J. Service Science & Management, 2010, 3 : 51 -66
doi:10.4236/jssm.2010.31006 Published Online March 2010 (http://www.SciRP.org/journal/jssm)
51
Evolution or Revolution of Organizational
Information Technology – Modeling
Decision Makers’ Perspective
Niv Ahituv1*, Gil Greenstein2
1Faculty of Management, Tel Aviv University, Tel Aviv, Israel; 2Faculty of Technology Management, Holon Institute of Technology,
Holon, Israel; *Corresponding Author.
Received December 28th, 2009; revised January 28th, 2010; accepted February 21st, 2010.
ABSTRACT
This paper suggests a new normative model that attempts to analyze why improvement of versions of existing decision
support systems do not necessarily increase the effectiveness and the productivity of decision making processes. More-
over, the paper suggests some constructive ideas, formulated through a normative analytic model, how to select a
strategy for the design and switching to a new version of a decision support system, without having to immediately run
through a mega conversion and training process while temporarily losing productivity. The analysis employs the in-
formation structure model prevailing in Information Economics. The study analytically defines and examines a system-
atic informativeness ratio between two information structures. The analysis leads to a better understanding of the per-
formances of decision support information systems during their life-cycle. Moreover, this approach explains norma-
tively the phenomenon of “leaks of productivity”, namely, the decrease in productivity of information systems, after they
have been upgraded or replaced with new ones. Such an explanation may partially illuminate findings regarding the
phenomenon known as the Productivity Paradox. It can be assumed that the usage of the methodology that is presented
in this paper to improve or replace information structure with systematically more informative versions of information
structures over time may facilitate the achievement of the following major targets: increase the expected payoffs over
time, reduce the risk of failure of new versions of information systems, and reduce the need to cope with complicated
and expensive training processes.
Keywords: Decision Analysis, Decision Support Systems, Productivity and Competitiveness, Information Technology
Productivity, the Productivity Paradox
1. Introduction
A major and continuing problem in the information te-
chnology (IT) profession is the high rate of failure of
new information systems (IS) or upgraded versions of
them. From a rational point of view it may be assumed
that IS professionals usually analyze and design IS “pro-
perly”. But is it really so? Are they aware of the possibil-
ity of limits in perception among IS users, especially
decision makers? Do they realize that “improvement” of
decision support information systems might lead some-
times to a result opposite to what has been expected,
namely degradation in the level of the productivity of the
firms, since new and unfamiliar decision rules have not
been fully implemented and adopted by the decision-
makers?
This article suggests a new normative model that at-
tempts to explain that improvement of versions of exist-
ing information systems do not necessarily increase the
effectiveness and productivity of decision making proc-
esses. It also suggests some constructive ideas, formu-
lated through a normative analytic model, how to select a
strategy of switching to new version of a system, without
having to immediately run through a mega training pro-
gram, and to take a risk of losing productivity.
The methodological and theoretical foundations for the
analysis presented here anchor in the literature on infor-
mation economics. The earliest mathematical model pre-
senting the relaying of information in a quantitative form
was that of Shannon [1]. The model distinguished be-
tween two situations:
1) A noise-free system—a univalent fit between the
transmitted input data and the received signals;
2) A noisy system—the transmitted input data (denot-

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 52
ing a state of nature) are translated into signals probabil-
istically.
In assigning an expected normative economic value to
information, some researchers made use of Microeco-
nomics and Decision Theory tools [2]. The combination
of utility theory and the perception that information sys-
tems can be noisy led to the construction of a probabilis-
tic statistical model that accords to an information system
the property of transferring input data (states of nature) to
output (signals) in a certain statistical probability [3–5].
This model, which delineates a noisy information system,
is called the information structure model. It is based on
the assumption that a system is noisy but it does not ex-
amine the nature of the noise. This paper expands the
analysis by examining some patterns of noise. The con-
sequences of that analysis are then demonstrated.
Over the years significant research was conducted to
explore aspects of the phenomenon termed by Simon [6]
as “bounded rationality”1 and its main derivative—sa-
tisficing behavior. Some of its aspects were presented
comprehensively by Rubinstein [7]. Ahituv and Wand [8]
showed that when satisficing is incorporated into the
information structure model, there might be a case where
none of the optimal decision rules will be pure anymore
(unlike the results of optimizing behavior).
Ahituv [9] incorporated one of the aspects of bounded
rationality into the information structure model: the in-
ability of decision-makers to adapt instantaneously to a
new decision rule when the technological characteristics
of the information system, as expressed by the probabili-
ties of the signals, are suddenly changed. Moreover, Ahi-
tuv [10] portrayed a methodology in which decision
support systems are designed to act consistently during
their lifecycle (in accordance with a constant decision
rule). He suggested that this decision rule (that was an
optimal decision rule in a previous version of the infor-
mation system) guarantees improvement of expected
outcomes, although it is not necessarily the optimal deci-
sion rule for later versions of this information system.
This study presents a conceptual methodology that
combines aspects of bounded rationality [9,10] dealing
with a rigid decision rule and the life-cycle of informa-
tion systems, with elements of rational behavior pre-
sented in the information structure model [5].
The article raises some questions: Is it possible to im-
prove an existing information system without adopting a
new decision rule? What are the analytical conditions
that enable a “smooth” (without much disturbance) up-
grading or replacement of an information system? In a
decision situation where two information structures are
activated probabilistically, and one of them is generally
more informative than the other, are there analytical con-
ditions encouraging to enhance the percentage of usage
of the superior system?
A normative framework is suggested to cope with es-
sential processes (e.g.: implementation processes, correc-
tion of bugs, or upgrading of versions) during the life
cycle of a decision support system [11]. By defining and
analyzing a new informativeness relationship - “the sys-
tematic informativeness ratio”, this paper demonstrates
situations where decision-makers are equipped with par-
tial information. Through these cases, it is explained how
to assure a “smooth” implementation of new or upgraded
information systems, as well as how to reduce the in-
vestment in implementation activities.
Moreover, It is shown that the existence of this new
relationship (ratio) between two information structures
enables to improve the level of informativeness without
the awareness and the involvement of the users (the deci-
sion makers).
“The systematic informativeness ratio approach”, whi-
ch is presented and analyzed for the first time in this pa-
per, contributes to better understanding of various as-
pects of the “productivity paradox” [12–14]. Furthermore,
it portrays a methodology that suggests how to deal with
some aspects of the “productivity paradox” which were
explored in earlier studies [15,16].
The next section summarizes the information structure
model and the Blackwell Theorem [5]. It describes the
motivation to use convex combinations in order to de-
scribe processes during the life cycle of a decision sup-
port system. Section 3 describes, analyzes, and demon-
strates a new informativeness relationship between two
information systems—“the systematic informativeness
ratio”. Section 4 explores the existence of systematic
informativeness ratio between un-noisy information str-
uctures. Section 5 presents some implications that could
be extrapolated to noisy information structures. The last
section provides a summary and conclusions, and pre-
sents the contribution of the study and the directions it
opens for further research. Proofs of the theorems and
lemmas appear in the appendix.
2. The Basic Models
2.1 The Information Structure Model and
Blackwell Theorem
The source model employed in the forthcoming analysis
is the information structure model [5]. This is a general
model for comparing and rank ordering information sys-
tems based on the rules of rational behavior.2
The information structure model enables a comparison
of information systems using a quantitative measurement
reflecting their economic value. An information structure
1Simon termed the human decision-making process, which is affected
by bounded rationality as “satisficing”, and the decision-maker in ac-
cordance as a “satisficer” (aims to be satisfied with his or her decision).
This is in contrast to the perception of the decision-maker under rational
behavior assumptions in “classical” Utility theory who is an “optimizer”
(aims to achieve the best out of his or her decision)

