Niche width theory reappraised

Journal of Mathematical Sociology © 1997 OPA (Overseas Publishers Association)1997, Vol. 22(2), pp. 201-220 Amsterdam B.V. Published under licenseReprints available directly from the publisher under the Gordon and Breach Publishers imprint.Photocopying permitted by license only Printed in India.

NICHE WIDTH THEORYREAPPRAISED

JEROEN BRUGGEMAN*,†

Center for Computer Science in Organization and Management,(CCSOM), University of Amsterdam, Sarphatistraat 143,

1018 GD Amsterdam, The Netherlands

Niche width theory is a part of organizational ecology, which predicts for given envi-ronmental conditions whether specialist or generalist organizations will do better. Thesepredictions are based on a mathematical model, but the model itself does not matchthe verbal argument, which in turn is ambiguous. In an attempt to improve the theory,I formalize it into logic. As a result, implicit information becomes explicit, explanatorysupport for the conclusions is added, and the theory becomes more parsimonious. Thekey advantage of logic is its strict definition of logical inference, which makes possiblethorough checks on soundness and on consistency.

KEY WORDS: Logic, Formalization, Organizational ecology

1 INTRODUCTION

According to organizational ecology, organizations "live" in an envi-

ronment that determines whether specialism or generalism is a better

strategy for organizational survival. A generalist is a Jack-of-all-trades;

a specialist maximizes the exploitation of a small set of environmental

opportunities. Organizational ecology addresses the distinction between

specialism and generalism under the heading "niche width theory"

* E-mail: [email protected].† A number of core ideas for this paper were developed in cooperation with Jaap

Kamps, Gábor Péli, and László Pólos. Many technical improvements of earlier draftsalso benefited from their comments. The author furthermore wishes to express his thanksto Kathleen Carley, Babette Greiner, Jelka Hopster, Michael Masuch, Breanndán ÓNualláin, and two anonymous referees. This research was supported by the NetherlandsOrganization for Scientific Research (NWO) through a PIONIER project awarded toMichael Masuch (grant # PGS 50-334). The author received SIR and STIR grants fromNWO and a TEMPUS grant from the European Community.

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202 J. BRUGGEMAN

(Hannan and Freeman, 1977; Freeman and Hannan, 1983). Nichewidth theory has received a fair amount of attention (Carroll, 1985;Grandori, 1987; Freeman and Hannan, 1987; Singh and Lumsden,1990; Carroll and Hannan, 1995; Briiderl et al, 1996).

The aim of this paper is to elucidate Hannan and Freeman's nichewidth theory by formalizing it. The formalization of this fragment ofthe theory is part of a concerted effort to formalize all parts of organiza-tional ecology and to establish new relations between them (Péli et al,1994; Péli and Masuch, 1997; Péli, 1993). This work is part of a growingtrend towards more formal organizational theorizing (Masuch andLaPotin, 1989; Masuch and Warglien, 1992; Carley and Prietula, 1994).

Mathematics is the traditional language to achieve theoretical preci-sion, parsimony, and clarity (Coleman, 1990). Along with mathematicalformalizations, there are nowadays computer simulations to investigateproperties of complex models (Glance and Huberman, 1993; Masuch,1995). In addition, there is formal logic to investigate inferencing withintheories (Péli et al, 1994). Since I want to focus on the logical propertiesof the niche width theory, logic is the language of my choice. My inves-tigation was supported by a "theorem prover", a computer programthat automatically generates proofs, and so helps to determine whetheror not a theoretical explanation is sound (Ó Nualláin, 1993).1

The logical formalization consists of two distinct but related sets ofactivities. First, clarifications of the basic concepts are made, and in-formation needed to infer the conclusions is made explicit. This set ofactivities is called the rational reconstruction of the theory (Section 2).In this particular theory, the rational reconstruction also involvesdiscussing a mathematical model that Hannan and Freeman use(Section 2.2). Second, Hannan and Freeman's niche width theory isformalized in logic, and this is the formalization proper (Section 3).The rational reconstruction helps to formalize the theory, and theformalization helps to further develop the rational reconstruction.As a result, the underlying assumptions of the theory are more clearlyspecified, the meaning of basic concepts is more clearly described,the domain of theory more clearly delineated, and the consistencyof, and inferencing within the theory more thoroughly examined.

1 The computer output of the theorem prover can be obtained from the author uponrequest.

NICHE WIDTH THEORY 203

Furthermore, the formalization of niche width theory makes possiblea more parsimonious organization of its explanatory structure.

