THE MODEL

We study an organization, O, with $ n>1 $ members who may have different specialties. There are $ m\ge 1 $ different specialties in the society.Footnote ⁶ S denotes the set of all specialties in O; $ {S}_i $ , the set of agent i’s specialties. ( $ {S}_i $ may be empty. This represents someone without specialized training or education.)

Organization O exhibits an accountability structure: the performance of some or all members of O is reviewed by others in the organization.Footnote ⁷ We impose no assumptions about the review relation other than precluding self-review (i reviewing himself). Thus, multiple officials may review the same individual or two officials may review each other (i reviews j and j reviews i).Footnote ⁸ Reviews may occur on a fixed schedule or may be triggered by exogenous shocks. Indeed, the model represents neither the causal structure underlying a reviewee’s performance nor why i reviews j. It thus obtains generality at the expense of granularity.

An unconstrained review relation allows for all theoretically possible review structures. Not all such structures, however, are reasonable models of organizations. For instance, what distinguishes an organization from an arbitrary collection of individuals is that an organization’s members are linked with each other. In our conceptualization, a link is a review relation. Therefore, a set of people in which someone is not linked with anyone else is inconsistent with the very idea of an organization. Similarly, assuming that O can be partitioned into two or more sets such that nobody in one set is linked with anyone in the other would not make sense.

Formally, we represent an organization’s review structure by a directed graph whose nodes are members of O and edges are binary review relations.Footnote ⁹ For the above reasons, we assume that this directed graph is weakly connected: in its undirected graph, any two nodes are connected by a set of edges.

We have stated that we limit our analysis to organizations in which the performance of some or all of the officials is reviewed by others in O. This assumption reflects two issues confronting meritocratic societies. First, organizations should be allowed to adopt any review structure they deem effective: A strict hierarchy may work for the military but not for a policy analysis unit. For many optimal review structures, top officials will be unaccountable to others in the organization; hierarchies are like that. Such structures may, however, also illicitly benefit organizational members (corrupt officials may benefit from a scheme of the agency’s leader) while harming people outside the organization (citizens who must bribe corrupt bureaucrats). Meritocratic review, which would guard against corruption, would require establishing an independent organization with the power to review officials who are unaccountable inside their organizations. Indeed, in a meritocratic (i.e., modern) society, the performance of everyone—including government employees—must be regularly reviewed. This is, we believe, a central feature of the concept of a meritocracy. We call the meritocratic requirement that everyone be reviewed universal accountability.

Accountability can be defined and assessed from the perspective of one organization or from that of a society seen as a set of organizations. We first consider the case of a single organization. The following definition formalizes the criterion of universal accountability in this context. For obvious reasons, the definition excludes self-review.Footnote ¹⁰

Definition 1 requires that there is at least one j who reviews i. There may be several: for example, both Congress and the President could review a regulatory commission. Thus, universal accountability allows for separation of powers as well as other complex authority structures not limited to standard tree hierarchies where every subordinate has exactly one superior. Hence, a political elite comprised of legislators, judges, and a president or prime minister may constitute the top of a government’s review structure.

Of course, satisfying universal accountability means that somebody reviews members of the political elite. One possibility: legislators review judges and vice versa. This would be a review cycle. The following definition of a cycle is standard in graph theory.

These definitions allow us to state our first result. (For all proofs, see the Supplementary Material.)

Two comments are worth making. The first pertains to S, the organization’s set of specialties; the second, to O’s finiteness.

First, Theorem 1 holds for all possible sets of specialties in O—that is, all possible sets of individual repertoires. The degenerate case— $ {S}_i $ is empty for all i—is admissible, as is O having a rich assortment of specialties.

Second, obviously, all real institutions have finitely many members. Hence, precluding infinite organizations may seem pedantic. But there is a reason to emphasize the obvious: without it, the theorem is invalid. Designing review structures that satisfy both universal accountability and acyclicity is easy for organizations of infinite size. For example, number the organization’s members 1, 2, 3, … and have official $ i+1 $ review i. Then everyone’s performance is assessed, yet there are no cycles. Linking performance review to authority lets us state the property this way: institutions with infinitely many members always have new possible superiors. In finite organizations, one must eventually run out of them; it is exactly then that universal accountability and acyclicity become inconsistent. Hence, Theorem 1 could be restated as follows: there exist review structures which satisfy both universal accountability and acyclicity if and only if an institution has infinitely many members.

