The Geometry of Organizational Adaptation: Drift, Inertia, and Viability

THE GEOMETRY OF ORGANIZATIONAL ADAPTATION: DRIFT, INERTIA, AND VIABILITY

GAËL LE MENS, MICHAEL T. HANNAN, AND LÁSZLÓ PÓLOS

1. Introduction Why do organizations generally lose their competitive edge as they get older? Recent theory and research on the dynamics of audiencesand categories in markets shed some new light on issues of organizational obsolescence. ő Inertia and environmental drift lie at the core of theoretical thinking about organizational obsolescence (Barron, West, and Hannan 1994; Hannan 1998; Carroll and Hannan 2000). The basic story holds that environments drift, but aging organizations cannot adapt well to change. As a result, fitness declines with age at some point, and viability then declines with further aging. Prior theoretical work on this issue suffers two important limitations. First, it does not specify clearly what drift means and why it affects fitness. Second, it relies on very strong—possibly unrealistic—assumptions of imprinting and inertia. According to this line of reasoning (Hannan and Freeman 1977, 1989), organizations get pre-selected at time of founding to fit to prevailing environmental conditions but have little ability to adapt to changing conditions. We develop a model that seeks to improve these two aspects of the obsolescence argument. We clarify the notion of drift by building on new thinking about fitness, rooted in a model of what makes an offer appealing to an audience. And we relax the strong assumption about organizational inertia. Instead of assuming that organizations can never adapt their core features to changing environments, we propose that organizations do possess some adaptive capacity but growing inertial pressures degrade this capacity as organizations age.

Date: December 8, 2012. 1 THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 2

This paper develops a formal representation of this new view. It builds on recent theory and research on the role of categories in structuring markets (Hannan (2010) and Negro, Koçak, and Hsu (2010) provide reviews). This work treats categories as constructions by audiences. Audience members some- times label certain sets of producers/products and come to some agreement about what these labels mean. These meanings shape tastes and, therefore, the appeal of producers and their offerings to the audience. We emphasize the effects of variations over time in audience tastes and on responses to these changes by producer organizations. In particular we define environmental drift in terms of changes in the meanings that audience members associate with category labels. We argue that audience member’s tastes tend to shift over time and that aging organizations have trouble adapting their offerings to changes in tastes. This combination creates obsolescence with aging. The model of drifting tastes and producer inertia has a broad range of potential application (discussed in the concluding section). Nonetheless we build a detailed model for only one set of implications by narrowing the focus to organizational viability. Concentrating on a well-studied issue that has already been subject to numerous formalization attempts makes it easy to see how the new model differs from the alternatives advanced previously. In particular, we incorporate ideas about inertia into the framework relating organizational capital and fitness to viability developed by Le Mens, Hannan, and Pólos (2011). This allows us to derive some new predictions about age variations in organizational viability. The theory proposed here builds on previous attempts to unify conflict- ing theories of age dependence (Hannan 1998; Pólos and Hannan 2002, 2004; Hannan, Pólos, and Carroll 2007). This earlier work sought unification by postulating the existence of qualitatively different age periods—marked by a common age of ending of endowment and a common age of the onset of obsolescence—with distinct dynamics. We make the model more realistic by avoiding such assumptions about qualitative phases in the life course. We do not postulate a priori the existence of fixed ages of either the ending of endowment or of the onset of an obsolescence. Instead, we conceptualize and model THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 3

the underlying processes that can vary continuously over the organizational life course. The paper has three main parts. The ﬁrst builds on work on cultural resistance to organizational change to construct a model of age variations in adaptive capacity. The second part exposes the implications of this model to empirical test. The third part integrates this model in a broader theory of age dependence.

2. Paths of Organizational Change What matters for obsolescence in changing environments is the ability to keep pace with changing tastes by modifying aspects of offerings, which means adjusting architectures. A highly adaptive producer can alter aspects of its architecture quickly, which allows it to change its offering quickly and main- tain appeal to an audience whose tastes drift. So the concept of adaptive capacity plays a central role in our argument. Because we think of offerings as constrained by organizational arrangements, we begin with issues of speed of organizational change. Measuring the speed of change entails measuring elapsed time and the distance traveled. We define movements by organizations in the space of architectures (or organizational designs). This kind of effort requires attention to the geometry of adaptation, defining an architectural space and the position of a producer in the space.

Let vx(t) denote an organization’s actual position in a space of architectural 2 feature values at t; and let df : F −→ N be the distance between two positions in the space. As we explain in Section 7, distance in such a space can usefully be regarded as an edit distance (or, more generally, a transformation 0 distance). Let df (v, v ) tell the number of changes of feature values need to convert organization x’s position from v to v0. The pair of the set of positions and this distance function defines a graph space for an organizational architecture. As we explain below, the graph distance considers distance “as the crow files”—it ignores the cultural typography of the space. Just as the shortest THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 4 route in physical space might pass through a mountain, a sequence of organizational changes might encounter an analogous roughness (specifically cultural resistance). Actual paths of change might not take the shortest distance (due perhaps to the social typography of the space). A path is a sequence of one-step moves that begins in v and ends in v0. In general more than one path connects any two positions. Some paths involve cycles, e.g., a move from A to B to A to B to C. We are interested in the paths that have the cycles eliminated.

Notation (Paths of Change). Let G(v, v0) denote the set of acyclic paths connecting the positions v and v. The component paths, p ∈ G(v, v0), are 0 sequences of one-step transitions: p = hv, v1i, hv1, v2i,... hvk, v i. The length of the path p, in notation |p|, is the number of one-step transitions. Furthermore let δx(p, t) be a real-valued non-negative random variable that tells the length of time it takes the focal organization, x, to transit the path p beginning at time t.

Our argument requires long-distance jumps to be ruled out, that organizations can change the value of only one feature at a time. We impose this constraint formally in terms of the time it takes to transit a multistep path. In these terms, the constraint that only one feature can be changed at a time can be stated as follows.

Auxiliary Assumption 1. Organizations cannot make long-distance jumps in architecture space: the expected duration of the transit of a path is simply the sum of the durations of its one-step transitions.

A p, t, v , v , x [(p ∈ G(v , v )) → δ (p, t) = P δ (p , t)]. 1 2 1 2 x hv3,v4i∈p x v3,v4

3. Age and Resistance to Architectural Change A key postulate of Hannan and Freeman’s (1984) theory of structural inertia holds that structural reproducibility increases with age. We now situate this claim in contemporary theoretical arguments. We follow Hannan et al. (2007) (hereafter HPC) in considering the possibility that the members of the internal audience might resist certain transitions that are at odds with the organizational culture. They develop the argument for cascades of induced changes: THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 5

it normally takes longer for a unit to eliminate an induced violation of an architectural code when cultural resistance is stronger. We broaden the notion here to apply to any kind of attempt at changing any feature of organization. (In the sections that follow we narrow the focus to attempts at moving from one architectural conﬁguration to another.)

Cultural Resistance to Change. The core intuition is that cultural resistance slows change. A potentially valuable way to formalize this notion em- ploys the language of defaults or taken for granteds. Following prior work, we use a language of modalities to formalize these slippery notions. The key modalities for the present argument concern perception and taken-for- grantedness. We first introduce notations for these two domain-general concepts and specify how they relate to each other. Then we make apply them to the specific case of structural configurations. We introduce modal operators that are defined for an (arbitrary) audience member y and a sentence (formula) ϕ (Polós, Hannan, and Hsu 2010). We use the following notation for these operators:

• p y ϕ stands for “The focal agent y perceives that ϕ is the case.”

• d y ϕ stands for “The focal agent y takes for granted that ϕ is the case.”

Throughout this section we use ϕx(t) to refer some fact (or proposition) about the organization x at time t. We assume that the time structure of perceptions of organizational arrangements is granular (and the granular structure likely depends on the fact involved). Prior work with these modalities has hinted at the process of default formation but has not introduced a specific model. Here we try to fill this gap. Our reading of the intuitions behind the prior work is that an agent likely treats some “fact” as a default if over a period she repeatedly perceives that the fact remains true and does not perceive otherwise. We define exposure in two steps. First, we introduce a notation for the beginning of a temporal interval ending at t during which an audience member

has not perceived at any (granular) point that the fact ϕ is false. Let ty denote the time of entry of the member y into the focal organization. Then the start THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 6

of the interval of interest can be deﬁned as follows:  sup{u | (ty ≤ u < t) ∧ p y ¬ϕx(u)} if the set is nonempty; υyt = 0 otherwise.

