FUNDAÇÃO GETULIO VARGAS
ESCOLA de PÓS-GRADUAÇÃO em ECONOMIA
Lucas Panico de Lara
Delegated Coordination
Rio de Janeiro 2019
Lucas Panico de Lara
Delegated Coordination
Dissertação submetida à Escola de Pós-Graduação em Economia (EPGE) da Fun-dação Getulio Vargas (FGV) como requi-sito parcial para a obtenção do grau de Mestre em Economia.
Área de concentração: Teoria Econômica Orientador: Humberto Luiz Ataíde Mo-reira
Co-orientador: Andrés Mauricio Carvajal Escobar
Rio de Janeiro 2019
Dados Internacionais de Catalogação na Publicação (CIP) Ficha catalográfica elaborada pelo Sistema de Bibliotecas/FGV
Lara, Lucas Panico de
Delegated coordination/ Lucas Panico de Lara. – 2019. 60 f.
Dissertação (mestrado) - Fundação Getulio Vargas, Escola de Pós- Graduação em Economia.
Orientador: Humberto Luiz Ataíde Moreira. Coorientador: Andrés Mauricio Carvajal Escobar. Inclui bibliografia.
1. Delegação de autoridade – Modelos matemáticos. 2. Processo
decisório – Modelos matemáticos. 3. Economia matemática. 4. Econometria. I. Moreira, Humberto Ataíde. II. Escobar, Andrés. III. Fundação Getulio Vargas. Escola de Pós-Graduação em Economia. IV. Título.
CDD – 330.015195
Elaborada por Márcia Nunes Bacha – CRB-7/4403
Abstract
We consider a multi-dimensional model of delegation in which a principal makes a decision so as to coordinate two agents, each with private information of their own impacting the outcomes of the decision and with one agent having finer and the other having coarser information. We solve the model for a class of such cases and show that the principal is able to use each agent’s uncertainty about the other’s information as a screening device, leading to an optimal mechanism that is both more complex and more favorable to the principal than the simpler mechanisms in standard one-dimensional models of delegation. We also show that these results have consequences for the optimal flow and distribution of information inside organizations.
Contents
1 Introduction 4
1.1 Structure of this work . . . 6
2 Related Literature 6
3 The General Model 8
4 The One-Dimensional Model 10
5 The Multi-Dimensional Model 13
5.1 Simple mechanisms . . . 15 5.2 Reducing dimensionality . . . 15 5.3 The symmetric case . . . 21
6 Implementation and Information in Organizations 29
7 Concluding Remarks and Future Developments 36
List of Figures
1 Optimal one-dimensional mechanism when ˆθ2 = 1/2. . . 13
2 Optimal multi-dimensional mechanism in the symmetric case when δ = 1/2. 27 3 Optimal multi-dimensional mechanism in the symmetric case when δ = 1/4. 28 4 Optimal multi-dimensional mechanism in the symmetric case when δ = 1/100. 29 5 Incentive compatibility for agent two for the two-step implementation when
δ = 1/5. . . 34 6 Incentive compatibility for agent two for the two-step implementation when
1. Introduction
Organizations arise as agents collectively pursue shared interests that could not be satis-factorily done so in a decentralized manner such as through the market system. In the words of Arrow (1974, p. 33), “organizations are a means of achieving the benefits of collective action in situations where the price system fails”. The most familiar economic example is the business firm, but as Gibbons and Roberts (2013, p. 1) note, this also includes “consortia, unions, legislatures, agencies, schools, churches, social movements and beyond”. As organi-zations expand and evolve, however, they develop constituent parts which, in the pursuit of better performance, specialize in their activities. This division brings not only the specializa-tion of tasks, but also of interests and incentives. Such division often means misalignment, with each subdivision pursuing their own interests to the detriment of the collective interests of the organization as a whole.
The specialization of activities also leads to the specialization of knowledge and informa-tion. Subdivisions of an organization will often have privileged access to information that is closely related to their line of work. Due to misaligned interests and incentives, in situa-tions in which the organization must take a collective decision the outcome of which depends on information held separately by its parts, they may attempt to manipulate the outcome in their favor by strategic withholding or partial sharing of their information. In light of this phenomenon, organizations face the challenge of properly designing themselves so as to realign the incentives of its constituent parts and thus avoid the informational pitfalls of division and specialization. In this sense, the organization needs to coordinate its subunits so that they work for their collective welfare. Moreover, as the allocation of resources inside organizations is usually determined through other means and with other goals in mind, the use of monetary transfers as an instrument for incentivizing the sharing of information is often either prohibited or highly undesirable.
In this work, we model the situation described above as a multi-dimensional principal-agent model with no transfers. One action must be taken in an organization, with the outcome depending on a multi-dimensional state of nature and with the information on each of the dimensions being privately held separately by one of two agents. The agents are to be seen as the constituent parts of an organization. A principal wishes to coordinate both
agents by maximizing their joint welfare. This principal may be seen either literally, as an authority figure at the top of the organizational hierarchy, or figuratively, as an abstract representation of the collective interest of the organization.
This setup puts our work squarely in the literature on delegation, in which an uninformed principal must delegate a decision to an informed agent with his own interests without the use of monetary transfers. This literature stresses the trade-off that the principal faces between giving the agent discretion to use his private knowledge and putting limitations on his choice to mitigate the misalignment of interests. Since transfers are off-limits, the principal’s only instrument is to design the way in which the agent is allowed to choose the outcome that is to be realized. Once the principal commits to a mechanism, the agent is certain of which decision will be implemented for each of his announcements, which is equivalent to letting him make it. The traditional result in the literature consists in the principal putting boundaries on the agent’s possible choices and letting him freely choose among the decisions lying between such bounds. This is known as interval delegation.
The novelty in our model comes from the existence of multiple agents and the fact that information is dispersed among them. Here, a single decision must be jointly delegated to two agents. This leads to very different results in comparison to the above. We show that since the outcome now depends on the decisions of both agents, this creates an uncertainty for each of them about what the outcome will be implemented for each of their own announcements. The principal is able to exploit this uncertainty and use it as a screening device - one, however, that is also costly to her. This leads to optimal mechanisms that take more complex shapes than the ones in interval delegation. We solve the model for a class of tractable cases and show that the optimal mechanism is continuous and deterministic with no pooling. The principal steers the agents towards coordination by making heavier use of her additional instruments whenever the interests of the agents are more divergent. Since this use of uncertainty is also costly to the principal, though, we show that more divergent interests lead her to have greater losses.
The method of implementing the optimal mechanism in the one-dimensional model through interval delegation has no straightforward extension to our multi-dimensional model. This leads us to discuss the possibilities of implementation in the last part of this work. This,
in turn, opens up a discussion on the effects that the order of individual decisions has on what collective outcomes are feasible and on the way that the flow and distribution of information inside an organization may help or hamper coordination. We show that the order of decision rights should be balanced with regards to informational privilege that certain constituent parts of an organization may have. Also, we show that constituent parts with more aligned interests may be allowed to be informationally integrated with little loss whereas parts with starkly divergent interests may be best kept informationally separated for the collective good of the organization.
In one of their surveys on the field of Organizational Economics, Gibbons and Roberts (2015) present a number of questions which it aims to provide answers for. In particular, we highlight the following ones:
How are subunits within the firm defined, linked and coordinated? Where does decision-making on different issues occur within the organization? What are the behavioral and performance effects of delegation? What information is collected on different matters, by whom, to whom is it communicated, and how is it used? The present work helps to improve our understanding of all of these questions, most notably the first one, placing it firmly as a contribution to the theoretical study of organizations. 1.1. Structure of this work
The rest of this work is structured as follows. Section 2 relates our work to the existing lit-erature. Section 3 introduces the general model of delegated coordination. Section 4 presents the one-dimensional model and matches it to the existing literature on one-dimensional del-egation. Section 5 contains our main results with regards to the multi-dimensional model. Section 6 explains some consequences of our results. Section 7 concludes. All proofs are in the Appendix.
