Transfer of Authority within Hierarchies1 Pinghan LIANG2 Research Institute of Economics and Management Southwestern University of Finance and Economics January 6, 2012

1 I’d

like to thank the helpful comments from Eduardo Azevedo, Alessandro Bonatti, Joan de Marti, Mikhail Drugov, Guillaume Haeringer, Oliver Hart, Sjaak Hurkens, Yuichiro Kamada, Kohei Kawamura, Georg Kirchsteiger, Ines Macho, David Martimort, Jordi Masso, Hannes Mueller, David Perez, Nathanael Vellekoop, Li-an Zhou as well as the attendants in the workshops in UAB, Tilburg, Renmin, Peking, Shandong, ENTER Jamboree (2010), DMM (2010), ESNIE (2010), Stoney Brook Festival (2010), 2nd Brazilian Workshop of Game Theory, CGTEEA (2010), and HKEA (2010). 2 Research Insitute of Economics and Management, SWUFE. E-mail address: [email protected]. Fax: +86 28 87352178

Abstract This paper highlights the distribution of decision-making responsibility within a stylized minister/senior/junior hierarchy. The minister could delegate authority only one level lower to a biased uninformed senior, then the latter could further delegate to the informed junior. We characterized the delegation outcomes according to the subordinates’ biases. Results showed that when the senior’s personal interest was between the minister’s and the junior’s, by restricting the delegation set delievered to the senior, the minister could attain the best outcome as if she contracted directly with the informed party. The implications for recruitment policies, job contents and promotion decisions were explored. The loyal agent may be assigned to the lower position. If the minister cannot …nd out each subordinate’s bias, she may delegate the task to a properly biased senior who can in turn observe the informed party’s bias. Finally, within the hierarchy the minister sometimes preferred commanding via the senior to the delegation. Key words: Delegation, Cheap Talk, Mediator, Hierarchies JEL classi…cation codes: D72, D78, D82

The principles of o¢ ce hierarchy and of levels of graded authority mean a …rmly ordered system of super- and subordination in which there is a supervision of the lower o¢ ces by the higher ones. — — — Max Weber (1946, p. 214)

1

Introduction

Bureaucratic systems are typically organized as multi-layer hierarchies, in which the top-level decision maker needs a chain of intermediaries to command the front-line executive agents. It is usually that neither can communicate directly with the Decision Maker (DM), nor may the DM skip-level command the bottom-level subordinates. For example, in the governmental budgeting process, a senior bureaucrat receives the budget from the minister of the department, and the senior also assigns the budget to his immediate subordinates, who, in turn, do the same for their subordinates and so forth. In typical federal countries, e.g., U.S.A., the federal government usually cannot skip the state level to in‡uence the behavior of municipality or county governments. Even the business strategic planning process is hierarchically structured: the corporate headquarters make decisions on the amount of resources to allocate to each division, and the division manager decides the allocation of funds to each business unit within the division (Kotler, 2003). In this paper we are interested in the consequences of hierarchies. We want to address how hierarchies a¤ect the distribution of decision-making responsibility, and how positions within hierarchies are …lled? We investigate the strategic interaction in a three-layer hierarchy with one principal (minister, "she") and two agents (senior and junior, "he"), i.e., a chain. All players want to adapt to the state of the world, though they also have their own constant private bene…ts (bias), e.g., intrinsic preferences, compensation package, or personal career concerns, etc. This bias is assumed to be common knowledge. Only the junior o¢ cial has information about the true state. It is not feasible for the minister to directly contract with the junior. In the main text we concentrate on hierarchical delegation, which refers to that the sequence of delegating is top-down: the minister delegates noncontractable decision rights over a range to the senior, then the latter determines the extent of responsibilities delegated to the informed junior. In other words, the minister has limited commitment power in the sense that she could delegate only to someone one level underneath her1 . 1 The situation that enforceable contracts can only be written between certain parties also motivates the study of Kolotilin, Li and Li (2011) on optimal limited authority.

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In the hierarchy we fully characterize the interval delegation mechanism, which resembles budgeting or quantity control2 . We show that even in the presence of strategic intermediaries, the e¢ cient delegation set of two-tier hierarchies a la Holmstrom (1984) and Alonso and Matouschek (2008, henceforth AM) can be attained whenever the subordinates have like biases and the senior is less biased than the junior, e.g., the junior wants to overstate the demand more than the senior. The intuition is somewhat straightforward: compared with the minister, an intermediately biased senior prefers to expand the discretion of the junior; thus the minister can optimally restrict the span of control of the senior to avoid this deviation. However, whenever the subordinates have opposing biases 3 , or the junior is less upwardly biased, the senior prefers to shrink the junior’s discretion, so the e¢ cient delegation set is never implementable. We then use this characterization to shed lights on organizational design, i.e., optimal composition of the senior and the junior. We …rst study the situation that the minister can re-organize the department completely, namely the senior and the junior are chosen by her simultaneously. Surprisingly, we demonstrate that an o¢ cial closer to the minister sometimes may be placed in the junior position, provided that the other o¢ cial is not too biased. We then look at the recruitment of junior o¢ cials. It’s shown that a compromised candidate will be hired, i.e., the optimal junior would have the intermediate bias between the superiors. Finally, if we relax the assumption that the junior’s personal bene…t is common knowledge, then the minister may gain by delegating full authority to an acquainted biased agent who perfectly observes the preference of the informed agent, instead of delegating to the latter directly. Thus both hierarchies and complete delegation can arise endogenously thanks to the asymmetric information about the preference misalignments. Dessein (2002) …nds that in a two-layer hierarchy, with standard assumptions about the DM’s payo¤ and prior, delegation is optimal "whenever informative communication is possible" (p. 822). We then compare hierarchical delegation and hierarchical communication,4 i.e., ordering the junior what 2

Interval delegation is widely used in many areas under di¤erent forms, e.g., budgeting in managerial accounting, price ceiling for a regulated monopolist, liquidity requirement, or a cap on short-term debt in bank regulation. Another reason to concentrate on interval delegation is the enforcement problem. Think about the budgeting procedure: if the toplevel management prohibits certain funds ranges within the budget, the subordinates can get rid of these constraints by requesting more than his needs and freely disposing the remaining money (Holmstrom, 1984). 3 E.g., for a particular public project, the junior wants to overstate the needs while the senior wants to understate them. 4 Goltsman et al (2009, henceforth GHPS) uses the term "mediator cheap talk" to refer

