Introduction

In recent years, the concept of modularity has gained great visibility in a number of disciplines as diverse as management (e.g. Schilling, 2000), organizational sciences (e.g. Brusoni et al., 2001), economics (e.g. Langlois, 2002; Marengo et al., 2005), biology (e.g. Callebaut and Rasskin-Gutman, 2005), cognitive sciences (e.g. Fodor, 1983), American studies (e.g. Blair, 1988), architectural and engineering design (e.g. Alexander, 1969; Suh, 1990). Overall, modularity has been proposed as a powerful organizing principle of the evolutionary processes of both artificial and natural complex systems. Within social sciences, modularity principles have been applied to explain the evolution of products, organizational design and knowledge management, on the one hand; and the emerging patterns, and dynamics, of coordination and division of labour between firms, on the other.

It is often argued that, by adopting modular design strategies, firms can take responsibility for the design and development of separate modules. Thus, they can develop new products at a faster pace, as the integration of the final product is a matter of mix and match of 'black boxes' (e.g. Sanchez and Mahoney, 1996; Baldwin and Clark, 2000; Baldwin and Clark, 2001). This is made possible by advanced technological knowledge about component interactions that can be used to fully specify and standardize component interfaces and, therefore, to decouple the design of the product architecture (i.e. arrangement of functional elements) from the design of each module. Modularity, by simplifying design and development processes, would allow a greater division of labour across firms. As a consequence, firms can focus their capabilities on few modules or on the architecture. To fully exploit modularity, a 'grammar of action' (Argyres, 1999), or 'design rules' (Baldwin and Clark, 2000) have to be set in terms of neat and powerful routines that govern interfaces at product and organizational level. Modularity emerges as a specific kind of problem-solving strategy, which entails a specific pattern of knowledge and tasks partitioning.

On this basis, formal models start from the already set grammar, or rules, and focusses on the exploitative aspects of modular design principles (Baldwin and Clark, 2000). Such grammar of action sets the rules of interaction across module, but also defines the organization of problem-solving activities in terms of division of tasks, and their allocation to decoupled design teams. A modular architecture generates more options than an integral architecture because experimentation takes place at the level of the modules, rather than at the level of the entire product (Baldwin and Clark, 2000). The finer the detail with which modules' interfaces are defined, the faster the pace of experimentation. That is to say, by adopting a modular product design strategy firms can speed up the process of exploitation of a given (modular) architecture.

When Baldwin and Clark (2001) discuss the advantages of modularity in terms of option value, they assume that a modular design is able to generate more options, and thus more value. They do so analysing 'modularity as a financial force ... [that] adds options to and thereby increases the financial value of complex design' (Baldwin and Clark, 2001: 1, italics in original).

We approach instead modularity as a specific example of problem-solving strategy. From this complementary standpoint, it becomes interesting to analyse what kind of problems can be framed and solved in a modular fashion (as opposed to an integral one), what are the characteristics of the search spaces generated by alternative problem-solving strategies, and what are the properties of the solutions so identified in the presence of varying degrees of environmental uncertainty.

We can consider product development as an example of problem-solving activity regularly taking place in firms and also across firms. Modular product design means that a particular decomposition of the design problem has been already decided upon and independent teams can work independently on the various sub-problems without explicit coordination. Product architecture defines the interfaces among modules, and new designs of specific modules are accepted only if they improve the overall product performance given the current configuration of the remaining modules. This process will, in general, be subject to lock-in into local optima because potentially superior module designs might not be accepted as the other modules are not co-adapted. On the other hand non-modular or integral search modules and interfaces are not pre-defined and search takes place by varying potentially all components together and accepting new product if its performance is higher. It must be indeed pointed out that the latter model of integral search might not be entirely realistic as also in that firms do usually work upon temporary problem decomposition (Schaefer, 1999) and frequent consultations among team (Siggelkow and Levinthal, 2003); however, it is useful as a reference model of a pure integral strategy, in which potentially all design parameters are subject to search and evaluation.

More specifically, this paper develops a formal model, broadly inspired by Stuart Kauffman's work on fitness landscapes (Kauffman, 1993), in order to shed some light on two related questions concerning the dynamic properties of modular search strategies. First, given the competitive environment, we want to understand whether there is a trade-off between the 'speed' of search (enabled by modularity) and the 'breadth' of search (enabled by non-modular search strategies). Apparently, modular search strategies are indeed highly efficient in the short term (i.e. they provide 'higher value'), enabling fast searches within a predefined search space. However, these gains might disappear in the long term, as 'slower' (i.e. less modular) search strategies catch up and reach better solutions as they can explore broader search spaces, exactly because they rely on less tightly defined 'design rules'. Second, and more fundamentally, we want to explore the relationship between alternative search strategies and changing competitive environments. Does modularity pay off in the presence of fundamental uncertainty? This is a basic question because, in dynamic terms, modularity may entail some risks: the more defined is the grammar, and the search space it enables, the more structured, limited, and limiting is the search process. Firms may miss value-generating alternatives because they cannot escape the boundaries set by the existing modular design strategies.

This paper is organized as follows. The next section focusses on recent empirical research on modularity to highlight a few facts that demand an explanation and outlines our problem-solving perspective. The succeeding section introduces the model that helps us addressing a few key issues related to the dynamic properties of modularity. The penultimate section presents some results we obtain by simulating the model and, finally, in the last section we draw some conclusions.