Having explained why the finiteness of organizations is analytically important for the tension between universal accountability and acyclicity, we will not repeat this assumption. Henceforth, we assume all O’s are finite.

Our explanation of the difference between a finite and an infinite domain is an instance of a well-known infinite regress argument (cf. MacKay Reference MacKay1980, 112–7) that has been used, notably by St. Thomas Aquinas, to prove the existence of God. The problem with infinite regress explains why and how political philosophers have looked for ways to justify a hierarchical polity, with nobody reviewing the monarch at the top of society. Since universal accountability and hierarchy are inconsistent, justifying monarchy must depart from the assumptions of our model. One departure: assume that certain decision-makers are perfect and hence need no review. God is such a decision-maker. Suppose that people in a society, represented as a single institution with a monarch at the top, agree that universal accountability holds if every imperfect decision-maker is reviewed. Perfect decision-makers, who always do the right thing, need not be evaluated. Then one can easily prove that acyclicity and universal accountability are consistent if and only if a society’s top officials are reviewed by a perfect decision-maker.

Efforts to evade Theorem 1 abound in political history. Believing that a monarch is accountable to an ominiscient and benign God is comforting. (In many coronation ceremonies, monarchs are consecrated—associated with the sacred—when anointed.)

But there is problem: perfect decision-makers may not exist. Yet, as long as everybody believes that one exists, they can also believe that monarchy and accountability are consistent. Suppose that everyone in a society shares this belief system:

1. (1) God exists.

2. (2) God is perfect.

3. (3) God is accountable to nobody.Footnote ¹¹

4. (4) The king is accountable only to God.

5. (5) Everyone else in the society is accountable to the king.

Belief (5) could be indisputable: a grisly end awaits those who displease the king. Openly questioning (1), (2), or (3) would be equally dangerous: even today some monarchies punish atheism or apostasy by death. But although everyone in the society, possibly excepting a cynical king, believes (1), (2), and (4), nobody knows whether they are true. Thus, for secular people, the belief system of (1)–(5) provides no escape from Theorem 1.Footnote ¹²

Henceforth, we assume that all decision-makers are imperfect. Thus, we will be examining real organizations in modern cultures, and following Weber, we take “modern” to include “secular.”

MODERN AUTHORITY SYSTEMS

Theorem 1 implies that in modern societies something must give. Should we relax universal accountability or acyclicity? When answering this question, we should consider how badly a criterion is violated. For example, we might allow a violation of the accountability requirement if only minor positions are unreviewed. What seems critical is the significance of unaccountable positions. Even one unaccountable position could be dangerous if the office is important. In particular, tree hierarchies satisfy acyclicity and only one position is not reviewed. Hence, we could keep the number of unreviewed positions to a bare minimum while preserving acyclicity. But this one violation may ruin the entire organization.

The following definition distinguishes different positions in an organization. Obviously, some offices are more important than others.

Leaders review others but are not reviewed by anyone in the organization.

Typically, a leader reviews only a few subordinates. She can, however, hold her subordinates accountable for their review of their subordinates. We say that i indirectly reviews k if i reviews j and j reviews k. We will use this terminology henceforth: i indirectly reviews k means that there is some j, distinct from both i and k, who is reviewed by i and who reviews k. (This may be a long chain: many officials may be in-between i and k.) It will sometimes be natural to refer to a matched pair: direct as well as indirect review. Direct review is simply the fundamental binary review relation: i reviews j, without intermediates.

Because we make few assumptions about which review structures are admissible, requiring only that they be weakly connected and exclude self-review, a leader or other official may both directly and indirectly review somebody. In the above example, i could review k as well j (and as before, j reviews k). This often happens: for example, if k’s immediate superior, j, mishandles his review, then k can appeal to j’s superior, who could review k’s performance herself. In other circumstances, i may be able to review only j. (Nevertheless, j remains accountable to i for his review of k, so even in such cases one can reasonably say that i indirectly reviews k.)

With these concepts in hand, we can state the following corollary to Theorem 1.

