Rev. Econ. Design (2017) 21:273–290 DOI 10.1007/s10058-017-0206-8 ORIGINAL PAPER

Transfer of authority within hierarchies Pinghan Liang1

Received: 4 February 2014 / Accepted: 11 September 2017 / Published online: 16 September 2017 © Springer-Verlag GmbH Germany 2017

Abstract This paper studies delegation and communication in a model of three-tier hierarchy. There is an uninformed principal, and uninformed intermediary, and an informed agent. Under delegation the principal chooses an interval of actions to delegate to the intermediary, and the intermediary chooses a sub-interval from that interval to delegate to the agent. Under communication, the agent communicates with the intermediary, after which the intermediary communicates with the principal. We characterize the equilibrium outcomes under delegation and communication. We show that under delegation the principal can appoint a more biased individual to be the intermediary, and a less biased individual to be the agent. Furthermore, we demonstrate that the principal can prefer to communicate with the subordinates rather than delegate decision rights to them if the intermediary and the agent have opposing biased. Keywords Delegation · Hierarchies · Communication JEL Classification D72 · D78 · D82 The principles of office hierarchy and of levels of graded authority mean a firmly ordered system of super- and subordination in which there is a supervision of the lower offices by the higher ones. ——— Max Weber (1946, p. 214)

B 1

Pinghan Liang [email protected] Center for Chinese Public Administration Research/School of Government, Sun Yat-sen University, Guangzhou, China

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1 Introduction Modern bureaucratic systems are usually organized as multi-tier hierarchies. The superior not only enjoys decision-making right, but also may delegate this right to his subordinates, who, in turn, do the same for their subordinates, and so forth. It is quite common in organizations to impose caps on the amount of payments that a manager or an agent can use. Think about the budget allocation procedure in large firms: the front-line manager knows the exact amount of resources required for the success of a project, but he might exaggerate the demand. On the other hand, his superior (middlelevel manager) might over-estimate or under-estimate the real demand. Knowing these preferences misalignments, the top-level management needs to restrict the amount of resources available to the middle-level, e.g., set up a budget cap, or a bottom, or both. This paper investigates how hierarchies affect the optimal restrictions on the discretionary use of resources, and its implication for the selection of subordinates. We examine a three-layer chain with one principal (“she”), one intermediary, and one agent (“he” for both). All players want to match the action with the true state, though they also have their private benefits (bias), e.g., intrinsic preferences, compensation packages, or personal career concerns, etc. Only the agent would be informed about the realization of the true state. The principal can not directly contract with the agent. We concentrate on hierarchical delegation: the principal first chooses an interval of actions to delegate to the intermediary,1 then the intermediary chooses a sub-interval from that interval to delegate to the agent. In other words, the principal has limited commitment power in the sense that she can delegate only to someone one level lower.2 We characterize the equilibrium outcomes under delegation, for any possible pair of subordinates’ biases. We show that the intermediary’s bias would not affect the principal’s payoff if it lies between the principal’s and the agent’s. Otherwise, the intermediary would effectively impose extra constraints, either a tighter cap or a bottom, on the agent’s action set. This leads to efficiency distortions in sub-delegation. We then use this to study the optimal selection of intermediary and agent. A subordinate whose interest is closer to the principal is assigned to the lower position, provided that the difference of subordinates’ preferences is not large. This is because when the subordinates are not sufficiently different, the benefits in flexible decisionmaking from a closer agent dominates the distortion in sub-delegation from a farther intermediary. Further, since an agent closer to the principal improves the decisionmaking efficiency, but an agent closer to the intermediary reduces the distortion in sub-delegation, when the principal and the intermediary jointly select the agent, the optimal agent would be an individual whose bias depends on the weight of principal’s and intermediary’s payoffs in the organizational goal. Finally, we compare two 1 Since we concentrate on the discretionary use of resources, e.g., budgeting, it is natural to only focus on an interval of actions. Alonso and Matouschek (2007) also use the term “threshold delegation” to refer to this decision rule. 2 The situation that enforceable contracts can only be written between certain parties also motivates the study of Kolotilin et al. (2013) on optimal limited authority, and Liang (2013) on optimal sequential delegation.

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organizational modes: hierarchical delegation and hierarchical communication,3 i.e., ordering the agent what to do after hearing from him via the intermediary. The previous result that delegation always dominates communication whenever an informative communicative equilibrium exists (Dessein 2002) reverses. The intuition is that delegating decisions to an oppositely biased intermediary yields few benefits, as the distortions in sub-delegation monotonically increases with the preferences misalignment. However, there exists a hierarchical communication equilibrium, in which an intermediary biased against the agent can punish the agent’s mis-report by randomly sending messages to the principal. This relaxes the incentive constraints of the agent to reveal information, hence improving communication efficiency (Ivanov 2010). The remaining parts of this paper are organized as follows. In the following section we review the related literature. In Sect. 3 we lay out the basic model. Section 4 characterizes the equilibrium hierarchical delegation outcomes. Section 5 studies the optimal selection of subordination. Section 6 compares hierarchical delegation with hierarchical communication. Section 7 concludes. All proofs are relegated to the “Appendix”.

