#  Incentivizing Agents through Ratings††thanks: I am indebted to Bart Lipman,
Chiara Margaria, Juan Ortner, and Krishna Dasaratha for their invaluable help
and support throughout this project. I thank Nageeb Ali, Ian Ball, Nina
Bobkova, Tilman Borgers, Tan Gan, Marina Halac, Xiao Lin, Albert Ma, Teddy
Mekonnen, Dilip Mookherjee, Andy Newman, Axel Niemeyer, Jawaad Noor, Marco
Ottaviani, Harry Pei, Alex Smolin, Mark Whitmeyer, and participants at BC-BU-
Brown Theory Workshop, EC’24, and 35th Stony Brook Game Theory Conference for
their helpful comments.

Peiran Xiao  Department of Economics, Boston University. Email<EMAIL_ADDRESS>
(

[Link to the latest version] )

###### Abstract

I study the optimal design of ratings to motivate agents’ performance or
investment in product quality. The principal designs a rating scheme that maps
their quality (performance) to possibly stochastic scores. Agents have private
information about their abilities (costs of effort/quality) and choose their
quality. The market observes the scores and offers a wage equal to the agent’s
expected quality [resp. ability].

I first show that an incentive-compatible interim wage function can be induced
by a rating (i.e., feasible) if and only if it is a mean-preserving spread of
quality [resp. ability] in the quantile space. Thus, I reduce the principal’s
rating design problem to the design of a feasible interim wage. When
restricted to deterministic ratings, the optimal rating design is equivalent
to the optimal delegation with a participation constraint (AmadorBagwell2022).
Using optimal control theory, I provide necessary and sufficient conditions
under which lower censorship, especially a simple pass/fail test, is optimal
within deterministic ratings. In particular, when the principal elicits
maximal effort (quality), lower censorship [resp. pass/fail] is optimal if the
density is unimodal [resp. increasing]. I also solve for the optimal
deterministic ratings beyond lower censorship for general distributions and
preferences. For general ratings, I provide sufficient conditions under which
lower censorship remains optimal. In the effort-maximizing case, a pass/fail
test remains optimal if the density is increasing.

###### Contents

  1. 1 Introduction
  2. 2 The Model
    1. 2.1 Setup
    2. 2.2 Revelation Principle and Feasibility
    3. 2.3 Principal’s Problem
    4. 2.4 Notations and Assumptions
    5. 2.5 Discussion on Assumptions
  3. 3 Optimal Deterministic Ratings
    1. 3.1 Principal’s Problem
    2. 3.2 Equivalence to Delegation
    3. 3.3 When is lower censorship optimal?
      1. 3.3.1 Optimal exclusion level
      2. 3.3.2 Sufficient (and necessary) conditions
      3. 3.3.3 Intuition for the Results
    4. 3.4 Linear Delegation
      1. 3.4.1 Quality Maximization
      2. 3.4.2 A “Twin” Objective Function
    5. 3.5 Relationship to the Literature
    6. 3.6 Beyond Lower Censorship: Optimal Deterministic Ratings
    7. 3.7 Discussion on Transfers
  4. 4 Optimal General Ratings
    1. 4.1 Principal’s Problem
    2. 4.2 When are deterministic ratings optimal?
      1. 4.2.1 Quality Maximization
    3. 4.3 When are stochastic ratings optimal?
    4. 4.4 With a Fixed Transfer
  5. 5 Extension: Ability Signaling
    1. 5.1 Setup
    2. 5.2 Optimal Ratings without Transfers
    3. 5.3 Optimal Ratings with a Fixed Transfer
  6. 6 Conclusion

##  1 Introduction

In many economic applications, the principal seeks to motivate agents’
performance or investment in product quality, but monetary transfers between
them are not allowed or limited. Instead, the principal can design a rating (à
la Blackwell) that maps agents’ quality (performance) to possibly stochastic
scores. Agents have private information on their abilities—that is, their
costs of effort/quality provision—and choose their quality levels or opt out
of investing any effort. When the market rewards them based on the perception
of their endogenous quality (performance) or exogenous abilities, performance
ratings can provide reputational incentives to agents.

For example, in scenarios where students invest productive efforts to improve
their quality (human capital), a school designs a grading rule that reveals
students’ endogenous quality to the job market to incentivize learning or
human capital accumulation. A certifier (e.g., credit rating or government
agencies) designs a quality certification to capitalize on certification fees
while motivating firms’ investment in product quality.111 Aside from monopoly
certifiers who only cares about certification fees (AlbanoLizzeri2001),
regulator certifiers may maximize a weighted sum of firms’ profits and quality
(BizzottoHarstad2023).  Employers (e.g., pre-doc positions) may pay a fixed
wage to employees and induce efforts through performance ratings (e.g.,
recommendation letters) that reveal their performance and abilities to future
employers.222 Future employers may care about performance (in addition to
abilities) because of learning by doing.  In these examples, a competitive
market offers agents a wage or price equal to the expected quality (or
ability) based on the rating, whereas contingent transfers between the
principal and agents are prohibited by law or infeasible in practice. Such
restrictions on transfers can also arise from economic constraints, such as
incentive compatibility or budget balance. The certification fee, for
instance, usually has to be an upfront flat fee as required by law; meanwhile,
this is also because a flat certification fee is ex-post incentive compatible
in that the certifier has no incentives to tamper with the rating.333 In 2008,
New York Attorney General Andrew Cuomo reached an agreement with Wall Street’s
three major credit-rating agencies that they will receive upfront fees
regardless if they are hired to assign a rating. Before the agreement, credit
rating agencies only received fees if a deal was completed, so they had an
incentive to assign high ratings (Reuters2008).

What can the principal achieve with ratings instead of contingent transfers?
In this paper, I start with the case where the competitive market values the
agent’s endogenous quality (performance) and the principal cannot make any
transfer. First, I reduce the principal’s rating design problem to the design
of an incentive-compatible _interim_ wage function subject to a feasibility
constraint that it must be a mean-preserving spread of quality. Then, within
deterministic ratings, because the interim wage always equals the quality, the
optimal rating design problem is equivalent to the optimal delegation problem
with voluntary participation (AmadorBagwell2022). Using optimal control
methods, I provide necessary and sufficient conditions under which lower
censorship, especially a simple pass/fail test, is optimal within
deterministic ratings. In particular, when the principal elicits maximal
effort (quality), lower censorship [resp. pass/fail] is optimal if the density
is unimodal [resp. increasing]. Further, I solve for the optimal deterministic
ratings beyond lower censorship for general distributions and preferences.
Next, for general ratings, I provide sufficient conditions under which lower
censorship remains optimal. In the effort-maximizing case, a pass/fail test
remains optimal if the density is increasing. I also consider the ability
signaling case where the market values the agent’s exogenous ability instead;
in other words, the agent’s effort is manipulative rather than productive. In
addition, I extend the model to allow for a fixed (and contingent) transfer.

##### Literature Review.

A wide range of literature has studied the optimal design of ratings to
motivate agents. A strand of literature assumes market values the agent’s
_endogenous_ action—i.e., quality, effort, or performance (AlbanoLizzeri2001;
SaeediShourideh2020; SaeediShourideh2023; Zapechelnyuk2020;
RodinaFarragut2020; BoleslavskyKim2021). In other words, the agent’s effort is
productive. AlbanoLizzeri2001 consider a monopoly certifier who maximizes the
expected certification fee and find the fee-maximizing rating is a noisy test
that fully reveals the quality with some probability and outputs an (almost)
uninformative signal otherwise. In SaeediShourideh2020, the principal
maximizes a weighted average of the seller’s revenue. They also use an interim
approach and find that the low-quality optimal rating is a noisy test as in
AlbanoLizzeri2001, the high-quality optimal rating is fully revealing, and the
mid-quality optimal rating involves fully revealing, bunching, and then
partially revealing. Zapechelnyuk2020 studies the optimal _deterministic_
quality certification to incentivize sellers’ investment in product quality,
where the principal (i.e., regulator) has a state-independent preference for
quality arising from consumer surplus maximization and the consumer’s binary
decision. He assumes the distribution of the consumer’s outside option takes
the form of power functions so that the problem can be translated into a
_linear_ persuasion problem.444 By contrast, my method allows for state-
dependent preferences and does not involve the assumption to translate it into
a linear persuasion problem. See the methodology part later.  He shows
sufficient (but not necessary) conditions for quality assurance (i.e., lower
censorship) and pass/fail certifications to be optimal. RodinaFarragut2020
assume the principal maximizes the agent’s effort (quality) and characterize
the properties of the optimal deterministic grading rule if the distribution
is “sufficiently” concave, convex, or unimodal. My results encompass these
(principal’s) objective functions and are sharper in the effort maximization
case as I obtain the necessary and sufficient conditions for the optimal
deterministic ratings to be lower censorship, pass/fail, or fully revealing
above a minimum standard that every type reaches.555 Additionally, in
SaeediShourideh2023, the agent’s effort increases quality stochastically, so
effort and quality can no longer be used interchangeably, in contrast to other
papers where effort determines quality deterministically. In
BoleslavskyKim2021, the agent’s effort improves the _distribution_ of his
quality (referred to as “state” in their paper), in contrast to the
aforementioned papers in which effort directly improves quality.

Another strand of literature, however, assumes the market values the
_exogenous_ abilities (types) à la Spence1973’s (Spence1973) signaling model
and Holmstrom1999’s (Holmstrom1999) career concern model (Rayo2013;
Zubrickas2015; Rodina2020; HornerLambert2021; OnuchicRay2023;
CamboniNiuPai2024). Under this assumption, the agent’s effort is manipulative
(i.e., signaling) rather than productive. In Rayo2013 and Zubrickas2015, the
agents are rewarded based on the market’s perception of their abilities, and
the principal designs a deterministic rating to maximize the agent’s effort.
My results (in Section 5) extend these papers by allowing for stochastic
ratings and general objective functions. Similarly, in DubeyGeanakoplos2010,
students care about their ordinal status or rankings in class, and the
principal wants to induce them to exert a binary effort. In a dynamic career
concerns model, HornerLambert2021 show the effort-maximizing rating is a
linear function of past observations.

At the intersection of the two assumptions, OnuchicRay2023 assume the market
values learning efforts in addition to inherent ability, thereby allowing for
both productive and manipulative effort. They focus on the deterministic
ratings (i.e., monotone partitions) that maximize the expected tuition fee,
which is equal to the expected payoff (before fees) of the student with the
lowest prior at the ex-ante stage. They also characterize sufficient
conditions under which the optimal deterministic rating is lower censorship.
Their model is distinct from the previous papers (and this paper) in that
agents have heterogeneous priors. With common priors, the objective would be
equivalent to maximizing the agent’s effort because the expected ability would
be a constant component of the student’s expected payoff.

In addition to the aforementioned papers, the literature has investigated the
benefit of coarse grading. DubeyGeanakoplos2010 find when students care about
rankings in class, the optimal grading to motivate them is coarse (e.g., A, B,
C) if their abilities are “disparate”—i.e., a student can achieve a higher
score than one another even when he shirks and the other works.
HarbaughRasmusen2018 find coarse grading can be the most informative to the
receiver because it leads to more participation that can outweigh coarser
reporting.

The paper is also related to the literature on optimal delegation with limited
transfers (e.g. AlonsoMatouschek2008; AmadorBagwell2013). AmadorBagwell2013
and AmbrusEgorov2017 also consider nonmonetary incentives through money
burning. AmadorBagwell2022 study a delegation problem of regulating a
monopolist without transfers and characterize the sufficient conditions for a
price-cap allocation to be optimal. In contrast to the previous works, the
principal needs to respect the agent’s participation constraint because an
outside option must be included in the delegation set, which complicates the
problem by creating a jump in allocation at the cutoff type. Under the
restriction to deterministic ratings, I show the rating design problem is
equivalent to their problem and also improve upon their results, which allows
for a pass/fail test (corresponding to a bang-bang allocation where firms
either exit or set the price at the price cap) to be optimal.666 See, e.g.,
HalacYared2022 for the optimality of bang-bang solutions.
KartikKleinerVanWeelden2021 study delegation in veto-bargaining with an
outside option (status quo). They assume the principal (i.e., delegator) has a
state-independent preference and identify the necessary and sufficient
conditions for full delegation, no compromise (i.e., bang-bang), and interval
delegation to be optimal. My model setup (under deterministic ratings) is
closer to AmadorBagwell2022 (i.e., expertise-based delegation) as I allow for
state-dependent principal preferences and more general (non-quadratic) agent
preferences. On the other hand, in terms of the results, it is closer to
KartikKleinerVanWeelden2021 in that I also identify the necessary and
sufficient conditions for lower censorship (in particular, bang-bang) to be
optimal.

Relatedly, KolotilinZapechelnyuk2019 show that the delegation problem is
generally equivalent to a persuasion problem where the principal restricts the
agent’s _information_ , when the delegation mechanisms are deterministic and
the information structures are monotone partitions. Based on the equivalence,
they translate a linear delegation problem (with or without participation
constraints) to the equivalent linear persuasion problem and solve it using
methods in Kolotilin2018 and DworczakMartini2019. By contrast, the principal
in my model manipulates the information sent to the _market_ instead of the
agent, whereas the agent is privately informed, so the equivalence is distinct
from theirs. Closer to my model, Zapechelnyuk2020 also discovers the
equivalence between optimal quality certification and delegation. Assuming a
state-independent preference of the principal and a power function
distribution of the consumer’s outside option, he translates the problem into
a linear persuasion problem and solves it à la KolotilinZapechelnyuk2019 and
KolotilinMylovanovZapechelnyuk2022.

Another closely related paper is by Rappoport2022, who studies a delegation
problem when agents have career concerns in that the market rewards the agent
more when it perceives a higher type. However, in his model, the market
observes the agent’s action directly rather than through a rating designed by
the principal.

The methods used in reducing the rating design problem to the design of
interim wage functions are based on the characterization of the extreme points
of the mean-preserving spread of a function in KleinerMoldovanuStrack2021.
Moreover, the method used in solving the optimal _deterministic_ ratings stems
from the Lagrangian methods in the delegation literature (e.g.,
AmadorWerningAngeletos2006; AmadorBagwell2013) but improves upon the results
of AmadorBagwell2022 using optimal control tools (see also Xiao2023a). In the
quality/effort maximization case, it is also related to the linear
optimization under majorization constraints in KleinerMoldovanuStrack2021.
Although reminiscent of the techniques in the persuasion literature
(Kolotilin2018; DworczakMartini2019; KolotilinMylovanovZapechelnyuk2022), this
method tackles the delegation problem directly without invoking the
equivalence established by KolotilinZapechelnyuk2019, Furthermore, in contrast
to their methods, this method also allows the principal’s marginal payoff to
be nonlinear in the agent’s action, in which case the sender’s payoff is
nonlinear in the state (so that not only the posterior mean matters) in the
equivalent persuasion problem; therefore, the methods in the persuasion
literature (Kolotilin2018; DworczakMartini2019) may not apply. In addition, it
is not clear how their methods can be applied when stochastic ratings are
allowed, whereas my method can readily extend to general ratings through the
interim wage function (where the interim wage is not necessarily equal to but
rather a mean-preserve spread of quality in the quantile space).