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to do after hearing from him via the senior. The previous results, which favor delegation,5 are no longer valid in hierarchies. When the subordinates have opposing biases, for some range of preference misalignment, delegation strictly underperforms communication. The logic is not hard to see: on the one hand, compared with direct delegation, the payo¤ from hierarchical delegation decreases as the senior turns out to be oppositely biased; the senior will overly constrain the junior when sub-delegating. On the other hand, communication via an oppositely biased senior can be improved over that in two-tier hierarchies since he has the incentive to …lter information ‡ow, i.e., using a speci…c garbling of information (Ivanov, 2010). When the subordinates’preference misalignment is not too far apart, delegation is not valuable; hierarchical communication dominates. Even though this work addresses a chain, the results can shed light on more complex hierarchies. The conditions for attaining the e¢ cient delegation outcome in a three-level chain can be extended to an n-level one: whenever the agents’ bias changes monotonically with respect to their ranking, by giving a suitable ceiling, hierarchies of arbitrary depth will have the same extent of responsibility delegated at the bottom level as a two-tier hierarchy. As long as hierarchies are formed based on the considerations other than strategic information transmission, and there is no interdependence among the agents’actions, e.g., limited capacity of the principal (Williamson, 1967), information processing cost (Radner, 1993), heterogenous knowledge (Garicano, 2000) or con‡ict over hiring and promotion decisions (Friebel and Raith, 2004), our results still hold in a multiple subordination structure, i.e., a tree. The remaining parts of this paper are organized as follows. In the following section we brie‡y review the related literature. In Section 3 we lay out the basic model, and introduce the benchmark case of direct delegation, which serves as the e¢ ciency criterion. Section 4 characterizes the optimal interval delegation set in the hierarchy. Section 5 studies the hiring policy and promotion decisions in the hierarchy, and investigates delegation mechanism under asymmetric information about the junior’s bias. Section 6 compares delegation with communication in the chain. Section 7 discusses the results and makes conclusions. All proofs are relegated in Appendix. it. 5

GHPS show that a constrained delegation mechanism can attain the best outcome from any incentive-compatible mechanism a la Myerson (1981), thus strengthening the dominance of delegation over communication.

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2

Related Literature

Our work focuses on strategic behavior within a …xed hierarchical structure. We investigate the issue of organizational design without transfer and under unveri…able information; thus, this paper is closely related to the literature of cheap talk. Crawford and Sobel (1982, henceforth CS) demonstrate that full information revelation is impossible so long as there is preference misalignment between the DM and the informed party. Noting that as the con‡ict becomes less intense, the e¢ ciency of unconstrained delegation increases more than that of communication, Dessein (2002) shows that in the CS classical speci…cation the principal is better o¤ by completely delegating decisionmaking authority to the informed agent. Further, Dessein shows that it could be a welfare-improvement if the principal delegates to a properly biased uninformed intermediary, then the latter makes decisions based on the report from the informed agent. He precludes the intermediary’s ability to sub-delegate, and the DM’s discretion in restricting delegation set. In this work we allow the downstream transfer of decision rights in a ranking and focus on constrained delegation, which covers unconstrained delegation as a special case. Consequently, we show that a suitable hierarchical delegation mechanism will achieve the same performance as the optimal revelation mechanism corresponding to a two-tier hierarchy. A closely related work is the paper by Ambrus et al (2011), who compare the performance of the closed rule and the open rule in the ‡oor-committeelobbyist hierarchy. In terms of our work, the open rule is equivalent to hierarchical communication, while the closed rule resembles unconstrained delegation to a senior. They also don’t allow the committee to sub-delegate, e.g., rubberstamp the proposal of the lobbyist, even though the repeated interaction between the committee and the lobbyist may provide incentives for the former to stick to this decision rule (Alonso & Matouschek, 2007). Thus our results on optimal interval delegation are more general. There is a large body of literature that addresses the bene…ts of hierarchies in organizational economics. Largely motivated by the idea of bounded rationality, the works by Radner (1993), Bolton and Dewatripont (1994), and Geanakoplos and Milgrom (1991) are concerned with the role of hierarchies in facilitating information processing, and stress information aggregation within vertical structure. Based on the heterogeneity among agents, another related strand of literature since Garicano (2000) develops the "knowledge-based" hierarchy. Williamson (1967), Calvo and Wellisz (1978) and Qian (1994) focus on the optimal span of control. As a complement, in this paper we follow McAfee and McMillan (1995) and leave unexplained why the hierarchy exists, and examine the strategic behavior within a given hierarchical structure. 4

In other words, our main emphasis is on the consequences of the hierarchy, not on the structure or origin of the hierarchy. However, Section 5.2 also provides a strategic foundation for the hierarchy in the sense that delegation via an intermediary arises due to the asymmetric information over the extent of preference misalignment of the juniors. It’s also noteworthy that this endogenous hierarchy in part is in accordance with the argument of bounds on any individual’s span of control. Tirole (1986) explicitly introduces a supervisor into the principal-agent relationship to study multiple-layer hierarchies. In his paper the supervisor holds private information about the type of agent, and the highlight is the collusion between the supervisor and the agent. He establishes the equivalence between coalition-proof contract and giving ownership to a supervisor, who then subcontracts with a downstream agent. Mookherjee and Reichelstein (1997) study the optimal budget mechanism in 3-tier hierarchies. Landier et al (2009) explore the costs and bene…ts of the preference divergence among agents in a 3-tier hierarchy. Our work explores similar questions in a di¤erent set-up in which the physical transfer is not feasible and the information structure is exogenous, thus, our results regarding the e¢ ciency of subcontract and organizational design also di¤er. There is an emerging body of literature addressing hierarchical communication. Based on the CS model, GHPS establishes that by hiring a neutral mediator who optimally …lters information from the informed party, the DM could implement the outcome of the optimal universal mechanism. Ivanov (2010) and Ambrus et al (2010, henceforth AAK) demonstrate that this outcome can be implemented when the mediator’s bias is in the opposite direction of the informed agent. In a hierarchical structure we compare these results with the hierarchical delegation mechanism, show that there exists some range of preference misalignment in which communication outcome cannot be replicated by delegation mechanism, therefore hierarchical communication may strictly dominate hierarchical delegation. Our paper also contributes to the growing body of literature on mechanism design embodied in network structures (Renou and Tomala, 2011). In particular, we share with Renou and Tomala the motivation to look at the implementation of mechanism in a network structure. However, we di¤er from this line of research in addressing a speci…c organizational form: delegation, and stress that network architecture may constrain the choice of mechanism.