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 53
Q1 is said to be more informative than an information
structure Q2 if the expected payoff of using Q1 is not
lower than the expected payoff of using Q2. The expected
payoff is trace (
 *Q*D*U)3, where trace is an operator
that sums the diagonal elements of a square matrix. The
objective function for maximizing the expected compen-
sation is
 )( UDQtrace
Max
D
***4.
Let us examine a numerical example. Assume that an
investment company serves its customers by using a web
based information system. Let Q1 be an information
structure that predicts the attractiveness of investing in
various alternative channels. The IS supports the deci-
sion-making of the investors. For simplicity, suppose
there are three categories of states of nature: S1 - acceler-
ated growth (probability: 0.2), S2 - stability (probability:
0.6), and S3 - recession (probability: 0.2). Assume also
that there are three possible decisions: A1 - Invest in bank
deposits; A2 - Invest in stocks; A3- Invest in foreign cur-
rency; Q1 - The information system provides the follow-
ing signals: Y1- Accelerated growth is expected; Y2 - Sta-
bility is expected; Y3 - Recession is expected;











100
06.04.0
04.06.0
1Q
The compensations matrix U, which represents the
expected percentage of profit or loss, is described as fol-
lows: 











311
050
103
U

4.2
311
050
103
100
06.04.0
04.06.0
2.000
06.00
002.0
***
3,32,31,3
3,2,21,2
3,12,11,1
2
1











































DDD
DDD
DDD
U)DQtrace(
Max
D
where 










100
001
001
*
D5. Invest “A1”, while the signal is
Y1 or Y2. Invest “A3”, while the signal is Y3.
Given two information systems that deal with the same
state of nature and are represented by the information
structures Q1 and Q2, Q1 is defined as generally more in-
formative6 Given two information systems that deal with
the same state of nature and are represented by the in-
formation structures Q1 and Q2, Q1 is defined as generally
more informative.7 The rank ordering is transitive.8
Over the years, a number of researchers developed
analytical models to implement the concept of the infor-
mation structure model in order to evaluate the value of
information technology. Ahituv [10], demonstrated the
life cycle of decision support information system with
the model. Ahituv and Elovici [17] evaluated the value of
performances of distributed information systems. Elovici
2According to the information structure model, four factors determine
the expected value of information.
The a priori probabilities of pertinent states of nature. Let Sbe a
finite set of n states of nature: S={S1,..,Sn}. Let P be the vector of a
p
riori probabilities for each of the states of nature: P=(p1,..,pn).
The information structure – a stochastic (Markovian) matrix tha
t
transmits signals out of states of nature. Let Y be a finite set of n sig-
nals, Y={Y1,..,Ym}. An information structure Q is defined such that its
elements obtain values between 0 and 1, Q: SxY
 [0,1]. Qi.j is the prob-
ability that a state of nature Sii displays a signal Yj 1
1



m
j
i,j Q
The decision matrix – a stochastic matrix that links signals with the
decision set of the decision-maker. Let A be a finite set of k possible
decisions, A={A1,..,Ak}. Let D be the decision function. Similar to Q, D
is a stochastic (Markovian) matrix, namely, it is assumed that the deci-
sion selected for a given signal is not necessarily always the same. D:
Y
x
A  [0,1]
The payoff matrix – a matrix that presents the quantitative compensa-
tion to the decision-maker resulting from the combination of a decision
chosen and a given state of nature. Let U be the payoff function: U :A
x
S   (a combination of a state of nature and a decision provides
a
fixed compensation that is a real number). Ui,j – is the compensation
yields when decision maker decides –“Ai”, while state of nature “Sj”
occurs.
3Sometimes Q represents an un-noisy (noise free) information struc-
ture. In these cases Q represents an information function f, Yf:S [4].
Q is a stochastic matrix that contains elements of 0 or 1 only. This
means that for each state of nature the information structure will al-
ways act identically (will produce the same signal), although it is no
t
guaranteed that the state of nature will be exclusively recognized.
4When the utility function is linear, that is, the decision-maker is of the
type EMV [2], a linear programming algorithm may be applied to solve
the problem, where the variables being the elements of the decision
matrix D. It can be proved that at least one of the optimal solutions is in
a form of a decision matrix whose elements are 0 or 1 (a pure decision
rule), [5]. For numerical illustrations of the model, see [8–10].
5*
Dis a decision matrix which represent the optimal decision rules in
this decision situation.
6It should be noted that when we deal with two information functions
rather than structures we use the term “fineness” to describe the general
informativeness ratio [4].
7In terms of the information structure model, if for every possible
p
ayoff matrix U, and for every a priori probability matrix







)()( *D*U*QtraceMax*D*U*QtraceMax
DD 2
1, then Q1is generally
more informative than Q2, Denoted: .21 QQ  Blackwell Theorem
states that Q1 is generally more informative than Q2 if and only if there
is a Markovian (stochastic) matrix R such that Q1*R = Q2. Ris termed
the garbling matrix
8It should be noted that the general informativeness ratio is a partial ran
k
ordering of information structures. There is not necessarily rank orde
r
between any two information structures.

