Design of Decision-Making Organizations

MANAGEMENT SCIENCE informs Vol. 56, No. 1, January 2010, pp. 71–89 issn 0025-1909  eissn 1526-5501  10  5601  0071 ® doi 10.1287/mnsc.1090.1096 © 2010 INFORMS Design of Decision-Making Organizations INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. Michael Christensen, Thorbjørn Knudsen Strategic Organization Design Unit, Department of Marketing and Management, University of Southern Denmark, DK-5230 Odense M, Denmark {[email protected], [email protected]} S tarting from the premise that individuals within an organization are fallible, this paper advances the study of relationships between the organization’s decision-making structure and its performance. We offer a general treatment that allows one to analyze the full range of organizational architectures between extreme centralized and decentralized forms (often referred to as hierarchies and polyarchies). Our approach furthermore allows designers to examine the change in the overall reliability of the organizational structure as the number of actors within the organization changes. We provide general proofs that show how decision-making structures can be constructed so they maximize reliability for a given number of agents. Our model can be used directly for a qualitative assessment of decision-making structures. It is thereby useful for assessment of the many complicated hybrid structures that we see in actual decision-making organizations, such as banks, purchasing departments, and military intelligence. An application from a bank illustrates how our framework can be used in practice. Key words: organizational architecture; organizational design; decision making; evaluation History: Received June 15, 2007; accepted April 28, 2009, by Olav Sorenson, organizations and social networks. Published online in Articles in Advance November 6, 2009. 1. Introduction of decision makers is sufficient for an alternative to be approved—will tend to minimize the probability of rejecting a superior alternative—i.e., polyarchy reduces Type I errors. The essence of designing decision-making organizations is to choose a structure that most effectively reduces Type I and/or Type II errors as required by the organization’s task environment. We offer a general treatment of this problem. That is, we extend the decision structures considered by Sah and Stiglitz (1986, 1988) to include all possible organizations spanned by the hierarchy and polyarchy. We then show how organization designers can identify the structure that most effectively reduces Type I and/or Type II errors (given any number of available decision makers). Another literature, in information theory, has discussed a similar problem. An important theorem, the Moore and Shannon (1956a, b) theorem, has shown that perfect reliability of electrical circuits can be accomplished with imperfect components. It might seem that this theorem could be directly applied to organizations. However, it cannot because individuals at any level of a human organization can, in principle, be assigned the final decision-making authority. This violates a fundamental assumption of the original Moore-Shannon results, where such delegation is ruled out. This limitation shows up as the MooreShannon restriction, which rules out that lower-level members of an organization can be assigned the final decision-making authority (on behalf of the entire organization) to accept or reject a project. We therefore How does organizational design effect decision making? Decision-making organizations—such as boards, corporate headquarters, and purchasing departments—generally exist with various structures (Colombo and Delmastro 2008). At one pole, organizations are designed as hierarchies, with lower-level managers reporting to their immediate superiors. But within hierarchical layers, there can be considerable variation in the number of people that make independent, parallel decisions. As we approach the other extreme pole, managers reside in a flat structure, where they make decisions in parallel. Our aim here is to advance a general treatment of decision-making organizations that allows one to consider the full range of organizational architectures that lie between these two extreme forms. At a basic level of analysis, decision making can suffer from two possible errors: Type I errors of rejecting a superior alternative and Type II errors of accepting an inferior alternative. As shown in prior work (Sah and Stiglitz 1985, 1986, 1988), different organizational structures vary in their proclivity to make one type of error or the other. In particular, hierarchical structures, in which a proposal needs to be validated by successive ranks of the hierarchy to be approved, tends to reduce the likelihood that an inferior alternative will be adopted—i.e., hierarchy reduces Type II errors. In contrast, what Sah and Stiglitz (1986) term polyarchies—flat organizational structures in which approval by any one actor in a parallel series 71 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. 72 redo the proofs with different topological structures that are consistent with the way Sah and Stiglitz (1985, 1986, 1988) and others have characterized human organizations—as opposed to systems of relays and similar electric components. Our approach builds on the information processing perspective in economics. The properties that drive the design of decision-making structures are multifaceted, so it is not surprising that they have been investigated from many perspectives. Prior work on decentralization of information has provided robust results regarding the way organizational structures can be designed to optimize efficiency under capacity constraints (Marschak and Radner 1972, Radner 1993, Bolton and Dewatripont 1994, Van Zandt 1999). By contrast, Sah and Stiglitz (1985, 1986, 1988) and followers (Ioannides 1987, Koh 1992) have mainly focussed on human evaluation. We directly extend prior work initiated by Sah and Stiglitz (1985, 1986). That work laid the foundations for a systematic approach to designing reliable decision-making structures by comparing the screening properties of simple hierarchies against decentralized polyarchies (Koh 1992, 1994; Sah and Stiglitz 1988; Sah 1991; Visser 2000). The study of sequential decision making in hierarchies and polyarchies has been complemented by analysis of voting and other forms of decision making that happen simultaneously, such as committee decision making (Ben-Yashar and Nitzan 1997, Li et al. 2001, Sah and Stiglitz 1988). From a broader perspective, the many facets of organizational design have been studied in economics, management, and related fields. There is a huge body of work on organizational design, spanning multiple streams of literature and including the information processing stream (to which our work contributes), transaction cost economics, the decentralization of incentives stream (including principal-agent theory), the evolutionary and behavioral stream, and contingency theory (Colombo and Delmastro 2008 provide a useful review). By and large, however, the existing literature does not analyze the broad range of structures between hierarchies and polyarchies. Although the study of committee decision making included a broader range of decision-making structures, it is limited to voting and other forms of decision making that happen simultaneously (Sah and Stiglitz 1988). This paper advances the economic and managerial literature by offering a general treatment that allows one to consider the full range of organizational architectures between hierarchies and polyarchies. Thereby one can design decision-making structures that trade off Type I errors (of rejecting a superior alternative) and Type II errors (of accepting an inferior alternative) as the relative degree of hierarchy and polyarchy Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS shifts. Our approach furthermore allows designers to examine the change in the overall reliability of the organizational structure as the number of actors within the organization changes. We provide general proofs that show how decision-making structures can be constructed so they maximize reliability for a given number of agents. However, it is beyond the scope of the present work, and would be too hasty, to engage in a detailed treatment of costs and benefits that derive from alternative decision-making structures. The conclusion in §5 considers these limitations and how they can be lifted. Despite limitations, our model can be used directly for a qualitative assessment of decision-making structures, because error rates are correlated with costs and benefits. It is thereby useful for assessment of the many complicated structures that we see in actual decision-making organizations. This paper is organized as follows. In §2, we specify the basic model, including a very broad range of decision-making structures spanned by the hierarchy and polyarchy of Sah and Stiglitz (1985, 1986). Section 3 introduces the extremity theorem (Theorem 1), which identifies hierarchies and polyarchies of any size, n, as evaluation structures that bound all other structures of size n with respect to screening. Together with Theorem 2, it provides a building block that is necessary to establish the main result of the present paper. Two further results to be used in the main proof are established in §3. The first result (Theorem 3) identifies the decision structures that most effectively increase judgmental discrimination. The second result (Theorem 4) identifies the structures that most effectively remove bias. These results provide the components for a constructive proof that makes our analytical results readily available for practical applications. Our main result, the theorem of perfection (Theorem 5), is established in §4. A case study of credit evaluation in a bank illustrates how our analytical framework can be used in practice. Section 5 concludes. 