#####

The paper makes three main contributions to the literature. First, I provide a
unified framework to study the optimal rating design to motivate agents, in
which the agent’s effort can be either productive or signaling, and ratings
can be potentially stochastic. This yields more general results than the
existing literature (e.g., Zubrickas2015; RodinaFarragut2020). Second, I show
an equivalence between the optimal deterministic rating design and the optimal
delegation with a participation constraint (AmadorBagwell2022). Using optimal
control methods, I improve upon their results and provide necessary and
sufficient conditions for lower censorship, which corresponds to a price-cap
allocation in delegation. Additionally, through the equivalence established by
KolotilinZapechelnyuk2019, the results also contribute to persuasion
literature, especially in cases where the sender’s payoffs are nonlinear in
the state. Third, I introduce an interim approach to information design that
reduces the problem to the optimization over interim wage functions rather
than ratings themselves. This approach simplifies the analysis and provides a
new perspective on optimal rating design problems.

The rest of the paper is organized as follows. In Section 2, I introduce the
model setup and the feasibility theorem and present the principal’s problem.
Section 3 focuses on deterministic ratings, in which I show the equivalence to
delegation with a participation constraint and solve for the optimal
deterministic ratings. In Section 4, I consider the optimal general
(potentially stochastic) ratings and also extend the model to allow for a
fixed transfer (e.g., certification fee).777 In Appendix LABEL:transfer, I
show the design of ratings does not matter if contingent transfers were
allowed and extend the model to allow for a fixed transfer.  Section 5
explores the ability signaling case where the market values the agent’s
exogenous ability.

##  2 The Model

###  2.1 Setup

An agent (he) has a private type $\theta$, which has a continuous distribution
$F$ with bounded support $\Theta=[{\underline{\theta}},{\bar{\theta}}]$ and
continuous density $f(\theta)$. He can choose a quality level $q\in
Q\equiv[0,q_{\max}]$ at cost $c(q,\theta)$, which is twice continuously
differentiable and satisfies $c_{q}>0$, $c_{\theta}<0$, $c_{qq}>0$,
$c_{q\theta}<0$. Assume $c(0,\theta)=c_{q}(0,\theta)=0$.888 This assumption
can be weakened to: For any $\theta\in[\underline{\theta},\bar{\theta}]$,
there exists $\underline{q}(\theta)\in Q$ such that
$c(\underline{q}(\theta),\theta)=c_{q}(\underline{q}(\theta),\theta)=0$.

The principal has a utility function given by $y(q,\theta)$, which is twice
continuously differentiable and satisfies $y_{qq}\leq 0$ and $y(0,\theta)=0$
(see Assumption 1 for the assumption on $y_{q}$). Assume the principal does
not observe $\theta$, may (or may not) observe $q$,999 It does not matter
whether the principal observes $q$ as long as the rating takes it as input. If
the principal observes $q$, the rating is a garbling of it. Otherwise, the
rating reveals $q$ by outputting a score $s\in S$.  and cannot make contingent
transfers to the agent. Instead, she can design a rating (Blackwell
experiment) $\pi\colon Q\to\Delta(S)$ to reveal his quality $q$ and type
$\theta$ to the market and thus provide reputational incentives.

The market values the agent’s quality $q$ at $v=v(q)$, which satisfies
$v(0)=0$, $v^{\prime}(q)>0$, and $c^{\prime}(q)/v^{\prime}(q)$ is increasing
in $q$. Assume the market is competitive and offers the agent a wage equal to
the expected value of his quality. If the agent takes the test, the market
observes the score $s\in S$ and offers a wage
$\tilde{w}_{s}=\operatorname{\mathbf{E}}[v(q)|s]$. Thus, the agent’s interim
wage is
$\hat{w}(q)=\operatorname{\mathbf{E}}_{s\sim\pi(q)}[\tilde{w}_{s}]$,101010
When $v(q)=q$, the interim wage is the _interim posterior mean_ of quality. To
see this, denote by $\mu_{s}\in\Delta(Q)$ the posterior belief induced by $s$;
then $\tilde{w}_{s}=\operatorname{\mathbf{E}}_{\mu_{s}}[q]$ is the posterior
mean and
$\hat{w}(q)=\operatorname{\mathbf{E}}_{s\sim\pi(q)}[\operatorname{\mathbf{E}}_{\mu_{s}}[\tilde{q}]]$
is the _interim_ posterior mean.  and he chooses $q\in Q$ to maximize his
payoff $u(q,\theta)=\hat{w}(q)-c(q,\theta)$. If the agent does not take the
test, the market observes a null signal $s=\varnothing$ and offers a wage
$\tilde{w}_{\varnothing}$.

See Section 2.5 for the discussion on the model assumptions.

##### Timing.

The timing of the game is as follows.

  1. 1.

Agents privately know their types
$\theta\in[\underline{\theta},\bar{\theta}]$.

  2. 2.

The principal commits to a rating $\pi\colon Q\to\Delta(S)$.

  3. 3.

Agents choose quality $q\in Q$ and whether to take the test or not. If an
agent takes the test, the market observes a signal $s$ drawn from $\pi(q)$.
Otherwise, the market observes a null signal $s=\varnothing$.

  4. 4.

The market updates her belief of the agent’s quality to $\mu_{s}$ according to
the Bayes’ rule and sets the wage equal to the expected value, i.e.,
$\tilde{w}_{s}=\operatorname{\mathbf{E}}[v(q)|s]\equiv\operatorname{\mathbf{E}}_{\mu_{s}}[v(q)]$.

##### Solution Concept.

Weak Perfect Bayesian Equilibrium (wPBE) or Sequential Equilibrium. Because
the quality (effort) is costly, in any equilibrium, if an agent does not take
the test, he must choose $q=0$. Thus, in any equilibrium, the market’s belief
must be that he has chosen $q=0$, and the market will offer
$\tilde{w}_{\varnothing}=0$ because otherwise, the agents whose payoffs are
just above zero will deviate to $q=0$ and not take the test.

######  Lemma 1.

In any equilibrium, if an agent does not take the test, he chooses $q=0$, and
the market offers him $\tilde{w}_{\varnothing}=0$.

######  Remark 1.

Under the weaker assumption in Footnote 8, Lemma 1 becomes: In any
equilibrium, if an agent of type $\theta$ does not take the test, he chooses
$q=\underline{q}(\theta)$ such that $c(\underline{q}(\theta),\theta)=0$, and
the market offers him
$\tilde{w}_{\varnothing}=\underline{q}(\underline{\theta})$.

###  2.2 Revelation Principle and Feasibility

In the setup, I consider a _rating mechanism_ where the principal commits to a
rating $\pi\colon Q\to\Delta(S)$ and the agent chooses quality $q\in Q$. The
same allocation can be implemented by a _direct mechanism_ where each agent
reports his type $\theta$, and then must choose quality $q(\theta)$ and
receive a (possibly stochastic) score $s(\theta)$ drawn from $\pi(q(\theta))$.
Moreover, if an allocation is implementable by a direct mechanism, there also
exists a rating (or indirect) mechanism $\pi(q)$ that implements it. In other
words, choosing $q$ is equivalent to reporting $\theta$ (see
FudenbergTirole1991, Remark on pp. 257).

Indeed, the rating mechanism can be viewed as an implementation of the direct
mechanism that does not require the agent’s quality $q$ to be observable by
the principal, as long as it is taken as input by the rating $\pi$.

######  Definition 1.

A _rating mechanism_ is an indirect mechanism where the principal commits to a
rating $\pi\colon Q\to\Delta(S)$ and the agent chooses quality $q\in Q$.

Say a rating $\pi(q)$ _induces_ an interim wage $\hat{w}(q)$ if
$\hat{w}(q)=\operatorname{\mathbf{E}}_{s\sim\pi(q)}[\operatorname{\mathbf{E}}[v(\tilde{q})|s]]$.
Instead of optimizing over Blackwell experiments $\pi\colon Q\to\Delta(S)$ or
score functions $s\colon[\underline{\theta},\bar{\theta}]\to\Delta(S)$, it is
easier to work with the interim wage functions $\hat{w}:Q\to\mathbb{R}_{+}$
that can be induced by an experiment.

######  Definition 2.

An interim wage function $\hat{w}(q)$ is _feasible_ if there exists a
Blackwell experiment $\pi\colon Q\to\Delta(S)$ that induces it, i.e.,
$\hat{w}(q)=\operatorname{\mathbf{E}}_{s\sim\pi(q)}[\operatorname{\mathbf{E}}[v(\tilde{q})|s]]$.

It follows immediately that an allocation $q(\theta)$ is implementable by a
rating mechanism $\pi(q)$ if and only if it is implementable by a feasible
interim wage function
$\hat{w}(q)=\operatorname{\mathbf{E}}_{s\sim\pi(q)}[\operatorname{\mathbf{E}}[v(\tilde{q})|s]]$.
Furthermore, the equivalence extends to a (reduced-form) direct mechanism
$\\{q(\theta),w(\theta)\\}$ consisting of a quality function
$q\colon[\underline{\theta},\bar{\theta}]\to Q$ and an interim wage function
$w\equiv\hat{w}\circ
q\colon[\underline{\theta},\bar{\theta}]\to\mathbb{R}_{+}$.

######  Definition 3.

A (reduced-form) direct mechanism $\\{q(\theta),w(\theta)\\}$ is _feasible_ if
there exists a rating $\pi(q)$ such that
$w(\theta)=\hat{w}(q(\theta))\equiv\operatorname{\mathbf{E}}_{s\sim\pi(q(\theta))}[\operatorname{\mathbf{E}}[v(q)|s]]$.

######  Lemma 2.

An allocation $q(\theta)$ is implementable by the rating mechanism $\pi(q)$ if
and only if it is implementable by a direct mechanism
$\\{q(\theta),w(\theta)\\}$.

######  Remark 2.

The implementation result is reminiscent of the Laffont-Tirole cost
reimbursement model (see LaffontTirole1993, Chapter 1), where the optimal
allocation implemented by a direct mechanism (analogous to
$\\{q(\theta),w(\theta)\\}$) can also be implementable by an indirect
mechanism (analogous to $\hat{w}(q)$), and vice versa.

######  Remark 3.

The lemma implies providing a menu of tests
$\hat{\pi}(\theta)\in\Pi\equiv\\{\pi\colon Q\to\Delta(S)\\}$ makes no
difference. To see this, consider the direct mechanism consisting of
$q\colon[\underline{\theta},\bar{\theta}]\to Q$ and
$\hat{\pi}\colon[\underline{\theta},\bar{\theta}]\to\Pi$. This can be
replicated by constructing a rating such that
$\pi(q(\theta))=\hat{\pi}(\theta)(q(\theta))$ for all
$\theta\in[\underline{\theta},\bar{\theta}]$, which induces the same interim
wage
$w(\theta)\equiv\operatorname{\mathbf{E}}_{s\sim\pi(q(\theta))}[\operatorname{\mathbf{E}}[v(\tilde{q})|s]]=\operatorname{\mathbf{E}}_{s\sim\hat{\pi}(\theta)(q)}[\operatorname{\mathbf{E}}[v(\tilde{q})|s]]\equiv\hat{w}(q(\theta);\hat{\pi}(\theta))$.
For each $\theta$, $\hat{\pi}(\theta)$ will penalize off-path deviations to
$q$ that no types would choose in the direct mechanism so that they will never
be chosen in the rating mechanism either.

######  Lemma 3.

A direct mechanism $\\{q(\theta),w(\theta)\\}$ is incentive compatible if and
only if

  * •

$U(\theta)\equiv
w(\theta)-c(q(\theta),\theta)=-\int_{\underline{\theta}}^{\theta}c_{\theta}(q(x),x)dx+\underline{U}$,
and

  * •

$q(\theta)$ is increasing,

where
$\underline{U}=w(\underline{\theta})-c(q(\underline{\theta}),\underline{\theta})$.
The conditions also imply $w(\theta)$ is increasing.

An increasing allocation $q(\theta)$ is incentive-compatible but not
necessarily feasible because it may be unable to be induced by a rating. Now I
characterize the necessary and sufficient condition for the feasibility of an
incentive-compatible direct mechanism, which allows us to optimize over
interim wages induced by experiments rather than experiments themselves.

######  Theorem 4.

An incentive-compatible direct mechanism $\\{q(\theta),w(\theta)\\}$ is
feasible if and only if $w(\theta)$ is a mean-preserving spread of
$v(q(\theta))$ in the quantile space, that is,

  1. (i)

$\int_{\underline{\theta}}^{\theta}w(\theta^{\prime})\mathop{}\\!\mathrm{d}F(\theta^{\prime})\geq\int_{\underline{\theta}}^{\theta}v(q(\theta^{\prime}))\mathop{}\\!\mathrm{d}F(\theta^{\prime})$
for all $\theta\in[\underline{\theta},\bar{\theta}]$ (MPS),

  2. (ii)

$\int_{\underline{\theta}}^{\bar{\theta}}w(\theta)\mathop{}\\!\mathrm{d}F(\theta)=\int_{\underline{\theta}}^{\bar{\theta}}v(q(\theta))\mathop{}\\!\mathrm{d}F(\theta)$
(Bayesian plausibility, henceforth BP).

A similar version of the theorem is also proved in SaeediShourideh2020.111111
See also of Rodina2020 and SaeediShourideh2023 for a characterization theorem
when investing effort increases quality stochastically.  The theorem is
reminiscent of the symmetric version of Border’s theorem (Maskin-Riley
condition) in reduced-form mechanism design (see, e.g., Border1991;
MaskinRiley1984; Matthews1984; KleinerMoldovanuStrack2021). The proof is à la
KleinerMoldovanuStrack2021 (see Xiao2023, Theorem 1).

For simplicity, I assume hereafter that the market values $v(q)=q$, so the
feasibility condition becomes “$w(\theta)$ is a mean-preserving spread of
$q(\theta)$ in the quantile space.”

######  Corollary 1.

If $w^{\prime}(\theta)\leq q^{\prime}(\theta)$ on
$[\underline{\theta},\bar{\theta}]$, then (BP) $\implies$ (MPS), so
feasibility $\iff$ (BP).

###### Proof.

Because $w(\theta)-q(\theta)$ is decreasing, there exists a point
$\hat{\theta}\in[\underline{\theta},\bar{\theta}]$ such that $w(\theta)\leq
q(\theta)$ for all $\theta>\hat{\theta}$ and $w(\theta)\geq q(\theta)$ for all
$\theta<\hat{\theta}$ (i.e., $w$ single-crosses $q$ from above). Thus,
$D(\theta)\equiv\int_{\underline{\theta}}^{\theta}(w(\theta)-q(\theta))\mathop{}\\!\mathrm{d}F(\theta)\geq
0$ for all $\theta<\hat{\theta}$, and
$D^{\prime}(\theta)=(w(\theta)-q(\theta))f(\theta)\leq 0$ for all
$\theta>\hat{\theta}$. Because $D(\bar{\theta})=0$ (BP), we have
$D(\theta)\geq 0$ (MPS). ∎

Because FOC implies $w^{\prime}(\theta)=c_{q}(q,\theta)q^{\prime}(\theta)$,
the corollary implies that if $c_{q}(q,\theta)\leq 1$, then feasibility $\iff$
(BP).

######  Remark 4.

If $w^{\prime}(\theta)\leq q(\theta)$ on $[\underline{\theta},\bar{\theta}]$,
then $\hat{w}^{\prime}(q)\leq 1$ on $[q(\underline{\theta}),q(\bar{\theta})]$,
and one can construct a rating that induces $\hat{w}(q)$ à la
AlbanoLizzeri2001. Define $\hat{q}$ as the (unique) fixed point of
$\hat{w}(q)$, and define $\beta(q)=\frac{q-\hat{w}(q)}{q-\hat{q}}$ for
$q\neq\hat{q}$ and $\beta(\hat{q})=0$. Then, $\hat{w}(q)$ can be induced by
the rating $(\pi,S)$ consisting of
$S=[q(\underline{\theta}),q(\bar{\theta})]\cup\\{\text{pass},\text{fail}\\}$
and

$\pi(q)=\begin{cases}q&\text{ w.p. }1-\beta(q)\\\ \text{pass}&\text{ w.p.
}\beta(q)\end{cases},\quad\forall
q\in[q(\underline{\theta}),q(\bar{\theta})],$ $\pi(q)=\text{fail}\text{ w.p.
}1,\quad\forall q\notin[q(\underline{\theta}),q(\bar{\theta})].$

###  2.3 Principal’s Problem

The principal’s problem can now be formulated as a mechanism design problem
without standard flexible transfers but with a “quasi-transfer” $w(\theta)$
subject to the feasibility constraint.