Modularity and problem-solving in organizations

One of the fundamental contributions of recent research on modularity is the identification of a series of constructs and key relationships that allows studying the connection between what firms do, how they do it, and what they need to know in order to do it. Research on modularity reminds us about the complexity of the relationship between conceptualizations of firms as knowledge structures, and of firms as producers of goods and services. The need to disentangle this complex relationship is of paramount importance for both practical and theoretical reasons. First, in the context of increasingly globalized markets, ever more complex supply chains, and international manufacturing networks, corporate decision-making processes involve more and more actors, variables and criteria that lead to less and less transparency about who is deciding what, and on what basis. Second, and relatedly, the notion of 'means of production' has less and less to do with hardware, and more and more to do with information and specialized knowledge. Management tasks increasingly involve the monitoring, control, and coordination of a widening range of useful, but highly heterogeneous, scientific, and technological disciplines that are embodied in products of increasing complexity, in terms of components and functionalities.

Modularity, as a product and organizational design strategy, provides one possible way to understand how this complex relationship is governed by modern corporations. Modularity allows the decoupling of complex artefacts into less-complex, self-contained modules; each module, at the extreme, could become the sole business of specialist firms. Modularity makes complexity manageable by making possible to run experiments at the level of modules, rather than the entire artefact, and in parallel (Baldwin and Clark, 2001), by exploiting the coordinating properties of markets on which modules can be transacted. Moreover, modularity is 'tolerant of uncertainty' because particular elements of a modular design may be changed after unforeseen contingencies emerge, as long as the design rules are obeyed (ibid.).

The advantages of modularity seem to be particularly compelling in high technology settings, such as the aircraft engine industry. For example, two competing aircraft engine architectures are employed in the industry, namely two-shaft and three-shaft. According to industry experts, the three-shaft (launched by Rolls-Royce in the early 1970s) has turned out to be more effective in accommodating evolving customer requirements in terms of engine power due to its more modular architecture. In a three-shaft engine, the compression work is split across three compressors (low-, intermediate-, and high-pressure). Each compressor can be driven by its own turbine at its optimum speed. In a three-shaft design the mapping between functions and physical structures tends to be more one to one. A three-shaft engine, therefore, is more modular than a two-shaft engine. In a two-shaft design, in fact, the compression work is split between two compressors. The fan and the booster run on the same shaft. They rotate at relatively low speed to maintain the fan tip speed below supersonic. This limits the compression that can be achieved in the first part of the engine leaving high duty on the high-pressure compressor.

The embedded modularity of the three-shaft design enabled Rolls-Royce to exploit the same architecture, hence cutting the high development costs of new engines, to cater for a broader range of power requirements. The so-called thrust growth capability of the three-shaft has been much larger than the two-shaft's. The thrust ratings of the JT9D and CF6 two-shaft engine families go from 44,250 to 56,000 lb and from 40,000 to 64,500 lb, respectively. Due to the exhausted growth potential of the JT9D and CF6 engines and to increasing thrust requirements asked by airlines to power larger aircraft, Pratt & Whitney and GE Aircraft Engines had to develop new engine families, the PW4000 and the GE90, respectively. The three-shaft architecture has been instead characterized by higher thrust growth. Rolls-Royce stretched the same engine architecture (the RB211) to develop engines from 42,000 to 115,000 lb. The in-built growth capability of the three-shaft design – due to its more modular architecture – conferred Rolls-Royce a clear competitive advantage in terms of speed of development of new engines. In fact, they were able to introduce incremental changes in the original architecture by (mixing and matching components) to meet a wider variety of aircraft makers' needs than their competitors. Emblematic is the example of the Trent engine 500 that has become the sole engine for the Airbus A340-500/600.

However, as noted by Baldwin and Clark (2000), the adoption of modular design strategies brings about costs too. First, the creation and dissemination of design rules is a rather expensive activity. Experimenting and testing on different modules is also costly. Moreover, increasing division of labour among firms also entail the traditional costs associated with the use of the market system (i.e. transaction costs) as well as agency costs related to the hold-up problem. The costs of creating the design rules deserve particular attention. Developing modular architectures is more difficult than developing integral ones. Achieving modularity requires a very precise understanding of the product functionalities, how they are allocated to components, and how the components interact. Thus, the choice of product architecture should be related to a company's product strategy. Ulrich (1995) argued that if a company wants to stress product performances, then the most appropriate choice would be the integral architecture, since global performance characteristics are optimized through this type of architecture. On the other hand, companies wanting to emphasize product change and variety, flexibility and upgradeability, may well choose a modular architecture.

Besides, there are other costs related to developing a modular architecture. One must also consider the costs of the foregone opportunities that might have been exploited adopting different architectural choices. This problem is particularly cogent in an innovative context. The crucial point is the change of unit of selection that modularity imply with respect to an integral system: having 'fine' rather than 'coarse' units of selection makes the search process faster (selection operates on the finer scale of modules and therefore the selection environment is in a sense 'richer') but essentially 'local' and gets quickly locked into a local optimum, whereas in an integral system search is global, which implies that there is no lock-in, but search is much slower and in complex space there is a lot of wasteful search as non-sensical options can be generated.

