Optimal Delegation with Information Manipulation
Yichuan Lou∗Feb 28, 2021

Abstract
A principal delegates a decision to an agent, who has the capacity to process the relevant
information. The principal cannot process information herself, but can jointly control
the actions and the information available to the agent. I provide sufficient conditions
under which the optimal mechanism (i) attains a perfect alignment of incentives—
subject to the constraints on discretion and information, the agent plays exactly as
the principal would want him to; and (ii) belongs to a simple class of delegation anddisclosure rules, called monotone partitional rules, which specify a finite set of allowableactions from which the agent chooses his preferred one, and moreover, partition thestate space into finite intervals so that the agent only learns to which interval thetrue state belongs. I then turn to the uniform-quadratic case which permits an explicit
characterization of optimal mechanisms and clear comparative statics. Finally, I discuss
two applications: the regulation of a monopolist, and the self-control of a dynamicallyinconsistent individual.

∗

Department of Economics, The University of Tokyo, 7-3-1 Hongo, Tokyo, 113-0033, Japan. E-mail:yichuanlou@e.u-tokyo.ac.jp. For continual guidance, I am grateful to my advisor Alessandro Lizzeri
and my committee members Erik Madsen and Ennio Stachetti. For helpful comments and suggestions, I
thank Dilip Abreu, Sylvain Chassang, Boyan Jovanovic, Sam Kapon, Xiaoye Liao, Laurent Mathevet, David
Pearce, Jacopo Perego, Debraj Ray, and Ariel Rubinstein.

1

Introduction

Consider a principal who grants decision-making authority to an agent who is potentiallybetter informed but has different objectives. Previous work has explored two ways for the
principal to influence the agent’s behavior in order to improve decision-making in this environment. The first way is by specifying a permissible set of actions (a delegation set) fromwhich the agent may choose, which has been analyzed as a delegation problem (Holmström,
1977, 1984). The second way is by limiting the access to information to the agent, which
has been analyzed as a persuasion problem (Kamenica and Gentzkow, 2011). In this paper,
I combine these two ways of influencing behavior and ask how to simultaneously constrain
discretion and information of the agent. The main result identifies a strong form of incentive
compatibility: the principal’s and the agent’s incentives are fully aligned under the optimal
joint restriction on discretion and information. In addition, the optimal mechanism belongs
to a simple class of delegation and disclosure rules, called monotone partitional rules, thatfeature a finite set of actions and a monotone partitional information structure. Consequently, the agent only learns which partition element the true state lies in and then chooses
from the delegation set an action that best serves his interests.
There are many environments that share the modeling features described above. For
instance, a monopolist may collect and use consumer behavior data to segment the market
and to make targeted offers. To prevent excessive discrimination and to protect consumer
welfare, a regulator can adopt policy tools such as price control which limits the prices thatcan be charged, and data protection which limits the use of consumer data for discrimination. Similarly, in jury trial procedures, judges enforce detailed rules on what evidence
can and cannot be presented to jurors, and additionally jurors are delegated a particular
set of decisions they can make (e.g., the set of charges they can decide on). Finally, in thecontext of congressional oversight, congressional committees have the power only to inquire
into matters and to make decisions within the scope of the authority delegated to them by2their parent bodies. In all these settings, a principal delegates a permissible set of actions toan agent and additionally designs what information the agent has access to. The purpose ofthis paper is to provide an in-depth analysis for the joint control of actions and information.
I develop a stylized model in which a principal (she) relies on an agent (he) to make an
informed decision and cannot use contingent transfers.1 I begin my analysis by assumingthat the principal can jointly commit to a delegation set and an information structure. The
delegation set is a set of permissible actions from which the agent can freely choose. The
information structure is modeled as a Blackwell experiment similar to that of Kamenica and
Gentzkow (2011): a probability distribution over signal realizations as a function of the stateof nature. After privately observing a signal realization, the agent then takes an action fromthe delegation set. I focus on the following question: what is an optimal mechanism as a
combination of delegation and disclosure rules from the principal’s perspective?
I begin by establishing two important results that greatly simplify the analysis. First,
I show that the optimal mechanism is characterized by an alignment principle—subject tothe constraints on discretion and information, the agent plays as if he shared the principal’s
preferences. More specifically, if the signal realization were publicly observed, the two parties would choose the same action from the delegation set. In fact, they may disagree about
their most preferred actions, but given the well-adjusted noise of the signal and the moderatesparsity of the delegation set, both agree upon how to play optimally. An immediate implication is that the optimal delegation set must be finite, as the alignment property requires
the implemented actions to be isolated far away from each other (relative to the preference
divergence of the two parties).
Second, I show that the optimal mechanism features a finite delegation set and a monotone partitional information structure. Although the principal’s choice of signals is not
restricted a priori, in the optimal mechanism she partitions the state space into a finite disjoint union of intervals and pools each interval into a single signal realization. The principal,
1

There are many situations in practice, like those described above, that transfers are infeasible.

3

in effect, introduces noise into the signal by letting the agent observe only which partition
element the true state actually lies in. This feature greatly simplifies the characterization of
the optimal mechanism by allowing me to focus on a finite-dimensional optimization problem
over the cutoffs of intervals and the associated delegated actions.
Then, I explicitly characterize optimal mechanisms in a leading case of the model—the socalled uniform-quadratic case. This allows me to explicitly solve for the optimal mechanism
and in turn to perform a comparative analysis of the optimal discretion and information
bestowed on the agent. I show that the principal would always provide more discretion andmore information to a less-biased agent. Also, the principal necessarily obtains higher welfare
if the agent is less biased. In Section 6, I further investigate to what extent the comparative
statics results that hold in the leading case can be generalized beyond the uniform-quadratic
specification.
I next apply my results to two specific economic settings. First, I study the regulation of amonopolist who engages in both quality and price discrimination. In this context, a welfaremaximizing regulator jointly restricts prices (i.e., price control) and consumer information(i.e., data protection) available to a profit-maximizing monopolist. The information about
consumer characteristics can be used to offer different quality levels and prices to different
segments of the market. I provide conditions under which the optimal joint regulatory policy
takes the form of a finite set of prices and a finite market segmentation. Specifically, theaggregate market is divided into finite segments in such a way that in each segment, themonopolist freely sets a quality level and chooses a price (from the delegated set of prices)
so as to maximize his profits. The optimality of finite market segmentations implies that
consumer privacy should be moderately preserved.
Next, I apply my results to a self-control problem. In the primary environment I study,
the principal and the agent represent distinct individuals with conflict of interests. However,
my model also captures a behavioral setup in which the principal and the agent represent
the same individual at two points in time, who overweighs instantaneous rewards relative4

to future payoffs (e.g., hyperbolic discounting). As is well known, sophisticated, yet timeinconsistent individuals value commitment power (Strotz, 1995). Within a consumptionexternality environment (Carrillo and Mariotti, 2000), I suppose that the current self possesses not only hard commitment power in that she can restrict the consumption choices
available to the future self, but also soft commitment power in that she can restrict information (about the extent of the externality) available to the future self. I establish that people
may choose to tie their hands by reducing the set of available consumption choices and toremain partially ignorant about the magnitude of the externality.

2

Related Literature

Since the paper studies the joint design of delegation and disclosure rules, it brings together
the literatures on optimal delegation and Bayesian persuasion. The delegation literature
begins with Holmström (1977, 1984), who considers a setting in which a principal faces a
privately informed but biased agent and in which contingent transfers are infeasible. The
agent is then given discretion over actions (in the form of a set of permissible actions)because his private information is valuable to the principal; the principal limits the degreeof discretion because of preference misalignment. A key goal in this literature has been to
identify sufficient conditions under which the principal optimally defines the permissible set
as an interval. Melumad and Shibano (1991) solve the optimal delegation problem withquadratic preferences and a uniform distribution for the state. Following that, Martimort
and Semenov (2006), Alonso and Matouschek (2008), and Amador and Bagwell (2013) extend
the analysis by allowing for more general preferences and distributions, and provide optimal
conditions for interval delegation. Most of these studies consider the agent’s information
as exogenous, whereas my work contributes by considering the information structure as an
endogenous component. I am able to explore how the qualitative features of the optimaldelegation set are altered in the presence of information manipulation.

5

Also, my work is related to the literature on Bayesian persuasion (Rayo and Segal 2010;
Kamenica and Gentzkow 2011), where a sender chooses a signal and a receiver takes anaction after observing a realization from this signal. The choice of a signal is isomorphic
to my modeling of a disclosure rule, although the interpretation is different: my principal
controls the information but does not observe the realization. Within this literature, several
papers have also identified sufficient conditions for the optimality of monotone partitionalsignals in various settings; see, for example, Kolotilin (2018), Dworczak and Martini (2019),
Ivanov (2021), and Mensch (2021).
The behavioral formulation captured by my model relates this paper to the literatureon optimal provision of commitment devices for individuals with low self-control due totime-inconsistent preferences. Amador, Werning, and Angeletos (2006) study the problem
of commitment through the restrictions on actions. In the context of a consumption-savings
problem, they find that imposing a minimum level of savings (i.e., interval delegation) isalways a feature of the optimal solution. On the other hand, Carrillo and Mariotti (2000)
and Bénabou and Tirole (2002) study the problem of commitment through the manipulationof information. Both papers show that, in the presence of time inconsistency, Blackwellgarbling of future self’s information may increase the current self’s payoff. My paper thus
provides the first study of optimal joint provision of hard and soft commitment devices inself-control problems.

3

Model

3.1

Environment

There are two parties in the model on which I focus: a principal (she) and an agent (he).
The principal has a payoff function uP (y, θ) = θy − CP (y), while the agent has a payoff

6

function uA (y, θ) = θy − CA (y).2 The value of y represents an action, and the value of θ
represents a state.
I assume that θ has an absolutely continuous distribution µ0 with bounded supportΘ ≡ [θ, θ] ⊂ R. The action y is chosen from a large compact interval Y ⊂ R. For theremaining of the paper, I impose the following conditions on the primitives:
Assumption 1. The following hold:(i) Both the functions CP : Y → R and CA : Y → R are continuously differentiable andstrictly convex in y;(ii) The functions cP ≡ CP′ and cA ≡ CA′ satisfy cP > cA .
Condition (i) in Assumption 1 ensures that the principal’s preferred action yP (θ) ≡ c−1
P (θ)
and the agent’s preferred action yA (θ) ≡ c−1
A (θ) are both continuous and strictly increasingin θ ∈ Θ. Condition (ii) in Assumption 1 guarantees that the agent is always biased towards
higher actions relative to the principal. Note that for each θ ∈ Θ, the principal’s and theagent’s payoff functions, uP (·, θ) and uA (·, θ), are single-peaked, respectively, around yP (θ)
and yA (θ).
As I focus on decision-making with incomplete information, it is useful to identify theplayers’ preferences over actions when the state is unknown. Let µ be a posterior belief andm = Eµ [θ] be the associated posterior mean. Since uP and uA are linear in the state θ,both players’ expected payoffs depend on µ only through m. With slight abuse of notation,
I will write uP (y, m) = my − CP (y) and uA (y, m) = my − CA (y) as the principal’s and theagent’s expected payoffs given y and m, respectively. Thus, given the posterior mean m, the−1principal’s preferred action is yP (m) = c−1
P (m) and the agent’s is yA (m) = cA (m).