$\displaystyle\text{[P]}\quad\max_{q(\theta),w(\theta)}\int_{\underline{\theta}}^{\bar{\theta}}y(q(\theta),\theta)\mathop{}\\!\mathrm{d}F(\theta)$
(1)

subject to

$\displaystyle
w(\theta)-c(q(\theta),\theta)=-\int_{\underline{\theta}}^{\theta}c_{\theta}(q(x),x)\mathop{}\\!\mathrm{d}x+\underline{U}$
(IC-Env) (2) $\displaystyle q(\theta)\mbox{ increasing}$ (IC-Mon) (3)
$\displaystyle w(\theta)-c(q(\theta),\theta)\geq 0$ (IR) (4)
$\displaystyle\int_{\underline{\theta}}^{\bar{\theta}}w(\theta)\mathop{}\\!\mathrm{d}F(\theta)=\int_{\underline{\theta}}^{\bar{\theta}}q(\theta)\mathop{}\\!\mathrm{d}F(\theta)$
(BP) (5)
$\displaystyle\int_{\underline{\theta}}^{\theta}w(\theta^{\prime})\mathop{}\\!\mathrm{d}F(\theta^{\prime})\geq\int_{\underline{\theta}}^{\theta}q(\theta^{\prime})\mathop{}\\!\mathrm{d}F(\theta^{\prime}),\,\forall\theta\in\Theta$
(MPS) (6)

As in AmadorBagwell2013; AmadorBagwell2022, the standard approach (i.e.,
substituting the envelope constraint into the objective, maximizing the
resulting expression, and checking the monotonicity of the solution) is
infeasible here because there is no transfer function available to ensure that
the solution satisfies the incentive compatibility.

###  2.4 Notations and Assumptions

Assume the market values $v(q)=q$. Assume the cost is multiplicatively
separable, i.e., $c(q,\theta)=c(q)/\theta$, where $c^{\prime}(q)>0$,
$c^{\prime\prime}(q)\geq 0$, and $c(0)=c^{\prime}(0)=0$.

Define the agent’s quality choice under full revelation ($\hat{w}(q)=q$) as

$q_{f}(\theta)=\operatorname*{arg\,max}_{q\in Q}q-c(q)/\theta\iff
c^{\prime}(q_{f}(\theta))=\theta.$ (7)

Define the principal-optimal quality as
$q_{e}(\theta)=\operatorname*{arg\,max}_{q\in Q}y(q,\theta)$. I assume that
the agent has a downward bias, i.e., $q_{f}(\theta)\leq q_{e}(\theta)$.121212
When $q_{f}(\theta)>q_{e}(\theta)$, the principal will use a noisy rating
(i.e., a garbling of the fully revealing test such that
$w^{\prime}(\theta)<q^{\prime}(\theta)$) to implement $q<q_{f}(\theta)$.

######  Assumption 1 (Downward bias).

$y_{q}(q_{f}(\theta),\theta)\geq 0$ for all
$\theta\in[\underline{\theta},\bar{\theta}]$.

Define the quality for which $\theta$ is indifferent to opting out (by
choosing $q=0$) as $q_{i}(\theta)$, which is given by

$\theta q_{i}(\theta)-c(q_{i}(\theta))=0,\quad q_{i}(\theta)>0.$ (8)

By the convexity of $c(q)$ and $c(0)=0$, a unique $q_{i}(\theta)$ exists for
all $\theta\in[\underline{\theta},\bar{\theta}]$ and satisfies
$q_{i}(\theta)\geq q_{f}(\theta)$ with equality if and only if
$\theta=0$.131313 For general increasing $v(q)$, this is implied by
$c^{\prime}(q)/v^{\prime}(q)$ being increasing in $q$ and $v(0)=c(0)=0$.

Define $\theta_{c}(\theta)=q_{f}^{-1}(q_{i}(\theta))\equiv
c^{\prime}(q_{i}(\theta))$ as the type that would choose $q_{i}(\theta)$ under
full revelation. Note that $\theta_{c}$ extends the type space
$[\underline{\theta},\bar{\theta}]$ to $[\underline{\theta},+\infty)$, so it
is possible that $\theta_{c}(\theta)>\bar{\theta}$, in which case
$\theta_{c}(\theta)$ is a (hypothetical) type in $\mathbb{R}_{+}$ that
satisfies $q_{f}(\theta_{c}(\theta))=q_{i}(\theta)$.

Further, I assume the principal would still value an increase in quality for
low types $[\underline{\theta},\theta_{c}(\theta)]$ when his (IR) constraint
is binding, i.e., $q_{e}(\theta)\geq q_{i}(\theta)$ on
$[\underline{\theta},\theta_{c}(\underline{\theta})]$.

######  Assumption 2.

$y_{q}(q_{i}(\theta),\theta)\geq 0$ for all
$\theta\in[\underline{\theta},\theta_{c}(\underline{\theta})]$.141414
AmadorBagwell2022 implies a stronger assumption that
$y_{q}(q_{i}(\theta),\theta)\geq 0$ for all
$\theta\in[\underline{\theta},\bar{\theta}]$.

Assumption 2 can be replaced by a stronger assumption that
$\underline{\theta}=0$, in which case it holds trivially because
$q_{i}(0)=c^{\prime}(q_{i}(0))=0$. The assumption is a simplifying assumption
that ensures the first-best quality (subject to IR) for the lowest type is
$q^{FB}(\underline{\theta})=q_{i}(\underline{\theta})$ (i.e.,
$\underline{U}=0$) so that under deterministic ratings, the cutoff type always
gets a zero payoff regardless of exclusion (see Lemma 7).151515 Without this
assumption, in the case of
$q^{*}(\underline{\theta})<q_{i}(\underline{\theta})$, one can always find
$\underline{\theta}^{\prime}\in[0,\underline{\theta}]$ such that
$q_{i}(\underline{\theta}^{\prime})=q^{*}(\underline{\theta})$ and treat
$\underline{\theta}^{\prime}$ as the (hypothetical) cutoff type; the exclusion
region $(\underline{\theta},\underline{\theta}^{\prime})$ is the empty set as
$\underline{\theta}^{\prime}<\underline{\theta}$.

######  Example 2.1.

Assume $y(q,\theta)=\beta(\theta)q-\alpha c(q)$ with $\beta(\theta)>0$ and
$\alpha\geq 0$. Assumption 1 requires $\beta(\theta)2\geq\alpha\theta$.
Assumption 2 requires that $\beta(\theta)\geq\alpha c^{\prime}(q_{i}(\theta))$
on $[\underline{\theta},\theta_{c}(\theta)]$: assume $c(q)=q^{m}/m$ with
$m>1$; then it requires $\beta(\theta)\geq m\alpha\theta$ on
$[\underline{\theta},m\underline{\theta}]$.

The quality maximization case $y(q,\theta)=q$ satisfies both assumptions.

###  2.5 Discussion on Assumptions

##### The market values quality.

I assume the market’s value $v(q)=q$ depends on the endogenous $q$ and not
directly on $\theta$ to shut down signaling. This captures the cases in (i)
the school example when learning is productive rather than pure signaling,
(ii) the product certification example as the consumer only cares about the
product quality, and (iii) the employer-employee example when the performance
accumulates human capital or skills (learning by doing) that the future
employer also value. In Section 5, I consider the other extreme case that the
market values the type (ability) $\theta$ à la Spence1973’s (Spence1973)
signaling model (in the school example) and Holmstrom1999’s (Holmstrom1999)
career concern model (in the employer example).161616 Rayo2013 and
Zubrickas2015 study this case with $y(q,\theta)=q$ and restrict attention to
deterministic ratings, whereas I allow for general $y(q,\theta)$ and
stochastic ratings.  In other words, the agent’s effort is manipulative rather
than productive. This case will be simpler because the types are exogenous.

##### The misalignment of incentives.

I assume every agent has a downward bias, i.e.,
$y_{q}(q_{i}(\theta),\theta)\geq 0$ for all $\theta$. This is the case, for
example, when the principal internalizes only a small fraction $\alpha$ of the
cost, i.e., $y(q,\theta)=q-\alpha c(q,\theta)$, a special case being
$y(q,\theta)=q$, where the principal elicits maximal effort (quality). This
incentive misalignment can arise in the following applications:

  1. (i)

the school-student example when

    1. (a)

the school is concerned with the human capital accumulation $q$ (à la
Zubrickas2015) due to the social benefits or spillovers;

    2. (b)

the school maximizes the student’s placement outcome—the expected wage, which
is equal to the expected quality (i.e.,
$\operatorname{\mathbf{E}}[w(\theta)]=\operatorname{\mathbf{E}}[q(\theta)]$ by
BP);

    3. (c)

the student faces a short-run (overestimated) cost or disutility that is
$1/\alpha$ times higher than the social (true) cost;

    4. (d)

the school charges a tuition fee at the ex-ante stage (before students learn
their types) equal to the students’ expected payoff, whereas their parents who
pay the tuition internalize only a fraction $\alpha$ of the cost (à la
OnuchicRay2023);171717 This is distinct from a fixed transfer mentioned above
at the interim stage (after the agents observe their types).

  2. (ii)

the product certification example when the certifier is a regulator who
maximizes a weighted sum of firms’ profit and quality (à la
BizzottoHarstad2023), i.e., $\operatorname{\mathbf{E}}[\alpha
q(\theta)+(1-\alpha)U(\theta)]=\operatorname{\mathbf{E}}[q(\theta)-\alpha
c(q(\theta),\theta)]$;181818 To see this, $\operatorname{\mathbf{E}}[\alpha
q(\theta)+(1-\alpha)U(\theta)]=\operatorname{\mathbf{E}}[\alpha
q(\theta)+(1-\alpha)(w(\theta)-c(q(\theta),\theta))]=\operatorname{\mathbf{E}}[q(\theta)-\alpha
c(q(\theta),\theta)]$. 191919 In Zapechelnyuk2020, the regulatory certifier
has a state-independent preference for quality that arises from maximizing the
surplus of consumers who make a binary decision (buy or not).

  3. (iii)

the employer-employee example when the employer elicits maximal effort $q$.

##### The role of (no) transfers.

I rule out transfers to focus on the role of ratings in providing incentives.
With contingent transfers $t(\theta)$, the design of ratings no longer matters
because $t(\theta)$ can provide the same incentives in place of $w(\theta)$
through redistribution (see Appendix LABEL:transfer:contingent). I also
consider a _fixed_ transfer that affects the optimal rating design through the
agent’s participation constraint for deterministic ratings (Appendix
LABEL:transfer:deterministic) and stochastic ratings (Section 4.4 and Appendix
LABEL:transfer:general), as well as for the alternative case à la Spence1973
(Section 5.3).

##### Rating only depends on quality.

I assume the distribution of rating scores, $\pi(q)$, depends only on the
quality $q$ and not on the type $\theta$. This is a restriction on the
mechanisms; potentially, the rating can also depend on the agent’s type, i.e.,
$\pi:q\times\theta\to\Delta S$ (see LiQiu2024; Perez-RichetSkreta2024). Perez-
RichetSkreta2024 show that under some assumptions, it is without loss of
generality to restrict attention to “score-based mechanisms” that do not
depend on the agent’s type.

##### Multiplicatively separable cost.

This assumption rules out another commonly used cost function:
$c(q,\theta)=c(q-\theta)$, that is, a type-$\theta$ agent can invest effort
$e\geq 0$ at the cost $c(e)$ to achieve quality $q=\theta+e$,202020 This cost
function has been used in, among others, LaffontTirole1993; AugiasPerez-
Richet2023; Perez-RichetSkreta2022.  which I explore in Appendix
LABEL:generalcost. When the market values quality $q$ (type $\theta$), this
effort is productive (manipulative). By contrast, in the main specification
$c(q,\theta)=c(q)/\theta$, the effort is implicit, which can be interpreted as
identical to the quality $q$. In Appendix LABEL:generalcost, I show that the
results can extend to this cost function $c(q,\theta)=c(q-\theta)$ and general
cost functions.

##  3 Optimal Deterministic Ratings

###  3.1 Principal’s Problem

In this section, I restrict attention to _deterministic_ ratings $\pi\colon
Q\to S$, which induces a combination of ($q^{\prime}(\theta)=0$) and fully
revealing ($q(\theta)=q_{f}(\theta)$) intervals; in either case, the interim
wage (quality) is $w(\theta)=q(\theta)$.

######  Lemma 5.

Under deterministic ratings, for each
$\theta\in[\underline{\theta},\bar{\theta}]$, the induced quality scheme
satisfies either $q^{\prime}(\theta)=0$ or $q(\theta)=q_{f}(\theta)$, so the
interim wage function $w(\theta)=q(\theta)$.212121 See also Zapechelnyuk2020.

Assume $c(q,\theta)=c(q)/\theta$ and rewrite the agent’s utility as
$u(q,\theta)=\theta q-c(q)$. Substituting $w(\theta)=q(\theta)$ into the
principal’s problem [P], we have

$\displaystyle\text{[P']}\quad\max_{q(\theta)}\int_{\underline{\theta}}^{\bar{\theta}}y(q(\theta),\theta)\mathop{}\\!\mathrm{d}F(\theta)$
(9)

subject to

$\displaystyle\theta q(\theta)-c(q(\theta))\geq 0$ (IR) (10)
$\displaystyle\theta
q(\theta)-c(q(\theta))=\int_{\underline{\theta}}^{\theta}q(x)\mathop{}\\!\mathrm{d}x+\underline{U}$
(IC-Env) (11) $\displaystyle q(\theta)\mbox{ increasing}$ (IC-Mon) (12)

###  3.2 Equivalence to Delegation

Under deterministic ratings, because $w(\theta)=q(\theta)$, the principal’s
problem [P’] is equivalent to the delegation problem where the principal
delegates the choice of $q$ to agents (by restricting their choice set) with
voluntary participation (because an agent can always choose $q=0$ and not to
take the test), as studied in AmadorBagwell2022.

######  Proposition 1.

The optimal rating design problem under deterministic ratings is equivalent to
the optimal delegation problem under deterministic mechanisms with
participation constraints (AmadorBagwell2022): a fully revealing [resp.
pooling] region corresponds to an unrestricted [resp. singleton] delegation
set.

######  Remark 5.

Roughly, Proposition 1 says that restricting the _market’s_ information is
equivalent to restricting an agent’s choice set. KolotilinZapechelnyuk2019
study the persuasion problem of an uninformed _agent_ and show restricting an
_agent’s_ _information_ about the state is generally equivalent to restricting
his _choice set_ of actions.

As in delegation problems, (IC) implies that the quality scheme implementable
by deterministic ratings must satisfy the following properties.

######  Lemma 6 (MelumadShibano1991; AlonsoMatouschek2008).

The quality scheme $q(\theta)$ implementable by deterministic ratings must
satisfy the following conditions on $[\underline{\theta},\bar{\theta}]$ at any
discontinuity $\hat{\theta}$:222222 See MelumadShibano1991, Proposition 1 and
AlonsoMatouschek2008, Lemma 2.