2. The Model Following the approach developed by Sah and Stiglitz (1985, 1986, 1988), we study organizations whose members are decision makers (agents) that evaluate a set of alternatives (projects) and make a binary choice (screening). We refer to such organizations as decisionmaking organizations or, more briefly, as evaluation structures. The decision-making organizations under study face the problem of choosing which projects to accept and which to reject. It is a notable feature of the model that final acceptance is equivalent to making a commitment of unavoidable economic consequence. Figure 1 shows three decision-making structures: a three-member hierarchy, a three-member polyarchy, Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS Figure 1 Example of Evaluation Structures INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. Hierarchy Polyarchy Self-dual hybrid, G * Notes. A three-member hierarchy, a three-member polyarchy, and a selfdual graph G∗ . Full lines are acceptance edges and dashed lines are rejection edges. and finally a special five-member structure, G∗ , known as a self-dual structure. The decision-making structures consider projects from an initial distribution (I). Individuals either accept or reject projects as they pass through the decision-making organization. Solid lines indicate that accepted projects are passed on, and dashed lines show rejected projects that are passed on. The projects can be passed on to other agents, or a decision can be made on the part of the whole organization to either accept or reject a project. The contrast between the hierarchy and the polyarchy is clear. The hierarchy is a conservative structure that preserves the strictest possible filtering as projects move to the top of the organizational hierarchy. In contrast, the polyarchy is a permissive structure that preserves the loosest possible filtering. Theorem 1 shows that the hierarchy and polyarchy are indeed the extreme structures with respect to reducing either Type I or Type II errors. It is also interesting to note that the self-dual graph G∗ is composed of a single agent with an acceptance edge to a two-member polyarchy and a rejection edge to a twomember hierarchy. That is, optimistic evaluations are checked by a hierarchy, whereas pessimistic evaluations are checked by a polyarchy. This construction serves to promote the symmetric reduction of Type I and Type II errors that is characteristic of self-dual structures (as explained in more detail below). Projects are independent and identically distributed as a random variable x . Each project is represented by a vector of signals, x, about the true quality of the project. These signals include all elements that are relevant if the decision-making organization accepts a project. There is a map, V , from the vector of perfect 73 signals onto a scalar economic value, a net income that is obtained by the decision-making organization if the project is accepted. This scalar valuation is a measure of the true net income of a project, including all the relevant benefits and costs associated with undertaking a project. The costs of making the decision are not included in the income because they are endogenous to the evaluation structure.1 The task of the individual decision maker within the decision-making organization is to evaluate projects. An evaluation is a binary choice—whether to accept or reject a project—made on the basis of the vector of perfect signals about true project quality. The individual decision maker is characterized by an ability to evaluate projects. This ability is captured in the agent screening function, f , which maps each project onto a probability that the decision maker accepts the project. An omniscient decision maker would not make a single error of judgment. Such a decision maker would process all the signals about the true project quality without error. The omniscient decision maker therefore accepts projects with economic value V ≥ 0 <0 with probability f = 1 (0). That is to say, the agent screening function of the omniscient decision maker, f , is the functional composition of the Heaviside step function and the economic value, f ≡   V . All human decision makers are fallible; they make errors of judgment. Errors in judgment come from a variety of sources, including noisy signals, incomplete processing of information, and defective information processing. Errors in judgment reduce the ability to make choices in the sense of discriminating between projects with positive and negative economic value. In the case of maximal error, the agent has no discriminating ability; the decision maker simply processes the signals about project quality by flipping a coin, f = 1/2. At the opposite pole, the agent has perfect discriminating ability; the decision maker is omniscient and assigns signals about project quality to acceptance or rejection in a deterministic way (projects with positive value are accepted with probability 1, and projects with negative value are rejected with probability 1). In the general case, the level of errorin the agent’s processing of signals is the measure x f x − f x dx. There are n members in an evaluation structure. The task of the evaluation structure is to decide which projects to accept and which to reject. Its objective is to maximize income V net of evaluation costs or, in some cases, to minimize the incidence of Type I and Type II errors. The present effort is focussed on the 1 The costs of making the decision depend on the size of the organization (the number of agents), the levels of pay, the choice of compensation method (e.g., fixed salary or pay per evaluation), and the possibility of economies of scale with respect to evaluation. INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. 74 design of reliable decision-making organizations in the sense that they minimize the incidence of Type I and Type II errors subject to the constraints of the number of available evaluators and their ability.2 We show how this problem can be solved with respect to internal structure and system size as choice variables, holding constant the decision-making ability. Consider a system of fallible agents that are homogeneous in their decision-making ability.3 Each individual evaluator has access to two distinct types of communication channels; one is used when a project is accepted and the other in case of rejection. It is the availability of both of these communication channels that allows the evaluators to make independent deliberate choices that are characteristic of human agents. It is possible for an evaluator to individually accept or reject a project on behalf of the organization without interference by other members. The human agent can further choose how projects are sent to other agents and who the possible receivers are. An important background matter here is as follows. An individual at any level of a human organization can, in principle, be assigned the final decisionmaking authority (on behalf of the entire organization) to accept or reject a project. This is ruled out in the classical Moore-Shannon formulation of how to improve reliability in electrical circuits (Moore and Shannon 1956a, b) and more recent work on network reliability (Lynn et al. 1998; Balakrishnan and Rao 2001). This is because the Moore-Shannon formulation is limited to relays and similar electric components that cannot individually block an electric current. For brevity, we refer to the preceding restriction as the Moore-Shannon restriction. Formally, the MooreShannon restriction excludes edges from lower-level individuals to external rejection nodes. This has the immediate implication that there will be inherent dissimilarities between actual human organizations and organizations of electrical components. Apart from being a useful concept with which to contrast human organizations, our approach reuses and upgrades the techniques used in Moore and Shannon’s (1956a, b) proofs. Because of the Moore-Shannon restriction, however, we build on the very different topological structures that are consistent with the way Sah and Stiglitz (1985, 1986, 1988) and others have characterized human organizations. The decision-making organization is modeled as a graph. Each node represents a decision maker, and 2 In the case of perfect decision-making ability, the organization has no effect because there is no single error; all projects with positive income would be accepted and all projects with negative income rejected. 3 As mentioned in the appendix, some issues of heterogeneity in screening ability can be handled within the current framework. Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS each edge represents a channel of communication. We study homogeneous graphs (one type of agent, A) with two types of edges (accept/reject). The entry and the exit of the projects are determined by the way the internal structure is connected to three external nodes: (1) the initial portfolio (I) containing the distribution of projects x ; (2) the final portfolio (F), where the accepted projects are implemented; and (3) the termination node (T), where the rejected projects are dumped (see Figure 1). The design of the evaluation structure involves specification of edges that connect members, specification of edges that connect the internal structure to external nodes (I, F, T), and specification of rules determining how many times a decision maker can evaluate the same project (truncation rule). The generalization of the agent screening function f to the level of a specific architecture G is the graph screening function fG . If the organization contains only a single agent A, then obviously fG = fA = f , as G = A. The graph screening function is an aggregation rule that assigns individual decisions of acceptance and rejection to any structure. It can be viewed as a generalization of the aggregation rules that have previously been used to model decision making by committees (Ben-Yashar and Nitzan 1997, Sah and Stiglitz 1988). In mathematical reliability theory, the graph screening function is known as the system reliability (Lomnicki 1973) or the reliability function of a network (Carlsson and Grenander 1966). It is often useful to express the graph screening fG in terms of the reduced graph screening function FG that operates not on the vector of signals regarding the project but solely on the scalar acceptance probability,  = f x. In the case of homogeneous ability, the graph screening function is a polynomial in , commonly known as the reliability polynomial.4 The reliability polynomial is the reduced graph screening function, and the relation between the screenings is again that of functional composition, fG = FG  f . 3. Fundamental Organization Structures Some organization structures have a fundamental role in reducing error and thereby improving the quality of decision making. Sah and Stiglitz (1985, 1986) identified two archtypical evaluation structures: hierarchies and polyarchies. These structures also play a fundamental role in our generalization of the MooreShannon result. Their primary role is to move a screening function to the desired part of the project distribution—hierarchies move the screening function to the “right,” whereas polyarchies move the screening function to the “left” (see Figure 2). The purpose In the case of j levels of heterogeneous ability, j = fj x, the graph screening function is a multinomial in j . 4 Christensen and Knudsen: Design of Decision-Making Organizations 75 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. Management Science 56(1), pp. 71–89, © 2010 INFORMS of moving a screening function is to remove bias. However, pure hierarchies (polyarchies) can overshoot the desired move, so it is sometimes necessary to use a slightly modified version of these graphs. This introduces a new set of graphs, called stair graphs. Stair graphs actually resemble stairs in that they are constructed with a few polyarchical (hierarchical) “interruptions” of longer hierarchical (polyarchical) sequences. Stair graphs serve the same purpose as the so-called ladder graphs in the original proof (Moore and Shannon 1956a, b). To improve reliability at a desired point in the project distribution, the graph must be steepened. This is achieved by graphs that are similar to self-dual committees, i.e., committees with an uneven number of members that use a simple majority vote (Sah and Stiglitz did not consider the self-dual property of committees). The notion of a self-dual graph refers to a decision-making structure that will induce a symmetrical graph screening function, i.e., a symmetrical reduction of Type I and II errors. We are considering sequential decision making, so we identify a set of self-dual structures that represent a sequential processing of projects (in the same way self-dual committees do for voting procedures). Self-dual graphs serve the same purpose as the so-called hammock graphs in the original proof. The theorems of this paper are proved in the appendix, and all the proofs are constructive to not only guarantee the existence of almost-perfect organizations but to show exactly how such organizations are obtained. These construction methods are directly applicable in real business cases, as explained after Theorem 2, and further illustrated in §4.3. The theorems of this section therefore serve two purposes. They are building blocks for the main perfection theorem and its extension (presented in §4). And they are guides on how to improve discriminating ability and remove bias (illustrated in §4.3). 