2. Related Literature
There exists by now a rich literature on delegation that sprang from Holmstr¨om (1984), almost all of it focusing on one-dimensional cases. Some notable examples are the works of
Melumad and Shibano (1991), Alonso and Matouschek (2008) and Kov´ac and Mylovanov (2009). The main contribution of our work consists in expanding this literature to the case with multiple agents, each having private information of their own. It is worth mentioning that in the case of cheap talk instead of delegation, the case of a multi-dimensional state of nature and multiple agents has been considered in the work of Battaglini (2002). However, even in this work it is assumed that each agent observes all dimensions of the state of nature, in contrast to our model in which each dimension is privately observed by one agent.
The general model that we will first consider and will later simplify in order to solve was first presented by Carrasco and Fuchs (2018). They propose a Divide and Discard (DD) mechanism, which through a sequential algorithm is capable of progressively approximating the region in which the types of both agents lie, with the approximation being more precise the closer the types are to each other. The proof of the optimality of such mechanism, however, remains an open question. Our work is to be seen as a contribution to the better understanding of their model by starting with simpler cases. Fleckinger (2008) also considers the same general model and proposes the same mechanism as they do, but also does not prove optimality.
The main works in multi-dimensional delegation so far are Koessler and Martimort (2012) and Frankel (2016). The first mirrors ours in the sense that in their model two decisions must be delegated to a single agent, while in ours a single decision must be delegated to two agents. We are also heavily methodologically indebted to their approach to dealing with multi-dimensionality. We follow their methods of reducing a multi-dimensional problem of mechanism design to a Calculus of Variations problem that is amenable to advanced tech-niques from the optimization literature. Nevertheless, the particular nature of our model leads to a different solution with different interpretations and different comparative statics. The second considers multiple decisions being delegated to a single agent with multidimen-sional information, but with each decision being associated to a single dimension of the information.
Che, Dessein, and Kartik (2013) consider a cheap talk model with multi-dimensional information and then extend their model to a case of commitment to a mechanism with no transfers. However, in their model all dimensions of information are revealed to a single
agent. Furthermore, they consider only a finite set of possible decisions.
Most recently, Amador and Bagwell (2018) consider a model of delegation with money burning and apply it to what they call a “cooperation model”, in which a principal wishes to maximize the joint welfare of two agents. Despite the apparent similarity, there are two major differences that separate our work from theirs. First, in their model the state of nature is one-dimensional, so that no questions of multi-dimensionality arise. Second, they designate one of the two agents as the one to whom the decision will be delegated, while the other, called a “partner”, does not participate in this process, even if his welfare is taken into consideration in the elaboration of the optimal mechanism. In this way, they avoid the two main issues in our model: a multi-dimensional information that is dispersed among agents and the fact that one decision must be jointly delegated to two agents.
3. The General Model
In this section we present the general model of delegated coordination as elaborated by Carrasco and Fuchs (2018).
A single action, represented by a real number a must be taken in an organization. The effects of such action depend on an unknown two-dimensional state of nature θ = (θ1, θ2).
Two agents in the organization are privately informed about one dimension of θ. Agent i = 1, 2 is informed about θi, which is commonly known to be independently distributed
according to the uniform distribution over the [0, 1] interval. That is,
θi ∼ U [Θi], i = 1, 2
with Θ1 = Θ2 = [0, 1].
Both agents have quadratic and single peaked preferences around their private informa-tion, which from now on we refer to as their types, so that each agent wants to match the action to be taken to their own type:
ui(a, θi) = − (a − θi) 2
, i = 1, 2.
A principal, uninformed about the state of nature, has the right to take the action, oversees both agents and wants to coordinate them. That is, her preferences are given by
uP (a, θ1, θ2) = u1(a, θ1) + u2(a, θ2) ,
which may equivalently be taken in the following form, which makes it clear that she desires to match the action to the average of the two types:
uP (a, θ1, θ2) = − a − θ1+ θ2 2 2 .
There is no communication between the agents.1 The principal may not make any mon-etary transfer to either agent in order to incentivize them.
Due to her ignorance of the state of nature, the principal may either always take a single action based on her prior belief or she may delegate the decision to the agents by commiting to a delegation mechanism.2 From the Revelation Principle, such mechanisms may be taken in the form of a function a belonging to the space A (Θ1 × Θ2) of measurable functions
defined on Θ1× Θ2 and assuming values on A, a compact interval containing [0, 1]:3
a : Θ1× Θ2 → A
The best such mechanism induces each agent to truthfully reveal their type while taking, for each combination of types, the best possible action for the principal in an ex-ante manner. Hence, it solves the problem
max a∈A (Θ1×Θ2) E(θ1,θ2)− (a(θ1, θ2) − θ1) 2− (a(θ 1, θ2) − θ2)2
s.t. Eθ−i− (a(θi, θ−i) − θi)
2 ≥ Eθ−i −a(ˆθi, θ−i) − θi 2 ∀ ˆθi, θi, i = 1, 2
In this work, we first take Θ2 = {ˆθ2}, and then Θ2 = {θ2, θ2} with θ2 < θ2.
1The consequences of letting the agents communicate are unclear. If they collude to maximize their joint
welfare, then they would be doing exactly what the principal wants anyway. Due to the misalignment of their incentives, however, it is difficult to see how they would agree to such an arrangement, which is precisely what motivates the problem to begin with.
2It is worth mentioning that the multi-dimensionality of the problem makes the penalty for taking the
first option greater than is the case in one-dimensional models.
3We consider only deterministic mechanisms. For all cases treated here, it will be shown that stochastic
4. The One-Dimensional Model
Consider the model presented in the previous section with the exception that now θ2 ∈
Θ2 = {ˆθ2} ⊂ [0, 1]. That is, agent two is commonly known to be of type θ2 = ˆθ2. In this
situation, therefore, only agent one’s type is unknown and thus agent two is left with no informational bargaining power. From this, it follows that in this setting the mechanism is written entirely as a function of agent one’s type, i.e., it is a function a : Θ1 → A belonging
to the space A (Θ1) of measurable functions defined on Θ1 and assuming values on A.
The principal’s problem is then given by
max a∈A (Θ1) Eθ1 − a(θ1) − θ1+ ˆθ2 2 !2 s.t. − (a(θ1) − θ1)2 ≥ − a(ˆθ1) − θ1 2 , ∀ ˆθ1, θ1.
Notice that this is a quite standard one-dimensional model. The principal’s (non-constant) bias4 b is given by b(θ1) = − θ1 2 + ˆ θ2 2.
Notice that agent two has been entirely absorbed by the principal’s bias. This is because, despite the fact that the mechanism depends only on agent one’s type, the principal still takes agent two into account.
We mostly follow Kov´ac and Mylovanov (2009) in characterizing the solution to this version of the problem.5
We begin by defining the functions
U1(θ1, ˆθ1) = −
a(ˆθ1) − θ1
2
4It is more standard in the literature to consider the agent’s bias. In our settings, however, it is more
convenient to consider the principal’s.
5We do not consider stochastic mechanisms since, as is shown in Kov´ac and Mylovanov (2009), they are
and
U1(θ1) = U1(θ1, θ1).
As is standard, we say that a mechanism is incentive compatible if it satisfies the restric-tions on the problem above, i.e., if it induces the agent to always truthfully reveal their type. Traditional envelope arguments then give us the following result.
Lemma 1. A one-dimensional mechanism a is incentive compatible if and only if i. a is non-decreasing
ii. for all θ1 ∈ [0, 1],
U1(θ1) = U1(0) + 2
Z θ1
0
(a(s) − s)ds. (1)
For future reference, we also register the following alternative but equivalent character-ization of incentive compatibility, presented by Alonso and Matouschek (2008), which will be useful to contrast with incentive compatibility in the multi-dimensional case. We define a−(θ1) = limθ˜1→θ−1 a(˜θ1) and a+(θ1) = limθ˜1→θ+1 a(˜θ1), which are well-defined by monotonicity.