Corollary 1 tells us that leaders come at a high price for acyclic organizations: although they directly or indirectly monitor the performance of everyone else in the organization, they themselves are not reviewed by anyone in the organization. This violates the meritocratic requirement of universal accountability. There is, however, a simple way to implement universal accountability: have leaders review each other. In a triumvirate, for instance, Caesar reviews Pompey, Pompey reviews Crassus, and Crassus reviews Caesar.

Although a reform turning three leaders into a triumvirate joined by a review cycle makes everyone in the organization formally accountable, it would replace the problem of unaccountable leaders by that of an unaccountable group.

A triumvirate’s review structure is vulnerable to collusion: leaders could become oligarchs whose bad performance is reviewed only by each other. Hence, though universal accountability is satisfied, the problem of unaccountable decision-makers persists: the top group remains unaccountable to everyone else in the organization and it still controls the entire organization. A triumvirate is but one example of a self-governing oligarchy (Michels Reference Michels1915) sitting atop an otherwise hierarchical organization.

Before examining solutions to this problem, we must reflect on what makes an organizational group an oligarchy. In addition to the convention that oligarchies must have at least two members—an “oligarchy” of one person is usually called an autocracy—we believe that three properties are crucial. (None of these depends on the set of specialties.) The first property is straightforward: an oligarchy rules the rest of the organization.

(1) If a nonempty group $ G\subset O $ is an oligarchy, then all members of O who are not in G are reviewed by someone in G.

Note that the definition is meaningful only if G is a proper subset of O. Thus, we preclude “an oligarchy of all.” All-inclusiveness obviously violates the common connotation of “oligarchy.”

The second property of an oligarchy, also noted above, is equally obvious: oligarchs are not accountable to anyone outside their group.

(2) Nobody in G is reviewed by anyone in O outside of G.

These properties reinforce each other: together they represent oligarchic dominance. A group characterized by (1) and (2) is likely to collude at the expense of others in the organization who cannot call it to account.

However, although properties (1) and (2) are necessary for identifying an oligarchy, they are insufficient. A group satisfying (1) and (2) may include people who are reviewed by an inner clique but who cannot review anyone in the clique. They thus have little if any power. Consider a feudal farm owned by two brothers, 1 and 2, who employ a supervisor, 3, and jointly manage a set of peasants, $ 4,\dots, n $ (Figure 1). The owners form the oligarchy’s inner clique; although the supervisor is a member of the group ruling the organization—the group of 1, 2, and 3 satisfies properties (1) and (2)—he is not in the clique. The owners form the ruling oligarchy that controls the entire organization. (The brothers monitor each other; hence, universal accountability is satisfied. And, because nobody else reviews them, property (2) holds.) The third property refines the meaning of oligarchy accordingly.

Figure 1. Governing Groups versus Oligarchies

Although the group of 1, 2, and 3 in Figure 1 satisfies properties (1) and (2), this by itself says nothing about the group’s internal review structure. This is why properties (1) and (2) fail to identify a proper oligarchy in Figure 1. While the three-person group satisfies (1) and (2), so does its subgroup of 1 and 2; hence, 3, fully controlled by 1 and 2, should be removed from a proper oligarchy. 1 and 2 form the organization’s true oligarchy: this set cannot be further reduced to any subset that satisfies properties (1) and (2). This minimality—the essence of oligarchy—is codified by Property (3).

(3) No proper subset of G satisfies properties (1) and (2).

Properties (1), (2), and (3) together define “oligarchy” in the context of our formal model.

Apart from the convention of not calling one person who satisfies properties (1)–(3) an oligarchy—we call such a person the autocrat of O—and precluding $ G=O $ , Definition 4 imposes no size criterion on oligarchies. Though oligarchy is often taken to mean a small group of individuals (e.g., a triumvirate), there is a good reason to eschew that property: the peril of oligarchy may persist in large cases. As Alexander Hamilton said: “Give all power to the many, they will oppress the few. Give all power to the few, they will oppress the many” (Chernow Reference Chernow2004, 233). Hence, although in standard political theory the term oligarchy is reserved for small ruling groups and “tyranny of the majority” for large ones, our definition ignores this distinction. At the core of our concept of oligarchy is the commonality Hamilton identified. (Social choice theorists [cf. Sen Reference Sen2017, 291] do not limit the size of “oligarchy” for precisely the same reason.)