  1. (i)

$q(\hat{\theta}{-})-c(q(\hat{\theta}{-}),\hat{\theta})=q(\hat{\theta}{+})-c(q(\hat{\theta}{+}),\hat{\theta})$,

  2. (ii)

$q(\theta)=q(\hat{\theta}{-})$ for
$\theta\in[q_{f}^{-1}(q(\hat{\theta}{-})),\hat{\theta})$ and
$q(\theta)=q(\hat{\theta}{+})$ for
$\theta\in(\hat{\theta},q_{f}^{-1}(q(\hat{\theta}{-}))]$,

  3. (iii)

$q(\hat{\theta})\in\\{q(\hat{\theta}{-}),q(\hat{\theta}{+})\\}$,

where $q_{f}^{-1}(\cdot)=c^{\prime}(\cdot)$.

Figure 1: $q(\theta)$ implementable by deterministic ratings when
$c(q)=q^{2}/2$

Figure 1 illustrates the quality $q(\theta)$ implementable by deterministic
ratings. For example, if $\underline{\theta}=0$ and
$c(q,\theta)=q^{2}/2\theta$, then $q(\theta)$ must be an extreme point of the
mean-preserving contraction (MPC) of $q_{f}(\theta)$. In general, by the
convexity of $c(q,\theta)$ in $q$, Lemma 6 implies that $q(\theta)$ must be a
contraction of $q_{f}(\theta)$ (but not necessarily mean-preserving) on
$[0,\bar{\theta}]$ (even when $\underline{\theta}>0$).

######  Lemma 7.

There exists a cutoff type $\theta_{0}\in[\underline{\theta},\bar{\theta}]$
such that $q(\theta)=0$ for all $\theta\leq\theta_{0}$ and $q(\theta)>0$ for
all $\theta\geq\theta_{0}$. Under the optimal rating, the cutoff type
$\theta_{0}$ always chooses $q^{*}(\theta_{0})=q_{i}(\theta_{0})$ (or $q=0$,
which he is different to) and therefore gets zero payoff; in particular,
$\underline{U}=0$.

###### Proof.

The first part follows from (IR) and (IC). For the second part, if
$\theta_{0}>\underline{\theta}$, $q^{*}(\theta_{0})=q_{i}(\theta_{0})$ follows
from Lemma 6 or (IC). If $\theta_{0}=\underline{\theta}$, suppose
$q^{*}(\underline{\theta})<q_{i}(\underline{\theta})$; then increasing
$q^{*}(\underline{\theta})$ to $q_{i}(\underline{\theta})$ will increase
$q^{*}(\theta)$ for
$\theta\in[\underline{\theta},q_{f}^{-1}(q_{i}(\underline{\theta}))]$ and
still entails no exclusion. By Assumption 2, this increases the principal’s
payoff; a contradiction. ∎

Despite the equivalence to delegation, I use a different approach to solve the
principal’s problem [P’] to characterize the solution under weaker sufficient
conditions; see Appendix LABEL:Hamiltonian:deterministic (see also Xiao2023a).

###  3.3 When is lower censorship optimal?

By Proposition 1, the deterministic rating design problem is equivalent
AmadorBagwell2022 without the fixed production cost: lower censorship
corresponds to the price-cap allocation in their model, and a pass/fail test
corresponds to a degenerate delegation interval (conditional on
participation).232323 See AmadorBagwellFrankel2018 for degenerate delegation
intervals but without voluntary participation.  However, I improve upon their
results and obtain weaker sufficient conditions (and also necessary in the
linear delegation case; see Section 3.4) using the optimal control methods.
With the weakened conditions, lower censorship with exclusion, or even a
pass/fail test, can be optimal. 242424 See Appendix LABEL:AB for the
comparison and technical details. See also Xiao2023a.

######  Definition 4.

_Lower censorship_ is a deterministic rating $\pi\colon Q\to S$ with a minimum
standard $q_{0}\in Q$ that fully reveals the quality $q$ if $q\geq q_{0}$ and
gives a fail otherwise. In other words, $\pi(q)=\begin{cases}q,&\text{ if
}q\geq q_{0},\\\ \text{fail},&\text{ otherwise}.\end{cases}$

######  Definition 5.

A _pass/fail test_ is a deterministic rating $\pi\colon Q\to S$ with a minimum
standard $q_{0}\in Q$ that gives a pass if $q\geq q_{0}$ and a fail otherwise.
In other words, $\pi(q)=\begin{cases}\text{pass},&\text{ if }q\geq q_{0},\\\
\text{fail},&\text{ otherwise}.\end{cases}$

Lower censorship with minimum standard $q_{0}$ can be characterized by the
induced quality scheme $q(\theta)$ (or interim wage $w(\theta)=q(\theta)$). A
pass/fail test can be viewed as a special case of lower censorship where
$q_{0}$ is so high that no type chooses $q(\theta)>q_{0}$ in the induced
quality scheme. A _fully revealing test_ is a special case of lower censorship
where $q_{0}=0$.

Figure 2: $q(\theta)$ induced by lower censorship (and pass/fail)

If the optimal rating induces a quality scheme that entails no exclusion, then
type $\underline{\theta}$ chooses $q_{i}(\underline{\theta})$, according to
Lemma 7. Thus, the optimal minimum standard $q_{0}\geq
q_{i}(\underline{\theta})$. Moreover, any $q_{0}>q_{i}(\bar{\theta})$ cannot
be optimal because it is strictly dominated by having no minimum standard
($q_{0}=0$). Therefore, whenever lower censorship is optimal, its minimum
standard must satisfy $q_{0}\in Q_{0}\equiv\\{q_{0}\in
Q:\underline{\theta}\leq c(q_{0})/q_{0}\leq\bar{\theta}\\}$, for which there
exists a cutoff type $\theta_{0}\in[\underline{\theta},\bar{\theta}]$ such
that $q_{i}(\theta_{0})=q_{0}$ (i.e., $\theta_{0}=c(q_{0})/q_{0}$).252525 For
$q_{0}=0$, $\theta_{0}=\lim_{q_{0}\to 0}c(q_{0})/q_{0}=0$.

For any cutoff type $\theta_{0}\in[\underline{\theta},\bar{\theta}]$, recall
that $\theta_{c}(\theta_{0})\equiv
q_{f}^{-1}(q_{i}(\theta_{0}))=c^{\prime}((q_{i}(\theta_{0})))$ is the type in
$\mathbb{R}_{+}$ that would choose $q_{i}(\theta_{0})$ under full revelation.
Define
$\theta_{L}(\theta_{0})=\min\\{\theta_{c}(\theta_{0}),\bar{\theta}_{0}\\}$ to
be the start of the fully revealing region. If
$\theta_{L}(\theta_{0})<\bar{\theta}$, we have
$q_{i}(\theta_{0})=q_{f}(\theta_{L}(\theta_{0}))$. Hence, we have the
following equivalence between lower censorship and the induced quality scheme.

######  Lemma 8.

The rating is lower censorship with minimum standard $q_{0}\in Q_{0}$ if and
only if the induced quality scheme is $q^{*}(\theta)=\begin{cases}0,&\text{ if
}\theta\in[\underline{\theta},\theta_{0})\\\ q_{i}(\theta_{0}),&\text{ if
}\theta\in[\theta_{0},\theta_{L}(\theta_{0})]\\\ q_{f}(\theta),&\text{ if
}\theta\in(\theta_{L}(\theta_{0}),\bar{\theta}]\end{cases}$ where
$q_{i}(\theta_{0})=q_{0}$ and
$\theta_{0}\in[\underline{\theta},\bar{\theta}]$.262626 By convention, $[x,x)$
and $(x,x)$ represent the empty set.

####  3.3.1 Optimal exclusion level

By Lemma 8, lower censorship with minimum standard $q_{0}\in Q_{0}$ is
uniquely characterized by the cutoff type $\theta_{0}$. Thus, to find the
optimal $q_{0}^{*}$, we need to find the optimal $\theta_{0}^{*}$.

The principal’s expected payoff is

$Y=\int_{\theta_{0}}^{\theta_{L}(\theta_{0})}y(q_{i}(\theta_{0}),\theta)\mathop{}\\!\mathrm{d}F(\theta)+\int_{\theta_{L}(\theta_{0})}^{\bar{\theta}}y(q_{f}(\theta),\theta)\mathop{}\\!\mathrm{d}F(\theta).$
(13)

By definition, $c(q_{i}(\theta_{0}))=\theta_{0}q_{i}(\theta_{0})$ implies

$(\theta_{c}(\theta_{0})-\theta_{0})q_{i}^{\prime}(\theta_{0})=q_{i}(\theta_{0})\implies
q_{i}^{\prime}(\theta_{0})=\frac{q_{i}(\theta_{0})}{\theta_{c}(\theta_{0})-\theta_{0}}>0$
(14)

where $\theta_{c}(\theta_{0})\equiv c^{\prime}(q_{i}(\theta_{0}))$. Thus, a
higher cutoff $\theta_{0}$ results in a higher minimum standard
$q_{i}(\theta_{0})$, thereby increasing the principal’s expected payoff $Y$ in
the bunching region $[\theta_{0},\theta_{L}(\theta_{0})]$. Denote the average
marginal increase per $q_{i}(\theta_{0})$ by

$A=\frac{1}{\theta_{c}(\theta_{0})-\theta_{0}}\int_{\theta_{0}}^{\theta_{L}(\theta_{0})}y_{q}(q_{i}(\theta_{0}),\theta)\mathop{}\\!\mathrm{d}F(\theta).$
(15)

The optimal $\theta_{0}^{*}$ must satisfy the Kuhn-Tucker FOC272727 See also
the jump (switching) condition in the optimal control in BrysonHo1975 and
Clarke2013.

$Y^{\prime}(\theta_{0})=A\cdot
q_{i}(\theta_{0})-{y(q_{i}(\theta_{0}),\theta_{0})}f(\theta_{0})\leq
0,\quad(\theta_{0}^{*}-\underline{\theta})Y^{\prime}(\theta_{0})=0.$ (OPT)

In words, increasing the cutoff $\theta_{0}$ leads to a marginal decrease in
the principal’s expected payoff of
${y(q_{i}(\theta_{0}),\theta_{0})}f(\theta_{0})$ in the exclusion region (due
to more exclusion) and a marginal increase of $A\cdot q_{i}(\theta_{0})$ in
the bunching region (due to a higher minimum standard $q_{i}(\theta_{0})$).
The optimal cutoff is when these two marginal effects balance out each other.
The marginal effects due to a higher $\theta_{L}(\theta_{0})$ always cancel
out each other in the bunching and fully revealing regions.

######  Observation 1.

If $f(\theta)$ is decreasing and $y_{q\theta}(q,\theta)\leq\kappa$, then
$Y^{\prime}(\theta_{0})\leq 0$, so no exclusion is optimal (i.e.,
$\theta_{0}^{*}=\underline{\theta}$).282828 The proof is in Appendix
LABEL:proof:obs1.

####  3.3.2 Sufficient (and necessary) conditions

By convention, the CDF $F(\theta)=1$ for all $\theta\geq\bar{\theta}$.
Therefore, we have $F(\theta_{c}(\theta_{0}))=F(\theta_{L})$ because
$\theta_{L}(\theta_{0})=\min\\{\theta_{c}(\theta_{0}),\bar{\theta}\\}$.
Accordingly, we can extend the domain
$\Theta=[\underline{\theta},\bar{\theta}]$ to
$[\underline{\theta},\theta_{c}(\theta_{0})]$ if
$\theta_{c}(\theta_{0})>\bar{\theta}$ when necessary. For example, the
multiplier $A$ defined in equation (15) can be written as
$A=\frac{1}{\theta_{c}(\theta_{0})-\theta_{0}}\int_{\theta_{0}}^{\theta_{c}(\theta_{0})}y_{q}(q_{i}(\theta_{0}),\theta)\mathop{}\\!\mathrm{d}F(\theta).$

Define $\kappa=\inf_{q,\theta}\\{-y_{qq}(q,\theta)/c^{\prime\prime}(q)\\}$ and

$r(\theta|q)=y_{q}(q,\theta)f(\theta)-\kappa(\theta-c^{\prime}(q))f(\theta)-\kappa(F(\theta)-F(\theta_{0})),$
(16)

and, slightly abusing notations,

$r(\theta)=r(\theta|q^{*})=\begin{cases}y_{q}(0,\theta)f(\theta)-\kappa\theta
f(\theta)-\kappa(F(\theta)-F(\theta_{0})),&\text{ if
}\theta\in[\underline{\theta},\theta_{0})\\\
y_{q}(q_{i}(\theta_{0}),\theta)f(\theta)-\kappa(\theta-\theta_{c}(\theta_{0}))f(\theta)-\kappa(F(\theta)-F(\theta_{0})),&\text{
if }\theta\in(\theta_{0},\theta_{c}(\theta_{0})]\\\
y_{q}(q_{f}(\theta),\theta)f(\theta)-\kappa(F(\theta)-F(\theta_{0})),&\text{
if }\theta\in(\theta_{L}(\theta_{0}),\bar{\theta}]\end{cases}$ (17)

and
$R(\theta)=\int_{\underline{\theta}}^{\theta}r(\tilde{\theta})\mathop{}\\!\mathrm{d}\tilde{\theta}$.
Recall that $\theta_{c}(\theta_{0})=c^{\prime}(q_{i}(\theta_{0}))$.

The function $R(\theta)$ can be viewed as the characteristic function of the
maximization problem and the proposed solution $q^{*}(\theta)$, which is
determined by the distribution $F(\theta)$, objective $y(q,\theta)$, the cost
function $c(q)$, and the proposed solution $q^{*}(\theta)$ (which is
determined by $\theta_{0}$ in lower censorship). On the pooling region
$[\underline{\theta},\theta_{c}(\theta_{0})]$, it is also useful to define

$\displaystyle L(\theta|\theta_{0})$
$\displaystyle=\frac{R(\theta_{0})-R(\theta)}{\theta_{0}-\theta}$
$\displaystyle=\begin{cases}\frac{1}{\theta_{0}-\theta}\left[\int_{\theta}^{\theta_{0}}y_{q}(0,\tilde{\theta})f(\tilde{\theta})\mathop{}\\!\mathrm{d}\tilde{\theta}-\kappa\theta(F(\theta_{0})-F(\theta))\right],&\text{
if }\theta\in[\underline{\theta},\theta_{0}),\\\
\frac{1}{\theta-\theta_{0}}\left[\int_{\theta_{0}}^{\theta}y_{q}(q_{i}(\theta_{0}),\tilde{\theta})f(\tilde{\theta})\mathop{}\\!\mathrm{d}\tilde{\theta}-\kappa(\theta-\theta_{c}(\theta_{0}))(F(\theta)-F(\theta_{0}))\right],&\text{
if }\theta\in(\theta_{0},\theta_{c}(\theta_{0})],\end{cases}$

which is the slope of the line connecting $\theta_{0}$ and $\theta$ on
$R(\theta)$. In particular, the multiplier
$A=L(\theta_{c}(\theta_{0})|\theta_{0})$.303030 Note that the multiplier à la
AmadorBagwell2022, denoted by $A_{AB}$, is
$A_{AB}=L(\theta_{L}(\theta_{0})|\theta_{0})\geq A$, with equality if and only
if $\theta_{c}(\theta_{0})\leq\bar{\theta}$.  Note that $r(\theta)$ may be
discontinuous at $\theta_{0}$, in which case $R$ will have an inward (concave)
kink at $\theta_{0}$, as shown in the Lemma 9.

