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A stochastic model of information value
InformationProcessing& Management,Vol.31, No. 4, pp. 543-554, 1995 Elsevier Science Ltd. Printedin Great Britain
Pergamon
0306-4573(95)00008-9
A STOCHASTIC
MODEL
OF INFORMATION
VALUE
JOHN A. A. SILLINCE Sheffield University ManagementSchool, 9 Mappin Street, Sheffield S 1 4DT, England (Accepted in final form December 1994) Abstract--Faced with a potentially vast amount of information, how is one to decide what
is worth searching and when a piece of information is valuable? With small quantities of information semantic concepts are useful--the relevance and significance of a piece of information can be assessed easily. But with large quantities non-semantic clues are needed, and ones which require less rich (and therefore easier to scan) information. The approach of this paper is to deduce quantitative (rate, flow) and qualitative (value) attributes of information form topological measures of organizational information channels. Keywords: Information value, Stochastic model, Organization, Network, Market, Transaction costs.
INTRODUCTION The investigation of measures of information continues to attract a high degree of interest both among researchers and practitioners hoping to find better methods of estimating relevance and value in the processes of retrieval and use. Some have used the concept of levels of processing (Hayes, 1993) or have used the Ives and Learmouth Life Cycle Model to understand information use (Suomi, 1993), others have suggested that information only exists as changed user behaviour (Paisley, 1980), while others have concentrated on the amount of information transferred in the form of signals (Guiasu, 1977). Can the value of information be quantified? We make the unfashionable claim that in one sense it can. This is in the sense that if we get the information we want we get positive value, whereas if we get information we do not want (and have to use effort to decide on its irrelevance) or if we exert effort to request information and do not receive it, then we get negative value. Now consider information passing round an information system. Some of the information is requested, and other information is not. Depending on the probability of information transfer from one person to another person within the system, the value of information for a particular person will vary. This information value will also be affected by whether or not the information system is an organization, within which information transfer can be regarded as costless, or a market, in which information transfer has a price (Williamson, 1975; Ouchi, 1980; Ciborra, 1987; Ciborra & Olson, 1989; Suomi, 1993). And if the information system is an organization, the information value for a particular person will depend on whether the organization is a hierarchy (information transfers between subordinates must be via a superior) or network (no constraints on information transfer).
MODEL In order to model the information channels provided by any organization, we shall use three people, because three is the smallest number which can still yield interesting topological possibilities. If information produced by person x is relevant to person y, then we shall say that 543
544
John A. A. Siilince
)
X
Z
y
x
y
/
(a)
x...__.~
y
S
Z
Z
(b)
(c)
Fig. 1. (a) Related information. (b) Alternative information. (c) Unrelated information.
x produces information for y, denoted x--'y. We place no conditions on what constitutes relev a n c e - t h a t is not a necessary part of the model. Now consider a subsystem comprising the three people x, y and z, where z is the person who is the information destination and x is the information source, and where y is somehow involved too. Consider the three cases in Fig. 1. We begin with x, and consider what are the ways in which information can be passed from x, involving y, to z. Related information [Fig. l(a)] means that if y is given information by x then something of value is added to y, and something of value is also passed from y to z, and so this information type increases the value of information produced by x for z. For example, x says "Oil is probably located in this grid square" and y then adds "The probability of finding oil is greater on shale than on gritstone". Alternative information [Fig. l(b)] means that information produced by x is useful to z, and that the same information is provided by y for z, and so although x is not crucial to adding value to z, the information produced by x corroborates the information produced by y, and so there is an increase in value. For example, x and y both say "Oil is probably located in this grid square". Unrelated information [Fig. l(c)] means that even though information produced by x is useful to y, because information produced by y is not useful to z there is no effect of x on z. For example, x says "If there is water under this grid square then it is probable that there is oil there too" and y then adds "No evidence exists one way or the other about water under this grid square". Each of the arrows (representing information passed as messages) in Fig. 1 will be ascribed a probability in what follows. A simplifying assumption is that this probability is the same for all messages. One of the factors which is relevant to the influence of these three information types on information value is the likelihood that such information types will occur in any given quantity of information. What follows is a model for dealing with this problem. We shall assume that the future behaviour of the process is dependent only on its present state S, at time t where t > t~ > t2 > t3> . . . and not on how it reached that state so that
P [S, = slS,~ =sl, S,2=s2, St3=s3 . . . . ]= P [S, =slS,, =s d.
(1)
and also behaviour is homogeneous:
P[Sm÷,=jlSm=il=p~n)
(2)
i.e. that the transition probability p~7) depends only on the length of the time interval n and not on the starting time m. From property (1) it can be deduced that
P[S, =jlS, = 0 = 2
P[S, =jlSr=k]P[Sr=klSm=i],
(3)
k
for any r such that m < r < n. In continuous time (1) becomes
P[S(t)=jIS(h)=i]= p~j(h, t)
(4)
A stochastic model of informationvalue
545
and (2) becomes
(5)
po(h, t)=pij(t-h) and (3) becomes
po(h, t)= £
p~k(h, u) . pkj(u, t), h < u < t.
(6)
k
Homogeneity enables us to denote (6) as
po(t + r)= Z
p,k (t ) "pkj( r),
(7)
k
For a small time interval r = &, and separating out change from no change,
po(t + & ) = pij(t ) . pjj( 6t ) + £
Pik(t ) "pkj( & ).
(8)
k,.)