2 Related literature We investigate the issue of organizational design under conditions of no transfer and unverifiable information. In this environment, Crawford and Sobel (1982, henceforth “CS”) demonstrate that full information revelation is impossible so long as there is preference misalignment between the principal and the informed agent. The following research looks for welfare-improving mechanisms. Dessein (2002) notes that as the preference divergence becomes smaller, the efficiency of unconstrained delegation increases more than that of communication. Hence, he shows that in the CS classical specification, whenever informative communication is possible, the principal is always better off by delegating all decision rights to the informed agent. Further, he shows that it can be a welfare-improvement if the principal delegates to a properly biased, uninformed intermediary, and then this intermediary communicates with the informed agent and makes decisions, because an intermediary closer to the agent can improve communication efficiency. Since he employs the incomplete contract approach, either the intermediary’s sub-delegate behavior or the principal’s discretion in restricting the delegation set is precluded. In this paper we allow the sequential transfer of decision rights, and focus on constrained delegation. Hence, Dessein’s results on the benefits of delegation to an intermediary, as well as the dominance of delegation, reverse. Our paper explore the sequential transfer of delegation set, hence, players other than the principal restrict the delegation set and choose payoff-relevant action. Two recent papers (Krahmer and Kovac 2016; Tanner 2014) explore sequential delegation in principal–agent context, the principal first chooses a menu of delegation set, and the agent chooses a set from the menu, then select an action to implement. This sequential arrangement is meaningful since the agent may hold private information about something more than the realization of true state, in pre-action stage, e.g., distribution of the state (Krahmer and Kovac 2016), agent’s preferences (Tanner 2014). In our paper the 3 In Goltsman et al. (2009) it is referred to as “mediator cheap talk”.

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rationale for sequential arrangement is not the private information, but the principal’s limited commitment. This paper is closely related to Ambrus et al. (2013a, b) and Liang (2013). In the environment of legislative decision-making, Ambrus et al. (2013a, b) consider how to optimally appoint an intermediary to collect information from a biased agent. They show that when the bias of the informed party is not large, it is better to employ the closed rule, which resembles the unconstrained delegation to an intermediary. They also derive the optimal bias of the intermediary. If the bias of the agent is large, their comparison between unconstrained delegation and hierarchical communication also favors the latter: it is better to hire an oppositely biased intermediary and adopt the open rule, which is equivalent to hierarchical communication. Our intuition is similar, but we focus on constrained delegation: as long as the preferences of an intermediary lie between those of the principal and those of the agent, the intermediary would not affect the principal’s welfare. Moreover, our comparison between hierarchical delegation and hierarchical communication would also be stronger, since our paper allows the principal to restrict the intermediary’s set of actions, instead of an “All or Not” delegation decision. Liang (2013) investigates the optimal design of delegation via a strategic intermediary, and shows that the optimal delegation set contains a “hole” when the intermediary and the agent are of opposing biases. The main difference is the focuses. This paper focuses on the implications of delegating an interval of actions for the selection of subordinates, and consider the comparison between delegation and communication mechanism. Some papers explore the strategic behavior in fixed hierarchies. Tirole (1986) explicitly introduces a supervisor into the principal–agent relationship to study multiple-layer hierarchies. In his paper, since the supervisor holds private information about the type of agent, it can be optimal to give ownership to a supervisor, who then subcontracts with a downstream agent. Mookherjee and Reichelstein (1997) study the optimal budget mechanism in three-tier hierarchies. The key difference in our paper is that physical transfer is not feasible. Thus, the basic tradeoff, as well as the results regarding the efficiency of subdelegation and the composition of subordinates, also differ. Our paper also compares hierarchical delegation with hierarchical communication. The construction of hierarchical communication outcome is established by previous research, which highlights the communication efficiency improvement from adding an intermediary between the principal and the agent. Based on the CS specification, Goltsman et al. (2009, henceforth “GHPS”) derive the maximal ex-ante payoff a principal can attain under the universal mechanism, subject to the agent’s incentive compatibility constraint to truthfully report, as well as the principal’s incentive compatibility constraint to follow the agent’s report. They establish that a principal can attain the optimal outcome by hiring a neutral intermediary who optimally filters information from the informed agent. Further, Ivanov (2010) and Ambrus et al. (2013a, b henceforth “AAK”) demonstrate that this outcome can be implemented when an intermediary is biased against the informed agent, since an oppositely biased intermediary can balance the agent’s incentive to exaggerate and extract more truthful reports. This naturally leads to the question about comparing the benefits of communication via a

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Fig. 1 Timeline under hierarchical delegation

strategic intermediary with the performance of delegation via a biased intermediary. Our paper provides some answers to this question.

3 Model There are three players: a principal (denoted as player “P”), an intermediary, and an agent (denoted as players “M” and “A”, respectively). The payoff of all players takes the quadratic loss function as the leading example in CS:4 U (θ, y, bi ) = − (θ − y + bi )2 , i = P, M, A

(1)

Their payoffs depend on the state θ ∈  = [0, 1], (e.g., the demands for a project), the action undertaken y ∈ Y = [0, 1], (e.g., the size of the project), and their private benefits bi . Each player wants to minimize the distance between the action undertaken and his ideal state-dependent ideal action, which is θ + bi for player i. Without loss of generality, we normalize the personal benefit of the principal to zero and use b M , b A to measure the preference misalignment between the subordinates and the principal. For the sake of simplicity, we use “Ui (θ, y)” to refer to the utility of player i. The principal has the right to take a decision. Only the agent will be informed about θ . He is able to implement the decision, but he can not communicate directly with the principal. The principal also cannot assign decision rights directly to the agent. In other words, the intermediary has full control over the information transmission and delegation between the principal and the agent. The intermediary and the principal have uniform prior beliefs on . Figure 1 describes the timing of hierarchical delegation (“HD” henceforth) game. The strategy of P is specifying a closed interval of actions Y M ⊂ Y that M will be restricted to take. M’s strategy is to choose a closed interval of options Y A ⊂ Y M and allow A to choose any actions within it. Finally, A becomes informed about θ , and chooses an action y ∈ Y A . Since sequential delegation is made before the agent learns the state, the principal and the intermediary make delegation decisions under their expectations of the agent’s actions. We will employ the Perfect Bayesian Equilibrium concept to investigate players’ behavior. We will also only consider the pure strategies.5 4 This specification is standard and widely used in the literatures on communication and organizations, e.g.,

Krishna and Morgan (2001), Goltsman et al. (2009), Blume et al. (2007), Kovac and Mylovanov (2009), and Ivanov (2010). 5 As GHPS show, the optimal mechanism in this typical environment is a deterministic mechanism. Hence,

we could restrict our attention to pure strategies in this delegation game.

