt
the data stream is to make it less than perfectly
in
of y, as
shall operationalize
distributed with
of
present
the
condition
random variable with distribution f (k|y).
a
f (k|y)
assuming
by
that
it
We
normally
is
mean y (the most likely condition) and variance v„ (a
measure of the data unreliability).
Similarly, we may represent
with distribution
fw(nly)
=
the manager's norm as
random
We shall operationalize ^mCi) by assuming that it
normally distributed
with
from
two
Uncertainty
uncertainty).
sources:
an
about
data stream should behave.
the
for
the
incomplete understanding of the
relationship between the data stream and the situation and
necessary
is
mean u^ (the most likely norm) and variance
v^ (a measure of the manager's
unreliable data
variable
Since the norm is independent of the datum,
f^(n).
ffjCn)'
norm derives
a
missing
or
conditional estimation of how the
Later in this paper, we shall
explore
the
economic impact of an information system that facilitates a decrease in
the first
source
of
uncertainty, and hence
a
decrease in v
,
through
managerial learning.
The difference is the sum of two normally distributed random variables,
Thus, it is also normally distributed
k and -n.
variance v„ +
K
with
mean
V-^u
a"d
v.,
N
f(d|y)
-
N(y -
u^^.v^^
+v^)
(19)
17
Notice that all uncertainty as
sources:
unreliability
the
of
difference
the
to
from
two
stream and the uncertainty
data
the
derives
about the norm.
The manager's interpretation of any particular difference,
depends upon not only the size of the difference, but also
by p(s Id),
upon the association of
represent the
the
manager's
difference
perception
absolute value of the correlation
situation,
represented
r
of
association
the
between
situation.
the
to
the
by
stream
data
We
may
r,
the
and
the
is a measure of the proportion of the variability in the
situation that is explainable by variation in the data stream.
If there is no association whatsoever, then r
the prior probability
of
being
=
control.
in
and p(s_|d)
there
If
association between the data stream and the situation (r
difference is
an
indicator
exact
the
need
enough
intervention
for
p'(s»),
is
perfect
1),
then the
of the state of the situation.
some point, d^, the difference is large
certain of
=
=
(state
that
the
s.).
manager
Below
d_
difference is too small for intervention and the manager is certain
being in control (state Sq)
.
is
the
of
Then p(sQ|d) is a step function of d.
1
d < d
"^
p(s Id) =r
(20)
°
d > d^
For values of r between zero and one, p(s^!d) has
two extremes.
At
This is diagrammed in Figure 2.
values
between
the
18
Figure 2
difference
The difference d^ may now be interpreted as the
For P'(s
)
=
p'(s
distribution of
p(s
not change his prior judgement:
would
the manager
)
d
=
1/2,
we
would
expect
is symmetric about d
,
d^
that
then d
Id
=
0.
which
for
)
p'(s
=
the
If
is also the expected
paper,
we
shall
difference prior to the receipt of y.
Later in this
develop an explicit formulation for d
in terms of other primitives.
For intermediate r values, p(s |d) as a function of d is
linear, but
point [d
,
it
may
).
probably
not
be approximated by a linear function, through the
p'(s )], as shown in Figure
3.
19
?(s.\i'i
Figure
The slope of the
depends upon
m'
,
intermediate
3
section
approximation
linear
the
A simple function for this slope,
and a scale factor.
r
of
is:
A
r
m'
=
A >
-m
(21)
d^ 1-r
A/d» is simply a scale factor.
required property
zero for r
-m' so
=
that
positive.
that
it
This function for
primitives
slope
monotonically negative in
is
and negative infinity for r
all
the
in
the
=
1.
has
r and
We have defined
subsequent
equations
the
equals
m
as
will
be
m
20
The general function for p(s
,'d),
with m, d
,
and P'(s
as parameters,
)
may now be written as:
d <
dQ- -.(I-p'(Sq))
(22)
m
1
1
pCs^ld) =r
p'(sQ)-m.(d-dQ)
dQ- -.(I-p'(Sq))
m
d >
<
dQ+ -.P'(Sq)
m
d <
dQ+ --P'CSq)
m
Applying equations (19) and (22) to (15) we get;
M
1
P(sQ!y)
P'(Sq)
=
*(z + \
(1-P'(s q)))
-*(^Z -
17
(
-.(i-p'(s^))
^N^.Z. Lfz .^--.(I-pUSq))"]
Z
-Jz
.p'(s„)]| -
{EXpf— [Z
where:
V +v'
N
•Ik
k'^n'"
- $ Z +
(23)
-.P'(3,
=
.
and
*
-
---!--.p.(Sq)J
EXpf— [Z
+
is the cumulative standard
(l-p'(s ))3)
normal
dis'n.