Lemma 2. An incentive compatible one-dimensional mechanism a must satisfy the following: i. a is weakly increasing;
ii. if a is strictly increasing and continuous in (θ10, θ100), then a(θ1) = θ1 for all θ1 ∈ (θ
0
1, θ
00
1);
iii. if a is discontinuous at ˜θ1, then the discontinuity must be a jump discontinuity that
satisfies:
(a) u1(a−(˜θ1), ˜θ1) = u1(a+(˜θ1), ˜θ1),
(b) a(θ1) = a−(˜θ1) for θ1 ∈ [max{0, a−(˜θ1)}, ˜θ1) and a(θ1) = a+(˜θ1) for θ1 ∈ (˜θ1, min{1, a+(˜θ1)}],
(c) a(˜θ1) ∈ {a−(˜θ1), a+(˜θ1)}.
Particular attention is to be given to item (ii), which says that an incentive compatible mechanism that is continuous and strictly increasing over an interval must coincide with the identity over that interval, which means that it must implement agent one’s preferred action whenever the state of nature is between those values. In other words, it must completely hand over power to the agent.
We now observe that the regularity condition that is assumed in Kov´ac and Mylovanov (2009, p. 1381) is satisfied in our setting. This allows us to use their results to characterize the solution to the one-dimensional problem.
Remark 1. In the one-dimensional model, the regularity condition in Kov´ac and Mylovanov (2009, p. 1381) is satisfied.
We may now proceed to present the optimal mechanism in this setting.
Proposition 1. In the one-dimensional model, given the type of agent two ˆθ2 ∈ [0, 1], the
optimal mechanism is given by
aθˆ2(θ 1) = 2ˆθ2 3 , if 0 ≤ θ1 ≤ 2ˆθ2 3 θ1, if 2ˆ3θ2 ≤ θ1 ≤ 2ˆθ23+1 2ˆθ2+1 3 , if 2ˆθ2+1 3 ≤ θ1 ≤ 1 .
As is usual in one-dimensional models of delegation, this mechanism can be interpreted as the principal letting agent one freely choose any action among the ones in the delegation interval aθˆ2([0, 1]) = " 2ˆθ2 3 , 2ˆθ2+ 1 3 # .
The fact that the optimal mechanism limits the available options of agent one - a standard result - has a specific interpretation in our setting. Remember that our model is to be interpreted as a mechanism that coordinates two agents. In this one-dimensional case, the decision is entirely delegated to agent one, since he’s the only one to have private information. However, the principal must not forget that agent one’s decisions also affect agent two’s welfare. Hence, the limitation enforced by the optimal mechanism may be seen as the principal offering agent two institutional protection against agent one’s discretion.
In what follows, we will use this one-dimensional result as a benchmark against which our multi-dimensional results will be compared. It will be particularly useful to register the one-dimensional solution for when ˆθ2 = 1/2 given by the delegation interval [1/3, 2/3], which
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 1: Optimal one-dimensional mechanism when ˆθ2= 1/2.
5. The Multi-Dimensional Model
We now move to the main section of this work. Consider the general model presented in Section 3, but with
θ1 ∼ U [0, 1] and θ2 ∼ U {θ2, θ2}
with θ2 < θ2, {θ2, θ2} ⊂ [0, 1].
We interpret this case as one in which agent one has fine information and agent two has coarse information. That is, agent two does not have full information about θ2 ∈ [0, 1], but
receives a signal such that, in posterior terms, he knows that E [θ2] = θ2 or E [θ2] = θ2,
and this is such that, on prior terms, P{E [θ2] = θ2} = P{E [θ2] = θ2} = 1/2. With this
interpretation in mind, we may treat agent two as possibly being of a low or high type. The multi-dimensional delegation mechanism is now given by a measurable function6
a : [0, 1] × {θ2, θ2} → A
However, instead of working with it in this form, we consider it instead as a pair of functions on A (Θ1) ×A (Θ1):
(a, a) : [0, 1] → A × A with the understanding that
a(θ1) = a(θ1, θ2) and a(θ1) = a(θ1, θ2)
Using this notation, the principal’s problem in this case is given by
max (a,a)∈A (Θ1)×A (Θ1) − Z 1 0 1 2(a(θ1) − θ1) 2 + (a(θ1) − θ2) 2 + 1 2 h (a(θ1) − θ1)2+ a(θ1) − θ2 2i dθ1 s.t. − 1 2(a(θ1) − θ1) 2 + (a(θ1) − θ1) 2 ≥ −1 2 a(ˆθ1) − θ1 2 +a(ˆθ1) − θ1 2 ∀ ˆθ1, θ1 − Z 1 0 (a(θ1) − θ2) 2 dθ1 ≥ − Z 1 0 (a(θ1) − θ2) 2 dθ1 − Z 1 0 a(θ1) − θ2 2 dθ1 ≥ − Z 1 0 a(θ1) − θ2 2 dθ1
The first thing to be noticed is that, with regards to both agents, for the mechanism to be incentive compatible it only needs to be so on average terms, with the average being taken with respect to the other agent’s type. Given a mechanism, the existence of another agent makes each of them uncertain about which action will be implemented for each announcement they make, since it depends on both announcements. Exploiting these uncertainties and using them as a screening device allows the principal to implement more mechanisms than would otherwise be the case. It is thus to be expected that optimal mechanisms in this multi-dimensional case will make good use of this possibility and take more complex forms than the ones in one-dimensional models. As will be clear from the optimal mechanism that we derive, this is indeed the case.
5.1. Simple mechanisms
Before starting to discuss the greater complexities of this multi-dimensional model, we will make some observations about a specific class of mechanisms. Notice that despite the much greater availability of mechanisms that the principal has at her disposal, she always has the choice of offering the agents a multi-dimensional mechanism that consists of two mechanisms that have the one-dimensional form. That is, she may always offer
a([0, 1]) = [γ
1, γ1] and a([0, 1]) = [γ2, γ2]
with 0 ≤ γ
1, γ1, γ2, γ2 ≤ 1, γi ≤ γi, i = 1, 2. We call mechanisms of this form simple
mechanisms.
It is evident that they are always incentive compatible for agent one. Thus, when using these mechanisms, the principal must consider only incentive compatibility for agent two. The most natural choice for a simple mechanism is to simply reproduce twice the optimal one-dimensional mechanism: a([0, 1]) = 2θ2 3 , 2θ2+ 1 3 and a([0, 1]) = 2θ2 3 , 2θ2+ 1 3
We’ll have occasion to discuss simple mechanism, their incentive compatibility as well as their relation to optimal multi-dimensional mechanisms and their implementation later on.
5.2. Reducing dimensionality
Our strategy in what follows is to bring this multi-dimensional problem as close as possible to a one-dimensional one, making it amenable to the techniques we will be using in order to obtain the optimal mechanism. Since agent one has a continuous type, while agent two has a discrete type, it is natural to attempt to push agent two into the background and focus on agent one. We thus re-write the model from the perspective of agent one.
The first step we take is to define two new functions of agent one’s type from those that characterize the mechanism, respectively the average action and the variance between the two actions:
a(θ1) = a(θ1) + a(θ1) 2 and σ(θ1) = (a(θ1) − a(θ1)) 2 4
These new functions allow us to write the mechanism as a pair (a, σ) of measurable functions, with σ being a non-negative function, that is:
(a, σ) : Θ1 → A × [0, κ]
with κ = (max A − min A)2/4.7
The principal now faces a trade-off between what action will be implemented, on average, for each of agent one’s types and by how much will the actually implemented action diverge from this average.