Generally speaking, modularity can indeed highly increase the number of options generated and the speed of search for each module by creating standard interfaces between modules, but cannot avoid the lock-in. Modularity will make the system climb the local optimum faster, but cannot make it jump to another, higher valued, local optimum (Fleming and Sorenson, 2001). This can only be done by changing the architecture of modules. However, if companies embedded in a modular network design their interface, and specialized capabilities, around the current product architecture, how can they learn about alternative architectures? '[L]earning about changes in the architecture of the product is unlikely to occur naturally' (Henderson and Clark, 1990: 28). Not only does learning about the architecture require dedicated efforts, but it also entails different kinds of organizations, people, and skills (Brusoni and Prencipe, 2001). Moreover, 'architectural knowledge can emerge only after an organization has developed sufficient experience with a problem to be able to fragment it into elements without losing critical information' (Henderson, 1992: 127). Once an organization recognizes an architectural innovation as such, it has to change its 'orientation from one of refinement within a stable architecture to one of active search for new solutions' (Henderson and Clark, 1990: 17).

This change of orientation is what 'systems integrating' firms can do. Systems integrators are companies that rely on wide and dispersed networks of suppliers of specialized components and capabilities, yet maintain broad and deep in-house capabilities. These are firms that 'know more than they make' (Brusoni et al., 2001) in order to be able to coordinate loosely coupled networks of suppliers, but also introduce new product architectures. The case of Fujitsu exemplifies the role played by systems integrator in the case of the hard disk drive network. Fujitsu successfully managed the introduction of a new product architecture, stemming from a major technological breakthrough embodied into the magneto-resistive head, a component that displaced the pre-existing mechanical-based technology. Relying on the modular architecture of the established product, Fujitsu, like other firms, relied on a decoupled network of external suppliers. However, unlike its competitors, Fujitsu continued to invest 'in systems knowledge and materials and component technology in its R&D labs' (Chesbrough and Kusunoki, 2001: 218). Fujitsu's systems knowledge went well beyond the range of products and components that the company produced in house. It enabled the firm to master the new, fast-moving technology and to navigate the dangerous waters of architectural innovation stemming from it. By knowing more than it needed for its own design and production, Fujitsu managed to avoid competency traps such as those described by Chesbrough and Kusunoki (2001) and Henderson and Clark (1990).

Brusoni et al. (2001) argued that cases like Fujitsu's show that decoupled, modular networks coordinated through markets and the exchange of codified knowledge (Sturgeon, 2002) are but particular cases of a more general model, which link firms' knowledge and production boundaries. They argued that truly modular networks could emerge only when product interdependencies are predictable and when the specialized bodies of knowledge are at the same stage of development. Interdependencies across components are predictable when a change in the design of one component entails a well-understood change in the design of other components and vice versa. The personal computer industry seems to fall into this situation (Langlois and Robertson, 1992). However, in the presence of product-level contingencies that cannot be fully predicted, 'co-ordinated activity is required to secure agreement about the estimates that will be used as a basis for action. Vertical integration facilitates such co-ordination' (Teece, 1976: 13). The automotive industry seems to provide an illustration to this case (Sako and Murray, 1999). Similarly, Davies (1999) studied the case of mobile phone systems, that is, products characterized by unpredictable interdependencies across components as well as imbalances at the technological level. He showed that under high technological and environmental uncertainty, tightly coupled organizations in which integrated firms maintain in house both the knowledge and the production activities involved in the design and production of their final products and component units, have a competitive advantage. The advantage of a 'single vendor solution' lies in the supplier's experience in delivering 'a verified system in which all the components work well together, and can be integrated, tested and ready for service more rapidly than is possible in multivendor solutions' (Davies, 1999: 120). Specifically, Davies argued that the key advantage of Ericsson, the world leader throughout the 1990s, was its constant involvement in both architectural and component innovations, as well as its efforts to control production costs. It is worth noting that Ericsson adopted such a broad innovation strategy while introducing a modular approach to system design, in which core systems – centrally designed – were then adapted by regional 'competence centres' (Edquist, 2003). However, this modular approach was accompanied, and enabled, by the development of in-house 'system competency', that is, the competencies to design, build, market, and support the entire system (McKelvey and Texier, 2000).

This is a key insight for understanding the dynamic trade-off implied by modular search strategies. Companies like Ericsson, operating under conditions of fundamental technological and environmental uncertainty, have not disintegrated to be replaced by modular networks of specialized innovators. Quite the opposite, they have zealously invested into both the exploitation of the current standard, and into explorative activities to shape the next generation standards. Given the incredibly rapid rate of change in the technologies, regulatory environments, and competitive landscapes, firms like Ericsson could not run the risk of remaining stuck in any one specific research trajectory. Hence, the need to be involved throughout the 1990s in all the major development efforts that led the industry from the 1G mobile phone systems, to the still recent launch of 3G. Firms that followed more specialized strategies lost their role of leaders despite their very early entry in the arena.

The above discussion informs the modelling exercise reported in the next section. First, we analyse formally the advantages brought about by modularity in terms of speed of adaptation to changing customer needs, highlighted by the Rolls-Royce case. We show that the speed of adaptation can give evolutionary advantages even though over-modular search strategies may not be the most efficient problem-solving strategy. Second, we explore the dynamic trade-offs of modular and integral problem-solving strategies under conditions of fundamental uncertainty. Building upon the evidence summarized above, which focus on the role played by 'systems integrators', we show that integral problem-solving strategies may provide a way out to firms caught by surprise by unexpected changes in their competitive environment. Modular problem-solving strategies instead prevent organizations from rapidly abandoning their established way of doing things.