'V^N
The last two terms are each approximately zero since for large r, m
very large
the
and
right
hand
whereas for small r, m is small
approximately zero.
equation (17)
we
multiplicand
and
the
left
is
is approximately zero,
hand
multiplicand
is
Thus, setting these two terms to zero and applying
arrive
at
an
explicit formulation for a manager's
assessment of the potential need for intervention.
—
,
21
d„
1-r
J V K +v N
Ar
y-u.,-d^
---N,,n— 0—
P(s^)
iV*^M
K N
(i-p'(s^))
(24)
I
.
y-^N-^0
----r-- + -—--- .
-P'(Sq),
1-r
(1-P'(s,))
Jv,,+v.
K
N
\
1-r
y-"N-^o
K
p'Cs^)
IV^
^NN
^''
The logic of this formulation is hardly transparent, but
without damaging its descriptive validity.
equation
of
*[(y-Uw-dQ)/4
-p'(s
)
make
approximations that will simplify the expression for p(s.)
two further
side
can
we
Vjr+vJI,]
and
monotonically
varies
(24)
and 0, as r varies from zero to one.
1
and
If
we
assume
that
the
proportional to r, then we have two simplifying
linearly
variation is
between
term varies monotonically between
second
the
The first term on the right
approximations:
(25)
----- -
+
—-—
]
'
1-r
.
n
(1-p'(s,))
9
=
Ar
V +v
K N
—
t:__r
I^
l^\^\
+ (i-r)(l
- * --^i=-\)
F^
J
and
(26)
1-r
«
#
']
Ar
Jvj^+vj^
r^
f^ ^
P'(s^)
t
F^u
^^n
^''
When equations (25) and (26) are applied to equation (24),
p(s
)
=
r,
.*f-:>-:-2]
K
r.»
+
(1-r)(1-p'(sQ))
1-r
it
yields:
(27)
N
+ (l-r)p'(s^)
(28)
]
22
To interpret
equation
probability p'(s
(28),
that
)
expected difference, d
receipt of
conditional distribution
NO
(y-u.,-d^)/i v„+v' is
K N
that
of
because
there
receipt
difference
which
d,
prior
a
of
is
the
After
mean
of the
the
y-iJij«
variance
has
datum.
Thus,
v„+v„.
measure of the number of standard deviations of
a
change in the expected difference.
Note that since the expected
value is u^+d_, the prior expected change is zero.
is a
is
situation is out of control, there is an
prior to the
,
expected
the
y,
the
note
*[(y-u„-d_)/J
datum
Vj.+v'
measure of the significance of the change in expected difference.
It is an assessment of the potential need for
the basis
intervention
solely
on
p'(s.) is the manager's prior assessment of
of the datum y.
the potential need for intervention and p(s
average
is an
)
these
of
two assessments, weighted by the representativeness of the data stream.
From equation (28) and knowledge that
we may derive an
difference.
have
we
the determination of p(s
probL p(s
=>
prob[ y
),
)
>
for P'(s
=
formulation
explicit
Since
d
for
d
=
p'(s
prior
the
,
)
=
1/2,
expected
chosen to ignore the effects of time upon
we should expect a priori that:
p'(s
)
]
1/2
=
(29)
+ d
fv +v' *"'(p'(s )) + u
N
U
K N
>
)
]
=
1/2
(30)
1
=>
1
F^Ejir^ *'(p'(sj)
-
I
=>
yM
=
K
N
NO
+ u„ + d.]
1
rv^^*''(P'(s^))
-^
Uj^
* d^
=
1/2
(31)
(32)
23
is the cumulative distribution of y and y
where F
value.
Applying our knowledge that d-
=
is the median datum
for p'(s^)
=
p'(s,)
1/2 to
=
equation (32) yields an explicit formulation of d^.
cIq
=
-Jv^
*"'(p'(s^))
does
Note that this formulation
distribution of
Applying
y.
(33)
require
not
equation
(33)
knowledge
any
of
the
(28) yields a final
to
formulation for p(s.)
p(s,)
y-"N
r.*
=
^-.