Next we define exposure over the interval from υyt to t. Here we face a choice. Should we define exposure simply in terms of the absence of contrary perceptions or in terms of repeated positive perceptions? Perhaps one initial perception of a fact with subsequent update is enough to drive the formation of defaults. We suspect that it generally takes more than the absence of updates, that repeated positive perceptions (with no negative updates) is more likely to lead to a default than the other scenario. So we define the period of exposure along these lines.

Deﬁnition 1. An agent’s exposure to an organizational fact at a given time is the total duration of the granules of positive perceptions of the fact since the last contrary perception or from the beginning of the agent’s membership if there was no negative perception.

P 0 e (ϕ, x, t) = 0 ϕ (t ) . y t ∈[υyt,t] p y x In this deﬁnition and what follows we use the standard notion [[·]] to represent the semantic value of a formula with 1 indicating “true” and 0 indicating “false.” So p y ϕx(t) takes the value of 1 if the agent y perceives at time point t that the proposition ϕ is true of x. So the summation in the foregoing equation gives the total duration of positive perceptions.

This deﬁnition sets experience with an organizational fact to zero at the beginning of a member’s involvement in the organization and at all subsequent times at which the members perceives that the factual situation does not hold. The maximum possible value equals the length of a member’s tenure in the organization in the case of continual perception of the fact. It turns out to be helpful in building a model to express the default formation process in terms of the probability that an agent has a default about an organizational fact at a time point. We think of this probability as the degree to which an agent takes a fact for granted. At its maximum of one, the agent treats the fact fully as a default; at its minimum of zero, the agent THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 7

treats the fact as purely accidental and changeable. Intermediate values of the probability indicate positions that shade from one extreme to another, in other words, degree of taken for grantedness. The basic intuition about taken-for-grantedness is that the probability of forming a default about some organizational arrangement increases monotonically with experience.

Postulate 1. The probability of forming a default about an organizational fact increases with exposure.

0 0 0 N x, y ∀t, t [(ey(ϕ, x, t) > ey(ϕ, x, t )) → Pr{ d y ϕx(t)} > Pr{ d y ϕx(t )}]. We argue that defaults, durable expectations, are treated more seriously than facts. Attempts at changing facts more likely sparks cultural resistance when they are taken for granted than when they are thought to be accidental and transitory. Following HPC, we formulate this argument in terms of culturally based resistance. Such resistance varies in intensity from mild grum- bling to unwillingness to cooperate in implementing change, to active eﬀorts to thwart the change, to leaving the organization. Let ry(ϕ, x, t) be a non- negative real-valued random variable that tells the intensity with which the agent y will resist any attempt to change ϕ at time t in organization x.

Postulate 2. Cultural resistance by agent to an attempt to change an organizational fact is stronger when the fact is taken for granted.

0 0 0 N x, y ∀t, t d y ϕx(t) ∧ ¬ d y ϕx(t ) → E{ry(ϕ, x, t)} > E{ry(ϕ, x, t )} . Obviously it follows that longer exposure leads to stronger resistance to change. We regard this implication as a formalization of Selznick’s (1957) claim that even practices chosen on purely technical grounds acquire a moral standing if kept in place long enough.

Proposition 1 (Selznick’s law). Resistance to an organizational change is an increasing function of exposure.

0 0 0 P x, y ∀t, t ,, [(ey(ϕ, x, t) > ey(ϕ, x, t )) → E{ry(ϕ, x, t)} > E{ry(ϕ, x, t )}]. The nonmonotonic quantiﬁer P expresses the consequence of an argument (a rule chain) that builds partly or wholly on generic (“normally” quantiﬁed) sentences. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 8

Proof. By the law of total probability,

E{ry(ϕ, x, t)} = ry(ϕ, x, t | d y ϕx(t)) · Pr{ d y ϕx(t)}

+ ry(ϕ, x, t | ¬ d y ϕx(t)) · (1 − Pr{ d y ϕx(t)}. From Postulate 2 we have that the level of resistance given a default exceeds that in the absence of a default, and from Postulate 1 that the probability that a fact is a default is higher at higher levels of exposure.

Aggregating to the Organizational Level. The distribution of exposure does not necessarily track organizational age closely. Turnover in personnel lowers exposure as we deﬁne it. So does change in (perceptions of) the facts. Indeed, if all the relevant facts change in an organization, it is tantamount to a new organization according to the view we advance here, whatever its age. Nonetheless, old organizations generality have members with longer tenure in the internal audience and more facts with long duration. [should we cite some things here?] We characterize the main case here by introducing an auxiliary assumption that relates age with the distribution of exposure to the fact ϕ. We use

F (z) = |{y | e (ϕ, x, t) ≤ z}/N , Eϕx(t) y x where Nx denotes the size of the internal audience in organization x. We refer to strict stochastic orderings using the standard notion:

F ≺:F ↔ ∀z [Pr{F > z} ≤ F > z}] Eϕx(t) Eϕx(t0) Eϕx(t) Eϕx(t0) ∧ ∃ z [Pr{F > z} < Pr{F > z}]. Eϕx(t) Eϕx(t0)

Auxiliary Assumption 2. The distribution of exposure to a fact among internal audience members at a later organizational age is strictly greater than at an earlier age.

A x ∀t, t0 [(a (t) > a (t0)) → F ≺ F ]. x x Eϕx(t0) Eϕx(t)

Above we deﬁne resistance at the individual level (as a binary variable).

Next to that we deﬁne resistance at the audience level. Let Rx(ϕ, t) be a random variable that records the average strength of cultural resistance in the THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 9

internal audience of producer x to an attempt to change ϕ at time t. P Rx(ϕ, t) = yry(ϕ, x, t)/Nx. Our assumption about the relation of organizational age and the distribution of exposure allows us to connect organizational age with resistance.

Proposition 2. The average strength of cultural resistance to a change in an organizational fact presumably increases with organizational age.

0 0 0 P x ∀t, t [(ax(t) > ax(t )) → E{Rx(ϕ, t)} > E{Rx(ϕ, t )}].

Proof. Auxiliary Assumption 2 relates organizational age to the distribution of exposure. As this distribution shifts to the left, audience members have higher or equal expected levels of exposure. Postulate 1 states that the probability of treating a fact as a default increases monotonically with exposure. With the change in the distribution of exposure, this probability either increases or remains the same over the audience members and consequently the default becomes more widespread. Postulate 2 holds that cultural resistance is stronger for defaults. So the strength of resistance is either higher (for at least one member) or the same at the later age. Aggregation over the members of the audience completes the chain. Organizational Age and the Speed of Structural Adaptation. The next link in the argument ties cultural resistance to the speed of architectural change. We adopt a simple modeling strategy. Instead of trying to characterize the durations of different changes at different ages, we compare the same change path at different ages. This lets us avoid having to model the choice of change path and keeps the focus squarely on the issue of resistance. Our core intuition is that cultural resistance lengthens the duration of a step in a change process.

Postulate 3. The strength of cultural resistance to an edit increases the expected duration of a one-step move.

0 0 0 0 N x ∀p, t, t , v, v [(p ∈ G(v, v )) ∧ (|p| = 1) ∧ (Rx(ϕ, t) > Rx(ϕ, t )) 0 → E{δx(p, t)} > E{δx(p, t )}]. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 10

This postulate and the foregoing argument have obvious implications for the speed of architectural change.

Proposition 3. The expected duration of the transit of a path is higher when it is undertaken at older age.

0 0 0 P x ∀p, t, t [(ax(t) > ax(t )) → E{δx(p, t)} > E{δx(p, t )}].

Proof. According to the assumption that long-distance changes are ruled out (Auxiliary assumption 1) and the fact that the same path is being compared at diﬀerent ages and therefore has same crow-ﬂy distance, the durations of the transit of the path at each age is a summation of the same number of durations. According to the rule chain behind Proposition 2, the expected resistance at each step is higher at the older age.

