The Emergence of Network Organizations in Processes of Technological Choice: A Viability Approach

Working Paper

The Emergence of Network Organizations in Processes of Technological Choice: a Viability

Approach

Jean-Pierre Aubin & Dominique Foray

WP-96-110 September 1996

slllASA

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria .

.

M.. Telephone: +43 2236 807 Fax: +43 2236 71313 o E-Mail: info@iiasa.ac.at

The Emergence of Network Organizations in Processes of Technological Choice: a Viability

Approach

Jean-Pierre Aubin & Dominique Foray

WP-96-110 September 1996

Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work.

VllASA

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria

kde

Telephone: +43 2236 807 Fax: +43 2236 71313 E-Mail: info@iiasa.ac.at

The Emergence of Network Organizations in Processes of Techno1oe;ical Choice: a

Viability ~ ~ p r o a c h

Jean-Pierre Aubin & Dominique Foray IIASA

A-2361 Laxenburg Austria and

I R I S - T S

UNIVERSITE

D E PARIS-DAUPHINE F-75775, Paris cx(16) France

Introduction

Traditionally, the analysis of emerging struct uses (technologies, conven- tions, etc.) starts fro111 the specification of micro-behaviors through local rules (sucll as "agent

i

chooses a technology s if tlle majority of agents has chosen this technology") i11 order to study tlle macroscopic evolution of the system. In this context, the network organization is given. T h e general purpose of this approach is to derive the collective consequences which can- not be estrapolatecl fro111 any kind of representative individual behavior.

See for instance [12, David, Foray & Dalle] and its bibliography.

In this paper, we shall follow the opposite track: instead of deriving solne illdeterininism from deterministic system, we rather detect some reg- ularities from indeterllliilistic micro-mecl~a~~isms.

In other words, our goal is to allow the system to discover all the network structures ^- described by influence matrices

among

the agents ^- wllich are dictated by tlle state of the systeln whenever viability constraints are iinposed on the system. To fix the ideas, we divide the constraints into two classes: the first one includes inclividual constraints while the second one clescribes interacting proxiinity effects through a loss function decreasing along tlle solu tio~ls.

Once tlle network structures known, we shall provides two classes of selectioll mecl~anisins of network structures, one mechanism being of a static nature, and the other one, dynarnic.

The dynamic mechanism provides the variations of the influence ma- trices. Among them, we shall choose the one with maximal inertia, called

"heavy evolution". We provide a definition of an "organizational niche", which is a networlc structure which regulates a noneinpty set of states.

We prove that heavy evolutio~ls enjoy the property of locking any network orgailizational niche as soon as a trajectory enters it.

In order to avoid mathematical technicalities, we just describe the model we propose and state the results which can be obtained without specifying illatllernatical assuinptions nor the exact forinulas, u ~ l ~ i c l ~ go beyond this i n t r o d u c t i o ~ ~ to the viability approach to the analysis of ernergiilg struc- tures. We refer to [4, Aubin] for more infol.inations on these rnatheinatical techniques.

The Network Constraints

We

start the description of tlle

nod el

with 1. n firms, labelled i = 1 , .

.

.

,

^n,

2. a finite d i n ~ e n s i o ~ ~ a l vector space Y of tecllrlologies described by z E Y. We denote by

X

:= Yn the space of tecllnological configurations

5 := ( z l , . . .

,

x,,) implemented by the n firms.

A

tech~~ological config- uration :c := ( z l , . . .

,

^s,) is an assignment of a tech~lology z j t o each fir111 j .

The constraints are defined a t

1. the level of individual agents,

by

cost or loss functions

IL;

: Y ^H Z, of tlle forrll

{xi E Y

I

ILi(zi) E

Mi

C Zi) describing tlle individual co~lstraillts of the firm.

2. the level of agent interactions, through a "proximity" function

Izo

:

X

^H ZO. These proximity effects are due to Marshallian externalities that affect for example the costs of screening ancl hiring workers:

Concretely, we postulate that for every firm, the relative wage costs of a worker of a given tecl~nological type is decreasi~lg if the number

of workers of that type currently eml~loyed by the enselnble of firms in the inunediate lleigllborllood of that firill is irlcreasing (see [12, David, Foray & Dalle] for a modelisatioil of a system with marshallian exterllalities using stochastic Ising models).

For instance, one can take Zo :=

R",

so that the j t h coml~onent / ~ ~ , ~ ( z ) describes the cost for firm j of tlle technological configuration x := ( x i , . - .

,

X,J.

In

summary, a viable evolutioll of tecllnological collfiguratiolls t H

z ( t ) is a time-tlepenclent tecl~nological configuration satisfying

so that the tecl~~~ological collfiguratioll shoulcl decrease exponentially to the lllasilllal proximity F satisfying I L ~ ( T ) =

0.

2 Influence Matrices Describing Network Or- ganizat ion

We furtller assuine known the dynalnical behavior of each firm j indepen- clently of the one of the other firms: It lllodifies the state of technology : c i ( t ) a t time t according to the differential equation

This is the dynalnical analogue of the classical static description of the 1)ehavior of agents through utility or cost functions.

