Information processing and organizational structure

Journal of Economic Behavior & Organization Vol. 36 (1998) 275±294

Information processing and organizational structure Stephen J. DeCanio*, William E. Watkins1 Department of Economics, University of California, Santa Barbara, CA 93106, USA Received 5 August 1997; accepted 4 December 1997

Abstract Standard economic theories of the firm (and other organizations) stress profit maximization as the foundation for derivation of predictable behavior. Yet statistical and case-study evidence continues to accumulate that real firms do not act as required by the neoclassical framework. Instead of being represented by ever more elaborate maximization models, the firm can be modeled simply as a network of information-processing agents. The actions of the firm are then a function only of the network structure and the information-processing capabilities of the agents. This approach can be used to explain a number of features of organizational behavior, including the process of technological diffusion. It also suggests that derivation of the optimal organizational structure may be computationally complex, with a number of implications for economic theory and policy development. # 1998 Elsevier Science B.V. All rights reserved JEL classi®cation: C63; D21; D23; L20; O30 Keywords: Theory of the ®rm; Organizational behavior; Technological diffusion; Computational complexity; Optimization

``It has long been known empirically to students of organization that one of the surest sources of delay and confusion is to allow any superior to be directly responsible for the control of too many subordinates.'' (Graicunas, 1933) 1. Introduction The theory of the firm (and of other organizations) is in the midst of a transformation. Conventional economic modeling has become so closely identified with the mathematics of constrained maximization that it has become difficult to tell whether this technique is * Corresponding author. Tel.: +1 805 893 3130; fax: +1 805 893 8830; e-mail: [email protected] 1 E-mail: [email protected] 0167-2681/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 7 - 2 6 8 1 ( 9 8 ) 0 0 0 9 6 - 1

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the basis of scientifically testable propositions or a self-contained language of discourse in the postmodern sense. While the assumption of profit maximization under neoclassical conditions provides enough structure to be a useful theory, it exhibits large gaps when used to explain the actual conduct and performance of companies. The failure of firms to adopt profitable energy-saving innovations (Koomey, 1990; Lovins and Lovins, 1991; Ayers, 1993; Jaffe and Stavins, 1993; DeCanio, 1993, 1998; Koomey et al., 1996), the unexplained empirical association between firms' economic characteristics and their willingness to join the EPA's Green Lights program2 (DeCanio and Watkins, 1998), and the apparent ability of some firms to transform environmental regulation into a competitive advantage (Porter, 1990, 1991; Porter and van der Linde, 1995a, b)3 are some of the instances in which standard profit maximization theory fails to conform to the evidence. It is fair to say more generally that the conventional theory of corporate behavior is challenged by a wide range of phenomena having to do with capital budgeting and technology choice (Fazzari et al., 1988; Calomaris and Hubbard, 1990; Arthur, 1994; Stole and Zwiebel, 1996). Profit maximization is also unsuitable as an approach to the behavior of government bureaucracies and non-profit organizations, and these constitute very large segments of the economic life of modern societies. Modern public choice theory recognizes that these organizations are staffed by self-seeking individuals, and that they exhibit many of the features of the large corporation, yet these non-market organizations do not maximize economic profits. It is not at all obvious how sharply the line should be drawn between ``market'' and ``non-market'' organizations, however. Public organizations have some motivation to control costs (because of budgetary oversight), and it is not always clear that very large private firms are continuously subject to the discipline of the market.4 The question of whether firms and other organizations can adequately be characterized as profit maximizers (or cost minimizers) has important policy implications. Even though the specialized economic literature recognizes the complexity and subtlety of organizational behavior, conventional methods used, for example, to forecast the costs and consequences of policies to reduce greenhouse gas emissions model the production sector as consisting of firms that maximize profits subject to their production technology constraints. The discrepancy between the cost estimates produced by these models and the ``bottom up'' assessments showing many technological opportunities that could reduce emissions at a profit is striking and has been noted before, along with the consequences of the opposing views for policy.5 An alternative to the pure maximization world-view was pioneered in the 1950s by Herbert Simon in the work that led to his Nobel prize. Simon emphasized the limitations on human computational capacity, and concluded that ``bounded rationality'' is the most 2 Green Lights is a voluntary pollution prevention program designed to encourage adoption of modern, costsaving lighting technologies. 3 For a negative assessment of Porter's hypothesis, see Palmer et al. (1995). 4 It is unlikely, for example, that the market for corporate control operates with complete efficiency. Many companies have constructed defenses against takeovers, and there is evidence that when states pass anti-takeover legislation, the value of firms headquartered in those states falls [see Karpoff and Malatesta, 1989 and the literature cited by them]. 5 See DeCanio (1997) for a summary and review.