######  Lemma 9.

$r(\theta_{0}{+})\leq r(\theta_{0}{-})$ for all
$\theta_{0}\in(\underline{\theta},\bar{\theta})$. The equality holds if and
only if $y_{qq}(q,\theta_{0})+\kappa c^{\prime\prime}(q)=0$ for almost every
$q\in(0,q_{i}(\theta_{0}))$.

Now I propose a condition on $r(\theta)$ that facilitates the bunching and the
exclusion regions.

######  Condition (L).

$\int_{\theta_{0}}^{\theta}r(\tilde{\theta})d\tilde{\theta}\geq
A\cdot(\theta-\theta_{0})$ for all
$\theta\in[\underline{\theta},\theta_{L}(\theta_{0})]$.

Condition (Condition (L)) can be separated into two conditions on the bunching
and the exclusion regions, respectively.

######  Condition (L1).

$L(\theta|\theta_{0})\geq L(\theta_{c}(\theta_{0})|\theta_{0})=A$ for all
$\theta\in(\theta_{0},\theta_{L}(\theta_{0})]$.

######  Condition (L2).

$L(\theta|\theta_{0})\leq L(\theta_{c}(\theta_{0})|\theta_{0})=A$ for all
$\theta\in[\underline{\theta},\theta_{0})$.

Graphically, the conditions mean the line $\ell$ connecting $\theta_{0}$ and
$\theta_{c}(\theta_{0})$ on $R(\theta)$ (which has a slope of
$L(\theta_{c}(\theta_{0})|\theta_{0})=A$) does not intersect with $R$ at any
other $\theta\in(\underline{\theta},\theta_{L}(\theta_{0}))$ other than
$\theta_{0}$. In other words, $\ell$ is the supporting hyperplane (line) of
the epigraph of $R(\theta)$ on
$\theta\in[\underline{\theta},\theta_{L}(\theta_{0})]$ (i.e.,
$\\{(\theta,r)\in[\underline{\theta},\theta_{L}(\theta_{0})]\times\mathbb{R}\colon
r\geq R(\theta)\\}$) containing $\theta_{0}$. As will be shown in Lemma 10, if
$\theta_{0}>\underline{\theta}$, $R(\theta)$ must be differentiable at
$\theta_{0}$, and thus $\ell$ must be tangent to $R(\theta)$ at $\theta_{0}$.

(a) $R(\theta)$ that satisfies condition (L)

(b) $R(\theta)$ that violates condition (L)

######  Observation 2.

If $\theta_{0}>\underline{\theta}$, then conditions (Condition (L1)) and
(Condition (L2)) imply $L(\theta_{0}{+}|\theta_{0})=r(\theta_{0}{+})\geq
r(\theta_{0}{-})=L(\theta_{0}{-}|\theta_{0})$.

######  Observation 3.

Condition (Condition (L1)) is weaker than condition (i) in AmadorBagwell2022
(see condition (LABEL:M1) in Appendix LABEL:sufficient), which is equivalent
to $L(\theta|\theta_{0})\geq A_{AB}=L(\theta_{L}(\theta_{0})|\theta_{0})$ for
all $\theta\in(\theta_{0},\theta_{L}(\theta_{0})]$, because
$\theta_{L}(\theta_{0})\geq\theta_{c}(\theta_{0})$. Consequently, their
condition forces a fully revealing region by ruling out the possibility that
$\theta_{c}(\theta_{0})>\bar{\theta}$ (e.g., when $r$ is increasing).

######  Lemma 10.

If $\theta_{0}>\underline{\theta}$, then condition (Condition (L)) implies
$r(\theta_{0}{+})=r(\theta_{0}{-})=A$ and $y_{qq}(q,\theta_{0})+\kappa
c^{\prime\prime}(q)=0$ for almost every $q\in(0,q_{i}(\theta_{0}))$.

Lemma 10 implies $r(\theta)$ is continuous at $\theta_{0}$ if
$\theta_{0}>\underline{\theta}$, so condition (Condition (L)) can be written
as $\int_{\theta_{0}}^{\theta}r(\tilde{\theta})d\tilde{\theta}\geq
r(\theta_{0})(\theta-\theta_{0})$ for all
$\theta\in[\underline{\theta},\theta_{L}(\theta_{0})]$ with equality at
$\theta=\theta_{c}(\theta_{0})$. Moreover, it shows condition (Condition (L))
is a somewhat strong assumption. I will relax this assumption in Appendix
LABEL:sec:weaker.

######  Observation 4.

Assume $\underline{\theta}=0$.313131 If $\theta_{0}=\underline{\theta}=0$, we
have $\theta_{c}(0)\equiv c^{\prime}(q_{i}(0))=0$ and $A=r(0)$ at
$\theta_{0}=0$.  Then, condition (Condition (L)) is equivalent to:

$\int_{\theta_{0}}^{\theta}r(\tilde{\theta})d\tilde{\theta}\geq
r(\theta_{0})(\theta-\theta_{0})\mbox{ for all
}\theta\in[\underline{\theta},\theta_{L}(\theta_{0})],\mbox{ with equality at
$\theta=\theta_{c}(\theta_{0})$.}$

Now I propose another condition (à la condition (ii) in AmadorBagwell2022)
regarding the fully revealing region.

######  Condition (M).

$r(\theta)\equiv y_{q}(q_{f}(\theta),\theta)f(\theta)-\kappa F(\theta)$ ($+$
constant) is decreasing in $\theta$ on
$(\theta_{L}(\theta_{0}),\bar{\theta}]$.

Graphically, conditions (Condition (L)) and (Condition (M)) the largest
concave function that lies below $R$ is given by
$convR=\begin{cases}\ell,&\text{ if
}\theta\in[\underline{\theta},\theta_{c}(\theta_{0})],\\\ R,&\text{ if
}\theta\in(\theta_{c}(\theta_{0}),\bar{\theta}],\end{cases}$ where $\ell$ is
the supporting line of $R$ on $[\underline{\theta},\theta_{c}(\theta_{0})]$
containing $\theta_{0}$.

######  Proposition 2.

If conditions (Condition (L)) and (Condition (M)) hold at some
$\theta_{0}^{*}\in[\underline{\theta},\bar{\theta}]$, the optimal
deterministic rating is lower censorship with cutoff type $\theta_{0}^{*}$.

In particular, if condition (Condition (L)) holds at some
$\theta_{0}^{*}\in[\underline{\theta},\bar{\theta}]$ such that
$\theta_{c}(\theta_{0}^{*})\geq\bar{\theta}$, the optimal deterministic rating
is a pass/fail test.

The proof is in Appendix LABEL:prooflcs and the intuition is in Section 3.3.3.
Under the two conditions, the optimal deterministic rating has a minimum
standard $q_{0}=q_{i}(\theta_{0}^{*})$ above which it fully reveals quality.
The induced quality scheme $q^{*}(\theta)$ features exclusion of the low types
$[\underline{\theta},\theta_{0}^{*})$, pooling of the intermediate types
$[\theta_{0}^{*},\theta_{L}(\theta_{0}^{*})]$, and full separation of the high
types $(\theta_{L}(\theta_{0}^{*}),\bar{\theta}]$.323232 Some of these
intervals can be empty if (i) $\theta_{0}^{*}=\underline{\theta}$, (ii)
$\theta_{L}(\theta_{0}^{*})=\bar{\theta}$, or (iii)
$\theta_{L}(\theta_{0}^{*})=\underline{\theta}$.  In particular, if
$\theta_{c}(\theta_{0}^{*})\equiv
c^{\prime}(q_{i}(\theta_{0}^{*}))\geq\bar{\theta}$ (so that
$\theta_{L}(\theta_{0}^{*})=\bar{\theta}$), or equivalently,
$q_{i}(\theta_{0}^{*})\geq q_{f}(\bar{\theta})$, the optimal deterministic
rating is a pass/fail test.

######  Corollary 2.1.

Assume $y_{qq}(q,\theta)+\kappa c^{\prime\prime}(q)=0$ on
$(\underline{\theta},\bar{\theta})$ so that $r(\theta)$ is continuous.333333
This ensures that $R(\theta)$ is differentiable at all possible cutoff
$\theta_{0}$ and is satisfied by, for example,
$y(q,\theta)=\beta(\theta)q-\alpha c(q)+d(\theta)$ (linear delegation); see
Section 3.4.  The optimal deterministic rating is lower censorship if
$r(\theta)$ is unimodal. In particular,

  * •

if $r(\theta)$ is increasing, it is pass/fail;

  * •

if $r(\theta)$ is decreasing, it has no exclusion (and is fully revealing if
$\underline{\theta}=0$).

###### Proof sketch.

A unimodal $r(\theta)$ with peak
$\hat{\theta}\in(\underline{\theta},\bar{\theta})$ satisfies conditions
(Condition (L)) and (Condition (M)) at some
$\theta_{0}\in(\underline{\theta},\hat{\theta})$ such that
$\theta_{c}(\theta_{0})\geq\hat{\theta}$. An increasing $r(\theta)$ satisfies
condition (Condition (L)) at some $\theta_{0}$ such that
$\theta_{c}(\theta_{0})\geq\bar{\theta}$ and thus satisfies condition
(Condition (M)) vacuously. A decreasing $r(\theta)$ satisfies conditions
(Condition (L)) and (Condition (M)) at $\theta_{0}=\underline{\theta}$. ∎

Figure 4: $q^{*}(\theta)$ for decreasing, unimodal, and increasing $r(\theta)$

######  Observation 5.

If condition (Condition (M)) holds at all
$\theta_{0}\in[\underline{\theta},\bar{\theta})$ à la AmadorBagwell2022, then
condition (Condition (L2)) can only hold (vacuously) at
$\theta_{0}=\underline{\theta}$, so exclusion is never optimal (except for the
knife-edge case where the principal is indifferent between some exclusion or
none). Therefore, a pass/fail test can never be optimal generically (unless
$\bar{\theta}\leq\theta_{c}(\underline{\theta})$ or the case of indifference).

####  3.3.3 Intuition for the Results

First, consider a small perturbation to $q^{*}(\theta)=q_{f}(\theta)$ on the
fully revealing region at
$\hat{\theta}\in(\theta_{L}(\theta_{0}^{*}),\bar{\theta})$. By Lemma 6, the
perturbation leads to a local _contraction_ of $q_{f}(\theta)$, given by

$q_{c}(\theta)=\begin{cases}q_{f}(\hat{\theta}-\varepsilon),&\text{ if
}\theta\in(\hat{\theta}-\varepsilon,\hat{\theta}),\\\
q_{f}(\hat{\theta}+\varepsilon),&\text{ if
}\theta\in(\hat{\theta},\hat{\theta}+\varepsilon).\end{cases}$

Condition (Condition (M)) ensures that a perturbation at any
$\hat{\theta}\in(\theta_{L}(\theta_{0}^{*}),\bar{\theta})$ is undesirable. For
example, when $y=q$ and $c(q)=q^{2}/2$, a perturbation to $q_{f}(\theta)$ at
$\hat{\theta}\in(\theta_{L},\bar{\theta})$ leads to $q_{c}(\theta)$ (in red)
on $(\hat{\theta}-\varepsilon,\hat{\theta}+\varepsilon)$ that is a mean
preserving (in the quadratic case) _contraction_ of $q_{f}(\theta)$, as shown
in Figure 5. This is undesirable if (and only if, in this case) $f(\theta)$ is
decreasing at $\hat{\theta}$, as required by condition (Condition (M)).

Figure 5: A perturbation (in red) to $q_{f}(\theta)$ at $\hat{\theta}$ on the
fully revealing region

Now consider a small perturbation to the optimal scheme $q^{*}(\theta)$ on the
pooling (exclusion and bunching) regions. By Lemma 6, the perturbation can
either lead to a _spread_ (earlier jump) or a _contraction_ (later jump) of
$q^{*}(\theta)$; in either case, the new quality scheme has one or more
different jumps.343434 Having no jump is equivalent to a jump at
$\underline{\theta}$.  For instance, the quality scheme due to the
perturbation at $\theta_{1}^{\prime}>\theta_{0}^{*}$ is given by

$q_{s}(\theta)=\begin{cases}q_{i}(\theta_{0}^{\prime}),&\text{ if
}\theta\in[\theta_{0}^{\prime},\theta_{1}^{\prime}),\\\
q_{i}(\theta_{1}^{\prime}),&\text{ if
}\theta\in[\theta_{1}^{\prime},\theta_{L}(\theta_{1}^{\prime})],\\\
q_{f}(\theta),&\text{ if
}\theta\in(\theta_{L}(\theta_{1}^{\prime}),\bar{\theta}].\end{cases}\mbox{ or
}q_{c}(\theta)=\begin{cases}0,&\text{ if
}\theta\in[\theta_{0}^{\prime},\theta_{1}^{\prime}),\\\
q_{i}(\theta_{1}^{\prime}),&\text{ if
}\theta\in[\theta_{1}^{\prime},\theta_{L}(\theta_{1}^{\prime})],\\\
q_{f}(\theta),&\text{ if
}\theta\in(\theta_{L}(\theta_{1}^{\prime}),\bar{\theta}].\end{cases}$

Condition (Condition (L)) implies $\theta_{0}^{*}$ is the optimal cutoff
(i.e., jump) and that no other jump(s) is desirable. For example, when $y=q$
and $c(q)=q^{2}/2$, a perturbation to $q^{*}(\theta)$ (in red) can lead to a
mean preserving (in the quadratic case) _spread_ or _contraction_ of
$q^{*}(\theta)$, as shown in Figure 6. Neither is desirable under condition
(Condition (L)), especially if $f(\theta)$ is unimodal with a peak on
$[\theta_{0}^{*},\theta_{L}(\theta_{0}^{*})]$, which is a sufficient condition
of condition (Condition (L)).

(a) A perturbation that leads to spread

(b) A perturbation that leads to contraction

Figure 6: A perturbation (in red) to $q^{*}(\theta)$ on the pooling regions

In particular, when $\theta_{c}(\theta_{0}^{*})\geq\bar{\theta}$ (i.e.,
$q_{i}(\theta_{0}^{*})\geq q_{f}(\bar{\theta})$), the optimal minimum standard
$q_{i}(\theta_{0}^{*})$ is so high that even the highest type will not choose
under a fully revealing rating, thereby leading to a pass/fail test in which
agents either choose $q=0$ (exclusion) or $q=q_{i}(\theta_{0}^{*})$
(bunching). If Condition (Condition (L)) holds at this $\theta_{0}^{*}$, any
perturbation (Figure 6) that leads to other jumps is undesirable. As noted in
Corollary 2.1, a sufficient condition is $r(\theta)$ being increasing. For
example, when $y=q$, an increasing $f(\theta)$ implies that a perturbation in
the fully revealing region (Figure 5) is always desirable, so the optimal
rating does not induce any fully revealing region; thus,
$\theta_{L}(\theta_{0}^{*})=\bar{\theta}$. Moreover, in the pooling regions, a
perturbation on the pooling region that leads to an earlier jump (Figure 6(a))
is undesirable if $f(\theta)$ is increasing. Therefore, the optimal
deterministic rating is a pass/fail test that induces an exclusion region and
a bunching region. Intuitively, the pass/fail test sets such a high minimum
standard that even the highest type needs to invest a lot in quality (higher
than under full revelation) to barely pass, at the cost of excluding more low
types. When there are increasingly more high types, this simple pass/fail test
induces the highest average quality.