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3

Model

A hierarchical organization (government department) is composed of three players: a minister (denoted as player DM ), a senior bureaucrat, and a junior bureaucrat (denoted as players sb and jb, respectively). The payo¤ of all players takes "quadratic loss function" as the leading example in CS6 : U ( ; y; bi ) =

(

y + bi )2 ; i = DM; sb; jb

(1)

Thus, their payo¤s depend on the state of the world 2 = [0; 1], (e.g., the demands for a public project), the action undertaken y 2 Y = [0; 1], (e.g., the size of the project), and their private bene…ts bi . Each player wants to minimize the loss from mis-adaptation to the true state, though to di¤erent extents, i.e., the ideal action for player i is + bi . Without loss of generality, we normalize the personal bene…t of the minister to zero and use bsb ; bjb to measure the preference misalignment between o¢ cials and the minister, e.g., the bureaucrats may want to overstate the budget to expand the subordinates. For the sake of simplicity, we use Ui ( ; y) to refer the utility of player i. The minister has the rights to take a decision, and only the junior will be informed about the true state . He is able to implement the decision, but he could not communicate directly with the minister. Neither the minister can assign decision-making rights directly to the junior. In other words, the senior has full control over the information transmission between the minister and the junior. The superiors (the senior and the minister) have uniform prior on . [Insert Figure 1] In Figure 1 we specify the timing of this game under hierarchical delegation. The minister …rst assigns two o¢ cials to the senior and the junior positions, and only the junior learns the true state. If the minister chooses hierarchical delegation (HD henceforth), she gives the senior full control over an interval of action Ysb Y . Then he can sub-delegate to the junior by granting him a menu of decisions Yjb Ysb . Finally, given , the junior will undertake his preferred action within Yjb . In other words, the junior is allowed to spend up to the budget speci…ed by the upper bound of Yjb . 6

This speci…cation is standard and widely used in the literatures on communication and organizations, e.g., Krishna and Morgan (2001), GHPS (2009), Blume et al. (2007), Mylovanov (2008), Kovac and Mylovanov (2009), Ivanov (2010).

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3.1

Benchmark: direct interaction

We start with the benchmark case that the informed junior can report to the minister directly, who then updates beliefs about the true state and makes a decision. Thus, we have the classical CS model, as Lemma 1 summarizes Lemma 1 (CS 1982) If the junior could send reports to the minister directly, then informative communication occurs if jbjb j 41 , and in equilibrium the induced set of actions consists of …nite elements unless bjb = 0. Proof. See Lemma 1 and Theorem 1 in CS. In this situation, the minister has imperfect commitment power and has to take the best responses to available information. She gains by keeping the ex post ‡exibility in decision making, but loses in motivating the informed party to provide information. Communication is informative if the minister would have a …ner partition about the state after hearing messages. It naturally leads to questioning whether and how the minister could improve her welfare. By delegating authority to the informed junior, the minister allows the informed party to undertake his preferred action within a speci…ed set. This is in e¤ect equivalent to making commitment to an outcome function in which the minister rubberstamps any recommendation from the informed agent whenever it falls within a boundary. She forges the control rights to facilitate the use of information. Dessein (2002) shows that unconstrained delegation dominates communication so long as the informed party is not too biased. Alonso and Matouscheck (2008) demonstrate that in two-tier hierarchies the optimal delegation set is always an interval of options or a single decision. GHPS further establishes that the optimal delegation mechanism a la Holmstrom (1984), in which the informed party is given control over an interval of Y , can implement the optimal arbitration rule, i.e., attain the best outcome from any incentive-compatible mechanism. In other words, optimal delegation can implement the outcome of direct revelation mechanisms (Myerson, 1981). Thus the optimal arbitration rule describes optimal delegation in minister-junior direct interaction. Lemma 2 If the minister can delegate authority to the junior directly, then 1 , and the action the optimal delegation set is Y = [0; 1 bjb ] if 0 < bjb 2 chosen is + bjb ; if 2 [0; 1 2bjb ] y( ) = 1 bjb ; otherwise On the other hand, if Y = [ bjb ; 1]

1 2

bjb < 0, then the optimal delegation set is

7

Proof. It’s a straightforward application of the Theorem 1 in GHPS (2009).

It is optimal for the minister to cap the junior’s choices against the direction of his bias. If the minister knows that the junior is biased towards spending too much on a project, he is given a budget and allowed to spend any amount up to that level, and vice versa. This optimal delegation set would serve as the e¢ cient delegation set for our analysis on hierarchical delegation. The expected utility for the minister under the e¢ cient delegation, thus, is:

EUDM =

Z

0

1 2bjb

b2jb d

Z

1

[

(1

1 2bjb

4 bjb )]2 d = ( bjb 3

1)b2jb

(2)

Various authors have provided characterizations of optimal delegation mechanisms, including using arbitration to resolve dispute (Melumad and Shibano, 1991), optimal interval delegation (AM 2008), veto-power of the decision maker (Mylovanov, 2008), and stochastic delegation (Kovac and Mylovanov, 2009). To concentrate on the issue of interest, we would focus on the widely-used interval delegation, in which the delegation set contains either one decision7 or an interval of decisions.

4

Hierarchical Delegation

Under hierarchical delegation, the minister cannot directly contact with the informed junior bureaucrat.8 We would focus on the case that jbsb j < 21 and bjb bsb 0 < bjb 12 , namely the junior has upward bias. We would use to represent the divergence of preferences between the junior and the senior. Hierarchical delegation can be best thought of as a two-stage game. In the …rst stage, the minister can only commit to transfer authority to someone one level lower or not, and she can put caps (bottoms) on actions. In the second stage, once the decision-making power is transferred downwards to This would be the ex ante optimal response EDM [ ] = 12 : The senior bureaucrats may actively prevent the direct link between his subordination and supervision. The behavior of Sir Humphrey Appleby, the permanent undersecretary in the …ctional Department of Administrative A¤airs in the BBC series Yes, Minister, vividly illuminates this point. 7

8

8

the senior, he is e¤ectively playing the role of the principal in a Dessein style delegation game. As Dessein (2002, Proposition 2) shows, even though communication becomes more informative as the preference misalignment between the DM and the informed party vanishes, the equilibrium partition becomes more coarse relative to the bias. Hence, as long as the junior is not too biased, the senior chooses delegation9 over communication. GHPS further shows that constrained direct delegation can implement the optimal universal mechanism. Therefore, whenever the minister delegates an interval of options, the senior prefers downstream delegation over cheap talk. Formally, we say a set of actions Y 0 is implementable if Y 0 Yjb Ysb , in other words, the actions undertaken should be in the interest of the junior, the senior, and the minister simultaneously. The implemented set is the junior’s interval of delegated decisions in that only he has the relevant information to choose the appropriate actions. The senior acts as a principal within an interval Ysb , and he will make delegation decision based on the di¤erence of bias . Therefore in hierarchical delegation the minister can a¤ect the behavior of the junior only through restricting Ysb . Lemma 2 immediately leads to the outcome that if Ysb is an interval or a point, so is Yjb . The minister thus will just limit the menu of options by imposing a cap and/or bottom. Then, we turn to characterize the implemented delegation set. Proposition 1 If > 0; the highest available option passed to the junior is yjb = minf1 ; supfy jy 2 Ysb gg This proposition demonstrates the consequence of con‡ict of interest between the minister and the senior. If the senior is granted with an unconstrained set of decisions, then he would choose his ideal set of delegated decisions. If the preference misalignment between him and the junior is smaller than that between the minister and the junior, relative to the minister, the senior always wants to expand the junior’s discretion, and the implemented delegation set ([0; 1 ]) is larger than the e¢ cient one ([0; 1 bjb ]), as shown in Figure 2. However, it also suggests that the minister could control this loss by truncating the delegation set to the senior, i.e., imposing the cap 1 bjb on Ysb . The e¢ cient delegation set thus could be implemented in a three-tier hierarchy. [Insert Figure 2] 9

The author acknolwedges an anonymous referee for this statement.