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appropriate way to describe a scientific behavioral grounding of economics. Individual agents are, in his view, to be thought of as ``satisficing,'' that is, doing the best they can to move in the direction of their goals (Simon, 1955, 1979 are representative examples of his approach. For a contemporary review article, see Conlisk, 1996). The bounded rationality view is in some general sense undeniably true, but its very generality has prevented it from displacing maximization as the dominant mode of theorizing in economics. Satisficing is too inclusive as a characterization of behavior. Bounded rationality per se rules out hardly any behavior ± seemingly irrational actions can be attributed to the unavailability or costliness of information, or to ubiquitous (but hardly ever measurable) ``transactions costs.'' It is necessary to be more specific about the limits to ``rationality.'' The growing literature on agent-based models offers one approach to this specification problem.6 In agent-based theory, the economic entities operate according to well-defined rules of behavior and on the basis of information available to them. No overreaching maximizing principal is invoked, although ``optimal'' patterns may emerge through selection and adaptation (Kauffman, 1989). Emphasis is on the calculations the agents actually are capable of performing, given their processing capacities and the data available to them. We propose a model of the adoption of innovations by firms that is in this tradition, and show that it provides a plausible explanation of some of the phenomena that are difficult to reconcile with the profit maximization story. 2. Agent-based models of firms and other organizations All of a firm's7 internal bureaucratic activities amount to information processing, but its interaction with the external material world can also be thought of in the same way. Nowadays, of course, many firms literally do nothing but manipulate data, but the information-processing view of firms' activities encompasses the more traditional activities of production and exchange of commodities. Production involves application of energy to decrease entropy locally, using ``blueprints'' consisting of information about the way the world works.8 Firms also process the information they receive from markets in the form of prices, orders, and other data, and they interact with customers and suppliers by exchanging information in the form of goods and services. A distinctive feature of the firm is that it consists of separate agents, and is not a unitary entity with a mind and will of its own. The individuals who make up the organization may have a variety of purposes, and these purposes may range from the strict rationality of homo economicus to a much more general suite of behavioral possibilities. 6 For a recent review, see Van Zandt (1996a). See Section 2 for a discussion of this literature in the context of the model introduced in Section 3. 7 Here and throughout, we will refer to the ``firm'' as an exemplar of an economic organization. Our theory applies to business organizations operating in a market arena, but it also carries over to non-market organizations and institutions. Speaking of ``firms'' will highlight the differences between the information-processing approach and the conventional theory. 8 Living organisms of all kinds utilize externally-supplied energy to increase the (local) complexity of the materials from which they are constituted. Even though the entropy of closed systems increases irreversibly, local decreases of entropy within the closed system are possible.

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The organization itself is defined by the pattern of information exchange among the agents. This pattern of communication between the agents can be described as a network of connected points (a graph), with different graphs corresponding to different organizational structures.9 It is possible to imagine each agent connected to all others (i.e., complete interconnectedness), a hub and spoke form with all agents accountable to a central administrator, a hierarchical tree structure, or any other potential pattern of linkages. Nothing intrinsically restricts the informational structure of the firm; in particular, informal channels of communication may exist alongside the formal lines of reporting and responsibility. The final element of the information model of the firm is the characterization of the processing capabilities of the agents. The agents' processing capabilities depend both on their own intrinsic abilities and on the capacities of the communication channels between them. In the simple model developed below, this capability will be modeled in terms of the agents' characteristics, but it should be kept in mind that the bandwidth of the communications channels is also potentially important. The stylized model developed here will have only single bits of information transmitted through the communication channels, but more complex models would be capable of separately treating agents' processing abilities and the capacities (and costs) of the network communications channels. It is evident that this model of organizations bears a strong resemblance to a computer network. This is no accident; computer networks are real-world embodiments of information-processing structures. As such, many of the characteristics of computer networks should have analogues in the world of human organizations. The theory of and accumulated experience with computer networks ought to have direct applicability to the theory of the firm. While we are convinced that this is true, it is not simple to make the translation. Computer networks have evolved to handle very specialized types of processing tasks, and computers and human agents are obviously quite different in their particular attributes. Models of firms as networks have appeared in the economics, sociology, and management literatures. Debackere et al. (1994) outlined a network approach to describing the dynamics of technology development. Radner (1992, 1993) has used a network-and-processing model to explore the role of hierarchy in management. He cites antecedents in computer science (Schwartz, 1980; Kruskal et al., 1990) and economics (Marschak and McGuire, 1971; Marschak and Radner, 1972; Marschak and Reichelstein, 1987; Keren and Levhari, 1983, 1989; Mount and Reiter, 1982, 1990, 1994). In Radner's papers, efficiency is measured by how long it takes a network of processors to solve associative problems, a class that includes addition or multiplication of numbers and finding a minimum or maximum. In this setup, a hierarchical structure in which no processor is ever idle is optimal. The associative assumption allows a hierarchical network to take full advantage of parallel processing. This line of inquiry has been 9 We use the term ``graph'' in the sense in which it is used in mathematical graph theory, with the agents as ``vertices'' and the channels of information flow the ``edges.'' The potential application of formal graph theory to the theory of organizations has been noted by Harary and Norman (1953), Kemeny and Snell (1962), Flament (1963), and Marshall (1971).