If the process is in state j at time t, pij(&) is the probability of no change in the interval (t, t + & ) and so 1-pjj(&) is the probability Aflt+ O(&) of some change in that interval, so that
1 -pjj(&) = aj& + O(&)
or lim 1- - p-j j ( & ) - a t 8, ~ o
&
(9)
so that the probability of a change depends on the length of the interval (t + &) and on the constant At which represent the rate of change in state j. The probability of a change occurring at x or y from k value units of information to some other value is Ak. The conditional probability of a change in z's information from k to j value units is denoted by qkj. Pkj is the absolute probability of value units of information at z changing from k to j items. So that Pkj = ( a k & + O ( & ) ) • qkj and from (8)
po(t + &) = pij(t)(1 - aj& - O ( & ) ) + £
p,k(t)(ak& + O (&))qkj(t) &
(10)
and so letting &-+O gives
p~ (t) = - Ajpo(t) + £
P,k(t)Ak. qkj
k,,j
with initial conditions
... f l , ~ = j P°(°)=IO, i #j
(11)
John A. A. Sillince
546
and where the matrix form is dP(t) - =P(t). R, dt where P(0) is the identity matrix and where R is of the form r~j= - a j rkj = A~hj, k,~j. Now consider again Fig. 1. Given a jth value unit of information arriving at source persons x or y, let the probability of new information being generated be aj for x or y. Let two exchanges count as one event, so that the probabilities for each of the three types of information flow are, relative to the sum a2+2a+a(1-a)=3a of these three types, a=/3a (related), 2a/3a (alternative), and a(1-a)/3a (unrelated). Because the probability of new information being generated is proportional to the number of information units j, the arrival rate is a t =j3A. Since related and alternative information increase information value from k to k + 1 units, and unrelated information reduces it to k - 1 units, the conditional probabilities are: a(A+2),j=k+l 3A q~J= a ( 1 - a ) , j = k _ 1 3,~ If information stock at time t is 1I (r) where
P[l-I(t)=n]=pn(t) then from (11), with initial information stock a, p'.(t) = -apo(t)
p](t)= -j3Apj(t)+[ ( j -
1)(A2+2A)] • p~_,(t)+[(j+ 1),~(1 -A)]
.pj÷,(t). j>a.
(12)
Using a generating function defined as
M(O, t ) = £ exp(jO)pi(t), j=o
pj(O)=O,j#a,pa(O)=l and multiplying the jth equation by exp(jO) and summing over j,
Z P;(t)exp(jO)=-Z 3jAp~(t)exp(jO)+Z [(J-1)(A2+2A)]'PJ-'(t)exp(jO) j=o
j=o
j=o
oo
+£ j=O
[(j+ 1)A(1-A)I.
pj+~(t)exp(jO)
(13)
A stochastic model of information value
547
which gives OM(8, t)= -3A ~ 0 +exp(8)(A2+2A) -ff~ aM +exp(- 8)A(1 -A) aM at aO
i.e.
OM(O, t)
OM c 3 ~ +A (8) ~ - =0
(14)
where A(0) = [3A - exp( 0)(a 2+ 2A) - exp( - 8)a ( 1- A)] and whose auxiliary equation is dt dO dM T =A i8) =-0- =0. Integrating for t and 8:
l(ZC)
t=-2AC~-~ln ~ -
+constant
A=X/A+2 -3
B=2X/-X + 2
(15)
/ 9 +4(A + 2)(A - 1)
4(-x Z=A exp(0)+B.
Becausepa(0)= 1 M( O, 0)=exp(aS)=C1
(16)
Z-C) C2= ~ exp(2AACt)
(17)
and from (15)
and setting t =0 in (17) the general solution is C(C2+1) Z- - (1 -C2)
(18)
so that substituting this value of Z into the left-hand side of (15), and using (16) the moment
548
John A. A. Sillince
generating function is given by: .
. ,,,
{C(C2+1)
M ( O, t )=exptato= ~-~l---C-f
(19)
so that Y =exp(0) dM(O, t)
/u,---
dt
=aya-ly
(20)
d2M( O, t)=aY"-I }"+a(a-- 1)Y"-2 Y dt
o~=d2M(O, t) dt 2
/z2.
Some results are given in Fig. 2. For low probabilities of arrivals at x or y [Fig. 2(a)] both mean and variance of information value reach peaks, and for both the slope is greater before the peak. In the period before the peak, during which mean information value at z is rising, new arrivals of information at x or y increase the chance of the existence of related or alternative information. But after the peak, new arrivals of information at x or y are more likely to increase unrelated information. For higher A [Fig. 2(b) and (c)] both variance and mean decline continuously over time. The decline in these values is because of the greater probability of unrelated information when ,~ rises.
OTHER MODELS
(1) No effect from alternative information It could be argued that alternative information does not increase information value, that in fact it has no effect at all. This would be the case, for example, in an automated system where corroboration is superfluous. If a valve gives a pressure reading there is no increase in
A stochastic model of information value
549
information value in having a second, duplicating, valve. In this case we have: A2
j=k+l
3A
j=k
2A
q~ =
3A
A(I - a )
j=k-l.
3A
(2) Hierarchy with superior requesting information In this case the information user is at the top of a hierarchy. For example, a general z requests information from a brigadier y. Let us suppose that z needs information from y with probability A (see Fig. 3). Also, y receives the same information supplied by x, a private, with probability #. So y is the middle man. In this case, the probability that information is needed and also available is A/z, that it is needed and not available is A(1-/z), that it is not needed and available is (1 - A)/.t, and that it is not needed and not available is (1 -A)(1 - / x ) . So
{
A/.L
j=k+l
qkj= (1-A)~+(1-A)(]-~)
j=k j=k-1.
A(1-/z)
1600 o
o
12,000
--
1400
(a)
1200
--
I000
--
.o
~ - - - _
>
.2
--
600
--
400
//
--\
8O00
_
/
/
. . . .