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Corollary 1 If two bureaucrats have like biases, and the senior is less biased, i.e., bjb > bsb > 0, then the e¢ cient delegation set Y can be implemented by imposing supfy jy 2 Ysb g = 1 bjb Therefore, in a three-tier hierarchy the minister still can replicate her optimal outcome Y in a two-layer hierarchy by appointing a less biased senior and removing extreme options from the delegation set Ysb . The expected payo¤ to the minister, thus, would be the one in (2). Compared with the results of Dessein (2002) and Ambrus et al (2011), we show that whenever the agent’s bias is increasing within the ranking, by placing appropriate caps on discretionary power, the minister can ensure that the second stage decision corresponds with the optimal delegation outcome. However, this result crucially depends on the senior being less biased than the junior. If he is more biased, i.e., the senior is interested in expanding subordinates, or two agents have opposing biases, i.e., compared with the junior, the senior wants to shrink the budget for a particular project, then relative to the minister, the senior strictly prefers to impose extra restrictions on the e¢ cient delegation set Y , and the e¢ ciency cannot be attained. Proposition 2 If the two bureaucrats have opposing biases or the senior is more biased than the junior, then the e¢ cient delegation set is not implementable, i.e., Yjb 6= Y : We illustrate the failure of implementable e¢ cient delegation set in the following …gures. In Figure 3(a), when these two agents have opposing biases, the di¤erence of bias between bureaucrats is even larger than that between the junior and the minister. As a result, the senior prefers to shrink Y by lowering the cap. The minister cannot force the senior to authorize the junior to take any decisions above bsb + 1 : Similarly, if bsb > bjb , the junior becomes downwardly biased relative to the senior, and the latter imposes a bottom on Y , as shown in Figure 3(b). Thus any decisions below bsb would not be delegated to the junior. [Insert Figure 3(a)] [Insert Figure 3(b)] In the following proposition, we summarize the characterization of the optimal implemented interval delegation set under di¤erent biases of agents. We should keep in mind that when either j j 1=2 or jbi j 1=2 fails, the minister prefers to take ex ante optimal action 12 ; the expected payo¤ thus 1 would be 12 . In other words, delegation is not valuable. 10

Proposition 3 In the three-layer hierarchy the optimal implemented delegation set prescribes the set of decisions available to the informed junior. Formally, it satis…es: 8 0 [0; 1 bjb ] , if 12 bjb bsb > > < [0; 1 + bsb ] , if bjb > 0 > bsb and 2 0; 21 Yjb = (3) [bsb ; 1 bjb ] , if bsb > bjb > 0 and j j 2 0; 21 > > : 1 , otherwise 2 and the minister’s expected utility is 8 if 12 bjb > bsb 0 ( 34 bjb 1)b2jb , > > < 8 3 4 3 2 2 b + b 4b b + b , if b > 0 > b and 2 0; 12 jb sb HD jb sb jb 3 sb 3 jb EUDM = 8 3 b + 4bjb b2sb b2jb , if bsb > bjb > 0 and j j 2 0; 21 > > 3 sb : 1 , otherwise 12

(4)

The expected payo¤ to the senior o¢ cial is 8 2bjb 2 (1 2bjb ) 0 b2jb + 3b2sb , if 12 bjb > bsb > > 3 > 1 4 3 2 < if bjb > 0 > bsb and 2 0; 2 +3 , HD 3 EUsb = 3 (bjb +bsb ) 2 > > (1 2b ) , if bsb > bjb > 0 and j j 2 0; 21 sb > 3 : 3 1 b2sb , otherwise 12 (5) In contrast with Dessein (2002) and Ambrus et al (2011), this proposition demonstrates that when the authority could be transferred from the senior to the junior, the senior’s bias would be irrelevant so long as it lies between the minister and the junior. Otherwise, the span of discretionary authority of the junior, i.e., the implemented delegation set, would be increasing (decreasing) with respect to the bias of the senior if it is downwards (upwards). Moreover, HD HD @EUDM @EUDM < 0 if b > b ; and > 0 if bsb < 0. Thus when the senior is sb jb @bsb @bsb more biased or oppositely biased, the more loyal the senior is, the larger the gains in delegation.

5

Organizational Design in Hierarchies

In this section we will use hierarchical delegation to study some issues of organizational design, i.e., how to …ll the positions in hierarchies. This involving question contains the optimal selection of the senior and the junior, i.e., hiring policy and promotion decisions. 11

The …rst question is given the preference misalignment between two of…cials, if the minister can re-organize the department, which one should be assigned to the higher position in the ranking? It di¤ers from promotion decisions in that it involves complete re-assignment of jobs, instead of promoting one bureaucrat while leaving the other unchanged. To deal with this question, without loss of generality, we assume that both bureaucrats have like biases, i.e., b2 > b1 > 0. The minister thus needs to balance between using the more biased agent 2 as the junior, in which the loss of control occurs in the high states, and using the less biased agent 1 as the junior, where the minister may gain in information use in the high states on the expense of loss in the low states. Surprisingly, we …nd out that the less biased agent will be assigned to the junior position if the other’s bias is not too far from his. Corollary 2 The loyal o¢ cial 1 will be assigned to the junior position if p 1 . b2 < 1+ 1+16b 8 The intuition is as follows: While a loyal senior provides the minister with better control of the informed party indirectly, loyalty in the junior position reduces the loss of control directly. The indirect gains in control due to a loyal senior cannot compensate for the direct loss, provided that the other agent is not too biased. Thus, given a pool of bureaucrats, the senior will not necessarily be more loyal, even if the minister has full control over promotion decisions within hierarchies.

5.1

Recruitment policy

In the real world, the top-level decision maker sometimes has very limited authority in choosing his immediate subordinates. For example, in the UK government, the permanent undersecretary in each department is the nonpolitical civil service head of a government department. They report to and advise the Secretary of State, and are answerable to Parliament, so the minister cannot promote a loyal agent to this position. On the other hand, the middle-level agent sometimes may a¤ect the recruitment of the immediate subordinates (juniors). For instance, in those federal countries, local governments are created by state constitutions. Therefore, the state government, instead of the federal government, usually has the authority to dissolve the local government council or mayor without any reason. In this subsection we explore the selection of the junior given the preference of the senior and the minister. Speci…cally, given the preference of the senior o¢ cial and a pool of candidates for the junior position, which one should …ll this 12