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extended and generalized in papers by Van Zandt and Radner (1996), Van Zandt (1996b, 1997) and Meagher and Van Zandt (1997). In Marschak's papers with McGuire and Reichelstein,10 hierarchical and nonhierarchical structures are studied with a focus on the cost of communication between agents and the informational needs of particular agents in order for them to perform their jobs efficiently. In these models, coordinated networks without duplication are efficient in the sense of minimizing the cost of communications. Costs are reduced by not having the agents relay messages they do not need. Unlike in our formulation, the informationprocessing capabilities of the agents are not modeled, and the task being performed by the organization is not that of adopting an innovation. In the sociology literature, Boorman (1975), White et al. (1976), Boorman and White (1976), Granovetter (1973), Burt (1992) and Uzzi (1996) (among others) have examined social organizations and interpersonal ties using network theory. In particular, a number of sociologists have explored the role of social networks in the diffusion of innovations. Valente's (1995) book is a recent example of this tradition that stretches back to the 1950s and 1960s in the work of Rapoport and Horvath (1961), Rapoport (1953, 1956, 1957), Coleman et al. (1966) and others. Yamaguchi (1994) has modeled the inefficiency of organizations in transmitting information as a function of network structure. Burt's model of organizational ``holes'' also bears some similarity to our approach. Burt argues that an individual's effectiveness is maximized by being embedded in non-overlapping subnetworks (that is, being in a network with ``holes'') in order to access the most nonredundant information possible.11 In the sociological models, however, the assumption typically is made that if a connection exists between two agents, information will flow between them automatically. The main difference of our approach is that we allow the agents' information-processing capabilities to influence the probability that information will move. This enables us to examine the relationships between agents' capabilities and network structure. The management literature offers a very early example of using the network approach to derive characteristics of organizational behavior, Graicunas (1933) article on the ``span of management.'' Based on counting the number of possible relationships between a manager and his or her subordinates, Graicunas' approach implies that it is impractical to manage more than five or six subordinates whose work interlocks (Urwick, 1938).12 10

In addition to the references given above, see Marschak and Reichelstein (1994, 1995). Burt says, ``Given a limit to the volume of information that anyone can process, the network becomes an important screening device. It is an army of people processing information who can call your attention to key bitsÐkeeping you up to date on developing opportunities, warning you of impending disasters'' [1992, p. 14]. 12 Two fascinating notes about Graicunas and his insight, one biographical (Bedeian, 1974) and one historical (Urwick, 1974), were published in the Academy of Management Journal in 1974. Graicunas' formula for the number of possible relationships between a manager and his or her subordinates is 11

C ˆ n…2nÿ1 ‡ n ÿ 1†; where n is the number of subordinates (Certo, 1986). Even before Graicunas' article was published, Sir Ian Hamilton, G. C. G., wrote in 1921 that `the average human brain finds its effective scope in handling from three to six other brains' (Hamilton, 1921). Graicunas spent most of his career in the private sector or government service, not academia. He probably served with the U.S. Office of Strategic Services (OSS) after World War II. In 1947, while on a trip to Moscow, he was arrested by the NKVD, and died on a hunger strike after being interrogated and tortured in prison.

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The ``computational complexity'' flavor of this argument has a great deal of similarity to the argument we will make below regarding the practical impossibility of determining the optimal organizational form.13 3. A model of innovation by firms 3.1. The basic framework The information processing organization we will examine has a very simple structure. Each ``agent'' is capable of being in one of two states, which can be designated as ``1'' or ``0'', so that each agent stores one bit of information. This bit may be thought of as whether or not the agent has adopted a profitable innovation, but the bit can also represent taking or failing to take any sort of action that has positive survival value for the firm. Connectedness between agents is also of the simplest form: each agent ``sees'' whether the agents it is connected to are in state 1 or state 0. A connection between two agents can be represented as a directed line connecting them. Mathematically, the organizational chart is a digraph. The content of the model arises from the interaction between the agents' information processing capabilities and the structure of their connectedness. The probability of an agent's switching from state 0 to state 1 is given by a function of the fraction of the number of the other agents it sees who are in state 1. The shape of this function will represent the processing capacity of the agents. Time passes in discrete intervals, and during each interval each agent can ``adopt'' the innovation (that is, go from state 0 to state 1) depending on the information available to it (what it observes about the states of the agents to which it is connected) and how it processes that information. Formally (omitting the time subscript for notational convenience), Pi …1 j 0† ˆ f ‰… j Yij †=Ti Š

(1)

for each time iteration. Here, Yijˆ1 if agent j is connected to agent i (in the sense that i sees j's state) and agent j is in state 1, while Yijˆ0 if agent j is connected to agent i and agent j is in state 0. Thus,  jYij is the number of agents connected to agent i that are in state 1. In Eq. (1), Ti is the total number of agents connected to agent i. The function f reflects information processing capability of the agents in the following way. If the agents' information processing capabilities are perfect, then all it will take for any agent to adopt the innovation is for it to see any other agent in state 1. The function f in this case would be described by f …x† ˆ 1; if x > 0 f …x† ˆ 0; if x ˆ 0:

(2)

In this case, the innovation will diffuse most rapidly through the organization if every agent is connected to every other one, because adoption by any one agent will immediately be transmitted to all other members of the organization, who will then all 13

These issues have been a concern of military managers for some time. When George Marshall became army Chief of Staff during World War II, he reduced from 61 to six the number of officials with direct access to his office (Overy, 1995, p. 273).