Model structure

Our model is made up of two elements: the problem space that is exogenously given and characterized by a given degree of difficulty (expressed in terms of sub-problem decomposability) and the problem-solving organization that searches in the problem space for superior solutions and tries to implement them. We assume that the organization is boundedly rational and therefore carries out its activities through a process of adaptive trial and error; at the same time, we also assume that this adaptive search is not purely random but is based on a (albeit possibly wrong) representation of the problem space.

Problem space

The problem space is an extension and generalization of Kauffman's NK model of fitness landscapes (Kauffman, 1993). A fitness landscape is simply a mapping from a vector characterizing an entity's form to a payoff value. The original structure developed by Kauffman postulated a random interaction structure where a given element interacted with K randomly specified other elements. In the spirit of Simon's work on nearly decomposable systems and building on the modelling approaches of Marengo et al. (2000) and Marengo and Dosi (2005), we characterize problem environments as potentially consisting of more structured patterns of interaction. In particular, we develop and use two notions of complexity of a problem environment, namely decompositions and near-decompositions, which give a precise indication of the degree to which the problem can be decomposed into independent or quasi-independent sub-problems (modules). As shown also in Frenken et al. (1999), Kauffman's 'K' can be a bad indicator of the decomposability of the problem: since blocks of epistatic interactions overlap and because of the randomness of fitness contributions, also problems with very low K values can be de facto non-decomposable.

More formally, the problem space is defined by N interdependent features¹ which, for simplicity and without loss of generality, can assume only two states, labelled 0 and 1. The set of features comprising the problem space consists of ℵ={x₁, x₂,..., x_N}, with x_i{0,1}. A particular configuration, that is a possible solution to the problem, is a string xⁱ=xⁱ₁xⁱ₂...xⁱ_N. The set of configurations is characterized as: X=x¹, x²,..., x^2N. The value, or fitness function, consists of a mapping from the set of configurations to the positive real numbers: . A problem is therefore defined by the couple (X,V).

As the size of the set of configurations is exponential in the number of components, whenever the latter is large enough, the state space of problem becomes much too vast to be extensively searched by agents with bounded computational capabilities. One way of reducing its size is to decompose it into sub-spaces.² Let be the set of indexes, and let a block be a non-empty subset of this set, and let |d_i| be the size of block d_i, that is, its cardinality.

We define a decomposition scheme (or simply decomposition) of the space ℵ as a set of blocks:

Note that a decomposition does not necessarily have to be a partition; that is, there may be some overlap among the particular decompositions d_i.

Decompositions structure the nature of the organizational and technological search process. Search for alternative basis of action does not take place on a holistic, system-wide basis but tends to be local and to approach different facets of the problem in a sequential manner (Cyert and March, 1956). In this spirit, a new configuration is generated and tested by picking a block d_jD at random and some (at least one and up to all) components in this block (and only in this block) are mutated, obtaining a new configuration x^h, which may differ from the original configuration xⁱ only in those components belonging to block d_i. If V(x^h)V(xⁱ), then x^h is retained and becomes the new current configuration; otherwise, x^h is discarded and xⁱ continues to be the current configuration.

We say that a decomposition scheme D^* is an optimal decomposition of the problem if multiple iterations of this search procedure are always able (after repeated random mutations) to locate the globally optimal configuration(s), starting from any initial configurations. That is, the scheme is such that there is no lock-in into sub-optimal configurations. In general, there can be many optimal decomposition.³ For instance, if D^* is an optimal decomposition, all decompositions that can be obtained by the union of some of its blocks will also be optimal decomposition. However, among the set of decompositions satisfying this criterion, we are particularly interested in the finest optimal decomposition(s), that is, the one(s) whose blocks have minimal cardinality. Blocks in the finest optimal decompositions represent the finest sub-problems into which the overall problem can be decomposed and still be optimally solved.

We can classify problems in terms of their finest optimal decomposition. In particular, the following types will be widely referred to in our subsequent analysis:

1. Non-decomposable problem, for which the finest optimal decomposition is the degenerate one: D^*={1, 2,..., N}
2. Nearly decomposable problems (Simon, 1981), whose finest optimal decomposition is made of non-disjoint (partially overlapping) blocks. Two cases are particularly interesting:
   • partially overlapping blocks, such as for instance: D^*={1, 2, 3, 4}, {3, 4, 5, 6}, {5, 6, 7, 8},
   • nested blocks, such as for instance: D^*={1}, {1, 2}, {1, 2, 3, 4},...,{1, 2, 3, 4, 5, 6, 7, 8}
3. Decomposable problems, whose optimal decomposition is made only of disjoint blocks. Furthermore, this decomposition of disjoint blocks can be:
   • coarse, if blocks are not all singletons (i.e. they contain more than one component) and
   • fine, if all blocks are singletons (i.e. they contain only one component).

Only in this last case is the problem 'simple' and optimally solvable through N separate local search processes and therefore fully modularizable.

Techno-organizational problem-solving

A decomposition scheme is a sort of template that determines how new configurations are generated and can therefore be tested by a selection mechanism. In large search spaces in which only a very small subset of all possible configurations can be generated and undergo testing, the procedure employed to generate such new configurations plays a key role in defining the set of attainable final configurations. Blocks in our model can be considered as a formalization of the notion of modules and decomposition schemes are a formalization of the notion of system architecture, which defines the set of modules in which a technological system or an organization are decomposed.

We will assume that boundedly rational agents can only search locally in directions that are given by the decomposition scheme: new configurations are generated and tested in the neighbourhood of the given one, where neighbours are new configurations obtained by changing some (possibly all) components within a given module.