###  3.4 Linear Delegation

Now I focus on a class of objective functions
$y(q,\theta)=\beta(\theta)q-\alpha c(q)+d(\theta)$ with $\beta(\theta)>0$ and
$\alpha\geq 0$, referred to as “linear delegation” in
KolotilinZapechelnyuk2019 because the principal’s marginal payoff from the
agent’s action is linear in (a transformation of) the agent’s action $q$. In
this case, the relative concavity
$-y_{qq}(q,\theta)/c^{\prime\prime}(q)=\alpha$ is constant ($\kappa=\alpha$),
so $y_{qq}(q,\theta_{0}^{*})+\kappa c^{\prime\prime}(q)\equiv 0$ (which makes
condition (Condition (L)) less restrictive; see Lemma 10). In this case, the
sufficient conditions in Proposition 2 are also necessary.

######  Proposition 3.

Assume $y(q,\theta)=\beta(\theta)q-\alpha c(q)+d(\theta)$. The optimal
deterministic rating is lower censorship with cutoff type $\theta_{0}^{*}$ if
and only if conditions (Condition (L)) and (Condition (M)) hold at some
$\theta_{0}^{*}\in[\underline{\theta},\bar{\theta}]$. In particular, it is
pass/fail if and only if condition (Condition (L)) holds at some
$\theta_{0}^{*}\in[\underline{\theta},\bar{\theta}]$ such that
$\theta_{c}(\theta_{0}^{*})\geq\bar{\theta}$.

###### Proof sketch.

The sufficiency is shown in Proposition 2. The necessity can be shown à la
(AmadorBagwell2013, Proposition 2) using perturbation arguments. ∎

When $y(q,\theta)=\beta(\theta)q-\alpha c(q)+d(\theta)$, the $r(\theta)$
function in equation (17) simplifies to

$r(\theta)=\beta(\theta)f(\theta)-\alpha\theta
f(\theta)-\alpha(F(\theta)-F(\theta_{0})),\mbox{ for all
$\theta\in[\underline{\theta},\bar{\theta}]$}.$ (18)

By Corollary 2.1, the optimal deterministic rating is lower censorship if
$r(\theta)$ is unimodal: pass/fail if $r(\theta)$ is increasing and has no
exclusion if (and only if) $r(\theta)$ is decreasing.

####  3.4.1 Quality Maximization

In some applications where the principal wants to elicit maximal quality or
effort (performance), her objective is $y(q,\theta)=q$. Thus, we have
$R(\theta)=F(\theta)$, $r(\theta)=f(\theta)$, and
$A=\frac{F(\theta_{L}(\theta_{0}))-F(\theta_{0})}{\theta_{c}(\theta_{0})-\theta_{0}}$.
In this case, the necessary and sufficient conditions become

  * •

Condition (Condition (L))$:{F(\theta)-F(\theta_{0})}\geq
A\cdot({\theta-\theta_{0}})$ for all
$\theta\in[\underline{\theta},\theta_{L}(\theta_{0})]$; and

  * •

Condition (Condition (M)): $f(\theta)$ is decreasing in $\theta$ on
$(\theta_{L}(\theta_{0}),\bar{\theta}]$

hold for some $\theta_{0}\in[\underline{\theta},\bar{\theta}]$. In particular,
$A=f(\theta_{0})$ if $\theta_{0}>\underline{\theta}$.

######  Corollary 3.1.

Assume $y(q,\theta)=q$. The optimal deterministic rating is lower censorship
if $f(\theta)$ is unimodal. In particular, it is pass/fail if $f(\theta)$ is
increasing; it has no exclusion and is fully revealing above the minimum
standard $q_{i}(\underline{\theta})$ if and only if $f(\theta)$ is decreasing.

See Figure 7 for illustration.

(a) Concave $F$

(b) Convex $F$

(c) Convex-concave $F$

Figure 7: Quality Maximization

######  Remark 6.

For $y(q,\theta)=\beta(\theta)q$ with $\beta(\theta)>0$, apply the results to
$r(\theta)=\beta(\theta)f(\theta)$.

######  Corollary 3.2.

The optimal deterministic rating is a pass/fail test if and only if there
exists $\theta_{0}\in[\underline{\theta},\bar{\theta}]$ such that
$\theta_{c}(\theta_{0})\equiv c^{\prime}(q_{i}(\theta_{0}))\geq\bar{\theta}$
and
${F(\theta)-F(\theta_{0})}\geq\frac{1-F(\theta_{0})}{\theta_{c}(\theta_{0})-\theta_{0}}(\theta-\theta_{0})$
for all $\theta\in[\underline{\theta},\bar{\theta}]$ (Condition (L)).

##### A linear-quadratic example.

Assume $y(q,\theta)=q$ and $c(q,\theta)=q^{2}/2\theta$.

##### Case 1.

$f$ is decreasing. The optimal quality is

$q^{*}(\theta)=\begin{cases}2\underline{\theta},&\text{ if
}\theta\in[\underline{\theta},\theta_{L}(\underline{\theta})]\\\
\theta,&\text{ if
}\theta\in[\theta_{L}(\underline{\theta}),\bar{\theta}]\end{cases}$

where
$\theta_{L}(\underline{\theta})=\min\\{2\underline{\theta},\bar{\theta}\\}$.
The optimal test has a minimum standard, entails no exclusion, and fully
reveals the quality above the standard; it pools low types
$[\underline{\theta},\theta_{L}(\underline{\theta}))$ and fully separates high
types $[\theta_{L}(\underline{\theta}),\bar{\theta}]$.353535 If
$\bar{\theta}\leq 2\underline{\theta}$, the variation in types is not large
enough to sustain a fully revealing region, so the optimal test degenerates to
a pass/fail test.

Because $F(\theta)$ is concave, we have
$\frac{F(\theta_{L}(\theta_{0}^{*}))-F(\theta_{0}^{*})}{\theta_{0}^{*}}\leq\frac{F(2\theta_{0}^{*})-F(\theta_{0}^{*})}{\theta_{0}^{*}}<f(\theta_{0}^{*})$.
By equation (OPT), the optimal exclusion $\theta_{0}^{*}=\underline{\theta}$.

##### Case 2.

$f$ is increasing. The optimal quality $q(\theta)$ is

$q^{*}(\theta)=\begin{cases}0,&\text{ if
}\theta\in[\underline{\theta},\theta_{0})\\\ 2\theta_{0},&\text{ if
}\theta\in[\theta_{0},\bar{\theta}]\\\ \end{cases}$

The optimal deterministic rating is pass/fail which has a minimum standard
that excludes low types and pools high types
($\theta_{L}(\theta_{0})=\bar{\theta}$).

The optimal exclusion $\theta_{0}^{*}$ is given by

$\frac{1-F(\theta_{0}^{*})}{\theta_{c}(\theta_{0}^{*})-\theta_{0}^{*}}=f(\theta_{0}^{*})\iff\frac{1-F(\theta_{0}^{*})}{f(\theta_{0}^{*})}=\theta_{0}^{*}\text{
(quadratic cost implies $\theta_{c}(\theta)=2\theta$)},$ (19)

which is unique because $f$ is increasing.

##### Case 3.

$f$ is increasing on $(\underline{\theta},\hat{\theta})$ and decreasing on
$(\hat{\theta},\bar{\theta})$.

$q^{*}(\theta)=\begin{cases}0,&\text{ if
}\theta\in[\underline{\theta},\theta_{0})\\\ 2\theta_{0},&\text{ if
}\theta\in[\theta_{0},\theta_{L}(\theta_{0})]\\\ \theta,&\text{ if
}\theta\in[\theta_{L}(\theta_{0}),\bar{\theta}]\end{cases}$

where $\theta_{L}(\theta_{0})=\min\\{2\theta_{0},\bar{\theta}\\}$. The optimal
test has a minimum standard that leads to exclusion and fully reveals the
quality above the standard.

Optimal exclusion $\theta_{0}^{*}$ is given by

$\frac{F(\theta_{L}(\theta_{0}^{*}))-F(\theta_{0}^{*})}{f(\theta_{0}^{*})}=\theta_{0}^{*}$
(20)

which satisfies $\hat{\theta}\in(\theta_{0}^{*},\theta_{L}(\theta_{0}^{*}))$.

##### Case 4.

$f$ is constant, e.g., $\theta\sim\text{Unif}\,[1,5]$, optimal quality schemes
include all three forms (and many more).

####  3.4.2 A “Twin” Objective Function

The sufficient condition in the quality maximization case ($y(q,\theta)=q$)
also holds for $y(q,\theta)=\theta q-c(q)/2$; in this sense, they are “twins.”

######  Corollary 3.3.

Assume $y(q,\theta)=\theta q-c(q)/2$. The same results in Corollary 3.1 hold:
the optimal deterministic rating is lower censorship [pass-fail/has no
exclusion] if $f(\theta)$ is unimodal [increasing/decreasing].

###### Proof sketch.

In this case, $r(\theta)=\theta f(\theta)/2-(F(\theta)-F(\theta_{0}))/2$.
Thus, $r(\theta_{0})=\theta_{0}f(\theta_{0})/2$ and $r^{\prime}(\theta)=\theta
f^{\prime}(\theta)/2\stackrel{{\scriptstyle sign}}{{=}}f^{\prime}(\theta)$. In
other words, $r(\theta)$ is increasing (decreasing) whenever $f(\theta)$ is
increasing (decreasing), so the same argument as Corollary 3.1 applies. ∎

######  Remark 7.

This result is consistent with KolotilinZapechelnyuk2019.

###  3.5 Relationship to the Literature

##### Relationship to the delegation literature.

I improve upon AmadorBagwell2022’s (AmadorBagwell2022) results and obtain
weaker sufficient conditions that are also necessary in the linear delegation
case (Section 3.4). The conditions are not only weaker due to a tighter
multiplier but also only need to hold at the _optimal_ exclusion level instead
of _all_ exclusion levels. Thus, lower censorship with exclusion, or even a
pass/fail test, can be optimal. See Appendix LABEL:AB for the comparison and
technical details.

KartikKleinerVanWeelden2021 do provide necessary and sufficient conditions for
interval delegation, including full delegation and no compromise
(corresponding to the pass/fail test) in delegation in veto-bargaining, but
they assume a state-independent principal preference and a quadratic agent
preference.

##### Relationship to KleinerMoldovanuStrack2021.

In the quality maximization case $y(q,\theta)=q$, if
$c(q,\theta)=q^{2}/2\theta$, any undominated incentive-compatible $q(\theta)$
is an extreme point of the mean-preserving contraction (MPC) of

$q_{c}(\theta)=\begin{cases}q_{i}(\underline{\theta})=2\underline{\theta},&\text{
if }\theta\in[\underline{\theta},\theta_{L}(\underline{\theta})]\\\
q_{f}(\theta)=\theta,&\text{ if
}\theta\in[\theta_{L}(\underline{\theta}),\bar{\theta}]\end{cases}$

(see Figure 1). Therefore, one can apply KleinerMoldovanuStrack2021 to prove
Corollary 3.1 (quality maximization). However, for general cost functions, $q$
is not an MPC of $q_{c}$, so their approach cannot be applied.

##### Relationship to the Persuasion Literature.

In the linear delegation case, KolotilinZapechelnyuk2019 translate it to an
equivalent linear persuasion problem and solve it using methods in
Kolotilin2018 and DworczakMartini2019. For example, Proposition 1 in
KolotilinZapechelnyuk2019 contains the same results as Corollary 3.3. By
contrast, the method used in this paper tackles the delegation problem
directly using the without invoking their equivalence result. This method uses
the optimal control approach that juxtaposes with the Lagrangian approach
developed by AmadorWerningAngeletos2006 and AmadorBagwell2013, which is also
reminiscent of the linear programming approach in Kolotilin2018 but differs in
terms of the program setup. In particular, this method allows for _nonlinear_
delegation, where the principal’s marginal payoff is nonlinear in (a
transformation of) the agent’s action. In that case, in the equivalent
persuasion problem, the sender’s payoff is nonlinear in the state (so that not
only the posterior mean matters), so their methods no longer apply.

###  3.6 Beyond Lower Censorship: Optimal Deterministic Ratings

In general, by the equivalence between deterministic rating design and
delegation (Proposition 1), the optimal deterministic rating can be
characterized by the quality scheme it induces. By Lemma 5, the induced
quality scheme may consist of an exclusion interval
$[\underline{\theta},\theta_{0}]$ and $k\geq 1$ (unexcluded) pooling and fully
revealing intervals $[\theta_{0},\theta_{1}],\ldots,[\theta_{k-1},\theta_{k}]$
with
$\underline{\theta}\equiv\theta_{-1}\leq\theta_{0}<\ldots<\theta_{k-1}<\theta_{k}\equiv\bar{\theta}$.
On each pooling interval $[\theta_{j-1},\theta_{j}]$, the quality is
$q(\theta)=q_{j-1}$, where $q_{k-1}>\ldots>q_{0}\geq q_{-1}\equiv 0$ satisfy
Lemma 6; in particular, $q(\theta)=0$ on the exclusion interval. On each fully
revealing interval, the quality is $q(\theta)=q_{f}(\theta)$.

The induced quality scheme must satisfy Lemma 6. For any two adjacent pooling
intervals $[\theta_{j-1},\theta_{j}]$ (on which $q(\theta)=q_{j-1}$) and
$[\theta_{j},\theta_{j+1}]$ (on which $q(\theta)=q_{j}$), Lemma 6 implies
$q_{j-1}-c(q_{j-1})/\theta_{j}=q_{j}-c(q_{j})/\theta_{j}$ at the jump
$\theta_{j}$ of $q(\theta)$. Each jump in $q(\theta)$ at $\theta_{j}$
corresponds to a minimum standard $q_{j}>0$ in the rating $\pi$. If two
pooling and fully revealing intervals are adjacent, $q(\theta)$ must be
continuous at the boundary $\theta_{j}$ of the pooling interval, i.e.,
$q=q_{f}(\theta_{j})$ on the pooling interval.

The lower censorship is a special case where $k=2$:
$[\underline{\theta},\theta_{0}]$ and $[\theta_{0},\theta_{1}]$ are the
pooling intervals, and $[\theta_{1},\bar{\theta}]$ is the fully revealing
interval; $q_{0}$ is the only minimum standard. At
$\theta_{0}\geq\underline{\theta}$, $q_{0}-c(q_{0})/\theta_{0}=0$ (i.e.
$q_{0}=q_{i}(\theta_{0})$); at $\theta_{1}\leq\bar{\theta}$,
$q_{0}=q_{f}(\theta_{1})$ (i.e. $\theta_{1}=\theta_{L}(\theta_{0})$).

######  Example 3.1.

Assume $c(q)=q^{2}/2$ and $\Theta=[0,5]$. Then an implementable quality scheme
is $q(\theta)=\begin{cases}0;&\theta\in[0,1]\\\ 2;&\theta\in[1,3]\\\
4;&\theta\in[3,4]\\\ \theta;&\theta\in[4,5]\end{cases}$, which has two jumps
(see Figure 8), so the rating that induces it has two minimum standards
$q_{0}=2$ and $q_{1}=4$.