2 SPECIALISTS AND GENERALISTS INCHANGING ENVIRONMENTS

The core of niche width theory consists of nine comparative state-ments (the predictions) about two populations,2 one of generalist andone of specialist organizations. Each of these statements concerns aperiod of observation in which one of these populations is favored byselection,3 given certain environmental conditions characterized by theconcepts grain size, variability, and dissimilarity.

2.1 Specialists and Generalists

Specialists and generalists are characterized by their routines, whichare "patterns of activity which can be invoked repeatedly by membersand subunits" (Hannan and Freeman, 1989, p. 76). Each routine (orset of complementary routines) matches a certain resource configura-tion (a combination of resources). Routines tolerate minor deviationsfrom a configuration without mismatch.4 The most important re-source for organizations are customers and their demand for organiza-tional output. The resource concept is taken broadly: all kinds ofsocial-economic conditions fall under this heading. A specialist elabo-rates its routines enabling the organization to obtain a good perfor-mance over a narrow range of resource configurations which matchits routines. In other words, a specialist has a narrow niche.5 It

2 Organizations are classified according to core features (Hannan and Freeman,1984). All organizations with the same core features have the same form, and all organ-izations of the same form are in one population. Organizational structures are said tobe inert and reorganization risky (Hannan and Freeman, 1984). Therefore, most organ-izations wil keep their form during their lives and thus will stay in the same population.

3 A frequently used operational definition of "population x is favored by selection topopulation y" is as follows: population x has a lower disbanding rate than populationy (Freeman and Hannan, 1983, p. 1127). Then the founding rates of both populationsmust be equal, and greater than zero.

4 I dichotomize "match" and "mismatch" of a routine and a resource configurationto state the argument more clearly. A more subtle distinction is not needed for theinvestigation of Hannan and Freeman's nine predictions.

5 A niche is a set of resource configurations on which a given organization, or popu-lation, can sustain itself or grow, provided that the amount of resources is sufficient.

204 J. BRUGGEMAN

thereby runs the risk that its routines will mismatch a new resourceconfiguration as soon as the environment changes. A generalist doesnot elaborate its routines, it spreads itself thin, as it were, such that itperforms moderately well over a broader range of resource configura-tions, and so has a broad niche. A generalist has the advantage ofhaving several parallel routines simultaneously, which may matchdifferent resource configurations. Niche theory itself does not tell inwhat manner organizational performance is important, so I make useof additional information from another part of organizational ecology(Hannan and Freeman, 1984), to establish a bridge between the giveninformation about routines and the predictions. Hannan and Freemansay that organizations are chiefly evaluated on two criteria: the reli-able production of goods or services, and the rational account foractions and decisions. Elaborated routines enable organizations toproduce more reliably and to account for their actions more rationally(Hannan and Freeman, 1984). I interpret this as follows. A specialistwhose routines match a resource configuration has higher reliabilityand accountability than a generalist whose routines match, and bothof them have higher reliability and accountability than organiza-tions whose routines mismatch. Reliability and accountability are as-sessed over a relatively long period (at least a year) prior to a point ofmeasurement.

Niche width theory compares organizations which fulfill the socalled principle of allocation. Under this principle, all organizationshave similar potential capacity, only it is used differently: specialistsare good at one thing, whereas generalists are moderately good atseveral things. Due to the principle of allocation, the theory cannotcompare a multinational to a local drugstore. The principle of alloca-tion thus limits the domain of the theory.

2.2 Environment

The organizational environment, conceptualized as a resource con-figuration, changes with time. Hannan and Freeman capture a chang-ing environment in terms of a sequence of discrete patches. Each patch

So, the niche provides a possibility of growth, not a guarantee. A population willdecrease in number if the resource configurations that are beneficial to the populationare only sparsely available.


is an instance of a resource configuration that has a certain dura-tion; for this duration, the same resource configuration is available toorganizations. In the somewhat simplified set-up of the theory, thereare only two kinds of patches, which alternate during a period ofobservation. The period of observation is supposed to last severalyears at least, and the patches are sub-periods within the period ofobservation. The two kinds of patches can have either long or shortdurations. An observation period during which the patches are onaverage short (highly frequent changes) is called fine grained, the casewherein patches are long is called coarse grained (see Figure 1). Grainsize is relative to the life expectancy of organizations of a certain form(Hannan and Freeman, 1977, p. 952). The concept grain size is notfurther elaborated in the theory. Seen from the life expectancy per-spective, mismatch can come in two different ways: the patch lengthis within the interval that an organization can survive on slack re-sources, or the patch length is excessively long and the organizationdies. I interpret both fine and coarse grain to be within the survivalcapabilities of the average organization, because the theory does notdiscuss extinction of an entire population.