An interpretation. The model can be interpreted as follows. The two parties equally2

In fact, one can extend the analysis to more general payoff functions. Suppose that uP (y, θ) = B(θ)y −CP (y) + DP (θ) and uA (y, θ) = B(θ)y − CA (y) + DA (θ), where B(·) is strictly increasing. First, I can discard
DP (θ) and DA (θ) without loss of generality because these two terms do not enter into the players’ strategic
considerations. Moreover, I can simply redefine the state as ω ≡ B(θ).

7share the benefits generated by an action y for any given state θ, which is equal to θy. The
action y, however, renders different levels of cost to the principal and the agent: CP (y) andCA (y), respectively. Moreover, the agent incurs a lower marginal cost from higher actions,
i.e., cA (y) < cP (y), and is thus biased toward actions that are too high from the principal’s
perspective.

Two specifications. The set of preferences described above entails two particular specifications. The first specification is the case of quadratic preferences with a constant bias, which
is widely-studied in the literature on cheap talk and delegation (for example, Crawford andSobel 1982; Melumad and Shibano 1991; Alonso and Matouschek 2008). In this specification,22and the agent’s payoff is − (y−θ−b), where y represents thethe principal’s payoff is − (y−θ)
22action taken by the agent and b > 0 represents the agent’s positive bias. This formulation
is equivalent to letting uP (y, θ) = θy − 21 y 2 with yP (θ) = θ, and uA (y, θ) = θy − ( 21 y 2 − by)with yA (θ) = θ + b, and is therefore a special case. I explicitly solve for optimal mechanisms
in this specification in Section 5.
The second specification considers the temptation and self-control problem of consumption with time-inconsistent preferences due to hyperbolic discounting, analyzed by Carrillo
and Mariotti (2000). There is an individual with a horizon of three periods t = 0, 1, 2.
At period t = 1, before taking a consumption decision x which generates an instantaneous
utility u(x), self -1 (the agent) learns about the probability ω of a negative externality e(x)on the welfare of self -2. The welfare for self -1 from periods t = 1, 2 with externality probability ω is then u(x) − βωe(x), where 0 < β ≤ 1. In contrast, the welfare for self -0 (theprincipal) from periods t = 1, 2 is u(x) − ωe(x). The discounting parameter β reflects thesalience of the present.3 The formulation is equivalent to uP (y, θ) = θy − e(u−1 (y)) anduA (y, θ) = θy − βe(u−1 (y)) by setting θ = ω1 and y = u(x), and hence is a special case as
3This setup represents a three-period version of quasi-hyperbolic discounting where benefits precede thecost, generating an intrapersonal conflict between selves. Examples include smoking, drinking, spending
freely, etc., in which individuals tend to excessively satiate immediate gratification at the expense of future
welfare.

8

well. I consider this specification in greater detail in Section 8.

3.2

The game

Prior to the realization of θ ∈ Θ, the principal can jointly commit herself to a delegation
set and an information structure. The delegation set is denoted by D, which can be either
countable, finite or infinite, or uncountable. The information structure σ consists of a signal
realization space S and a measurable mapping σ : θ → ∆(S) that assigns to each state θ aprobability distribution dσ(·|θ) over S. The cost of every information structure is identicaland set equal to zero.
The timeline of the game is as follows:(i) The principal publicly selects a delegation set D ⊂ Y and an information structure σ.
(ii) A pair (θ, s) is drawn according to the distributions µ0 and σ, respectively.
(iii) The agent privately observes s and chooses an action y ∈ D.

3.3

Obedient recommendation mechanisms

Fix a delegation set D. In general, the principal could follow any rule for generating theagent’s private information. However, a revelation principle-type argument establishes that
I can restrict attention to information structures in which signal realizations are action recommendations that will be obeyed by the agent. It is therefore immediate that, instead ofchoosing a delegation D and an information structure σ separately, the principal can without
loss of generality choose an obedient recommendation mechanism M, which consists of a delegation set D ⊂ Y , and an action recommendation function (also an information structure)σ : Θ → ∆(D), and moreover, satisfies the obedience constraint (OB). In particular, OB is
the requirement that when the agent is recommended to take action y ∈ D according to σ,

9he would follow his recommendation y rather than choose any other available action ŷ ∈ D.4
I describe the condition formally as follows.
Given a recommendation mechanism M = (D, σ) and the prior belief µ0 , every action
recommendation y ∈ D induces a posterior belief µσy and the associated posterior meanmσy = Eµσy [θ]. A recommendation mechanism M = (D, σ) is said to be obedient if
uA (y, mσy ) ≥ uA (ŷ, mσy ) ∀y, ŷ ∈ D.

The principal’s problem is thusˆ ˆ
MaximizeuP (y, θ) dσ(y|θ) dµ0 (θ)Θ

Yamong all obedient recommendation mechanisms. M is said to be optimal if it solves theabove problem.
All proofs are relegated to the Appendix.

3.4

Remarks

A key feature of my model is that the principal fully controls the quality of the agent’s private
information without learning its content. One might wonder why the principal relies on abiased agent to process the information and to make a biased choice subsequently. Why can’t
the principal simply assigns herself an information structure that informs herself perfectly
and then chooses her first-best action? After all, there is no cost to adopt a particular
information structure. To address this question, I offer three scenarios in which a principal
may not be able to become informed herself. She therefore relies on some agent to gather,process, and use the information.
In the first scenario, the principal indeed has unlimited access to information but lacks
4
The notion of obedience here is weaker than that in Bergemann and Morris (2016), which imposes noaction restrictions, and therefore the agent is free to choose any action.

10the time and resources to acquire information herself. Such a scenario fits in well with theapplication to organizations in which top management has formal authority over data accessand usage, but is often too busy to have the time to process every piece of information.
Top management thus relies on lower management to analyze data and to make informed
decisions.
In the second scenario, the principal has unlimited access to information as well but faces
technological constraints to observe the realization of information. In the earlier temptationand self-control problem of consumption, the current self can self-restrict information gathering about the unknown consumption externality, but cannot observe it herself because itis realized in the future.
In the third scenario, the principal not only has no access to information but also has nocontrol over what information may be available to the agent. Instead, the principal has theability to limit the degree of information the agent’s action can effectively condition on. In
the example of regulating a monopolist, considered in Section 7, even the regulator does notdirectly possess consumer data, she can impose limits, by law, on the monopolist’s use of
consumer data for discrimination. As a result, the monopolist might obtain excessive information on consumers himself, but by law, he is prohibited from using it for discriminatoryoffers.

4

Characterizing Optimal Mechanisms

In this section, I derive a general structure of optimal mechanisms and use it to simplify theprincipal’s problem into a finite variable optimization problem. In particular, Section 4.1
states that local OB is sufficient to imply full OB. Section 4.2 provides a sufficient condition
under which the principal finds it optimal to fully align the agent’s incentives. I then use
this insight to prove the optimality of a monotone partition in Section 4.3.
For the clarity of analysis, I assume for now that the delegation set D is finite and consists11

of N actions. I label it by {y1 , . . . , yN } ⊆ Y with yi < yi+1 . Later I will show that thisfiniteness restriction can be dropped without loss of generality. A mechanism M = (D, σ)
is called finite if D is finite.

4.1

Preliminary observations

Recall that the agent’s interim preference over actions depends solely on the posterior mean of
the state. Therefore, the single-peakedness property of uA (y, θ) with respect to y is inheritedby the interim preference. Taking advantage of this observation, I can show (Lemma 1 below)that local OB is sufficient to establish full OB.
Lemma 1. Let M = (D, σ) be a recommendation mechanism that implements N actions.
Then local OB is sufficient to imply full OB.
Lemma 2 (optimality of compromises). Let M = (D, σ) be an optimal obedient recommendation mechanism that implements N actions. Then it must satisfyyP (mi ) ≤ yi ≤ yA (mi )∀i = 1, . . . , N.

Lemma 2 says that the implemented action in the optimal mechanism is always a compromise—it lies between the principal’s preferred action and that of the agent. The logic isroughly as follows. Suppose that an action y is implemented outside the compromise region[yP (m), yA (m)] for a given posterior mean m. Then a shift of action y in the direction closer
to [yP (m), yA (m)] will strictly benefit the principal without violating OB. An implicationof Lemma 2 is that the agent’s local downward OB must be slack. This is because yi−1 isfarther to the left of the peak yA (mi ) than yi . Then single-peakedness of uA with respect toy implies that action yi is strictly preferred by type yi to action yi−1 . As a consequence, theprincipal does not need to consider such downward deviation by the agent.
This set of results from Lemma 1 and Lemma 2 allows me to focus on the relaxed programwith only the local upward OB:
12

max

ˆ "X
N

(D,σ)

Θ

#
uP (yi , θ)σ(yi |θ) dµ0 (θ)

i=1

subject touA (yi , mi ) ≥ uA (yi+1 , mi ) ∀i = 1, . . . , (N − 1).

(OBi,i+1 )An equivalent, yet convenient, way of writing (OBi,i+1 ) ismi ≤CA (yi+1 ) − CA (yi )yi+1 − yi∀i = 1, . . . , (N − 1),which says that the agent’s marginal benefit of higher actions is smaller than his averagerate of change of cost from implementing action yi to action yi+1 . Hence a type yi agent has
no incentive to deviate to take action yi+1 .

4.2

Alignment principle

In this subsection, I introduce an incentive compatibility notion of aligned mechanisms. In
an aligned mechanism, all relevant agent types play exactly as if they were maximizing theprincipal’s payoff. An immediate implication is that the principal would also find it optimal
to be obedient if she observed the mechanism’s recommendation. The formal definition ofaligned mechanisms is as follows.
Definition 1. An obedient recommendation mechanism M = (D, σ) is called aligned if, for
any type y ∈ D,y ∈ arg max uP (ŷ, my ) ∩ arg max uA (ŷ, my ).5
ŷ∈D

ŷ∈D

In an aligned mechanism, the players’ incentives are fully aligned. Given the optimal
action restriction together with the information restriction, there is no way to improve thedecision-making procedure—a reallocation of decision authority from the agent back to the5

The notion of aligned mechanism in this paper is similar to that in Frankel (2014), who considers the
scenario where a principal delegates multiple decisions to an agent.

13

principal would not affect actions that are implemented.
Due to the single-crossing property of preferences and the agent’s upward bias, the following result ensures that only local downward OB is effectively needed for the principal tobe obedient.
Lemma 3. Let M = (D, σ) be an obedient recommendation mechanism that implements N
actions. Then it is aligned if and only if

uP (yi , mi ) ≥ uP (yi−1 , mi ) ∀i = 2, . . . , N.