4. Architectures and Offerings 4.1. The challenge. In this paper a deductive (and predictive) theory is developed concerning the relationship between organizational age and adaptive capacity. To expose this theory to empirical testing it is worth assuming that organizations typically realize their potential, i.e. they adapt as speedily as their adaptive capacity allows them to. With this simplification it becomes sufficient to measure how fast the organizations actually adapt. But the term "adaptation" is ambiguous, it can refer to the speed by which the organization anticipate, and adjust its structure and culture to, the expectations of the internal and external audience. It can also refer to the changes in the portfolio of offerings of products, and services the organization introduces to shape, or to follow the drifting tastes of the (external) audience. Even though the theory developed relies on organizational considerations and therefore is about the velocity of organizational changes, the existing empirical tests (almost) exclusively rely on the adaptation of offering portfolios. This is because it is hard to tell when internal change processes start or finish. For example the detrimental process effects of reorganizations may linger around long after the change project is "officially" completed. Newly introduced processes struggle to gain taken-for-grantness, the feeling of se- curity undermined by the change might lower loyalty, extinguish citizenship, THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 11

decrease engagement for a considerable period of time. Combined with the fact that organizations often start new reorganizations before they fully recovered from the hangover of the previous reorganization. On the other hand product launches, augmentation of the services portfolio with new oﬀerings, etc. are by nature public events, and typically come with a sharply deﬁned date.

4.2. Substantive considerations. Due to the mismatch between the theoretical developments, and the empirical tests further explicit considerations are required to rectify this discrepancy. • To produce a particular offering not all organizational structures are suitable, and some can only be produces (for technical, or more in- terestingly for cultural reasons) with a small number of organizational arrangements. Therefore, if the organizations intends to adapt externally, and the current arrangement is not suitable for the new offering, adjustments of the arrangement becomes necessary, i.e. the external adaptation can only be achieved when these adjustment is completed. • Of course there are multiple suitable arrangements, and one of them might prepare the organization for future adaptive steps better than others. However this might not be known to the decision makers at the time they choose one of the alternatives. The simplifying assumption we offer here is that whatever strategy the organizations use in select- ing the new, suitable arrangement is determined by institutionalized organizational factors that remain constant throughout (long periods of) organizations life. For example They might follow a satisficing strategy, and chose the nearest suitable arrangement. • Whatever arrangement is suitable is not externally given, but is also defined by organizational factors. We posit that organizations typically assess how radically the novel element of their offering departs from their existing portfolio of offers, and find organizational arrangement suitable if they require similarly radical alterations. This might be interpreted as a result of the consideration that the more radically new the augmentation of the offer portfolio is the more radical organizational change it legitimizes, and managers seek to maximize legitimacy to gain managerial freedom. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 12

4.3. The formal model. To capture the implications of these considerations we offer a formal model. In what follows we assume that both the organizational arrangements and the product offering portfolios are points in a respec- tive feature value space, and both updating the portfolio, and reorganizations are moves between points in these spaces. What we described above in terms alternative suitable arrangements for a portfolio augmentation and alternative offerings that can be produced by the same organizational arrangement means that even though these two feature value spaces are connected by suitability conditions these conditions can not be modeled by functions: these are merely binary relations allowing for some contingencies. Let Σ be a feature value space for organization x while Ω is the feature value space for the offer portfolios of x. We assume that there is distance defined

between any two positions in both feature value spaces, δΣ and δΩ respectively. Suppose furthermore that R is a binary relation on Σ, i.e. R ⊆ Σ × Σ. For

the sake of convenience (x1, x2) ∈ R will be written in the more customary

format (x1Rx2), and will be interpreted as x2 is directly accessible from x1.

Similarly S ⊆ Ω × Ω, and (o1, o2) ∈ S if and only if (o1So2). Now the scene is set to spell out the conditions for the suitability relation.

Deﬁnition 2 (The bisimilarity relation between feature value spaces). Let Ξ be the union of two relations one between Σ and Ω, and one between the binary relations on Σ and the binary relations on Ω. We call Ξ a bisimilarity

relation between (Σ, δΣ, R) and (Ω, δΩ, S) if the following conditions are met: (1) RΞS

(2) ∀x1, x2, o1[x1Rx2 ∧ x1Ξo1 → ∃o2[o1So2 ∧ x2Ξo2]] −1 −1 (3) ∀x1, o1, o2[o1So2 ∧ o1Ξ x1 → ∃x2[x1Rx2 ∧ o2Ξ x2]] −1 (4) ∀x1, x2, o1[δΣ(x1, x2) = k → ∀02[(01So2 ∧ o2Ξ x2) → δΩ(o1, o2) = k]] It there exists a bisimilarity relation between

5. Age and Adaptive Capacity Our basic intuition is that producers face limits on the speed of change of architectural features. Some can make extensive changes rapidly, and, as just THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 13

noted, this ability likely changes over time. Adaptive capacity (over an interval) can be defined as a radius in the feature space that bounds the change over the interval. The idea is that feature values can be altered freely; producers can move in the architectural space. But the total distance of these moves cannot exceed the radius. We build this construction by defining an upper bound on the possible distance between the characteristics of an offering at the beginning and end of the interval. We define this notion explicitly by treating adaptive capacity as a time- varying state variable, a real-valued, positive function that records the speed with which a producer can reshape the features of its offering: ρ : x × t → + R , a function that takes x and t and returns ρx(t). The following meaning postulate provides an inductive definition.

Meaning Postulate 1. A producer’s adaptive capacity creates a limit on the speed of change in its oﬀering.

0 0 t0 0 N x ∀t, t [(τx ≤ t ≤ t ) → d(oxt, oxt ) ≤ t ρx(s)ds ]. ´ The core claim of theories of obsolescence is that adaptive capacity declines with age.

Theorem 1. An organization’s adaptive capacity declines with age.

0 0 0 P x ∀t, t [(ax(t) > ax(t ) → ρx(t) < ρx(t )].

Proof.

6. Age and Innovation: An Empirical Analysis We now test the major implication of this argument, Theorem 1. In addition to conducting this test, we also explore the substantive value of the geometric approach we propose. We reanalyze data used in an exemplary study of organizational aging and innovation (Sørensen and Stuart 2000).

7. The Geometry of Audience Tastes and Producer Offerings Having shown that the geometric-based argument about age and adaptive capacity sharpens analysis, we now build a formal theory that builds a deeper THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 14

structure for relating adaptive potential to issues pertaining to the evolution of tastes in an audience and organizational obsolescence. We begin by introducing formalisms to describe the similarity of offerings and schemas and the similarity of schemas at different points in time. Then we use the new formalisms to define drift in tastes and organizational adaptation. And finally, we draw some implications from these basic assumptions for the evolution of organizational capital. This allows us to tie the argument about inertia and drift to age-variation in organizational viability. In developing the new theory, we build on the framework delineated by HPC. We consider a generic producer that operates in an (unspecified) category and tries to capture resources controlled by members of the audience for that category. The relevant audience in this part of the paper is external to the producer. It consists of actual and potential customers, actual and potential organizational members, and more generally, any individual, organization, or governmental agency that controls resources useful to the organization and also takes an interest in the category.1

Schemas and Offerings. Categorization processes associate meanings with labels. In the line of work we follow, the term category refers a label with consensual meaning. For instance, if “Sushi restaurant” is a category label (to an American audience), then members of this audience largely agree on what it means for a producer to qualify as a typical instance. For this example, the relevant features likely include various aspects of the menu offerings and mode of service, e.g., raw fish prepared in sight of the clientele by a skilled artisan and served with specific kinds of rice served on lacquered plates in min- imalist “Japanese-style” room (Carroll and Wheaton 2009). To give another