The probability of the occurrence of one kind of patch in a period, orequivalently, the relative duration of the two kinds of patches, is cap-tured by the notion of variability. Variability can change independentlyfrom grain size, (except in the extreme case of a stable environment inwhich there is only one patch). If one type of patch dominates, theobservation period is said to have low variability (Figure 2): one of thetwo types of patches has a high ratio of occurrence and the other a lowratio of occurrence. If both patches are, on average, equally long, theobservation period has high variability (both sequences in Figure 1).

FIGURE 1 A Fine Grained Period on Top, a Coarse Grained Period Below.

FIGURE 2 Low Variability.

206 J. BRUGGEMAN

In the description of variability, I closely followed Levins (1968)and Roughgarden (1979) on which Hannan and Freeman base theirniche theory. However, Hannan and Freeman define variability as thevariance of a temporal series about its mean (Freeman and Hannan,1983, p. 1119). This definition is not needed for the theory and leadsto undesired consequences. Figure 3 shows that this definition per-mits two sequences to have the same variance although one sequencehas high variability and the other low variability. Therefore I discardthe definition and maintain the concept of variability as given byLevins and Roughgarden.

The patches of each type separately are not necessarily exactly equalin length. For instance, the black patches in Figures 1 and 2 have somevariance of patch lengths about the mean of the black patches. Thesame holds for the white patches. (If there are two kinds of patches theneach type of patch has the same variance about its own mean.) Thevariance of patch lengths for each type of patch plays no (explicit) rolein the theory. This variance is supposed to be fairly low, otherwise apopulation might become extinct in an incidentally very long patchwhich happens to mismatch the routines of its constituent organizations.

Along with variability and grain size, the theory has a third param-eter to characterize the environment during a period of observation: thedissimilarity of the two kinds of resource configurations with respectto the niche widths of the organizations under comparison. My inter-pretation of dissimilarity for the binary patch world under considera-tion is then as follows. The routines of specialists can only match oneresource configuration; any other resource configuration mismatches.Generalists have a broader niche than specialists, but their number ofroutines is limited too (Hannan and Freeman, 1984, p. 155). If the tworesource configurations are similar, then both match the routines ofgeneralists (though not at the same time). If the resource configurationsare dissimilar, one of the two resource configurations mismatchesthe routines of generalists. For the comparison of organizations at thepopulation level, it is important to notice that organizations in one

I low variability

I high variability

FIGURE 3 Low Variability on Top, High Variability Below.


population "have very nearly the same fundamental niche" (Hannanand Carroll, 1992, p. 30), which means that all organizations in thesame population match the same resource configuration(s).

The theory compares only those organizations whose routines matchat least one of the two resource configurations in an observation period.Organizations that mismatch both resource configurations do not standa chance of survival. In low variability conditions (see Figure 2), onlyorganizations that match the relatively long patches are taken into ac-count. Other organizations that match only the short patches would inthis theory always be outcompeted by organizations which match thelong patches, or both kinds of paches, and are not considered.

I have now discussed the key concepts of the theory, namely special-ists, generalists, the principle of allocation, and the environment interms of variability, grain size, and dissimilarity. Having discussed amathematical model that Hannan and Freeman use, I introduce formallogic, and then provide the logical formalization of niche width theory.

2.3 Mathematical Model

Hannan and Freeman not only provide a verbal argument to explainthe relative advantage of generalists or specialists in given environ-mental conditions, they also use a mathematical model from biology(Levins, 1968; Roughgarden, 1979). Why then use formal logic here ifa mathematical model already exists? After summarizing the model Iwill show that it cannot make the required predictions by giving twocounterexamples.

The mathematical model has three dimensions. The first is the en-vironment, on which two types of patches are represented as two dif-ferent values, ex and e2. Generalism and specialism are defined on thisdimension. If an organization is a generalist, it has a wide niche,denoted as a broad interval. A niche, or interval on the environmentaldimension, means that the organization in question has routines thatmatch resources denoted as values in that interval (and mismatchenvironmental values outside the interval). A specialist, then, has anarrow interval. A trait of organizations from a given population isthe second dimension of the model.6 A trait can be, for instance, the