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It is noteworthy that there are many outcome-equivalent PBE, e.g., the mapping from the set of states to the set of actions by the agent are the same. First, from the perspective of A, there are infinitely many redundant actions which would not be chosen by the agent in any state, or be precluded by the intermediary in sub-delegation. For instance, the option y  = b A strictly dominates any lower options for any θ . Thus, if y  ∈ Y A , then including or excluding any subset of [0, b A ) from Y A would not affect any players’ equilibrium payoffs, since they would not be chosen by the agent for any realization of state. Second, there are also infinitely many actions, even if they are added to Y M , that would not be chosen by the intermediary when choosing Y A . Therefore, we will concentrate on the minimal outcome-equivalent delegation set where any action in it is strictly chosen in some state, and unchosen actions are removed. Finally, we use the outcome of principal–agent direct delegation as the efficiency benchmark. As Holmstrom (1984) establishes, if A is allowed to execute actions, decision-making becomes more flexible, and P’s welfare is improved. However, since A’s ideal action differs from P’s, P also suffers from the loss of control. The optimal delegation set balances the gains from flexible decision-making and the loss of control. It for P to delegate A with all options within [0, 1 − b A ] whenever b A ∈   is1optimal 0, 2 , and to delegate a single action { 21 } when b A > 21 . GHPS further show that this delegation scheme attains the best outcome of a universal mechanism a la Myerson (1981). Since any actions lower than y A = b A are dominated for A, we refer the set Y A∗ = [b A , 1 − b A ] as the efficient delegation set from P’s perspective. We establish the following lemma useful for further analysis. Lemma 1 In any Perfect Bayesian Equilibrium, the minimal outcome-equivalent delegation set specifies that the principal would not include any actions y > 1 − b A and y < b A in the delegation set.

4 Hierarchical delegation We would focus on the case that |b M | ≤ 21 and 0 ≤ b A ≤ 21 , namely the agent is upwardly biased. We would use  ≡ b A − b M to represent the divergence of preferences between M and A. To keep sub-delegation between M and A meaningful, we assume || ≤ 21 . Under hierarchical delegation, once the decision-making right over Y M is transferred to M, he acts as if he is a principal within an interval Y M , and he will make delegation decision based on . Therefore, in HD P can affect A’s behavior only through restricting Y M . The principal suffers from double loss of control: in addition to a biased agent who distorts the decision towards his ideal, in the sub-delegation stage a biased intermediary shapes the delegation set towards his preferred one. The optimal delegation set balances these two distortions against the gain in flexible decision-making. The main insights are: depending on the divergence of preferences between M and A, M might prefer relaxing or tightening the constraints imposed by P. P can forbid M’s relaxing behavior by restricting M’s set of options. Hence, the distortion in sub-delegation is eliminated when M’s bias lies between P ’s and A’s. However, if M wants to impose extra constraints on A, increasing M’s set of options won’t help, and P cannot reduce sub-delegation distortion.

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We will solve this game by backward induction. The first task for us is to characterize some properties of the sub-delegation from M to A. The lemma 2 in Liang (2013) states that if Y M is a connected set, then on any interval of Y , the optimal delegation set Y A assigns a connected set of decisions on it, e.g., an interval, or one decision, or an  set. In other words, M has no incentives to remove an interval of actions  empty ym, yn from Y A , in which ym , yn ∈ Y A . Since we address the interval delegation, e.g., Y M is a closed interval of options, we could only use the upper bound and lower bound of Y A to characterize the optimal set. Lemma 2 establishes that conditional on whether A is upwardly or downwardly biased relative to M, M will optimally impose a cap/bottom on A’s delegation set.   Lemma 2 1. If  ≥ 0, in any PBE we have Y A = y A , y A , in which y A = min{1 −  + b M , max{y |y ∈ Y M }} and y A = max{b A , min{y |y ∈ Y M }}.   2. If  < 0,in any PBE we have Y A = y A , y A , in which y A = min{1, max{y |y ∈ Y M }}, y A = max{b M − , min{y |y ∈ Y M }} If A is upwardly biased relative to M, A is always inclined to take an action higher than M’s ideal point. On the one hand, for the low state, the gain from flexible decisionmaking outweighs the distortion due to preferences misalignment, and M will allow A to take his ideal actions. On the other hand, this trade-off reverses in the high state, hence M imposes a tight ceiling. If A is downwardly biased, since he always tends to take a lower action than M’s ideal whenever  < 0, the distortion in decision-making is the most salient in the low state. Hence, M would like to restrict A’s discretion when θ is small, and give full discretion when θ is high. Then, if M is granted an unconstrained set of decisions, he will choose his ideal delegation set. If  < b A , relative to P, M always wants to expand A’s discretion. For example, if b M > 0,  ≥ 0 , and M is granted with Y M = Y , then by Lemma 2 the delegation set ([b A , 1 −  + b M ]) is strictly larger than the efficient one Y A∗ = [b A , 1 − b A ]. However, it also suggests that P could control this distortion by truncating M’s delegation set, i.e., imposing the cap 1 − b A on Y M . The efficient delegation set Y A∗ thus could be replicated in a three-tier hierarchy. However, when b M < 0 and  ≥ 0, 1 −  + b M < 1 − b A always holds; and when  < 0, b M −  > b A always holds, Y A∗ is no longer replicable in these situations. Hence, the bias of M entails additional loss on P. Proposition 1 is our main result, in which we summarize the characterization of the optimal interval delegation set. We should keep in mind that if P orders the ex ante 1 , hence this restricts the scope of optimal action 21 , the expected payoff would be − 12 M’s and A’s biases that makes delegation valuable. The detailed description of the design of optimal delegation mechanism, as well as the proof, is left to the appendix.