Among all the decomposition schemes of a given problem, we are especially interested in those for which the global optimum becomes reachable from any initial configuration. One such decomposition always exists, and is the degenerate decomposition D={{1, 2, 3,..., N}} for which of course there exists only one local optimum and it coincides with the global one. But obviously we are interested in – if they exist – finer decompositions and in particular in those of minimum size. These latter decompositions represent the maximum extent to which the search space can be subdivided into independent modules coordinated by a simple selection mechanism, with the property that such selection processes can eventually lead to optimality from any starting condition. On the contrary, even finer decompositions will not in general (unless the starting configuration is 'by chance' within the basin of attraction of the global optimum) allow decentralized selection processes to optimize.

Minimum size decomposition schemes can be found recursively with the following procedure that we describe informally⁴:

Let us re-arrange all the configurations in X by descending rank X={x⁰, x¹,..., x^{2N - 1}} where xⁱxⁱ⁺¹.

The minimum size decomposition can be computed as follows:

1. Start with the finest decomposition D⁰={{1}, {2},..., {N}}.
2. Check whether x⁰P(xⁱ, D) xⁱ i=1, 2,..., 2^N-1, that is, if there is a path leading to the global optimum from every other configuration for decomposition , if yes STOP.
3. If no, build a new decomposition D¹ by union of the finest blocks for which condition (2) was violated and go back to (2).

Let us finally provide an example for illustration. Consider a system of three binary components and imagine having a selection (value) landscape described by the following table:

If the system is fully modular (i.e. there are three modules {1}, {2} and {3}) and the current state is 001, then search will always be locked into the local optimum 010 and never reach the higher value solution 100. To see this, just notice that there is no one-mutation and value-increasing path leading from 001 to 100: for instance, the first module, which is initially set to 0 can never switch to its optimal configuration 1 because switching to 1 always decreases value given the other modules. In order to ensure that maximum value can always be achieved one needs coarser modules: for instance, in this example, the finest possible set of modules is composed by two modules: {1,2} and{3}.

Near decomposability

When building a decomposition scheme for a problem, we have looked so far for perfect decomposability. In other words, we require each single block to be optimized independently from the others. In this way, we are guaranteed to decompose the problem into perfectly isolated components, which can be solved independently. This is however very stringent a requirement: even when interdependencies are rather weak, but diffused across all components, we easily tend to observe problems for which no perfect decomposition exists.

One can soften the requirement of perfect decomposability into one of near-decomposability: one no longer requires the problem to be decomposed into completely separated sub-problems, that is, sub-problems that fully contain all interdependencies, but only wants sub-problems to contain the most 'relevant' interdependencies while less-relevant ones can persist across sub-problems. In this way, optimizing each sub-problem independently will not necessarily lead to the global optimum, but to a 'good' solution. In other words, we construct near-decompositions that give a precise measure of the trade-off between decentralization and optimality: higher degrees of decentralization and market coordination, and therefore higher speed of adaptation, can be obtained at expenses of the optimality of the solutions that can be reached.

Let us re-arrange all the configurations in X by descending rank X={x⁰, x¹,..., x^{2N - 1}} where xⁱxⁱ⁺¹, and let X={x⁰, x¹,..., x^-1} with 02^N-1 be the ordered set of the best configurations.

We say that X is reachable from a configuration yX and for decomposition D if there exist a configuration xⁱX such that xⁱP(y, D).

We call basin of attraction (X, D) of X for decomposition D the set of all configurations from which X is reachable. If (X, D)=X we say that D is a -decomposition for the problem.

-decompositions of minimum size can be found algorithmically with a straightforward generalization of the above algorithm, which computes minimum size decompositions schemes for optimal decompositions.

Higher degrees of decomposition and decentralization can be attained by giving up optimality and allow providing a precise measure for this trade-off. In order to provide an example, we generated random problems⁵ of size N=12 all characterized by |D|=12 (i.e. they are not decomposable). Figure 1 shows the sizes of the minimum size decomposition schemes as we vary the number of acceptable configurations (average on 100 random landscapes).

Figure 1.

Near decomposability.

Full figure and legend (32K)

It shows that sharp reductions of complexity and time of search⁶ can be obtained by accepting sub-optimal 'satisficing' solutions. Thus there is a trade-off between optimality and speed of search, which has interesting implications that will be examined in the next section.

Speed and optimality of search strategies

The evolutionary advantage of excess modularity

So far, we have characterized a system in terms of its decomposability, now, using this toolbox we are able to construct a problem whose structure is perfectly known and then test the relative efficiency of different search strategies. We will concentrate on the comparison among search strategies based upon different degrees of modularity. A search strategy consists of a rule that produces a new configuration starting from a current one: if the new configuration is better than the previous one it is retained, otherwise it is discarded.

In general, fully modular search strategies, that is, those in which each component is optimized independently of the others, are not optimal (Kauffman, 1993) as they can locate the globally optimal configuration only if there are no interdependencies among components.

In Frenken et al. (1999), the properties of other search strategies based upon coarser modules are analysed. In fact, we can consider a search strategy that divides the N components of the configuration into modules, each containing a given number of components, say S.⁷ The generalized S-search strategy consists of choosing one module (instead of a single component) and mutating one or more components in the module.