3.1. Extremity of Hierarchies and Polyarchies This section introduces a theorem that identifies hierarchies and polyarchies of any size, n, as evaluation structures that bound all other structures of size n with respect to screening. As described above, hierarchies are extreme because they are the structures that filter out most information on the way to the top (at the expense of loosing good projects in the process). In contrast, polyarchies are the extreme permissive structures that allow the most information through (at the expense of promoting low-quality projects). The extremity theorem provides a necessary building block that enables us to provide a general tool for the design of reliable decision-making organizations (§4). A simple version of the theorem is presented here. The proof is in §A.1 of the appendix, together with a number of extensions. Theorem 1. Let n for any positive integer n be the set of all homogeneous graphs that can be constructed from agents of type A, such that the maximal evaluation count is no more than n. Let Pn ∈ n be the polyarchy of n members with reduced graph screening function FPn , and let Hn ∈ n be the hierarchy of n members with reduced graph screening function FHn . Any graph G ∈ n with reduced graph screening FG satisfies FHn ≤ FG ≤ FPn  (1) The proof, which appears in the appendix, is inductive on maximal evaluation count, i.e., the maximal number of evaluations that is necessary for the organization to finally accept a project. For intuition, note that the addition of a new member to a hierarchy can either soften or enhance extreme sceptical evaluation (i.e., minimize Type II error). If the new member is put in a position that extends the hierarchy, extreme sceptical evaluation is enhanced. Otherwise, it is softened (i.e., becomes less extreme). Similar reasoning captures the addition of new members to a polyarchy. In extreme situations, when there are only projects with positive (V > 0) or negative (V < 0) income, the n member polyarchy (hierarchy) will therefore dominate any other structure. Provided the costs of making the decision in an evaluation structure are not prohibitive, the n member polyarchy (hierarchy) will also dominate the individual agent.5 Even if this result is rather trivial, it holds with remarkable generality (see §A.1), even to the extent of lifting the homogeneity assumption and letting the agents creatively manipulate the projects. According to the extremity theorem, finite hierarchies (polyarchies) map the agent screening of any project closer to 0 (1) than any other structure with the same number of agents. The minimal number of agents required to reduce the incidence of Type I and Type II errors to some minimal desired level is provided in Theorem 2. The remainder of the paper shows how decision-making organizations that include both hierarchical and polyarchical elements can be designed to minimize both Type I and Type II errors. Theorem 2. Given any threshold 0 <  < 1 and a point 0 ∈0 1 , the number of agents n in a hierarchy such that FHn  ≤  ∀  ∈ 0 0 , is n≥ log  log 0 (2) 5 In situations with more realistic project distributions that include both positive and negative income, the optimal evaluation structures are hybrids that include both hierarchies and polyarchies. Christensen and Knudsen: Design of Decision-Making Organizations 76 Management Science 56(1), pp. 71–89, © 2010 INFORMS and the number of agents n in a polyarchy such that FPn  ≥ 1 −  ∀  ∈ 0 0 , is log   log1 − 0  (3) Proof. The result follows trivially from the graph screening functions of the n member hierarchy Hn and polyarchy Pn : FHn  = n and FPn  = 1 − 1 − n   (4) To illustrate, consider a CEO who wants to hire a new manager, knowing that the available managers are too optimistic. The CEO’s reservation value is set to = 0 (as shown in Figure 2). Further, assume that available managers have a screening function like Christie’s in Figure 2. The CEO knows he must use a hierarchical structure to reduce or perhaps even remove the optimistic bias. How large should the hierarchy be? In Christie’s case, = 0 and 0 = f   ∼ 088. In contrast, for an unbiased (imperfect) screening function, 0 = 050. Assume that the CEO’s target is to build a structure that achieves FHn  ≤  = 055; i.e., the CEO accepts a tolerance of 10% from the target of the unbiased screening function of 0.50. According to Theorem 2, the CEO finds that a hierarchy of n ≥ log / log 0 ∼ log 055/ log 088 ∼ 468 will meet his target. A comparison of the four- and five-member hierarchy leads to the conclusion that the resulting unbiased evaluation structure is a hierarchy employing five managers, all with a bias similar to Christie’s. The CEO has now designed an unbiased evaluation structure from biased members. Its screening function is shown in Figure 2. Note that if Christie’s bias had been pessimistic, the CEO should Figure 2 Removal of Bias and Further Improvement of Judgmental Ability 1.0 0.8 Christie F(),  = f (x) INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. n≥ have chosen a flat, polyarchical structure to remove the bias. We have illustrated how a decision-making bias can be removed such that the incidence of Type I and Type II errors is balanced around the CEO’s reservation value, . By choosing an appropriate design, including both hierarchical and polyarchical elements, it is possible to achieve further improvements that reduce both Type I and Type II errors. To achieve this, the CEO must use a self-dual evaluation structure (a formal characterization and proof for optimality are provided in §3.2). Such an evaluation structure achieves a symmetric improvement of a screening function. The self-dual graph used here will also appear in our general proof. To use this self-dual graph, the CEO hires five unbiased managers and places them in the structure, G∗ , as shown in Figure 1. If the only available managers are of Christie’s type, another possibility is to design a structure where each of the nodes in the self-dual graph is a five-member hierarchy. As shown in Figure 2, Christie’s unbiased screening function is further steepened. As a result, such decision-making organizations make better evaluations than organizations using managers of similar ability on an individual basis. Our general proof shows how many agents are required to achieve a given level of reliability, given their inherent ability. In principle, the self-dual graph could be used repeatedly to approach perfect screening. For each repeated use, the decision-making organization increases by a factor 5; very quickly, this organization would grow to unrealistic proportions. The important point, however, is that significant incremental improvements can be achieved with a small number of agents. Even if we abstract from decision-making costs here, it should be clear how our approach identifies cost components and tradeoffs. The relevant question from the point of view of a practical application is what the marginal costs and gains from a redesign of the decision-making organization (gains versus personnel costs and reorganization costs) are. We provide further illustration from a real-world example from a bank. Christie’s bias removed 0.6 0.4 Christie’s screening further improved 0.2 0 –3 –2 –1 0  1 2 3 X 3.2. Optimality of Self-Dual Committees We now characterize decision structures that are optimal with respect to increasing judgmental discrimination. The aim is to identify the most effective symmetrical graph screening function, i.e., the decision-making structure that most effectively reduces both Type I and II errors. Such a symmetrical screening function is known as a self-dual graph. At the limit, a self-dual graph will approach a step function; i.e., projects below some criterion are accepted with probability 0 and projects above this criterion will be accepted with probability 1. That is, we identify the optimal self-dual graph. Christensen and Knudsen: Design of Decision-Making Organizations 77 Management Science 56(1), pp. 71–89, © 2010 INFORMS INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. Theorem 3. For any positive integer n, let n be the set of all self-dual graphs that can be constructed from agents indistinguishable with respect to project screening, such that the maximal evaluation count is no more than n. The slope of the screening polynomials at  = 1/2 of the graphs in n cannot exceed that of FGn∗  = n  Bi  i i=n+1/2 −1i−n+1/2 n + 1 Bi =  n + 1/22 i  with n − 1/2 i − n + 1/2  (5) and at least one graph Gn∗ ∈ n has this reduced graph screening. Here, is the Gamma function. Interestingly, the screening polynomial given by Equation (3) matches the screening of a self-dual committee (Sah and Stiglitz 1988). Such a committee always consist of odd n members and consensus k = n + 1/2, i.e., the simple majority rule. The proof in §A.2 of the appendix shows how to build a graph with the optimally discriminating screening without using consensus rules. These are the graphs with the steepest graph screenings FGn∗ 1/2 = n + 1 1 2n−1  n + 1/22 (6) in the middle of the reduced screening interval. This proof has two stages. First, the polynomial view finds the optimal reduced screening function with respect to discriminating ability. Then the topological view shows by construction that there exists a unique graph with the found optimal screening polynomial. For intuition, note that the self-dual graph G∗ shown in Figure 1 has symmetric checks on optimistic and pessimistic evaluations. That is, optimistic (pessimistic) evaluations are checked once before they are accepted (rejected) by the organization. Note further that a reversal in a decision leads to an additional check. For example, if an optimistic evaluation is rejected, there will be an additional check. But this additional check is omitted when the first two agents are consistently positive. This construction serves to promote a symmetric reduction of Type I and Type II errors that is characteristic of self-dual structures. The proof establishes that there exist unique selfdual structures that correspond to optimal screening polynomials. 