Now we assuine t h a t , clue to tlle interactioll collstraillts (the Marsllallian effect), tlle solutions to the deceiltsalized system (1) do not llecessarily satisfy the above constraillts and satisfycing property.

Therefore, sollle regulation mechanism should b e designed. We propose t o investigate a networli organizatioll describecl through a graph matrix

W

of influence weights wi of fisin j on firm i. The case when

&:

⁼

0

describes the situation where firm i does not take into consideration the behavior of

firm j. When wi

>

0, firm

i

displays an apish behavior towards f i r ~ n j. The case whell w!

<

0 denotes a n antitlletical behavior of fir111 i toward fir111 j.

We underline that these weights are not a priori

predetermined

proba- bilities of interactions. Our aim is precisely to let them emerge a posteriori from the confrontation of the dynamics and the constraints.

In

this context, a network organization is described by an influellce matrix (wllich, by the way, can be regarded as the matrix of a graph)'.

The firllls inodify their autonomous dyna~nical behavior by integrating the behavior of the other firms through their influence weights. We choose for silnplicity a linear illteractio~l of the for111

In

a Inore (:ompact form, it call be written in the forill

wilere W (t ) denotes the time-dependent influence matrix

3 Organizational Niches

One can ilnpose a given network organization, described by a given influence nlatrix W , and study the properties of the dy~lalllical system:

For instance, we call 100li for the set E ( W ) of equilibria T of the above systerns, solutions to the equation W g ( c ) =

0,

their stability property, their basin of at traction, ancl the depenclence of these i terns with respect to

I/,

using for illstallce bifurcation theory.

We shall not follow this course in this paper. However, for studying later lock-in properties, we illtroduce the concept of "organizational niche"

' T h e m a t h e m a t i c a l techniques used in this s t u d y have been devised in [3, Aubin] in t h e framework of neural networks a n d cognitive systerns. T h e y have been a d a p t e d t o a n economist context in [4, Aubin] in t h e framework of "connectionist complexity".

N ( W ) of the influence nlatrix W: It is the viability kernel of the clifferential equation x f ( t ) = W g ( x ( t ) ) , i.e., the largest

subset

of states satisfying the constraints which is viable under tlle differential equation x f ( t ) = W g ( x ( t ) ) . It is also equal to t h e set of initial states xo from which the solution to clifferential equation x f ( t ) = Wg(x(t)) is viable.

In other worcls, starting from a state in the organizational niche, the solution to the system organized according the influence nlatrix W satisfies the al~ove constraints forever.

4 How Network Organization Evolves

But ^o71,e can reverse the questioning and, insteat1 of studying the properties of a given network organization, look for all network organizations com- patible with the constraints in the following sense: find (time-dependent) illfluence nlatrices

W(t)

such that, fronl any initial state satisfying tlle constsaints, there exists a vial~le solution (x(.), W ( . ) to tlle ~~aranletrizecl tliffesential equation

x f ( t ) = W ( t ) g ( x ( t ) ) (4) i.e., a solution such that

s ( t )

satisfies tlle constraints for ever.

The basic viability theoren1 (see for instance [2, A u l ~ i n ] ) applietl to this situation provitles t h e feed back map

R

associatillg with each tecllnological configuration :c a subset R ( x ) of influe11c.e lllatrices W . The system is viable if and only if R ( x ) is not empty for every technological configuration .x sntisfying the con~traints. I n this case, the evolution of viable solutions . r ( t ) obeys the regulation law

In other. wo~.cls, the feecll~a.cl< map

R

assigns t o every technological con- figura.tion the set of networl< orgallizations "vial~le" with respect to the constra.ints.

In tlle favorable case, the set R ( x ) of viable influence ~natrices nlay colltaiil nlore than one nlatrix. Actually, the larger this set, tlle more robust, since it allows for errors.

So, the question of selecting ii1fluenc.e matrices arises, and Inany scenasii c,an be consitlered.

We sliall desc,sibe two prototypes of selectioil mechanisms, one "s tatic"

,

and tlle o ther one, "dynamic".

5 Minimizing a Static Complexity Index

Tile static one iilvolves a complexity i n d e x 5 f a ~letworli organization de- scribed by a n influence inatrix

W.

It is defined 1)y the distance between the irifluence inatrix

W

and the unit inatrix 1, whicll clescribes the dec.en- tralized situation. T h e idea is to regarcl the clecentralizecl situation as the s i n l ~ l e s t one, ant1 thus, t o regard a network organizatio~l as co~llplex as it is fa,r froni this siillplest situation. One clan the11 compute for eac.11 z the vial,le iliatrix W 0 E R ( : c ) wllich is the closest to the unit illatsix ^- hence tlie simplest ^- and to show that despite its lack of continuity, a solutiou to the diffesential equation

still exists.