Proposition 1 In the hierarchical delegation, in any PBE, the optimal (minimal outcome-equivalent) delegation set prescribes the set of decisions available to the informed agent as:

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⎧ [b A , 1 − b A ] , ⎪ ⎪ ⎨ [b , 1 − b + 2b ] , A A M YA = ⎪ − b , 1 − b [2b A A] , ⎪ ⎩ 1 M { 2 },

if 21 ≥ b A ≥ b M ≥ 0 if b A ≥ 0 > b M ≥ b M if b M ≥ b M > b A ≥ 0 otherwise

(2)

In which b M is defined by the non-positive real root of the following equation: 8 4 1 − b2A + b3M − 4b2M b A + b3A = − 3 3 12

(3)

and b M is defined by the non-negative real root of the following equation: 8 1 − b3M + 4b A b2M − b2A = − 3 12

(4)

As the first line of (2) shows, M’s bias doesn’t affect delegation outcome so long as it lies between P and A. In effect, as Liang (2013) suggests, this result could be extended to a chain with arbitrary depth. If all M and A are like-biased, and they are ranked according to their biases, e.g., the agent is more biased than the lowest intermediary, and the lowest intermediary is more biased than the second lowest one, etc., then the principal could impose a tight ceiling, which is determined by the agent’s bias, on the delegation set to the highest intermediary. By Lemma 2, no intermediary has any interest to impose extra constraints, and the efficient delegation outcome is still attained. It is straightforward to derive P’s expected payoff with respect to the biases of M and A, and we conduct some comparative statics in the following remark.     ∂ EU H D Remark 1 If b M ∈ b M , 0 , then we have ∂b MP ≥ 0. If b M ∈ b A , b M , then ∂ EU PH D ∂b M

≤ 0.

Thus, P’s expected payoff decreases with respect to the absolute value of b M . The intuition is: given A’s preference, an overly biased intermediary will impose a extra bottom on Y A , which becomes higher when the bias of M increases. Meanwhile, an oppositely biased intermediary will impose an extra cap lower than P’s ideal on Y A , and as M’s bias becomes more negative, this cap is even lower. Both entails distortions in sub-delegation. Hence, an intermediary closer to P would reduce the distortion in sub-delegation either through lowering the extra bottom, or through making the extra cap closer to P’s ideal ceiling. M’s expected payoff also varies with respect to A’s bias. However, it turns out that there might exist a non-monotonic relationship between them. Remark 2 characterizes this. ∂ EU H D

Remark 2 1. If b M ∈ [0, b A ], then ∂b AM ≤ 0.   ∂ EU H D 2. If b M ∈ b M , 0 , then ∂b AM ≤ 0.   ∂ EU H D 3. If b M ∈ b A , b M , then ∂b AM ≤ 0 if b A ≥ b M (1 − 2b M ); and b A ≤ b M (1 − 2b M ).

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HD ∂ EU M ∂b A

≥ 0 if

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The Part 1 and 2 of Remark 2 show that when M is moderately biased or oppositely biased, since the preferences misalignment between M and A and that between P and A is in the same direction, a more biased agent deteriorates the expected payoff to M and P simultaneously. However, Part 3 suggests that an overly biased intermediary can gain from a more biased agent if A is not very biased. The intuition is that: as the agent becomes increasingly biased, though M loses since P will impose a lower ceiling on Y M , the preference divergence between M and A also becomes smaller. Hence, M gains by setting a lower bottom y A . When A is not very biased (b A ≤ b M (1 − 2b M )), the latter effect dominates the former one, and M gains from a more biased agent.

5 Selections of subordinates in hierarchies In this section we use the characterization about hierarchical delegation to study the optimal selection of subordinates, i.e., how to fill the positions in hierarchies. The first question we address is: under the hierarchical delegation game, assume that there are two individuals who have like biases, i.e., b2 ≥ b1 ≥ 0, who should be assigned to the intermediary position, and who should be assiened to the agent position? Therefore, the game now runs like this: First, the principal fills the positions in the hierarchy with individual 1 and 2; then the principal delegates Y M to the appointed intermediary, who further delegates Y A to the appointed agent. Finally, the agent learns the realization of true state and chooses an action y ∈ Y A . Our intuition is: if individual 2 is appointed to the lower position, P suffers from the distorted decision-making in high state, where the conflict of interest between P and A is the most severe. But if individual 1 is appointed to the lower position, then the more biased individual 2 occupies the higher position. On the one hand, P gains in flexible decision-making in high state from a loyal agent (a less biased one). On the other hand, since an overly biased intermediary imposes an extra bottom on A’s action set, P suffers from the distortions in sub-delegation in low state. When the difference of preferences between two individuals is not large, the gain in flexible decision-making outweighs the distortions in sub-delegation, and the less biased individual will be assigned to the lower position. Proposition 2 formally states this result. Proposition 2 Under the hierarchical delegation game, the principal strictly prefers √ 1+ 1+16b1 ). If b1 = b2 assigning individual 1 to be the agent if and only if b2 ∈ (b1 , 8 or b2 =

√ 1+ 1+16b1 , 8

the principal is indifferent between these two individuals as

whom to become the agent. If b2 > individual 2 to be the agent.