Variance
6000
--
4000
--
\ \
\
>
.o
- - //
\ x
\ \
¢
o~...~----:~ tffi20 triO
Variance
\
Mean
/
t=0
Mean
. . . .
\
//
200 .z.
_=
•(b) \
/
800
10,000
I
I
I
t=40 t=30
I
tffi60 t=50
I
I
0 t=O
_=
tffiS0 t=70
2000
t=90
t=10 t=5
tffi20 t=15
Time ~
t=25
tffi30
t=40
t=35
Time
12,00t)
7--
(c)
Mean
__\
.2 lo,~o
. . . .
\
Variance
k
"~ o ee
80011 \
6OO0
\
>
=
._= es
~ ¢~
--
4000
\
2000
o triO
tffi5
tffi 1 0
t = 15
tffi20
Time Fig. 2. (a) Model results for A=0.01, 0 = - 1 0 , a=3. (b) Model results for A=0.1, 0---10, a=3. (c) Model results for A=0.02, 0 = - 10, a=3. IPH 31-4-G
t=45
550
John A. A. Sillince
Z requested
3/
supplied~ x Fig. 3. Hierarchy with superior requesting information.
(3) Hierarchy with subordinate requesting information from another subordinate In a hierarchy, subordinates must communicate via a superior. For example, a tank battalion wants to communicate with an artillery battery via headquarters. Here a subordinate z must get information from another subordinate x, by going through a superior y (see Fig. 4). So, z needs information (with probability /x), y requests it on his behalf (with probability A), and it is available at x (with probability/z). The good result (j=k + 1) for z is that z needs, y requests, and it is available at x, with probability /,t2A2/(l+~h.-]/,). The bad results ( j = k - l ) are first, that z needs, y does not request, and it is available at x, with probability / x 2 A ( 1 - A ) / ( l + / z h - / x ) , second, that z needs, y requests, and it is not available at x with probability/zh2(1- bt)/(1 + / x h - / z ) , and third, that z needs, y does not request, and it is not available at x with probability / z h ( 1 - / x ) ( 1 - A ) / ( 1 + / z h - / z ) . The indifferent result, ( j = k ) is that z does not need and it is available at x with probability / x ( 1 - / z ) / ( l + / z X - / z ) , and that z does not need, and it is not available at x with probability (1-/z)2/(1 + / z h - / z ) . So
qkj=
A2~2/(1 +/xh - / z )
j = k+ 1
(1 - / z ) / ( 1 + / z k - / z )
j=k
/zh(2/z + 1 -/.th)/(1 +/xh - / z )
j=k-1.
(4) Network Here the constraint that subordinates communicate via a superior is removed. Now z can request information directly from x. For example, a journalist requests information from a photographer without going through an editor. For there to be an increase in information value to z, information is either available from x with probability A/z, or it is not available from x but x gets it for z from y, with probability A2(1-/.t)/.t. For there to be a decrease in information value, either information is requested but is not available with probability A2(1-/z) 2 or it is
Y
Z
X
T
Fig. 4. Hierarchy with z requesting information from x to y.
A stochastic model of information value
551
available but not requested with probability/z2( 1 - A) 2. /~), + A2( 1 - p,)/.t, h2(1 +/.,2--/.,) +/z(/., + A-- 2/zh)
j=j+l
q~= /~2(1 - ~)2 +/2,2(1 - A)2 X2( 1 +/x 2-/~) +/~(/z + A - 2/xh)
j=k-1.
(5) Market with prices Here the freedom of the network prevails, but each piece of information has a price• This may be a market price, or it may be a retrieval cost. For example, in order to search for patents, hierarchical status is irrelevant, but retrieval cost is important. This can be modelled by using a probability Pk that z is prepared to pay for the kth information item. The effect of this on the conditional probabilities in sub-section 4 are that for j = k - 1 there are two extra terms, the probability that information is requested, is available, but z is not prepared to pay, which is (1 -p)A/x, and the probability that information is requested, is not available from x but x gets it from y, but that z is not prepared to pay, which is ( l - p ) A # ( l - / x ) .
FUTURE WORK The simple triads are supposed to be smallest possible cases useful for studying organizational characteristics. But the question remains how the scaling up can be done. Figure 1 shows that information value added at z depends on information being provided both by x and y (not just by one of them)• Let P[X,] be the probability of X giving information at time t, then the probability of information value added at Z is
e[z,] =P[X, + Y,] so that
{z}={x}*{y} by convolution• Figure 5 shows a three-tier hierarchy and the following n - l - f o l d convolution of convoluted pairs generalizes to n - 1 tiers
{x211} = {Xl.l*lx.2} Ix3.} = {{x,. }*{x,12} }*l {Xl2,* 1x~22} } = { {x~j, }* {x,j~ } 1(2~ {X..l } = {x3. }'1x312} = { Ix., }*{x.~} }'1 {x~2~}* {x1221 }* = {{x.~ 1" 1x1321 }'1 {Xl4, }*{x~.21 } ~_~{ {Xlj I },{XIj2} }(3)
{Xojl } = { IXj~l }* {x~j2} }c.-,. Another approach is a theoretical axiomatic one, offering the possibility for the generation and proving of related theorems. For example, Egghe and Rousseau (1993) have set out a framework for understanding the statistical distributions of information items from sources,
552
John A. A. Sillince
•
.2,/,..,,.