1We make two major simplifications to keep the argument and the notation as simple as possible. First we assume that each producer specializes in the space of categories, that it bears only the focal category label. Among other advantages, this restriction lets us avoid the complicated matter of aggregating fit in multiple categories to come up with an overall measure of fitness. Second, we assume that each producer specializes in resource space, that it operates at a single (unspecified) social position by targeting a relatively homogeneous sub-audience. (We do not represent positions formally.) Following standard sociological arguments, we assume that the audience members at a social position have similar tastes. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 15

example the relevant features of “beer,” likely include color, transparency, several taste dimensions, country of origin, type of producer (e.g., large brewery, microbrewery) and so forth (Carroll and Swaminathan 2000). The cognitive sciences generally use the concept of schema to represent this kind of cognitive-cultural model of meaning. The term schema refers to a cognitive representation of a label on several dimensions (those that an

audience member regards as relevant to the meaning of a label). Let fn =

{f1, f2, . . . fn} be the set of n relevant features; and let Γ = r1 × · · · × rn denote the set of n-tuples of the values of the relevant features. An agent’s schema for a label can be represented as a nonempty, closed, and bounded subset of Γ that contains exactly the patterns (n-tuples) of feature values that conform to her meaning. Let C(Γ) denote a set of nonempty subsets of Γ, and let p denote the set of audience members at the focal social position, and t be a set of time points. In formal terms, a schema for a label maps pairs of an audience member and a time point to a nonempty subset of the Cartesian product of the ranges of the set of relevant features: σ : p × t −→ C(Γ) 6= ∅. This function takes a triplet of a label, an audience member y and a time point t, and returns σ(y, t) = syt. Note that σ(y, t) is a subset—not an element— of Γ, because a schema need not be a single vector of feature values. Social schemas often allow several combinations. Producers frequently make multiple offers in a given market (e.g., automobile models) or a menu of options (e.g., degree programs in a university), which makes the offer a set. However, the models get very complex if we allow this kind of realism. In the interest of clarity, we restrict our attention producers making single offers, e.g., one car model or one curriculum.

Notation (Schemas and Oﬀers). We use an informal notation for schemas in the interest of simplicity. Whenever we refer to a set syt, we intend that this be understood to mean that the agent y associates a schema with the focal label at the time point t and σ(y, t) = syt. We denote the (schematized) oﬀer

of the producer x at time t as oxt. As we noted above, we treat the schema- relevant oﬀer of a producer as a point in the n-dimensional space supporting

the schema of the audience member: oxt ∈ Γ. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 16

Research in cognitive psychology and sociology reveal that agents often perceive objects as varying in the degree to which they ﬁt category labels (Hannan 2010). A useful way of representing partial memberships formally treats labels and categories as fuzzy sets. According to this interpretation, audience members (implicitly) assign grades of membership in labels to producers and

their products. Such a grade-of-membership function, denoted by µσ(x, y, t), tells how well the producer x fist the schema σ for the focal label from the perspective of an audience member y at a time point t. Fit increases with the proportion of the object’s feature values (as perceived by the audience member) that lies within the ranges given by the schema. (Below we provide a precise definition.) What does it mean for an offering to fit a schema? A schema points to a set of acceptable n-tuples of feature values. High typicality means having values in this acceptable set. An offering whose features meet all the constraints of a schema has a very high grade of membership, whereas an offering that meets them only partially has a somewhat diminished one.

Distance in a Sociocultural Space. The mathematical description of the structure of the set of schemas is not yet rich enough for modeling the fit of offerings to schemas and of changes in schemas over time. We must introduce a geometric interpretation: a socio-cultural space and a distance measure. The dimensions of the space are the features that audience members find relevant for the category: an N-dimensional space of feature values. Choice of a measure of distance depends on the type of features. At one extreme, the features might be real-valued, e.g., engine displacement, horse- power, miles per gallon, and so forth. hen it is natural to define the distance between offers and schemas as Euclidean distance. Alternatively, the relevant features are qualitative, as in the example of “Sushi restaurant.” In some cases the relevant features are binary, as in the distinction between “public” and “private” universities. In other cases, qualitative features have multiple ranges, e.g., a restaurant might serve any combination of breakfast, lunch, and dinner or a bank might offer any subset of a list of services. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 17

If the relevant feature are real-valued, then we set the distance measure to be Euclidean distance. But what about the qualitative case? Schemas for qualitative features change by elaboration of possible values of a feature (e.g., changes in technology often provide new possibilities), deletion of features that are no longer relevant, and addition of features. Offers can be regarded as (n-long) strings of discrete feature values, and schemas can be regarded as sets of such strings. A general conception of distance that applies to comparisons of such strings is called edit distance. Perhaps the simplest such measure is the Hamming distance, which counts the number of edits (replacements) needed to transform one string to the other. Unfortunately this simple measure is defined only for strings of the same length. We need to allow the elaboration or shrinking of the set of relevant features. A more general edit distance that does the job is Levenshtein distance. This measure counts the number of replacements, deletions, and insertions need to transform one string to another. This satisfies the metric proper- ties of symmetry and satisfies the triangle inequality (which is important in our derivations). Because this measure is well defined as the length of the strings increases without bound, it also satisfies the unboundedness condition needed for characterizing drift. Therefore interpreting distance for qualitative schemas and offerings as Levenshtein edit distance allows the qualitative and quantitative cases to be given a uniform treatment.2

Distance between offerings and schemas. Let o denote an offering and s denote a schema. To clarify what we mean by the distance between o and s, we utilize the standard definition of a distance between a point and a set as the smallest distance between the point and any element of the set: −→ d (o, s) ≡ inf d(o, s), s∈s where d(·) denotes one of the chosen distance metrics, either Euclidean distance or Levenshtein distance in this paper.

Distance between schemas. Because a schema is a set of n-tuples of feature values and not just one particular n-tuple, characterizing the distance between

2We defer treatment of the more complicated mixed qualitative and quantitative case. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 18

schemas requires a generalized measure of distance. The directed Hausdorff −→ distance, D, generalizes the distance between points to a distance between sets in a metric space (Burago, Burago, and Ivanov 2001). In particular, the directed Hausdorff distance from one set to another is a function between + 0 C(Γ) × C(Γ) and R such that for all s, s ∈ C(Γ): −→ −→ D(s, s0) = sup{ d (s, s0)}. s∈s This formula tells that the directed distance identifies the point in s that lies furthest from s0 and computes the distance of this point from the nearest point in s0.3 Because this measure is asymmetric, it is not a metric.

Distance, Similarity, and Grades of Membership. Having a high grade of membership in an agent’s schema means having (perceived) feature values that are highly similar to the schema. Similarity is obviously inversely related to distance. But how? The answer to this question depends on how audience members aggregate information about (perceived) fit and lack of fit with their schemas. For example, they might give equal weight to each feature in judging fit, as assumed by Hsu, Hannan, and Pólos (2011), or they might give more weight to certain diagnostic features, and so forth. A powerful result from cognitive psychology lets us sidestep the aggregation issue. Recall that the offer is a point and the distance between the offer and the schema is the distance from the nearest point in the schema. Therefore, we can draw on Shepard’s (1987) “universal law” that posits a negative exponential relationship between the perceived similarity of a pair of stimuli (objects here)

and their distance in the psychological space. Let simy(oxt, s) denote a real- valued function that gives y’s perception of the similarity of oxt and syt. This function maps from triplets of audience members, oﬀerings, and time points to [0,1].

Postulate 4. The (perceived) similarity of an oﬀering and a schema is a negative exponential function of the distance between them. −→ −k d (oxt,syt) N x, y ∀t [(k > 0) ∧ (σ(y, t) = syt) → simy(oxt, syt) = e ]. −→ 3If s is included in s0 (in the sense of set inclusion), D(s, s0) = 0, even if the inclusion is strict. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 19

Because similarity and distance are both real-valued functions (in the Eu- clidean case) or functions to the natural numbers (in the case of edit distance), the foregoing formula is satisfied by a unique value of k. In the formulas that follow, k refers to this parameter. Existing theory does not provide any details about the determinants of grades of membership in the meaning of a label. Instead, it refers to fit to the audience member’s schema. Here we sharpen this notion by stating it in the language of similarity as follows. Postulate 5. An offering’s grade of membership in an agent’s schema for a label equals the perceived similarity of the offering and the schema.