(4.1)

Likewise, (4.1) can be equivalently written asmi ≥CP (yi ) − CP (yi−1 )yi − yi−1∀i = 2, . . . , N.

(4.2)

P (ŷ)
Let ∆CP (y, ŷ) ≡ CP (y)−Cdenote the principal’s average rate of change of cost from actiony−ŷŷ to y. From the principal’s perspective, (4.2) thus requires that the marginal decrease inbenefit is greater than the marginal decrease in cost from action yi to yi−1 .
Let CDiff ≡ CP −CA denote the difference in costs. The next assumption provide sufficientconditions under which optimal mechanism is aligned.
Assumption 2. The following hold:(i) The agent’s cost function CA satisfies CA′′′ ≤ 0;′′(ii) The cost difference function CDiff satisfies CDiff≥ 0.

Condition (i) in Assumption 2 ensures that the agent’s preferred action yA (θ) = c−1
A (θ) is′convex for all θ ∈ [θ, θ]. Condition (ii) in Assumption 2 together with CDiff= CP′ − CA′ > 0

implies that the cost difference between Principal and Agent increases faster for higher
realizations of the state.
The following is a sufficient result:
14

Proposition 1. Let M = (D, σ) be an optimal obedient recommendation mechanism thatrecommends N actions. M is aligned if Assumption 2 holds.
Proposition 1 indicates that it is always optimal for the principal to design a mechanismthat completely align the agent’s incentives with her own and hence avoid any interim conflict. The players may disagree about their preferred actions, but given the informationaland discretionary constraints of the mechanism they agree on how to play optimally.
The optimality of aligned mechanisms turns out to be depending on the curvature ofplayers’ preferred actions. To see the intuition, suppose by contradiction that the mechanism
is not aligned. By (4.2) there must exist an action recommendation yi conditional on which
the principal strictly prefers action yi−1 over yi . In this case, the principal may wish tocoarsen the original mechanism by collapsing the two recommendations yi−1 and yi intoa single recommendation ŷ which takes a moderately small value (hence desirable to theprincipal). In turn, the modification reduces the informativeness of the mechanism andconsequently the actions implemented become less responsive to the state, but at the sametime, it mitigates the agent’s incentive to take undesirable actions (specifically in this case,yi ) and hence improves efficiency in the allocation of actions from the principal’s perspective.
′′≥ 0, I ensure that the agent’s preferred action yA (θ) = c−1
By requiring CA′′′ ≤ 0 and CDiffA (θ)

is convex in θ, and that Agent is less biased for lower states. As an implication, the agent’s
incentive to deviate to higher actions (specifically, yi+1 ) will be greatly mitigated when theposterior mean drops from mi down to m̂. By combining the two recommendations yi−1 andyi , the principal is thus able to implement an action ŷ small enough without violating OB,
which delivers a welfare improvement for the principal.
Assumption 2 will be maintained for the remainder of the paper.

4.3

Monotone partitional mechanism

In practice, information disclosure often takes the form of monotone partitional signals—
pooling, if present, is only between adjacent states. For example, schools adopt grading15

policies that contain only letter grades. Financial rating agencies also classify assets into
different categories according to their riskiness.
Formally, a recommendation mechanism M = (D, σ) is called monotone partitional if
the state space Θ = [θ, θ] is partitioned into a finite number N of intervals {(θi−1 , θi ]}N
i=1
with θ0 = θ and θN = θ, and the agent observes only the interval that contains the realizedstate θ.
One apparent benefit of a monotone partitional mechanism is that it can be identified
by two simple components: (i) the cutoffs 0 = θ0 < θ1 < · · · < θN −1 < θN = 1; (ii)the delegated actions y1 < · · · < yN . For a given realized state θ, a monotone partitionalNmechanism M = {(θi )Ni=1 , (yi )i=1 } recommends the agent to take action yi with probability

1 if θ ∈ (θi−1 , θi ].
The following two theorems play an important role in my analysis. They establish that
it is without loss of generality in focusing on mechanisms that partition the state space
into a finite set of intervals and recommend different actions in different intervals. Thesefeatures greatly simplify the characterization of the optimal mechanism by reducing it to a
finite dimensional optimization problem over the cutoffs of intervals and the corresponding
actions.
Theorem 1. For any obedient recommendation mechanism that implements N actions, there
exists another obedient recommendation mechanism M = (D, σ), such that M is a monotone
partition and entails a weakly higher payoff for the principal.
The result that the optimal mechanism (in the class of finite mechanisms) is a monotone
partition stems from the fact that both parties in my setting prefers to take higher actions
for higher realizations of the state of nature. Since the two parties’ incentives are fullyaligned in the optimal mechanism, intuitively there is no reason for the principal to induce“mismatched pairs” of states and actions.
I have so far restricted the mechanism to be finite. Below, I augment the space of
mechanisms to allow for information structures with an arbitrary number of realizations, and16

show that an optimal mechanism exists and it is finite. Using this together with Theorem
1, I can now restrict attention to the class of finite monotone parititional mechanisms.
Theorem 2. An optimal obedient recommendation mechanism exists and it is finite.
The main idea of the proof follows from the approximation argument in Proposition 1 ofBergemann and Pesendorfer (2007), that the principal’s payoff from any given mechanismwith an arbitrary number of delegated actions can be approximated arbitrary well by a finite
mechanism. In fact, it can be shown that the principal’s set of payoffs generated by finite
mechanisms is dense in the set of payoffs generated by all arbitrary mechanisms.

17

5

Uniform-Quadratic Case

The results in the previous section tell us a great deal about optimal mechanisms but do
not fully describe them. In this section I first provide an explicit characterization of optimalmechanisms for the case of quadratic preferences mentioned in Section 3.1, under a further
assumption that µ0 is uniform on Θ = [0, 1]. I then use this result, together with the bias
parameter b, to discuss comparative statics on the agent’s discretion and information in
Section 5.2. Finally I conclude this section by analyzing the welfare effects of information
manipulation on this delegation problem in Section 5.3.

5.1

Optimal partitional structure

I now turn to the study of optimal mechanisms for the uniform-quadratic case. As noted in
Section 3.1, the quadratic formulation can be equivalently expressed as uP (y, θ) = θy − 12 y 2with yP (θ) = θ, and uA (y, θ) = θy − ( 12 y 2 − by) with yA (θ) = θ + b. Moreover, since′′= 0, Assumption 2 holds
CA (y) = 12 y 2 − by and Cdiff (y) = by ensure that CA′′′ = 0 and CDiff

here and therefore Proposition 1 and Theorem 1 apply.
The following property will play an important role in the analysis.
N
Definition 2. A monotone partitional mechanism M = {(θi )Ni=1 , (yi )i=1 } features alternating

symmetry if
θi − θi−1 = θj − θj−1

and yi − θi−1 = yj − θj−1

for both odd (even) i, j ∈ {1, 2, . . . , N }.
An illustration of this property is illustrated in Figure 1. It can be seen that all oddnumbered (even-numbered) elements are essentially “symmetric”, in the sense that they havethe same interval length, and the respective actions are located at the same positions relative
to the associated intervals. The next lemma shows that the property indeed holds in theuniform-quadratic case.
18

θi−1

yi

θi+1

θi

yi+2 θi+2

Figure 1: Alternating symmetry.

Lemma 4. In the uniform quadratic specification, the optimal mechanism satisfies the property of alternating symmetry.
Lemma 4 significantly simplifies the characterization of the optimal mechanism, because
it allows me to restrict attention to the first two elements; the rest of elements are merely“replications” of the first two. Moreover, the alternating symmetry structure also helps
identify a finite upper bound, denoted N (b), on the size of an aligned monotone partitional
mechanism. To see it, combining (4.1) and (4.2), and rearranging terms yields

yi+1 − yi−1 ≥ 2b ∀i = 2, . . . , (N − 1).

Together, (5.1) and the property of alternating symmetry imply N (b) ≡

(5.1)

1
2b

+

16
.
2b

I am now ready to give an explicit characterization of optimal mechanisms in two steps.
First, building on the previous observation that one only needs to focus on the first twoNelements, I obtain the suboptimal mechanism {(θi )Ni=1 , (yi )i=1 } for each size from one throughN ≤ N (b) by solving a simple linear program.7 Second, the size of the optimal mechanism,
denoted N ∗ (b), can be determined by simply comparing the principal’s expected payoffs
under the resulting suboptimal mechanisms for each size N ≤ N (b). The explicit expressionof the optimal mechanism is summarized by the following proposition.
N
Proposition 2. Suppose that M = {(θi )Ni=1 , (yi )i=1 } is an optimal obedient recommendation

mechanism.
67⌊x⌋ (⌈x⌉) is the largest (smallest) integer that is smaller (greater) than or equal to x.
See Appendix.

19

(1) If N is odd, then

θi =




 (i−1)+(N −i)θ1 ,for i odd,
 i(1−θ1 ) ,for i even,

N −1

N −1

with θ1 =1−(N 2 −1)b+

yi =




b + (i−2)+(N −i+1)θ1 ,for i odd,b + (i−1)−(i−1)θ1 ,N −1

for i even,

N −1

√

(N +1)2 (N −1)2 b2 +1.
N +1(2) If N is even, then

θi =

i
,
N

yi = b +i−1
N∀i = 1, . . . , N.

Figure 2 displays the sizes N ∗ (b) of optimal mechanisms together with the corresponding
upper bounds N (b) for different values of b > 0. The figure indicates that the principal
does not always prefer the partition with the largest possible number of elements, which,as previously noted in the Introduction, reflects the principal’s trade-off between amount of
information and efficiency of action allocations. The reason behind is that the mechanismwith the highest number of elements, in spite of the most amount of information it generates
in the first place, can result in the distortion of decision-making at the later stage. Indeed,
the principal has to compromise between the two conflicting objectives of providing theagent with precise information and implementing actions sufficiently close to her optimal
ones. It turns out that this trade-off, as illustrated in Figure 2, is resolved in favor of using
mechanisms with intermediate sizes of partitions.

20

Size

Residual Variance
0.025

20
0.02

N (b)
15

0.015
10

0.01

5

0.005
N ∗ (b)

0

0

b
0

0.1

0.2

0.3

0.4

0.5

Figure 2: Optimal size of partitions
and upper bounds.

5.2

b
0

0.1

0.2

0.3

0.4

0.5

Figure 3: Informativeness of optimalmechanisms.

Comparative statics

In search for the optimal mechanism, the principal’s problem is twofold: how much information the agent should possess, and how much discretion the agent should have. In thissubsection, I examine comparative statics on these two features with respect to the divergence of the principal’s and the sender’s preferences.