•
O
Fig. 5. Three-tierhierarchy.
using previous results such as Lotka's Law, the Lorenz curve, and the 80/20 rule. What their framework lacks, however, is any model of the user's organization--i.e, the destination and routeing of items, which our paper introduces. So far, the formulation models the positive and negative value of information flow around an information system. There is no consideration of analysis, judgement, or decision, crucial components in according added value (Bawden, 1990). The model here assumes that merely getting the right information is all that is involved. It is, however, possible to weight the importance of the users (here known as persons x, y, and z) or the importance of the decisions, by means of larger or smaller increments in the value of j. So a person x who was twice as important as person y could give rise to increments of the form j = k + 2 or j=k - 2 . It is misleading to think of information as being drawn through an organization or market for final use by a unique person. The information would probably be useful in many ways to many people. More detail is required within these different cases of organizational topology, together with more types. For example, it is necessary to generalize from price to transaction cost, so that these ideas can be linked to transaction cost theory within the economics of the firm (Williamson, 1975). Transaction cost theory is able at a theoretical level to suggest the reasons why individuals choose to transact within free markets or within organizations, depending upon levels of uncertainty, the need for secrecy and information retrieval costs. Incentives to exploit information differences opportunistically shrink when the people concerned place transactions in a single organisation. Such internal organizational relationships may enhance information coding, the convergence of expectations, and auditing control, though at greater costs than when price alone can moderate the exchange between people (Osbom & Baughn, 1990). The theory
A stochastic model of informationvalue
553
has been applied to information technology with the speculation that IT brings down transaction costs to such an extent that organisations are needed less, so that much work formerly kept within the boundaries of the organization can be outsourced or market tested. However, transaction theory cannot of itself solve the basic problem of putting a price on information--agreeing on a price for information is problematic unless a buyer knows what the information is---once that knowledge is disclosed the buyer need not pay for it (Anderson & Gatignon, 1986; Calvet, 1981). This problem of information valuation through market mechanisms may also enhance a preference for transaction forms providing high control, such as bringing the production and storing of that information within the organization's boundaries (Osborn & Baughn, 1990). The formulation suggested above is intended as a solution to this lack of any theory of information value. Recent work has applied transaction cost ideas to game theory (e.g. Crampton, 1991). Much of this work is based upon the evolution of information value over time (with varying discount rates simulating decaying value) and how it is managed (e.g. contracts vs long term relationships within markets; explicit rules vs trust within organizations). This work and the current models are directly linked, since Fig. 2 shows the different effects of information arrival probability on information value over time. The formulation suggested above requires empirical testing in order to discover which organizational or market forms are most able to deliver high information value. The advantage of the current approach is that it emphasises the characteristics of the user and his organizational/market environment, rather than abstract notions of information, which are apt to mean little unless applied in specific contexts and for specific uses (Bawden, 1990). Thus, any information system will not be only an abstract topological structure, but may consist of some parts (e.g. R&D, marketing) where task uncertainty is relatively high, or other parts (e.g. gaining competitor intelligence) where costs of information retrieval are high, (slanting information transfers towards within-organization flows according to transaction cost theory). It is therefore fully within the scope of what Taylor (1982) has called a "user-driven" (rather than "content-driven" or "technology-driven") model of information.
CONCLUSION
The formulation enables investigation of qualitative (value) variables in the different topologies which organizations represent. How to measure information value has previously been ignored or dealt with historically (e.g. Suomi, 1994) yet the ability to monitor current levels of information value is much more useful. Such an ability opens up the possibility of investigating questions such as the differences between hierarchical and non-hierarchical organizations. This is very topical as studies are suggesting that organizations are changing away from hierarchical structures with many levels towards network-based ones with much fewer levels, and structured around fluid teams which form and dissolve as and when specific tasks present themselves (Feeny et al., 1989; Burnes, 1992; Hodgkinson, 1990). Moreover particular forms of information technology enable particular types of topology--as electronic mail makes the formation of networks much easier. The relationship between information technology and organizational topologies have been suggested within firms (Ciborra, 1987; Sproull and Kiesler, 1986; Malone et al., 1987). Bakos (1991) has suggested a relationship for inter-organizational contexts too---he distinguishes between bilateral links between firms who already have an established connection and multilateral links between firms who have no preestablished connection, within an electronic marketplace. This suggests that not only are previously accepted information channels losing relevance, but also that better theoretical tools are needed of judging what kinds of information channel are of value. These studies bring out another aspect (ignored here) of information flows within organizational topologies-that particular network nodes and particular topologies give more power in terms of control over access to and price of information.