N x, y ∀t [(σ(y, t) = syt) → µσ(x, y, t) = simy(oxt, syt)]. The foregoing argument connects grades of membership and the distance between offering and schema. Proposition 4. An offering’s grade of membership in an agent’s schema for a label is a negative exponential function of the (perceived) distance between the offering and the schema. −→ −k d (oxt,syt) P x, y ∀t [(σ(y, t) = syt) → µσ(x, y, t) = e ]. Proof. The nonmonotonic logic we use (Pólos and Hannan 2004) preserves first-order logic’s so-called cut rule: the formulas N x[ϕ(x) → ψ(x)] and N x[ψ(x) → χ(x)] jointly imply the provisional theorem P x[ϕ(x) → χ(x)]. This follows from a cut rule applied to Postulates 4 and 5. This small argument has an important strategic implication: we need not specify how the agents aggregate their perceptions of fits over features to characterize a grade of membership. The three key concepts—distance, similarity, and typicality (grade of membership)—all concern an agent’s perceptions. Hu- mans presumably find it easier to make similarity judgments between objects and schemas than to provide explicit accounts of how they combine features in forming these judgments. However, from the researcher’s perspective it seems easier to operate in the domain of distances, because this lends itself to geometric representation. We formulate the postulates and assumptions of the foregoing theory mostly in terms of distances in the sociocultural space. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 20

Distance and Intrinsic Appeal. This construction of a distance between an offering and a schema can be used to draw implications about the intrinsic appeal of an offering to an audience member. Offerings that fit closely to the audience’s schema(s) can be said to fit their aesthetics and to have high intrinsic appeal. Fitting an agent’s schema provides benefits only when the agent regards the category positively. Hence we consider only positively valued categories, those for which having a high grade of membership makes an offer intrinsic appealing to audience members. Instead of conditioning every formula with complicated constructions that instantiate these restrictions, we state the restriction globally as an auxiliary assumption.

Auxiliary Assumption 3. All members of the audience associate schemas with the label at all time points; moreover the agents attach positive valuation to the label at all time points.

When we refer to this concept in the formulas, αey(x, t) denotes a function mapping to [0, 1] that tells the intrinsic appeal of the offering to the audience member y at time t (at the unspecified social position to which the producer P specializes); and αex(t) = y∈pαy(x, t) denotes the total intrinsic appeal at the target social position. With these background and notational considerations in hand, we can derive a relationship between the distance between an offering and a schema and the intrinsic appeal of an offering to an audience member.

Proposition 5. For positively valued categories, the intrinsic appeal of an oﬀering decreases with the (perceived) distance between the oﬀering and the agent’s schema for the label.

0 −→ −→ 0 P x, y ∀t, t [( d (oxt, syt) < d (oxt0 , syt0 )) → αey(x, t) > αey(x, t )], and

P x, y ∀t [lim −→ αy(x, t) = 0]. d (oxt,syt)→∞ e Proof. In the case of a positively valued category, a higher grade of membership yields higher intrinsic appeal, and intrinsic appeal is zero for producers/products with zero grade of membership. Proposition 4 states that grades of membership in labels are given by a negative exponential function of the THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 21

distance of the offering from the focal audience member’s schema for the label. The term in the antecedent of the first part of the proposition that tells that one offering is closer to the schema than another offering guarantees that the grade of membership in the category is higher for the closer offering. Then it follows by a cut rule that the intrinsic appeal of the closer offer is higher. The second part of the proposition follows from the continuity of the distance function. As the distance from the offer to the schema converges toward zero, the producer/offering’s grade of membership in the category also converges to zero. At the limit, the intrinsic appeal of the offering is zero by Proposition 4.

8. Drifting Tastes We have developed this geometric representation of a space of schemas and offerings to specify precisely what it means for tastes to drift. Changing tastes presumably entail changes in meanings, in schemas. If what used to be pleasing about a full-fledged “automobile” no longer pleases critics and consumers, this undoubtedly signals that the meaning of “automobile” has changed. What it took to be regarded as an acceptable instance during the time of the hegemony of the Ford Motor Company’s Model T and Model A would no longer qualify. Consumers have come to expect that an automobile possesses many features that they lacked in that earlier era. The now prevailing schema refers to many characteristics that were unknown previously. Tastes and associated schemas in this domain have changed over time. Analyses of the effect of drift and inertia must consider two clocks: one records the passage of time for the audience and the category (historical time), and the other tells the time elapsed since an organization’s founding (organizational age). Notation (Age and Historical Time). Throughout we denote the historical clock by t, and we denote the time of an organization’s founding by τx. With

this notation the age of organization x at historical time t, is given by ax(t) =

t − τx. We condition formulas as holding for time points beginning at time τx and evaluate functions and predicates at various time points t ≥ τx. Although changes in tastes might be cyclical, the main case for modeling obsolescence involves what Hannan (1998) called drift. In this section, we THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 22 examine the consequences of drifting tastes. We say that the taste of the members of an audience drifts over time when the distance between their current schemas for the label and their earlier schemas becomes arbitrarily large once enough time has elapsed.

Definition 3. A category’s meaning to an audience drifts iff the directed Hausdorff distance between the schemas of the audience members for a label at an earlier point in time and their schemas at a later point in time increases monotonically with the length of the interval separating the two time points.

0 0 −→ 0 drift ↔ ∃ d [(d > 0) ∧ N y ∀t, t [(t < t ) → D(syt, syt0 ) ≥ d(t − t)]].

Notice that the definition of drift concerns only schemas; it does not depend on any producer’s offering. Although audience tastes certainly change in response to cultural trends, demographic change, and even to changes of the offerings of other producers in the category, we do not model how and why tastes change. While such a modeling exercise would be interesting in itself, it lies beyond the scope of this paper. Therefore, we simply assume that tastes drift, without worrying about what causes the drift.

9. The Dynamics of Organizational Viability We build on a model of organizational evolution proposed by Le Mens et al. (2011) (hereafter LHP), itself constructed on the basis of a theoretical framework summarized in HPC. According to this model, the hazard of failure of a producer is driven by its stock of organizational capital (see also Levinthal (1991)). An ample stock of (financial and social) resources buffers the organization from failure; but a small stock provides little buffer causing failure hazards to be elevated. Given this negative monotonic relation between organizational capital and viability, the hazard falls (rises) when the producer experiences a net inflow (outflow) of resources. Whether the hazard of failure increases or decreases with aging therefore depends on the sign of the net flow of resources from the audience. The key construct that characterizes an organization’s ability to garner resources is fitness. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 23

Fitness depends on the actual appeal of a producer’s offering, which itself depends on the intrinsic appeal of the offering and on the producer’s engagement with the relevant audience. In what follows, we reprise the prior contributions by providing the formal definitions of these concepts and sketching how they relate to each other. Where necessary, we complement the existing theoretical framework by sharpening existing definitions. LHP did not explicate the dynamics of intrinsic appeal. Instead, they ex- plored the influence of variations in engagement on the dynamics of viability. With the framework developed in the previous sections, we can formulate a refined model that also takes into account the variations of the intrinsic appeal that result from the combined effect of the evolution of audience tastes and limits on organizational adaptation.

Intrinsic Appeal, Engagement and Actual Appeal. An offering has intrinsic appeal to an audience if it fits their tastes. However, even those offerings that do fit tastes (schemas) will generally not gain actual appeal unless their producers engage the audience and make their offerings available in an appropriate way. The work on which we build relies on a relatively weak relationship con- nection intrinsic appeal and engagement to actual appeal. Specifically LHP assumed simply that an offering gains appeal when its producer increases engagement. Here we introduce a more specific relationship that will support our analysis. We propose that (1) the actual appeal of an offer equals a por- tion of its intrinsic appeal; (2) the ratio of the actual appeal to intrinsic appeal normally increases with the producer’s engagement (with respect to that offering), and (3) a sufficiently high level/quality of engagement makes the actual appeal of an offering arbitrarily close to its intrinsic appeal.

We represent this imagery as follows. Let x(t) denote a non-negative real- valued function that records the level/quality of a producer’s engagement with respect to its current oﬀering at the target social position and g denote a function that is an increasing mapping from the non-negative real numbers into the interval [0, 1]. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 24

Meaning Postulate 2. Increased engagement raises the ratio of actual to intrinsic appeal. αy(x, t) N x, y ∀t [(τx ≤ t) ∧ (αey(x, t) 6= 0) → = g(x(t) ∧ ( lim g() = 1)]. αey(x, t) →∞ Note that this postulate imposes the restriction that low intrinsic appeal guarantees low actual appeal, no matter the engagement.