Put it more precisely, the steps of a generalized S-search strategy are: (1) choose randomly one of the N/S modules; (2) choose randomly an integer number Z in the range [1,S]; (3) choose randomly Z bits among the S of the chosen module; (4) switch the state of the selected bits; (5) test the fitness of the newly produced string; and (6) accept the new string if it produces a higher fitness than the current string, or reject it otherwise.⁸

We can draw a parallel between the complexity of the problem (decomposability) and the search strategy (modularity). For example, suppose that we consider a problem whose minimal decomposition is: {{1, 2, 3, 4, 5}, {6, 7, 8, 9, 10},...,{N-4, N-3, N-2, N-1, N}}. Obviously, an S-strategy based upon the same modules is always able to reach the optimal configuration. Instead, lower dimensional S-strategies (finer modules) are usually locked into local optima. Higher level S-strategies (larger modules) may be able to find the global maximum, but they take longer.

Figure 2 shows a simulation on a random problem à la Kauffman (1993) of three populations of 100 agents each, that independently search a random problem of size N=40 and optimally decomposable in 10 modules of size 5 each, starting from the same (randomly drawn) initial configuration. The three populations adopt search strategies based upon modules of size, respectively, 1, 5, and 10. In other words, the first class of agents are over-modularized, the second use optimal modules, and the third are under-modularized.

Figure 2.

Average fitness values of three populations searching on a problem space with N=40. The first population (black series) adopts S=1, the second S=5 (red series), and the third S=10 (green series).

Full figure and legend (39K)

All agents with the 'optimal' strategy at the end of the simulation have managed to reach the global maximum, having explored 10,000 configurations (a portion of less than 1/10¹⁰ of the total number of configurations). Instead, none of the agents in the other two populations manage to reach the optimum in the same time. Over-modular agents in the first population quickly get stuck in different local optima, from which are unable to unlock. The third population, though moving continuously up-hill, is very slow, since they explore a much larger portion of the search space.

This property of the S-strategy derives from an analytical result that shows that the maximum number of strings required to be tested in order to select with certainty the maximum fitness is a linear function of 2^K+1 (Frenken et al., 1999).

However, if we consider the initial steps of the simulations, reported in Figure 3, we can see that the first population of agents adopting the finest modules possesses a big initial advantage over the 'optimal' strategy, lasting for many periods, although its agents are doomed to be stuck in a local optima. Why this temporary advantage? The reason lies in the quicker response of the modular strategy (S=1) in respect of the more integrated ones. The strategy aiming at testing smaller variations of the current string is able to test a higher number of possibilities, quickly improving the fitness in the beginning of the search.

Figure 3.

First 300 steps of the same results shown in Figure 2.

Full figure and legend (44K)

The advantage of the more modular strategy in the first population for the initial period may produce interesting dynamic properties. Suppose for instance that different search strategies compete in a market-like selection environment, where fitness values indicate market performance and are correlated to the probability of surviving competition and being imitated. The selection mechanism can be simply represented, for instance, by the removal of a fixed number of agents (the ones with the lowest fitness values) and their replacement with 'copies' of the best ones, where a copy is a new agent adopting the search strategy of the copied agent (i.e. the same modularity), and being assigned an initial random configuration. Figure 4 shows the number of agents in each population with the same settings as above, with the selection mechanism acting every 50 iterations.

Figure 4.

Number of agents in the three populations with S=1 (black), S=5 (red), and S=10 (green). Selection applies every 50 time steps, replacing 20 agents.

Full figure and legend (33K)

Clearly, the first population can exploit its initial advantage, while the 'optimal' strategy (and even more so less modular) do not have the time to unfold their superiority. In other terms, while strategy with S=5 is globally optimal, under the selection pressure it can become evolutionary dominated by a modular strategy that, though bound to long-run sub-optimality, quickly locates and climbs 'satisficing' local optima. This property gives the modular strategy an evolutionary edge over integrated strategies.

These results show that there exists a trade-off between speed and optimality. Aiming at the optimal solution of a problem entails the necessity to take into account all the interdependencies among components. However, this enlarges enormously the space to be searched and therefore the time required to explore it. Conversely, an over-modular approach focussed on the exploration of each component independently from the others, may be doomed to be limited in the maximum performance that can be finally obtained, but has the advantage of providing quick-and-dirty improvements that, in a highly competitive environment, may be the key to evolutionary success. Competitive selection pressure provides a strong drive towards excess, higher-than-optimal modularization.

Volatility and the revenge of integrated systems

We have seen that a modular search strategy enjoys an evolutionary advantage both when the problem space is actually decomposable and also when it is not so, because of higher speed of adaptation. There are, however, cases when an integrated search strategy outperforms modular ones: this happens, contrary to the current wisdom,⁹ when the environment is highly volatile.

In this section, in fact, we show that when the fitness values of configurations change rapidly, then even if at any moment in time the problem space is fully decomposable, modular strategies are rapidly outperformed by integrated ones. A modular search strategy consistently climbs up from its current position with 'steps', which are smaller, the finer the modules. On the contrary, an integrated search strategy can 'jump' to locations far away from the current one. In a stable environment, the former strategy is more effective as it quickly climbs a local optimum while the latter spends a long time wandering around the problem space. But in a highly volatile environment, it can happen that an agent finds itself in a location with very low fitness (the bottom of a 'well' or trough) – in this case local short steps will be too slow a strategy for climbing out of the low-fitness area, while long jumps have a high probability of quickly finding a higher fitness level.¹⁰ Conversely, when agents are in high-fitness location, the integrated strategy has a much lower probability of bringing a further improvement than a modular one, but the latter can only bring small improvements.