Comparative statics on information. I measure informativeness as the expected residual
variance of the state: E[variance(θ|µ)]; a lower value of E[variance(θ|µ)] implies more precise
N
posterior beliefs in expectation. Formally, a mechanism M = {(θi )Ni=1 , (yi )i=1 } is said to beN̂
more informative than another M̂ = {(θ̂i )N̂
i=1 , (ŷi )i=1 } if
N ˆ θiX
i=12(θ − mi ) dθ ≤θi−1N̂ ˆ θ̂iXi=1(θ − m̂i )2 dθ.

θ̂i−1This measure of informativeness is a natural one given the partitional information structure
and the uniform prior assumption, because the posterior distribution of the state now is fully
characterized by its mean and variance.

21

Figure 3 depicts a monotonic relationship between the agent’s constant bias b and theinformativeness of the corresponding optimal mechanism. In particular, it shows that a
principal prefers to provide a greater amount of information if the agent’s bias parameter b issmaller. This result might seem obvious in the uniform-quadratic specification. However, theimpact of preference divergence on amount of information can be ambiguous in the generalenvironment. On the one hand, the expected residual variance might not be an appropriate
measure of informativeness when the prior distribution is not uniform. On the other hand,when preferences are less divergent the principal might also provide less information and still
be able to steer the agent’s incentives toward her own favor.

Comparative statics on discretion. I measure discretion by the cardinality of delegationN
sets. Formally, a mechanism M = {(θi )Ni=1 , (yi )i=1 } is said to grant more discretion thanN̂another mechanism M̂ = {(θ̂i )N̂
i=1 , (ŷi )i=1 } if N > N̂ . Figure 2 illustrates the comparative

statics on the agent’s discretion: there is less discretion when the bias parameter b is bigger.
This result confirms the standard intuition that an agent gets more discretion the morecongruent he is with the principal.

5.3

Value of Information Manipulation

What is the value of limiting the agent’s information in delegation problems? In this subsection I address this question and compare the players’ payoffs in optimal delegation with andwithout information manipulation. When there is no information manipulation, the agent isassumed to have complete information about the state of nature. In this case, the principal
simply faces a perfectly-informed agent.
Figure 4 displays the payoffs to the principal under optimal delegation with information
manipulation compared to that without information manipulation. It is obvious that thepayoffs are higher with information manipulation, because a full information structure canbe viewed as a special informational policy under information manipulation. The value of22keeping the agent partially informed simply reflects the aforementioned trade-off between
quality of information and efficiency of action allocations.
It is interesting to note that the principal’s gain from information manipulation is nonmonotonic in the degree of bias. Clearly, for b small, the principal’s payoff is close toher first-best level and the gains from information manipulation are small. As b increases,
distorted action allocations become more severe and there is more scope for information
manipulation to fix this incentive problem by aligning preferences. However, as b becomes
sufficiently large, resorting to information manipulation improves this situation, but the costof aligning incentives through limited information also increases. When b ≥ 12 , the abilityof limiting information is of no value whatsoever as the principal then optimally takes anuninformed action.
Figure 5, on the other hand, compares the payoffs to the agent in the two cases. It illustrates that in general the agent does not always achieve more information rents by possessing
more private information. In fact, there are some regions of sufficiently large bias b in which
an imperfectly-informed agent (possessing principal-optimal partitional information) is better off than a perfectly-informed agent.8 As an implication, ignorance may generate strategic
benefits for the agent even though private information guarantees the agent a positive rent
under the optimal delegation scheme. Intuitively, when the agent’s bias becomes sufficiently
large (as long as b ≤ 21 ), by restricting his information in a manner that increases the distance
between posterior means, it is possible to reduce the agent’s incentive to take higher actions
for intermediate types, and thus increase the principal’s willingness to delegate more highactions.
To illustrate, consider the case where b ∈ ( 52 , 21 ). It follows from Proposition 2 that, withinformation manipulation, the principal optimally selects a binary partitional information
structure with cutoffs { 12 , 1}, and a delegation set of {b, b + 21 }. The reader can easily
verify that the agent’s expected payoff is 21 (b2 + b + 41 ). In contrast, without information
8

This phenomenon is also confirmed by Roesler and Szentes (2017) in the setting of static monopoly
pricing.

23

manipulation, the principal’s optimal delegation set takes the form of an interval: [b, 1 − b],9which delivers expected payoff 21 (− 83 b3 + b2 + b + 13 ) to the agent. These imply, as Figure
5 shows, that the agent strictly prefers the binary partitional information structure underwhich the principal offers a delegation set including a sufficiently high action b + 12 , whichallows the agent to respond to higher states.
UA

UP
0

Optimal delegation withinformation manipulation

Optimal delegation withinformation manipulation

0

−0.02
−0.1
−0.04
Optimaldelegation

−0.2
−0.06
Optimaldelegation
−0.3

−0.08
b
0

0.1

0.2

0.3

0.4

0.5

b
0

Figure 4: Principal’s payoffs.

0.1

0.2

0.3

0.4

0.5

Figure 5: Agent’s payoffs.

9

In the uniform-quadratic case with perfectly-informed agent, the optimal delegation set takes the form
D = [b, 1 − b] if b ≤ 21 and D∗ = { 12 } if b > 12 . This characterization was first given by Melumad andShibano (1991).
∗

24

6

General Comparative Statics

It is natural to ask at this point to what extent the comparative statics results that hold in
the uniform-quadratic case can be generalized. This section establishes that the comparative
statics on discretion and on the principal’s payoffs still carry over to the general model.
To do so, I introduce the concept of an agent A with CA being uniformly less-biased thananother agent Ã with CÃ if cA (y) ≥ cÃ (y) for each y ∈ Y . Recall that yA (θ) = c−1
A (θ) andyÃ (θ) = c−1(θ). It is straightforward to see that yÃ (θ) ≥ yA (θ) > yP (θ); namely, yA (·) liesÃ
between yÃ (·) and yP (·). Note that this notion of preference divergence encompasses that of
the uniform-quadratic case, which is captured by a single parameter b.
I am now ready to generalize the comparative statics results on the agent’s discretionand the principal’s payoffs.
Theorem 3. The principal always grants more discretion to the agent who is uniformly
less-biased than another agent.
Theorem 3 extends the observation in Section 5.2 that the principal optimally offers finerpartitions to a more congruent agent, and as a result, offers more discretion. (I use “finer”
informally here.)
Next, it is easy to establish that the principal necessarily obtains higher payoffs if theagent is less biased. Consider moving from CÃ to a uniformly less-biased CA . LettingN
M = {(θi )Ni=1 , (yi )i=1 } denote an recommendation mechanism, it is immediate thatmi ≤CA (yi+1 ) − CA (yi )CÃ (yi+1 ) − CÃ (yi )⇒ mi ≤
yi+1 − yiyi+1 − yi∀i = 1, . . . , (N − 1).

This means the feasible set of obedient mechanisms with CA is a superset of that with CÃ ,so the principal’s expected optimal payoff is weakly higher facing agent CA than agent CÃ .
This result is stated formally in the next theorem.
Theorem 4. The principal always prefers the agent who is uniformly less-biased.
25

7

Application: Monopoly Regulation

In this section, I apply my results to the classic problem of regulating a monopolist who is privately informed (Baron and Myerson 1982). In contrast to the standard literature, I supposethat (i) contingent transfers between the regulator and the monopolist are not feasible; (ii)in addition to restricting prices (price control policy) as in Alonso and Matouschek (2008),
the regulator can restrict the monopolist’s access to private information about consumers(data protection policy).
I solve for the optimal joint regulatory policy using the results of Theorem 1 and 2. Ishow that (i) the optimal price control policy takes the form of a finite set of prices, which is
strikingly different from the price cap regulation, as is commonly observed in practice and intheory (see, e.g., Alonso and Matouschek 2008; Amador and Bagwell 2020); (ii) the optimaldata protection policy takes the form of monotone partitions: adjacent consumer types arepooled together in the same market segments.
I then compare the qualitative features of optimal joint regulatory policy with two benchmarks: one where there is only price control, and one where there is only data protection.

7.1

Joint regulation

A monopolist sells a good to a consumer.10 The monopolist can choose a quality level a
and a unit price y to the consumer, who, in turn, chooses a quantity level q to maximize aquadratic utility function:1u(y, a, q, θ) = (θ + λa)q − yq − q 2 ,2which yields consumer demand q(y, a, θ) = θ + λa − y. One can interpret θ ∈ [0, 1] as thebaseline willingness to pay for the first unit of the product and henceforth refer to θ as the10This framework of price-quality discrimination is also adopted in Argenziano and Bonatti (2020), whoprovide a microfoundation for consumers’ privacy preferences in a dynamic environment.

26

consumer’s type. The parameter λ represents the monopolist’s marginal value of the qualityof the product.
The monopolist has a constant marginal cost of producing quantity q that I normalize tozero and a fixed per-consumer cost of producing quality a. The monopolist seeks to maximize
his profit uA (y, a, θ) = yq − 12 a2 . Fix a price y, the monopolist’s optimal quality level can be
written as a∗ (y) = λy. Substituting a∗ into uA yields the monopolist’s payoff
1uA (y, θ) = θy − (2 − λ2 )y 2 ,21
where yA (θ) = 2−λ2 θ is the profit-maximizing price.

The regulator’s payoff, then, is the sum of the profit and the consumer surplus
11uP (y, θ) = λ2 θy − (1 + λ2 − λ4 )y 2 + θ2 ,2
22

where yP (θ) = 1+λλ2 −λ4 θ is the welfare-maximizing price. To ensure that yA (·) > yP (·) I assume that λ ∈ (0, 1). Observe that a monopolist with λ is uniformly less-biased than anothermonopolist with λ′ if λ ≥ λ′ . Therefore, the parameter λ fully measures the monopolist’s
upward bias.
The regulator and the monopolist share a prior µ0 about θ. I suppose the regulator
can employ two regulatory policies at the same time: one is price control, which specifies aset of prices D available to the monopolist; the other is data protection, which is formallyan information structure (equivalently, a randomized price recommendation function) σ :[0, 1] → ∆(D) that provides the monopolist with information about the consumer’s private
type. The monopolist privately observes a realization from σ, and then he chooses a quality
level a, and sets a price y from D to maximize profit uA (y, θ).
By Theorem 1 and 2, the optimal joint regulatory policy (D, σ) features a monotone
partition. That is, the optimal price control D is a finite set of prices; and the optimal dataprotection policy partitions the state space [0, 1] into finite intervals and the monopolist only27

observes which interval the consumer’s type actually lies in.
Notice that the monopolist’s pricing problem is essentially a problem of third-degree pricediscrimination, where different realizations of the information play the role of market segments. Thus, the single-consumer setup introduced above is equivalent to a setup where
there is a continuum of consumers, each of whom has a private type θ, which is drawn
independently according to µ0 . With this interpretation, an information structure σ thencorresponds to a market segmentation which is a division of the aggregate market into different markets. To see why, observe that any information structure σ must lead to a distribution´
τ over posteriors µ ∈ ∆([0, 1]) satisfying µ dτ (µ) = µ0 . Thus, a probability measure τ canbe viewed as a market segmentation, with the interpretation that τ (µ) is the proportion of
the population in market segment µ.
As a result, with a continuum of consumers, I can reexpress the regulator’s data protection
policy as a market segmentation. Accordingly, a monotone partition corresponds to a “finite
monotone segmentation”, which splits consumers into finite groups and each group only
consists of consumers with similar types. The monopolist then offers different quality levels
and different prices to different segments. One implication of the optimality of finite market
segmentations is that the consumers’ privacy is protected moderately, which is between noprotection (i.e., perfect segmentation with each segment containing consumers of a singletype) and full protection (i.e., no segmentation with a uniform quality level and a uniform
price).11

7.2

Comparing Regulatory Policies

I compare the key features of optimal joint regulatory policies with two benchmarks: one
with only price control and one with only data protection.