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John A. A. Sillince REFERENCES
Anderson, E., & Gatignon, H, A. (1986). Modes of foreign entry: A transaction cost analysis and propositions. Journal of International Business Studies, •7(3), 1-25. Bakos, J. Y. (1991). Information links and electronic marketplaces: The role of interorganisational information systems in vertical markets. Journal of Management Information Systems, 8(2), 31-52. Bawden, D. (1990). Value added information systems and services. In C. Oppenheim, C. L. Citroen, & J. M. Griffiths, Perspectives in information management (pp. 157-159). London: Bowker Saur. Burnes, B. (1992). Managing change. London: Pitman. Calvet, A. L. (1981). A synthesis of foreign direct investment theories and theories of the multinational firm. Journal of lnternational Business Studies, 12(1), 4-59. Ciborra, C. (1987). Reframing the role of computers: The transaction costs perspective. Office, Technology and People, 3,1. Ciborra, C., & Olson, M. H. (1989). Encountering electronic work groups: A transaction costs perspective. Office, Technology and People, 4(4), 285-298. Crampton, E C. (1991). Dynamic bargaining with transaction costs. Management Science, 37(10), 12211233. Egghe, L., & Rousseau, R. (1993). Evolution of information processes and its relation to the Lorenz dominance order. Information Processing & Management, 29(4), 499-513. Feeny, D., Earl, M., & Edwards, B. (1984). IS arrangements to suit complex organisations (1) an effective IS structure. OXIIM, Templeton College, Oxford, RDP 89/4. Guiasu, S. (1977). Information theory with applications. New York: McGraw-Hill. Hayes, R. M. (1993). Measurement of information. Information Processing & Management, 29(1 ), l-l 1. Hodgkinson, S. L. (1990). Distribution of responsibility for IT activities in large companies. OXIIM, Templeton College, Oxford, RDP 90/5. Malone, T., Yates, J., & Benjamin, R. (1987). Electronic markets and electronic hierarchies. Communications of the ACM, 30, 484--497. Osborn, R. N., & Baughn, C. C. (1990). Forms of interorganisational multinational alliances. Academy of Management Journal, 33(3), 503-519. Ouchi, W. G. (1980). Markets, bureaucracies, and clans, Administrative Science Quarterly, 25, 129-141. Paisley, W. (1980). Information and work. In B. Dervin & M. J. Voigt (Eds), Progress in communication sciences (Vol. 2), Norwood, N.J.: Albex. Sproull, L., & Kiester, S. (1986). Reducing social context cues: Electronic mail in organisational communication. Management Science, 32, 1492-1512. Suomi, R. (1993). A transaction cost view of decision support. Behavior & Information Technology, 12(4), 228-237. Suomi, R. (1994) What to take into account when building an inter-organizational information system. Information Processing & Management, 30(1), 151-159. Taylor, R. S. (1982). Value added processes in the information life cycle. Journal of the American Society of Information Scientists, 33, 341-346. Williamson, O. E. (1975). Markets and hierarchies: Analysis and antitrust implications. New York: Free Press.
Pergamon
0306-4573(95)00008-9
A STOCHASTIC
MODEL
OF INFORMATION
VALUE
JOHN A. A. SILLINCE Sheffield University ManagementSchool, 9 Mappin Street, Sheffield S 1 4DT, England (Accepted in final form December 1994) Abstract--Faced with a potentially vast amount of information, how is one to decide what
is worth searching and when a piece of information is valuable? With small quantities of information semantic concepts are useful--the relevance and significance of a piece of information can be assessed easily. But with large quantities non-semantic clues are needed, and ones which require less rich (and therefore easier to scan) information. The approach of this paper is to deduce quantitative (rate, flow) and qualitative (value) attributes of information form topological measures of organizational information channels. Keywords: Information value, Stochastic model, Organization, Network, Market, Transaction costs.
INTRODUCTION The investigation of measures of information continues to attract a high degree of interest both among researchers and practitioners hoping to find better methods of estimating relevance and value in the processes of retrieval and use. Some have used the concept of levels of processing (Hayes, 1993) or have used the Ives and Learmouth Life Cycle Model to understand information use (Suomi, 1993), others have suggested that information only exists as changed user behaviour (Paisley, 1980), while others have concentrated on the amount of information transferred in the form of signals (Guiasu, 1977). Can the value of information be quantified? We make the unfashionable claim that in one sense it can. This is in the sense that if we get the information we want we get positive value, whereas if we get information we do not want (and have to use effort to decide on its irrelevance) or if we exert effort to request information and do not receive it, then we get negative value. Now consider information passing round an information system. Some of the information is requested, and other information is not. Depending on the probability of information transfer from one person to another person within the system, the value of information for a particular person will vary. This information value will also be affected by whether or not the information system is an organization, within which information transfer can be regarded as costless, or a market, in which information transfer has a price (Williamson, 1975; Ouchi, 1980; Ciborra, 1987; Ciborra & Olson, 1989; Suomi, 1993). And if the information system is an organization, the information value for a particular person will depend on whether the organization is a hierarchy (information transfers between subordinates must be via a superior) or network (no constraints on information transfer).
MODEL In order to model the information channels provided by any organization, we shall use three people, because three is the smallest number which can still yield interesting topological possibilities. If information produced by person x is relevant to person y, then we shall say that 543
544
John A. A. Siilince
)
X
Z
y
x
y
/
(a)
x...__.~
y
S
Z
Z
(b)
(c)
Fig. 1. (a) Related information. (b) Alternative information. (c) Unrelated information.
x produces information for y, denoted x--'y. We place no conditions on what constitutes relev a n c e - t h a t is not a necessary part of the model. Now consider a subsystem comprising the three people x, y and z, where z is the person who is the information destination and x is the information source, and where y is somehow involved too. Consider the three cases in Fig. 1. We begin with x, and consider what are the ways in which information can be passed from x, involving y, to z. Related information [Fig. l(a)] means that if y is given information by x then something of value is added to y, and something of value is also passed from y to z, and so this information type increases the value of information produced by x for z. For example, x says "Oil is probably located in this grid square" and y then adds "The probability of finding oil is greater on shale than on gritstone". Alternative information [Fig. l(b)] means that information produced by x is useful to z, and that the same information is provided by y for z, and so although x is not crucial to adding value to z, the information produced by x corroborates the information produced by y, and so there is an increase in value. For example, x and y both say "Oil is probably located in this grid square". Unrelated information [Fig. l(c)] means that even though information produced by x is useful to y, because information produced by y is not useful to z there is no effect of x on z. For example, x says "If there is water under this grid square then it is probable that there is oil there too" and y then adds "No evidence exists one way or the other about water under this grid square". Each of the arrows (representing information passed as messages) in Fig. 1 will be ascribed a probability in what follows. A simplifying assumption is that this probability is the same for all messages. One of the factors which is relevant to the influence of these three information types on information value is the likelihood that such information types will occur in any given quantity of information. What follows is a model for dealing with this problem. We shall assume that the future behaviour of the process is dependent only on its present state S, at time t where t > t~ > t2 > t3> . . . and not on how it reached that state so that
P [S, = slS,~ =sl, S,2=s2, St3=s3 . . . . ]= P [S, =slS,, =s d.