Thus, in our results, “oligarchy” applies to a group of any size. What defines an oligarchy are only properties (1), (2), and (3).Footnote ¹³

Before we are ready to use the concept of oligarchy in the following results, we should note two features of oligarchies that do not follow transparently from Definition 4. First, could an organization have more than one oligarchy? Second, does the definition imply anything about an oligarchy’s internal power structure? Can oligarchs hold asymmetric positions within their group? They could certainly control different parts of the organization. These parts may but need not overlap. A diarchy may have a division of power: one oligarch controls most of the government, but the secret police is controlled by the other. Similarly, can oligarchs play asymmetric roles in the review structure?

Answers to both questions are provided by the following lemma, which states that our definition identifies an oligarchy uniquely and that oligarchs must have symmetric review relations. (Lemma 1 is crucial for proving a theorem that follows.)

Having analyzed the meaning of oligarchy, we are ready to revisit the problem of accountability. A strictly hierarchical organization with a single leader is run by an autocrat. If instead of a single leader an otherwise identical organization has two who review each other, then everyone in the organization is reviewed; hence, universal accountability is satisfied. But turning from one unaccountable leader to a diarchy has little if any significance for others in the organization: they can be exploited by the unaccountable diarchy. The example’s general point is of fundamental importance to society. If a country’s military, for instance, is a hierarchy, then the top generals form an oligarchy. Coalitions of cronies could flourish, thereby weakening accountability. Hence, it is reasonable to search for review structures that eliminate oligarchies. Doing so turns out to profoundly affect an organization’s authority system.

Given that oligarchy G is a proper subset of O and nobody in G is reviewed by anyone in O and outside of G, the following lemma is straightforward.

What is much less obvious is the next lemma.

Thus, eliminating oligarchies and autocracies produces a pervasive violation of acyclicity: the organization exhibits cycles everywhere. From a traditional perspective, this is bad: authority systems should be hierarchical, hierarchy is inherently asymmetrical, and an asymmetric structure is acyclic. But we used the word “traditional” intentionally, to contrast it with “modern.” Holding everybody accountable (by other humans) is a modern idea, conceptually as well as temporally.

It is good to know, however, that ubiquitous cycles do the job: they are not only necessary for eliminating oligarchies and autocracies, by Lemma 2 they are also sufficient.

Together, these lemmas yield the following characterization of organizations in which all groups and all individuals are accountable.

Theorem 2 tells us that an appropriately demanding sense of performance accountability (citizens are accountable to each other—complete reciprocal accountabilityFootnote ¹⁴) is equivalent to a vital feature of democracy: eliminating autocracy and oligarchy. And, because performance accountability is central to meritocracy, it follows that democracy and meritocracy are tightly linked.Footnote ¹⁵ Thus, in this important sense, democratic political institutions are modern and fully modern governments are democratic.

Together, Theorems 1 and 2 are depicted by Figure 2. (The figure displays just three review structures—autocracy, oligarchy, and democracy—because Theorem 2 implies that these are the only possible types.)

Figure 2. Relations between Review Structures and Accountability

Note: Tree hierarchy: one leader; everyone else has exactly one superior.

Theorem 2 points to an important tension between hierarchical authority and democracy. Specialists may insist that hierarchy (acyclicity, more generally) is essential if their organization is to do its job (e.g., defend the country) effectively. This claim is plausible when organizational efficacy requires coordinating the actions of many people or telling them to do dangerous things. This point is not limited to the military. Constraining how organizations structure their review systems may be counter-productive: constraints may not only impair organizational problem-solving; they might violate the technical autonomy of organizations expected in modern societies. Hence, the general issue is how to let organizations figure out which internal procedures and structures work well, given their tasks, while satisfying the meritocratic criterion of holding everyone in society accountable for their performance.

If a group is not accountable to anyone inside their organization, they may still be accountable to members of a different organization. Indeed, this other organization may be created in order to review people like them. Regarding the military, a set of generals, who form an oligarchy within their hierarchical bureaucracy, may be held accountable by civilians elsewhere in the government. Indeed, a government’s executive branch can be thought of as a network of organizations linked by review relations that make individuals occupying positions in one part of the bureaucracy subject to review by another part of the organization. Therefore, let us think of a government’s executive branch as a set of organizations and focus on the case of the military—an organization of great historical importance which, given its coercive potential, poses vital issues for democracy. For hierarchical review structures, the following observation holds.