6 Minimizing a Dynamic Complexity Index

Tile ~ 1 ~ n a l l i c a . l one coilsists in differentiating tlie regulation law. Appeali~lg to set-valued ailalysis (see for i~lstailce [6, Aul)in & F'ra~~ko~slia.]), one ca.11

' ~ 1 i ~ s i c . i s t s have a t t e m p t e d to measure "con~plexity" in various ways, through the con- ct>])t. of Clausius's entropy, Shannon's information, the degree of regularity instead of randomness, "11ierar~hic.al con-~plexity" ill the display of level of intel.actions, "grammati- cal complexity" measuring the language to describe i t , temporal or spatial corriputatioi~al, measuril~g th e computer time or the amount of computer irlernory needed t o describe a system, e t c .

One can also measure other features of c~onnectionist complexity through the sparsity of tlie connection matrix, i.e., the number - or the position - of entries which are equal to zero or "small". T h e sparser such a connection m a t r i x , the less coinplex t h e system.

Each colnponent of a system which can evolve independently in the absence of con- st,raints, must interact each other in order t o maintain the viability of the system imposed by its e n v i r o n n ~ e n t . Is not complexity meaning in the day-to-day language t h e labyrinth of connections between t h e components of a living organism or organization or system ? 1s not t,he purpose of complexity t o sustain t h e colist,raints s e t by the environment and its growth parallel to the increase of the web of constraints ?

derive from the regulation law

(5)

a tlifferential inclusion of tlie form

wliicli, together with tlie original systelii

(4),

specifies the evolution of both tlie tech~iological co~ifigurations

x(

t

)

and tlie influence rnatrix W(t

).

One can regard a nor~li 11 W1(t)

J J

of tlre velocity W1(t) of tlre i~iflue~lce inat six as a

dyna mical complexity index.

The larger this dynanlical corn- plesi ty index, tlre fastest tlie conriectioiiist complexifica.tion of tlre ~ietwork orga~iization. Hence, the question arises to select tlie velocity v O ( s , W )

wi

tlr

~ l i i ~ i i ~ l i a l ~ i o r ~ i i in tlre subset R1(z, W ) of viable velocities of tlie influe~lce niatrices. Oiie can prove that tlie systelii of differential equations

1.)

s l ( t )

=

W(t)g(x(t)) ii) W1(t)

=

vo(x(t), W ( t ) )

1la.s solutions (:c(t), W ( t ) ) , whicli are ilaturally viable. They are called

"heavy solutions".

7 The Lock-In Property

"Heavy solutions" have tlie property of locliing-in organizational niches:

If, for sollie ti~iie T , tlie solutio~i enters a organizational niche, i.e.,

if

.r:(T)

E

N ( W ( T ) ) , tlien for all t > T , the technological configurations call be regulatecl by the constant influence matrix W ( T ) , i . ~ . , accorcling to tlie cliffei.ent ial equation

'v'

t > T , s l ( t )

=

W(T)g(x(t))

allcl tlie solution will remain in the organizational ~iiclie of W ( T ) :

t,

t 2 T, xl(t)

E

N ( W ( T ) )

Tliis is another ~iletal>lior of tlie lock-in property of organizational ~iiclies.

8 Conclusions

I11 this paper, we liave tried to propose

a

modeling strategy, the purpose

of wliicli is to allow a d y ~ i a ~ n i c systeni of tecli~iological c1ioic.e~ to discover

and select the network structures (that is to say the particular influence nlatrix characterizing the set of interacting agents), whicll are compatible with the viability constraints generated 11y a particular technological con- figuration. As very well descsibecl in [S, Cohendet], the streanl of works focussing on enlergent structures within the context of stocllastic interac- tions among agents continuously increases the conlplexity which is assigned to tile micro-behaviors, in orcler to clescribe more complex trajectories of nlacroscopic evolution. Agents are allowed to cleviate from the norinative rules [[9, Dalle]; percolation l>rol)al~ilities are introclucecl ill order to allow sonle subsystenls to keep isolatecl ailcl so not infecteel 11y the choices of the ma.jority ( [ l l , David & Foray]); a super-agent, provicling t o each one the sa.nle infor~lla.tion, can be coi~sirlerecl ([9, Dalle]; the parameter clescribing the strengtll of interactions ca.n 11e clla~lgecl ([12, Davicl, Foray & Dalle]);

last but not least some lcinds of learning capacities are attril~utetl t o tlle agents, allowing the111 to acljust their l~ellaviors with respect t o what they learn in the course of their recurrent clecisions ( [ I , Arthur]). Of course, we want not to claim that such eserc-.ises are evolving towards a deacl- locl;. However, such a conlplexity increase on the side of incliviclual ant1 collective I>ehaviors l ~ y no illeails allow this approach to escape from a cle- terministic logic: each clecision center or agent possess ex ante a program of a.ctions/rea.ctions, certainly complex 11ut ultimately invariable along the life of the collective systenl (see [ l o , Dalle

8.

Foray], for a discussion of tlle status of incliviclual rationality ill stochastic models of interactions).

In this pa.per, we have l>rol>osetl a clearly opl,oserl vision. Tlle inclividual psogranls of actions/reactions are not known ex ante. They are rather the object of inquiries, the emerging and lock-in structures, clerivecl from tlle viability constraints and the selection nlechanisms. This paper only ps(~vides a first step of this research program wllicll will 11e continued in the near f11t1u.e.

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