3.3. Stair Graphs The final family of graphs used in the main proof is stair graphs. A graph is a stair graph if and only if it has one entry point and no loops and each agent has at most one channel of communication to another agent. The generic stair graph is therefore just a chain of agents arranged as a hierarchy whose last agent rejects to a polyarchy whose last agent accepts to a hierarchy, etc. As in the family of committees of various degrees of consensus, hierarchies and polyarchies are also the extremes of the family of stair graphs. The stair graphs have been chosen for (at least) three reasons. First, they have monotonous graph screening polynomials (because there is no branching in the structure), which is a crucial property if any of the work of Moore and Shannon (1956a, b) is to be reused in the current setup. Second, the size equals the maximal evaluation count. Third, they are very effective at removing bias, as the following theorem shows. Theorem 4. For any 0 < 0 < 1 and any 0 < < d ≡ min0 1 − 0 , a stair graph G exists with no more than n = log /2/ log d (7) agents satisfying the relation: FG 0 −  < 1/2 < FG 0 +  (8) The proof goes by sequential construction, ensuring convergence at the desired point 0 as the number of agents is increased. The proof is found in §A.3 of the appendix. The intuition is that hierarchies are the most effective way to move a screening function to the “right,” and polyarchies are most effective in moving a screening function to the “left.” Because pure hierarchies (polyarchies) can overshoot the desired point 0 , including a polyarchical (hierarchical) element can always achieve the necessary correction. In effect, this leads to construction of stair-like graphs with a few polyarchical (hierarchical) “interruptions” of longer hierarchical (polyarchical) sequences. More precisely, the theorem shows that any screening function can be modified via a stair graph to cross from mainly rejecting to mainly accepting projects at any 0 with an arbitrarily low tolerance . Picking the shift point to occur within the set of projects of zero value, V = 0, will remove any bias of the original screening under the assumptions of perfect and positive correlation between screening and project value. 4. Approaching Perfection Our approach reuses and upgrades the techniques used in Moore and Shannon’s (1956a, b) proofs. We cannot build directly on the original proofs because they were limited to structures comprised of electrical components. As previously mentioned, this limitation shows up as the Moore-Shannon restriction, which rules out that lower-level members of an organization can be assigned the final decision-making authority (on behalf of the entire organization) to accept or reject a project. We therefore redo the proofs with different topological structures that are consistent with the way Sah and Stiglitz (1985, 1986, 1988) and others have characterized human decisionmaking organizations. Because most readers are probably not familiar with Moore and Shannon’s (1956a, b) original proofs, it is useful to briefly summarize the main result and assumptions. First, starting from a single component, they showed how the graph can be incrementally improved when additional components are added. Second, they showed (see Theorem 6) that it is possible to build a graph out of unreliable electric components (relays) with a screening performance that deviates arbitrarily little from perfection at the infinite limit. The original results build on the following three assumptions: (1) the agents are able to discriminate between projects with positive and negative value; (2) the agents are more likely to accept projects with positive than with negative value, i.e., the correlation between project value and agent screening is perfect and positive; and (3) the graph screening function is monotone (see Moore and Shannon 1956a, b, Equation (4)). Because we break with the Moore-Shannon restriction to characterize (human) decision-making organizations, we redo the original proof with new topological structures and thereby introduce a new formalism. It is important to note that our new graph formalism lifts assumption (3) so that our results hold even if the graph screening is nonmonotone. This is necessary because the assignment of final decisionmaking authority to lower-level members will often lead to nonmonotone graph screening. We proceed as follows. First, we establish Moore and Shannon’s (1956a, b) result for decision-making organizations, i.e., human organizations that are built of individuals who can, in principle, be assigned the final decision-making authority. This result is established under assumptions (1) and (2). Second, we show that it is possible to build reliable decision-making organizations of unreliable human agents even if assumption (2) is dropped. That is to say, if only the agents are able to discriminate between projects with positive and negative value, a reliable decision-making organization can be designed. The surprising implication is that it is possible to design reliable organizations even when individuals are severely misguided such that they confuse negative project value with a positive value and vice versa. 4.1. Perfection: The Single-Step Function We now establish Moore and Shannon’s (1956a, b) result for (human) decision-making organizations under assumptions (1) and (2). The projects can be divided into two disjoint sets according to the sign of their value. A perfect positive correlation between Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS Figure 3 A Graph Screening Function That Separates Disjoint Sets of Project Value on the Basis of Screening Outcomes in a Point 0 Satisfying Condition (9) 1.0 1–  F() INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. 78 0.5  0 0 0 – 0 0 + 1  screening outcomes and project value ensures that the corresponding sets are disjoint and ordered in a similar way. The proof operates in the reduced screening function space. As a consequence, the task of constructing a graph screening that deviates arbitrarily little from perfection can be described as constructing a graph whose reduced screening polynomial F shifts from a probability arbitrarily close to 0 to a probability arbitrarily close to 1 within an arbitrarily narrow interval around some point 0 . The point of the desired shift, 0 , in the graph screening function is any point that separates the two disjoint sets relating to screening outcomes. At this point, it is useful to define perfection at the level of decision-making organizations, i.e., the perfect graph screening function. The perfect graph screening function is identical to the perfect agent screening function, f x. The deviation from perfection can be expressed by two parameters, and . As shown in Figure 3, the parameter defines the interval around 0 . The parameter  defines the deviation from the extreme screening outcomes of 0 and 1. The error rates of the screening polynomial should not exceed  outside the interval 0 − 0 + : F 0 −  <  and F 0 +  > 1 −  (9) The boundary points, 0 ∈ 0 1, are trivially dealt with by polyarchies and hierarchies of increasing size, as shown in the extremity theorem (see Theorem 2). For any other 0 , we now present a theorem that includes Moore and Shannon’s (1956a, b) perfection theorem as a special case: Theorem 5. Given any position 0 < 0 < 1 for the shift in graph screening, any threshold 0 <  < 1/2, and any Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS radius 0 < < min0 1 − 0 , then an architecture can be constructed from no more than   log 5/log11/8 1 log /2 · nck ≤ 25 · logmin0 1 − 0  2 log 5/log 2  log3 · (10) log3/4 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/.  agents whose reduced graph screening polynomial fulfills condition (9). If  > 1/3, the number of agents to achieve the required level of reliability in Theorem 5 reduces to   log5/log11/8 1 log /2 ·  (11) nck ≤ 5 · logmin0 1−0  2  The major obstacle in the generalization of the original proof (Moore and Shannon 1956a, b) originates in the fact that human decision-making structures can delegate decision rights to lower hierarchical levels. Including this characterization of human organizations in the formalism will often lead to nonmonotonous reduced graph screening functions. Because the original proof only goes through under the assumption of monotonicity (see the fundamental Equation (4) of Moore and Shannon 1956a, b), it does not hold for all social and economic decision-making organizations. Moreover, the specific graphs used in the original proof cannot be applied here because they are not general enough to include social agents that have the powers to reject or accept a project on behalf of the organization. To overcome these obstacles, we extend Moore and Shannon’s (1956a, b) proof by including entirely new structures whose members can be assigned the final decision-making authority to accept or reject a project. Our result includes Moore and Shannon’s (1956a, b) original proof as a special case.6 6 For comparison, the original bound of Moore and Shannon translated to the present setup is   log 9/log3/2 1 log /2 · logmin0 1 − 0  2  √ log 9/log 3 log 8 ·  √ log1/ 2  nms ≤ 81 · As can be seen, performance is always better for human decisionmaking structures than for electrical circuits (as treated by Moore and Shannon). This is because our proofs build on an assumption that has relaxed the Moore-Shannon restriction. Because this gives us fewer constraints upon optimization or upon incremental change, performance will be better (or at least no worse) than with the Moore-Shannon formalism. 79 Sketch of Proof. The proof can be found in §B of the appendix, and it uses a technique similar to that of Moore and Shannon (1956a, b) for constructing an explicit graph. To ease the exposition, the proof mainly elaborates on the extensions that are necessary to generalize Moore and Shannon’s (1956a, b) proof to encompass graphs that include human agents with powers to reject and accept projects on behalf of the organization.7 This paper uses the technique of constructing an explicit graph provided by Moore and Shannon (1956a, b). As in Moore and Shannon (1956a, b), the proof will, for reasons of mathematical convenience, be provided in a construction process with three steps, referred to as the opening, middle game, and end game. The opening game consists of finding an architecture with n members such that the point 0 is the position of the shift in the reduced graph screening function. That is to say, the graph screening function is moved over such that it intersects the value 1/2, F 0 −  < 1/2 and F 0 +  > 1/2. The middle game consists of steepening the graph by recursive expansion (thus increasing n) such that F 0 −  < 1/4 and F 0 +  > 3/4. Recursive expansion is the procedure of replacing each agent in a selfdual and highly discriminating graph with a copy of the architecture found in the opening game. The end game then consists of further steepening the graph by recursive expansion to obtain the required level of reliability with a total of n agents such that condition (9) is fulfilled. The solution to the general problem is a total of s expansions of a simple graph having as agents the architecture found in the opening game. This procedure is known as composition. The intuition behind the proof relates organizational performance (in terms of error reduction) to the properties of decision-making graphs. A special class of graphs has the property that the graphs will steepen the slope of the screening function when they are recursively expanded; i.e., each node is replaced with a copy of the graph itself. These graphs are referred to as self-dual graphs. With each expansion, more agents are added and the decision-making structure becomes more discriminating. The boundary points are dealt with by applying the extremity proof (Theorem 1); i.e., the hierarchy is used to steepen the screening function at the “right” boundary and the polyarchy is used in a similar way at the “left” boundary. Finally, note that the extremity proof also shows that the most effective way of moving a screening function to the desired part of the project distribution is to use a hierarchy (move to the 7 Moore and Shannon’s (1956a, b) proof is readily available in the original as well as in reprint (Sloane and Wyner 1992). Christensen and Knudsen: Design of Decision-Making Organizations INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. 