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 54
et al [18] used this method to compare performances of
Information Filtering Systems. Ahituv and Greenstein
[15] used this model to assess issues of centralization vs.
decentralization. Aronovich and Spiegler [19] use this
model in order to assess the effectiveness of data mining
processes.
The model was expanded to evaluate the value of in-
formation in several aspects: the value of a second opin-
ion [20]; the value of information in non-linear models of
the Utility Theory [21]; analyzing the situation of case
dependent signals (the set of signal is dependent on the
state of nature, [22]); a situation of a two-criteria utility
function [23]. The model was also implemented to
evaluate empirically the value of information in postal
services [24], and in analysis of Quality Control methods
[25,26].
2.2 The Use of Convex Combinations9 of
Information Structures to Represent
Evolution during Their Life-Cycle
A possible reason why we should consider probabilistic
combination of information systems is the existence of
decision support systems that use Internet (or intranet)
based search engines. These engines can retrieve infor-
mation from several information sources, and produce
signals accordingly. The various sources are not always
available.
Information sources are essential for the proper sur-
vivability of competitive organizations. As a result, the
importance of proper functioning of information systems
is increasing. When a certain source in unavailable it is
possible to acknowledge the users about it by alarming
them with a special no-information signal [15]. Another
option, which is presented in this paper, is to consider
implementation of a “mixture” of information systems.
For example: suppose there is “a state of the art” organ-
izational information center that can serve, during peak
times only 90% of the queries. How will the rest 10% are
served? One alternative is to reject them.10 Another one is
to direct those queries to a simpler (perhaps cheaper)
information system whose responses are less informative.
This leads to consider probabilistic usage of information
systems that can be delineated by a convex combination.
The analysis focuses on the convex combination of
information structures reflecting a probabilistic employ-
ment of a variety of information systems (structures),
where the activation of each one of them is set by a given
probability. The various systems react to the same states
of nature and produce the same set of signals.11
The mechanism of convex combinations12 of informa-
tion systems is employed in an earlier research by Ahituv
and Greenstein [15] which analyses the effect of prob-
abilistic availability of information systems on produc-
tivity, and illuminates some aspects of the phenomenon
that are termed as “the productivity paradox” [12–14].
3. The Systematic Informativeness Ratio
3.1 Definition of the Systematic
Informativeness Ratio
As mentioned in Section 2, when an information struc-
ture Q1 is more informative than an information structure
Q2 irrespective of compensations and a priori probabili-
ties, a general informativeness ratio exists between the
two of them [5].
If an information structure Q1 is more informative than
Q2 when the optimal decision rule of Q2 is employed, and
given some certain a priori probabilities of the states of
nature, then under some assumptions on the payoffs, an
informativeness ratio under a rigid decision rule is de-
fined between them [9].13
9The convex combination of information structures was discussed in
earlier studies. Marschak [4] notices that the level of informativeness o
f
convex combination of two information structures (denoted Q1 and Q2)
which produce the same set of signals is not equivalent to the level o
f
informativeness of using Q1 with a probability p and Q2 with the com-
p
lementary probability (1-p).
10Sulganik [27] indicates that a convex combination of information
structure could be used to describe experimental processes (with a
p
robability p of success and (1-p) of failure). For example, he investi-
gates the convex combination of two information structures: one pre-
sents perfect information and the other one no-information (its rows are
identical).
11It should be noted that in case that the two information structures do
not produce the same set of signals, the non-identical signals can be
represented in by columns of zeroes respectively [9].
12A convex combination of two information systems is defined as fol-
lows: Let Q1 and Q2 be two information structures describing informa-
tion systems. Let S={S1,…,Sn} be their set of the states of nature. Let
Y={Y1,…,Ym} be their set of signals. When a decision situation is given
let p the probability that Q1 will be activated, and (1-p) that Q2will be
activated. Since, decision makers do not aware which information
structure is activated, Q3, the weighted information structure, is repre-
sented by a convex combination of Q1 and Q2.
Q3= p* Q1 + (1-p) * Q2
13Given two information systems that deal with the same states of na-
ture, produce the same set of signals, and are represented by the infor-
mation structures Q1 and Q2 respectively, Q1 will be considered more
informative than Q2 under a rigid decision rule if its expected payoff is
not lower than that of Q2 for the following conditions:
.,..,1, nii  Let )),(( iikU the single maximum payoff when the
state of natureiSoccurs.
Denote:
*max(((),); *min(((),);uUkiiuUkii
i i
 0max(,); 0min(,);
,() ,()
uUki uUki
ikk i ikk i

 
t
(k(n))1(n,(k(1)),..,1(1,
*
1)q(qq t
)q(qq (k(n))2(n,(k(1)),..,2(1,
*
2, respectively.
;*
*
0uu   ;*
*0uu 
 ;000 uu  
The theorem which is proved by Ahituv [9], states that:
If 0
*
2
* *
1
*)(
  qq
t then Q1 more informative than Q2with
regard to ΠandU .
The ratio will be denoted: 21QQ
R

The informativeness ratio under a rigid decision rule is a partial rank
ordering of information structures.

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 55
Assume those two informativeness ratios can be con-
ceptually combined to a new informativeness ratio: Let
Q1 and Q2 be two information structures that deal with
the same state of nature and produce the same set of sig-
nals. Q1 will be considered systematically more informa-
tive than Q2 if for any decision situation (for any a priori
probabilities vector- and any payoff matrix-U), its
expected payoff is not lower than that of Q2 while Q1
operates under an optimal decision rule of Q2. In terms of
the information structure model, this is presented herein-
after by Definition 1.
Definition 1: Let Q1 and Q2 be two information struc-
tures representing two information systems operating on
the same set of states of nature S = {S1,…,Sn} and pro-
ducing the same set of signals Y = {Y1,…,Ym}. Q1 is de-
fined systematically more informative than Q2, denote
21 QQS
, if for any decision situation (irrespective of
payoffs and a priori probabilities) Q1 is more informative
than Q2 under an optimal decision rule of Q2.
It means that if Q1 is systematically more informative
than Q2, then for every decision situation14 there exists an
optimal decision rule of the inferior information structure
Q2, that can be used with the superior information struc-
ture Q1, and guarantees at least the optimal outcomes of
using Q2.
Mathematically it looks this way:
2
22
21
22
{max()}()
((*)) ())
QQ
Q
D
DDQ ,traceΠ*Q *D*U
MaxtraceQ*D*UtraceΠ*Q *D*U
 

where 2 {max( )}DQ - denotes the set of optimal deci-
sion rules, when Q2 is activated in this specific decision
situation.
In contrast to the general informativeness ratio, in the
systematic informativeness ratio the information struc-
ture Q2 can be replaced with the superior systematically
information structure Q1, without an immediate aware-
ness of the decision makers (the users), since the decision
rule does not necessarily have to be changed instantane-
ously. It means that when the systematic informativeness
ratio exists between two information structures, at least
the same level of expected payoffs is guaranteed when
the superior15 information structure is activated. Hence
the decision maker does not have to adopt a new optimal
decision urgently.
Let us now examine the informativeness ratio between
two information systems from the point of view of “sm-
ooth” implementation. This is presented in Definition 2.
Definition 2: Let Q1 and Q2 be two information struc-
tures representing two information systems operating on
the same set of states of nature S={S1,…,Sn} and produc-
ing the same set of signals Y={Y1,…,Ym}. Assume Q1 is
generally more informative than 2 Q. A smooth imple-
mentation of Q1 instead of Q2 is defined if for any level
of usage p

 10


p 221 *)1(* QQpQp  .
The importance of this ratio is that in any probabilistic
level of usage of the superior information system Q1, the
mean of the expected payoffs (compensation) that the
decision-makers gain is not less than that achieved by
using only the inferior information system. It contributes
to a smooth implementation of the superior information
structure Q1.
In our study we argue that those definitions (1 & 2) are
equivalent. Theorem 1 proves analytically the equiva-
lence of Definition 1 and 2.
Theorem 116
Let Q1 and Q2 be two information structures operating
on the same set of states of nature S = {S1,…,Sn}, and
producing the same set of signals Y = {Y1,…,Ym}. Then
21 QQ S