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adopt the innovation. If the innovation is introduced anywhere in the organization in time cycle 1, it will be fully diffused throughout the organization in time cycle 2. Now, consider a more general form of the f function. Let information processing capacity be represented by a logistic-type function, such that the probability of an agent's switching to state 1 is very low unless the agent observes a substantial fraction of those to whom he is connected already in state 1. That is, for agent i, f …xi † ˆ 1=‰1 ‡ eÿ‰xi ÿ…a=c†Š=…b=c† Š ÿ 1=‰1 ‡ e…a=b† Š

(3)

where xi is defined as in Eq. (1) above. This modified logistic function14 requires a bit of explanation. The second term on the right-hand side of Eq. (3) is subtracted to guarantee that the function is zero when xi is zero. The parameter c is introduced to represent the processing power of the agents in a particularly simple way. For large c, the f function is located close to the vertical axis and the agents have a high probability of making the transition from the 0 state to the 1 state if even a small fraction of the agents they are connected to are in the 1 state. For smaller c values, the f function is shifted to the right. This means that a relatively larger fraction of the agents seen have to be in the 1 state to induce a transition. Functions with different c values never cross, and all originate at zero. Thus, the processing power of the agents is unambiguously represented by c in a monotonic way.15 For small c (low processing power), a situation of complete connectedness can actually slow down the process of innovation, because even if an improvement is adopted by one member of the organization, the non-linearity of the f function will impede its diffusion. The ``signal'' of adoption by someone in the organization will be drowned out by the ``noise'' of the non-adopters. Different f functions (with aˆ0.2 and bˆ0.025) are shown graphically in Fig. 1. What organizational change can increase the speed of adopting the innovation in the case where individual agents' abilities to process information is weak? One possibility for improvement is to break up the organization into teams so that individual agents are connected only to a limited number of others. If this is done, then adoption of the innovation by one member of the team will lead to relatively rapid adoption of the innovation by the whole team, because the logistic function shows an increasing probability of adoption as x increases. In the limit, if information processing is very poor, the ideal organizational form would be for each agent to receive information from only one other. In that case, the number of time periods required for an innovation to be adopted by all the agents would be just equal to the number of agents in the organization. This time period would be shorter than if any of the agents were multiply connected, 14 With cˆ1 and without the second term on the right side of Eq. (3), f would be the ordinary logistic cumulative distribution function with the mean and variance of the underlying probability density function being a and b2 2/3 respectively (Mood et al., 1974). 15 The c parameter was chosen as the representation of agents' processing capabilities rather than a simple horizontal shift of the logistic to avoid certain messy mathematical details. First, it is important that the function pass through the origin. If f does not start at 0, each agent would have a discontinuous increase in its probability of transition for x > 0, and the height of the ``jump'' would depend on the parameters a and b. On the other hand, if a simple logistic with different values of a but constrained to originate at zero were used, the functions would cross for some values of x. This would make the relative processing power of the agents dependent on x, which obscures the relationships being explored. The solution to these technical difficulties is to specify the function with a ``stretch'' parameter c as is done in Eq. (3).

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Fig. 1. Logistic Transition Functions.