Actual Appeal, Organizational Fitness, and the Hazard of Failure. As we move to analyze ﬁtness, we need to consider the actual appeal of an

oﬀer to the audience members (at the focal social position). Let αy(x, t) be a real-valued function that tells the appeal of the oﬀering of the producer in the market for the category to the audience member y at time t and let P αx(t) = y∈p αy(x, t).

Definition 4. An organization’s fitness, relative to the other producers in the category, is its share of the total appeal among the offerings in the category at the unique position it targets. (HPC Definition 9.1 specialized to one social position) αx(t) ϕx(t) = αx(t) + Ax(t)

where Ax(t) denotes the total appeal of all of the focal producer’s competitors. P That is, αx(t) + Ax(t) = xαx(t).

A producer has high fitness when its offer is relatively attractive to audience members. Therefore, a producer with high fitness can generally accumulate resources. A “fitness threshold” regulates the pattern of resource flows. If a producer’s fitness lies below the threshold, then its stock of resources shrinks and its viability falls; otherwise its stock of resources grows and viability improves. Within this framework, a producer’s long-term cost structure determines the fitness threshold. Because this result plays such a central role in the arguments developed below, we restate it here (without restating the proofs). And since the result requires assuming that the amount of resources devoted by the audience to the category remain constant over time, we formulate this condition as an auxiliary assumption that will be assumed to hold throughout. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 25

Auxiliary Assumption 4. The total resources available to the category stays constant over time.

We analyze viability in terms of the hazard of organizational failure. Let

the hazard of failure for the producer at time t be denoted as hx(t). (We do not regard other types of exits, such as voluntary acquisitions and mergers, as failure events.)

Proposition 6 (LHP Proposition 1). A producer’s failure hazard presumably decreases with age if its ﬁtness exceeds its cost-structure threshold fx and increases with age if its ﬁtness falls below fx.

10. The First Theory Stage: Organizational Learning and Age-Dependence As mentioned above, the ﬁrst stage of the theory emphasized the eﬀect of growth in engagement over the life course. It attempted to translate a general line of argument, initially proposed by Stinchcombe (1965), that performance increases with aging as an organization gains experience and organizational members learn to operate with one another and with the institutional environment. This learning argument was integrated into the framework just sketched via a postulate that claims that the level/quality of engagement increases with age.

Postulate 6. A producer’s level/quality of engagement normally rises with age.

Under the scenario sketched to this point (which will be revised when we bring considerations of obsolescence into the picture), the actual appeal of an oﬀering increases with its producer’s age.

Proposition 7 (LHP Proposition 2). The actual appeal of an oﬀering presumably rises with the age of its producer.

This proposition implies that a producer’s actual appeal converges to some −→ limiting value α x (because it is a bounded and increasing function of time). THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 26

The ﬁrst stage of the theory applies to environments in which the focal producer experiences competitive stability.4 This restriction can be implemented formally by invoking the following predicate.

Deﬁnition 5. A producer’s environment is characterized by competitive stability, in notation cs, if the sum of the actual appeals of its competitors in the focal category remains constant and positive.

cs(x) ↔ ∃ Ax∀t [(τx ≤ t) → Ax(t) = Ax > 0].

Imposing this condition (by adding the foregoing predicate to the antecedent in the formulas that follow) gives results that relate ﬁtness at founding to the failure hazard in a setting in which actual appeal improves with age.

Proposition 8 (LHP Proposition 3). Under conditions of competitive stability, a producer’s ﬁtness presumably increases with age and becomes close to −→ −→ −→ α x/( α x + Ax) ≡ ϕ (x), which is called long-run ﬁtness.

This ﬁrst theory stage culminates in the following result.

Proposition 9 (LHP Theorem 1).

A. If a producer’s ﬁtness at founding exceeds the threshold, then its organizational capital presumably grows with age and its failure hazard presumably falls with age—a liability of newness.

B. If a producer’s fitness at founding falls below the threshold but its long- run fitness exceeds it, then organizational capital presumably first diminishes and then grows with age and its failure hazard presumably first rises and then falls—a liability of adolescence.

C. If both ﬁtness at founding and long-run ﬁtness lie below the threshold, then a producer’s organizational capital presumably diminishes with age and its failure hazard presumably rises with age—a liability of aging.

4For a discussion of what happens when this assumption is relaxed, see Le Mens et al. (2011). THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 27

11. The Second Theory Stage: Drifting Tastes and Cultural Resistance Now we incorporate considerations of drift and cultural resistance into the theory summarized in the previous section. Doing so leads to some different conclusions. These differences come not from withdrawing premises of the first theory stage but instead from using specificity relations to control inferences when available argument chains lead to opposing conclusions. That is, the integration follows the basic principle of nonmonotonic logic that specificity considerations control inferences when different arguments point in different directions. The first step in the new argument examines actual appeal. Recall that no amount of engagement can compensate for negligible intrinsic appeal (Meaning postulate 2). Therefore a result parallel to Proposition 10, which connects drift to intrinsic appeal, also holds for actual appeal. Drift coupled with age-related cultural opposition drives actual appeal to an arbitrarily low level at old age.

Proposition 10. If the meaning of a (positively valued) label drifts and a producer experiences age-related cultural resistance, then the actual appeal of its oﬀering goes to zero as it becomes old.

P x, y ∀t [drift → lim(t−τx)→∞ αy(x, t) = 0 ].

Proof. Meaning postulate 2 implies that the intrinsic appeal is an upper bound to the actual appeal. Proposition 10 implies that the intrinsic appeal converges to zero when time becomes large. Because actual appeal is non-negative, it also converges to zero when the organization’s reaches old age. In classical ﬁrst-order logic the claim that actual appeal converges to zero in case of drift and inertia would stand in contradiction with the claim expressed by Proposition 7: actual appeal increases with age.5 By working outside the classical ﬁrst-order logical environment we can avoid this contradiction. In

5The contradiction arises because, according to the rule chain supporting this proposition, the producer’s offering has positive actual appeal for at least one time point say t1: α(t1) > 0. ∗ Now let = α(t1)/2. Due to the limit construction for this , there must also exist some t ∗ ∗ such that ∀t2 [(t2 > t ) → α(t2) < α(t1)/2]. Then ∀t3 [(t3 > max{t1, t } → α(t3) < α(t1)/2]. 0 0 But this result contradicts the claim of Proposition 7: ∀t, t [(t1 < t2) → αx(t ) < αx(t)]. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 28 nonmonotonic logic, when two or more rule chains of comparable specificity have opposing implications and one chain is more specific than the rest, then this most specific chain serves as the basis for inference. In other words, the claim represented by this most-specific rule chain is proven. The premises that lead to Proposition 10 are specific to the scenarios involving drift. Therefore, the rule chains leading to this corollary are more specific than the ones that warrant Proposition 7. We can now formulate an obsolescence theorem by connecting the foregoing corollary with viability. The rule chain supporting the proof uses the following: given that the speed of drift remains above a positive constant and a producer’s adaptive capacity declines monotonically with age, the intrinsic appeal of its offer converges to zero. If competitive pressure remains positive, declining intrinsic appeal will cause fitness to fall below and stay below the fitness threshold. Beyond the date at which the threshold is passed, organizational capital depletes and the hazard of mortality rises with aging.

Proposition 11 (Obsolescence with categorical drift). Given categorical drift for a positively valued category, age-related inertia, and competitive stability: the hazard of failure increases with age after some age.

0 0 0 P x ∃ q ∀ t, t [(τx < q ≤ t < t ) ∧ drift ∧ cs(x) → hx(t) < hx(t )].

Proof. Again specificity considerations make an important difference. Parts A and B of Proposition 9 tell that failure hazards decline with age at older ages. The formula in this theorem states the opposite. As in the proof of Proposition 10, we rely on the fact that definitions and premises are stated as generic (“normally”) statements (rules with exceptions) and use specificity considerations to control the clash between the two arguments. The rule chains supporting the three sub-theorems of Proposition 9 are relatively non-specific (once we take into account that the antecedents in the sub-theorems include terms about fitness relative to the threshold that, when taken together, include all of the relevant possibilities). The rule chain that supports the current theorem (sketched below) begins with drift. The set of situations that satisfy these conditions is a proper subset of those that satisfy the conditions given by the first term in the rule chain for Proposition 9. It is, therefore, more THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 29

speciﬁc, and, as such, overrides the argument of Proposition 9 when schemas drift. The rule chain that yields this theorem goes as follows. Deﬁnition 5 guar-

antees that Ax remains above a positive ﬂoor. Proposition 10 implies that actual appeal presumably becomes arbitrarily small when enough time has elapsed since the producer’s founding. Therefore, ﬁtness presumably shrinks

toward zero. This implies that there exists a time q such that ϕx(t) < fx if t ≥ q. Proposition 6 implies, in turn, that the hazard of presumably rises for t ≥ q.