6 Traits play a role neither in Hannan and Freeman's verbal predictions nor in mylogical formalization.

208 J. BRUGGEMAN

degree of formalization of procedures (Freeman and Hannan, 1983,p. 1119). Different organizations can have different degrees of formali-zation, i.e., different values on the trait dimension. A population oforganizations is depicted as an interval on the trait dimension contain-ing these values. One could interpret the width of this interval as ameasure of population generalism (Freeman and Hannan, 1983,p. 1118). This particular interpretation of population generalism dif-fers from the definition in the remainder of the paper, and also differsfrom Hannan and Carroll's (1992, pp. 30, 159), wherein a generalistpopulation consists of generalists - all having the same niche definedon the environmental dimension - and wherein a specialist populationconsists of specialists - under the same restrictions. The third dimen-sion is fitness (operationalized as survival chance) of individual organi-zations in a population. For each of the two environmental values, et

and e2, there is one optimal trait value, ̂ and y2, respectively, on thetrait dimension. Organizations with other trait values are less fit. Fora fluctuating environment there is an optimal trait value, y*, some-where between y1 and y2. Using the mathematical model one cancompute the optimal trait value in a given population, given two en-vironmental values (eu e2, and their distance apart), grain size (fineor coarse), and variability (high or low). The result can be graphicallyapproximated by using so called "fitness sets" (Freeman and Hannan,1983, pp. 1124-25). By computing the optimal trait, the model satis-fies its purpose in biology, and, in principle, could also serve a purposein sociology. But it cannot predict the optimal degree of organiza-tional specialism or generalism (even if the model were expanded topredict the optimal width of intervals on the trait axis). Neither in caseof a dichotomy between specialism and generalism can it predictwhich of these two forms has a higher fitness in given environmentalconditions. For one reason or another, Hannan and Freeman confusethe environmental dimension and the trait dimension. They then con-clude - without further argument - that the predictions of the modelare about generalism versus specialism, which would only be the caseif the two dimensions were meaningfully related. I will now give twocounterexamples to show that this is not the case. The first counter-example is about restaurants, the second about supermarkets.

Suppose that for a population of restaurants, the trait is the averageprice for a course on the menu. The environmental dimension is the


price for a course that customers are prepared to pay. Presume thateach restaurant in this population offers a small menu with low vari-ance about the mean price. Each of the restaurants is a specialist.The population, however, is a generalist if it contains a broad rangeof restaurants, from cheap fast food up to posh haute cuisine, eachattracting different groups of consumers from the environment. Imag-ine a second population, of supermarkets. Suppose that the averageprice of consumer goods is fairly similar for all supermarkets. Thenthe range of traits in the population is narrow, and so the populationis specialist. However, each of the organizations in this population hasa broad price range of many different goods, and so is a generalist.Both counterexamples show that the relation between the trait dimen-sion and the environmental dimension is arbitrary and differs fromcase to case. The niche width theory contains no information to relatethese two dimensions.

A formalizer who wants to elucidate the theory can now go intwo directions. One direction is to modify the mathematical modelsuch that it can make the required predictions. This is done else-where (Bruggeman and Ó Nualláin, 1995), but the modified model'spredictions do not conform to Hannan and Freeman's. The otherdirection is to take Hannan and Freeman's predictions as a tar-get to derive, which I do in my logical formalization. I interprettheir verbal theory in a way that enables me to construct the theoryanew.

3 LOGICAL FORMALIZATION "

The core of the niche width theory is formalized into first-order logic,abbreviated to FOL, which is the most widespread logic (Gamut,1991). The formalized core of the theory consists of three kinds ofstatements: assumptions, which are contingent statements about theworld, meaning postulates, which give (partial) descriptions of con-cepts, and theorems, which are conclusions logically inferred from as-sumptions and meaning postulates. The inferencing was done by atheorem prover computer program (Fitting, 1990; Ó Nualláin, 1993).The consistency of the theory was determined by asking the theoremprover to derive a falsehood from all of the statements of the theorytogether. Since any statement, including a false one, can be derived

210 J. BRUGGEMAN

from an inconsistent set of statements, the success of this trial wouldprove the theory to be inconsistent.

I use the following classes of symbols to make sentences in FOL.The meaning of a sentence in FOL is the composition of the meaningsof its parts.

• Name constants for objects in the domain of the theory. For in-stance, "ex" and "e2" are names for resource configurations e¡ ande2, respectively.

• Variables which range over the objects in the domain. (The objectsare, among others, populations, observation periods, and resourceconfigurations.)

• Predicate constants for properties of, or relations between objects.For instance Pop(x, t) is a predicate relating organizational popula-tion x to time t. Another example is the "greater than" symbol, >.Table 1 contains the predicate constants used.

• One function symbol, ra(x, t), which takes a population x as its firstargument, a time / as its second argument and gives the reliability/accountability of x at time t as its function value.