The superiority of integrated strategies is the outcome of two factors: probability of fitness improvements and expected size of fitness improvements. Concerning the former, a modular strategy is only slightly more likely to provide a fitness improvement when applied to low-fitness points than to high-fitness ones. Instead, integrated strategies are much more likely to provide improvements when the current fitness is low and very unlikely when it is high. To confirm this statement, in Figure 5 we show the percentage of successful mutations produced in two populations using a modular and a integrated search strategy, respectively. The figure is obtained from simulation performed with N=1000 and K=0. The two populations are made of 50 agents each, running for 10,000 time steps, each time step all agents are relocated in the same (randomly chosen) point.

Figure 5.

Average percentage of agents producing a successful mutation as a function of the average fitness values for two populations: the first applies a one-bit search strategy (highly modular); the second applies a N-bit search strategy (integral).

Full figure and legend (120K)

As to the size of the improvement, the integrated strategy is likely to provide large improvements when starting from low-fitness locations, while it will provide low improvements when the starting location has high fitness. The modular approach will provide small improvements in both circumstances, as shown in Figure 6, obtained under the same conditions as Figure 5.

Figure 6.

Average gain from successful mutations as a function of the average fitness values.

Full figure and legend (96K)

This result contradicts the common wisdom that modularity provides robustness against volatile environments. In fact, it shows that the opposite is true. In highly volatile environments, the systemic recombination allowed by integrated and non-modular search strategies allow for radical change that is more effective in highly uncertain and volatile environments than the local search enabled by modular strategies.

This result is confirmed also when the volatility concerns only part of the system and agents know which parts are affected. That is, we use a shock that modifies randomly the fitness provided by half of the available dimensions, instead of modifying all of them. When the shock occurs, agents know which half of their environment has been affected and direct the mutation efforts to repair that portion of their current activities.

For this second experiment we defined two populations of agents, where the integrated strategy can modify either the first or the second half of the string. We allowed agents to apply their strategy for 20 time steps, after which a shock produces the modification of the fitness contributions provided by half of the bits. Immediately after the shock, agents attempt a mutation choosing one (for modular strategies) or more (for integrated ones) of the bits affected by the shock.

The results do not change sensibly, although in this experiment modular agents were allowed to climb a stable environment for some time, and the shock affected only half of the environment. It can be noticed that the 'clouds' of points is in fact centred above the expected fitness value of 0.5, since agents are able to climb somewhat from the average level during the 20 stable periods.

Figures 7 and 8 report the same statistics seen above for this new set up, considering only the time steps immediately after the shocks. Figure 8 reports the average gain from successful mutations while Figure 7 reports the percentage of successful mutations for integrated and modular strategies.

Figure 7.

Percentage of successful mutations for two populations of 'modular' and 'integrated'. N=100, K=50, and a shock affecting either the first or the second 50 dimensions occur every 20 steps. Agents 'know' which half of the environment underwent the shock and attempt to repair that part. Data used for the figure concern only the periods just after a shock.

Full figure and legend (144K)

Figure 8.

Average gain from successful mutations for two populations of 'modular' and 'integrated' agents with random changes of fitness contributions on half of the components. Same conditions as in Figure 7.

Full figure and legend (113K)

Discussion and conclusions

Taking a complementary perspective to Baldwin and Clark's option value view of modularity, this paper has attempted to single out the advantages and disadvantages of problem-solving strategies in relation to the time horizon of the search and the volatility of the environment. Simulation results show that modular search strategies are particularly efficient in the short term. This is due to the fact that modular search strategies are relatively better at quickly climbing a local optimum. In other words, a highly modular design favours a rapid exploitation of the local improvements allowed by the design itself. Speed of search and adaptation and incremental innovation are therefore important advantages of modularity. Our results also show that there is a trade-off between speed of search and breadth of search. In the long term, integral search strategies can reach higher performance by not being stuck into the local optima determined by a fixed (and inevitably sub-optimal) design. However, if the system is made of many components that interact in a complex and largely not understood fashion, such integral search is haunted by complexity limits and will be evolutionary inferior in a selective environment.

Short-term advantages may lead to sustainable evolutionary advantages at population level. That is to say, populations of organizations that rely on over-modular problem-solving strategies may come to dominate populations of organizations that rely on the 'right' pattern of modularization. In a rather speculative manner, one might argue that this result captures a key feature of the competitive struggle in the PC industry. The open and modular architecture of IBM-compatible PCs has given Microsoft (and Intel) a great competitive advantage over Apple with its proprietary and more integrated architecture. Despite some experts' opinion that Apple can deliver better technical solutions, the so-called WIntel architecture definitely dominates the market.

We also analysed the effectiveness of alternative search strategies in relation to different characteristics of the competitive environment and in particular its stability or volatility. Simulations showed that in stable environments, modular search strategies are more effective because of the above-mentioned fast rate of adaptation and local improvement. For example, one might argue that this is the situation enjoyed by Rolls-Royce in the 1980s and 1990s, that is, two decades during which the civil aviation industry went through a phase of continuous and stable growth. Whether the competitive advantage built upon the modular architecture of the three-shaft engine is defendable in a more turbulent environment is to be seen. Our simulation results would suggest some form of scepticism.