(1)
and also behaviour is homogeneous:
P[Sm÷,=jlSm=il=p~n)
(2)
i.e. that the transition probability p~7) depends only on the length of the time interval n and not on the starting time m. From property (1) it can be deduced that
P[S, =jlS, = 0 = 2
P[S, =jlSr=k]P[Sr=klSm=i],
(3)
k
for any r such that m < r < n. In continuous time (1) becomes
P[S(t)=jIS(h)=i]= p~j(h, t)
(4)
A stochastic model of informationvalue
545
and (2) becomes
(5)
po(h, t)=pij(t-h) and (3) becomes
po(h, t)= £
p~k(h, u) . pkj(u, t), h < u < t.
(6)
k
Homogeneity enables us to denote (6) as
po(t + r)= Z
p,k (t ) "pkj( r),
(7)
k
For a small time interval r = &, and separating out change from no change,
po(t + & ) = pij(t ) . pjj( 6t ) + £
Pik(t ) "pkj( & ).
(8)
k,.)
If the process is in state j at time t, pij(&) is the probability of no change in the interval (t, t + & ) and so 1-pjj(&) is the probability Aflt+ O(&) of some change in that interval, so that
1 -pjj(&) = aj& + O(&)
or lim 1- - p-j j ( & ) - a t 8, ~ o
&
(9)
so that the probability of a change depends on the length of the interval (t + &) and on the constant At which represent the rate of change in state j. The probability of a change occurring at x or y from k value units of information to some other value is Ak. The conditional probability of a change in z's information from k to j value units is denoted by qkj. Pkj is the absolute probability of value units of information at z changing from k to j items. So that Pkj = ( a k & + O ( & ) ) • qkj and from (8)
po(t + &) = pij(t)(1 - aj& - O ( & ) ) + £
p,k(t)(ak& + O (&))qkj(t) &
(10)
and so letting &-+O gives
p~ (t) = - Ajpo(t) + £
P,k(t)Ak. qkj
k,,j
with initial conditions
... f l , ~ = j P°(°)=IO, i #j
(11)
John A. A. Sillince
546
and where the matrix form is dP(t) - =P(t). R, dt where P(0) is the identity matrix and where R is of the form r~j= - a j rkj = A~hj, k,~j. Now consider again Fig. 1. Given a jth value unit of information arriving at source persons x or y, let the probability of new information being generated be aj for x or y. Let two exchanges count as one event, so that the probabilities for each of the three types of information flow are, relative to the sum a2+2a+a(1-a)=3a of these three types, a=/3a (related), 2a/3a (alternative), and a(1-a)/3a (unrelated). Because the probability of new information being generated is proportional to the number of information units j, the arrival rate is a t =j3A. Since related and alternative information increase information value from k to k + 1 units, and unrelated information reduces it to k - 1 units, the conditional probabilities are: a(A+2),j=k+l 3A q~J= a ( 1 - a ) , j = k _ 1 3,~ If information stock at time t is 1I (r) where
P[l-I(t)=n]=pn(t) then from (11), with initial information stock a, p'.(t) = -apo(t)
p](t)= -j3Apj(t)+[ ( j -
1)(A2+2A)] • p~_,(t)+[(j+ 1),~(1 -A)]
.pj÷,(t). j>a.
(12)
Using a generating function defined as
M(O, t ) = £ exp(jO)pi(t), j=o
pj(O)=O,j#a,pa(O)=l and multiplying the jth equation by exp(jO) and summing over j,
Z P;(t)exp(jO)=-Z 3jAp~(t)exp(jO)+Z [(J-1)(A2+2A)]'PJ-'(t)exp(jO) j=o
j=o
j=o
oo
+£ j=O
[(j+ 1)A(1-A)I.
pj+~(t)exp(jO)
(13)
A stochastic model of information value
547
which gives OM(8, t)= -3A ~ 0 +exp(8)(A2+2A) -ff~ aM +exp(- 8)A(1 -A) aM at aO
i.e.
OM(O, t)
OM c 3 ~ +A (8) ~ - =0
(14)
where A(0) = [3A - exp( 0)(a 2+ 2A) - exp( - 8)a ( 1- A)] and whose auxiliary equation is dt dO dM T =A i8) =-0- =0. Integrating for t and 8:
l(ZC)
t=-2AC~-~ln ~ -
+constant
A=X/A+2 -3
B=2X/-X + 2
(15)
/ 9 +4(A + 2)(A - 1)
4(-x Z=A exp(0)+B.
Becausepa(0)= 1 M( O, 0)=exp(aS)=C1
(16)
Z-C) C2= ~ exp(2AACt)
(17)
and from (15)
and setting t =0 in (17) the general solution is C(C2+1) Z- - (1 -C2)
(18)
so that substituting this value of Z into the left-hand side of (15), and using (16) the moment
548
John A. A. Sillince
generating function is given by: .
. ,,,
{C(C2+1)
M ( O, t )=exptato= ~-~l---C-f
(19)
so that Y =exp(0) dM(O, t)
/u,---
dt
=aya-ly
(20)
d2M( O, t)=aY"-I }"+a(a-- 1)Y"-2 Y dt
o~=d2M(O, t) dt 2
/z2.
Some results are given in Fig. 2. For low probabilities of arrivals at x or y [Fig. 2(a)] both mean and variance of information value reach peaks, and for both the slope is greater before the peak. In the period before the peak, during which mean information value at z is rising, new arrivals of information at x or y increase the chance of the existence of related or alternative information. But after the peak, new arrivals of information at x or y are more likely to increase unrelated information. For higher A [Fig. 2(b) and (c)] both variance and mean decline continuously over time. The decline in these values is because of the greater probability of unrelated information when ,~ rises.