Once an optimal mechanism (a∗, σ∗) is obtained, it is straightforward to verify that we may recover the equivalent optimal mechanism (a∗, a∗) by taking8
a∗(θ1) = a∗(θ1) − p σ∗(θ 1) a.e. and a∗(θ1) = a∗(θ1) + p σ∗(θ 1) a.e. (2)
It is worth registering, in anticipation, that this will imply
a∗(θ1) ≥ a∗(θ1) a.e. (3)
The next step is to characterize incentive compatibility for each agent in terms of these new functions. We begin with agent one. It is possible to re-write his utility function in terms of (a, σ) so that we may define
U1(ˆθ1, θ1) = −
a(ˆθ1) − θ1
2
− σ(ˆθ1).
Notice that because of the quadratic nature of agent one’s preferences, and because we have written the new functions from his perspective, his payoff depends only on the average action and the variance, assuming a quasi-linear form. In this form, the average action assumes
7We put this bound so that every (a, σ) mechanism may be mapped back into a (a, a) mechanism. Since
A is bounded, if we allowed σ to assume arbitrarily large values, this would not be possible.
8Taking a(θ
1) = a(θ1) +pσ(θ1) and a(θ1) = a(θ1) −pσ(θ1) creates an additional loss term for the
the main “matching” role, with the agent wanting to match it to his type. Meanwhile, the variance serves as a cost, playing a role similar to that which transfers do in other models, with the caveat that it due to the non-negativity of the variance, it must always be indeed a cost.
This new form also makes explicit our earlier comments on the differences in incentive compatibility between one-dimensional models of delegation and our multi-dimensional one. The principal now has two instruments at her disposal that she can use to screen and incen-tivize the agents. For agent one, even if he is certain about what action will be implemented, on average, for each message ˆθ1 he may send, which is given by a(ˆθ1), he is uncertain about
the specific action, i.e., whether it will be a(ˆθ1) or a(ˆθ1), which is to be seen as a cost given
by σ(ˆθ1). The principal may then exploit this uncertainty by using σ as a screening device,
bringing a and a closer or pushing them further apart, even for the same values of a. This opens up new possibilities in terms of what mechanisms are implementable. Of course, she must also be aware of the effects that this will have for agent two, which is something that we will explore when discussing his incentive compatibility constraints.
We now define
U1(θ1) = U1(θ1, θ1)
and are ready to characterize incentive compatibility for agent one. Lemma 3. Incentive compatibility for agent one is equivalent to:
i. The expected payoff function for agent one U1 being absolutely continuous with a first
derivative defined almost everywhere such that at any point of differentiability:
˙
U1(θ1) = 2(a(θ1) − θ1). (4)
and
ii. The average action a being non-decreasing and thus also having a first derivative defined almost everywhere and such that at any such point of differentiability:
˙a(θ1) =
¨ U1(θ1)
The first condition above is a traditional envelope condition, which for our purposes is best expressed in derivative form. The second one reinforces the differences in incentive compatibility brought by multi-dimensionality. The monotonicity condition now needs to apply solely to the average action, rather than to the mechanism as a whole. We now warn the reader that we will mainly concern ourselves with the first condition, leaving the second one on monotonicity in the background, left to be checked when the appropriate moment arrives.
The results in Lemma 3 allow us to recover from a given absolutely continuous U1 the
equivalent (a, σ) mechanism given by:
a(θ1) = θ1+ ˙ U1(θ1) 2 (6) σ(θ1) = −U1(θ1) − ˙ U2 1(θ1) 4 ≥ 0 (7)
We now proceed to characterize incentive compatibility for agent two in terms of the new functions that have been established. Writing them in terms of the (a, σ) mechanism and simple calculations yield the equivalent restrictions:
Z 1 0 (a(θ1) − θ2) p σ(θ1)dθ1 ≥ 0 (8) and Z 1 0 θ2− a(θ1) pσ(θ1)dθ1 ≥ 0 (9)
In order to interpret these, note that they divide Θ1 into the following regions:
R1 = {θ1 ∈ Θ1 : a(θ1) ≤ θ2}
R2 = {θ1 ∈ Θ1 : a(θ1) ≥ θ2}
R3 = {θ1 ∈ Θ1 : a(θ1) ≤ θ2}
R4 = {θ1 ∈ Θ1 : a(θ1) ≥ θ2}
R1 counts negatively towards the restriction on θ2 while R2 counts positively towards it,
whereas R3counts positively towards the restriction on θ2while R4counts negatively towards
it.
In a point-by-point analysis, R1 is the region in which agent two sees the average action
a as being ”too low” when he is of type θ2 and R4 is the region in which he sees a as being
”too high” when he’s of type θ2. As such, these are precisely the regions in which he might
have an incentive to misreport his type - overreport in the case of θ2 and underreport in the case of θ2 - so as to bring the outcome closer to his desired value. This point-by-point
consideration is then multiplied by the standard deviation √σ because the points in which σ is high are the ones in which it makes a bigger difference to misreport his type, whereas if σ has a small value at that point it does not lead to a much different outcome. Agent two thus takes a weighted average over this point-by-point analysis and decides if he’s truthfully reporting or not.
From this we may conclude that in order to make the mechanism incentive compatible for agent two, the principal must keep the values of the average action a inside regions R2
and R3 as much as possible while also having σ(θ1) high for those values of θ1 in which
a(θ1) ∈ R2 ∩ R3 and low otherwise.
Our next step is to define two new variables, the average type of agent two and the spread of his types:
ˆ θ2 =
θ2+ θ2
2 ∈ [0, 1] and ∆θ2 = θ2− θ2 > 0. (11)
We now register the following result, which is in accordance with usual results in mech-anism design models with a discrete-type agent.
Lemma 4. Given a mechanism (a, σ) that is incentive compatible for agent two, if σ is not identically zero almost everywhere, then at most one of the incentive compatibility restrictions for agent two may be binding.
Our final step is to write the principal’s problem entirely in terms of the new variables and functions we have defined. Manipulating the expression for the principal’s loss and
substituting our new variables whenever appropriate, we can define9 1 2 " a(θ1) − θ1+ θ2 2 2 + a(θ1) − θ1+ θ2 2 2# = Λ(θ1, U1(θ1), ˙U1(θ1)) := −U1(θ1) + θ1 − ˆθ2 2 ˙ U1(θ1) − ∆θ2 2 s −U1(θ1) − ˙ U2 1(θ1) 4
and note that we may also write the incentive compatibility restrictions for agent two in terms of our new variables. This allows us to write the principal’s problem as a Calculus of Variations problem with restrictions over the space of absolutely continuous functions over the [0, 1] interval: min U1∈AC[0,1] Z 1 0 Λ(θ1, U1(θ1), ˙U1(θ1)) dθ1 s.t. Z 1 0 θ1+ ˙ U1(θ1) 2 − ˆθ2+ ∆θ2 2 !s −U1(θ1) − ˙ U2 1(θ1) 4 dθ1 ≥ 0 Z 1 0 θ1+ ˙ U1(θ1) 2 + ˆθ2− ∆θ2 2 !s −U1(θ1) − ˙ U2 1(θ1) 4 dθ1 ≥ 0 We first note a few difficulties with this problem:
i. Λ is actually not defined over the entire space AC[0, 1]. For any absolutely continuous function U such that 4U (θ1) > − ˙U (θ1)2 for some θ1 ∈ [0, 1], Λ(θ1, U (θ1), ˙U (θ1) is
unde-fined. If this happens in any set of positive measure, the objective functional will not be defined for such U . This, however, is a minor problem. Since the term inside the square root in the formula of Λ is actually the variance σ, the functions for which this problem arises are not actually considered in our model. Therefore, we will mostly ignore this from now on.
9The equalities are to be understood as being equalities except for constants which do not affect the
ii. More importantly, Λ is non-differentiable. The derivative of Λ(t, s, v) is undefined when-ever 4s = −v2. For this same reason, it is actually not even Lipschitz. This is a more
delicate problem that we will have to take into account.
We have managed to almost completely remove agent two from our problem, writing all our variables in terms that put agent one at the center of our model. Unfortunately, though, this problem has still proven to be simply too untractable so far, particularly due to the form of the restrictions. We now proceed to solve a particular class of cases which will prove to be much more tractable.