221 *)1(*,10, QQpQppp 



This theorem shows that the two ratios which have been
defined above are identical. Replacement (or improvement,
or upgrade) of an information structure with a more sys-
tematically informative, information structure than it,
guarantees smooth implementation, and vice versa.
Moreover, from the aforementioned equivalence it is
understood that during a smooth implementation of the
superior information structure1 Q17, we do not have to
adopt a new decision rule, and we can stick to an optimal
decision rule we used in the past with the inferior infor-
mation structure2 Q. In fact this theorem sets a new nor-
mative perspective that defines the necessary and suffi-
cient conditions for the ability to implement a superior
information structure smoothly without immediate inter-
ference in the routine work of decision makers.
Using this method facilitates information systems pro-
fessionals to plan systems under the assumption that
during a certain transition period the decision-makers
may act identically and stick to the same decision-rule
[10]. The existence of this informativeness ratio reduces
the criticality of an urgent implementation process.
3.2 A Framework to Examine the Existence of
the Systematic Informativeness Ratio
In order to identify the existence of a systematic infor-
mativeness ratio between two information structures
when one of them is generally more informative than the
other, we would analyze a special case in which the
number of signal and the number of states of nature are
identical. In this case, the identity square matrix I is a
complete and perfect information structure. We will try
to find out whether I is systematically more infor-
mative than any other square stochastic matrix of similar
14A given set of a-priori probabilities - Π, and a given utility matrix
-.U
15Systematically more informative than the other.
16The proof is provided in the appendix.
17Since 221 *)1(*,10, QQpQppp  , 21 QQ S
.

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 56
dimensions.
The motivation to do this is provided by Lemma 1.
Assume two information structures Q1 and Q2 act on the
same set of states of nature, and respond with the same
set of signals, and ,*12 RQQ  where R is a stochastic
matrix (Blackwell Theorem’s condition). In Lemma 1 it
is shown that the existence of systematic informativeness
ratio between I and R sets a pre-condition (sufficient
condition) to the existence of the general informativeness
ratio between Q1 and Q2.
Lemma 118
Let Q1 and Q2 be two information structures operating
on the same set of states of nature S = {S1,…,Sn}, and
producing the same set of signals Y = {Y1,…,Ym}. As-
sume that Q1 is generally more informative than Q2, im-
plying that Q2= Q1*R, where R is a stochastic matrix [5].
If ,01, *(1)*pp pIpRR,
Then 221 *)1(*,10, QQpQppp 


From Lemma 1 it can be shown that if Q1 is generally
more informative than Q2, namely Q2 = Q
1*R (R is a
stochastic matrix) and S
I
R (I is systematically more
informative than R) then 21 QQS
 (Q1 is systematically
more informative than Q2).
3.3 The Monotony of the Systematic
Informativeness Ratio
The following lemma deals with the improvement of the
accuracy level of information systems by enhancing the
probability to receive perfect information.
Lemma 219
Let I be an information structure that provides perfect
information. Let Q be any information operating on the
same set of states of nature 1{}nSS,..,S

and producing
the same set of signals Y = {Y1,…,Yn}.
If: for Qp)*Q(,p*Ip 110 (every convex
combination of I and Q is generally more informative
than Q)
Then:
q)*Q(q*Ip)*Q(,p*Ipqq,  1110
Conclusion:
If: Qp)*Q(,p*Ipp, 

110
Then:
q)*Q(q*Ip)*Q(,p*Ipqq,p,



 1110
This lemma proves the monotony of the systematic
informativeness ratio. Actually it is shown that an im-
provement in the accuracy level of information (ex-
pressed by increasing the probability of perfect informa-
tion) is positively correlated with the general informa-
tiveness ratio of a convex combination.
3.4 The Systematic Informativeness
Ratio – An Example
We continue the example of Q1, an Information system
for choosing an investment option, which was first dem-
onstrated in Section 2.











100
06.04.0
04.06.0
1Q
Suppose it is intended to replace the information sys-
tem with an improved one, Q2: 










100
09.01.0
01.09.0
2Q
Due to technological and organizational limitations,
e.g.: inability to implement the system simultaneously all
across the organization and the need to monitor carefully
the system’s performances, the system is implemented
step by step.
By using some of the lemmas and theorems presented
above, it can be demonstrated that the information struc-
ture Q2 is systematically more informative than Q1.
Let Q0 be an information system: 










100
05.05.0
05.05.0
oQ
Let Q3 be an information structure, which represents
perfect information.










100
010
001
3Q
Since,

 0003 **)1(*,10, QQQpQppp  20




















































100
05.05.0
05.05.0
100
05.05.0
05.05.0
*
100
05.05.0
05.05.0
*)1(
10 0
010
001
*pp
then, from Theorem 1 it is clear that Q3 is systematically
more informative than Q0.
Let us present Q1 and Q2 as convex combination of Q3
and Q0.
032 *2.0*8.0
100
09.01.0
01.09.0
QQQ 











18The proof and an example are provided in the Appendix.
19The proof is provided in the Appendix.
20We use the information structure Q0, as a garbling (stochastic) matrix,
either.

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 57






















100
05.05.0
05.05.0
*2.0
100
010
001
*8.0


































100
05.05.0
05.05.0
*8.0
100
010
001
*2.0
*8.0*2.0
100
06.04.0
04.06.0
031 QQQ
According to Lemma 2 Q2 is systematically more in-
formative than Q1. Table 1 demonstrates the implication
of the existence of the systematic informativeness ratio
between Q2 and Q1.
This example illustrates that if an upgrading of an in-
formation system is based on the implementation of later
versions of it which are systematically more informative
than the earlier versions, then sticking to the old and fa-
miliar decision rule will not harm productivity.
The principle of developing information systems to be
systematically more informative provides the luxury of
training and on-site implementation which is “life-cycle
independence”. It facilitates the implementation of a new
version of information system or insertion of minor
changes, without the immediate awareness of the deci-
sion makers. Therefore, organizations can schedule the
optimal timing of wide training processes. That is in con-
trary to the usual situation, when the scheduling of train-
ing and on-site implementation might interfere with other
organizational considerations and requirement (e.g.: pe-
riodically tasks). Moreover, development of new deci-
sion support systems without adopting this principle may
explain, normatively, “leaks of productivity”. In other
words it may explain the decrease in user performance of
information systems, although they have been improved.
This degradation in the expected outcomes while using
improved information systems can be attributed to the
inability of users to adapt immediately to new decision
rules.
4. The Systematic Informativeness
Ratio – A Noise Free Scenario.
4.1 Conditions for Existence of the Systematic
Informativeness Ratio
Historically, the starting point for analyzing the value of
information in noisy information structures was the ana-
lysis of the value of information in noise free information
structures. These are also termed information functions
[4,5]. Following this approach, we will start with a sim-
ple presentation of the informativeness ratio between in
formation functions, which could be classified as un-
noisy information structures.
In order to identify the existence of a systematic in-
formativeness ratio between two information functions, a
new aggregation ratio (a fineness ratio that keeps orders
of signals) between information functions is defined,
hereinafter.
Definition 3
Let If be the identity information function. Let
}{ n,..,SSS1

be its set of states of nature and
}{ n,..,YYY 1

the set of signals If produces. Hence,
YS fiiI

)( .
Assume for simplicity that {1} S ,..,n, {1} Y,..,n

then I()
f
ii

Table 1. Comparison of two information structures, one of them is systematically more informative than the other
The current information
structure
The improved information
structure
Information structure  