+
*
(p'(s^))l
]
L.v^+v^
This descriptive model of a manager's
founded upon
finding.
led us
assessment
theory
of
(l-r).p'(s^)
(34)
of
been
p(s.)
has
process of problem
the
overcomes the two difficulties of the Bayesian model
It
to
descriptive
Pounds'
+
reject
that
approach as descriptively invalid.
that
First, it
does not require that the manager have knowledge of f(y|s_) or f(yls.).
Second, it incorporates the theory of Kahneman and Tver sky [1972,
1973]
and Swieringa, et
prior
al^
probabilities in
proportion
managers
that
[1976],
to
tend
to
representativeness
the
ignore
of
the data
stream.
This model
approach.
has
It
further
a
important
extensive attention
and
empirical
the
Bayesian
the
same
form,
1956] That
model
has
received
validation, both in the laboratory
and the field. [Slovic and Lichtenstein
taken
over
of the same form as the Brunswik lens model of human
is
information processing. [Brunswik 1952,
model has
advantage
1971] It is remarkable that
our
even though the development has not
24
relied upon Brunswikian concepts of human information processing.
THE VALUE OF THE INFORMATION SIGNAL FOR MONITORING
From equations (4), (7), and (34), we may derive an
rule for
when
explicit
decision
manager chooses to intervene into the situation.
a
rule is, intervene (choose
a
)
The
if:
1
a- +*
r.I^ -—^7
+*'(p'(s
(p'(s,))J
))j + (l-r).p'(s^)
C/L
>
(35)
or.
.,
y
>rv^Ml*
I
(C/L) - (1-1
l-r)p'(s.)"]
--1
.,
-*"[p'(s^)]
. u^ . d^
(36)
(37)
The a priori value
comparing the
of
the
information
signal
can
be
.
1981] It is assumed that
p'(s
)
< pT,
manager was not prepared to intervene.
=
p(s
by
expected value of outcomes when the signal is used minus
the expected value of outcomes without the signal [Demski
V(y)
measured
and choose a ).u(s ,a
)
that
without
1972;
the
Treacy
signal
the
We have:
+ p(s
and choose a ).u(s ,a
+ p(Sq and choose a ).u(s ,a,) + p(s
)
(38)
and choose a ).u(s ,a.)
- p'(s^).u(3Q,aQ) - p'(s^).u(s^,aQ)
=
p(s
and y
<
y'^).u(3 ,a
+ p(Sq and y > y
)
+ p(s
and y
<
).u(sQ,a^) + p(s^ and y
- p'(sQ).u(sQ,aQ) - p'(s^).u(s^,aQ)
y'^).u(s
>
,a
(39)
)
y ).u(s ,a
)
)
25
Let us define F(y|s
f(y|s_) and
)
f(y|s,)
and F(y|s
)
cumulative
as the
respectively.
These
distributions
distributions
latter
precisely those that we earlier argued were difficult to
generate
Nevertheless, they are useful and usable primitives for a
theory of information signal value.
=
are
and
primitives in a descriptive theory of managerial judgement.
invalid as
V(y)
of
prescriptive
We may now write:
F(yTlsQ).p'(sQ).u(SQ.aQ) + FCy^l
+ [1-F(y^|SQ)].p'(sQ).u(sQ,a^) +
.p
(s^) .u(s^ .a^)
^
)
[
1-F(y'^| s^
s
•
)
].p
'
(40)
(s^) .u(s^ ,a
^
- p'(sQ).u(sQ,aQ) - p'(s^).u(s^,aQ)
=
[1
-
- F(y'^ls^)].p'(s^).[u(s^,a^)
[1
-uCs^.a^)]
- F(y^|sQ)].[1 - p'(s^)].[u(sQ.aQ)
Using equations (5) and (6) we may substitute
(41)
-u(sQ,a^)]
for
the
utilities
and
arrive at the simple formulation:
V(y)
=
(L - C).[1
- F(y'^!s^)].p'(s^) - C.[1 -
F(y'''|
s^) ].p (s^)
(42)
•
Unlike the information economics model, this value can be negative.
signal can lead
a
A
manager astray.
THE VALUE OF AN INFORMATION SYSTEM
In economics,
as a
the common characterization of an information
collection
of
Thus, the value of an
expected value
information signals. [Marschak 1971;
information
system
is
exactly
of its information signals, V(y).
observed
that
little
of
the
Demski
equal
Treacy
is
1972]
the
to
In a recent study
sixteen executive level information systems [Rockart and
it was
system
of
1981]
value of an information system
26
derives from information signals
situation.