5.3. The symmetric case We now consider Θ2 = 1 2 − δ, 1 2+ δ
with 0 < δ ≤ 1/2. That is, the possible types of agent two are symmetrically dispersed around 1/2. Naturally, in terms of the variables we have been working with this means that θ2 = 1/2 − δ, θ2 = 1/2 + δ, ˆθ2 = 1/2 and ∆θ2 = 2δ.
The following definition will be instrumental in allowing us to obtain the optimal mech-anism in this case.
Definition 1. The mechanism (a, a) is anti-symmetric if
a(θ1) = 1 − a(1 − θ1) ∀ θ1 ∈ [0, 1]. (12)
This anti-symmetry property is to be seen as the mechanism being invariant under a change of variables from θ1 to 1 − θ1 and from θ2 to θ2 and a shift in its values. In the
symmetric case, it ties a and a together in a way which actually turns out to have an important connection with the way that θ2 and θ2 are tied together in the symmetric case.
Our next result reveals the particular tractability of the symmetric case.
Lemma 5. In the symmetric case, if the mechanism (a, a) is anti-symmetric, the incentive compatibility constraints for agent two are equivalent and reduce to
Z 1 0 a(θ1) dθ1 ≥ Z 1 0 a(θ1) dθ1. (13)
The importance of Lemma 5 comes from the fact that the condition expressed in equa-tion (13) is actually trivial, thanks to the property that we have observed in equaequa-tion (3). This means that any optimal anti-symmetric mechanism automatically satisfies the incentive compatibility constraints for agent two.
Thus, Lemma 5 provides the key that we use to solve the model in the symmetric case. If we obtain an anti-symmetric solution then, because of Lemma 5, it will satisfy the restrictions for agent two, regardless of whether we actually considered them when solving the problem or not. Our strategy then is to solve the principal’s problem ignoring these restrictions and if the solution we obtain is anti-symmetric, then they will be satisfied.
Following this, we are faced with the problem
min U1∈AC[0,1] Z 1 0 −U1(θ1) + θ1− ˆθ2 2 ˙ U1(θ1) − ∆θ2 2 s −U1(θ1) − ˙ U2 1(θ1) 4 dθ1 (C)
Before characterizing the solution to this problem, we first register two results. First, the existence of such a solution, and second, the fact that this solution will be deterministic, since stochastic mechanisms are not optimal.
Proposition 2. There exists a unique solution to C, which is deterministic.
We may now proceed to investigate the optimality conditions that this solution must satisfy.
Proposition 3. The necessary and sufficient condition for optimality that a solution U1∗ to C must satisfy is the existence of an absolutely continuous function p that satisfies
i. The Euler inclusion: for almost all θ1 ∈ [0, 1]
( ˙p(θ1), p(θ1)) ∈ ∂Λ(θ1, U1∗(θ1), ˙U1∗(θ1))
ii. The transversality conditions:
p(0) = p(1) = 0.
At any point of differentiability, this is equivalent to U1∗ satisfying iii. The Euler-Lagrange equation:
∂ ∂U1 Λ(θ1, U1(θ1), ˙U1(θ1)) = d dθ1 ∂ ∂ ˙U1 Λ(θ1, U1(θ1), ˙U1(θ1)) . (14)
iv. The transversality conditions:
∂ ∂ ˙U1 Λ(θ1, U1(θ1), ˙U1(θ1)) θ1=0 = ∂ ∂ ˙U1 Λ(θ1, U1(θ1), ˙U1(θ1)) θ1=1 = 0. (15)
We are now finally ready to obtain the optimal mechanism in the symmetric case by solving C. Using the optimality conditions presented in Proposition 3, we may obtain the optimal U1∗that solves problem C, from which we may recover the optimal (a∗, σ∗) mechanism according to equations (6) and (7) and then, finally the optimal (a∗, a∗) mechanism according to equation (2). We register the properties that this mechanism has in our main result below. Theorem (Optimal Mechanism for the Symmetric Case). In the symmetric case, the optimal multi-dimensional mechanism (a∗, a∗) has the following properties:
i. a∗ and a∗ are continuous functions given by
a∗(θ1) = θ1+ ˙ U1∗(θ1) 2 − ∆θ2 3 U1∗(θ1) U∗ 1(θ1) + µ∗ (16) and a∗(θ1) = θ1+ ˙ U1∗(θ1) 2 + ∆θ2 3 U1∗(θ1) U∗ 1(θ1) + µ∗ (17) with U1∗ and µ∗ as below, and such that a∗(θ1) < a∗(θ1) for all θ1 ∈ [0, 1].
ii. U1∗ is an everywhere strictly negative continuously differentiable function that is strictly increasing over [0, ˆθ2), strictly decreasing over (ˆθ2, 1], with ˙U1∗(ˆθ2) = 0 that is symmetric
U1∗(θ1) = U1∗(1 − θ1) for all θ1 ∈ [0, 1]
with
˙
U1∗(θ1) = − ˙U1∗(1 − θ1) for all θ1 ∈ [0, 1]
and that satisfies the differential equations
˙ U1(θ1) = 2 s −U1(θ1) − ∆θ2 2 9 U1(θ1) U1(θ1) + µ∗ 2 over [0, ˆθ2] and ˙ U1(θ1) = −2 s −U1(θ1) − ∆θ2 2 9 U1(θ1) U1(θ1) + µ∗ 2
over [ˆθ2, 1] for some µ∗ ∈
−∆θ22 36 , 0 . iii. It is anti-symmetric: a∗(θ1) = 1 − a∗(1 − θ1) for all θ1 ∈ [0, 1].
iv. The average action is continuous and strictly increasing everywhere:
˙a∗(θ1) > 0 a.e.
has ˆθ2 as a fixed point
a∗(ˆθ2) = ˆθ2
and is such that
a∗(θ1) 6= θ1, if θ1 6= ˆθ2.
v. The variance is a continuously differentiable function that is strictly positive everywhere:
and such that ˙σ∗(θ1) < 0, if θ1 < ˆθ2 = 0, if θ1 = ˆθ2 > 0, if θ1 > ˆθ2 .
A number of conclusions can be drawn from the properties stated in the theorem above. The main one is the fact that the average action is continuous and strictly increasing and yet coincides with the desired action of agent one on a single point only - the one which, on average, both agents actually agree on which action should be implemented. This is in stark contrast with the one-dimensional case in which, as previously discussed in Lemma 2 (ii.), must delegate full power to the agent whenever the mechanism is continuous and strictly increasing. This means that the principal is capable of exercising greater control over the agents thanks to the fact that each agent is uncertain about which action will be implemented since it depends on the decisions of both agents.
A second important observation is the fact the there are no pooling regions. The mech-anism is capable of implementing different actions for different states of nature. This is another consequence of the greater capacity of screening the agents that the principal has due to the uncertainty that the agents are subject to because of each other. Thus, the multi-dimensional mechanism is more refined than one-multi-dimensional ones, assuming more complex shapes that can account for a greater amount of contingency.
Given that the variance is the instrument that allows the principal to exercise control over the uncertainty that the agents face, it is to be expected that she will make good use of it so as to exploit its screening effects. This is indeed the case, since the variance is strictly positive everywhere. It is also smaller when the type of agent one is close to the average type of agent two, which is when the agents’ incentives are relatively more aligned on average, making coordination easier to achieve. Thus, the principal makes greater use of uncertainty for cases in which agents’ interests are more divergent, exploiting this additional instrument that multi-dimensionality allows for in order to ease coordination. Interestingly, this is actually the opposite of what was presented in our discussion of incentive compatibility for agent two. That is, the principal is able to satisfy the restrictions by controlling the average
action alone. The anti-symmetry of the optimal mechanism means that a∗(θ1) ≤ θ2 if and
only if a∗(1 − θ1) ≥ θ2, which means, in terms of our discussion of regions 1 to 4 presented
in equation (10), that regions 1 and 4 actually cancel each other out.