The timing of the onset of obsolescence depends on the speed of drift and the organization’s adaptive capacity (its initial level and the rate of decay). This formulation provides an important substantive advantage over extant theories of obsolescence. It does not require postulating, a priori, the existence of a population-specific age of onset of obsolescence. Indeed the new theory operates strictly at the organizational level and can accommodate heterogeneity among the producers in a population in terms of the speed with which inertial forces come into play. But how can we define the onset of obsolescence? One possibility attends to the relative speeds of adaptation and drift. Once adaptation speed falls below the drift speed and stays there, the producer has lost its alignment with the audience. Then it is only a matter of time as to when the intrinsic appeal of its offerings starts to fall, followed by the decline of actual appeal, fitness, and finally organizational capital and the corresponding rise in the failure hazard, as expressed in the foregoing theorem.

Deﬁnition 6. The onset of obsolescence for the producer x is the minimal time, ωx, such that the speed of drift in taste exceeds its adaptive capacity for all t > ωx.

ωx = inf{q | (τx ≤ q ≤ t) → ρx(t) < d}.

Definition 3 and the theory of age-related cultural resistance establish that ωx is well defined and finite under conditions of drift. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 30

With this deﬁnition we know what is inevitably coming after adaptive speed stays persistently below the velocity of drift.6

12. Reconciling the Two Theory Stages We naturally want to integrate the two theory stages. However, we lack the knowledge required to do so in a completely general way. The full set of premises gives rise to lines of argument leading to opposing conclusions even for the period before obsolescence, and these opposing arguments do not have a clear specificity order. Two issues need to be addressed to understand the theoretical implications of this apparent impasse. First, before an audience’s taste drifts beyond what a producer’s adaptive capacity can accommodate, a drifting schema might move toward the offering, before drifting away. Our construction does not rule out the possibility that a producer adapts is offering to “track” the drift; it also does not require this. This feature of our modeling strategy rules out a claim that intrinsic appeal necessarily increases over the early life course. This means that we cannot derive predictions about the complete time paths of fitness and organizational capital. Second, even if we introduce a way to resolve the first problem (as we will try below), we face another complication. Recall that the first theory stage reveals that it makes a crucial difference whether long-run fitness surpasses the cost-structure threshold fx. The long-run view appears justified because the argument implies that actual appeal increases with age up to a limit (on the basis of increasing engagement, given that the variations of intrinsic appeal are unspecified in that theory stage). Bringing obsolescence and cultural resistance into the picture makes the long-run approach uninformative. This is because intrinsic appeal approaches zero in the limit, which means that actual appeal

6This definition does not tell exactly when these misfortunes materialize. One might consider the alternative of looking at the persistently decreasing stock of organizational capital (and the resulting decline in viability). The an alternative definition of the time of the onset of obsolescence is the beginning of the period of a monotonically increasing failure hazard. The alternative provides a gives a sharp definition of the time of onset of obsolescence; but it loses the desirable feature of expressing inevitability. Besides, it is plagued by the difficulties of accurately measuring organizational capital (see LHP). Thus it seems unrealistic to believe that one can know exactly when a monotonic decrease sets in. This reasoning motivates our choice of the definition proposed in the main text. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 31

also falls to zero in the limit. And it makes a decisive difference in the time path of capital (and thus of the failure hazard) whether engagement has had time to increase to a level high enough to bring fitness above the threshold before obsolescence kicks in. These uncertainties arise because we made very weak assumptions about the functional relations between the constructs of the theory. (This is because we believe that the current state of knowledge does not support more precise assumptions about the matters under study.) Lacking assumptions about functional relations and associated parameters, we cannot make predictions about the predicted time paths of change. Nonetheless, we can make progress and gain new predictions by narrowing the scope of the argument. We do so in a way that identifies a condition that, if satisfied, allows the two theory stages to be integrated. The condition that separates the cases concerns the alignment of the offering with the drifting schemas. We introduce a predicate that tells that a producer tracks drift over the period before obsolescence gains sway; and we invoke the predicate in the antecedents of formulas to limit the scope of the argument to this well-behaved case.

Definition 7. A producer experiences pre-obsolescence alignment with the audience for a category iff the distance between its offering and the schemas of audience members does not increase with age before the onset of obsolescence.

0 0 −→ −→ al(x) ↔ ∀t, t , y [(τx ≤ t < t ≤ ωx) → d (oxt, syt) ≥ d (oxt0 , syt0 )].

With the assumption of alignment and increasing engagement due to organizational learning, we can show that fitness rises over the early life course. Depending on the conditions at founding, this leads to different patterns of age-dependence. The following theorems summarize our theoretical integration. Specifically they integrate the predictions about early aging (consistent with Proposition 9) and about old age (consistent with Proposition 11) under the constrained scenario involving pre-obsolescence alignment by relying on the integrative capability afforded by nonmonotonic logic (and specificity considerations). THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 32

Unfortunately, none of these theorems makes predictions for the full lifetimes of organizations. The gap in predictions concerns a period immediately

following the onset of obsolescence, for an age interval [ωx, ωx + q], where q is a positive constant. Fitness can still increase after the onset of obsolescence, even though it will ultimately decline. This is because actual appeal can still

increase for a time after ωx if engagement rises fast enough to oﬀset the initial decline in intrinsic appeal.7

Notation. We simplify the formulas by using a summary predicate that in- stantiates the theoretically relevant conditions:

Θ(x) ↔ drift ∧ al(x) ∧ cs(x).

Consider first the scenario with fitness at founding above the threshold in parallel with Proposition 9A.) Introducing considerations of drift and age- related cultural resistance yields a pattern not seen in the first theory stage: U-shaped age-dependence in the hazard.

Theorem 2 (Constrained uniﬁcation: Part A). Given the scope conditions

stated in Θx: if a producer’s ﬁtness at founding exceeds the threshold, then its hazard of failure presumably initially decreases with age up to a point and then increases with age thereafter.

P x ∃ q ∀ t1, t2, t3, t4 [Θx ∧ (ϕ(τx) > fx) ∧ (τx ≤ t1 < t2 ≤ ωx ≤ q ≤ t3 < t4)

→ (hx(t1) > hx(t2)) ∧ (hx(t3) < hx(t4))]. Proof. This theorem (and the two that follow) applies to situations where drift ∧ al holds. Because those situations are more speciﬁc than the situations of applicability of Proposition 7 (actual appeal increases with age), any argument whose supporting rule-chain makes use of that proposition cannot hold. In particular, Proposition 8 (which was crucial in deriving Proposition 9) does not hold for all ages in the setting we consider. Therefore, we cannot directly rely on Proposition 9 to derive results about what happens during the early phase of organizational lifetimes. The situations of application are also

7Also, we cannot show that intrinsic appeal declines for sure after the onset of obsolescence due to complexities associated to the fact that schemas are sets rather than just points in the sociocultural space. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 33

more speciﬁc than those invoked in the general drift and inertia theory (which forms a key part of the rule chains behind Proposition 10 and Proposition 11). Therefore, we cannot use those results unchanged either. The condition of drift invoked in the antecedent guarantees that the onset

of obsolescence, ωx, is well defined (ωx < ∞). The chain rule that links the antecedent with the first term in the consequent applies to times before ωx. Postulate 6 holds that engagement increases at all ages within the range being considered; and the definition of al(x), Definition 7, and Proposition 5 jointly imply that intrinsic appeal does not decrease in this age range. If engagement increases and intrinsic appeal does not decrease, then actual appeal rises (Meaning postulate 2). Given the restriction to stable competition (Defini- tion 5), this rule chain warrants the claim that fitness rises with age. This implies that fitness remains above the fitness at founding level ϕ(τx), which exceeds the threshold fx by stipulation. This, in turn, implies that the failure hazard decreases with age.