TABLE 1Predicate Constants and Function Symbols

Predicate and Functionsymbols

Variability(high, ó)

Bl(t,o)Spec{s, t)Gen(g, t)PAM(x,r)

Pop(x, t)

Er(x,t)

Fa(x,y,o)

ra(x,t)

Description

High variability in observation period o; low andhigh are name constants

Observation period o has two resource configurationsr¡ and r2

Time / belongs to observation period oPopulation s at time point / consists of specialistsPopulation g at time point / consists of generalistsThe principle of allocation appliesResource configuration r matches the routines of

the organizations in population xPopulation x exists at time point t (it has at least

one member)The organizations in population x at time point /

have elaborated routinesIn observation period o, population x is favored by

selection to population yThe reliability/accountability function has a

population x and a time point t as its arguments


• Two quantifiers, "for all..." (V), and "there exists a..." (3).• Five logical connectives- "not" (-i)- "or" (V) and "and" (A)- "if-then" (=>) and "if-and-only-if " («-)

which are listed in decreasing priority. For instance, AAB=>C,means: if A and B are true, then C is true, so, {A AB)=> C.

3.1 Principle of Allocation

Now these symbols are "put to work". First, the principle of alloca-tion is formalized because it is the basic constraint of the niche widththeory. The principle of allocation is formalized into meaning postu-lates, both for specialists and for generalists. Specialists are character-ized by elaborated routines which match one resource configuration,and generalists by non-elaborated routines which match two resourceconfigurations. I name these resource configurations e^ and e2. For-mally, if the principle of allocation PA holds, then a population s ofspecialists exists at time t, Spec(s, t), if and only if the specialists inthat population have elaborated routines at that time, Er{s,t), andthe routines match resource configuration ex but not e2: M(s,e¡) and—i M(s, e2). Meaning Postulate 2 states that under the principle of al-location, PA, the organizations in a population g of generalists at timet, Gen{g, t), do not have elaborated routines, —i Er(g, t), but their rou-tines match both resource configurations et and e2: M(g,el) andM(g, e2). The fact that their routines do not match any other resourceconfiguration is not formalized - the logical derivations do not needthis fact.

MEANING POSTULATE 1 VJ, / (PA => (Spec(s, t) <» Er(s, t) A M(s, q) A —iM(s,e2))).

MEANING POSTULATE 2 V̂ r, t (PA =*(Gen(g, t)o—iEr(g,i)AM(g,el)AM(g,e2))).

Under the principle of allocation, Hannan and Freeman comparea population of specialist organizations to a population of generalistorganizations. They do so for eight different patterns of environmental

212 J. BRUGGEMAN

Fine grainCoarse grain

TABLE 2Hannan and Freeman's Predictions

Similar

Low var.

(1) specialist(2) specialist

High var.

(3) generalist(4) generalist

Dissimilar

Low var.

(5) specialist(6) specialist

High var.

(7) specialist(8) generalist

(polymorph)

change - see Table 2 - plus a stable environment, and predict which ofthese two populations is favored by selection. The stable environment,which is case 9 in the formalization below, is left out from the tablebecause it is unrelated to any of the three environmental parameters.7

3.2 Low Variability and Similar Patches

In case 1, the observation period o has two resource configurations el

and e2: Op(o,e1,e2). The observation period has fine grain, similarpatches, and low variability, Variability{low,o). The e2 patches arerelatively short compared to the ey patches (like in Figure 2). Theorganizations in the specialist population have elaborated routineswhich match the e1 resource configuration, and the organizations inthe generalist population have non-elaborated routines and can bene-fit from both resource configurations. Assumption 1 is stated in ageneral fashion. A population containing organizations with elabo-rated routines matching one available resource configuration mostof the time is compared to a population containing organizationswithout elaborated routines matching one of the two resource con-figurations all of the time. If this holds, then the first populationwill have higher reliability and accountability than the second. Thisreflects Hannan and Freeman's idea that specialists outcompete gen-eralists over the range of resource configurations to which they havespecialized for as long as the environmental variation remains withinthat range (Hannan and Freeman, 1977, p. 950). When the "bad"patches are short relative to the "good" patches, organizations can

7 Table 2 reflects Hannan and Freeman's verbal argument. However, due to twomisprints in a table in their book (1989, p. 311), acknowledged by Freeman (1995,personal communication), two predictions in that table contradict their verbal argu-ment. Cases 1 and 2 in my table are thus modified with respect to corresponding casesin the table in their book.


survive on slack resources such that their reliability and accountabilitywill not be dramatically affected (Hannan and Freeman, 1977, p. 948).