Finally, we observe that agent one can expect to have a higher payoff the closer his type is to the average type of agent two. In a sense, agent one is punished for having divergent interests from the other agent, in expected terms, and rewarded whenever their interests coincide. This is another point that emphasizes the fact that coordination is easier for cases of aligned incentives and harder when they are misaligned.
We now register a few examples. Figures 2 and 3 show, respectively, the optimal mech-anism for the cases in which δ = 1/2 and δ = 1/4. These examples show that the optimal mechanism combines characteristics of the one-dimensional mechanism with the new pos-sibilities opened up by multi-dimensionality. Indeed, notice that a and a put boundaries on what actions may be implemented. This is particularly pronounced in the case when δ = 1/2. a is such that only actions not too far from θ2 = 0 may be implemented and the opposite happens for a, only actions not too far from θ2 = 1 may be implemented.
This is reminiscent of the role played by the boundaries of the delegation interval in the one-dimensional model: the implemented outcome is not allowed to diverge too far from the type of agent two. When δ becomes smaller and the types of agent two become closer to ˆ
θ2 = 1/2, the boundaries are put both upwards and downwards, again in a similar fashion
to the one-dimensional mechanism. On the other hand, the novel characteristics created by the multi-dimensionality of the mechanism are expressed in the distance between a and a that the principal uses to control the variance, and thus the uncertainty, and the curvature that a, a and a exhibit, which allows for more refined screening and no pooling intervals.
Figure 4 shows the optimal mechanism when δ = 1/100. Notice that, as expected, when δ → 0, the optimal mechanism converges to the one-dimensional mechanism when agent two is known to be of type ˆθ2 = 1/2. Compare with Figure 1.
Our final step in this section is to investigate the comparative statics of ∆θ2 over the
principal’s loss. That is, is it to the advantage of the principal to have the types of agent one closer or farther from each other? This is not immediately clear. On the one hand, if θ2 and θ2 are far apart, it creates greater uncertainty for agent one and thus gives the principal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 2: Optimal multi-dimensional mechanism in the symmetric case when δ = 1/2.
more room to use the screening possibilities that this creates. On the other, however, using these screening devices is actually costly for the principal. Any uncertainty that she creates for agent one also partially affects her own payoff due to her interest in the joint welfare of both agents. If the cost of screening out-weights its benefits, it is then better for the principal to have the types of agent two closer to its average. In some way this is to be expected. The entire motivation of the coordination problem comes from the fact that each agent has private information of their own. With θ2 and θ2 being close to a known average,
part of the principal’s lack of knowledge is compensated, leading to a better outcome from her perspective. Our next result shows that this is indeed the case.
First, we introduce the principal’s loss as a function of ∆θ2:10
10We have actually reintroduced a term that we have ignored thus far, because, even though it was
irrelevant for obtaining the optimal mechanism, it depends on the spread and thus affects the principal’s payoff and our comparative statics.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 3: Optimal multi-dimensional mechanism in the symmetric case when δ = 1/4.
L (∆θ2) = Z 1 0 Λ∆θ2(θ1, U ∗ 1(θ1), ˙U1∗(θ1)) + ∆θ2 2 8 dθ1
in which we have reminded the reader that the Lagrangian depends on ∆θ2. Using this
function and our results so far we can prove the following.
Proposition 4. The principal’s expected loss L (∆θ2) is a strictly increasing function of
∆θ2.
This result highlights one of the major differences between our model and one in Koessler and Martimort (2012). More specifically, it puts the spotlight on the differences generated by whether multi-dimensionality comes from multiple actions or multiple agents. In their model, a greater spread between the agent’s biases towards each action had a positive effect for the principal. The gains coming from greater screening possibilities out-weight the losses from the cost of such screening. In ours, the opposite happens, the necessity for greater screening leads to a loss to the principal that is not compensated by better information
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 4: Optimal multi-dimensional mechanism in the symmetric case when δ = 1/100.
revelation.
6. Implementation and Information in Organizations
In this section we discuss the more practical consequences of our results. A strength of the one-dimensional model of delegation is its rather straightforward interpretation: the principal may implement the optimal mechanism via a delegation set, i.e., by letting the agent freely choose among a constrained set of possible actions. This simple and elegant implementation is actually often observed in many organizational settings. In our multi-dimensional setting, in which one action is being jointly delegated to two agents, however, it is much more complicated to formulate some kind of direct implementation that can be interpreted as having practical applicability. Therefore, in what follows we give a first attempt at surmounting these difficulties and presenting a more practical implementation of the optimal mechanism seen in our theorem.
be a useful first step in analyzing implementation possibilities. First, straightforward calcu-lations lead to the following.
Remark 2. In the symmetric case, the simple mechanism a([0, 1]) = 2θ2 3 , 2θ2+ 1 3 = 1 − 2δ 3 , 2 − 2δ 3 and a([0, 1]) = 2θ2 3 , 2θ2+ 1 3 = 1 + 2δ 3 , 2 + 2δ 3
is incentive compatible for agent two if and only if δ ≥ 1/10.
We thus assume δ ≥ 1/10 in what follows. We also observe that if δ is sufficiently small, the principal has the reasonable option of using the one-dimensional mechanism for the average type of agent two:
a([0, 1]) = a([0, 1]) = " 2ˆθ2 3 , 2ˆθ2+ 1 3 # = 1 3, 2 3
which by our previous discussion actually approximates the optimal mechanism in general. Observe that these simple mechanisms can be implemented in a two-stage setup as follows. Agent two first chooses one of the two intervals, a([0, 1]) or a([0, 1]). Then, agent one is left to choose freely among any actions belonging to the interval chosen by agent two. That is, the decision on which action should be taken eventually rests with agent one. However, agent two first exercises a preliminary control over the other agent’s possible actions. In a sense, this is an expansion from our discussion about the one-dimensional mechanism. Remember that, in the one-dimensional case, the boundaries put on the delegation interval were interpreted as institutional protection that the principal offers agent two against agent one’s discretion. Despite not actually participating in the mechanism, agent two could rest assured that agent one’s decision would not lie too far from his own type. A similar logic applies here, but now that agent two is given greater agency, he may choose the limitations that best suit him by choosing the delegation interval that is best for his type.
Since the simple mechanism is incentive compatible for agent two from our assumption, he will choose a([0, 1]) when θ2 = θ2 and a([0, 1]) when θ2 = θ2. Since simple mechanisms
are always incentive compatible for agent one, he will behave in the manner familiar from the one-dimensional model. That is, he will choose his type whenever it is available from the intervals and will choose one of the extremes whenever his type is outside the intervals.
Notice that in this process, the type of agent two is actually revealed to agent one. When faced with the delegation interval, he knows whether agent two is of a low or high type. Thus, there is a gradual revelation of information in this implementation. Once agent two makes his choice, all uncertainty for agent one dissipates. This is why a mechanism (a, a) that is to be implemented in this way must be such that both a and a are individually incentive compatible for agent one. Needless to say, this means that this form of implementation loses half of one of the main elements of the multi-dimensional mechanism: the additional screening possibilities opened up by uncertainty from the perspective of agent one.
There is also a specific direction in which the information flows: from the agent with coarser to the one with finer information. Agent two is kept in the dark at the moment of his decision, but is forced to reveal his private information to agent one. Hence, this implementation induces a distribution of information that is more favorable to agent one. He’s not only given greater control over the final outcome, but is also given full information at the time of his decision. Any implementation that leaves the final direct choice to agent one must be such that these two elements are present, the decision rights and informational advantage. This is because agent one has finer information and, therefore, the choices that are presented to him must reflect this, leading to delegation sets that allow for a lot of discretion, even if this discretion is limited according to agent two’s choice.
Given that simple mechanisms are never optimal, it is no surprise that this implemen-tation can be improved. Therefore, we now consider the implemenimplemen-tation of the optimal mechanism and explore the possibilities of implementing it.