100
06.04.0
04.06.0
1Q











100
09.01.0
01.09.0
2Q
The matrix of a priori probabilities for the states of nature + The matrix of compensation (percentage):











2.000
06.00
002.0












311
050
103
U
The Decision rules:
A1 – Invest in Bank Deposits
A2 – Invest in stocks
A3 – Invest in foreign currency 










100
001
001
D











100
001
001
D











100
010
001
D
Expected compensation (percentage) 2.4 2.4 3.12

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 58
Let
F
be a set of information functions that acts on
the same decision environment of If.
)( )(()( kkgtheni,kigiforiigFg  )
Since there is an isomorphism between the representa-
tion of information functions and a set of information
structures representing them,
F
can be defined analogi-
cally as a set of un-noisy information structures.
)111 1111  k,ki,ki,i Ftheni,,kFifor(FFF
Following that If is equivalent (for example) to
I
,
an information structure that produces perfect information
I
is represented by the identity Matrix of the order nxn.
In fact
F
is a complete set of information functions
that could be termed as aggregations of If. If Fg

its
way of transforming of states of nature into signals does
not contradict the way If transforms states of nature
into signals. In other words it can be said that If is a
higher resolution of every information function belong-
ing to the set
F
.
Theorem 2 sets the necessary and sufficient conditions
for the existence of systematic informativeness ratio be-
tween I (the identity information structure), and any
non-noisy information structure:
Theorem 221
Let F1 be an information function. Let ,..,SSSn}{ 1
be its set of states of nature. Let }{n,..,YYY 1 be its set
of signals. Let I be the identity information function,
which represent perfect information, then 1
F
I
S
 if and
only if 1
F
F.
It is shown in Theorem 2 that for every information
function (non-noisy information structure) F1, the neces-
sary and sufficient condition that
I
is systematically
more informative than F1, is that 1
2
1
F
F
. In fact,
four equivalent conditions are found as we will show
below.
Let 1
F
be an information function. Let
,..,SSSn}{ 1be its set of states of nature. Let
}{n,..,YYY 1be its set of signals. The following condi-
tions are equivalent.22
1) 1FI S 
2) F-p)*F(p*Ipp,11110 
3)
F
F
1
4) 1
2
1
F
F

The equivalence of the first and second conditions
(which was demonstrated earlier by Definition 1 and 2
respectively) was proven by Theorem 1. Conditions 1
and 2 are not specific to un-noisy information structures,
and can hold for any type of structure. In contrast to
Conditions 1 and 2, the third and fourth conditions are
relevant only to un-noisy information structures. By us-
ing those two latter conditions we can explicitly classify
un-noisy and diagonal information structures into two
separate classes:
1) Structures that the identity information structure is
systematically more informative than them,
2) Structures that the identity information structure is
not systematically more informative than them.
4.2 The Implications of the Systematic
Informativeness Ratio – An Example
In the example that follows, two scenarios are presented,
analyzed, and compared. The first scenario: upgrading an
un-noisy information structure F1 to I—the identity in-
formation structure, while I is systematically more in-
formative than F1.
The second scenario: upgrading an un-noisy informa-
tion structure F2 to I—the identity information structure,
while I is not systematically more informative than F2.
We use the situation of choosing an investment alter-
native, that was shown earlier in Section 3, except the
fact that the current information structures are F1 or F2
respectively.
Suppose an un-noisy information structure that pre-
dicts the attractiveness of an investment in various chan-
nels is installed. This information structure does not dis-
tinguish between S1 – accelerated growth, and S2 – stabil-
ity. It is intended to replace the information system with I
- an information function that provides perfect informa-
tion: 










100
010
001
I
Due to technological and organizational limitations,
e.g.: inability to implement the system simultaneously all
across the organization and the need to monitor carefully
the system’s performance, the system is implemented
step by step.
First Scenario: The existing information structure is
F1.
2
11
11
100100 100
100 100*100
001001 001
100
100
001
FF
FFF














Since
F
F

1, it can be shown from Theorem 2 that
1
FI S . Table 2 demonstrates that while the probability
21The proof is provided in the Appendix.
22While, Theorem 1 proves the equivalence between expressions 1&2,
Theorem 2 proves the equivalence between expressions 3&4, and then
p
roves the equivalence between expressions 1&3.

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 59
for perfect information increases, the expected compen-
sation increases too.
Second Scenario: Suppose the decision situation is
identical to the previous one, but instead of F1 the exist-
ing information structure is F2 where:
 











































2
2
22
001
001
001
010
001
001
*
010
001
001
010
001
001
F
FF
FF

2
Since FF 
2, according to Theorem 2 I is not sys-
tematically more informative than F2.
Table 3 demonstrates that although the probability of
perfect information increases the level of informativeness
declines.
The comparison between those two scenarios is dem-
onstrated in Figure 1:
By observing the aforementioned example it can be
concluded that, when a new (improved) information sys-
tem is systematically more informative than the current
information system two important goals are achieved:
1) “Decision situation independence” -The ability to
implement the information system step by step and to
improve the level of informativeness is guaranteed.
2) “Life-cycle independence” -The ability to imple-
ment the information system without interfering the users
(the decision makers) and while existing expected out-
comes are guaranteed (without the necessity to start
training and testing processes).
5. Towards Assessing the Systematic
Informativeness Ratio between Noisy
Information Structures – the Dominancy
of Trace
A characteristic of F is that its diagonal elements are
(weakly) dominant (in accordance with Definition 4).
From Theorem 3 it can be shown that this characteristic
is a necessary condition for existence of the systematic
informativeness ratio between I and Q:
Theorem 334
Let I (the identity matrix), and Q be two information
structures. ,..,SSSn}{ 1is the common set of states of
nature, and }{n,..,YYY 1 is their same set of signals
they produce.
1i,i j,i
S
I
Qi,i,..,n,QQij  (The diagonal el-
ements are weakly dominant in each and every column).
This theorem implies that the dominancy of the di-
agonal elements in each and every column of an infor-
Table 2. Expected compensation in various levels of prob.
for perfect information (1st scenario)
Expected
compen-
sation
The prob-
ability to
receive F1
The probability
to receive I.
Characteristics
of the Decision
situation
2.6 1 0
2.76 0.8 0.2
2.92 0.6 0.4
3.08 0.4 0.6
3.24 0.2 0.8
3.4 0 1
A-priory
probabilities:
(0.2,0.6,0.2)
Perfect
information