Pure price control (full information). In this benchmark, the regulator imposes no11

For the privacy implications of consumer data collection, I refer the reader to the survey by Acquisti,
Taylor, and Wagman (2016).

28

restrictions on the monopolist’s access to consumer privacy, and thus the monopolist isperfectly informed about each consumer’s type. The regulator’s search for the optimal
price control policy can then be formulated as an optimal delegation problem in which theagent (monopolist) has perfect information. Most of the delegation literature has focusedon characterizing sufficient conditions for interval delegation to be optimal. For instance,in the present regulatory environment, it follows immediately from Proposition 4 in Alonsoand Matouschek (2008) that there exists a λ∗ ∈ (0, 1) such that for all λ ∈ (λ∗ , 1), i.e., themonopolist’s bias is sufficiently small, the optimal price control policy is a price cap, below
which the monopolist can set any prices.

Pure data protection (full discretion). In this benchmark, the regulator imposes norestrictions on prices so the monopolist has full discretion. The regulator’s seaerch for the
optimal data protection policy can then be formulated as a Bayesian persuasion problem.
Following Kamenica and Gentzkow (2011), I can then easily solve for the optimal information
structure by representing the regulator’s (sender’s) payoff as a function of the monopolist’s
q √(receiver’s) beliefs. It can be shown that when λ ∈ ( 3−2 5 , 1), full disclosure is optimal, i.e.,there is quality and price discrimination against each consumer type and hence no protectionq √of consumer privacy. When λ ∈ (0, 3−2 5 ), no disclosure is optimal, i.e., there is no quality
and price discrimination and hence full protection of privacy.12

Qualitative differences. By comparing the optimal joint regulatory policy with the twobenchmarks, I obtain two striking qualitative insights. First, with additional data protection,
significantly less pricing flexibility is delegated to the monopolist. Specifically, the regulator
optimally grants only a finite set of prices in the presence of data protection, rather thanan interval of prices (i.e., a price cap) in the full information benchmark. Second, when themarginal value of the quality λ is sufficiently large, i.e., the monopolist’s bias is sufficiently
12

For the welfare effects of third-degree price discrimination, see, for instance, Pigou (1920), Robinson(1933), Schmalensee (1981), Varian (1985), Aguirre, Cowan, and Vickers (2010), Cowan (2016).

29

small, the social welfare is enhanced when the regulator moderately protects consumer privacy, but that cannot happen without controlling prices at the same time.

8

Application: Temptation, Information, and Discretion

In this section, I examine the consumption decision of a dynamically inconsistent individualfor goods that provide an immediate benefit and a delayed cost, and ask how self-restricting
information and discretion can serve as complementary commitment devices and alleviate
the effects of temptation.
Consider an individual who lives for three periods t ∈ {0, 1, 2}. In period 1 he decides theamount x of good he is going to consume. Consumption generates an instantaneous utility
equal to u(x). However, it also exerts a negative externality in period 2 by an amount e(x)with magnitude ω. I assume that the individual’s preferences exhibit time-inconsistency, dueto quasi-hyperbolic discounting. The intertemporal payoffs from the perspective of self -0
and self -1 from periods t = 1, 2 with externality shock ω are respectively given by

U0 (x, ω) = β[δu(x) − δ 2 ωe(x)] and U1 (x, ω) = u(x) − βδωe(x),where u is a concave utility function, e is a convex externality function, and both are strictlyincreasing and continuously differentiable. The value of δ represents the standard discountfactor. Since the main focus here is on the impacts of time inconsistency, I assume thatδ = 1. The value of β is the hyperbolic adjustment that reflects the momentary salienceof the present. When β is lower, the temptation for current consumption is higher. To
highlight the demand for commitment devices, I assume β ∈ (0, 1]. As a result, self -1’s
excessive preference for the present leads to overconsuming from self -0’s perspective. For
analytical tractability I restrict attention to a specific class of functional forms.

30

Assumption 3. The following hold:1−γ(i) The utility of consumption satisfies u(x) = x1−γ for 0 ≤ γ ≤ 12 ;(ii) The externality of consumption is linear, i.e., e(x) = x.
Letting y = u(x), self -0’s and self -1’s payoffs can be re-written as
1

1

U0 (y, θ) = θy − (1 − γ) 1−γ y 1−γ

11

and U1 (x, θ) = θy − β(1 − γ) 1−γ y 1−γ ,(8.1)

where θ is defined to be the inverse of ω.13
As is well known, a sophisticated and dynamically inconsistent individual values commitment power (Strotz, 1995). I consider two commitment devices that are commonly studied
in the literature. A hard commitment device ties the hands of self -1 by reducing the setof available consumption choices; a soft commitment device keeps self -1 ignorant about themagnitude of the externality through selective exposure to information.
I analyze the optimal combination of the hard and the soft commitment devices. One caneasily examine that the utility specification in (8.1) satisfies Assumption 1 and 2, and hence
can be formally captured by the general analysis in Section 4. In particular, the alignment
principle from Proposition 1 implies that self -0 would optimally eliminate the disagreementwith self -1 under self-restricted consumption and information. Moreover, I can easily use
Theorem 2 to show that the optimal joint commitment features a finite consumption set anda monotone partitional information structure.

9

Conclusion

I have studied an optimal delegation problem in which the principal has full control over theagent’s information environment. The principal’s problem is to jointly select a delegation
set and an information structure that maximize her expected payoffs.
13

I have removed β in the expression of U0 (y, θ) without loss of generality.

31

Notable features of the optimal mechanism include: (i) the players act as if they shared
identical preferences given the chosen information and discretion; (ii) the optimal information
structure takes the form of monotone partitions.
For future research it would be interesting to extend my model by allowing for the possibility of contingent monetary transfers. In the design of static decision-making processes,
there are roughly three incentive schemes available to align players’ interests: delegation,
persuasion, and monetary transfers. By employing all the three incentive instruments, I
would be able to analyze the three-way interaction among allocation of authority, provisionof information, and commitment to transfers in optimal mechanisms, and moreover, to study
the extent to which players’ conflicts can be reduced.

32

A

Appendix

A.1

Proof of Lemma 2

Let M = (D, σ) be an optimal obedient mechanism that implements N actions. I first show thatyP (mi ) ≤ yi for all i = 1, . . . , N . Suppose instead that there exists an action yi ∈ D such thatyi < yP (mi ). I split the analysis into two cases, depending on whether type yi+1 ’s local downward
OB is binding.
)−CA (yi )Case 1. Suppose uA (yi+1 , mi+1 ) > uA (yi , mi+1 ). It then follows that mi+1 > CA (yyi+1=
i+1 −yi∆CA (yi+1 , yi ). Consider now a perturbation in which the principal replaces yi with a slightly bigger
action yi↑ = yi + ε with ε > 0, while keeping all other delegated actions fixed. I argue thatthe principal must be strictly better off and hence a contradiction to optimality. To see this, let´1
pi = 0 σ(yi |θ) dµ0 (θ) denote the absolute probability of realization yi . The perturbation onlychanges the principal’s payoff for type yi . The change in payoff ishihi∆UPσ = −pi [mi yi − CP (yi )] + pi mi yi↑ − CP (yi↑ ) = pi (yi↑ − yi ) mi − ∆CP (yi↑ , yi ) > 0

given that ε > 0 is small and yi < yP (mi ). The only thing left to verify is that OB is preserved. By
Lemma 1 I only need to check type yi−1 ’s (OBi−1,i ), type yi ’s (OBi,i−1 ) and (OBi,i+1 ), and typeyi+1 ’s (OBi+1,i ), that is,mi−1 ≤ ∆CA (yi↑ , yi−1 ),(OBi−1,i )∆CA (yi↑ , yi−1 ) ≤ mi ≤ ∆CA (yi+1 , yi↑ ),(OBi,i−1 + OBi,i+1 )∆CA (yi+1 , yi↑ ) ≤ mi+1 .

(OBi+1,i )Since yi↑ > yi , both (OBi−1,i ) and (OBi,i+1 ) are automatically satisfied. As for (OBi,i−1 ) and′ (y ) < C ′ (y ) <(OBi+1,i ), observe that they must be slack due to the fact that ∆CA (yi↑ , yi−1 ) < CAi
P i

mi , and ∆CA (yi+1 , yi ) < mi+1 , respectively.

33

Case 2. Suppose uA (yi+1 , mi+1 ) = uA (yi , mi+1 ). It then follows that mi+1 = ∆CA (yi+1 , yi ).
Consider now a small increase in yi and a small decrease in yi+1 by setting yi↑ = yi + ε and↓↓yi+1= yi+1 − δ with ε, δ > 0 and ∆CA (yi+1, yi↑ ) = ∆CA (yi+1 , yi ). The change in the principal’s

payoff is thenhi∆UPσ = − pi [mi yi − CP (yi )] + pi mi yi↑ − CP (yi↑ )hi↑↑− pi+1 [mi+1 yi+1 − CP (yi+1 )] + pi+1 mi+1 yi+1
− CP (yi+1
)
h
i
h
i
↓
↓
=pi (yi↑ − yi ) mi − ∆CP (yi↑ , yi ) + pi+1 (yi+1− yi+1 ) mi+1 − ∆CP (yi+1, yi+1 ) > 0↓
given that ε > 0 is small, mi > CP′ (yi ), and mi+1 = ∆CA (yi+1 , yi ) < ∆CP (yi+1, yi+1 ). I amleft to verify type yi−1 ’s (OBi−1,i ), type yi ’s (OBi,i−1 ) and (OBi,i+1 ), type yi+1 ’s (OBi+1,i ) and(OBi+1,i+2 ), and type yi+2 ’s (OBi+2,i+1 ):mi−1 ≤ ∆CA (yi↑ , yi−1 ),(OBi−1,i )↓∆CA (yi↑ , yi−1 ) ≤ mi ≤ ∆CA (yi+1, yi↑ ),

(OBi,i−1 + OBi,i+1 )↓↓∆CA (yi+1, yi↑ ) ≤ mi+1 ≤ ∆CA (yi+2 , yi+1),

(OBi+1,i + OBi+1,i+2 )↓∆CA (yi+1, yi+2 ) ≤ mi+2 .