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because the non-linearity of the f function would lead to long delays in adoption if any non-adopters were observed. Thus, for very high information-processing capability, complete connectedness is the best organizational form, while for very poor processing capability, a completely hierarchical organization would be best. For intermediate levels of information processing capability, there will be some optimal structure and degree of connectedness that will minimize the expected length of time it takes for an innovation to spread through the organization.16 The fact that an optimum organizational chart exists does not mean that it is easy or even possible to specify what it might look like, however. The number of graphs rises very rapidly with the number of agents making up the organization, as Table 1 shows. It was proved by Cayley in 1889 that there are nnÿ2 distinct labeled trees on n vertices (Wilson, 1985). Enumerations of the number of simple graphs rely on the Polya Enumeration Theorem, but there is no easy formula giving the number of connected simple graphs as a function of organization size (Trudeau, 1976; Polya and Read, 1987 (1937)). In our model with a non-linear transition probability function, it seems unlikely that explicit algebraic expressions for the optimal structure can be derived for organizations of even moderate size, or even that an algorithm can be found that is capable of finding the optimal structure in a reasonable finite amount of time. In other words, determination of the optimal organizational structure is in all likelihood computationally intractable. More formally, we conjecture that the problem of finding the optimal organizational structure is at least as difficult as the class of NP-complete problems.17 Issues of computational 16 For the moment, we will defer the question of the decision-making rule for the organization and proceed on the basis that the innovation is adopted by the entire organization when everyone has shifted to state 1. It is unlikely that the flavor of the results would be altered by use of another plausible decision rule, such as adoption of the innovation by the CEO (defined as the individual at the top of the organizational hierarchy), adoption by a majority of the ``board of directors'' (defined as the top level of the organization's hierarchy), etc. 17 Time complexity is defined as the number of computational steps required to solve a problem as a function of its ``size.'' For example, if n is the number of vertices of the graph of the organizational chart and an algorithm for finding the optimal structure could reach a solution in fewer than np steps (where p is an integer), the problem would be said to be solvable in ``polynomial time.'' If the problem could only be solved by an algorithm requiring pn steps, it would be of ``exponential'' time complexity. Problems that cannot be solved in polynomial time are considered to be intractable. Computational problems that are solvable in polynomial time are classified as belonging to the class P. For example, linear programming is known to be in P. Problems for which candidate solutions can be checked in polynomial time are classified as being NP. The main open problem in computational complexity theory is whether P is a proper subset of NP. Hundreds of problems are known which belong to NP, but for which no algorithm capable of solving them in polynomial time is known. These include the famous Traveling Salesman and Knapsack problems, as well as a large number of problems in graph theory (such as whether a given graph has a Hamiltonian Circuit). The inability of mathematicians to find polynomial time algorithmic solutions to these problems provides support for the conjecture that NP includes many problems ``harder'' (in the sense of computational complexity) than those in P. The hardest NP problems form an equivalence class known as the ``NP-complete'' problems, but there are also problems more complex than those in NP. Most of the known NPcomplete problems of graph theory involve determining whether some particular graph has a given property (such as whether it has a Hamiltonian circuit). Our problem involves finding, for a given organizational size, the graph (or digraph) having minimum expected adoption time. As such, it does not appear to be readily transformable into one of the standard problems known to be NP-complete. The time required to check all graphs of size n is exponential in n, and we can see no easy way to reduce the search; hence we conjecture that the problem of finding the optimal organizational chart is at least NP-complete. Further investigation of this question will be the subject of future research. A rigorous yet accessible introduction to the subject of computational complexity is Garey and Johnson (1991).

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Table 1 Number of possible graphs for organizations of small size Type of graph Connected simple graphs Labeled trees Connected simple digraphs

Size of organization nˆ2

3

4

5

6

7

8

1 1 2

2 3 13

6 16 199

21 125 9364

112 1296 2106

853 16,807 9108

11,117 262,144 21012

Source: Wilson (1985), reprinted by permission of Addison Wesley Longman Ltd. A labeled tree is a graph in which the vertices (agents) are individually named; a digraph is a graph in which the direction of the information flow between agents is specified. The number of graphs listed here are the number that are not ``isomorphic'' to one another, that is, the number having a distinct structure.

complexity have begun to surface in current thinking about optimization and rationality. For example, Spear (1989) has shown that in an environment in which agents do not have perfect knowledge of the state of nature, rational expectations equilibria cannot be learned. Spear's result is an application of GoÈdel's incompleteness theorem to the theory of computable functions, and hence the computational complexity (in this case, impossibility) of rational expectations is even greater than that of NP-complete problems. In a similar vein, Board (1992) shows that it is unlikely that agents can form rational expectations in complex economies. Even if complete characterization of optimal structures is computationally intractable in our model, it is possible to make progress in understanding the interactions between organizational structure, processing capability, and time to adoption of an innovation, through numerical simulations of the characteristics of particular organizations. We will specify an organization of constant size (nˆ128) and examine the performance of different organizational forms as agents' processing power varies. We examined four organizational structures. The completely connected organization, in which every agent sees every other agent, has been described above. A second type is the simple hierarchy depicted in Fig. 2, in which four managers oversee subordinate groups of five. The agents are represented by the small shaded squares, and connections between them by lines. The subordinate groups are completely connected, and one person in each group is connected to the manager of the group. The four managers are completely connected, and each one supervises several subordinate groups. (Not all the subordinate groups are shown; the dotted lines represent connections from managers to the groups not shown in Fig. 2.) In all cases, communication is two-way, so that each member of a connected pair sees the other. This allows the graph to be drawn without arrows representing the direction of communication on each of the edges. For this structure, we assumed that the innovation first occurs at random to one of the lowest-level employees; in the other structures, the innovation can occur anywhere in the organization. The third type of structure we examined was a seven-dimensional hypercube. This is a non-hierarchical, symmetrical organizational form in which each of the vertices of the hypercube is connected to its ``neighbors''. Mathematically, each agent is represented by

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Fig. 2. Partial Organizational Chart: Four Managers, Groups of Five.