100
010
001
Partial
information










100
001
001
Table 3. Expected compensation in various levels of prob-
ability of perfect inf. (2nd scenario)
Expected
compen-
sation
The prob-
ability to
receive F2
The probability
to receive I
Characteristics
of the Decision
situation
2.6 1 0
2.52 0.8 0.2
2.4666 0.666 (2/3) 0.333 (1/3)
2.56 0.6 0.4
2.84 0.4 0.6
3.12 0.2 0.8
3.4 0 1
A-priory
probabilities:
(0.2,0.6, 0.2)
Perfect
information










100
010
001
Partial
information










010
001
001
mation structure is a necessary condition for the exis-
tence of a systematic informativeness ratio between the
identity information structure (which represents complete
information) and the non-identity information structure.
This casts a preliminary condition for the existence of the
informativeness ratio.
The following example demonstrates, by using Theo-
rem 3, that the systematic informativeness ratio is not
always transitive.
1
100
100
001
F








,2
100
010
010
F









Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 60
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91
Th e pr obabilit y of perfe ct infor mation
expected outcome
F1
F2
Figure 1. A comparison between the two scenarios
11
100
02507502501 0
001
1001 00
07510 0075 025 0
0010 01
Q.*I.*F.*
.*. .




 




 
 
 
 

 
 
 
 

21 2
100
*0406075 025 0
001
QQ.*I .*F..*









100 100
0401 00601 0
001 010
100100 100
075 025 0010075 0250
0010 06 04006 04
.* .*
.. *..
.. ..

















 
 
 
 

 
 
 
 
Since
F
F
1, from Theorem 2 it is concluded that
1Q
I
S
. Moreover, since
F
F

2, from Lemma 1 it is
concluded that21 QQS
. However, from Theorem 3, it is
concluded that since2,2 2
2,3
2QQ , I is not systematically
more informative than Q2.
Since the systematic informativeness ratio is not al-
ways transitive, when there is a multi stage implementa-
tion and improvement program during the life-cycle of a
decision-support information system and the informa-
tiveness ratio of this information system can be improved
systematically, the preservation of systematically infor-
mativeness ratio is not automatically guaranteed during
the whole life-cycle of information system. Hence, the
importance of a long-range perspective arises. This can
be achieved in one of two ways, depending on the ability
to guarantee whether the last version of information
structure can be systematically more informative than
any previous version, or only superior to its predecessor
version:
When the systematic informativeness ratio can be ob-
tained between each and every two sequential versions of
an information system during its lifecycle, then a long-
range plan of the versioning mechanism is required. This
could guarantee that the latest version of an information
system will be systematically more informative than any
of the previous versions. Moreover, it will guarantee a
growth (or at least stability) in expected outcomes during
the lifecycle of the decision support system, without
alerting the decision makers. Hence, implementation and
training processes between versions of the information
system become less critical.
When the systematic informativeness ratio can be
achieved only between the last version of an information
system and its predecessor version, then a training and
implementation plan is required. However, the existence
of systematic informativeness ratio between consequent
versions reduces the costs and lowers the criticality of the
implementation and training processes.

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 61
6. A Summary and Conclusions
This paper analytically examines and identifies the sys-
tematic informativeness ratio between two information
structures. The methodological approach presented here
may lead to a better understanding of the performances
of decision support information systems during their
life-cycle.
This approach may explain, normatively, the pheno-
menon of “leaks of productivity”. In other words it may
explain the decrease in productivity of information sys-
tems, after they have been improved or upgraded. This
degradation in the expected outcomes can be explained
by the inability of the users to adapt immediately to new
decision rules.
It can be assumed that the usage of the methodology
that was presented in this paper to improve or replace
information structure with systematically more informa-
tive versions of information structures over time may
facilitate the achievement of the following major targets:
1) Increase the expected payoffs over time.
2) Reduce the risk of failure of new information sys-
tems as well as new versions of information systems.
3) Reduce the need to cope with complicated and ex-
pensive training processes during the implementation
stages of information systems (as well as the implemen-
tation of new versions of the systems). Moreover, some-
times this process can be completely skipped during the
installation of a new version of an information system.
The paper analyzes the conditions for the existence of
a systematic informativeness ratio between I -the identity
information structure which represents complete infor-
mation, and another information structure. In the case of
non-noisy information structures the necessarily and suf-
ficient conditions for existence of the systematic infor-
mativeness ratio between I and a second information
structure are set and proved comprehensively. As a result,
some necessary and sufficient conditions are set, proved
and demonstrated for the noisy information environment
as well.
Further research can be carried out in some directions:
1) Exploration of additional analytical conditions for
the existence of the systematic informativeness ratio be-
tween I, the identity information structure and noisy in-
formation structures.
2) Classification of cases where the systematic infor-
mativeness ratio inheritably exists by using the condi-
tions those are set so far.
3) Devising empirical methods to examine the impact
of using the principle of developing decision support
information systems is systematically more informative
over time, on the performance of decision-makers, as
well as on their perceived satisfaction from using those
systems.
4) Designing empirical studies (experiments, case
studies and surveys) to validate the theoretical analysis
provided here.
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Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 63
Appendix
Theorem 1:
Let Q1 and Q2 be two information structures operating
on the same set of states of nature S = {S1,…,Sn}, and
producing the same set of signals Y = {Y1,…,Ym}. Then
21 QQ S
  221 *)1(*,10, QQpQppp 
First, Lemma 1.1 is proven.
Lemma 1.1:
Let 1 Qand 2 Q be two information structures de-
scribing information systems. Let S = {S1,…,Sn} be their
set of the states of nature of 1 Q and 2 Q. Let
Y={Y1,…,Ym} be their set of signals. Then for any given
decision situation described by (a matrix of a-priori
probabilities of states of nature), U (a matrix of utilities
or compensations), A (a set of decisions), where

max 2pure -D(Q) is the set of optimal decision rules when
Q2 is used, there exists >0, such that if 0<p, and Dp is
an optimal decision rule of the Information structure
121p* Q(p)* Q
Then

max 2ppure -DD (Q).
Proof (of Lemma 1. 1):
1) It can be assumed that every optimal decision rule is
a convex combination of pure decision rules [10]. So we
try to find the optimal decision rule of
121p* Q(p)* Q in the set of the optimal pure deci-
sion rules of Q2,

max 2ppure -DD (Q).
2) Let k be the number of possible decisions in this
given decision situation. This means that there are km
pure decision rules, denoted 1m
kD ,..,D.
Let

p
ureD the full set of the possible pure decision
rules for this given decision situation
3) If

max 2
p
ure -pureD(Q)D, that means that every
pure decision rule is an optimal decision rule it is obvious
that

max 2ppure -DD (Q).
4) So, assume that

max 2
p
ure -pureD(Q)D
.
5) Hence:
 
max 2purepure -D\D (Q)

6) Let’s calculate for every pure strategy Di the fol-
lowing values: 11*iiV trace(ΠQ*D*U),
22*iiV trace(ΠQ*D*U)
7) 12*1 itrace(Π(p* Q(p)* Q)*D*U)) 
8) 1
2 1
i
i
p*trace(Π*Q*D*U))
(p)* (trace(Π*Q *D*U))

 
9) 121iip*V (p)*V
10) Let’s define in this specific decision situation:



11
max i
D
VMaxV - The (optimal) expected
value when using the information structure Q1.