Given that universal accountability is a necessary feature of fully modern systems, Corollary 2 implies that an internally hierarchical military—the prototypical form—is modern only if there is civilian control. Footnote ¹⁶ (The same point holds for the police and internal security services.) In a fully modern society, everyone’s performance is reviewed by somebody. If the generals succeed in preserving internal hierarchy, for which there are probably good instrumental reasons, then nobody in the military will review them.Footnote ¹⁷ Hence, only outsiders can do the job. Of course, insiders usually regard outsiders as amateurs. They are often right. But this complaint of an organization’s leaders can be self-serving: they may be trying to be the only unaccountable officials in their organization. If the insiders’ stereotype of outsiders is correct—outsiders are, in fact, amateurs—then this fact and Corollary 2 together imply that the choice is necessarily difficult: either the organizations’ leaders are reviewed by possibly incompetent amateurs or they remain unaccountable. Clemenceau chose the first alternative, famously saying, “war is too important a matter to be left to the generals.” Universal accountability remains a crucial desideratum, especially for the coercive bureaucracies.

This leads directly to our next topic: the tension between accountability and specialized knowledge.

ACCOUNTABILITY, SPECIALIZATION, AND WEBERIAN TENSIONS

The tension between accountability and expertise is not peculiar to the civilian control of the military. It is a common issue in most contemporary organizations: bureaucracies (Dahlström and Lapuente Reference Dahlström and Lapuente2022) as well as firms. Two reasons are as simple as they are fundamental. First, modern organizations tend to be knowledge-intensive institutions: for example, responding effectively to a pandemic requires public health departments with specialists in virology and epidemiology. Second, acquiring the necessary knowledge is costly (Gailmard and Patty Reference Gailmard and Patty2013); it takes years of study and training (Feltovich, Prietula, and Ericsson Reference Feltovich, Prietula, Ericsson, Ericsson, Hoffman, Kozbelt and Williams2018).

Further, because becoming competent in a knowledge-intensive domain is time-consuming, one cannot be a specialist in many fields.Footnote ¹⁸ Complex organizations typically employ different kinds of specialists and have areas of expertise populated by some but not all of its members: for example, some military officers specialize in war on land; others, at sea.

The problem is conspicuous in cabinets where a President reviews department secretaries with specialties unfamiliar to the President. The President could try to finesse this by only selecting people with his background. For most U.S. presidents, that would restrict appointees to people with backgrounds in law and electoral politics. But this maneuver merely pushes the problem down a level in the hierarchy. For example, Dick Cheney, a Defense Secretary under Reagan, was a career politician without military experience. Consequently, President Reagan did not have to review a secretary who had specialized military knowledge—but Secretary Cheney faced that problem. Hence, even if nobody in the Office of the Secretary of Defense had a repertoire bigger than the President’s, they had to review the performance of career military specialists.

This is related to Weber’s argument that contemporary bureaucracies, in authoritarian systems as well as democracies, cannot function effectively without specialized knowledge and skills—usually the province of specialists. This suggests that holding all officials accountable requires that some specialists must be reviewed by amateurs. (A now-salient possibility: the specialists are AI agents.)

We now analyze this issue formally. Consider an organization, $ O=\{{i}_1,\dots, {i}_n\} $ , with $ n>1 $ members. For any $ i\in O $ , $ {S}_i $ denotes i’s set of specialties; for example, data scientist, helicopter pilot, and lawyer. With this framework, we can give a precise definition of situations in which hierarchical authority and the authority of expertise conflict versus those where they do not.

Obviously, i’s reviewing of j is free of Weberian tensions if both were trained exclusively in the same specialty. It is equally obvious that i’s review of j could be problematic if each has mastered one specialty which are different or if i has no specialty while j does. Hence, we begin our search for review structures that are free of Weberian tensions by examining circumstances in which everyone in O has mastered one specialty. Given that many positions require just one specialty, this is a natural starting point.