A population predicate Pop{x, t) means that a population x existsat a time t; a "belongs to" predicate, Bt{t, ó), means that time t belongsto observation period o. Assumption 1 is stated generally for all pairsof resource configurations,8 hence I use variables, rl and r2, insteadof the constants ex and e2. The reliability/accountability9 of a popula-tion, x, at a time, t, is the function value of the reliability/account-ability function ra{x, t).

ASSUMPTION 1 Vx,y,t,r1,r2,o (Op(o,r1,r2)AVariability(low,o)ABt(t, o) A Pop(x, t) A Er(x, t) A M(x, rt) A - i M(x, r2) A Pop(y, t) A ~iEr(y, t) A M(y, r¿ A M(y, r2) =* (ra(x, i) > ra(y, t))).

A natural language transcript of Assumption 1 is as follows: for allx,y,t,rltr2,o, if o is an observation period with resource configura-tions rx and r2, and the variability in o is low, and a time t belongs too, and x is a population at time t with organizations having elaboratedroutines, and these routines match resource configuration r t and notr2, and y is a population at time t with organizations with non-elab-orated routines, and these routines match resource configurations rt

and r2, then the reliability/accountability of x at t is higher than thereliability/accountability of y at t.

Niche width theory does not contain enough information to con-clude at this point which population is favored by selection. For-tunately, another part of the theory contains information I can use:"Selection in populations of organizations in modern societies favorsforms with high reliability of performance and high levels of account-ability" (Hannan and Freeman, 1984, p. 154). In my formalizationof this assumption, a population x is favored over a population y

8 If there is variability in a period of observation, either high or low, the two resourceconfigurations r1 and r2 are not equal; in low variability conditions, the r1 patches arethe relatively long ones.

9 Because reliability and accountability always appear to "move together" in organi-zational ecology (Hannan and Freeman, 1984), I treat them as if they were one factorof organizational performance (here aggregated at the population level). This simplifiesthe formalization, but one may always replace the single function by two distinct oneswithout altering the conclusions.

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during an observation period o, Fa{x,y,o), if at all times t belongingto the observation period, x has higher reliability/accountability than y.

ASSUMPTION 2 yix,y,t,o,rí,r2 (Op(o,r1,r2)ABt(t,o)A Pop(x,t)APop(y, t) A (ra(x, t) > ra{y, t)) => Fa(x,y, o)).

Meaning Postulate 3 has only a "bookkeeping" function: if x is apopulation of specialists at time /, Spec(x, t), or if x is a population ofgeneralists at time /, Gen(x, t), then x is a population at that time,Pop{x,i).

MEANING POSTULATE 3 Vx, t (Spec(x, t) V Gen(x, t) => Pop{x, /)).

Theorem 1 states that if variability is low and resource configura-tions are similar, then specialists are favored by selection. This theoremis deduced from Assumptions 1 and 2, and Meaning Postulates 1-3.

THEOREM 1 Vt,s,g,o (Op(o,e1,e2)AVariability(low,o)ABt(t,o)APA A Spec(s, t) A Gen(g, t) => Fa(s, g, o)).

In case 2, the observation period has low variability, coarse grain,and similar patches. The environmental conditions differ from theprevious case with respect to the patch lengths, which are now longdue to the grain size. However, the ratio of occurrence of the patchesis the same (as is the matching of resources configurations and rou-tines) and therefore specialists can enjoy the same advantages, relativeto generalists, as in case 1. With the same argument as in the first case,specialists are favored by selection. Theorem 1, which generalizes overgrain size, also covers case 2. The theorem corresponds to Hannanand Freeman's conclusions for these two cases.

3.3 High Variability and Similar Patches

In case 3, variability is high, patches are similar, and grain size isfine. Both kinds of patches come in the same ratio, so each type ofpatch occurs half of the time (like Figure 1). The specialists' routinesand a resource configuration match half of the time, whereas all thetime one of the resource configurations and routines of generalists match.If a population containing organizations with elaborated routines that


mismatch the available resource configurations half of the time iscompared to a population containing organizations without elabor-ated routines, but which match the available resource configura-tions all the time, then the latter has higher reliability/accountability.Reliability/accountability is assessed over a longer period prior to as-sessment containing several patches. One good patch for specialists ina row of equally long bad patches will not lift their overall reliability/accountability above that of generalists.