11
max 2
i
D
i Dpure-(Q)
VMaxV

 - The (optimal) expected
value when using the information structure Q1, when the
set of decision rule is limited to the optimal set of pure
decision rules when using the information structure Q2.



2max 2
max 2
i
D
i Dpure-(Q)
VMaxV

 - The (optimal) expected
value when using the information structure Q2.



22
max 2
i
D
i Dpure-(Q)
VMaxV

 - The (optimal) expected
value when using the information structure Q2, when the
set of decision rule is limited to the non-optimal set of
pure decision rules when using the information structure
Q2.
11) According to expression (4) 

 purepure- D)(QD  2max
Hence: 0 222 max 

)V-(VVΔ
12) Moreover: 0
111 max 

)V-(VVΔ
13) Let’s examine for
  )(QD\D
D
ipure-pure 2max

when
it is not an optimal decision rule of p*Q1+(1-p)*Q2. We
try to identify a small value of p that will always give
max212111 p)*V(p*Vp)*V(p*V ii




. In fact, the
purpose is to find an “environment” of Q2 where an opti-
mal decision rule of Q2 is also an optimal decision rule of
p*Q1+(1-p)* Q2.
14) From (9) it is concludes that
2121 11 max p)*V(p*Vp)*V(p*V ii 


15) Let’s examine weather exists:
max max121211p*V( p)*Vp*V(p)*V


16)




221
222211 maxmaxmax
V)VVp*(
-VV-VVp*-VVp*
ΔΔΔ




21
2
0VV
V
p
ΔΔ
Δ


That’s according to (10), (11)
0021201 VV,VV ΔΔΔ,Δ
17) Let’s pick: 10
21
2

 VV
V
ε
ΔΔ
Δ
18) And in this environment (for every



p0) at
least one optimal decision rule of Q2 is an optimal deci-
sion rule of p*Q1+(1-p)* Q2
Q.E.D (Lemma 1.1)
Proof (of the theorem itself):
First direction: assume that for every decision situa-
tion:
1) ))(trace(ΠMax
U)*D*Q*trace(Π, )}(QDD
2
D
Q12max{Q22
*D*U*Q


where max 2
{()}DQ is the set of optimal decision rules

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 64
when Q2 is used in this specific decision situation.
Then implies:
2) 12Π1
D
M
ax(trace(* (p* Q(p)* Q)*D*U)) 
3)
1
2
((Π)
(1) (())
D
M
axp* trace* Q*D*U
p*traceΠ*Q *D*U


4) 12
22 1
Q
Q
p* trace(Π*Q*D *U)
(p)* trace(Π*Q *D*U)


5) 22
22
1
Q
Q
p* trace(Π*Q *D*U)
(p)* trace(Π*Q *D*U)

 
6) 2
D
Max(trace(Π*Q *D*U)) (first direction is
proven)
Second Direction:
7) 12201 1p,p, p* Q(p)* QQ
8) According to the lemma there exists >0, such that
if 0<p1<1, then exists Dp1 an optimal decision rule of
the IS: 11121p*Q(p )*Q that implies

1max2pure -DpD(Q ).
9) Let DQ2 this optimal decision rule

)(QDD pure-Q2max2
10)



)*D*U))Q)p(*Q(p(trace(Max *
D
2111 1*Π
11)

 *U)))*D)*Qp(*Q(p(trace(ΠQ22111 1*
12) 2
2
1
1
2
1
Π
1Π
Q
Q
p*trace(*Q*D* U)
(p)trace(*Q*D* U) 
13) *U)*D*Qtrace( Q22Π (According to (7))
14) 2
2
12
12
Π
1Π
Q
Q
p* trace(* Q*D*U)
(p)trace(*Q*D*U)


15) From (12), (13), (14) 21ΠQtrace(*Q*D* U)
22ΠQtrace(*Q*D* U)
That is correct for every decision situation (any given
Π and U)
Q.E.D
Lemma 1
Let Q1 and Q2 be two information structures operating
on the same set of states of nature S={S1,…,Sn}, and pro-
ducing the same set of signals Y={Y1,…,Ym}. Assume that
Q1 is generally more informative than Q2, implying that
Q2= Q1*R, where R is a stochastic matrix [5].
If ,01, *(1)*pp pIpRR ,
then 221 *)1(*,10, QQpQppp 
Proof:
1) According to the second condition of Blackwell’s
theorem [5] for every p there exists Rp, where Rp is a
stochastic matrix of the order nxn.
(p * I + (1-p) * R)*Rp = R
2) Therefore: Q1*(p * I + (1-p) * R)*Rp =Q1 * R
3) Hence:


221 **)1(* QRQpQp p


Q.E.D.
Lemma 2:
Let I be an information structure that provides perfect
information. Let Q be any information operating on the
same set of states of nature ,..,SSSn}{ 1 and produc-
ing the same set of signals }{n,..,YYY 1.
If: for Qp)*Q(,p*Ip 



110 (every convex
combination of I and Q is generally more informative
than Q)
Then:
01(1) (1)q,qp,p*Ip*Q q*Iq*Q

 
Proof:
1) According to the 2nd condition of Blackwell’s theo-
rem, pp RR ,

is a stochastic matrix, such that:
(p*I+(1-p)*Q) *Rp=Q
2) Since, p
qpq
pq, *I*R
pp







 is a stochastic
matrix.
3) Let’s examine:
[(1)][]p
qpq
p*I-p*Q **I*R
pp



4)
[1 ]
[(1)]
p
qq
p** I(-p)*Q** I
pp
pq
*p*I -p*Q*R
p




5)
1
q-p*qp q
q* I*Q*Qq* I
pp
q- p*qp-q*Qq*I (q)*Q
p

 


6) According to the 2nd condition of Blackwell’s theo-
rem )5(
011 1q,qp,p*I (p)*Qq*I(q)*Q

 
Q.E.D
Theorem 2:
Let F1 be an information function. Let ,..,SSSn}{1

be its set of states of nature. Let }{n,..,YYY 1be its set
of signals. Let I be the identity information function,
which represent perfect information, then 1FIS  if and
only if 1
F
F

First 3 lemmas are demonstrated and proven:
Lemma 2.1
Let fI be the identity information function. Let
,..,SSSn}{ 1

be its set of states of nature. Let
}{n,..,YYY 1

be its sets of signals. ()
I
ii
f
SY.
Without loosing generality (for the sake of simplicity)
Assume {1} S,..,n

, {1} Y,..,n

.