a distinct 7-tuple Aˆ(a1, a2, a3, a4, a5, a6, a7), with aiˆ1 or 0. There are 128 such 7-tuples (27ˆ128). Two agents A and B in the hypercube are connected (by a two-way edge) if all the elements of A and B but one are identical. Thus, each agent is connected to seven others in this structure. The fourth organizational form we examined was a random structure in which each agent sees five others selected at random. The agents ``seen'' are sampled without replacement, to ensure that each agent sees exactly five others. In this organizational form, the digraph nature of the communication network matters; if Agent i sees Agent j, it is not required that Agent j also see Agent i. In addition, we test each network prior to running the simulation on it to make sure that it is connected, in the sense that there are no groups entirely isolated from each other (if there were, an innovation could fail to diffuse through the entire organization). In practice, this restriction appears to make little difference, so long as a path to every other agent can be traced from the particular agent first adopting the innovation. To compare the efficiency of different organizational structures, we simulated the progress of an innovation through each structure 500 times, and computed the sample means and standard deviations of the time (measured by number of cycles) to full adoption for each structure for different values of c. The sample statistics for time to adoption over 500 trials are listed in Table 2. The simulations were coded in Mathematica, Version 3.0 (Wolfram, 1996). The code is available from the authors on request. The sample statistics computed may properly be considered to be based on random samples drawn from underlying populations. Standard hypothesis tests can resolve differences in these means to a high degree of confidence. In the discussion that follows, we omit the language of formal statistical hypothesis testing because the differences in sample means are in most cases quite large relative to the standard deviations of the sample means.

6 7 6.036 0.008

8 8 8 0

Hierarchy: min four managers, max groups of five mean s. d. of mean

min max mean s. d. of mean

min max mean s. d. of mean

Hypercube

Random Connections, each agent sees five others

5 7 5.452 0.024

2 3 2.048 0.010

5 8 5.448 0.025

8 8 8 0

6 7 6.034 0.008

3 14 4.946 0.059

c ˆ100 9

min max mean s. d. of mean

Completely connected organization

5 7 5.454 0.024

8 8 8 0

6 7 6.036 0.008

4 17 5.728 0.092

8

Value of c parameter

5 7 5.438 0.024

8 8 8 0

6 7 6.040 0.009

4 33 7.172 0.146

7

5 7 5.450 0.024

8 8 8 0

6 7 6.038 0.009

4 38 9.194 0.196

6

Table 2 Cycles to adopt for four representative firm types (nˆ128; 500 runs each)

5 7 5.458 0.023

8 8 8 0

6 7 6.034 0.008

4 38 11.526 0.247

5

3

2

1

0.9

0.8

0.7

0.6

5 8 5.484 0.025

8 8 8 0

6 7 6.026 0.007

5 8 5.484 0.026

8 8 8 0

6 8 6.104 0.014

5 8 5.466 0.025

8 8 8 0

6 12 7.130 0.040

8 19 11.79 0.082

6 18 8.634 0.071

10 30 15.408 0.143

7 22 11.408 0.113

12 50 20.908 0.247

8 11 65 115 16.264 27.164 0.261 0.574

14 19 73 163 30.95 51.854 0.419 0.829

6 17 29 48 97 150 21 379 506 825 1401 2115 8.150 92.23 148.57 234.15 377.95 636.75 0.069 2.373 3.778 5.639 8.532 14.91

5 6 11 19 42 44 67 86 63 116 198 400 615 588 671 1152 17.598 26.266 51.418 140.14 163.24 201.16 241.27 297.07 0.462 0.680 1.341 3.059 3.606 4.394 4.987 6.422

4

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3.2. Discussion First, compare the completely connected organization to the more bureaucratic, hierarchical structure with four managers overseeing groups of five. It is apparent that for high processing capability (large c), complete connectedness is superior. The theoretical limit for how quickly the organization adopts the innovation is two cycles; one for introduction of the innovation and the second for its adoption by everyone else. This theoretical limit is almost achieved in the completely connected case for the high value of cˆ100 in Table 2. However, as c declines, the hierarchical structure adopts the innovation more rapidly than the completely connected one. For the range cˆ7 through cˆ0.9, the time to complete adoption is shorter for the managed hierarchy than for the completely connected organization. Another way of describing this result is to say that improved processing can support a ``flattening'' of the hierarchy. This is shown by the superiority of the completely connected (totally flat) organization over the hierarchical form for high values of c(8). At the same time, the hierarchical structure exhibits a constant time to adoption for all c greater than a threshold (which occurs at around cˆ4). The time to adoption is about six cycles for all c4, which is the minimum time to adoption in this structure even if all its members are perfect processors. This means that if the organizational structure is fixed and hierarchical, there is a point at which there is no payoff from increasing the processing capability of the members. The reason is that the existence of ``bureaucracy'' slows down the diffusion of innovation even for more capable agents. Because of the paucity of connections, it is necessary for an innovation to be accepted by the team in which it is first introduced, then be accepted by the manager overseeing that team, from which it diffuses through the management level and finally goes down to the other lowlevel teams.18 These steps take time, even if each individual agent adopts the innovation more quickly because of enhanced processing capability. Thus, bureaucracy can overcome the non-linear effects of ``information overload'' when information-processing capability is weak, but impedes the process of diffusion as processing power increases. There is another feature of the trade-off between bureaucracy and informationprocessing capability that is evident from Table 2. Consider further the comparison between the completely connected organization and the firm organized into teams of five with four managers. At the lowest end of information-processing capability (c0.7), the completely connected organization again becomes superior to the bureaucratic one. This might be characterized as the ``democracy effect.'' As processing power declines, the teams in the bureaucracy become very slow to adopt the innovation. The completely connected organization's members are slow to react also, but once they do, the innovation diffuses through the organization unimpeded by structural barriers. The bureaucratic organization, on the other hand, is slowed both by the inertia of its members and by the limited number of channels through which information can flow.19 18

That this process takes a minimum of six steps can be verified by counting through the cycles, assuming that agents adopt the innovation as soon as they see it. 19 As Winston Churchill said, ``[m]any forms of government have been tried, and will be tried in this world of sin and woe. No one pretends that democracy is perfect or all-wise. Indeed, it has been said that democracy is the worst form of Government except all those other forms that have been tried from time to time'' (Augarde, 1991 citing Churchill in Hansard, November 11, 1947).