The important part of Theorem 3 is the necessity component. Government departments that have only one specialty are quite rare now. Hence, Weberian tensions are to be expected in domains where learning a specialty is so demanding that people master just one in their career.

Although many governmental positions probably require only one kind of specialized training, some (e.g., military lawyers) require more than that. Our next theorem shows that relaxing this assumption still yields a sharp result for review structures which have an autocrat.

Recall that an autocrat is in effect an oligarchy of one. Thus, i is the autocrat of O if nobody in O reviews i and everybody else in O is reviewed, directly or indirectly, by i. Tree hierarchies have these features.

The next result, Theorem 4, analyzes a relation between hierarchical authority and knowledge-based authority for review structures with an autocrat. Theorems 3 and 4 complement each other: the former restricts agents’ repertoires (everyone has mastered exactly one specialty) while allowing for arbitrary review structures; the latter does the opposite.

Because Theorem 3 assumes that all agents have singleton repertoires, relations among their repertoires are simple: either i and j are the same kind of specialist, whence their repertoires are identical, or their specialities differ, so their repertoires are disjoint. Since Theorem 4 allows agents to have mastered multiple specialties, their repertoires may overlap in various ways. To create order out of this complexity, the concept of a class of specialists is useful.

Several properties of classes of specialists, which are partial orders induced by the set-inclusion relation, are worth noting. First, everyone in O with the same set of specialties belongs to the same class of specialists. (Setting $ k=i $ or $ k=j $ in the definition implies that.)

Second, every organization has at least one class of specialists.Footnote ¹⁹ Third, the set of repertoires of everyone in a class of specialists has a unique maximal element.Footnote ²⁰

The last property implies that within a class of specialists, one can always construct a review structure that is free of Weberian tensions: let those in the class with the maximal set of specialties review everybody else in that class. More generally, follow the hierarchical structure among the agents’ repertoires implied by Definition 7. Consider, for example, the seven-person organization of Figure 3. That authority structure is free of Weberian tensions because it corresponds to the hierarchy of repertoires.

Figure 3. An Organization with a Single Class of Specialists

Reviews across classes of specialists are more complicated.

Because nobody can be reviewed by someone from a different class of specialists if there is only one class of specialists, Proposition 1 matters only when there are multiple classes. In such circumstances, to avoid Weberian tensions, it is necessary (not sufficient) that agents who belong to just one class of specialistsFootnote ²¹ are unaccountable to anyone in the organization who is from a different class.

Although simple, Proposition 1 identifies a property that is central to the appearance of Weberian tensions. Our next major result, Theorem 4, puts this property to work.

Analyzing review relations between classes of specialists is more complicated in the setting of Theorem 4 than in the stark world of Theorem 3. It is helpful to visualize an agent’s repertoire as a bundle of specialties attached to a node, just as we have done in Figure 3. Review relations are the graph’s edges; they can be constructed independently of agents’ repertoires. However, for an edge to represent a review relation without a Weberian tension, a reviewee’s repertoire must be a subset of his reviewer’s repertoire.

For an autocrat-governed organization to be free of Weberian tensions, obviously the autocrat must have the maximal repertoire of the (unique) class of specialists. We call such an agent a top specialist of that class.

The important part of Theorem 4 concerns necessity. It tells us that if an organization has multiple classes of specialists, then every review structure with an autocrat exhibits Weberian tension somewhere.Footnote ²² Contemporary governments include a wide array of specialists, from aeronautical engineers to X-ray technicians, which differ too much to be linked by the set-inclusion property of Definition 7. Hence, it is extremely unlikely that such institutions have only one class of specialists. Today the necessary condition in Theorem 4 is not satisfied even by modest-sized governments. This also holds for nongovernmental organizations. Therefore, if an organization has a chief executive, we should anticipate Weberian tensions.

Theorem 4 is not restricted to democracies; it holds for all autocratic organizations. Whether a particular organization—say, a government’s executive branch—is embedded in a democratic or an authoritarian polity is irrelevant: the result holds for both. Of course, the interpretation of Theorem 4 depends on the political context of the government’s executive branch. Because in democracies an autocrat of the executive branch is accountable to voters, one would probably call him or her a chief executive. In contrast, an autocrat of that branch who is not reviewed by anyone outside the bureaucracy is a dictator.