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 65
Let
F
be the set of information functions (without
garbling of signals) that fI is systematically more infor-
mative than each one of them:
))()(()(kkgtheni,kigiforiigFg  (
Let F,gg21 and g1 is finer than g2 then:
)())((1 212 igigg,..,n,i,i 
Proof (of Lemma 2. 1):
1) Let’s check all the possible situations, given:
))()(()( kkgtheni,kigiforiigFg (
2) 1st Case: 12212 () ()(()) ()
g
igii ggigii 
3) 2nd Case: 12 2
21 2
()()()
(()) ()
g
igikigk
kggigk k


4) 3rd Case: 12 212
()()( ())()
g
ii,giki ggigik 
5) 4th Case: 121
()()()
g
iki,giigkk 
6) Since g1 is finer than g2: 2()
g
ki
7) Hence: 21 22 (5) (6)(( ))()( ),ggigkigi
8) 5th Case:
12 12
()()( )()
g
iki,gi jigkk,gj j 
9) Moreover, since g1 is finer than g2: 2()
g
kj

10) Hence: 21 22 (8) (9)(( ))()( ),ggigkjgi
11) It is proved for any possible situation that:
)())((1 212 igigg,..,n,i,i 
Q.E.D (Lemma 2.1)
Following that an Adaptation to the information struc-
ture model is concluded straight forward: Let fI be the
identity information function. Let ,..,SSSn}{ 1be its set
of states of nature. Let }{n,..,YYY 1be its sets of signals.
iiI Y)(Sf .
Without loosing generality (for the sake of simplicity)
Assume {1} S,..,n, {1} Y,..,n. Let
F
be the set
of information functions (without garbling of signals) that
fI is systematically more informative than each one of
them:
Let F21g,gand g1 is finer than g2 .g1 is equivalent
to 1G, and g2 is equivalent to 2G.
Then 221 *GGG 
Lemma 2.2:
Let F1, F2 and F3 be information structures. F1 repre-
sents information functions accordingly. Let
,..,SSSn}{ 1be their set of states of nature. Let
}{n,..,YYY 1be their set of signals.
123
12 3
01
(1 )
F
FFp, p,
Fp*F p*F


Proof (of Lemma 2. 2):
1) First direction - Assume: 321 FFF then nec-
essarily: 321 110 p)*F(p*FF,pp,  .
2) Second direction - Assume:
12 301(1)p,p,Fp*Fp* F . without loos-
ing generality suppose (on the negative form) there
exists an index i,ji,j FFi,21

.
3) Then (by calculating):
312
1()0
1
i, ji, ji, jF*F-p*F
p


. It is a contradiction.
Q.E.D (Lemma 2.2)
Lemma 2.3
Let F1 be an information structure, which represents
information function. Let ,..,SSSn}{ 1be its set of
states of nature. Let }{n,..,YYY 1

be its set of signals.
Then: 2 111
F
FF F


Proof (of Lemma 2.3):
1) 1st direction: From the definition of
F
, and
Lemma 2.1 it is obvious that 2 111
F
FFF 
2) 2nd direction: Assume (on the negative form)
2
11
F
F

and 1
F
F

3) Let’s examine f1, which is described by
F1.




111
f
:,..,n ,..,n
4) 1
F
F

, Hence there exist an index i, f1(i)=j, f1(j)  j
5) f1(i) = j = f1(f1(i))  j , a contradiction.
6) Hence, 1F
F

Q.E.D (Lemma 2.3)
Proof of the theorem:
1) 1st direction: it is clear from Lemma 2.3 that
111
1111
((1))
(1 )
F
Fq*I-q*F*F
q*F-q*F*FF



2) Hence: 1101 (1)q,qq*I-q*FF


3) From Theorem 1 it is proven that,
11 101 (1)S q,qq*I-q*FFIF

 
4) 2nd direction: From Theorem 1 it is proven that,
11 101 (1)S q,qq*I-q*FFIF

 .
5) According to the 2nd condition of Blackwell’s theo-
rem [5], there exists a stochastic matrix R,


11(1 )
F
q*I-q*F *R .
6) 11(1 )
F
q*R-q*F *R


7) From Lemma 2.2 it is obvious that: R=F1
8) Moreover, from Lemma 2.2 it is understood that
F1*R = F1.
9) Hence: 2
11
F
F

.
Q.E.D
Theorem 3
Let I (the identity matrix), and Q be two information
structures. ,..,SSSn}{ 1

is the common set of states of
nature, and }{n,..,YYY 1

is their same set of signals
they respond with.
1i,i j,i
S
IQi,i,..,n,QQij (The diagonal

Evolution or Revolution of Organizational Information Technology – Modeling Decision Makers’ Perspective 66
elements are weakly dominant in each and every col-
umn).
Proof:
1) Suppose (on the negative way): There exists an in-
dex j jiQQ, j,ii,i  then 01  ΔQQ i,ij,i where
(without loosing generality) j,i
Q is the maximal element
in the i column.
2) Let’s examine a specific decision situation: suppose
there are n possible decisions, and
1
1k,k k,k,...,n, Π
n
 .
Let’s define U (the utility matrix) as follows:
11
,
0
r,s
r,r,...,n, s,s,...,n
A,(A Δ),rj,si
A-Δri,si
UΔ,ri,sj
,else
 








3)
   
1
D
D
Max traceΠ*Q*D*U*MaxtraceU*Q*D
n

4)


1
11
0
r,s
i,s
n
r,m m,sj,si,s
m
r,r,...,n,s,s,...,n Q*U
A
*Q ,rj
Q*U Δ*QA Δ*Q, ri
,else

 








5)
  


2
i,ij,i i,i
j,i i,i
i,i i,i i,i
Q*UA* QQ*UΔ*QA Δ*Q
Δ*Q ΔAΔ*QA*Q Δ
 

6) Suppose D1 represents the optimal decision rule.
1(5) 1
i,i
D.
7) Moreover,
 
1
1
n
i,i
j,m
m
mi
Q*UA* -Q




.
8) From (5), (7) it is derived that:
  

2
1
2
11
1
i,i i,i
trace Π*Q*D*U* A*-QA*Q
Δ
n
*A Δ
n


9) Moreover, 1110
i,ii, j
DD


10) Hence,



1
1
11
i, ji,i
trace Π*I*D*U* UU
n
*ΔAΔ*A
nn

 
11) It means that:

 
11trace Π*I*D*Utrace Π*Q*D*U
12) Under this decision situation Q is precisely more
informative than I. Hence I is not systematically more
informative than Q.
Q.E.D