OTHER MODELS
(1) No effect from alternative information It could be argued that alternative information does not increase information value, that in fact it has no effect at all. This would be the case, for example, in an automated system where corroboration is superfluous. If a valve gives a pressure reading there is no increase in
A stochastic model of information value
549
information value in having a second, duplicating, valve. In this case we have: A2
j=k+l
3A
j=k
2A
q~ =
3A
A(I - a )
j=k-l.
3A
(2) Hierarchy with superior requesting information In this case the information user is at the top of a hierarchy. For example, a general z requests information from a brigadier y. Let us suppose that z needs information from y with probability A (see Fig. 3). Also, y receives the same information supplied by x, a private, with probability #. So y is the middle man. In this case, the probability that information is needed and also available is A/z, that it is needed and not available is A(1-/z), that it is not needed and available is (1 - A)/.t, and that it is not needed and not available is (1 -A)(1 - / x ) . So
{
A/.L
j=k+l
qkj= (1-A)~+(1-A)(]-~)
j=k j=k-1.
A(1-/z)
1600 o
o
12,000
--
1400
(a)
1200
--
I000
--
.o
~ - - - _
>
.2
--
600
--
400
//
--\
8O00
_
/
/
. . . .
Variance
6000
--
4000
--
\ \
\
>
.o
- - //
\ x
\ \
¢
o~...~----:~ tffi20 triO
Variance
\
Mean
/
t=0
Mean
. . . .
\
//
200 .z.
_=
•(b) \
/
800
10,000
I
I
I
t=40 t=30
I
tffi60 t=50
I
I
0 t=O
_=
tffiS0 t=70
2000
t=90
t=10 t=5
tffi20 t=15
Time ~
t=25
tffi30
t=40
t=35
Time
12,00t)
7--
(c)
Mean
__\
.2 lo,~o
. . . .
\
Variance
k
"~ o ee
80011 \
6OO0
\
>
=
._= es
~ ¢~
--
4000
\
2000
o triO
tffi5
tffi 1 0
t = 15
tffi20
Time Fig. 2. (a) Model results for A=0.01, 0 = - 1 0 , a=3. (b) Model results for A=0.1, 0---10, a=3. (c) Model results for A=0.02, 0 = - 10, a=3. IPH 31-4-G
t=45
550
John A. A. Sillince
Z requested
3/
supplied~ x Fig. 3. Hierarchy with superior requesting information.
(3) Hierarchy with subordinate requesting information from another subordinate In a hierarchy, subordinates must communicate via a superior. For example, a tank battalion wants to communicate with an artillery battery via headquarters. Here a subordinate z must get information from another subordinate x, by going through a superior y (see Fig. 4). So, z needs information (with probability /x), y requests it on his behalf (with probability A), and it is available at x (with probability/z). The good result (j=k + 1) for z is that z needs, y requests, and it is available at x, with probability /,t2A2/(l+~h.-]/,). The bad results ( j = k - l ) are first, that z needs, y does not request, and it is available at x, with probability / x 2 A ( 1 - A ) / ( l + / z h - / x ) , second, that z needs, y requests, and it is not available at x with probability/zh2(1- bt)/(1 + / x h - / z ) , and third, that z needs, y does not request, and it is not available at x with probability / z h ( 1 - / x ) ( 1 - A ) / ( 1 + / z h - / z ) . The indifferent result, ( j = k ) is that z does not need and it is available at x with probability / x ( 1 - / z ) / ( l + / z X - / z ) , and that z does not need, and it is not available at x with probability (1-/z)2/(1 + / z h - / z ) . So
qkj=
A2~2/(1 +/xh - / z )
j = k+ 1
(1 - / z ) / ( 1 + / z k - / z )
j=k
/zh(2/z + 1 -/.th)/(1 +/xh - / z )
j=k-1.
(4) Network Here the constraint that subordinates communicate via a superior is removed. Now z can request information directly from x. For example, a journalist requests information from a photographer without going through an editor. For there to be an increase in information value to z, information is either available from x with probability A/z, or it is not available from x but x gets it for z from y, with probability A2(1-/.t)/.t. For there to be a decrease in information value, either information is requested but is not available with probability A2(1-/z) 2 or it is
Y
Z
X
T
Fig. 4. Hierarchy with z requesting information from x to y.
A stochastic model of information value
551
available but not requested with probability/z2( 1 - A) 2. /~), + A2( 1 - p,)/.t, h2(1 +/.,2--/.,) +/z(/., + A-- 2/zh)
j=j+l
q~= /~2(1 - ~)2 +/2,2(1 - A)2 X2( 1 +/x 2-/~) +/~(/z + A - 2/xh)
j=k-1.
(5) Market with prices Here the freedom of the network prevails, but each piece of information has a price• This may be a market price, or it may be a retrieval cost. For example, in order to search for patents, hierarchical status is irrelevant, but retrieval cost is important. This can be modelled by using a probability Pk that z is prepared to pay for the kth information item. The effect of this on the conditional probabilities in sub-section 4 are that for j = k - 1 there are two extra terms, the probability that information is requested, is available, but z is not prepared to pay, which is (1 -p)A/x, and the probability that information is requested, is not available from x but x gets it from y, but that z is not prepared to pay, which is ( l - p ) A # ( l - / x ) .