√ 1+ 1+16b1 , the principal strictly prefers assigning 8

The second question we highlight is: under the hierarchical delegation game, given the preferences misalignment between M and P, as well as a pool of candidates for the agent position, which candidate should be the optimal agent? We assume the optimal H D, selection decision is to maximize the organizational goal β EU PH D + (1 − β) EU M in which β is the weight of principal’s expected payoff. First, if M is downwardly biased (b M ≤ 0), both M and P want to cap A, and both gain as A’s bias becomes smaller. The smaller the bias of the agent, the better for M and P.

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When M is upwardly biased (b M > 0), a more biased agent (b A > b M ) leads to the distortion in decision-making in the high state, which is harmful for both M and P. Therefore, both of them seek a compromised agent who is no more biased than M. Within this scope, an agent closer to the principal can improve the flexible decision-making in the high state, while an agent closer to the intermediary would reduce the distortion in sub-delegation in the low state. The optimal bias of agent will balance these two forces. The exact degree of A’s bias depends on the weight of M ’s and P  s payoffs in the organizational goal. Proposition 3 Under the hierarchical delegation game, suppose b M ≥ 0, and the weight of principal (intermediary) in the organizational goal is β (1 − β), then the ideal agent’s bias is b∗A = βbA + (1 − β) bA , in which 1. bA = 2b2M is the principal’s optimal candidate if she selects the agent alone, e.g., β = 1; 2. bA = b M (1 − 2b M ) is the intermdiary’s optimal candidate if he selects the agent alone, e.g., β = 0. This proposition shows that the optimal bias of A is the linear combination of P’s and M’s ideal agent’s bias. When the intermediary has no voice in selection, e.g., β = 1, the principal will choose an agent who is less biased than the intermediary. If the intermediary decides the selection, e.g., β = 0, he will also choose an agent whose bias lies between him and the principal. Further, we explore the difference in ideal agent’s bias for P and M. Remark 3 Comparing bA with bA , we have if b M ≤ 41 , then bA < bA ; if b M > 41 , then bA > bA . We could see that when M is modestly biased, P prefers recruiting an agent closer to herself because the gains in flexibility exceed the distortion from sub-delegation. However, if M is very biased, even if P could make the recruitment decision alone, she would flatter the very biased intermediary by recruiting a candidate closer to the latter. This proposition could derive predictions that can be empirically tested. Iyer and Mani (2012) examine the Chief Minister/District Politician/Bureaucrat hierarchy in India, and find out that when the Chief Minister and the District Politician are elected from different parties, the Chief Minister is more likely to change the bureaucrat to seek for compromise.

6 Comparison with hierarchical communication Here we compare the outcome of hierarchical delegation with that of communication. Under the communication mode, the intermediary is delegated with any authority. We would like to address this question: given the biases of players in the hierarchy, should the principal choose delegation or communication? We show that since an oppositely biased intermediary deteriorates delegation outcome, but improves the communication efficiency, communication will dominate delegation for some range of preferences misalignment.

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Hierarchical communication (“HC” henceforth) shares the same vertical structure with hierarchical delegation, but differs in that P keeps decision rights and the information flows upwardly. Thus, the timing is that the informed agent first sends an unverifiable message from the signal space S = [0, 1] to M, and the latter reports m from the messages space M = [0, 1] to P. We restrict attention to strategies represented by measurable mappings. The reporting strategy for A is μ A :  → S, and for M is μ M : S → M. P forms a posterior about the state after receiving m, and commands M to undertake her ideal action y = E [θ |M ]. In the end, M will pass this command to A, and the latter undertakes it. We would solve this hierarchical communication game with Perfect Bayesian Equilibrium. To get rid of the multiple equilibria problem common in cheap talk game,6 we would focus on the most informative equilibrium, i.e., the ex ante Pareto-dominance one. AAK show that an intermediary cannot improve information transmission compared with direct P − A communication in any pure strategy equilibrium. However, as Ivanov (2010), Ambrus et al. (2013a, b), and AAK (2013) show, there exists some mixed-strategy PBE that an intermediary improves P’s expected payoff. In this equilibrium, upon receiving A’s report “s”, a biased intermediary might induce P’s two distinct actions “y  ”, “y  ”. For instance, if compared with A, M prefers investing less on a specific project, while A wants to expand it, then upon hearing the financial needs from A, in the mixed strategy PBE it would be optimal for M to randomly under-state the needs. Anticipating M’s downward distortion, A realizes that that over-statement might lead to less investment. M’s randomizing reports effectively constitute an implicit punishment on A’s overstatement, and ease the incentive constraints of A to reveal information.7 Besides, only an oppositely biased intermediary has the incentives to undertake this strategy to punish A’s mis-report. However, as AAK demonstrate, it is very complicated to fully characterize this mixed strategy equilibria, and a certain type of mixed strategy PBE can be nonmonotonic in M’s bias. Hence, here we would use the partial characterization of mixed strategy PBE and focus on the range of b M such that a two-interval mixed-strategy equilibrium could be induced. To concentrate on the issue of interest, we will highlight the situation where M is moderately biased or oppositely biased,8 i.e., b M ≤ b A . We would compare the principal’s expected payoffs under the optimal communication equilibrium and that under the optimal delegation.9 6 Babbling equilibrium (Uninformative communication equilibrium) always exists in the cheap talk game, in which the principal would choose his ex ante optimal action E [θ ] = 21 . 7 This intuition is also explored by Blume et al. (2007), in which the noise in communication invalidate

the monotonicity condition of action with respect to the messages, hence encouraging the agent to report more truthfully, and increasing the principal’s welfare. 8 The situation for an overly biased intermediator could be established analogously. However, the range

of b M such that HC dominates HD would be much smaller, and the two-interval mixed strategy PBE equilibrium could not improve over the direct communication. 9 It is noteworthy that here we compare mixed-strategy communication equilibrium payoffs with the pure

strategy delegation payoffs. This is without loss of generality. As demonstrated by GHPS, and Kovac and Mylovanov (2009), the optimal mechanism in this environment is a deterministic mechanism. Besides, pure strategy communication also cannot performs better than mixed-strategy communication (Ambrus et al. 2013a, b).