position? It turns out that it will depend on the power of the superiors making the recruitment decisions. First, since bjb 0, if the subordinates have opposing biases, i.e., bsb < 0, both the minister and the senior want to restrict the discretion of the junior in the high states. Thus, the candidate with the smallest bias is always recruited as the junior regardless of who makes the hiring decision. If the senior is upwardly biased (bsb > 0), then if the minister hires a more biased junior (bjb > bsb ), she will lose in adaptation in the high states, and her payo¤ (the …rst part of (4)) would decrease with respect to the bias of the junior. Hence, she would like to hire a junior no more biased than the senior. When the junior is less biased than the senior, a junior closer to the minister can facilitate adaptation in the high states, while a junior closer to the senior would reduce the loss from extra restrictions in the low states. From the perspective of the minister, the optimal junior will balance these two forces. Take the …rst-order condition of the third part of (4) with respect to bjb , we see that the optimal junior is a compromised candidate, b0jb = 2b2sb . Similarly, for the senior, his expected payo¤ (the …rst part of (5)) strictly decreases when the preference misalignment of a more biased junior becomes even larger. Thus, he also seeks a compromised candidate. By di¤erentiating the third part of (5) with respect to bjb and equalizing it to zero, we have that if the senior makes the recruitment decision, his ideal candidate has bjb " = bsb (1 2bsb ). Compare b0jb with bjb ", we found that 1 , b0jb < bjb ", the when the senior bureaucrat is modestly biased, i.e., bsb 4 minister prefers to recruit a junior closer to herself in that the gains in better adaptation exceeds the loss from the extra restrictions. When the senior is very biased, i.e., bsb > 14 , we have b0jb > bjb ". For the minister the loss due to a strategic senior outweighs the gains from a loyal junior, thus, she will ‡atter the senior by recruiting a junior closer to him. If the superiors, i.e., the minister and the senior, have joint control over the recruitment decision, the recruitment of the junior will depend on the total expected payo¤ of the minister and the senior. To focus on the issue of interest, we assume that bjb ; bsb 2 0; 21 , i.e., the o¢ cials have like biases. We will look at the optimal bias of junior, bjb . Since both parties prefer hiring a compromised candidate, we could concentrate on the case that 0 < bjb < bsb We use < 1 to denote the weight of the minister in the total welfare, e.g., is a measure of the bargaining power of the minister in recruitment decisions. By summing the third parts of (4) and (5) together we have 8 W = [ b3sb + 4bjb b2sb 3

3

b2jb ] + (1

)[ 13

2

3

(1

2bsb )

(bjb + bsb )3 ] (6) 3

dW We take the …rst order condition of (5). By db = 0 we have bjb = jb 2 0 2(2 1)bsb +(1 )bsb = bjb +(1 ) bjb ". This is a compromise between the minister and the senior in the sense that their ideal biases enter the optimal choice in accordance with their bargaining power. When the superiors have equal bargaining power, i.e., = 12 , then the optimal junior will lie in the exact middle of his superiors, i.e, bjb = b2sb . The analysis above is summarized in the following proposition:

Proposition 4 The minister and the senior have di¤erent ideal candidates for the junior position. The bias of the junior who maximizes the weighted welfare will lie between the superiors. To recruit an ideal junior, the minister has to balance the e¢ ciency loss from the extra constraints in the low states imposed by the senior, and the gains from better adaptation in the high states. As a result of the joint recruitment decision of the superiors, these two parties will recruit a compromised junior.

5.2

Uncertainty about preferences misalignment

In the preceding analysis we assume that the biases of all players are common knowledge. In the real world, the minister of a department, as an elected politician, may not have enough time and/or resources to learn each o¢ cial’s personal bene…ts from implementing a project. Alternatively, the minister could now consider supervising a number of properly biased senior such that each senior supervises a su¢ ciently small number of juniors and could perfectly learn their biases. In this subsection we will show that hierarchy is desirable when the minister cannot observe the informed party’s bias, but can hire a senior whose type she can observe and who can in turn observe the informed party’s preferences. Formally, we assume that the minister perfectly observes bsb and has a prior over bjb , F (bjb ). To make the problem tractable and without loss of generality, we specify that the subordinates have like biases, i.e., bsb > 0 10 , and F is a uniform distribution over 0; 12 , i.e., the minister only knows the direction of the junior’s bias, but not the exact extent of it. On the other hand, the senior observes bjb perfectly. Since the junior is of upward bias, the minister will restrict Ysb by imposing a cap y, then the senior makes the 10

If the subordinates have opposing biases, i.e., the senior is downwardly biased, then he always wants to preclude more options from the junior’s set of decisions, so any binding cap could not strictly bene…t the minister, and she should delegate full control to the senior.

14

downward delegation decision. Using the logic leading to (3), the expected payo¤ to the minister will be in the following form:

EUDM (y) =

8 > > < > > :

3

(2bsb bjb ) b3jb b2jb (1 2bjb + 2bsb ) + ; if 12 > bjb > 1 + 2bsb y 3 b2jb y + 23 b3jb + 31 (1 y)3 , if 21 > 1 + 2bsb y > bjb bsb 3 (2bsb bjb ) (1 y)3 2 bjb (y 2bsb + bjb ) ; if bsb > bjb > 0 3 3 (7)

When the senior’s bias is between the minister’s and the junior’s, he prefers to expand the junior’s discretion. Thus, if his ideal ceiling is lower than y, in other words, the junior’s bias is far away from him, the senior can completely implement his preferred outcome by imposing the cap 1 + 2bsb bjb . The …rst row in (7) shows the welfare of the minister in this situation. Otherwise, if the senior’s preferred cap is higher than y, which means the junior is closer to him, the restriction imposed by the minister is binding and the implemented delegation set will end up with y; the expected payo¤ to the minister thus will be represented as the second row in (7). Moreover, if the junior is less biased than him, wanting to add an extra restriction for the low states, the third row shows the expected payo¤ to the minister in this situation. The minister gains from ceiling y if the junior is more biased than the senior, since it limits the senior’s tendency to delegate excess authority. However, when the junior is like-minded this binding cap precludes possible bene…cial adaptation in the high states, which may be relegated to the junior. The optimal delegation mechanism balances loss of control and gains in knowledge. The precise cap depends on which scenario above is more likely to occur. Furthermore, the like-minded senior provides the minister with better control of the junior. In the limit case where bsb = 0, i.e., the senior is perfectly aligned with the minister, he can be granted full control of actions. Formally, we calculate the expected payo¤ to the minister with uncertainty about the junior’s bias, and …nd out that it is optimal for her to relegate an unconstrained delegation set when the senior is not too biased.11 Proposition 5 If the minister has a uniform prior about the junior’s bias over the support 0; 21 , then: 1. When delegating to a senior, It is optimal for her to set a ceiling y such 11

Certainly the quantitative result is obtained by the speci…cation of the minister’s prior. However, the intuitions remain.