In highly volatile, fast-changing environments in fact, modular search strategies are shown to have a high probability of being trapped into low-fitness zones of the landscape and, since they can only change locally, they take too long to get out. Integral search strategies on the contrary perform search on a broader spectrum and can therefore jump out of low-fitness zones of the landscapes in which sooner or later everybody will fall in a highly volatile environment. In our view, this result captures the key role played by systems-integrating companies. For example, evidence from the mobile phone industry suggests that systems integrators need to remain involved in exploratory research that look beyond the boundaries set by current architectures to be able to lead the process of development of successive generations of mobile telephony systems. The narrow, but fast, search enabled by highly modular problem-solving strategies might lock incumbents into sort of competency traps from which they cannot escape if changing environments require more radical re-design of the system.

The line of reasoning presented in this paper needs be extended in several directions. Two seem most promising to us. First, in this paper we have conceptualized, and modelled, organizations as 'pure' problem solvers. While problem-solving is a fundamental activity performed by organizations, it is not the only one. It is necessary to build models capable of providing richer characterizations of firms' behaviour. This is not just a quest for realism. In a previous paper, we argued that decisions to outsource production (and other functions) are different from decisions to outsource technological knowledge. In other words, there is a gap between firms' production and knowledge boundaries (Brusoni and Prencipe, 2001; Brusoni et al., 2001). Such a gap is fundamental to explain how incumbents manage to react to rapid technical change, like Fujitsu did in the hard disk drive industry. We need to develop new families of models capable of distinguishing, and cope with the joint dynamics, of the division of labour and that of knowledge. A recent paper by (Dosi et al., 2003) begins to explore these issues.

Second, and related, we need to explore further the relationship between firms' organization and environmental changes. Specialization and loose coupling are often advocated as a suitable way of organizing business in the context of fast environmental changes. For example, Levinthal (1997) argued that '[t]ightly coupled organizations can not engage in exploration without foregoing the benefits of exploitation' (p. 949). However, cases like Fujitsu and Ericsson, show that some incumbents can actually manage both exploration and exploitation activities. In our view, to make organizational sense of these issues one need to clearly distinguish between the organization of manufacturing activities, and the organization of more knowledge-intensive activities. Moreover, learning and manufacturing processes are embedded in dense networks that link manufacturers of the final product to suppliers of components and specialized knowledge. In innovative, fast-changing environments, it becomes more and more difficult to pinpoint 'firms' (whether systems integrators or mere assemblers) as the correct unit of analysis. Problems are solved 'socially', and understanding how problem-solving strategies unfold within communities of specialists that cut across firms' boundaries is a challenge to both practitioners and scholars.

Concerning product development, the analysis presented in this paper should be expanded in order to account for some important elements that are still missing in our model. First, comparing modular and integral search in product development is in some sense comparing two quite radically different search processes, which are also normally occurring at different stages. In modular search, the architecture of the product is kept constant, whereas in integral search firms are looking for better product and for better designs at the same time, and search also provides information about interdependencies and attainable performance goals (Ulrich and Eppinger, 1995). In a richer model, therefore, we should compare a world in which firms and teams only search for better performing modules within a given architecture, starting from a much higher level of understanding of the design problem, to one in which they also search for better architecture and more appropriate performance standards (Ethiraj and Levinthal, 2004). More radically, product development involves also the very re-definition of the search landscape as interdependencies are to some extent modified in the process. In the preliminary model, we investigated in this paper, the 'true' modularity of the problem is exogenously given and kept constant, and different search strategies are evaluated on this same performance landscape. In reality, on the contrary, firms often make a deliberate effort to reduce the amount of interdependencies in the product development problem.

Finally, in product development modular and integral search strategies also entail quite different mechanisms of knowledge accumulation. The question is whether highly specialized and relatively independent teams can outperform generalists in being able to change the interdependencies (Postrel, 2002), or generalist knowledge and systems integrators are always needed in order to indicate the directions of search and modify product architecture (Gavetti and Levinthal, 2000; Brusoni et al., 2001).

Notes

¹ In this paper, we refer essentially to informational and technical interdependencies and not to those arising from incentives. An investigation of the relationship between the two within a model of the same family can be found in Dosi et al. (2003).

² A decomposition can be considered as a special case of a search heuristic. Search heuristics are in fact ways of reducing the number of configurations to be considered in a search process.

³ See Marengo and Dosi (2005) for a more formal and detailed account of the properties of optimal and sub-optimal decompositions and for an algorithmic procedure that computes them.

⁴ The complete algorithm is quite lengthy to describe in exhaustive and precise terms. Its Pascal and C++ implementations are available from the authors upon request. See also Marengo and Dosi (2005) for a more formal treatment of decompositions and their properties.

⁵ Here problems are random NK landscapes à la Kauffman (1993) with N=12 and K=5.

⁶ Every reduction of 1 in the size of the decomposition schemes implies that the number of solutions to be tested and the expected time of search are cut down by one-half.

⁷ Without loss of generality and for the sake of simplicity, we assume that each module contains exactly S components and, therefore, that N/S is an integer number.

⁸ Obviously, the S-search strategy with S=1 is the one-bit mutation.

⁹ Baldwin and Clark (2000) for instance maintains that modularity is especially advantageous in uncertain and changing environments. See also Langlois (2002).

¹⁰ A more formal treatment of this proposition can be found in Valente (2003), where it is shown that when fitness levels rapidly change, the expected gains of integrated strategies are higher than those of modular strategies.

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Acknowledgements

We thank Koen Frenken and two anonymous referees of this journal for helpful suggestions. The usual caveat applies.