Consider, then, an optimal mechanism (a∗, a∗) and consider the following two-stage im-plementation. Agent one is presented with a menu of choices
({a∗(θ1), a∗(θ1)})θ1∈[0,1]
with each choice being a pair of possible actions and thus outcomes. The actual implemented outcome, however, is chosen by agent two. That is, agent one first chooses a pair among the possible ones presented by the optimal mechanism. Once this choice is made, agent two must pick one of the two actions in the pair chosen by agent one. We first compare this with the implementation of simple mechanisms presented above and then discuss its feasibility.
The main difference between this implementation and the one we previously discussed is the order in which agents are asked to make their decision. Whereas before agent two made his choice first and agent one second, now agent one is the first one to choose and then agent two is the one to make the final choice.
In this process, information now flows from the agent with finer information to the one with coarser information. It is now agent two who has complete information about the state of nature at the moment of his decision and who actually implements the final outcome. In contrast with the implementation of simple mechanisms, this method of implementing the optimal mechanism provides a greater balance between informational refinement and decision power. Agent one is given a greater set of options to choose from, as must be the case since his information is infinitely refined - he must be given more options to correctly apply his knowledge. However, his uncertainty about the consequences of his actions are now left intact. Agent two is given the ultimate decision power, but since his information is coarse, his set of options is much simpler. Because the privileges and powers of each agent are better balanced in this way, it is actually possible to implement the optimal mechanism, which leads to greater coordination.
As far as feasibility goes, that is, whether this form of implementation will lead to both agents truthfully revealing their private information in their choices, notice that we only need to worry about agent two. Indeed, from the perspective of agent one, this form of imple-mentation is actually identical to the traditional simultaneous direct impleimple-mentation from mechanism design. He is asked to choose a value ˜θ1 ∈ [0, 1], which maps to {a∗(˜θ1), a∗(˜θ1)},
but is left uncertain whether the actual final outcome will be a∗(˜θ1) or a∗(˜θ1), since this
depends on agent two. Because the optimal mechanism is constructed to be incentive com-patible for both agents, this means that this implementation is incentive comcom-patible for agent one. Things, however, are rather more delicate with regards to agent two. At the moment that he is asked to make a decision he becomes aware of what type agent one is based on the set of choices {a∗(θ1), a∗(θ1)} that is presented to him. That is, he acquires full
information. Therefore, all the effects of uncertainty that were used to create the optimal mechanisms are undone for agent two by this form of implementation, just as was the case for agent one with the implementation of simple mechanisms. It might be possible, thus, for
this form of implementation to not be incentive compatible for agent two, something that we now investigate.
Incentive compatibility for agent two for this form of implementation is given by the conditions: (a∗(θ1) − θ2) 2 ≤ (a∗(θ1) − θ2) 2 ∀ θ1 ∈ [0, 1] and a∗(θ1) − θ2 2 ≤ a∗(θ1) − θ2 2 ∀ θ1 ∈ [0, 1].
Naturally, these are far more stringent conditions than the ones that were considered when obtaining the optimal mechanism. It might thus be expected that this form of imple-mentation will simply not be feasible. Perhaps surprisingly, however, this is not the case. For some cases this implementation is actually incentive compatible for both agents. This might be related to the fact that, in the symmetric case, because of anti-symmetry, the incentive compatibility restrictions agent two are trivialized and are thus satisfied with significant slackness.
Notice first that any pair {a∗(θ1), a∗(θ1)} such that
θ2 ≤ a∗(θ1) ≤ a∗(θ1) ≤ θ2
is incentive compatible, since agent two has simply no reason to choose an even higher (respectively lower) outcome when he is of a low (respectively high) type. Therefore,
θ2 ≤ a∗(θ1) ≤ a∗(θ1) ≤ θ2 ∀ θ1 ∈ [0, 1]
implies incentive compatibility. This is the case for δ = 1/2 (see Figure 2). Our final result shows that this is the case around 1/2.
Proposition 5. There exists an ε > 0 such that the optimal mechanism may be implemented by the two-stage implementation described above if δ ∈ (1/2 − ε, 1/2].
It is useful to contrast Propositions 4 and 5. Together they imply that on the one hand, having the possible types of agent two closer to each other gives a higher payoff to the principal and on the other, having them far apart facilitates implementation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Figure 5: Incentive compatibility for agent two for the two-step implementation when δ = 1/5.
Figures 5 and 6 show the graphs of the functions
(a∗(θ1) − θ2) 2 − (a∗(θ1) − θ2) 2 and a∗(θ1) − θ2 2 − a∗(θ1) − θ2 2
for δ = 1/5 and δ = 1/8, respectively. The first one is shown as a dashed line and the second as a dashed-dotted line. There’s incentive compatibility if both graphs are above the x-axis. Note that incentive compatibility breaks down precisely for values of θ1 nearer to the
extremes of the [0, 1] interval. This is when agent one’s type is far from the average type of agent two and thus, as we have argued before, coordination becomes more difficult and the principal needs to make greater use of her instruments to induce it. When δ is sufficiently small, the types of agent two are close to the average and the fact that agent one’s type is far away from this average has greater weight. Since this form of implementation does not
fully utilize all the instruments that at the principal’s disposal - namely the uncertainty of agent two about agent one - some kind of problem was to be expected. Nevertheless, even for cases in which total incentive compatibility breaks down, it is still possible to have partial compatibility. For values of θ1 for which both graphs are above the x-axis, the principal is free
to reveal agent one’s decision to agent two and follow through with the implementation. It is worth noting that this happens for values in which there is greater alignment of incentives for both agents on average, that is, when θ1 is close to ˆθ2.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.02 0 0.02 0.04 0.06 0.08 0.1
Figure 6: Incentive compatibility for agent two for the two-step implementation when δ = 1/8.
Our discussion so far suggests that there are ways in which information should flow and be distributed inside organizations when trying to coordinate multiple agents via a collective choice that is to be implemented through successive individual decisions. Agents with more specialized knowledge should make their own decisions earlier in the process while agents with less specialized decisions should come later. Additionally, these results have consequences for informational policy inside organizations. Since the uncertainty is what allows the principal to implement the optimal mechanism, it follows that in some cases it might be beneficial for
an organization to not fully inform its constituents on the decisions and activities of each other, at least temporarily. In particular, a policy of greater openness may be pursued in situations in which incentives are more aligned. That is, if the interests of some parts of an organization diverge starkly, it might be optimal to keep them informationally separated, while parts with shared interests may be allowed to also share information. That more aligned incentives facilitate coordination is obvious enough, but the results that we have presented in obtaining the optimal mechanism and our discussion on implementation reveal an informational aspect to the problem that ties the issue of coordination to the more traditional themes of Information Economics.
7. Concluding Remarks and Future Developments
We have investigated what mechanisms are capable of coordinating multiple agents in an organization when each one of them privately holds information that has an impact on the outcome of a decision. This has been studied in the context of a multi-dimensional model of delegation. We have solved a class of cases when one agent has fine information while the other has coarse information. The optimal mechanism exploits the uncertainty that each agent has about the other’s information. This is more accentuated when the agents have more divergent interests and less when their interests coincide. The shape of the optimal mechanism is more complex than the one for one-dimensional delegation: there is no pooling and no intervals in which full delegation occurs. The use of the additional instruments has been shown to be costly to the principal, so that forcing her to make greater use of them leads to greater losses.
We have discussed the possibilities of implementing the optimal mechanism we have obtained, leading to a discussion on the way that information and decisions rights in orga-nizations are interrelated. In particular, we have shown that a better balancing of decision rights and informational privilege among the constituent parts of an organization can lead to better coordination. We have also shown a connection between greater alignment of interests and information sharing. When subdivisions of an organization have convergent interests, they may be allowed to share information without introducing great obstacles to coordi-nation. Coordinating subdivisions with more sharply diverging interests, however, requires
keeping them informationally separated.