15

as y=

1,pif bsb < 14 3 , if bsb > 1 6

1 4

2. If this p minister can directly delegate to a junior, she will choose y = 3 . 1 6 3. When the senior’s bias is small, it’s optimal for the minister to delegate to him, instead of delegating to a junior directly. Part 1 says that the cap varies with respect to the bias of the senior. Actually, unconstrained delegation could be the equilibrium outcome provided that the preference divergence of the senior is not too large. Part 2 says that if the minister can directly delegate to the informed junior and is uncertain about his preferences, she will make a delegation decision based on her expectation. Part 3 compares direct delegation to an unknown junior with indirect delegation thorough a knowledgeable acquainted strategic senior, and demonstrates that delegating to this senior may bene…t the minister more. In our opinion, Part 1 provides a rational for the unconstrained delegation via intermediaries (Dessein 2002, Ambrus et al 2011): when the DM is uncertain about the preference misalignment of the informed party, and the intermediary is not too biased, it will be bene…cial to delegate all decision rights to this intermediary who has the relevant information. Again information gains ex post (more ‡exible implemented delegation set) outweighs loss of control ex ante (the senior has an unconstrained set of decisions). The complete delegation mechanism thus arises as the optimization outcome of the minister. Moreover, this provides an additional rationale for the emergence of hierarchical architecture. Part 3 argues that if the senior’s preferences are su¢ ciently closer to that of the minister, it is welfare-enhancing to delegate to a knowledgeable senior, instead of directly delegating to an unknown informed junior. Due to the resource constraint, the minister cannot discover all subordinates’preference misalignment, she will optimally forgo supervising the unacquainted informed party, and rely on an immediate subordinate to control the front-line agent instead. The asymmetric information between the senior and the minister leads to not only complete delegation, but also the emergence of hierarchy. It is worthy noting that the minister cannot bene…t from adding a pre-play communication stage in which the o¢ cials are requested to report the junior’s 12 bias bf jb . That’s because once the preference of the senior is known, the 12

The author thanks David Martimort for raising this point.

16

optimal delegation mechanism (Proposition 3) prescribes that the minister only imposes a ceiling on Ysb , which is determined by bjb . Therefore, if the positions are …lled, both the junior and the senior have a common interest in convincing the minister that there is no preference misalignment of the junior, i.e., bf jb = 0. Thus, both reports are uninformative. If the minister attempts to create a con‡ict of interest between the senior and the junior to induce truth-telling by promising to promote the junior if the reported bf jb < bsb , then this announcement would hardly be credible in that by Corollary 2, it is not always in the interest of the minister to assign a more loyal agent to the senior position.

6

Comparison with Hierarchical Communication

We have investigated the performance of hierarchical delegation and its implications for personnel issues. In reality, however, usually the selection of o¢ cials is not made by the direct superior. For example, in the U.S. system of separate powers, the executive (the President) appoints the administrative agency managers, e.g., FDA, while the Congress dictates policy and oversees its implementation. The administrative agency directly monitors the companies in its respective …eld, e.g., pharmaceutical companies. In terms of our model, the decision maker (Congress) can select neither the mediator (FDA) nor the informed agent (the pharmaceutical company). As an alternative, the Congress would respond by varying the level of oversights (Warren, 2008). In this section we ask the question: if the minister has no in‡uence on recruitment and promotion decisions, when will she choose delegation over communication? Formally, given the preference of o¢ cials, we compare hierarchical delegation with hierarchical communication. This involves the allocation of authority in hierarchies: should the senior be granted some decision-making responsibilities, or just act as an information gate-keeper? We discover that the inability to access the informed party may reverse the conclusion in previous literature that the DM is always better o¤ by delegating. Hierarchical communication (HC henceforth) shares the same vertical structure with hierarchical delegation, but di¤ers in that the minister concentrates all control in her hands and the information ‡ows bottom-up. the informed junior …rst delivers a message from the signal space S to the senior, and the latter in turn reports m from the message space M to the minister. The strategy for the junior is jb : ! S, and for the senior is 17

M . The minister forms a posterior about the true state after m : S ! receiving m, and orders the senior to undertake her ideal action y = E [ jM ]. In the end, the senior will pass this command to the junior. Under hierarchical communication, in equilibrium the senior can only convince the minister on which partition the state lies in, i.e., the minister takes a decision as the optimal response to a certain partitions structure. To get rid of the multiple equilibria problem common in cheap talk game,13 we would focus on the most informative equilibrium, i.e., the ex ante Pareto-dominance one. Ivanov (2010) shows that with a properly oppositely biased mediator, the DM could attain the optimal outcome of a universal communication mechanism a la Myerson (1981), in which a disinterested third party can make commitments. The junior’s incentives to overstate information would be o¤set by the senior’s incentives to understate it. Take the governmental decision-making example to illustrate the mechanism. If the senior prefers less expenditure on a speci…c project than the minister, while the junior wants to expand it, then upon hearing from the junior, it would be optimal for the senior to randomly distort the message downward. Anticipating this response of the senior, the junior realizes that there is no monotonic relationship between the minister’s decisions and his reports, her incentive to overstate thus reduces. Consequently, he would like to speak out more honestly14 . Lemma 3 For any bjb 2 [0; 12 ], there exists a senior with bias bsb 2 ( 2bjb ; 0] HC = and an equilibrium in the game with this senior that provides EUDM 1 b (1 bjb ), the upper bound of expected payo¤ from a universal mecha3 jb nism. However, if bsb 2 [0; bjb ), the indirect communication could not improve upon direct cheap talk. Proof. See Ivanov (2010) Theorem 1 and Lemma 4. This lemma demonstrates that e¢ ciency is improved only when two agents have opposing biases. In equilibrium the properly downwardly biased senior will randomize between the lowest two actions. The upwardly biased senior (0 < bsb < bjb ) could not improve e¢ ciency since mixing messages is not in his best interest. We could compare the maximum payo¤ in communication 31 bjb (1 bjb ) with the outcome of hierarchical delegation. Since

HD @EUDM @bsb

0 whenever bsb < 0, if there is a b0sb such that

13

Babbling equilibrium (Uninformative communication equilibrium) always exists in the cheap talk game, in which the DM would choose his ex ante optimal action E [ ] = 12 14 This intuition is also explored by Blume et al (2007), in which noise in communication invalids the monotonicity condition of action with respect to the messages and increases DM’s welfare.

18

HD EUDM (b0sb ) = 13 bjb (1 bjb ), we could say that for any bsb > b0sb , hierarchical delegation strictly dominates hierarchical communication in the sense it can attain higher payo¤ than the upper bound of the expected payo¤ under any communication mechanism. As a result, b0sb turns out to be a nonlinear function of bjb , which is formally de…ned by the following equation:

b2jb

(1 +

2b0sb )

+

bjb )3

(2b0sb 3

5b3jb + = 3

1 bjb (1 3

bjb )

(8)