Figure 8: An implementable $q(\theta)$ with two jumps

Beyond lower censorship, the optimal control method can still be applied to
solve for the optimal deterministic rating in general. On the technical side,
the Hamiltonian multiplier $\mu=0$ must hold at any
$\theta_{j}\in(\underline{\theta},\bar{\theta})$ and on any fully revealing
interval. Moreover, the switching condition needs to hold at any
$\theta_{j}\in(\underline{\theta},\bar{\theta})$ at which $q(\theta)$ is
discontinuous. To characterize the sufficient conditions, define

$r_{j}(\theta|q)=y_{q}(q,\theta)f(\theta)-\kappa
f(\theta)(\theta-c^{\prime}(q))-\kappa(F(\theta)-F(\theta_{j-1})).$ (21)

By Lemma 6, on each the pooling interval $[\theta_{j-1},\theta_{j}]$ where
$q(\theta)=q_{j-1}$, $q_{f}(\theta)$ must single-cross with $q_{j-1}$ at a
unique $\hat{\theta}_{j}\in[\theta_{j-1},\theta_{j}]$. Furthermore, for any
pooling interval $[\theta_{j},\theta_{j+1}]$ (on which $q(\theta)=q_{j}$),
there exists an adjacent pooling interval ($[\theta_{j+1},\theta_{j+2}]$ or
$[\theta_{j-1},\theta_{j}]$).363636 If $\theta_{0}=\underline{\theta}$,
$[\theta_{-1},\theta_{0}]=[\underline{\theta},\underline{\theta}]$ is
considered as a degenerate pooling interval.

######  Condition (L-j).

On any two adjacent pooling intervals $[\theta_{j-1},\theta_{j}]$ and
$[\theta_{j},\theta_{j+1}]$,
$\int_{\theta_{j}}^{\theta}r_{j}(\tilde{\theta}|q(\tilde{\theta}))d\tilde{\theta}\geq
A\cdot(\theta-\theta_{j})$ for all
$\theta\in[\hat{\theta}_{j-1},\hat{\theta}_{j+1}]$, with equality if
$\theta\in\\{\hat{\theta}_{j-1},\hat{\theta}_{j-1}\\}$ and
$\theta\in(\underline{\theta},\bar{\theta})$.

######  Condition (M-j).

On any fully revealing interval $[\theta_{j},\theta_{j+1}]$,
$r_{j}(\theta)\equiv r_{j}(\theta|q_{f}(\theta))$ is decreasing in $\theta$.

######  Proposition 4.

The rating characterized by pooling and fully revealing intervals with
endpoints
$\underline{\theta}\equiv\theta_{-1}\leq\theta_{0}<\ldots<\theta_{k-1}<\theta_{k}\equiv\bar{\theta}$
is the optimal deterministic rating if conditions (Condition (M-j)) and
(Condition (L-j)) holds for all $j=1,\ldots,k-1$.

######  Corollary 4.1.

Assume $y(q,\theta)=q$ or $y(q,\theta)=\theta q-c(q)/2$. If $f$ is bimodal,
the optimal deterministic rating has at most two minimum standards (e.g., low
pass and high pass).373737 In other words, the induced $q(\theta)$ has at most
two jumps as in Figure 8.

###  3.7 Discussion on Transfers

See Appendix LABEL:transfer for the case with a contingent or a fixed
transfer. With contingent transfers $t(\theta)$, the design of ratings no
longer matters because $t(\theta)$ can provide the same incentives in place of
$w(\theta)$ through redistribution. As soon as the envelope equation and
Bayesian plausibility
$\operatorname{\mathbf{E}}[w(\theta)]=\operatorname{\mathbf{E}}[q(\theta)]$ is
substituted into the objective, the problem reduces to a classical mechanism
design problem with transfers (e.g., BaronMyerson1982; LaffontTirole1993,
Chapter 1), where the optimal $q^{*}(\theta)$ has no jumps, in contrast to the
case without transfers.383838 On the technical side, this is because, with
contingent transfers $t(\theta)$ as a control variable, there are no longer
pure state constraints, and the maximized Hamiltonian is strictly concave in
$q$, which implies the state variable $q$ has no jumps (KamienSchwartz1971).

In Appendix LABEL:transfer:deterministic, I explore a fixed transfer
$t\in\mathbb{R}$, which can be interpreted as the principal charging a fee (as
a monopoly certifier) or providing a subsidy (as a school) to agents.

##  4 Optimal General Ratings

###  4.1 Principal’s Problem

Now we allow for stochastic rating schemes so that $w(\theta)=q(\theta)$ is no
longer necessary. Instead, the constraints (MPS) and (BP) must hold. With a
multiplicatively separable cost function $c(q)/\theta$, the principal’s
problem [P] is

$\displaystyle\text{[P]}\quad\max_{q(\theta),w(\theta)}\int_{\underline{\theta}}^{\bar{\theta}}y(q(\theta),\theta)\mathop{}\\!\mathrm{d}F(\theta)$
(22)

subject to

$\displaystyle\theta
w(\theta)-c(q(\theta))=\int_{\underline{\theta}}^{\theta}w(x)dx+\underline{U}$
(IC-Env) (23) $\displaystyle q(\theta)\mbox{ increasing}$ (IC-Mon) (24)
$\displaystyle\theta w(\theta)-c(q(\theta))\geq 0$ (IR) (25)
$\displaystyle\int_{\underline{\theta}}^{\bar{\theta}}w(\theta)\mathop{}\\!\mathrm{d}F(\theta)=\int_{\underline{\theta}}^{\bar{\theta}}q(\theta)\mathop{}\\!\mathrm{d}F(\theta)$
(BP) (26)
$\displaystyle\int_{\underline{\theta}}^{\theta}w(\theta^{\prime})\mathop{}\\!\mathrm{d}F(\theta^{\prime})\geq\int_{\underline{\theta}}^{\theta}q(\theta^{\prime})\mathop{}\\!\mathrm{d}F(\theta^{\prime}),\,\forall\theta\in\Theta$
(MPS) (27)

Define
$D(\theta)=\int_{\underline{\theta}}^{\theta}(w(\theta^{\prime})-q(\theta^{\prime}))\mathop{}\\!\mathrm{d}F(\theta^{\prime})\geq
0$. In the optimal control problem, I add $D$ as a state variable subject to
$D(\theta)\geq 0$ (MPS) with the Lagrangian multiplier $\lambda(\theta)$ and
$\dot{D}=[w(\theta)-q(\theta)]f(\theta)$ with the Hamiltonian multiplier
$\Lambda(\theta)$. By construction, $\dot{\Lambda}=-\lambda$. See Appendix
LABEL:Hamiltonian:general for details.

###  4.2 When are deterministic ratings optimal?

The optimal rating is deterministic if and only if (MPS) holds with equality
for all $\theta$. For the fully revealing region, we have the following
results.

######  Condition (N1).

$N_{1}(\theta)=[y_{qq}(q_{f}(\theta),\theta)/c^{\prime\prime}(q_{f}(\theta))+y_{q\theta}(q_{f}(\theta),\theta)]\theta+y_{q}(q_{f}(\theta),\theta)[1+\theta
f^{\prime}(\theta)/f(\theta)]$ is decreasing in $\theta$.

######  Observation 6.

Assume $y(q,\theta)=\beta(\theta)q-\alpha c(q)+d(\theta)$. Then,
$N_{1}(\theta)=(\beta^{\prime}(\theta)-\alpha)\theta+(\beta(\theta)-\alpha\theta)[1+\theta
f^{\prime}(\theta)/f(\theta)]$. When $y(q,\theta)=q$, $N_{1}(\theta)=1+\theta
f^{\prime}(\theta)/f(\theta)$.

######  Lemma 11.

The optimal rating is fully revealing on $\Theta_{f}\subseteq\Theta$ only if
$N_{1}(\theta)$ is decreasing on $\Theta_{f}$. If the optimal deterministic
rating is fully revealing on $\Theta_{f}$, then the condition is also
sufficient.

In the pooling region of lower censorship, we have the following results.

######  Condition (N2).

$N_{2}(\theta)=A/f(\theta)+\kappa\theta+\kappa(F(\theta)-F(\theta_{0}))/f(\theta)$
is decreasing in $\theta$.393939 If
$(\theta_{0},\theta_{L}(\theta_{0}))=(\underline{\theta},\bar{\theta})$ (i.e.,
bunching without exclusion), the condition can be relaxed to
$\tilde{N}_{2}(\theta)=\kappa\theta+\kappa F(\theta)/f(\theta)$ is decreasing.

Now I provide sufficient conditions for the optimal rating to be lower
censorship.

######  Proposition 5.

The optimal rating is lower censorship if the optimal deterministic rating is
lower censorship for which conditions (Condition (N1)) and (Condition (N2))
hold in the fully revealing and pooling regions, respectively.

###### Proof sketch.

Use the same multipliers as in the deterministic ratings (where
$D(\theta)\equiv 0$). Conditions (Condition (N1)) and (Condition (N2))
guarantee the Lagrangian multiplier on $D\geq 0$ is positive. ∎

######  Corollary 5.1.

The optimal rating is a pass/fail test if the optimal deterministic rating is
pass/fail and condition (Condition (N2)) holds on
$[\underline{\theta},\bar{\theta}]$.

####  4.2.1 Quality Maximization

Now I focus on the case where the principal maximizes total quality/effort,
i.e., $y(q,\theta)=q$. The optimality of a pass/fail test is robust to
stochastic ratings.

######  Corollary 5.2.

Assume $y(q,\theta)=q$. Then, the optimal rating is pass/fail if $f$ is
increasing.404040 Another sufficient condition is both
$\theta_{0}^{*}=\underline{\theta}$ (i.e., no exclusion) and
$\bar{\theta}\leq\theta_{c}(\underline{\theta})$ (i.e., variation in types is
not large enough to sustain a fully revealing region).414141Because $w$ is a
mean-preserving spread of $q$ in the quantile space, one may be tempted to
apply the results in KleinerMoldovanuStrack2021, but it would not work as the
constraints make the objective no longer linear.

The optimality of a fully revealing test is less robust to stochastic ratings.

######  Corollary 5.3.

Assume $y(q,\theta)=q$. Then, the optimal rating is fully revealing if and
only if (i) $\underline{\theta}=0$ and (ii) both $f(\theta)$ and $\theta
f^{\prime}(\theta)/f(\theta)$ are decreasing.

######  Remark 8.

$N_{1}(\theta)\equiv\theta f^{\prime}(\theta)/f(\theta)$ is the _elasticity_
of $f(\theta)$, which is decreasing if and only if
$f^{\prime\prime}(\theta)\leq
f^{\prime}(\theta)^{2}/f(\theta)-f^{\prime}(\theta)/\theta$. If $f(\theta)$ is
decreasing, this is equivalent to
$f^{\prime\prime}(\theta)/f^{\prime}(\theta)\geq
f^{\prime}(\theta)/f(\theta)-1/\theta$, which relates to the relative
concavity of $F(\theta)$ and $-f(\theta)=-F^{\prime}(\theta)$.

######  Remark 9.

Condition (ii) is satisfied by uniform, exponential, and Pareto distributions.
In fact, for most unimodal distributions (e.g., normal), $\theta
f^{\prime}(\theta)/f(\theta)$ is decreasing if the density $f(\theta)$ is
decreasing.

The Pareto distribution has a strictly decreasing $f(\theta)$ and a _constant_
$\theta f^{\prime}(\theta)/f(\theta)$, so it is “almost” a counterexample.
Thus, one can construct a counterexample by adding even more disparity to the
Pareto distribution to make the decreasing density $f(\theta)$ even more
convex (thereby increasing $f^{\prime\prime}(\theta)/f^{\prime}(\theta)$)
compared to $F(\theta)$.

######  Remark 10.

If $y(q,\theta)=\beta(\theta)q$ is linear in $q$, the condition for pass/fail
becomes $\beta(\theta)f(\theta)$ is increasing; condition (i) for fully
revealing becomes (i’) both $\beta(\theta)f(\theta)$ and
$\beta^{\prime}(\theta)\theta+\beta(\theta)[1+\theta
f^{\prime}(\theta)/f(\theta)]$ are decreasing.

Say lower censorship is _proper_ if the bunching and the fully revealing
regions are both nonempty. The optimality of proper lower censorship is even
less robust because Condition (Condition (N2)) requires $f$ to be _constant_
on the bunching region $(\theta_{0}^{*},\theta_{L}(\theta_{0}^{*}))$.

######  Corollary 5.4.

Assume $y(q,\theta)=q$ and the optimal deterministic rating is proper lower
censorship with cutoff $\theta_{0}^{*}$. Then, it remains optimal among all
ratings if (i) $f$ is increasing on $(\underline{\theta},\theta_{0}^{*})$ and
constant on $(\theta_{0}^{*},\theta_{L}(\theta_{0}^{*}))$, and (ii) both
$f(\theta)$ and $\theta f^{\prime}(\theta)/f(\theta)$ are decreasing on
$(\theta_{L}(\theta_{0}^{*}),\bar{\theta})$.

###  4.3 When are stochastic ratings optimal?

Stochastic ratings expand the set of incentive-compatible quality $q(\theta)$.
Under deterministic ratings, $q(\theta)$ can only be an extreme point of
$\text{MPC}(q_{f})$ on the participation region—either constant or
$q_{f}(\theta)$ (see Figure 1). Stochastic ratings lead to a more flexible
$q(\theta)$.

The optimal rating is stochastic only if (MPS) does not bind on some regions.
If (MPS) does not bind on $[\theta_{1},\theta_{2}]\subseteq\Theta$, the
multiplier $\Lambda(\theta)$ is constant on $[\theta_{1},\theta_{2}]$. If
$q^{\prime}>0$ on $[\theta_{1},\theta_{2}]$, we have

$c_{q}(q(\theta),\theta)=1-y_{q}(q(\theta),\theta)/\Lambda+\left[\frac{F(\theta_{2})-F(\theta)}{f(\theta)}+\frac{\Gamma(\theta_{2})}{\Lambda
f(\theta)}\right]c_{\theta q}(q(\theta),\theta)$ (28)

for general cost functions (see Appendix LABEL:generalcost). If
$c_{q}(q(\theta),\theta)\leq 1$ for all $\theta\in\Theta$, then
$w^{\prime}(\theta)=c_{q}(q(\theta),\theta)q^{\prime}(\theta)\leq
q^{\prime}(\theta)$ on $\Theta$. By Corollary 1, (BP) $\implies$ (MPS), and
therefore (MPS) does not bind.

Below is an example where (MPS) is binding on some regions.

##### Running example (linear-quadratic).

Assume $y(q,\theta)=q$ and $c(q,\theta)=q^{2}/2\theta$.
$\theta\sim\text{Unif}\,[1,5]$. If (MPS) does not bind so that
$\Lambda(\theta)$ is constant and $\dot{q}>0$, we have
$q(\theta)=\frac{1-1/\Lambda}{5}\theta^{2}\equiv K\theta^{2}$,
$w(\theta)=\frac{2}{3}K^{2}\theta^{3}-\frac{1}{6}K^{2}+\underline{U}$.
Substituting it into (BP) we have $\underline{U}=0$ and $\Lambda=-1$, that is,

$c_{q}(q(\theta),\theta)=2+\frac{1-F(\theta)}{f(\theta)}c_{\theta
q}(q(\theta),\theta),\mbox{ which is $>1$ for some $\theta$}.$

Thus, $q(\theta)=0.4\theta^{2}$ and
$w(\theta)=\frac{8}{75}\theta^{3}-\frac{2}{75}$, but they violate (MPS).
Therefore, (MPS) must be binding on some regions.

##### Intuition.

Intuitively, if $q^{\prime}(\theta)>0$, we can go back to the $q$-$\hat{w}$
space and write
$c^{\prime}(q^{*}(\theta))/\theta=\hat{w}^{\prime}(q^{*}(\theta))\equiv
w^{\prime}(\theta)/q^{\prime}(\theta)$. A stochastic rating scheme allows
$c^{\prime}(q^{*}(\theta))/\theta=\hat{w}^{\prime}(q^{*}(\theta))>1$ on some
regions, resulting in a higher $q(\theta)>q^{f}(\theta)$. The benefit of a
stochastic rating scheme is twofold. On the one hand, it leads to a more
flexible $q(\theta)$ which can be closer to (or even coincide with) the
principal-optimal $q_{e}(\theta)$. On the other hand, if the principal-optimal
$q_{e}$ is unattainable (i.e., $q_{e}(\theta)>q_{i}(\theta)$), it leads to
higher $q(\theta)$ in regions where $\theta$ has high density at the expense
of lower $q(\theta)$ in regions where $\theta$ has low density, which is more
desirable for the principal under some distributions.