ASSUMPTION 3 Vx,y,t,rur2,o (Op(o,r1,r2)AVariability(high,o)ABt(t, 6) A Pop{x, t) A Er(x, i) A M{x, r t) A~i M(x, r2) A Pop(y, t) A~iEr{y, i) A M(y, rj A M(y, r2) =s> (ra(y, t) > ra(x, t))).

Assumption 3 conforms to Hannan and Freeman's idea that if theenvironmental resources match specialists only occasionally, then theywill be selected against (Hannan and Freeman, 1977, p. 950). General-ists are favored by selection, because their accountability and reliabil-ity are higher. Theorem 2 is inferred from Assumptions 2 and 3, andMeaning Postulates 1-3.

THEOREM 2 Vi, s, g, o (Op(o, elf e2) A Variability(high, 6) A Bt(t, o) APA A Speeds, t) A Gen(g, t) => Fa(g, s, ó)).

Case 4 features high variability and similar patches, like in case 3,but differs for grain size, which is coarse: long bad patches will affectthe reliability and accountability of specialists more strongly than theshort bad patches in the fine grained period (case 3), therefore thesame conclusion - also contained in Theorem 2 - holds aforteriori.

3.4 Dissimilar Patches

The next four cases are: (5) low variability, fine grain, and dissimilarpatches; (6) low variability, coarse grain, and dissimilar patches;(7) high variability, fine grain, and dissimilar patches; (8) high vari-ability, coarse grain, and dissimilar patches. All four cases can besummarized by observing that the routines of both specialists andgeneralists match one resource configuration in an observation period,and do so for the same duration.

216 J. BRUGGEMAN

When grain size increases from fine to coarse, and resource con-figurations are dissimilar, then both specialists and generalists have toendure increasingly longer patches which mismatch their routines.They have to wait longer until they can replenish their resources ina subsequent good patch. Because both organizational forms areaffected by this state of affairs, and roughly equally badly, the relativeoutcome does not change. In general, populations containing organi-zations with elaborated routines have higher reliability and account-ability than populations containing organizations with non-elaboratedroutines, if the routines of the organizations of both populationsmatch the available resource configurations equally long. Note thatthis assumption is weaker than Assumption 1, wherein the populationwithout elaborated routines has the advantage of matching both re-source configurations. The fact that in Assumption 4 both popula-tions mismatch one resource configuration is expressed as follows.In the observation period, one of the two resource configurations, r3,is not equal to any of the resource configurations that match thepopulations x and y, so —i {rl = r3) and ~n (r2 = r3). Both populationsonly match the other occurring resource configuration, ru althoughpopulation y would have matched resource configuration r2 if it hadoccurred.

ASSUMPTION 4 Vx,y,t,ri,r2,r3,o (Op(o,rur3)A—\ (r1=r3)A—\(r2 = r3) A Bt(t, o) A Pop(x, t) AEr(x, t) A M(x, r t ) A~i M(x, r2) APop(y, t) A-i Er(y, t) A M(y, rj A M(y, r2) => (ra(x, t) > ra(y, t))).

Theorem 3 now follows, capturing cases 5-8: specialists are favoredby selection. Theorem 3 is derived from Assumptions 2, 4, and Mean-ing Postulates 1-3.

THEOREM 3 V/,s,g,r3yo (Op(o,eltr3) A —i (e^=r3) A~i (e2=r3) ABt(t,o) ASpeeds, t) A Gen(g, t) A PA=> Fa(s, g, o)).

Theorem 3 correspond to Hannan and Freeman's conclusions forcases 5-7, but not for case 8. Coarse grained, dissimilar, and highvariability periods (case 8) could in principle make it possible for twokinds of specialists to exist simultaneously, each of them specialized toone of the two types of patches (Hannan and Freeman, 1977, p. 953).Such simultaneously occurring specialist organizations are interpreted


by Hannan and Freeman as holding companies that contain severalspecialists. Holding companies are "super" generalists with elaboratedroutines - due to their specialists - that match several different re-source configurations. Obviously, such holding companies may flourishin the real world, but they violate the principle of allocation of thetheory: the capacity of holding companies is higher than of generalistsand of specialists which fulfill the principle, so holding companies can-not be incorporated into the theory as it stands. Note that holdingswould probably also fare better in high variability, coarse grained, simi-lar environments, and perhaps fare better in all high variability environ-ments, although this is not in line with Hannan and Freeman's writings.

3.5 Stable Environment

Case 9 is a stable environment. Instead of two different resource con-figurations, as in the previous cases, there is only one resourceconfiguration which lasts the entire observation period; this fact isformalized by putting one resource configuration at both resourceconfiguration slots in the observation period predicate: Opip,rur^).A population whose organizations have elaborated routines thatmatch an available resource configuration all the time has higher ac-countability and reliability than a population whose organizationshave non-elaborated routines of which one set of routines matches theresource configuration all the time.