(OBi+2,i+1 )Observe that (OBi−1,i ), (OBi,i+1 ), (OBi+1,i ), and (OBi+2,i+1 ) still hold by the construction of yi↑↓
and yi+1. Moreover, (OBi,i−1 ) and (OBi+1,i+2 ) are slack because ∆CA (yi↑ , yi−1 ) < ∆CA (yi↑ ) <↓∆CP (yi↑ ) < mi , and mi+1 = ∆CA (yi+1 , yi ) < ∆CA (yi+2 , yi+1).

Proceeding in a similar fashion as above, I can also show that yi ≤ yA (mi ) for all i = 1, . . . , N .

A.2

Proof of Proposition 1

Let M = (D, σ) be an optimal obedient mechanism that implements N actions. Assume thecontrary, i.e., that there exists a type yi with 2 ≤ i ≤ N such that uP (yi , mi ) < uP (yi−1 , mi ), or
equivalently, mi < ∆CP (yi , yi−1 ).
First, I must have i ̸= N . Otherwise the principal can derive a net benefit from joining thetype yi−1 and type yi into a single type associated with action yi−1 .

34Second, I claim that (OBi−1,i ) must be binding, that is,mi−1 = ∆CA (yi , yi−1 ).

(A.1)

Suppose not. Then I can perturb the information structure σ by subtracting a small portion fromrealization yi , and adding it to realization yi−1 . More formally, the perturbed information structure
σ ′ is the same as σ except thatσ ′ (yi |θ) = (1 − ε)σ(yi |θ),σ(yi−1 |θ) = σ ′ (yi−1 |θ) + εσ(yi |θ).

By doing so, the principal is strictly better off since she prefers action yi−1 over yi , conditional on
realization yi . Moreover, (OBi−1,i ) is still satisfied given the hypothesis and that ε > 0 is small. A
contradiction.
Before proceeding to the contradiction argument, note that type yi ’s (OBi,i+1 ) can be writtenasmi + δi = ∆CA (yi+1 , yi )(A.2)

with δi ≥ 0. With this notation, the contradiction assumption mi < ∆CP (yi , yi−1 ) can be rewritten
as ∆CP (yi , yi−1 ) > ∆CA (yi+1 , yi ) − δi , or equivalently,∆CDiff (yi , yi−1 ) > ∆CA (yi+1 , yi ) − ∆CA (yi , yi−1 ) − δi .

Moreover, define action ȳ ∈ Y as the solution toρmi−1 + (1 − ρ)mi + (1 − ρ)δi = ∆CA (yi+1 , ȳ),pi−1where ρ = pi−1
+pi .

Now consider a modification M̂ = (σ̂, D̂) of the mechanism M as follows:

35

(A.3)

(i) The realization space (delegation set) D̂ is given byD̂ = {y1 , . . . , yi−1 , ŷ, yi+2 , . . . , yN },where the value of ŷ will be specified later.
(ii) σ̂ is described byσ̂(y|θ) =




σ(y

i−1 |θ) + σ(yi |θ),
σ(y|θ),if y = ŷ,
if y ∈ D̂ \ {ŷ}.

There are two exhaustive cases to consider:

Case 1: ȳ ≤ yi−1 . In this case, let ŷ = yi−1 . To check the obedience of M̂, it suffices to verify
that type ŷ’s local upward OB constraint holds,m̂ = ρmi−1 + (1 − ρ)mi = ∆CA (yi+1 , ȳ) − (1 − ρ)δi ≤ ∆CA (yi+1 , yi−1 ).

It follows that the principal is strictly better off under M̂ by the hypothesis that she strictly prefers
action yi−1 over yi on the support of realization yi . A contradiction.

Case 2: ȳ > yi−1 . In this case, let ŷ = ȳ. To verify obedience, it is sufficient to verify that bothtype yi−2 and type ŷ are upwardly obedient. For type yi−2 , I havemi−2 ≤ ∆CA (yi−1 , yi−2 ) < ∆CA (ŷ, yi−2 ).

For type ŷ, I havem̂ + (1 − ρ)δi = ρmi−1 + (1 − ρ)mi + (1 − ρ)δi = ∆CA (yi+1 , ȳ) = ∆CA (yi+1 , ŷ).

36

(A.4)

Additionally, replacing mi−1 by ∆CA (yi , yi−1 ) and mi by ∆CA (yi+1 , yi ) − δi in (A.4), I obtain∆CA (yi+1 , ŷ) = ρ∆CA (yi , yi−1 ) + (1 − ρ)∆CA (yi+1 , yi )(A.5)

yi −yi−1
Since I have assumed ȳ > yi−1 , it follows that ρ ∈ (0, yi+1
−yi−1 ). And I must have ŷ → yi fromyi −yi−1
below as ρ → 0, and ŷ → yi−1 from above as ρ → yi+1
−yi−1 .

To prove that the modification makes the principal better off, I begin by expressing the difference in the principal’s payoff between the original mechanism M and the modified mechanism M̂
(normalized by pi−1 + pi ) as
UP (M̂) − UP (M)=[m̂ŷ − CP (ŷ)] − ρ[mi−1 yi−1 − CP (yi−1 )] − (1 − ρ)[mi yi − CP (yi )]pi−1 + pi=m̂ŷ − ρmi−1 yi−1 − (1 − ρ)mi yi− ρ[CP (yi ) − CP (yi−1 )] − [CP (yi+1 ) − CP (yi )] + [CP (yi+1 ) − CP (ŷ)]=m̂ŷ − ρmi−1 yi−1 − (1 − ρ)mi yi− ρ∆CP (yi , yi−1 )(yi − yi−1 ) − ∆CP (yi+1 , yi )(yi+1 − yi ) + ∆CP (yi+1 , ŷ)(yi+1 − ŷ).
(A.6)
Using the OB constraints (A.1), (A.2), (A.4) to substitute in for mi−1 , mi , m̂, and further replacing∆CP with ∆CA + ∆CDiff , (A.6) simplifies to
UP (M̂) − UP (M)=[∆CA (yi , yi−1 ) − ∆CA (yi+1 , yi )]ρ(yi+1 − yi )pi−1 + pi− ∆CDiff (yi , yi−1 )ρ(yi − yi−1 )− ∆CDiff (yi+1 , yi )(yi+1 − yi )+ ∆CDiff (yi+1 , ŷ)(yi+1 − ŷ)− (1 − ρ)δi ŷ + (1 − ρ)δi yi .

37

Substituting with the contradiction assumption (A.3) yields
UP (M̂) − UP (M)> − ∆CDiff (yi , yi−1 )ρ[yi+1 − yi−1 ] + CDiff (yi ) − CDiff (ŷ)
pi−1 + pi(A.7)− (1 − ρ)δi ŷ + (1 − ρ)δi yi − ρδi (yi+1 − yi )I next show that the RHS of (A.7) is weakly positive. Note that by the previous steps, limρ→0+ ŷ =yi and limρ→[(yi −yi−1 )/(yi+1 −yi−1 )]− ŷ = yi−1 . Using this and (A.7), I obtainlim

ρ→0+ or[(yi −yi−1 )/(yi+1 −yi−1 )]−UP (M̂) − UP (M)= 0.
pi−1 + pi

Implicitly differentiating (A.5) with respect to ρ, I get thatŷ ′ (ρ) = −∆CA (yi+1 , yi ) − ∆CA (yi , yi−1 )∆CA (yi+1 ,ŷ)
∂ ŷ

, and

∂ 2 ∆CA (yi+1 ,ŷ) ′ŷ (ρ)[∆CA (yi+1 , yi ) − ∆CA (yi , yi−1 )]∂ ŷ 2ŷ ′′ (ρ) =.
(∆CA (yi+1 , ŷ))2(A.8)

Recall that by Condition (i) in Assumption 1, ŷ ′ (ρ) < 0, and by Condition (i) in Assumption 2,ŷ ′′ (ρ) ≥ 0. Consider the second derivative of (A.7) with respect to ρ:′′′ŷ ′′ + δi [ŷ ′ − (1 − ρ)ŷ ′′ ].
(ŷ ′ )2 − CDiff−CDiff′This second derivative is weakly negative given the fact that ŷ ′ (ρ) < 0, ŷ ′′ (ρ) ≥ 0, CDiff
> 0,
′′
CDiff
≥ 0 (Condition (ii) in Assumption 2). Hence, the RHS of (A.7) is concave, and therefore,
P (M)together with (A.8), is weakly positive. This result implies that UP (M̂)−U> 0 and thus yields
pi−1 +pi

a contradiction.
Notice that the finiteness restriction does not play any role in the above argument. So thealignment principle holds generally in the class of discrete mechanisms (i.e., D is discrete).

38

A.3

Proof of Theorem 1

Let M = (D, σ) be an optimal obedient mechanism that implements N actions. Suppose bycontradiction that M is not a monotone partition. It follows that there must be two actions
yj < yk in D and a cutoff x ∈ θ such that the conditional probabilities of the lower and upper
segment have strictly positive probabilities, i.e.,
ˆ xˆ θdµσj > 0,

θ

ˆ θ

ˆ x
dµσk > 0,dµσj > 0,xθdµσk > 0.
x

I consider two exhaustive cases separately.

Case 1: Type yk ’s (OBk,k+1 ) is binding for any 2 ≤ k ≤ (N − 1). In this case, consider a local
perturbation in the information structure σ:(a) On the interval [θ, x), subtract an absolute weight ε proportionally from recommendation yk ,and distribute the weight ε proportionally to recommendation yj . Specifically, for each stateθ ∈ [θ, x), the conditional probability switching from recommendation yk to yj is given byσ(yk |θ)ε.
θ σ(yk |θ) dµ0´x(b) On the interval (x, θ], subtract an absolute weight ε proportionally from recommendation yj ,and distribute the weight ε proportionally to recommendation yk . Specifically, for each stateθ ∈ (x, θ], the conditional probability switching from recommendation yj to yk is given by´θ

σ(yj |θ)ε.

x σ(yj |θ) dµ0(c) On the interval [θ, θ], subtract an absolute weight ν proportionally from recommendationyk−1 , and distribute the weight ν proportionally to recommendation yk . Specifically, for eachstate θ ∈ [θ, θ], the conditional probability switching from recommendation yk−1 to yk is39

given by´θ

σ(yk−1 |θ)ν,

θ σ(yk−1 |θ) dµ0

where the value of ν is determined by restoring the binding (OBk,k+1 ) under the perturbed
mechanism.
To simplify, the perturbed mechanism M̂ = (σ̂, D̂) can be described as(i) the delegation set D̂ is the same as D.
(ii) σ̂ satisfiesσ̂(y|θ) =where
σ̂(yj |θ) =σ(y|θ),


if y ∈ D̂ \ {yj , yk },σ̂(yj |θ),






σ̂(yk |θ),σ̂(yk |θ) =if y = yk ,σ(yk |θ)ε
σ(yj |θ) + ´ x− ´θσ(yj |θ)ν

σ(yj |θ)ε
σ(yj |θ) − ´ θ− ´θσ(yj |θ)ν

,if θ ∈ [θ, x),,

if θ ∈ (x, θ],σ(yj |θ)ν

σ(yk |θ) − ´ x σ(yk |θ)ε
+ ´θ
,

σ(y
|θ)
dF
(θ)
k
θ
σ(y |θ) dF (θ)

if θ ∈ [θ, x),σ(yj |θ)ε
σ(yk |θ) + ´ θ

if θ ∈ (x, θ].