These
signals
are
about
interest.
model" of
organization.
the
a
computer
a
of
a
manager through
terminal
to
monitor
The information systems were valued, instead,
sharpening
for their assistance in
condition
usually available to
other channels long before he sits at
situations of
present
the
a
manager's
norms,
"mental
As Ackoff[1967] earlier noted, managers
rarely suffer a lack of relevant information, but an
irrelevant information.
his
Information
systems
are
overabundance
of
valued not because
they provide more signals, but because they allow a manager
derive
to
more value from existing signals, through a sharpening of his norms.
Sharper norms can be achieved through two means:
variance
smaller
v
N
(more certainty)
and
more accurate mean
(more accuracy).
u.,
We shall
N
examine the effects of decreasing the variance upon V(y), the value
of
a signal.
Consider the first derivative of V(y) with
respect
to
v
If
.
this
N
derivative is
negative,
then
the
uncertainty of the norm is decreased.
value of
The
a
signal increases as the
value
of
an
information
signal that causes this decrease in uncertainty is equal to the gain in
value of the information signal.
^V(y)
Mathematically, we have:
^V(y) iyT
(43)
[C.f(yT!sQ).p'(SQ) - (L-C).f(yTl3^).p'(s^)]
(C/L) - (l-r).p'(s^)
1
.[
2lv^
K
N
$
t [p'(s^)]
(44)
27
The second multiplicand of
p'(s
)
p^.
<
equation
(^V(y)/^
Therefore,
(4U)
)
always
is
positive
since
if the first multiplicand is
<
negative.
C.f(yTlsQ).p'(sQ) - (L-C).f(yT|s^).p'(s^)
<
(45)
(L-C).f(s^lyT) - C.fCs^lyT)
>
(46)
Equation (46) is satisfied as long as the threshold datum
T
y
,
of
action,
model is greater than the Bayesian, normative
descriptive
the
for
n
threshold datum, y
,
which is defined such that:
(L-C).f(s^!y^) - C.f(sQ!y^)
From equations
and
(36)
decreasing
Therefore,
decreases.
beyond the point at which y
Vj.
y
decreases
There is
B
T
=
as
Vj^
uncertainty about the
manager's
a
norm decreases the threshold datum for action.
in decreasing
T
that
note
we
(37),
(47)
=
y
•
a
diseconomy
It is an empirical
question as to whether information systems are ever capable of reducing
v„ enough to produce such diseconomies.
CONCLUSIOMS
We have developed a descriptive model of
decision using
a
general
the
intervention
managerial
framework drawn from information economics.
This model provides the basis for an investigation of the value
information signal
context.
and
of
an
information
We have explored the value
provides not
unique
of
an
system
in
information
a
of
an
monitoring
system
that
signals on events, but assistance to a manager in
sharpening his "mental model" of the
data
stream
under
observation.
28
Results suggest that tehre are at times diseconomies in this role.
The models of managerial behavior and of information value are far from
complete.
No attempt has been made to consider
situation with
multiple
distributing his
situations.
signals
attention
or
of
a
manager's
different,
across
monitoring
the
difficulty
alternative approach
developments have
interrelated,
but
step.
It
since
an
excellent
example
making.
decision
the
of
unnecessary
complexity induced by the standard information economics approach.
models
and
an
considerably under the massive weight of its
slowed
provides
provides
augmentation
That approach is in need of
overly general conceptualizations of information and
in
in
normative Bayesian approach to
standard
the
to
information analysis.
results
a
These matters are still under development.
Nevertheless, this work is an important first
Hilton[1980]
of
which
theory
operationalization
defy
It
and,
therefore, exist as untestable truisms.
The solution to these difficulties lies in specialization.
context.
knowledge about the
improve
and
Through
particular
sharpen
our
models
There are no claims that they apply
in this paper apply to monitoring.
outside this
The
specialization
context,
models.
When
such
we
are able to apply
we
as
Pounds'
build
upon
work,
relevant
non-mathematical works, which, after all, are usually far ahead of
models, both can only gain.
to
our
29
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Ackoff,
V.
R.
,
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The Conceptual Framework
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,
of
Psychology
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,
Management
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,
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14
N.
14,
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tf
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7267 029
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