The higher dashed curve OB in Figure 4 describes this indi¤erence line. For any bsb above this curve, delegation strictly dominates the communication mechanism. Unfortunately, there is a non-monotonic relationship between the existence of the k-action mixed strategy equilibrium and the senior’s bias. AAK note that although two-action mixed-strategy equilibrium requires that bsb is "su¢ ciently" far below zero, three-action mixed equilibrium can exist when both bjb and bsb are close to 0. This non-monotonicity in hierarchical communication substantively limits complete characterization of the relative ef…ciency of these two modes. However, we still can compare the lower bound of the expected payo¤ from informative communication with hierarchical delegation. We will look at the range of values of bsb for the existence of the two-interval equilibrium in HC mode and valuable HD mode. We relegated in Appendix B the construction of the two-action mixed strategy equilibrium under HC. At equilibrium, when the junior reports the true state in the 2nd lowest interval, the senior distorts information by recommending, with some probability, that it is in the lowest one. Consequently, the minister shifts the lowest action upwardly, which makes it more attractive for a larger range of the junior’s low bias. For the lower bound of HD, note that any valuable HD requires the following inequality being satis…ed: b2jb (1 + 2bsb ) +

bjb )3

(2bsb 3

+

5b3jb 3

1 12

(9)

Figure 4 illustrates this comparison. The vertical axis (horizontal axis, resp) is the bias of the senior (junior, resp). The region ABCE represents the area where two-partition equilibrium in HC exists.15 In other words, in this region communication can be informative. If the bias of the senior lies in the area below the line BF, the payo¤ from hierarchical delegation is 15

It’s de…ned by 0

bsb

1 4

and bjb

1 2

19

bsb

bjb 3

1 6:

1 lower than 12 . Thus, for the minister, when the subordinates’ biases are not too far apart, in the shadow triangle BCD, communication dominates delegation. For a slightly biased senior (bsb higher than OB), delegation 1 but strictly dominates communication. For a very biased senior (bsb 4 1 , i.e., all value below the line EC) and a less biased junior, delegation 2 weakly dominates.

[Insert Figure 4] Therefore, if two o¢ cials have like biases, then the minister should delegate the senior with limited discretionary power. On the other hand, if two bureaucrats have opposing biases, the minister may keep control rights and restrict the role of the senior to information processing. An important conclusion in Dessein (2002) is that whenever there is informative communication in CS speci…cation, delegation dominates communication. Moreover, as GHPS has shown, under direct interaction constrained delegation can implement the second-best outcome. As a consequence, by constrained delegation the minister could do no worse than cheap talk. Hence it seems that the minister should always engage in delegation instead of communication. However, we show that this result would not hold within threelayer hierarchies, i.e., when the minister cannot access the informed party directly. Because the minister cannot directly access the junior, i.e., she cannot observe the message s, nor decide Yjb , she has to take the incentives of the senior into account when making decisions based on m. She is reluctant to delegate control rights or rubberstamp the proposal from the subordinates not due to commitment cost or communication cost, as the literature suggests. Instead, it is her inability to interact with the informed party directly. As a result, the possible welfare-enhancing solution in organizational design is not o¤ering commitment device or adding interaction stage. Network architecture, in particular the depth of hierarchies, matters in providing proper incentives for an informed agent. This result also suggests that job design is related to hiring policy since the loyalty of the senior determines the content of this job. The elected politician may retain control over decisions if she cannot replace the bureaucrats. Thus, even though a hierarchy as a structural feature is well de…ned, the job content of each position still depends on the authority of the minister over personnel management.

20

7

Concluding Remarks

Within a bureaucratic system, the span of discretionary authority decreases top-down, and the bottom-level agents are usually empowered with substantial discretions from the superiors. For instance, in even the most centralized countries, the local government always has some autonomy over the money it receives. To understand these phenomena, this paper studies optimal interval delegation in multi-tier hierarchies. We contribute to the literature by providing a complete characterization of the interval delegation set, and establishing conditions for attaining e¢ ciency in hierarchies. Moreover, we show that in contrast with the previous results, when the DM can restrict the menu of delegated options, direct delegation to an informed party weakly dominates delegation via an intermediary, and in hierarchies informative communication may dominate delegation. Furthermore, we also demonstrate that hierarchies and complete delegation may arise in response to asymmetric information about the types of the subordinates. This framework shows that the human resources management practices within a bureaucratic system, i.e., job design, promotion and recruitment policy, are interrelated. Some predictions of this paper are consistent with our daily experiences, and receive empirical supports. Iyer and Mani (2009) examine the Chief Minister/District Politician/Bureaucrat hierarchy in India, and …nd out that when the Chief Minister and the District Politician are elected from di¤erent parties, the Chief Minister is more likely to change the bureaucrat to seek for compromise. Our paper also shed lights on vertical decentralization within a multi-layer government. One implication is that the empowerment of local governments (job content in terms of our work) depends on the way they are chosen in the positions (recruitment policy). Therefore, a unitary state might delegate substantial real power (…scal resources, administrative discretions, etc.) to the local level since the central government has more controls over the selection of bottom-level bureaucrats. A new dataset prepared by Ivanyna and Shah16 looks at multi-tier government and highlights decentralized local governance, i.e., localization. They show that some federal countries, like India, has a very low localization level since responsibilities are simply shifted to the intermediate tier, instead of strengthening local decision making, and the highest localization occurs in some unitary countries, e.g., Nordic countries. A more systemic empirical study awaits for future research. In this paper we stress interval delegation, which is in widespread use, e.g., quantity control, budgeting, etc. However, for the oppositely biased 16

See Ivanyna and Shah (2011) for some brief information about this dataset.

21

or more biased senior, interval delegation may not be the generic optimal delegation mechanism. Actually optimal delegation in this situation may require removing some intermediate decisions from the delegation set to the senior. The investigation of the optimal delegation mechanism in multi-tier hierarchies is left for future research.

A

Appendix A

Proof of Proposition 1 Since Yjb Ysb , the highest action available to the junior yjb cannot exceed supfy jy 2 Ysb g, the highest action in Ysb . On the other hand, by Lemma 2, if the senior has an unconstrained delegation set from the minister, then the size of downstream delegation set relies only on the relative di¤erence of bias . Thus the implemented ceiling also could not be higher than 1 . Proof of Corollary 1 We look at the interval delegation, thus, the minister can focus only on the ceiling of Ysb . Since < bjb , 1 > supfy jy 2 Ysb g = 1 bjb , by truncating Ysb with the highest decision of Y , 1 bjb , the junior would be limited to take any action lower than 1 bjb , which is exactly what the e¢ cient delegation set prescribes. Proof of Proposition 2 1. We …rst study the case of a downwardly biased senior and show that the e¢ cient delegation set is not implementable. Since bjb > 0 > bsb , > bjb > 0. Suppose in contrast that the e¢ cient delegation set Y = [0; 1 bjb ] is implementable, then we should have Y Ysb , i.e., there is an interval larger than [0; 1 bjb ], which belongs to the senior’s delegation set. Because the relative di¤erence of interest is and 1 + bjb < 1 bjb = supfy jy 2 Y g, by Proposition 1 the senior would like to only delegate [0; 1 + bsb ] to the junior, provided that his set of delegated decisions is Ysb . Hence we get the contradiction needed. 2. Now we check the case of a more upwardly biased senior. Hence < 0, and according to Part 2 in Lemma 2, the senior would impose a bottom line on Yjb , while Y requires that the lower bound is 0. Since the senior’ prefers to impose a lower bound, the minister could not implement Y that allows the junior to take actions in the low states. Proof of Corollary 2