Despite our results, there remains a lot of work to be done in the model of delegated coordination. The asymmetric case is still unsolved and we now briefly explain our next steps in attempting to solve it. As we have shown, at most one of the incentive compatibility restrictions for agent two will be binding. We have also shown, however, that these restric-tions are trivialized in the symmetric case. Therefore, we can expect them to be satisfied with considerable slackness. Barring any abrupt discontinuities in the solutions, it is to be expected that introducing a small amount of asymmetry will not suddenly make one of the restrictions binding. It may thus be possible to continue to solve the model ignoring the restrictions for a larger class of cases then we considered thus far. Pushing asymmetry to the limit will also allow us to study which restriction becomes binding, which is a crucial step in facilitating the process of obtaining the solution for the asymmetric case.
Beyond the case when agent two has two types, we may begin to investigate the cases when his type is one among a finite number of them. We believe that our results for the cases of two types, both the symmetric and asymmetric, will be instrumental in analyzing these new cases. If we take the possible types of agent two to be symmetrically dispersed around the [0, 1] interval, we know that each type will have a symmetric opposite, which we know will create no problems in terms of incentive compatibility. For the other types, we expect that incentive compatibility will only matter in one direction, which is to be revealed by our future study of the asymmetric case. Ultimately, naturally, our objective is to solve the general model of delegated coordination in which both agents types lie in a continuum. Another possible way of expanding our model in future research is to consider the types of agents one and two as being distributed according to other distributions than the uniform one. This extension might be seen as a change of variables in the preferred actions of the agents. That is, it might be subsumed into the case in which agents one and two want to implement the actions r1(θ1) and r2(θ2) respectively instead of simply matching the action
to their types.
In a recent commentary, Holmstr¨om (2016, p. 20) urges economists to study the internal organization of firms. He argues that
strategic objectives. Firms frequently restructure themselves, not just by moving people around but also by changing reporting structures, job descriptions, and remarkably the whole internal architecture of the firm. In all cases, decision rights are at the center: they are being regrouped and reallocated to gain more effective incentive and communication structures, much like an army regroups its troops in response to changing objectives.
Notwithstanding the fact that his comments were made in relation to the theory of Property Rights and Incomplete Contracts, we actually believe that our model of delegated coordina-tion helps bringing to light many of the phenomena described in them. As such, we believe that continued work on this model is important in creating a better understanding of how firms in particular but also organizations in general, shape themselves in order to coordinate their constituent parts so as to protect the collective interests that originally lead to their formation.
Appendix A.
Proof of Lemma 1. See Kov´ac and Mylovanov (2009, Lemma 1). Proof of Lemma 2. See Alonso and Matouschek (2008, Lemma 2).
Proof of Remark 1. In the context of Kov´ac and Mylovanov (2009, p. 1380), it is straightforward to check that we have
g(θ1) = 1 + ˆ θ2 2 − 3 2θ1
which is non-decreasing everywhere, thus satisfying the desired condition.
Proof of Proposition 1. Following Kov´ac and Mylovanov (2009, p. 1383), we have
G(θ1) = Z θ1 0 g(˜θ1) d˜θ1 = − 3 4θ 2 1 + θ1 1 + ˆ θ2 2 ! . Since G(1) = 1/4 + ˆθ2/2, simple calculations yield
{θ1 ∈ [0, 1] : G(θ1) ≥ θ1} = " 0,2ˆθ2 3 # and {θ1 ∈ [0, 1] : G(θ1) ≥ G(1)} = " 2ˆθ2+ 1 3 , 1 # . So that we have α0 = max{θ1 ∈ [0, 1] : G(θ1) ≥ θ1} = 2ˆθ2 3 and β0 = min{θ1 ∈ [0, 1] : G(θ1) ≥ G(1)} = 2ˆθ2+ 1 3 .
Noticing that α0 < β0, applying Kov´ac and Mylovanov (2009, Proposition 1) gives us the
optimal mechanism presented.
Proof of Lemma 3. We need only note that u1 and Θ1 have all the required properties
required to apply the results from Milgrom and Segal (2002).
Proof of Lemma 4. Noting that θ2 = ˆθ2− ∆θ2/2 while θ2 = ˆθ2 + ∆θ2/2, we can rewrite
equations (8) and (9) respectively as Z 1 0 ˆθ 2− a(θ1) p σ(θ1) dθ1 ≤ Z 1 0 ∆θ2 2 p σ(θ1) dθ1 and Z 1 0 a(θ1) − ˆθ2 p σ(θ1) dθ1 ≤ Z 1 0 ∆θ2 2 p σ(θ1) dθ1.
Or we may put them together as follows:
− Z 1 0 ∆θ2 2 p σ(θ1) dθ1 ≤ Z 1 0 a(θ1) − ˆθ2 p σ(θ1) dθ1 ≤ Z 1 0 ∆θ2 2 p σ(θ1) dθ1.
− Z 1 0 ∆θ2 2 p σ(θ1)dθ1 < Z 1 0 ∆θ2 2 p σ(θ1)dθ1
we may then conclude that at most one of the weak inequalities presented in the incentive compatibility restrictions for agent two can be an equality.
Proof of Lemma 5. We show this for one of the restrictions. The other is analogous. Note that the restriction for agent two when he is of type θ2 may be written as
Z 1 0 a(θ1)2 − a(θ1)2 dθ1 ≥ Z 1 0 2θ2(a(θ1) − a(θ1)) dθ1.
Now using that, in the symmetric case, θ2 = 1/2 − δ, we write it as
Z 1 0 a2(θ1) − a(θ1) − a2(θ1) − a(θ1) dθ1 ≥ Z 1 0 −2δ (a(θ1) − a(θ1)) dθ1.
From this we use the anti-symmetry property and a change of variables to obtain the result.
Proof of Proposition 2. We first prove existence. Our objective is to use Clarke (1990, Theorem 4.1.3). Our method of proof is a close adaptation of the one for Koessler and Martimort (2012, Lemma 3). From our Lagrangian
Λ(t, s, v) = −s +t − ˆθ2 2 v − ∆θ2 2 r −s − v 2 4 we define the extended-value Lagrangian
L(t, s, v) = Λ(t, s, v), if s ≤ −v42 +∞, if s > −v42 .
Thanks to the continuity in all its variables and convexity of Λ in (s, v), L is lower semicontinuous in all its variables and also convex in (s, v), easily satisfying the required conditions in Clarke (1990, p. 167), so we have that
i. L is L × B measurable.11
ii. For each t ∈ [0, 1], the function L(t, ·, ·) is lower semicontinuous. iii. For each (t, s), the function L(t, s, ·) is convex.
Our Hamiltonian is then defined over [0, 1] × R × R as
H(t, s, p) = sup v∈R {pv − L(t, s, v)} = sup v∈R ( pv + s + ˆ θ2− t 2 v + ∆θ2 2 r −s − v 2 4 ) .
Now if s ≤ −v2/4, the supremum is achieved for
v = 8 p + ˆ θ2− t 2 ! v u u t −s 16p + θˆ2 2 2 + ∆θ2 2 . So that we have H(t, s, p) = s + s −s 2 16p + θˆ2−t 2 2 + ∆θ2 2 , if s ≤ 0 −∞, if s > 0 .
The subadditivity of the square root gives us that for all (s, p)
H(t, s, p) ≤ |s| 2 + ∆θ2 p|s| 2 + 4 p + θˆ2− t 2 p|s| 2
and using the fact that p|s| ≤ 1 + |s|/2 and the triangle inequality leads us to
H(t, s, p) ≤ ∆θ2+ 4|p| + 2|ˆθ2 − t| + |s|
1 + ∆θ2
2 + 2|p| + |ˆθ2− t|
so that defining the continuous and thus summable functions
µ(t, p) = ∆θ2+ 4|p| + 2|ˆθ2− t|
and
11This means that it is measurable with respect to the σ-algebra generated by products of Lebesgue