80 Management Science 56(1), pp. 71–89, © 2010 INFORMS “right”) or a polyarchy (move to the “left”). If hierarchies or polyarchies “overshoot” the movement, a modified stair graph is used (as explained earlier). If necessary, the graph can be adjusted so the desired level of reliability is achieved. This is done by nesting the graph in a hierarchy or polyarchy (or a modified stair graph). The proof then ensures that the desired level of reliability (in terms of minimizing both Type I and Type II errors) is achieved by adding the fewest possible agents. This means that the proof can be used for design purposes, i.e., to achieve the maximal incremental improvement or to build the most reliable structure, given the available number of individuals. Below, we use an application from a bank to illustrate how this can be done in practice. 4.2. Perfection: The Multistep Function Theorem 5 shows that decision-making organizations can be designed to deviate arbitrarily little from the limit of perfect reliability under all the three abovementioned assumptions. This result is now established in the more general case. We require the minimal assumption that agents are able to discriminate between disjoint sets of positive and negative project value, but drop the two remaining assumptions. We do not require that the correlation between project value and agent screening is perfect, positive, or even different from zero. That is to say, we do not require a monotone agent screening function, which is an advantage because we are able to design decision-making organizations that can repair serious human error. Second, we do not require a monotone graph screening function. When we replace relays by human agents, who have powers to individually accept or reject projects on behalf of the decisionmaking organization, the graph screening polynomial is not necessarily monotone even if the agent screening is monotone. To the best of our knowledge, this problem has not been recognized (or solved) in previous research. In the following, we establish a new version of Moore and Shannon’s (1956a, b) theorem of perfection that is valid even in the case of nonmonotone agent screening or nonmonotone graph screening or both. Assume that agents are merely able to discriminate between projects with positive and negative value. That is, let projects fall within disjoint sets: A− ∩ A+ =  (12) Here it will be assumed that the agent screening function and the value mapping are both piecewise continuous, which imply that A− ∪ A+ is at most split into a finite number of intervals. Theorem 6. Given any threshold 0 <  < 1, a series of m (odd) shift points in reduced space 0 < 1 < 2 < · · · < m < 1 (13) and a radius 0 < < mini i+1 − i /2, a graph can be constructed whose screening will jump from below  to above 1 −  (and back alternatingly) within of the i ’s. Sketch of Proof. A graph with the postulated screening function is built from the single-step functions of Theorem 5 using a sufficiently small  and reusing . As Figure 4 shows, the graphs shifting at the required appearences are lined up into a hierarchy, starting with 1 closest to entry. The ones shifting from 0 to 1 must reject to the termination node, and the rest must reject to polyarchies (this follows from Theorem 1) large enough to ensure almost certain acceptance as required by the threshold. In case 1 ≤ or m ≥ 1 − , the first or last single-step graph, respectively, must be replaced by a suitable polyarchy or hierarchy according to Theorems 1 and 2. Assuming that polyarchies (and hierarchies, if needed) complete the required shift within the same  and as the generic single-step graphs, it is easy to show that the total graph will have a graph screening that meets the required threshold  if  ≤ /m is used. Finally, the assumption regarding the polyarchies is satisfied (again according to Theorem 2) by picking n ≥ log  / log1 − 1 + . In practice, a graph displaying the desired multistep screening function can be obtained by first finding single-step graphs using a stricter  <  but reusing . Then these graphs are arranged into a hierarchy, with every other subgraph rejecting to a suitably large polyarchy, as shown in Figure 4 (three jumps). An example of a multistep graph screening function obtained from this procedure is shown in Figure 5 (five jumps). Figure 4 Graph for Multistep Function I S 1 S2 S3 F Pn T Notes. Social and economic decision-making organizations may have nonmonotonous screening capabilities as a function of project appearance, which allow multistep functions shifting at 1  2  3  (generalizable to any odd number of shifts). These properties can be utilized to create more general architectures which screen perfectly whenever the appearence of good and bad projects fall within disjoint sets. Full lines are acceptance edges and dashed lines are rejection edges. Christensen and Knudsen: Design of Decision-Making Organizations 81 Management Science 56(1), pp. 71–89, © 2010 INFORMS Figure 5 Example of a Multistep Function with  = 1/20,  = 1/54, and Shifts 3/18 9/18 11/18 13/18 15/18 as Constructed by the Method Devised in the Proofs of Theorems 5 and 6 PI = 1.0 F () INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. PII = 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0  4.3. V x1 − fG xx dx (14) where x is the distribution of projects, and the probability of a Type II error (accepting a bad project) is 0.8 0 0 their ability. The probability of a Type I error (rejecting a good project) is8 An Example: Organizing Credit Evaluation in Bank F This example is based on a case study of bank F. It has a number of local branches, where credit advisors evaluate applications from business clientele. The evaluations result in immediate approval, rejection or, as is often the case, referrel to a credit officer in the bank’s central credit unit. The credit officer can approve, reject, refer the project to the next layer, or in some cases consult with a colleague at the same level. A number of advisers work at that level, and one of these may finally approve or reject the project. Or the project may be pushed on to its final station, the head of the central credit unit. We are here considering applications of modest size (approximately $1 million) that occur rather frequently (we considered a sample of 209 recent credit applications). The common measure of the efficacy of a bank’s credit evaluation is the number of defaults, a term that refers to the frequency of losses (error rates). The bank has a good estimate of Type II errors (defaults) but little information on Type I errors (rejecting good applications). The error rate for this bank is approximately 0.5%. The official policy of this bank is to “caution all evaluators to be mindful of the balance between risk and reward.” In practice, this translates into a conservative policy of evaluating a number of indicators that are thought to correlate with risk of default. The bank’s objective is to minimize the incidence of Type I and Type II errors subject to the constraints of the number of available evaluators (system size) and 1 − V xfG xx dx (15) To derive specific prescriptions it is useful to consider the agent’s discriminating ability against the system’s tolerance of error; i.e., what is the tolerance for uncovered losses on a credit application? The system’s tolerance of error is the maximal absolute project value for which errors have consequences that can be ignored. A system that tolerates no error whatsoever requires a zone of uncertainty that is zero. Such a system requires employment of an omniscient decision maker, f x, but these are not on the job market. The zone of uncertainty defines the range of project values where Type I and Type II errors will be committed (like a confidence interval in statistical theory). In bank F, the system necessarily has some zone of uncertainty. The aim is to design the credit evaluation system so this zone excludes credit applications with unacceptably high losses. From a design perspective, it is further useful to make a distinction between judgmental bias and ability. Judgmental bias is a deviation from symmetric screening around the point of zero project value. An employee may systematically be overoptimistic and accept credit applications that turn out to be loosing propositions. What is more likely is that an employee may have a “healthy” conservative bias that tends to exclude good applications. A biased agent screening function is skewed. If the skew is significant, the system’s tolerance of error may no longer include the agent’s zone of uncertainty. The agent’s judgmental ability is a different matter. Even an unbiased agent may have too little discriminating ability, which means that the agent’s zone of uncertainty exceeds the system’s tolerance for error. There are two complementary approaches to improving the system. The first is simply to hire evaluators with superior ability and replace evaluators with intolerable performance. The second concerns the design of the evaluation system, given the present level of judgmental ability. To illustrate the application of our theory, we consider how bank F could dramatically improve credit evaluation by choosing a design that both removes 8 Percentages (conditional probabilities) are readily obtained from these quantities, and a performance measure can easily be constructed by a suitable weighting of the errors. Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS judgmental bias and increases the discriminating ability. We actually conducted an experiment in bank F to extract a screening function from 40 randomly selected credit evaluators. A mixture of 12 indicators that the bank commonly uses to evaluate credit applications was selected, and a number of fake applications were constructed. The fake applications had known quality and frequency, so it was possible to extract the average screening function. It was sigmoid and it was fitted to the function y = 050 + tanhx − 306/543 with less than 0.5% unexplained variance. In the following example, we model credit applications as the scalar value, V x = x. Consistent with the evidence from bank F, we use a sigmoid agent screening function 1 + tanhx − x0 /x f x = 2 Figure 6 Repairing a Biased Screening Function 1.0 A G (4 agents) G′ (20 agents) 0.8 G : Unbiased F(),  = f(x) INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. 