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Proposition 4 Comparing the principal’s payoff under HD and HC, we have:     1. If b M ∈ b M , b A , HD dominates HC, in which b M is defined by 

3

2b M − b A 5b3 1  + A = − b A (1 − b A ) − b2A 1 + 2b M + (5) 3 3 3     2. For any b A ∈ 0, 21 , we could find a bM ∈ max{− 41 , b A − 21 }, b3A − 16 such   that if b M ∈ max{− 41 , b A − 21 }, bM , HC dominates HD. The second part is the key, which says that for a certain scope of the intermediary’s bias, the principal may prefer communication to delegation even when informative communication is feasible. To derive the threshold bM , we first establish the range of b M such that a two-interval mixed-strategy equilibrium exists. Then we compare P’s payoffs under HC and HD, respectively, and establish the existence of bM by the theorem of intermediate value. It is noteworthy that bM in effect represents a “lower bound” of M’s bias where HC dominates HD. Since the type of mixed strategy PBE can be non-monotonic of b M , e.g., the range that three-interval PBE exists might not overlap with the range of two-interval mixed strategy PBE, we could not derive the maximal HC payoff corresponding to each pair of (b A , b M ). Therefore we could not directly compare the maximal payoff under any HC equilibrium  with that under HD. Hence, it is unclear about the comparison results when b M ∈ bM , bM . An important conclusion in Dessein (2002) is that in a two-layer hierarchy, with the specification similar to CS, delegation is optimal “whenever informative communication is possible” (p. 822). Moreover, as GHPS has shown, under direct interaction the constrained delegation can attain the best outcome of an universal mechanism. As a consequence, by imposing appropriate constraints on the delegation set, P could do no worse than communication. However, we show that this result would not hold within three-layer hierarchies. The main intuition behind this reverse is: when M is oppositely biased, P ’s loss under HD is monotonically increasing when M becomes more biased, since HD entails the oppositely biased intermediary with more discretions to distort the decision-making by imposing an extra tight cap. On the other hand, under HC a properly oppositely biased intermediary might constrain A ’s mis-reporting incentive, and reveal more information to P, hence facilitating P’s flexible decision-making. Therefore, an oppositely biased intermediary could raise P’s expected payoff in HC over direct communication (Ivanov 2010). Therefore, though direct constrained delegation always dominates direct communication, it is possible to find out a range of b M such that HC dominates HD.

7 Conclusion This paper studies optimal interval delegation in multi-tier hierarchies. We provide a complete characterization of the interval delegation set, and use it to study the optimal

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selection of subordinates. Moreover, in contrast with the previous results, when the principal can constrain the scope of delegated options, direct delegation to an informed party weakly dominates delegation via an intermediary, and in hierarchies informative communication can dominate delegation. It would be interesting to investigate whether the irrelevance of a moderately biased intermediary still holds for a more general environment. Since it is established that capping the delegation set is a part of optimal delegation when the utility is nonuniform-quadratic or the prior follows other distributions (Alonso and Matouschek 2008), we think it is not very difficult to extend our results to these contexts. However, due to the limit of space, we leave the formal proof for future research. Even though this work addresses a simple principal/intermediary/agent chain, the results can be extended to more complex hierarchies. 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., information processing cost (Radner 1993), heterogenous knowledge (Garicano 2000), or conflicts over hiring and promotion decisions (Friebel and Raith 2004), our results still hold in a multiple subordinates structure, i.e., a tree. Acknowledgements The author acknowledges the useful comments from the Associate Editor and two referees. The comments from various people and seminars are acknowledged. Funding was provided by National Natural Science Foundation of China (Grant No. 457 71503208).

A Appendix Proof of Lemma 1 We construct a set of delegated options to A, [b A , 1 − b A + ξ ], in which ξ ≥ 0. A would choose action θ + b A for any state θ ≤ 1 − 2b A + ξ , and the constant action 1 − b A + ξ otherwise. The expected payoff to P under the new  1+ξ −2b A 2 1 b A dθ − 1+ξ −2b A [θ − (1 − b A + ξ )]2 dθ . Take delegation set is U P = − 0 the derivatives of U P with respect to ξ , we have ∂U P2 ∂2ξ

∂U P ∂ξ

= −b2A + (b A − ξ )2 . Since

< 0, the optimality requires ξ = 0. Thus P would not extend the upper bound from 1 − b A to 1 − b A + ξ . On the other hand, since in the lowest state θ = 0 A would undertake y = b A , including any actions lower than b A would not increase P’s expected payoff. Hence, we get the results.  Proof of Lemma 2 1. First we show that the upper bound y A = min{1 −  + b M , max{y |y ∈ Y M }}. Since Y A ⊂ Y M , A’s highest available action available y A cannot exceed max{y |y ∈ Y M }. On the other hand, if M is delegated with an unconstrained delegation set, e.g., Y M = Y , then he acts as if a principal within Y M . By Lemma 1, he could attain his best payoff by imposing an ideal cap 1 −  + b M . Thus we have this result. Then we turn to the lower bound y A = max{b A , min{y |y ∈ Y M }}. Clearly, since A’s ideal action when θ = 0 is b A , any decision lower than b A is strictly dominated. And since Y A ⊆ Y M , we have this result. 2. If M is delegated with a unrestricted set of options, we study his ideal delegation set Y A = [x, z], x ≥ b A . The expected payoff to M is U M =