With these ideas in mind, Theorem 4 implies the following corollary.

Thus, authoritarian systems cannot evade the problems created by the review of highly specialized agents by less or differently specialized ones.Footnote ²³ An important contemporary example: Putin, like Clemenceau and Wilson, never served in the active military.

The restrictiveness of Theorem 4’s necessity component is revealed by the following property of a single class of specialists.

Given Remark 1’s restrictiveness, probably most contemporary agencies have multiple classes of specialists.

Remark 1 and Theorem 4 apply not only to entire organizations but also to their parts. Consider, for example, the Department of Health and Human Services in the U.S. executive branch. This department includes eight major divisions: for example, the Offices of Infectious Diseases, of Disease Prevention, and of Women’s Health. Because each division has an autocrat, an assistant secretary, Theorem 4 holds for these units. Hence, there are no Weberian tensions inside, say, the Office of Disease Prevention if and only if this part of HHS has just one class of specialists. Similarly, Remark 1 implies that the Office of Disease Prevention has just one class of specialists if and only if the repertoire of someone in that unit equals the union of all the repertoires in that unit.

Thus, Weberian tensions are not restricted to relations between politicians and bureaucrats: they also arise within bureaucracies. (Again, this holds for bureaucracies in autocracies as well as those in democracies.) Indeed, in the important special case of tree hierarchies, where every subordinate has a single superior, every subunit is headed by an official who is effectively the autocrat of that subunit. For such structures, the recursive nature of Theorem 4 becomes apparent.

Thus, complaints about, for example, civilian control of the military reflect only the tip of an iceberg. In modern bureaucracies, the same issue—specialists being reviewed by people who lack some of the reviewees’ domain-specific knowledge—is repeated down the line, probably well into middle-sized parts of agencies.Footnote ²⁴

However, Theorem 4 is confined to review structures where there is an autocrat. It does not hold for all review structures. Figure 4 provides an example. The organization has two classes of specialists: one has repertoires $ \{(1,\hskip2pt 2),(1)\} $ ; the other, $ \{(1,\hskip2pt 3),(1)\} $ . This example shows that despite multiple classes of specialists, one can construct a review structure without Weberian tensions.

Figure 4. Review Relations without Weberian Tensions

The example in Figure 4 illustrates an important property: overlapping classes of specialists. The only other possibility is nonoverlapping (disjoint) classes of specialists. Figure 5 displays an organization with disjoint classes of specialists: one is (1, 2), (1), and (2); the other, (3, 4), (3), and (4). The figure shows that constructing review structures within each class of specialists that are free of Weberian tensions is easy. Making it work between classes is more difficult. Indeed, Proposition 1 says it is impossible. Since its review structure must be a weakly connected graph, as is required by the concept of an organization, somebody in class $ {C}_1 $ would have to review somebody in $ {C}_2 $ or vice versa. (The review structures in Figure 5 have no Weberian tensions, but they represent authority systems in two independent organizations.) The first two possibilities are weakly connected review structures, but both exhibit Weberian tensions: for example, if somebody in $ {C}_1 $ reviews anybody in $ {C}_2 $ , the reviewee has a specialty that the reviewer lacks.

Figure 5. Disjoint Classes of Specialists and Review Structures without Weberian Tensions

This example leads to the next result.

Theorem 5 is based on a concept of a separable set of classes of specialists.

For example, suppose organization $ {O}_1 $ has two classes of specialists. One class is people with repertoires of $ (1,2) $ and $ (1) $ ; individuals in the other have repertoires $ (2,3) $ and $ (3) $ . This set of classes of specialists is separable: nobody’s repertoire is in both classes. Organization $ {O}_2 $ also has two classes of specialists. One is composed of people with repertoires $ (1,2) $ , $ (1) $ , and $ (2) $ ; the other, those with repertoires $ (2,3) $ , $ (2) $ , and $ (3) $ . This set of classes of specialists is non-separable: people with repertoire $ (2) $ belong to both classes.Footnote ²⁵

Bureaucracies with multiple classes of specialists—the typical case today—which are free of Weberian tensions have some odd and possibly troubling properties. Theorem 6 identifies several of these.

Recalling that an oligarchy is defined as the minimal group whose members (1) review everyone outside the group and (2) are not reviewed by anyone outside the group makes it easier to understand the result.