θ σ(yk |θ) dF (θ)

x σ(yj |θ) dF (θ)

andif y = yj ,

θ σ(yj |θ) dF (θ)θ σ(yj |θ) dF (θ)

jθx σ(yj |θ) dF (θ)+ ´θ

σ(yj |θ)ν

θ σ(yj |θ) dF (θ),(iii) the value of ν is set to beν=
where mLk =L
mH
j − mkε,mk − mj´xσ0 θ dµk denotes the posterior mean conditional on the lower segment of signal

yk , and mH
j =

´1

σ
x θ dµj

denotes the posterior mean conditional on the upper segment ofsignal yj .
First, OB remains to hold under M̂. To see this, notice that for i ̸= j, k, type yi ’s (OBi,i+1 )is automatically satisfied, which is directly inherited from that of M. For type yj , the conditional40

posterior mean´θθ θσ̂(yj |θ) dµ0m̂j = ´θθ σ̂(yj |θ) dµ0HH(pj − ν)mj + εmLmL
k − εmjk − mj== mj +
εpj − ν
pj − ν

is smaller than yet close to mj for ε > 0 small enough. Hence, (OBj,j+1 ) is satisfied. For type yk ,the conditional posterior mean´θθ θσ̂(yk |θ) dµ0m̂k = ´θθ σ̂(yk |θ) dµ0=

H
pk mk − εmL
k + εmj + νmj= mk
pk + ν

stays the same by construction. An immediate implication of the above equality is that type yk ’s(OBk,k+1 ) is still binding under M̂.
Second, I argue that the perturbed mechanism M̂ strictly improves the principal’s payoff. In
order to see this, I can represent the principal’s change in payoff byˆ x
UP (M̂) − UP (M) =θσ(yk |θ)εdµ0
θ σ(yk |θ) dµ0[uP (yj , θ) − uP (yk , θ)] ´ x

ˆ θ

+
x

σ(yj |θ)εdµ0
θ
x σ(yj |θ) dµ0[uP (yk , θ) − uP (yj , θ)] ´

ˆ θ
+
θ

[uP (yk , θ) − uP (yj , θ)] ´σ(yj |θ)ν

θθ σ(yj |θ) dµ0

dµ0

=ε

L
mH
j − mk{mk (yk − yj ) − [CP (yk ) − CP (yj )]}mk − mj=εLmH
j − mk[uP (yk , mk ) − uP (yj , mk )]mk − mj(A.9)≥0,
where the inequality follows from the fact that the optimal mechanism must be aligned from
Proposition 1. If uP (yk , mk )−uP (yj , mk ) > 0, (A.9) yields UP (M̂)−UP (M) > 0, which contradicts
the optimality of M. If uP (yk , mk )−uP (yj , mk ) = 0, then the value of j must be equal to (k−1) due
to the alignment principle. In this case, consider an alternative perturbation M̃ by pooling typeyj (= yk−1 ) and type yk into a single type associated with some action ŷ. Analogous arguments to
those used in the proof of Proposition 1, the perturbation would weakly increase Principal’s payoff.

41

Case 2: Either type yk ’s (OBk,k+1 ) is not binding for 2 ≤ k ≤ N , or k = N . This case can betreated by solely applying the “swapping” procedure (a)–(b) from Case 1. It is straightforward to
see that OB is satisfied in the perturbed mechanism M̂ = (D, σ̂) given that ε > 0 is small enough.
Doing so, the principal’s marginal payoff change can be expressed as
UP (M̂) − UP (M)L
H
H=[uP (yj , mLk ) − uP (yk , mk )] − [uP (yk , mj ) − uP (yj , mj )]
ε→0
ε
lim

L
=(yk − yj )(mH
j − mk ) > 0,

which, again contradicts the hypothesized optimality of M.
Combining Case 1 and Case 2, I obtain that the principal can without loss focus on the classof monotone partitions.

A.4

Proof of Theorem 2

I prove the theorem in two steps. In the first step, I show that in the class of discrete mechanisms(i.e., D is discrete), the optimal mechanism exists and it is a finite monotone partition. In thesecond step, I prove that the finite monotone partitional mechanism is also maximal in the class of
mechanisms with an arbitrary number of delegated actions.
First, using an argument analogous to the one used by Theorem 1, I obtain that the monotone
partitional mechanism (not necessarily finite) is also maximal within the discrete class. To proveexistence, invoking Theorem 1 and Lemma 5 below allows me to focus on the space of finite monotone partitional mechanisms and denote by z = (x1 , . . . , xN , y1 , . . . , yN ) ∈ [θ, θ]N × [yP (θ), yA (θ)]Na typical element of this space. Here N is the upper bound established in Lemma 5. It is easy to seethat the space is compact under the Euclidean metric. By continuity of uP (y, θ) and Weierstrass’
theorem, an optimal mechanism exists.
Second, to prove the optimality of finite monotone partitional mechanisms in the class of mechanisms with an arbitrary delegation space, it suffices to show that for any payoff generated byan arbitrary obedient mechanism, I can construct a sequence of finite mechanisms such that theassociated payoff sequences converges to the given one (proof postponed in Lemma 6). To see thesufficiency, suppose instead no finite mechanism obtains the payoff supremum. Then by Lemma

42

6, there exists a sequence of finite obedient mechanisms {(Dn , σn )}∞n=1 such that |Dn | < |Dn+1 |,UP (Dn , σn ) < UP (Dn+1 , σn+1 ), and limn→∞ UP (Dn , σn ) attains the supremum. But by the firststep, there must exist n∗ < ∞ and |Dn∗ | ≤ N such that for all n > n∗ , UP (Dn , σn ) ≤ UP (Dn∗ , σn∗ ),a contradiction.
Lemma 5. Let M = (D, σ) be an optimal obedient mechanism that is discrete. Then, it must be
finite.
Proof of Lemma 5. Let yi−1 , yi , yi+1 be three adjacent actions induced in the mechanism. By obedience condition, it follows that mi ≤ ∆CA (yi+1 , yi ). By the alignment principle, it follows thatmi ≥ ∆CP (yi , yi−1 ). Combining the two inequalities and using the fact that cP > cA yields∆CP (yi+1 , yi ) − δ ≥ ∆CA (yi+1 , yi ) ≥ mi ≥ ∆CP (yi , yi−1 ),where δ ≡ inf y∈[yP (θ),yA (θ)] cP (y) − cA (y) > 0. Then by the mean value theorem, there existyl ∈ (yi−1 , yi ) and yh ∈ (yi , yi+1 ) such thatcP (yh ) − cP (yl ) = ∆CP (yi+1 , yi ) − ∆CP (yi , yi−1 ) ≥ δ.
Since c′P > 0 is continuous, this gives an upper bound K to the value of c′P attainable on[yP (θ), yA (θ)]. Thus, the above inequality yields

K≥

cP (yh ) − cP (yl )δ≥,yh − ylyh − ylwhich then implies
yi+1 − yi−1 > yh − yl >δ≡ ε > 0.
K

Using the fact that the delegation set is bounded by yP (θ) and yA (θ) from Lemma 2, this inequality implies that the size of any aligned and discrete mechanism must be bounded above by
l
m
P (θ)
2 yA (θ)−y+ 1, which completes the proof.
εLemma 6. Suppose that M = (D, σ) is an obedient mechanism with an arbitrary D. Then, there

43exists a sequence of finite mechanisms such thatlim UP (Dn , σn ) = UP (D, σ).

n→∞Proof of Lemma 6. Because the principal’s payoff depends on posterior beliefs only through theposterior mean, her expected payoff from an arbitrary mechanism M = (D, σ) can be written asˆ θ
UP (D, σ) =uP (y(m), m) dG(m),θwhere G is the cumulative distribution over posterior means, and y(m) ∈ D is the corresponding
action recommendation that induces m. Thus, every pair (D, σ) can be equivalently described by
the corresponding pair (G, y), which I use for the rest of the proof.
Take an arbitrary positive integer n. Let {B1n , B2n . . . , Bnn } be a partition of the posterior mean
space [θ, θ] such that
Bkn = (bnk−1 , bnk ] ∀k = 1, . . . , n,knwith bnk = n−kn θ + n θ and b0 = θ. Let M denote the support of distribution G and define a sequencenof posterior means {mnj }Jj=1 by

´
Bn

k(j)
mnj = ´m dG(m)

dG(m)
n
Bk(j)

n
n
such that k(j) is the j th partition element Bk(j)that satisfies Bk(j)∩ M ̸= ∅. By construction, I

can define a discrete distribution Gn by

Gn (m) =





0





for m ∈ [θ, mn1 ),G(bnk(j)−1 ) for m ∈ [mnj−1 , mnj ),






1
for m ∈ [mnJn , θ].

nLet mnj and mnj denote respectively the lower bound and the upper bound of Bk(j)∩ M . Define

44

y n : M → Y asy n (m) =y(mn )

if m ∈ [mnj−1 , mnj ) or [mnJn , θ] and UA (y(mnj )|mnj ) ≥ UA (y(mnj )|mnj ),y(mn )
j

if m ∈ [mnj−1 , mnj ) or [mnJn , θ] and UA (y(mnj )|mnj ) < UA (y(mnj )|mnj ).

j

The above construction allows me to obtain a sequence of finite mechanisms {(Gn , y n )}∞n=1 . Noticethat (Gn , y n ) satisfies the obedience condition. Also, both Gn and y n (·) are simple functions which
uniformly converge to G and y(·), respectively.
Now I am ready to show that the sequence of principal’s expected payoffs from {(Gn , y n )}∞n=1
converges to that from (G, m). To see this, notice that both uP (y n (m), m) and Gn (m) are uniformly
bounded, and converge uniformly to uP (y(m), m) and G(m), respectively. By bounded convergence
theorem, it then follows that
ˆ θ
lim

n→∞ θ

ˆ θ
uP (y n (m), m) dGn (m) =uP (y(m), m) dG(m),θwhich completes the proof.

A.5

Proof of Lemma 4

Proof of Lemma 4. This is an immediate consequence of Lemma 7 and 8 below.
N
Lemma 7. Let M = {(θi )Ni=1 , (yi )i=1 } be an optimal mechanism in the uniform-quadratic case.