ASSUMPTION 5 Vx,y,t,r1,r2,o (Op(p,rurl)ABt(t,o)APop(x,t)AEr(x, t) A M{x, rx) A ~i M(x, r2) A Pop{y, t) A ~i Eriy, t) A M(y, r t) A

, r2) =>(ra(x, i) > raiy, *))).

Theorem 4 is deduced from Assumptions 2 and 5, and MeaningPostulates 1, 2, and 3. Specialists dominate in a stable environment.

THEOREM 4 Vt,s,g,o(Op(o,e1,e1)ABt(t,ó)ASpecis,t)AGenig,t)A

4 DISCUSSION AND CONCLUSIONS

The niche width theory has now been formalized into first-order logic.The original theory was ambiguous, and a mathematical model im-ported from biology mismatched the theory. Despite these difficulties,

218 J. BRUGGEMAN

eight out of nine of Hannan and Freeman's conclusions could beexplained in a fairly straightforward manner in the logical representa-tion of the theory. The only exception is the prediction for an envi-ronmental sequence that has coarse grain, high variability, anddissimilarity of resource configurations (case 8). In this one case myconclusion differs from Hannan and Freeman's because their conclu-sion contradicts a basic tenet of the theory, the principle of allocation.

Surprisingly, the concept of grain size is never needed to derive atheorem. The formal representation holds for any grain size betweenfine and coarse. When comparing a specialist to a generalist popula-tion, grain size does not alter the relative outcome of selection forthese two populations. Therefore the formal representation can bestated more parsimoniously by discarding the concept grain size.Grain size is important though, because long bad patches have astronger impact on a population than short bad patches. In badpatches which are very long relative to the lifetime of organizations, apopulation may become extinct. Extinction is still outside the domainof the niche width theory; it would require a more complex formaliza-tion than the one presented here.

Three underlying assumptions of the theory have become explicit.First, the theory is about periods in which there are only two kinds ofpatches (or a stable period with one patch) and these two kinds ofpatches must obey a restriction imposed by the concept of variability:the variance of each type of patch must be fairly low, or else a popula-tion would become extinct in one incidental very long "bad" patch.Second, the theory is about those specialists and generalists whichmatch at least one of the two resource configurations in an observa-tion period; in low variability conditions, both organizational formsmust match the relatively long patches. Third, generalists cannot makeuse of both of their alternative (sets of) routines at the same time. Theprinciple of allocation, which limits the comparison of organizationsto those of similar capacity, and these three assumptions taken to-gether clarify the constraints on the domain of the theory.

The logical formalization was kept relatively simple by concentrat-ing on those assumptions necessary in the derivations of the theorems.Hannan and Freeman did not provide sufficient explanatory supportfor their conclusions, so I had to state explicit assumptions aboutmagnitudes of combined effects, here conceptualized as match versus


mismatch, and elaboratedness versus non-elaboratedness, of routines.Necessary for the conclusions are assumptions like the ones statedabove, or similar ones of equal "strength" about the magnitudes ofcombined effects. But there are other ways of deriving the con-clusions.10 My assumptions 1, 3, 4, and 5 are in line with the originaltheory, but they are not stated literally there. These assumptions arestatements about longer parts of an observation period, i.e., longerthan one patch. Future research may focus in detail on individualpatches to assess the magnitudes of the positive and negative effectsfor various patch lengths and organizational forms. Much about themagnitudes of these effects is yet unknown. Future research may re-fine the theory about the internal structure of organizations towardsdegrees of elaboratedness of routines and investigate the match ofroutines and resource configurations, or may come up with a differentconceptualization where flexibility and learning play a role.

Hannan and Freeman's niche width theory has been given an ex-plicit argument formalized in logic. The meanings of basic notionssuch as variability, dissimilarity, and the principle of allocation havebeen elucidated, as well as the domain of the theory. Some informa-tion was obtained from another part of organizational ecology:Assumption 2, for instance, which establishes a link between account-ability/reliability and environmental selection. Last but not least, byformal logic and theorem proving software, the inferencing within,and consistency of the theory have been more thoroughly examinedthan is commonly done in sociology. Perhaps in the future, logic andtheorem provers for theory construction will become as widespread asstatistics and statistical software already are for data analysis.

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10 Péli showed that the conclusions can also be derived from a different set of assump-tions (Péli, 1997).

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Niche width theory reappraised

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