The second term in the consequent applies to ages after ωx + q. After the onset of obsolescence, the general argument about drift and inertia is not constrained by the initial alignment requirement (the deﬁnition of al(x), Def- inition 7, binds only until ωx). This means that the rule chain behind Propo- sition 11 does not get overridden in this age range (once intrinsic appeal falls enough to overwhelm a possible increase in engagement after ωx. Therefore, the second term in the consequent holds.

Next consider what happens when fitness at founding lies below the threshold. Recall that Proposition 9 holds that long-run fitness relative to the threshold decisively shapes the pattern of age-dependence in failure hazards. The first theory stage does not consider variations in intrinsic appeal, and it as- sumes that engagement rises with age. Because this construction implies that actual appeal increases with age up to a limit, it is natural to define long-run fitness as a limiting construction in that setting. In the second theory stage, such a limiting construction does not work, as we noted above. With drift and cultural resistance in the picture, fitness falls in the long run for all producers. So, we must proceed in another way. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 34

Our analysis reveals that we can consider two distinct cases depending on

fitness at the onset of obsolescence, φ(ωx). We first consider a producer whose fitness begins below the threshold but then surpasses it before obsolescence rules, which parallels Proposition 9B.)

Theorem 3 (Constrained Uniﬁcation: Part B). Given the scope conditions

stated in Θx: if a producer’s fitness at founding lies below the threshold but its fitness at the onset of obsolescence exceeds it, then its hazard of failure first increases with age to a point, then declines with age until the onset of obsolescence, and then (perhaps after some gap) increases with further aging.

P x ∃ q1, q2 ∀ t1, t2, t3, t4, t5, t6 [Θx ∧ (ϕ(τx) < fx) ∧ (ϕ(ωx) > fx)

∧ (τx ≤ t1 < t2 ≤ q1 ≤ t3 < t4 ≤ ωx ≤ q2 ≤ t5 < t6)

→ (hx(t1) < hx(t2)) ∧ (hx(t3) > hx(t4)) ∧ (hx(t5) < hx(t6))].

Proof. As in the proofs of the previous theorem, q can be chosen such that the rule chain behind Proposition 6 does not get overridden by considerations

of drift and inertia in the interval [τx, q1]. More specifically, the antecedent supplies that initial fitness lies below the threshold and that fitness exceeds

the threshold at ωx. The restriction to initial alignment ensures that intrinsic appeal does not decrease before ωx (according to Definition 7). Then the assumption that engagement increases at all ages implies that fitness increases monotonically with age before ωx. Therefore, there must be a time point such that fitness first reaches the threshold and thereafter remains above it for the remainder of the pre-obsolescence period. Fitness lies under the threshold at all ages before this time point and above the threshold at all ages after this time point and before ωx. The obvious choice would set q1 to the time at which fitness first reaches the threshold. By this construction, the hazard of failure presumably rises with age before q1 and declines with age after q1 (as implied by Proposition 6). The chain rule that links the antecedent with the last term in the consequent remains the same as for Theorem 2, given that the antecedent states that q2 falls on or after the time of onset of obsolescence. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 35

In the ﬁnal case both ﬁtness never passes the threshold. Introducing considerations of drift and cultural resistance yields a weaker version of the prediction from Proposition 9C.): the hazard of failure increases with age during a youth- ful period and in old age, but there is no prediction for some intermediate age range.

Theorem 4 (Constrained uniﬁcation: Part C). Given the scope conditions

stated in Θx: if both fitness at founding and fitness at the onset of obsolescence lie below the fitness threshold, then the hazard of failure presumably rises with age over the early age and over a later age range.

P x ∃ q ∀ t1, t2, t3, t4 [Θx ∧ (ϕ(τx) < fx) ∧ ϕ(ωx) < fx)

∧ (τx ≤ t1 < t2 ≤ ωx ≤ q ≤ t3 < t4) → (hx(t1) < hx(t2)) ∧ (hx(t3) < hx(t4))]. Proof. The rule chain behind Proposition 6 does not get overridden by considerations of drift in the interval [τx, ωx]. And, as for Theorem 2, ﬁtness increases monotonically with age before ωx. Therefore it remains at least as low as ϕ(ωx), which is below the threshold. Therefore, the hazard of failure increases with age before ωx. The chain rule that links the antecedent with the last term in the consequent remains the same as for Theorem 2, given that the antecedent states that q falls on or after the onset of obsolescence.

13. Conclusion Our formulation of the similarity of an offering and a schema as inverse to their distance in a metric space allows us to specify precisely what it means for tastes to drift and for a producer to lose the ability to adapting to this. These ideas have received only a sketchy treatment in earlier work, which limited the potential for empirical study of the key parameters. Of course, a price must be paid for gaining such precision. We had to limit the analysis to cases in which schemas contain only range restrictions on functions onto a metric space, that is, where the features are quantitative. It will be important to learn how limiting this restriction is from a substantive point of view. We narrowed our analysis to treat the case of drifting tastes at a single homogeneous social position. An obvious next step would generalize the model THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 36 to allow heterogeneity on the audience side, to take multiple positions into account. Our model will, when suitably adapted, allow precise identification of the conditions that favor growing homogeneity of tastes within an audience (Koçak, Hannan, and Hsu 2009) those that create divergence (Hannan, Pólos, and Carroll 2010; Pontikes 2012; Smith 2011). Other patterns of change in tastes over time can be addressed as well. For instance, early organization niche theory contains a model of cyclic environmental change, one in which the state of the environment shifts back and forth between configurations (Freeman and Hannan 1983). Our model can potentially be used along with contemporary renditions of niche theory to explore the implications of various patterns of change. The story of obsolescence with drift bears a resemblance to the notion of a competency trap. In the classic version of that story (Levitt and March 1988), producers who have developed high competence with a production technique (by extensive learning by doing), are reluctant to switch to a superior technique with which they have no experience. In the short-run such a switch would degrade performance: high competence with the inferior technique yields better performance than does low competence with the superior one. (And, presumably, superior techniques continue to appear.) The parallel with our theory is that the first theory stage holds that performance increases with age due to age-dependent improvements in engagement. So organizations get better at what they do. But the second theory stage tells that what they do becomes less and less appealing to the audience. Their offerings lose alignment with the audience due to drifting tastes. So the common imagery is that older (more experienced) producers get trapped. But there are important differences. For one thing, the competency trap operates on the producer side—it is a story about learning and technical choice. In our theory, the causal action lies on the audience side: the internal audience resists culturally incompatible moves and the external audience’s tastes change. Moreover, only the high performers get trapped in the producer learning story. According to the obsolescence-drift story, all producers suffer from drift. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 37

Our model of cultural resistance ties it to exposure to existing organizational arrangements. What happens if an old organization does manage to overcome resistance and move over a paths of changes in the architectural space? In our model, this sets exposure to zero for each of the features on the path, which means that resistance to further changes in those features is low. This breaks the tie between organizational age and the distribution of exposure. So the inertial force weakens. In the terms of our theory, the producer’s adaptive potential might rise. What does this mean for the producer? The theory we propose does not provide a coherent answer to this question. Considerations of organizational learning (as in the competency trap story) and stability of organizational routines (Hannan and Freeman 1984; Barnett and Carroll 1995) suggest that performance is degraded when organizations change their structures. Our formulation ties quality/quantity of engagement to organizational age, not to the duration of the structural arrangements. This part of the argument does not allow performance to degrade with change. Re- vising this part of the argument seems essential to addressing whether organizations that undertake extensive change might catch up with drifting tastes. THE GEOMETRY OF ORGANIZATIONAL ADAPTATION 38

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Gaël Le Mens: Universitat Pompeu Fabra, Ramon Trias Fargas, 25–27, 08005 Barcelona, Spain E-mail address: [email protected]

Michael T. Hannan: Graduate School of Business, Stanford University, 655 Knight Way, Stanford CA 94305–7298, USA E-mail address: [email protected]

László Pólos: Durham Business School, Durham University, Mill Hill Lane, Durham DH1 3LB, UK E-mail address: [email protected]