Considering that contemporary bureaucracies typically have multiple classes of specialists, Theorem 6 establishes that avoiding Weberian tensions exacts a heavy price: an oligarchy governs the organization. The oligarchy is a coalition of guilds: every class of specialists has at least one “representative”—a top specialist of that class—in the governing coalition. Moreover, the absence of Weberian tensions implies, via Proposition 1, that these representatives of different guilds are not accountable to each other. Nor are they accountable to an autocrat: Theorem 6 assumes that O has multiple classes of specialists, yet its review structure is free of Weberian tensions, a combination which precludes autocracy (Theorem 4). Thus, the plans of different classes of specialists can be at cross-purposes. (In World War II, the Japanese army and navy had sharply different war aims which were never reconciled.) This is coordination failure on a grand scale. It is not an effective way to run large complex bureaucracies.

An oligarchy could appoint an administrator with the authority to coordinate the classes of specialists below him and who would be accountable to the oligarchy. This is approximately the structure of publicly traded corporations (the chief executive officer is formally accountable to the corporate board) and of U.S. cities which have a top manager accountable to the city council.

But this structure cannot evade the constraints identified by Theorem 4. A city manager with that kind of authority is in effect an autocrat of most of the city government: she reviews all the bureaucrats. Hence, Theorem 4 holds for this part of the organization. Therefore, its review structure is free of Weberian tensions only if that part of the organization has just one class of specialists and the city manager is a top specialist of that class. Thus, the problem cannot be finessed by creating a role of top administrator.

Finally, Theorem 6 has implications for authoritarian systems that are ruled by a group (a situation not covered by Corollary 3). By Theorem 6, if a government’s executive branch has multiple classes of specialists and is free of Weberian tensions, then it is run by an oligarchy. Hence, if this group is not accountable to anyone outside the government, then it is accountable to nobody. This is an authoritarian system, governed by a group. On the other hand, if the executive branch’s oligarchy is accountable to a mass electorate, then Weberian tensions must appear. This trade-off is inescapable.

Table 1 summarizes what we know about the appearance of Weberian tensions, depending on the type of review structure and the distribution of specialized knowledge in O. (The table’s cells are mutually exclusive and collectively exhaustive.)

Table 1. Which Review Structures Exhibit Weberian Tensions?

Note: None: No review structures in this cell exhibit Weberian tensions. Some: Some review structures in this cell exhibit Weberian tensions. All: All review structures in this cell exhibit Weberian tensions.

The leftmost column represents the simplest kind of distribution of specialized knowledge: everyone in the organization has the same specialty. Obviously, therefore, Weberian tensions cannot appear in any review structure in cells 1, 4, and 7. These tensions become more common as we move from left to right in the table, that is, as the knowledge structure becomes more complex.Footnote ²⁶

Although Table 1 gives a useful overview, its cells report only qualitative information about the frequency of Weberian tensions. Our last major result provides quantitative information about all possible organizational categories except for oligarchies with non-separable classes of specialists.

1. (i) If the review structure is an autocracy, or an oligarchy with separable classes of specialists, then it has at least $ c-1 $ Weberian tensions.

2. (ii) If a review structure is a democracy, then there are no Weberian tensions if agents’ repertoires are homogeneous, and at least c Weberian tensions if repertoires are heterogeneous.

Thus, for these types of review structures, the new knowledge or skills embodied in new classes of specialists strictly increase the minimal number of Weberian tensions exhibited by the authority structure. This is a sharp statement about the relation between specialized knowledge and problematic performance reviews.Footnote ²⁷

Theorem 7 points up an important difference between autocracy and democracy. If O has only one class of specialists, then by Theorem 4 one can construct an autocratic review structure that is free of Weberian tensions. This is not, however, generally true of democratic systems. The reason is fundamental. The structure of a class of specialists with heterogeneous repertoires is inherently asymmetric. In contrast, democratic review is inherently symmetric: all citizens in such systems are connected by review cycles (Theorem 2). Napoleon’s regime is a good example of these structural differences. Under him the French military was much more meritocratic than it had been in the Ancien Regime and Napoleon warranted being at the top of a military prestige hierarchy, but as Emperor he was unaccountable.