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Simple hierarchies may have advantages over complete connectedness when information processing capability is weak, but other forms of restricted connectedness can be even better than the four-manager hierarchy shown in Table 2. First consider the Hypercube, shown as the third organizational form in Table 2. For very high processing capability, the theoretical minimum number of cycles required for the organization to completely adopt the innovation is eight, and this minimum is reached for values of c greater than or equal to 2. Thus the highly structured but non-hierarchical hypercube is inferior to the hierarchically structured organization for high processing capability agents (values of c3). On the other hand, the hypercube is quite methodical in diffusing the innovation. For organizations with high processing power agents, it takes exactly eight cycles for the innovation to diffuse in the simulations we ran. For low-processingcapability agents, the hypercube is superior to both the completely connected organization and the structured hierarchy. For low-capability agents, there is a great advantage in reducing the amount of information overload and doing so in a systematic, non-hierarchical manner. It might be thought that this kind of symmetric non-hierarchical connectedness that reduces information overload would be the best organizational form, but that does not appear to be the case. For an organization with random connections between the agents performs better (on average) than the hypercube for the range of c values shown in Table 2. The last set of statistics in Table 2 show the results for just such an organization. It is theoretically possible (though unlikely) for an entire organization of this type to adopt in only two cycles, if every agent happens to observe the first innovator, although this never happened in the 500 simulations we performed. However, because each group is a random sample drawn from the whole organization, it is likely that everyone will observe the innovation after a small number of cycles. Thus, with high processing power, the number of cycles required for complete diffusion is low. For cˆ100, the average number of cycles to achieve complete diffusion is about 5.5 for the organization in which each agent sees five others. This random connectedness is inferior to the completely connected organization for high values of c (c9) but is superior to every other organizational form shown in the table for lower values of c. This result highlights the advantages of widespread interlocking as opposed to a rigid bureaucracy. The randomly connected organization is superior to the structured organizations (both hierarchical and non-hierarchical) for the range of processing power shown. While bureaucracy can overcome the information overload that comes with complete connectedness, it imposes a friction that impedes the diffusion process. Random connectedness captures the advantage of reducing the information overload (so that the non-linearity of the f function does not make adoption by the group extremely unlikely), while avoiding the slowdown of diffusion caused when the diffusion has to proceed through ``channels'' in a bureaucratized structure. Of course, this result is consistent with the well-known feature of real organizations that informal communications are extremely important and contribute to the efficient operation of the organization. It is also likely that it depends on the simplifying assumption that all agents have equal processing capability. Whether hierarchical structures can be advantageous if agents have differing abilities is a subject for future research.

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We do not know whether it is possible to determine the optimal structure, even in the simple case of identical agents. As noted earlier, the combination of the very large number of possible graphs and the non-linearity of the transition function f make either closed form or polynomial time algorithmic solutions unlikely. Even if some kind of sparse but optimally connected structure could be specified for some particular value of c (or more generally, for a particular transition probability function), it is certainly the case that a different c would be associated with a different optimal structure. Experimental results (not shown in Table 2) show that for randomly connected organizations, the average speed to adoption may be greater or less than the average speed for organizations in which each agent sees five others, depending on the value of c. That is, there is no degree of random connectedness that is superior for all levels of processing capacity. In addition, the time to complete adoption varies considerably for different randomly connected organizations. Some of this variance is due to the randomness of adoption by individual agents, but some ``random'' organizations may also be better than others at adopting the innovation. Table 2 shows that the variance in time to adoption is greater for the randomly connected organization than for either the hierarchy or the hypercube for high levels of processing capability. This suggests that some ``randomly connected'' organizations are more efficient than others. We expect that the average speed to adoption for an organization is related to graph-theoretic characteristics of its network, such as the circumference and diameter of the graph, or the average status and eccentricity of the nodes.20 The simulation methods developed here should allow for statistical testing of these and other hypotheses about the relationship between organizational structure, the processing power of the agents, and the organization's speed in adopting innovations. Also, it must be kept in mind that the structures we have been discussing are very simple, and that the amount of information being processed is quite small (each agent stores only one bit). In real life, the information-processing capability of each agent would be different; the information-processing tasks would be more demanding; and the size of the relevant organizations would be larger. A typical large multinational firm can have 40,000 employees or more. We plan to explore the effects of generalizations that take account of these factors in subsequent work. 4. Conclusions and directions for further research A number of tentative conclusions can be drawn from the numerical simulations of the model in Section 3. 1. For an organization of any given size consisting of agents with a given degree of information-processing capability, the efficiency of the organization varies with its structure. Also, the efficiency of any particular structure varies with the processing power of the agents in the organization. 20

Circumference is the length of the longest cycle in a graph; eccentricity of a node is the distance to a node farthest from that node; diameter is the maximum eccentricity of the nodes of a graph; and the status of a node is the sum of the distances to every other node.