FUTURE WORK The simple triads are supposed to be smallest possible cases useful for studying organizational characteristics. But the question remains how the scaling up can be done. Figure 1 shows that information value added at z depends on information being provided both by x and y (not just by one of them)• Let P[X,] be the probability of X giving information at time t, then the probability of information value added at Z is
e[z,] =P[X, + Y,] so that
{z}={x}*{y} by convolution• Figure 5 shows a three-tier hierarchy and the following n - l - f o l d convolution of convoluted pairs generalizes to n - 1 tiers
{x211} = {Xl.l*lx.2} Ix3.} = {{x,. }*{x,12} }*l {Xl2,* 1x~22} } = { {x~j, }* {x,j~ } 1(2~ {X..l } = {x3. }'1x312} = { Ix., }*{x.~} }'1 {x~2~}* {x1221 }* = {{x.~ 1" 1x1321 }'1 {Xl4, }*{x~.21 } ~_~{ {Xlj I },{XIj2} }(3)
{Xojl } = { IXj~l }* {x~j2} }c.-,. Another approach is a theoretical axiomatic one, offering the possibility for the generation and proving of related theorems. For example, Egghe and Rousseau (1993) have set out a framework for understanding the statistical distributions of information items from sources,
552
John A. A. Sillince
•
.2,/,..,,.
•
O
Fig. 5. Three-tierhierarchy.
using previous results such as Lotka's Law, the Lorenz curve, and the 80/20 rule. What their framework lacks, however, is any model of the user's organization--i.e, the destination and routeing of items, which our paper introduces. So far, the formulation models the positive and negative value of information flow around an information system. There is no consideration of analysis, judgement, or decision, crucial components in according added value (Bawden, 1990). The model here assumes that merely getting the right information is all that is involved. It is, however, possible to weight the importance of the users (here known as persons x, y, and z) or the importance of the decisions, by means of larger or smaller increments in the value of j. So a person x who was twice as important as person y could give rise to increments of the form j = k + 2 or j=k - 2 . It is misleading to think of information as being drawn through an organization or market for final use by a unique person. The information would probably be useful in many ways to many people. More detail is required within these different cases of organizational topology, together with more types. For example, it is necessary to generalize from price to transaction cost, so that these ideas can be linked to transaction cost theory within the economics of the firm (Williamson, 1975). Transaction cost theory is able at a theoretical level to suggest the reasons why individuals choose to transact within free markets or within organizations, depending upon levels of uncertainty, the need for secrecy and information retrieval costs. Incentives to exploit information differences opportunistically shrink when the people concerned place transactions in a single organisation. Such internal organizational relationships may enhance information coding, the convergence of expectations, and auditing control, though at greater costs than when price alone can moderate the exchange between people (Osbom & Baughn, 1990). The theory
A stochastic model of informationvalue
553
has been applied to information technology with the speculation that IT brings down transaction costs to such an extent that organisations are needed less, so that much work formerly kept within the boundaries of the organization can be outsourced or market tested. However, transaction theory cannot of itself solve the basic problem of putting a price on information--agreeing on a price for information is problematic unless a buyer knows what the information is---once that knowledge is disclosed the buyer need not pay for it (Anderson & Gatignon, 1986; Calvet, 1981). This problem of information valuation through market mechanisms may also enhance a preference for transaction forms providing high control, such as bringing the production and storing of that information within the organization's boundaries (Osborn & Baughn, 1990). The formulation suggested above is intended as a solution to this lack of any theory of information value. Recent work has applied transaction cost ideas to game theory (e.g. Crampton, 1991). Much of this work is based upon the evolution of information value over time (with varying discount rates simulating decaying value) and how it is managed (e.g. contracts vs long term relationships within markets; explicit rules vs trust within organizations). This work and the current models are directly linked, since Fig. 2 shows the different effects of information arrival probability on information value over time. The formulation suggested above requires empirical testing in order to discover which organizational or market forms are most able to deliver high information value. The advantage of the current approach is that it emphasises the characteristics of the user and his organizational/market environment, rather than abstract notions of information, which are apt to mean little unless applied in specific contexts and for specific uses (Bawden, 1990). Thus, any information system will not be only an abstract topological structure, but may consist of some parts (e.g. R&D, marketing) where task uncertainty is relatively high, or other parts (e.g. gaining competitor intelligence) where costs of information retrieval are high, (slanting information transfers towards within-organization flows according to transaction cost theory). It is therefore fully within the scope of what Taylor (1982) has called a "user-driven" (rather than "content-driven" or "technology-driven") model of information.
CONCLUSION
The formulation enables investigation of qualitative (value) variables in the different topologies which organizations represent. How to measure information value has previously been ignored or dealt with historically (e.g. Suomi, 1994) yet the ability to monitor current levels of information value is much more useful. Such an ability opens up the possibility of investigating questions such as the differences between hierarchical and non-hierarchical organizations. This is very topical as studies are suggesting that organizations are changing away from hierarchical structures with many levels towards network-based ones with much fewer levels, and structured around fluid teams which form and dissolve as and when specific tasks present themselves (Feeny et al., 1989; Burnes, 1992; Hodgkinson, 1990). Moreover particular forms of information technology enable particular types of topology--as electronic mail makes the formation of networks much easier. The relationship between information technology and organizational topologies have been suggested within firms (Ciborra, 1987; Sproull and Kiesler, 1986; Malone et al., 1987). Bakos (1991) has suggested a relationship for inter-organizational contexts too---he distinguishes between bilateral links between firms who already have an established connection and multilateral links between firms who have no preestablished connection, within an electronic marketplace. This suggests that not only are previously accepted information channels losing relevance, but also that better theoretical tools are needed of judging what kinds of information channel are of value. These studies bring out another aspect (ignored here) of information flows within organizational topologies-that particular network nodes and particular topologies give more power in terms of control over access to and price of information.
554
John A. A. Sillince REFERENCES
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