22

If the more biased agent 2 is placed in the junior position, then as (4) shows, the expected payo¤ to the minister is 34 b2 1 b22 , while if the loyal agent 1 takes the junior position, the expected payo¤ becomes 83 b32 +4b1 b22 b21 . By comparing these two formula it is clearly bene…cial to assign the more biased agent 2 to the front line only if 4bp32 (1 + 4b1 ) b22 + b21 > 0. The 1 remaining reasonable root leads to b2 > 1+ 1+16b . 8 Proof of Proposition 5 By (7), we divide the discussion into two parts: a binding ceiling level y or not. 1. First, we study the case that the upper bound is always binding, namely y < 1 bjb + 2bsb whenever bsb < bjb < 12 . This condition translates into y < 12 + 2bsb . Thus, we would use only the second and third row of (7) Therefore, the ex ante expected payo¤ to the minister is Z

bsb

0

+

Z

1 2

bsb

"

"

1)3

(y 3

1)3

(y 3

b2jb (y

2bsb + bjb )

bjb )3

(2bsb 3

# 2 b2jb y + b3jb dF (bjb ) 3

#

dF (bjb(10) ) (11)

The …rst term is the expected payo¤ if the junior is less biased than the senior, where the senior will set a lower bound on Yjb . The second term represents the expected payo¤ whenever the junior is more biased than the senior, in which the senior further delegates to the junior up till y, the cap of his own authority. Take the …rst-order condition with respect to y, we have y = 1 hR 1 i 12 2 2 . Using the assumption that F is a uniform distribub dF (b ) jb 0 jb tion over 0; 21 we have y = 1

EU (y) =

p

3 6

and p

1 3 + 16 108

2b4sb

Check it with the initial condition y < 12 +bsb , we need bsb 2

h

1 4

(12) i p 3 1 ; . 12 2

2. Second, we study the case that the cap sometimes may not hold. In other words, y > 21 + bsb , thus the senior may also imposes his preferred ceiling. Now all the three rows in (7) will enter the expected payo¤ to

23

the minister, which can be expressed as # Z bsb " 3 (y 1)3 (2b b ) sb jb b2jb (y 2bsb + bjb ) dF (bjb ) + 3 3 0 # Z 1 y+2bsb " (y 1)3 2 3 2 bjb y + bjb dF (bjb ) + 3 3 bsb # " Z 1 2 (2bsb bjb )3 + b3jb 2 bjb (1 + 2bsb 2bjb ) dF (bjb ) 3 1+2bsb y The …rst term is the same as that in (10). The second term establishes that when the junior’s preference is not far away from the senior’s, the cap y binds on Yjb since the senior always prefers to expand the junior’s discretion. The third term shows that when the junior is farther from the senior, the senior will impose a cap from his own interest. The …rst order condition suggests that y = 1 + 2bsb or 1 + bsb , both are impossible since the highest action in Y is 1. However, since the second order condition is always negative, we know that the minister gains by setting a higher y. Hence the minister will choose the highest available ceiling y = 1. In other words, the minister will completely delegate to the senior. The expected payo¤ is 1 3b4sb 24 We return to check the initial condition and get bsb < 14 . EU (y) =

(13)

3. hFinally, since i the results in steps 1 and 2 have overlap when bsb 2 p 3 1 1 ; , we compare the expected payo¤ to the minister under 4 12 4 these two y and …nd out that y = 1 generates higher payo¤. Thus, we …nish the …rst part of this proposition. Now we turn to the second part. Suppose the minister can directly delegate to a junior, but she has only uniform prior over the junior’s bias. First, for a given upper bound y, the expected payo¤ to the minister is EUDM (y) =

Ry 0

bjb 2 bjb d

R1

y bjb

(y

)2 d =

b2jb y + 23 b3jb +

(y 1)3 3

R1 Then she will choose y to maximize 02 EUDM (y) dF (bjb ), the …rst order hR 1 i 12 p 3 2 2 = 1 condition turns out to be y = 1 b dF (b ) . Thus, the jb 6 0 jb 24

expected payo¤ under the optimal direct delegation mechanism is p 1 3 + EU (y) = 16 108

(14)

It is easy to see that (12) is always smaller than (14), thus we concentrate p

1 4

3 1 on comparing (13) and (14). Then, we have whenever bsb 144 324 0:1999, (13) is larger than (14). In other words, delegating to a knowledgeable acquainted senior dominates directly delegating to an unknown junior.

B

Appendix B

We follow AAK closely to construct the two-action mixed strategy equilibrium in hierarchical communication. We focus on the outcome-equivalent perfect Bayesian-Nash equilibrium that the upwardly biased junior reports s1 at any 2 [0; x] and s2 at any 2 (x; 1], the downwardly biased senior (bsb < 0) recommends action m1 to the minister after receiving message s1 , and mixes between m1 (with probability p) and m2 after getting message s2 . Finally the minister acts according to the report sent by the senior, y1 = m1 and y2 = m2 . Since the minister understands that x when she hears m2 , we need . Because the senior must be indi¤erent between sending m1 and m2 = 1+x 2 1+x m2 when he receives s1 , we have m1 = 2 + 2bsb . Similarly, the junior has to be indi¤erent between both actions at = x, so we get x = 1 2 . Thus, we substitute it into m1 and m2 , and solve m2 = 1 and m1 = 1 + 2bsb . By Bayesian Law, the minister must update her posterior in such a way that E ( jm1 ) = m1 , so we have x 2

Pr ( 2 [0; x] jm1 ) +

1+x 2

x x+(1 x)p 1 (1 2 )(1+4bsb ) : 8 bsb

Pr ( 2 [x; 1] jm2 ) = m1

where Pr ( 2 [0; x] jm1 ) =

Thus, we get p = By the feasibility requirement that p 2 [0; 1], we get 0 1 and (1 + 4bsb ). 2

1 ;b 2 sb

1 , 4

References [1] Alonso, Ricardo and Niko Matouschek (2007) Relational Delegation, RAND Journal of Economics, 38, 1070-1089. 25

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28

Figure 1. Timeline under hierarchical delegation

Junior

Senior

bureaucrat

bureaucrat

Y jb

minister

Ysb

Figure 2. Property of the optimal delegation set from the minister and the senior standpoint, respectively. The bold line depicts the action chosen as a function of state θ.

y

1

1− ∆ 1 − b jb

b jb bsb 0

θ 1 − 2b jb 1 − 2∆

1

Figure 3(a). The delegation set with an oppositely biased senior

y

Y* 1 − b jb 1 + 2bsb − b jb

b jb 0

bsb

θ

1

Y jb

Figure 3(b). The delegation set with a more upwardly biased senior

y

Y jb 1 − b jb 2bsb − b jb bsb

Y*

b jb 0

θ 1

Figure 4. The comparison of expected payoff under HD and HC modes.