Then, the following must holdyi − θi = yj − θj∀i, j = 1, . . . , N.

(A.10)

Proof. Notice that I only need show that (A.10) holds for any two consecutive intervals. To see
this, consider the following modified mechanism M(ε):(a) If ε > 0, then replace action recommendation yi+1 with yi throughout states θ ∈ [θi , θi + ε];
if ε < 0, replace action recommendation yi with yi+1 throughout states θ ∈ [θi + ε, θi ].

45

(b) Replace action yi+1 from D with ŷi+1 = yi+1 + ε.
It follows that obedience is satisfied by construction. Moreover, the principal’s incremental change
in payoff can be written asˆ θi +ε 


1 2
1 2
UP (M(ε)) − UP (M) =(θyi − yi ) − (θyi+1 − yi+1 ) dθ22θiˆ θi+1 
1 2
1 2+(θŷi+1 − ŷi+1 ) − (θyi+1 − yi+1 ) dθ
2
2
θi +ε
2


ε
ε
= (yi+1 − θi+1 )2 − (yi − θi )2 + [(yi − θi ) + (yi+1 − θi+1 )].
2
2(A.11)

Provided that M is optimal, I must have(yi+1 − θi+1 )2 = (yi − θi )2∀i = 1, . . . , (N − 1).

(A.12)

Otherwise, I can take the limit of ε to zero, and obtain

lim

ε→0+

UP (M(ε)) − UP (M)= (yi+1 − θi+1 )2 − (yi − θi )2 ,εorUP (M(ε)) − UP (M)= (yi − θi )2 − (yi+1 − θi+1 )2 ,
ε
ε→0−
lim

contradicting the optimality of M.
Also, observe that the second term of (A.11) must be weakly negative, i.e., (yi − θi ) + (yi+1 −θi+1 ) ≤ 0 due to the optimality of M. This observation, combined with (A.12), leads toyi − θi = yi+1 − θi+1

∀i = 1, . . . , (N − 1).

N
An immediate implication of the above proof is that any optimal mechanism M = {(θi )Ni=1 , (yi )i=1 }must satisfy yi ≤ θi for all i = 1, . . . , N , which I will use later. Notice also that, by Lemma 7,
N
an optimal mechanism {(θi )Ni=1 , (yi )i=1 } (with an abuse of notation) can be equivalently expressedN
as {(θi )Ni=1 , d}, where d = θi − yi for all i = 1, . . . , N . Fix M = {(θi )i=1 , d} and denote by46∆θi = θi − θi−1 the length of the ith interval. The principal’s ex-ante payoff from M isN ˆ θiX

1(θyi − yi2 ) dθ
2
i=1 θi−1N ˆ θiN ˆ θiXX
1 2 1 2
1 2=
(θyi − yi − θ ) dθ +
θ dθ
2
2
2
i=1 θi−1
i=1 θi−1


N
X
1
2
1
2
2
(θi − θi−1 ) d(θi − θi−1 ) − d − (θi − θi−1 ) +=2
3
6

UP (M) =

=

i=1
N
X
i=1

(A.13)

1ϕ(∆θi , d) + ,6

where ϕ(x, d) = − 16 x3 + 21 dx2 − 12 d2 x is strictly concave in x for x > 0.
Before proceeding to Lemma 8, let me first introduce the majorization relation and the maNjorization inequality. A sequence {xi }N
i=1 majorizes another sequence {x̂i }i=1 if

x1 + · · · + xi ≥ x̂1 + · · · + x̂i∀i = 1, . . . , N − 1.

andx1 + · · · + xN = x̂1 + · · · + x̂N .
N
Karamata’s inequality. Let sequences {xi }N
i=1 and {x̂i }i=1 be non-increasing and ϕ(·) denoteNa real-valued, concave function. If {x̂i }Ni=1 majorizes {xi }i=1 , thenNX

ϕ(x̂i ) ≤NX

ϕ(xi ).

i=1

i=1

I am now ready to state and prove Lemma 8.
N
Lemma 8. Let M = {(θi )Ni=1 , (yi )i=1 } be an optimal mechanism in the uniform-quadratic case.

Then, the local upward OB is binding.
Proof. There are two distinct cases: either N is odd, or N is even. I now show that the binding
condition is necessary for optimality in both cases.

47

N
Case 1. N is odd. Let M̂ = {(θ̂i )Ni=1 , (ŷi )i=1 } be an obedient mechanism that satisfies d = θ̂i − ŷiNfor some d ≥ 0. Then I can construct another mechanism M = {(θi )Ni=1 , (yi )i=1 } which satisfies d =θi −yi , and moreover, features binding local upward OB. It can be shown that θi = 1−(b+d)(N −i)for odd i, and θi = (b + d)i for even i, which implies that ∆θi = 1 − (b + d)(N − 1) for odd i, andN∆θi = (b + d)(N + 1) − 1 for even i. Denote by {xi }N
i=1 the non-increasing permutation of {∆θi }i=1 .

Consider next the mechanism M̂. For even i, OB condition implies θ̂i ≥ θi−2 + 2(b + d) ≥ · · · ≥(b + d)i = θi . For odd i, OB condition implies θ̂i ≤ θ̂i+2 + 2(b + d) ≤ · · · ≤ 1 − (b + d)(N − i) = θi .
I thus obtain ∆θ̂i = θ̂i − θ̂i−1 ≥ θi − θi−1 = ∆θi for even i and ∆θ̂i ≤ ∆θi for odd i. Likewise, INdenote by {x̂i }N
i=1 the non-increasing permutation of {∆θ̂i }i=1 .
N
It can be shown that {x̂i }Ni=1 majorizes {xi }i=1 , and hence, by Karamata’s inequality and
PN
P
1
1(A.13), I obtain UP (M) = N
i=1 ϕ(xi , d) + 6 ≥i=1 ϕ(x̂i , d) + 6 = UP (M̂).

N
Case 2. N is even. In this case, consider a mechanism M = {(θi )Ni=1 , (yi )i=1 } which satisfies

θi = N1 , yi = b + i−1
N for all i = 1, . . . , N . One can easily verify that M features binding local
upward OB. The next step is to show that the mechanism M is indeed payoff superior to all
obedient mechanisms of the same size N , and hence the claim in Lemma 8 follows.
Nˆ
To see this, consider an obedient mechanism M̂ = {(θ̂i )Ni=1 , (ŷi )i=1 } that satisfies d = θ̂i − ŷiˆ which, together
for some dˆ ≥ 0. The OB condition is equivalent to requiring θ̂i+1 − θ̂i−1 ≥ 2(b + d),P −11ˆwith the fact that N
i=1 (θ̂i+1 − θ̂i−1 ) = 1, implies d ≤ N − b. Now consider another mechanism
′
1
1
′
′
′ N
ˆ
M̂′ = {{(θˆi )Ni=1 , (ŷi )i=1 } with θ̂i = N and ŷi = N − d. It follows that M̂ is obedient. Thenan analogous majorization argument to that used to prove Case 1 can be applied to show that
UP (M̂′ ) ≥ UP (M̂). The argument is similar and hence omitted.
So all it needs to show is then that M yields a higher payoff than M̂′ . Note that by (A.13),
P
PN
1
1
1
1 1
1 ˆ
′
ˆ
UP (M) = N
i=1 ϕ( N , N − b) + 6 ≥i=1 ϕ( N , d) + 6 = UP (M̂ ) due to the fact that d ≤ N − band ϕ(x, d) is increasing in d.

48

A.6

Proof of Proposition 2

It follows from Lemma 4 that one only needs to focus on the first two partition elements. The
problem of finding the suboptimal mechanism for each size N ≤ N (b) reduces to one of the following
two trivial programs:(1) If N is an odd number, thenN +1max
θ1 ,θ2 ,y1 ,y2
2

ˆ θ1
0

N −1
uP (y1 , θ) dθ +2ˆ θ2
uP (y2 , θ) dθ

(A.14)

θ1subject to (i) y1 − θ1 = y2 − θ2 , and (ii) N 2+1 θ1 + N 2−1 θ2 = 1.
(2) If N is an even number, thenN
θ1 ,θ2 ,y1 ,y2 2

ˆ θ1

max

uP (y1 , θ) dθ +
0

N
2

ˆ θ2
uP (y2 , θ) dθ

(A.15)

θ1subject to (i) y1 − θ1 = y2 − θ2 , and (ii) N2 θ1 + N2 θ2 = 1.
Therefore, it is straightforward to verify that (2) and (2) are solutions to programs (A.14)
and (A.15), respectively.

A.7

Proof of Theorem 3

Fix an arbitrary agent Ã with cÃ . Define the associated set of uniformly less-biased agents bycÃ = {cA : cA (y) ≥ cÃ (y), ∀y ∈ Y }, where cÃ is endowed with the sup norm ∥ · ∥∞ . The induced
metric is then d(cA , cA′ ) = ∥cA − cA′ ∥∞ for any cA , cA′ ∈ cÃ .
Let M∗Ã of size NÃ∗ and M∗A of size NA∗ be the optimal mechanisms with cÃ and cA , respectively.
It follows from Theorem 4 that M∗A achieves a weakly higher expected payoff than M∗Ã , i.e.,
UP (M∗A ) ≥ UP (M∗Ã ).
To prove NA∗ ≥ NÃ∗ for any cA ∈ cÃ , note that it suffices to check “local monotonicity”, i.e.,NA∗ ≥ NÃ∗ for ∥cA − cÃ ∥∞ sufficiently small. Suppose by contradiction that NA∗ < NÃ∗ . Take M∗Ã,N ∗A∗
as the optimal mechanism of size NA∗ with agent cÃ and it is obvious that UP (M∗Ã ) > UP (MÃ,N
∗ ).
A

49

Let F easN (cA ) be the set of all obedient mechanisms of arbitrary size N with agent cA :

θi−1 + θi
CA (yi+1 ) − CA (yi )N
M = {(θi )N≤i=1 , (yi )i=1 } :2
yi+1 − yi.

By definition, it follows that F easN (cA ) is compact and continuous in cA . Since UP : [θ, θ]N ×[y, y]N → R is continuous, Berge’s Maximum Theorem implies that the optimal expected payoff
function
VPN (cA ) ≡max

M∈F easN (cA )UP (M)(A.16)

is continuous. Take M∗A,N to be an optimal solution for (A.16) by restricting the partition size to
be N . Thus, for sufficiently small ε > 0 with d(cA , cÃ ) < ε, there exists δ > 0 small enough suchthat UP (M∗A,N ) = VPN (cA ) ≤ VPN (cÃ ) + δ = UP (M∗Ã,N ) + δ. Setting N = NA∗ , one then obtains
UP (M∗A ) ≤ UP (M∗Ã,N ∗ ) + δ < UP (M∗Ã ). A contradiction.
A

50

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