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 x−b A 0

(θ − x + b M )2 dθ −

 z−b A x−b A

(b M − b A )2 dθ −

the derivatives of U M with respect to x and z, we ∂U 2

1

2 z−b A (θ − z + b M ) dθ . Take 2 2 M have ∂U ∂ x = − (b M − x) +  ,

since on the maxima we need ∂ 2 xM < 0, we have the optimal x = b M −  . On the 2 2 M other hand, ∂U ∂z = (1 − z + b M ) −  > 0 always holds for any z ≤ 1. Hence, he would delegate [b M − , 1] to A conditional on having an unconstrained delegation set. Besides, since Y A ⊆ Y M , the upper- and lower-bound of Y A could not exceed that  of Y M . We have the results. Proof of Proposition 1 We first formally describe the principal’s design of optimal delegation. Since P could only limit Y A by imposing a tight cap y A and/or bottom y A on Y M , we could write down P’s delegation problem as the follows, in which it is as if that P directly delegates to A, subject to the incentive-compatible constraints of M in sub-delegation: 

θ − yA

max −

y A ,y A

max{y A −b A ,0}

0

2

 dθ −

y A −b A

max{y A −b A ,0}

 b2A dθ

−

1 y A −b A

(θ − y A )2 dθ (6)

subject to  yA =  yA =

min{1, max{y |y ∈ Y M }}, min{1 −  + b M , max{y |y ∈ Y M }}, max{b M − , min{y |y ∈ Y M }}, max{b A , min{y |y ∈ Y M }},

if  < 0 if  ≥ 0

if  < 0 if  ≥ 0

(7) (8)

The objective function of P (6) considers the expected  parts.  payoff to P in three The first part is that when the true state lies within 0, max{y A − b A , 0} , it is of A’s best interest to execute the bottom y A , consequently the payoff to P for every

2 realization of state is − θ − y A . It is noteworthy that this part might be equal to   zero if y A ≤ b A . For any θ ∈ max{y A − b A , 0}, y A − b A , A always execute his ideal action θ + b A , and the payoff to P for every θ is −b2A . Finally, for any state above y A − b A , A has to execute the cap y A . The two constraints (7) and (8) replicate Lemma 1 and 2, pointing out that any delegation set has to be in M’s best interest. Directly replacing y A and y A in (6) with (7) and (8) leads to the optimal delegation set described in (2). Then we need to find out the range of b M that keeps delegation valuable, e.g., P’s expected payoff from delegation exceeds that from giving only one action to M and 1 . Following ( 2), we could calculate the principal’s expected payoff A, which is − 12 as the follows  2 ⎧4 if 21 ≥ b A > b M ≥ 0 ⎪ 3 bA − 1 bA, ⎪ ⎨ 2 8 3 4 −b A + 3 b M − 4b2M b A + 3 b3A , if b A ≥ 0 > b M ≥ b M EU PH D = (9) ⎪ − 8 b3 + 4b A b2M − b2A , if b M ≥ b M > b A ≥ 0 ⎪ ⎩ 31 M − 12 , otherwise

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The second line of (6) describes the situation of a downwardly biased intermediary, and it is straightforward that in this situation delegation is valuable if −b2A + 83 b3M − 1 . The next step is to determine the existence of b M that maintains 4b2M b A + 43 b3A  − 12 1 . this inequality. We define a function f (b A , b M ) = −b2A + 83 b3M − 4b2M b A + 43 b3A + 12 , b Using the implicit function theorem, we take the derivative of f = 0 with (b ) A M   2 1 2 M respect to b A . This leads to 4b M (b M − b A ) ∂b ∂b A = 2 b M − b A + b A . Since b A ≤ 2 and 0 > b M ≥ b A − 21 , we have

∂b M ∂b A

> 0. Furthermore, when b A = 0 we have  1  13 + = 0, whose non-positive real root is b M = − 32 . When f (0, b M ) =  8 1  1 2 b A = 2 , f 2 , b M = b M 3 b M − 2 = 0, whose non-positive real root is b M = 0.   Hence, for b A ∈ 0, 21 we could find a non-positive b M that satisfies f (b A , b M ) = 0, this defines b M . The third line of (6) describes the situation of an overly biased intermediary, and in 1 . Our remaining this situation delegation is valuable if − 83 b3M + 4b A b2M − b2A  − 12 task is to ensure that there always exists at least one b M > b A > 0 that maintains this inequality. Similar as before, we could write down an implicit function f (b A , b M ) = 1 . With a little abuse of terminology, we redefine b M = b A +t, − 83 b3M +4b A b2M −b2A + 12 1 . and rewrite the implicit function as f (b A , t) = − 83 (b A + t)3 +4b A (b A +t)2 −b2A + 12 Now the task becomes ensuring there is a t > 0 satisfies f (b A , t) = 0. Taking the derivative of f (b A , t) = 0 with respect to b A , we have ∂b∂tA < 0. When b A = 0,  1  13 1 , which has a non-negative real root t = 32 . solving f (b A , t) = 0 we have t 3 = 32 When b A = 21 , solving f (b A , t) = 0 we have 32t 3 + 24t 2 = 0, which has a nonnegative real root t = 0. Hence, within b A ∈ [0, 21 ) we always have a t > 0 that keeps  f (b A , t) = 0. This defines b M as we need.   Proof of Remark 1 If b M ∈ b M , 0 , then we take the derivative of the second line of b3M

1 32

∂ EU H D

(9), we have ∂b MP = 8b M (b M − b A ) ≥ 0.   If b M ∈ b A , b M , then we differentiate the third line of (9) and get −8b M (b M − b A ) ≤ 0.