82 0.6 0.4 0.2 G ′: Unbiased and steeper 0 –2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0 x (16) where x0 is an arbitrary bias and 1/f x0  = 2x is the zone of uncertainty. The tolerance xtol defines the interval −xtol /2 xtol /2 around zero, containing credit applications of little consequence (most of the risk is covered by collateral and other securities). We set the scale of the zone of uncertainty to unity, x = 1, and measure the system’s tolerance of error on this scale. That is to say, the agent’s zone of uncertainty coincides with the system’s tolerance when xtol = 2. If the system’s tolerance is significantly lower than the agent’s zone of uncertainty (xtol < 1/10), the agent is not sufficiently reliable. As the system approaches perfect reliability, → 1/2, xtol → 0, and f 0 → . One advantage of the three-step construction of Theorem 5 is that the first step, the opening game, removes any bias in the graph screening function.9 By application of the first step in the design of a reliable system, the graph screening function is moved over such that it intersects the value 1/2 with arbitrary precision. After application of this procedure, the screening of the system is unbiased. Figure 6 shows how a biased agent screening function can be repaired. With a few agents (four in the example), any level of judgmental bias can be removed. The example further shows how an unbiased system, G, can be steepened by applying the middle and end game. In the example, the unbiased graph G is steepened by one composition with the self-dual graph G∗ . This requires a system including a total of 20 agents. This example is a forceful demonstration that the procedures provided in the present study can be used 9 Still assuming a reasonable overlap between the zone of uncertainty and level of tolerance, the graph screening will not be much distorted, only shifted. If this assumption does not hold, the two remaining steps of the construction procedure must also be applied to (re-)steepen the screening function. Notes. Given a low tolerance xtol = 1/25, a fairly large bias x0 = −0 3 can be removed with just four agents organized in a stair graph (a two-member polyarchy in which the last agent accepts to another two-member polyarchy). Using the procedure outlined in the proof, the graph can be further steepened if the structure is designed from 20 agents using one composition. to achieve incremental improvements even for small decision-making organizations. In bank F, such design improvements may significantly decrease the unacceptable defaults. More generally, Table 1 shows the number of agents needed to produce an unbiased graph screening function. These numbers were obtained from the application of the opening game of Theorem 5. Table 1 shows improvements of reliability in terms of the reduced graph screening polynomial F . We assume that the bank employs fairly reliable agents in accordance with our experiment. A reliable agent has a high value of because its agent screening function maps alternatives onto a probability that is close to 0 (alternatives with negative value) or 1 (alternatives with positive value) outside the interval of little importance. Fewer agents are required to repair a system with a high level of . To increase the reliability of the system, any initial bias must first be removed by the procedure outlined in the opening game of Theorem 5. Further Table 1 Number of Agents Needed in the Stair Graph (from the Opening Game) to Remove Any Initial Bias 0 = 1/2 for Any Agent Screening 0 \ 0.0001 0.0010 0.0100 0.1000 0.4000 0.45 0.35 0.25 0.15 0.05 4 12 12 10 28 4 7 5 7 20 4 5 5 (4) (12) (1) (2) (2) (3) (5) (1) (1) (1) (1) (2) Notes. The numbers in parentheses are polyarchies. Values of 0 > 1/2 are obtained by symmetry through dual graphs. Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS Table 2 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. \ 0.0100 0.0050 0.0010 0.0005 0.0001 System Sizes Obtained by Explicit Construction According to Specified Thresholds  and Radii  for a Single Shift Graph with 0 = 1/2 0.400 0.450 0.490 0.495 0.499 9 9 27 27 27 3 9 9 9 27 1 3 3 3 9 1 1 5 3 3 1 1 1 3 3 improvements can then be achieved by increasing the system’s discriminating ability. The middle and end game of Theorem 5 provides the procedure to achieve a desired level of discriminating ability by improving an unbiased graph screening function. We now illustrate how the results of this paper can be used to accomplish this. Table 2 shows the number of agents that is necessary to steepen a symmetric (unbiased) screening function (0 = 1/2) to various desired levels of reliability. As Tables 1 and 2 show, it is possible to achieve dramatic improvement in reliability through the proper design of an evaluation structure. Even if a large number of agents is needed to reach a high level of reliability when starting from incompetent agents, it is always possible to obtain significant incremental improvement with a handful of agents, either by diminishing the bias or by narrowing the agent’s zone of uncertainty. Overall, our example from bank F illustrates how our model can be used to design decision-making organizations. In actual practice, the application in bank F led to a number of suggested improvements. Further obvious applications are those where a system of evaluators jointly considers a high number of comparable projects. Examples include insurance companies, military intelligence, and medical diagnostics. 5. Conclusion This paper advances a general treatment of decisionmaking organizations that allows one to consider the full range of organizational architectures between extreme centralized and decentralized forms—often referred to as hierarchies and polyarchies. Thereby one can design decision-making structures that trade off Type I errors (of rejecting a superior alternative) and Type II errors (of accepting an inferior alternative) as the relative degree of hierarchy and polyarchy shifts. We provide proofs that show how decisionmaking organizations can be constructed so they maximize reliability for a given number of agents. We also show how incremental improvement can be achieved when additional components are added. This allows organizational designers to examine the change in the overall reliability of the organizational structure as 83 the number of actors within the organization changes. Our model can directly be used for a qualitative assessment of actual decision-making organizations such as boards, corporate headquarters, purchasing departments, and other management teams. An application from a bank illustrated how our framework can be used in practice. Our contribution is to show how organizations in general can improve the quality of collective decisions. Our proofs have direct applicability but may also be used to develop algorithms that can answer specific concerns relating to performance trade-offs. Recent work by Csaszar (2009) provides a very useful step in this direction. Our results are derived from a new extended version of an important theorem in information theory, the Moore-Shannon (1956a, b) theorem. This theorem showed how systems can incrementally approach perfect reliability even though all components are imperfect. Although this theorem offers useful inspiration, it cannot be directly applied to the design of decision-making structures because it is restricted to systems of relays and similar electric components. The Moore-Shannon restriction has the immediate implication that there will be inherent dissimilarities between actual human organizations and organizations of electrical component. The most important dissimilarity originates in the fact that human decisionmaking structures can delegate decision rights to lower hierarchical levels. This observation breaks with the assumptions of the original proof. Our approach therefore builds on the very different topological structures that are consistent with the way Sah and Stiglitz (1985, 1986, 1988) and others have characterized human organizations. As a model of organizational decision making, a number of limitations should be considered. First, we abstract from decision-making costs. In practice, the costs of making the decision usually depend on the size of the organization (the number of agents), the levels of pay, the choice of compensation method (e.g., fixed salary or pay per evaluation), and the possibility of (some) economies of scale with respect to evaluation. Therefore, total personnel costs would enter as a constraint on the problem of designing reliable decision-making structures. It is fairly easy to extend our approach by including the relevant cost model. Our study of bank F further indicated that reorganization costs are often important from the point of view of a practical application, i.e., marginal gains from re-organization in terms of changes in error rates or revenue. The estimation of marginal revenue requires further specification of the project distribution (as we did for bank F). It obviously makes a huge difference if decision-making structures consider innovation projects where quality has an exponential INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. 84 distribution or credit applications where the distribution is Gaussian. A second limitation concerns our choice of focus. Prior work on decentralization of information has provided robust results regarding the way organizational structures can be designed to optimize efficiency under capacity constraints (Marschak and Radner 1972, Radner 1993, Bolton and Dewatripont 1994, Van Zandt 1999). The theory of decentralization of information has analyzed design of structures while largely abstracting from the problems relating to reliability that we have considered here. By broadening the scope of organizational structures that can be analyzed from a reliability perspective, it should be possible to advance a synthesis of the two perspectives. A third limitation concerns incentives. We have assumed that individual ability (as characterized by the screening function) is exogenously given. But incentives are commonly thought to stimulate effort also with respect to decision making. This implies that (the expression of) ability is often endogenous to incentives. Our model further shows that performance signals can be made less noisy by using appropriate evaluation structures. This implies that the level of noise in principal-agent models would become a choice variable (as opposed to the common assumption that noise is exogenous to the problem). If organizations have not optimized their evaluation structures, the implication is that they would undervalue the effect of incentivizing individuals (and overvalue the effect of group-level incentives). This line of query would merge the information processing view on organizations with the incentive stream (for a review, see Colombo and Delmastro 2008). In our view this is a very promising item on the agenda for future research. Fourth, we follow prior research by Sah and Stiglitz (1985, 1986, 1988) in abstracting from heterogeneity in ability. It would be interesting to gain further theoretical insights on the allocation of employees to positions in the decision-making structure when there is variation in ability. Intuitively, one would