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2. Introduction of hierarchy, breaking the organization down into teams, or otherwise limiting channels of information flow can have efficiency advantages for some ranges of agents' processing power. 3. Organizational structure may limit the efficiency gains obtainable by increasing the information-processing capabilities of members of the organization. 4. Non-hierarchical structures may be superior to hierarchies. Clearly, the simple information-processing framework outlined here is sufficient to provide content to the theory of the firm. The results obtained from numerical solution of the simple model also allow predictions to be made for firms operating in the real world. For example, it is a prediction of the model that a ``flattening'' of organizations (elimination of layers of management) would be one result of improved informationprocessing capability by the agents. Similarly, the model suggests that an organization's size may constrain its speed in adopting innovations. The flattening and downsizing that have been the most prominent features of the business world in recent years may fundamentally be unintended consequences of the computer revolution, as computers have multiplied the information-processing capabilities of employees.21 More generally, this way of thinking about organizations illuminates some of the deeper regularities of human and social behavior. The ubiquitous presence of structure and hierarchy in social organizations, sometimes referred to as the ``Iron Law of Oligarchy'' (Burnham, 1943 citing Michels, 1915),22 may be a consequence of the most fundamental information-theoretic features of human action. The connection is not necessarily a simple one, however, for unstructured organizations that handle the information overload problem can be superior to hierarchical ones. The limited processing capacity of individuals means that organizations that are able to structure intelligently their internal communications and limit the channels through which their members receive information will have a competitive advantage.23 This is opposite the usual ``more is better'' presumption of conventional economic theory. While it seems clear that more information-processing capacity is better than less, it is also clear that more raw information can be a barrier to productive change when processing capacity is limited. Also, the question of whether the necessity of structure implies the necessity of hierarchy is open. There are very many digraphs in which the number of connections are limited but which are not hierarchical in form. It should be possible to use simulation methods to evolve organizational structures with superior performance in adopting an innovation. Miller (1996) has demonstrated that a genetic algorithm can allow organizations to ``learn'' better structures for solving decomposable (associative) problems. 21 Brynjolfsson and Hitt (1996), citing Milgrom and Roberts (1990), Malone and Rockart (1991), Scott Morton (1991), and their own discussions with managers, state that ``recent economic theory has suggested that `modern manufacturing,' involving high intensity of computer usage, may require a radical change in organization'' (pp. 556±557). 22 The first English translation of Michels' Political Parties appeared in 1915. The original German version appeared in 1911 (Burnham, 1943). 23 In situations of extreme informational ``noisiness,'' such as combat, extreme hierarchy may be required. This is probably why the chain of command is so much more rigid in military organizations than in civilian ones.

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These findings have practical and policy implications. For example, the conclusion from some macroeconomic general equilibrium models that greenhouse gas emissions abatement policies will impose large costs on the economy rests on the assumption that the productive units in the economy are located on their production-possibilities frontiers (DeCanio, 1997). But considerations of computational complexity suggest that it may be impossible to determine the optimal structure of even moderately-sized firms so that improvements in performance will always be possible, especially in a technologically dynamic environment. The role of management (and policy) is constantly to seek ways to improve organizational performance, knowing that the task is unending. In a setting where the optimal organizational structure of the firm is too complex to compute, it is plausible that simultaneous improvement along more than one dimension of the firm's performance (e.g., profitability and environmental impact) can be achieved. Finally, this model and the others cited in the text point the way to a reconstruction of the theory of the firm and other organizations. Viewing the firm as an informationprocessing network yields sufficiently rich implications to warrant further exploration. Computing power has increased sufficiently to allow numerical solutions to models that are conceptually simple and relatively easy to implement, but which would resist analytical solution. Ironically, the increased processing power that is now available to researchers may make it possible to develop fully the insights offered by models of human organizations as information-processing networks.

Acknowledgements This work was partially supported by grants from the U.S. Environmental Protection Agency. Research assistance was provided by James E. Grefer. We have benefited from the input of Lisa Bernstein, Stanley Burris, Penelope Canan, Catherine Dibble, Barry Galef, John Gliedman, Rich Howarth, Jonathan Koomey, Glenn Mitchell, Nancy Reichman, Marc Ross, Nancy Samson, Alan Sanstad, and Jeffrey Williams. Earlier versions of this paper were presented at the Third International Conference on Computing in Economics and Finance at Stanford University, June 30±July 2, 1997, and at the Western Economic Association International 73rd Annual Conference, Seattle, July 9±13, 1997. The views expressed are those of the authors alone.

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