∂ EU PH D ∂b M

= 

Proof of Remark 2 From Proposition 1, it is straightforward to calculate the expected payoff to the intermediary, which is as the follows

HD EU M

⎧  2b A  2 2 2 ⎪ ⎪ − (1 − 2b A ) − 3 b A + 3b M , ⎪ ⎨ −2 + 4 3 , 3 = 3 2 (1 − 2b ) − (b A +b M )3 , ⎪ −  ⎪ M 3 ⎪ ⎩ 31 − 12 − b2M ,

if 21 ≥ b A > b M ≥ 0 if b A > 0 > b M ≥ b M if b M ≥ b M > b A > 0 otherwise

(10)

∂ EU H D

1. Take the derivative of the first line of (10) with respect to b M , we have ∂b AM = −2b A + 2b M + 4b2A − 8b A b M . This derivative is negative if and only if b M ≤ b A −2b2A 1−4b A .

Because b M ≤ b A , and have the result.

b A −2b2A 1−4b A

≥ b A always holds for any b A ≥ 0, we

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2. Take the derivative of the second line of (10) with respect to b M , we have   2 (2 − 1), since  ∈ 0, 21 , we have the result.

HD ∂ EU M ∂b A

=

∂ EU H D

3. Take the derivative of the third line of (10) with respect to b M , we have ∂b AM =   ∂ EU H D −2 2b2M − b M + b A . Thus, ∂b AM ≤ 0 iff 2b2M − b M + b A ≥ 0. which is translated to b A ≥ b M − 2b2M , we have the result.  Proof of Proposition 2 If the more biased individual 2 is placed in the agent position,  then as ( 9) shows, the expected payoff to P is 43 b2 − 1 b22 , while if the less biased individual 1 takes the agent position, the expected payoff becomes − 83 b23 + 4b1 b22 − b12 . By comparing these two formula, it is beneficial to P by assigning the more biased individual 2 to the agent position only if 4b23 − (1 + 4b1 ) b22 + b12 > 0. The √ 1+ 1+16b1 . It is straightforward to verify that 8   b1 ∈ 0, 21 .

remaining reasonable root leads to b2 > √ 1+ 1+16b1 8

≥ b1 always holds for any

Proof of Proposition 3 We use W to denote the weighted sum of P’s payoff and M’s payoff, in which β represents the weight of the principal. By summing the third parts of (9) and (10) together we have     3 8 3 (b A + b M )3 2 2 2 W = β − b M + 4b A b M − b A + (1 − β) −  (1 − 2b M ) − 3 3 3 (11) When β = 1, e.g, P makes the recruitment decision alone, we could take the first-order condition of (11) with respect to b A , and the compromised candidate is bA = 2b2M . When β = 0, e.g., M chooses A, by differentiating the (11) with respect to b A and equalizing it to zero, we have that his ideal candidate has bA = b M (1 − 2b M ). dW = 0 we have b∗A = 2(2β − 1)b2M + (1 − β)b M = βbA + If β ∈ (0, 1), by db A  (1 − β) bA . Proof of Proposition 4 1. When b M ∈ [0, b A ], as Ivanov (2010) and AAK suggest, an intermediary cannot improve information transmission over direct communication. On the other hand, HD can attain the best outcome of a universal mechanism. Hence, delegation dominates communication. Ivanov (2010) shows that with a properly oppositely biased intermediary, the principal could attain the best outcome from a universal communication mechanism a la Myerson (1981), which is − 13 b A (1 − b A ). We could compare this outcome with the ∂ EU H D

outcome of hierarchical delegation. Since ∂b MP ≥ 0 whenever b M < 0, if there is   a bM such that EU PH D bM = − 13 b A (1 − b A ), we could say that for any b M ≥ bM , HD strictly dominates HC in the sense it can attain higher payoff than the highest possible expected payoff under any communication mechanism. As a result, bM turns out to be a nonlinear function of b A , which is formally defined by (5) 2. We will look at the range of b M for the existence of the two-interval mixedstrategy equilibrium in H C. Under this equilibrium, an oppositely biased intermediary

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sends a message indicating the low interval after hearing the low state, and randomly sends two distinct messages, indicating the low (high) interval, respectively, when hearing that the state is high. Following the method in AAK, we have that the twopartition HC equilibrium exists if max{− 41 , b A − 21 } ≤ b M ≤ b3A − 16 . We restrict attentions to this region. The expected payoff to P thus is EU PH C = b A − (b A − 3b M ) (b A − b M ) − 2b M −

1 3

By combining it with the second line of (9), we have the ex-ante payoff difference between HD and HC as T = EU PH D − EU PH C =

8 3 4 3 1 b + b +3b2M +2b M −4b2M b A −4b A b M −b A + (12) 3 M 3 A 3

∂T It is straightforward to derive ∂b = 4b2A − (1 + 2b M )2 ≤ 0 since b A − 21 ≤ b M . A Hence, for any b M such that a two-interval HC equilibrium exists, as b A increases, relative to HD, the HC payoff increases. On the other hand, the derivative of T with ∂T = 4 (b M − b A ) (1 + 2b M ) + 2 (1 + b M ). Using the fact that respect to b M is ∂b M 1 ∂T ≥ 0. b M − b A ≥ − 2 , we have ∂b M  Now we need to find a b M implicitly defined by T = 0. Even though we could not analytically solve T = 0, we may still obtain some properties about bM by studying the value of T on the boundary of b M such that a two-interval HC  equilibrium  exists, e.g, b M = − 41 , b M = b A − 21 , and b M = b3A − 16 . Since T b M = − 41 =

  1 − 48 + 43 b3A − b4A ≤ 0, T b M = b A − 21 = −(b A − 21 )2 ≤ 0, T b M = b3A − 16 =

2   bA 1 80 23 ∂T − 3 6 9 b A + 9 ≥ 0, and ∂b M > 0, by the theorem of intermediate value,   we have that there exists a bM ∈ max{− 41 , b A − 21 }, b3A − 16 as an implicit function

of b A such that for any b M ≤ bM , we have T ≤ 0. Hence, HC dominates HD in this region. 

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