place the agent with the highest ability (steepest unbiased screening function) where the information flow is highest, i.e., at low levels in the organization. However, intuition may be misleading, because the result would depend on the definition of the project distribution. So, we need research that provides a detailed examination of the relation among the economic context (project distribution), individual ability, and decision-making structure. Fifth, we are abstracting from a rather fundamental aspect of decision-making organizations, namely, joint learning. Joint learning is the common situation where individuals influence each other’s learning opportunities. To the extent that individuals learn from experience, Christensen and Knudsen: Design of Decision-Making Organizations Management Science 56(1), pp. 71–89, © 2010 INFORMS decision-making structures will have a huge impact on ability. This is because different structures filter information in different ways. For example, a hierarchy filters out most information on the way to the top of the organizational hierarchy. This means that higher-level managers are deprived of learning opportunities. We would therefore expect that their ability to pass judgment would suffer. This line of inquiry leaves an important unanswered question: To what extent is ability a function of (formal and informal) organization structures as well as talent? Although our abstraction from decision-making costs and the economic context (project distributions and personnel costs) served to provide general insights, the methods outlined in the present paper provide answers to two fundamental questions: How many individuals of a certain ability are required to build a decision-making organization of a certain level of reliability? How can maximal incremental improvements be achieved when individuals of a certain ability are added to the decision-making organization? We thereby advance research on the relation between organizational design and the efficiency of managing. Our approach advances a reliability perspective that is complementary to prior work on the decentralization of information and invites elaborations on the relation between individual ability and organizational structure. Acknowledgments The authors thank Markus C. Becker, Winston T. H. Koh, Daniel A. Levinthal, James G. March, Roy Radner, Raaj Sah, Larry Samuelson, and Nils Stieglitz for discussions and helpful comments on previous drafts. The authors also thank Martin Krone Dahl for access to the case study of bank F. This research is supported by a grant from the Danish Social Science Research Council. Appendix. Proofs of Theorems A. Proofs of Extremity, Optimality, and Shifting The theorems of §3 are proved here. A.1. Proof of Extremity The theorem of extremity is remarkably general as discussed below. We first prove the theorem with homogeneous agents and then discuss extensions. Theorem 1. Let n for any positive integer n be the set of all homogeneous graphs that can be constructed from agents of type A, such that the maximal evaluation count is no more than n. Let Pn ∈ n be the polyarchy of n members with reduced graph screening function FPn , and let Hn ∈ n be the hierarchy of n members with reduced graph screening function FHn . Any graph G ∈ n with reduced graph screening FG satisfies FHn ≤ FG ≤ FPn  (17) Christensen and Knudsen: Design of Decision-Making Organizations 85 Management Science 56(1), pp. 71–89, © 2010 INFORMS Figure A.1 Effective Form (with Loops Unfolded) of the General Member of n+1 Immediately After Entry the weights during such a dispatching must take values in the unit interval and sum up to unity: a+ Ga1 m  aj = 1 and r + j=1 a1 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. Gam F am a I r Grm T r1 Gr1 Notes. The solid lines are acceptance edges, the dashed lines are rejection edges, and Gaj Grj represent the subgraph to be seen in case of acceptance (rejection) along edge j. Proof of Theorem 1. The proof runs by induction on the maximal evaluation count n. The basic step, n = 1, is trivial, as all graphs in 1 have the same screening function as P1 and H1 , both consisting only of one agent. The induction hypothesis consists of the assumption that the theorem holds for k for all k from 1 and up to some positive n. So by assumption, all graphs in k have their screening functions bounded by those of Pk and Hk . The recursiveness of the theorem will now be shown by construction. Consider any graph G ∈ n+1 of maximal evaluation count n + 1, and let m > 0 be the number of agents in the architecture. All directed edges from any node to any other node can be collected into two effective edges without affecting the screening capabilities, a rejection edge and an acceptance edge having weights equal to the collective chance of moving between the two nodes in case of rejection and acceptance, respectively. Furthermore, only graphs with one entry point from the external node I need to be considered, because other graphs will have screening functions that are linear combinations of these single-entry graphs. After having entered the graph at some specific node, the most general form of the graph is as shown in Figure A.1. Because m agents are in the structure, the number of possible agents that can receive the project after the initial evaluation is m at most (one of them being the first agent itself), regardless of the result of the agent screening. On arrival at the next agent (number j with probability aj in case of acceptance and rj for rejection), the effective subgraph to be seen by the project has a maximal evaluation count of n because one evaluation has already been spent. Thus, the subgraphs Gaj ∈ n and Grj ∈ n belong to the set of graphs for which the theorem holds because of the induction hypothesis. This is the cornerstone of the proof. There is a possibility that the project is terminated or ultimately accepted directly as a result of the first evaluation. These probabilities are denoted r and a, respectively. Because the project must leave the agent after evaluation, rj = 1 (18) j=1 The graph screening of G can be expressed recursively in terms of the entry agent and the subgraphs reached after the first evaluation.  m m   FG x = f x a+ aj FGa x +1−f x rj FGr x (19) j=1 rm m  j j=1 j Similar expressions can be obtained for the polyarchy FPn+1 x = f x + 1 − f x FPn x (20) and for the hierarchy FHn+1 x = f x FHn x (21) as well. The recursiveness of the theorem is now established by comparing Equation (19) to Equations (20) and (21). First,   m  FPn+1 x − FG x = f x 1 − a − aj FGa x + 1 − f x j j=1   m  · FPn x − rj FGr x ≥ 0 j=1 j (22) where the inequality follows from Equation (18) and the induction hypothesis, making both terms nonnegative. Finally,   m  FG x − FHn+1 x = f x a + aj FGa x − FHn x j j=1 + 1 − f x m  j=1 rj FGr x ≥ 0 j (23) is reached by similar arguments. Invoking the principle of mathematical induction, the theorem must hold for all n > 1.  Dynamic dispatching of projects can be included in the model by letting the weights of the channels of communication depend on the project and the path that it has seen so far, e.g., aj x. It thereby allows freelancing schemes for reusing agents and subgraphs to keep the total number of agents down (e.g., the consensus rule of committees), it can supply a truncation mechanism for potential infinite loops, and it allows organizations to have different modes of response triggered by certain project parameters. As mentioned, dynamic dispatching greatly extends the set of organizations that can be considered, because it adds more flexible structures with a realistic touch, usually by reducing size or cost. Despite the broader scope of such a model, the theorem of extremity still holds because it is unaffected by such arguments on the weights. Introduction of project transformations constitutes a major extension of the basic model. If agents are characterized by a (stochastic) transformation Tacc/rej y x, representing the probability that project x is transformed into Christensen and Knudsen: Design of Decision-Making Organizations INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. 86 Management Science 56(1), pp. 71–89, © 2010 INFORMS project y during the evaluation process leading to acceptance/rejection, then a wide range of new models can be constructed. It then becomes possible to model imperfect channels of communication as well as examine the effects of agents actively manipulating the projects. Regardless of whether the agents transform the project before or after evaluation/dispatching, the theorem of extremity still holds. Transformations may also lift eventual degeneracies between good and bad projects of equal appearence. Graphs built from agents that are heterogeneous in screening properties can even be treated if the graph screening performance is averaged over the distribution of different agents. This is a realistic assumption when screening performance is averaged over time in systems with high replacement rates. This would be the case if agents are regularly replaced by drawing new ones from the common agent distribution or if each agent is actually a department of individuals whose capabilities follow the given distribution. Long-time performances of organizations drawing employees from the same workforce can thus be treated, and even in these situations the theorem of extremity holds. Further extensions and a more rigorous treatment of the project selection formalism can be found in Christensen and Knudsen (2002). Theorem 2 is proved in the main text. A.2. Proof of Optimality Theorem 3. For any positive integer n, let n be the set of all self-dual graphs that can be constructed from agents indistinguishable with respect to project screening, such that the maximal evaluation count is no more than n. The slope of the screening polynomials at  = 1/2 of the graphs in n cannot exceed that of FGn∗  = n  Bi i i=n+1/2 Bi = n + 1 −1i−n+1/2  n + 1/22 i  with n − 1/2 i − n + 1/2  (24) and at least one graph Gn∗ ∈ n has this reduced graph screening. Proof of Theorem 3. This proof has two stages. First, the polynomial view approaches from the analytical side to find the optimal reduced screening function with respect to discriminating ability. Then the topological view shows by construction that a graph does exist having the found optimal screening. The Polynomial View. Graph screening functions are polynomials in the agent screening whose maximal order is the maximal evaluation count. Thus, with the number of agent evaluations limited to n, there are at most n + 1 coefficients to tune. And for every (nonsymmetric) constraint put on the polynomial, the self-duality constraint produces yet another. So the optimal screening polynomials should be sought within the family of odd evaluation count (assumed in the following unless otherwise stated), and at most n + 1/2 conditions can be specified. The optimality strived for here is a minimal deviation from the perfect (self-dual) screening,  − 1/2, the step function. The screening is required to stay close to 0 below  = 1/2 (and close to 1 above due to self-duality) and to change sharply around the middle of the unit interval. Clearly, the best way of achieving this is to require that the screening function be as flat as possible near the ends of the interval. In this way the screening stays closest to the extreme values for as long as possible, leaving as little as possible of the parameter space over which to perform the jump between these extremes. Moreover, this requirement ensures monotonicity, which again ensures that the screening value stays in the unit interval as it must for interpretation as a probability. From a polynomial perspective, the optimality requirement is that the screening and its first n − 1/2 derivatives are zero at  = 0: d i FGn∗  di =0 =0 with i ∈ 0 1    n−1  2 (25) Here the hypothetical optimal self-dual graph with maximal evaluation count n is denoted Gn∗ . The solution to conditions (25) has a derivative proportional to  to the power n − 1/2 and, by self-duality, 1 −  to the same power. Working out the normalization constant, the entire solution (3) can be obtained from the integral of FGn∗  = n + 1 n−1/2 1 − n−1/2  n + 1/22 (26) with vanishing constant as FGn∗ 0 = 0. The Topological View. The proof is carried out in the simplest of models, where each agent receives projects from a single predecessor only and each agent has only one successor on the acceptance and rejection side, respectively. Within this model the graphs are unique. More advanced models, for example, with complex dynamic rules, can always be simplified to fit within the simple model without affecting the screening properties. This is done by duplicating, or unfolding, agents appearing on multiple paths. However, the addition of extra rules may break the uniqueness property of the graphs. The existence (and uniqueness within the simple model) of graphs having the optimal screening is now proved by explicit construction. Because they are characterized by a maximal evaluation count n and concern sets of paths of certain lengths, a few general properties of the unfolded graphs are derived first. There is exactly one graph with fixed maximal evaluation count n that has exactly k accepts on every path to ultimate acceptance. Both existence and uniqueness can be shown by induction. Obviously 1 ≤ k ≤ n and the single agent covers the basic case of n = 1 = k. Assume now that the all the graphs exist up to n for all allowed k, and let n k denote each of these graphs. Then n + 1 n + 1 is constructed by letting a single agent accept to n k, which yields the hierarchy Hn+1 , and similarly n + 1 1 is constructed by letting a single agent reject to n 1, which yields the polyarchy Pn+1 . For all intermediate k the graph n + 1 k is constructed by letting a single agent accept to n k − 1 and reject to n k. Because the graphs at level n were unique, the acceptance and rejection branches are independent, and the subset of n + 1 graphs just constructed are all different, thus, the uniqueness must hold as well at level n + 1. Returning to the self-dual graphs of optimal screening, it is seen from the 0th derivative polynomial requirement (25) Christensen and Knudsen: Design of Decision-Making Organizations 87 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. Management Science 56(1), pp. 71–89, © 2010 INFORMS that there should be at least n + 1/2 accepts on all paths to the final portfolio (again with odd e). Furthermore, the self-dual variant of the same constraint dictates that there should be at most n + 1/2 accepts on all paths. Therefore, there must be exactly n + 1/2 accepts on every path leading to ultimate acceptance. This is just the special graph labeled n k = n + 1/2, the existence and uniqueness of which has just been proved above.  Although polynomials of even maximal evaluation count seem to miss a constraint compared to their odd counterparts, they are actually fully constrained as well as the self-duality requirement forces the coefficient of n to 0 for even n. Alternatively, an additional self-duality constraint not conflicting with optimality can be put at  = 1/2 where the generic dual constraints collapse into one, FGn 1/2 = 1/2. Consequently, there is nothing to gain with respect to screening capabilities by adding a single evaluation to an optimal decision structure with odd maximal evaluation count. A.3. Proof of Shifting Theorem 4. For any 0 < 0 < 1 and any 0 < < d ≡ min0 1 − 0 , a stair graph G exists with no more than n = log /2/ log d (27) agents satisfying the relation: FG 0 −  < 1/2 < FG 0 +  (28) Proof of Theorem 4. A sequence of stair graphs of increasing size is constructed in the following fashion. Start out with two empty dummy graphs G0↓ and G0↑ representing default strategies of rejection and acceptance, respectively. Let G1 = A be the graph consisting of a single agent. These graphs obviously have screening functions, FG0↓  = 0, FG0↑  = 1, and FG1  = . For each step in the sequence, if FGn 0  < 1/2, then (i) Gn↓ = Gn and Gn↑ = Gn−1↑ , (ii) to obtain Gn+1 add a new agent at rejection from the latest added agent; else (i) Gn↓ = Gn−1↓ and Gn↑ = Gn , (ii) to obtain Gn+1 add a new agent at acceptance from the latest added agent. This sequential construction has several properties ensuring convergence, such as is why this specific construction is chosen. Therefore, theorem 1 of the original proof, stating that FGn  1 − FGn FGn  > 1 1 −  (31) (except at the endpoints of the interval and if n = 1 where equality holds), can be be applied to prove by contradiction that FGn  intersects 1/2 within I whenever n ≥ log /2/ log d agents have been added. (32)  B. Proof of the Human Moore-Shannon Theorem and Extension The theorems of §4 are proved here. B.1. Proof of Human Moore-Shannon Theorem Theorem 5. Given any position 0 < 0 < 1 for the shift in graph screening, any threshold 0 <  < 1/2, and any radius 0 < < min0 1 − 0 , then an architecture can be constructed from no more than   log 5/log11/8 1 log /2 · logmin0 1 − 0  2   log3 log 5/log 2 · (33) log3/4  nck ≤ 25 · agents whose reduced graph screening polynomial fulfills condition (9). (29) Proof of Theorem 5. The present proof uses a technique similar to that of Moore and Shannon (1956a, b). The opening game accomplishes a shifting of the screening via stair graphs, which removes any initial bias. The middle and end game steepen the screening using graph composition, in which the previously found graph substitutes the agents of a highly discriminating and self-dual graph. The Opening Game. The opening game consists of finding an architecture with n members such that the graph screening function, FGn , intersects the value 1/2 within I ≡ 0 − 0 + . The main difficulty lies in finding a suitable graph. Moore and Shannon (1956a, b) used ladder graphs in their proof, but these graphs cannot be used when agents have powers to reject or accept a project on behalf of the economic system. A suitable graph, including such agents, is the stair graph that is built according to the proof of Theorem 4. Thus, the construction process is continued until n = log /2/ log d (34) d ≡ max0 1−0  < 1 (30) Although Equations (3) and (4) in Moore and Shannon (1956a) do not hold for all social and economic systems and certainly not for expansion around any arbitrary agent, they do actually hold for the above sequence of graphs, which agents have been added. The Middle Game. The middle game consists of steepening the graph by recursive expansion such that FGn 0 −  < 1/4 and FGn 0 +  > 3/4 are obtained. To accomplish this, begin from Gn↓ and Gn↑ , obtained in the opening game. From Gn↓ and Gn↑ select the graph G0 lying closest to 1/2 at 0 as a building block for further construction. Then select a self-dual graph to be used in recursive expansion of G0 . A self-dual graph in the reduced space, FGn↓  ≤ FGn  ≤ FGn↑  FGn↓ 0  < 1/2 1/2 ≤ FGn↑ 0  and FGn↑ 0 −FGn↓ 0  ≤ d n with Christensen and Knudsen: Design of Decision-Making Organizations 88 Management Science 56(1), pp. 71–89, © 2010 INFORMS Figure A.2 Self-Dual Graph G∗ Used in the Middle and End Game to Steepen the Graph Screening Function Until the Threshold  Is Met INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/. compositions. The End Game. Because FGs 0 −  < 1/4 and FG∗  < 32 for  < 1/4, the end game of the original proof can be applied directly, thereby showing that the desired threshold is reached in no more than   log3  s ≥ log log 2 (39) log3/4 F I T Note. Full lines are acceptance edges and dashed lines are rejection edges. where the graph screening is a function of the agent screening, is defined as10 FG  + FG 1 −  = 1. Recursively substituting the graph for the agents in a self-dual graph with a steep slope around  = 1/2 will further steepen the total graph screening, first to ensure that it falls outside 1/4 3/4 on I, then to ensure that it falls outside  1 − . A new sequence of graphs Gs  is obtained in this way. For each step, s, of this procedure of recursive expansion, the size of the graph increases. The self-dual graph G∗ used here is illustrated in Figure A.2; it is a single agent with an acceptance edge to a two-member polyarchy and a rejection edge to a two-member hierarchy.11 The choice of G∗ was based on the premise that it is the best (in the sense of steepening the graph screening) self-dual graph that can be obtained with a small number of agents (less than nine agents). This optimality is guaranteed by Theorem 3 as G∗ ≡ G3∗ . The reduced graph screening function of the self-dual graph G∗ is FG∗  = 2 3 − 2 (35) Initially FG0 0 −  < 1/2 − /2 and similarly FG0 0 +  > 1/2 + /2. Owing to the symmetry of the problem, it suffices to consider the lower end of the interval,  = 0 − . The technique of Moore and Shannon can be applied directly, but an even better bound can be obtained if the deviation s from 1/2 is observed as a function of composition count s, s+1 ≡ 1/2 − s − FG∗ 1/2 − s  = s 1/2 − 2s2  as long as s is below one-quarter, which is reached no later than s ≥ − log2 / log11/8 (38) (36) additional compositions with G∗ . The theorem is finally proven by counting the agents required by all the above steps, and a prefactor is added, because some agent or composition counts may not come out even.  B.2. Proof of the Multistep Theorem Theorem 6. Given any threshold 0 <  < 1, a series of m (odd) shift points in reduced space 0 <  1 < 2 < · · · <  m < 1 (40) and a radius 0 < < mini i+1 − i /2, a graph can be constructed whose screening will jump from below  to above 1 −  (and back alternatingly) within of the i ’s. Proof of Theorem 6. A graph with the postulated screening function is built from the single-step functions of Theorem 5 using a sufficiently small  and reusing . As Figure 4 shows, the graphs shifting at the required appearences are lined up into a hierarchy, starting with 1 closest to entry. The ones shifting from 0 to 1 must reject to the termination node, and the rest must reject to polyarchies (follows from Theorem 1) large enough to ensure almost certain acceptance as required by the threshold. In case 1 ≤ or m ≥ 1 − , the first or last single-step graph, respectively, must be replaced by a suitable polyarchy or hierarchy according to Theorems 1 and 2. Assuming that polyarchies (and hierarchies, if needed) complete the required shift within the same  and as the generic single-step graphs, it is easy to show that the total graph will have a graph screening meeting the required threshold  if  ≤ /m is used. 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