Below are more examples where stochastic ratings are optimal.

######  Example 4.1.

Assume $y(q,\theta)=q$, $c(q)=q^{2}/2$. Assume $\Theta=[1,3]$ and $f$ is
decreasing. Then, the optimal deterministic rating is lower censorship without
exclusion and with a minimum standard $q_{0}=2$, but it is not optimal among
general ratings. Here is a stochastic rating $\pi(q)$ that improves upon it.
Fix $\hat{q}>q_{0}$. Consider the following noisy test:

$\pi(q)=\begin{cases}q,&\text{ with probability }1-p(\theta)\\\
\hat{q},&\text{ with probability }p(\theta)\end{cases}$ (29)

Figure 9: A noisy test can potentially improve upon deterministic ratings

By partially pooling low types with high types, the principal can set a higher
minimum standard $q_{0}^{\prime}>q_{0}$ at the cost of incentives for high
types, which could benefit the principal only if $f$ is decreasing.

######  Example 4.2.

Assume $y(q,\theta)=q$ and $\underline{\theta}=0$. If $f(\theta)$ is
decreasing and $\theta f^{\prime}(\theta)/f(\theta)$ is not decreasing (e.g.,
a more “extreme” distribution than Pareto distribution), the optimal
deterministic rating is fully revealing, which is not optimal among general
ratings.

###  4.4 With a Fixed Transfer

To show further cases where stochastic (noisy) tests can be optimal, I
introduce a fixed transfer $P>0$ from the agent to the principal if he takes
the test (see Appendix LABEL:transfer:general for details), which can be
interpreted as a certification fee. I restrict transfer to be fixed because,
with contingent transfers, the design of ratings no longer matters as
contingent transfers can provide incentives through redistribution in place of
the ratings (see Appendix LABEL:transfer:contingent).

With a fixed transfer, Lemma 1 still holds: an agent takes the test if and
only if he chooses $q>0$; if he does not, the market offers
$w_{\varnothing}=0$.424242 Remark to Lemma 1 may not necessarily hold if the
cost function implies $\underline{q}(\theta)>0$ (and is nonconstant), that is,
agents can still have a strictly positive (and heterogeneous) quality without
investing any effort.  Denote $\mathbf{1}[q]=\mathbf{1}[q>0]$. The principal
is now concerned with the transfer $P\cdot\mathbf{1}[q]$ in addition to
$y(q,\theta)$.

######  Example 4.3 (Optimal certification by a monopoly certifier).

In AlbanoLizzeri2001 with a constant certification fee $P$, the agent’s
utility is
$U(\theta)=w(\theta)-c(q(\theta),\theta)-P\cdot\mathbf{1}[q(\theta)]$. The
principal only cares about the expected certification fee
$\operatorname{\mathbf{E}}[P\cdot\mathbf{1}[q(\theta)]]$, so
$y_{q}(q,\theta)=0$. Thus, ignoring (MPS) so that $\Lambda$ is constant, we
have

$c_{q}(q(\theta),\theta)=1+\frac{1-F(\theta)}{f(\theta)}c_{\theta
q}(q(\theta),\theta)<1,$

Thus, (MPS) is redundant, and the optimal rating is a noisy test.

######  Proposition 6 (AlbanoLizzeri2001; see also Xiao2023).

The monopoly certifier’s revenue-maximizing $q^{*}(\theta)$ is given by

$\displaystyle\begin{cases}c_{q}(q^{*}(\theta),\theta)=1+\frac{1-F(\theta)}{f(\theta)}c_{\theta
q}(q^{*}(\theta),\theta),&\text{ if }\theta\geq\theta_{0}^{*},\\\
q^{*}(\theta)=0,&\text{ otherwise},\end{cases}$

where the optimal cutoff is

$\theta_{0}^{*}=\sup\\{\theta\in[\underline{\theta},\bar{\theta}]:q^{*}(\theta)-c(q^{*}(\theta),\theta)+\frac{1-F(\theta)}{f(\theta)}c_{\theta}(q^{*}(\theta),\theta)\leq
0\\}.$

The optimal certification fee is

$\displaystyle
P^{*}=\operatorname{\mathbf{E}}\left[q^{*}(\theta)-c(q^{*}(\theta),\theta)+\frac{1-F(\theta)}{f(\theta)}c_{\theta}(q^{*}(\theta),\theta)\mid\theta\geq\theta_{0}^{*}\right],$

and the optimal rating is a noisy test characterized by the interim wage444444
AlbanoLizzeri2001 construct the optimal rating that fully reveals quality with
some probability and outputs an (almost) uninformative signal that one has
taken the test otherwise. See also Xiao2023.

$\displaystyle
w^{*}(\theta)=\begin{cases}\int_{\theta_{0}^{*}}^{\theta}c_{q}(q^{*}(\theta),\theta)q^{*\prime}(\theta)\mathop{}\\!\mathrm{d}\theta+P^{*},&\text{
if }\theta\in[\theta_{0}^{*},\bar{\theta}],\\\ w_{\varnothing}=0,&\text{
otherwise}.\end{cases}$

######  Example 4.4 (Optimal certification by a regulator).

Suppose instead the principal is a benevolent principal certifier who
maximizes a weighted sum of the certification fee and the firm’s (agent’s)
profit, that is, $\operatorname{\mathbf{E}}[\alpha
P\cdot\mathbf{1}[q(\theta)]+(1-\alpha)U(\theta)]$. Ignoring (MPS), we have

$c_{q}(q(\theta),\theta)=1+\frac{2\alpha-1}{\alpha}\cdot\frac{1-F(\theta)}{f(\theta)}c_{\theta
q}(q(\theta),\theta)$

When $\alpha\geq 1/2$, $c_{q}(q(\theta),\theta)\leq 1$ always holds, so (MPS)
does not bind. When $\alpha<1/2$ is small (the certifier cares enough about
the firm), (MPS) becomes binding.

######  Example 4.5 (Optimal certification by a regulator).

Suppose instead the benevolent certifier maximizes a weighted sum of the
certification fee and the average quality, that is,
$\operatorname{\mathbf{E}}[\rho
P\cdot\mathbf{1}[q(\theta)>0]+(1-\rho)q(\theta)]$. Thus,
$y_{q}(q,\theta)=\frac{1-\rho}{\rho}$. Ignoring (MPS) so that $\Lambda$ is
constant, we have $\Lambda=-1$ and thus

$c_{q}(q(\theta),\theta)=1/\rho+\frac{1-F(\theta)}{f(\theta)}c_{\theta
q}(q(\theta),\theta),\mbox{ which is $>1$ for some $\theta$}.$

If $\rho<1$, (MPS) is binding for sufficiently high $\theta$.

##  5 Extension: Ability Signaling

###  5.1 Setup

Now I consider the other extreme case where the market only values the agent’s
exogenous ability, $\theta$, à la Spence1973’s (Spence1973) signaling model
(in the school example) and Holmstrom1999’s (Holmstrom1999) career concern
model (in the employer example). In this case, the interim wage is
$\hat{w}(q)=\operatorname{\mathbf{E}}_{s\sim\pi(q)}[\theta|s]$. As before,
denote $w(\theta)=\hat{w}(q(\theta))$.

The lemmas for the equivalence to reduced-form direct mechanism and incentive
compatibility still hold. As for feasibility, a theorem similar to Theorem 4
holds.

######  Theorem 12.

An incentive-compatible direct mechanism $\\{q(\theta),w(\theta)\\}$ is
feasible if and only $w(\theta)$ is a mean-preserving spread of $\theta$ in
the quantile space, that is,

  1. (i)

$\int_{\underline{\theta}}^{\theta}w(\theta^{\prime})\mathop{}\\!\mathrm{d}F(\theta^{\prime})\geq\int_{\underline{\theta}}^{\theta}\theta^{\prime}\mathop{}\\!\mathrm{d}F(\theta^{\prime})$
for all $\theta\in[\underline{\theta},\bar{\theta}]$ (MPS’),

  2. (ii)

$\int_{\underline{\theta}}^{\bar{\theta}}w(\theta)\mathop{}\\!\mathrm{d}F(\theta)=\int_{\underline{\theta}}^{\bar{\theta}}\theta\mathop{}\\!\mathrm{d}F(\theta)$
(BP’).

In other words, an increasing $w(\theta)$ is feasible if and only if it is a
mean-preserving spread of $\theta$ in the quantile space. With deterministic
ratings, it can only be an extreme point of $\text{MPS}(\theta)$ in the
quantile space, which is referred to as a “truthful filter” in Rayo2013. For
stochastic ratings, a sufficient condition for feasibility is given below.

######  Corollary 2.

If $w^{\prime}(\theta)\leq 1$ on $[\underline{\theta},\bar{\theta}]$, then
(BP’) $\Rightarrow$ (MPS’), so feasibility $\Leftrightarrow$ (BP’).

Because the type $\theta$ is exogenous, the rating design problem is simpler
than the case where the market values the endogenous quality. On the technical
side, because (MPS’) and (BP’) do not involve the state variable $q(\theta)$,
the Hamiltonian becomes simpler as it does not involve pure state constraint.
Hence, in this section, I look for the optimal general (possibly stochastic)
ratings directly, without having to start by restricting to deterministic
ratings.

###  5.2 Optimal Ratings without Transfers

As in the previous sections, I start with the case without transfers. Because
the test is costless, taking the test is a strictly dominant strategy for
every agent (except the lowest type $\underline{\theta}$ who can be
indifferent), even if he invests no effort (i.e., $c(q,\theta)=0$). Therefore,
every agent participates in the test (although he may invest no effort), in
contrast to the previous sections. Consequently,
$w_{\varnothing}=\underline{\theta}$.

######  Lemma 13.

In any equilibrium, if an agent does not take the test, he must be the lowest
type $\theta=\underline{\theta}$ who chooses $\underline{q}$ such that
$c(\underline{q},\underline{\theta})=0$, and the market offers him
$w_{\varnothing}=\underline{\theta}$.

Assume the cost is multiplicatively separable (see Appendix
LABEL:generalcost:type for general cost functions). The principal’s problem is

$\displaystyle\max_{q(\theta),w(\theta)}\int_{\underline{\theta}}^{\bar{\theta}}y(q(\theta),\theta)\mathop{}\\!\mathrm{d}F(\theta)$
(30)

subject to (MPS’), (BP’), and

$\displaystyle\theta
w(\theta)-c(q(\theta))=\int_{\underline{\theta}}^{\theta}w(x)\mathop{}\\!\mathrm{d}x+\underline{U},$
(IC-Env) (31) $\displaystyle q(\theta)\mbox{ increasing}.$ (IC-Mon) (32)
$\displaystyle\theta w(\theta)-c(q(\theta))\geq\underline{\theta}\cdot\theta,$
(IR) (33)

Say a rating induces _full separation_ (i.e., fully revealing) if
$w(\theta)=\theta$.454545 This should not be confused with the fully revealing
test in previous sections that induces $\hat{w}(q)=q$ ($w(\theta)=q(\theta)$)
when the market values quality.  Define $q_{f}(\theta)$ as the quality scheme
under full separation, which is characterized by

$\displaystyle\hat{w}(q_{f}(\theta))\equiv w(\theta)=\theta,$ (BP)
$\displaystyle
q_{f}(\theta)=\operatorname*{arg\,max}_{q}\\{\theta\hat{w}(q)-c(q)\\}\iff\hat{w}^{\prime}(q_{f}(\theta))=c^{\prime}(q_{f}(\theta))/\theta,$
(FOC) $\displaystyle\underline{\theta}^{2}-c(q_{f}(\underline{\theta}))=0.$
(IR)

The first two conditions imply

$c^{\prime}(q_{f}(\theta))\cdot q_{f}^{\prime}(\theta)=\theta,$ (34)

which, along with the initial condition in (IR), determines
$q_{f}(\theta)$.464646 In the previous sections, the fully revealing test
induces $c^{\prime}(q_{f}(\theta))=\theta$.  I also maintain Assumption 1
(downward bias) that $y_{q}(q_{f}(\theta),\theta)\geq 0$. Denote

$J(\theta|q_{f})=\frac{y_{q}(q_{f}(\theta),\theta)}{c^{\prime}(q_{f}(\theta))}\theta-\frac{\int_{\theta}^{\bar{\theta}}y_{q}(q_{f}(x),x)/c^{\prime}(q_{f}(x))\mathop{}\\!\mathrm{d}F(x)}{f(\theta)}$
(35)

In the linear delegation case where $y(q,\theta)=\beta(\theta)q-\alpha
c(q)+d(\theta)$, the expression simplifies to

$J(\theta|q_{f})=\frac{\beta(\theta)}{c^{\prime}(q_{f}(\theta))}\theta-\frac{\int_{\theta}^{\bar{\theta}}\beta(x)/c^{\prime}(q_{f}(x))\mathop{}\\!\mathrm{d}F(x)}{f(\theta)}-\alpha\left(\theta-\frac{1-F(\theta)}{f(\theta)}\right)$
(36)

If the principal elicits maximal quality/effort—i.e., $y(q,\theta)=q$, then
$J(\theta|q_{f})=\frac{\theta}{c^{\prime}(q_{f}(\theta))}-\frac{\int_{\theta}^{\bar{\theta}}1/c^{\prime}(q_{f}(x))\mathop{}\\!\mathrm{d}F(x)}{f(\theta)}$.

######  Proposition 7.

The optimal rating induces full separation (i.e.,
$q^{*}(\theta)=q_{f}(\theta)$) if and only if $J(\theta|q_{f})$ is increasing
in $\theta$.

###### Proof sketch.

Rewrite the constraints and apply the optimal control methods to the
principal’s maximization problem. See Appendix LABEL:Hamiltonian:type for
details. ∎

######  Remark 11.

For $y(q,\theta)=q$, the result is consistent with Rayo2013 (which further
assumes $c(q)=q$) and Zubrickas2015 which restrict attention to deterministic
ratings. The proposition does not impose this restriction.

######  Remark 12.

When $y(q,\theta)=q$, the necessary and sufficient condition on
$J(\theta|q_{f})$ is reminiscent of that for the optimality of winner-take-all
contests in Zhang2024. Indeed, effort maximization in the ability signaling
model is similar to that in contests.

Proposition 7 provides a regularity condition that is necessary and sufficient
for full separation to be optimal. In particular, if $y(q,\theta)=q$ and
$c(q)=q$, full separation is optimal if and only if
$J(\theta)\equiv\theta-\frac{1-F(\theta)}{f(\theta)}$ is increasing.

######  Example 5.1.

Assume $y(q,\theta)=q$, $c(q)=q$ (as in Rayo2013), and $\underline{\theta}=0$.
Then $\hat{w}(q)=\sqrt{2q}$ and $q_{f}(\theta)=\theta^{2}/2$. The optimal
rating induces full separation $q^{*}(\theta)=\theta^{2}/2$ if and only if
$J(\theta|q_{f})=\theta-\frac{1-F(\theta)}{f(\theta)}$ is increasing.

######  Example 5.2.

Assume $y(q,\theta)=q$, $c(q)=q^{2}/2$, and $\underline{\theta}=0$. Then,
$\hat{w}(q)=q$ and $q_{f}(\theta)=\theta$. The optimal rating induces full
separation $q^{*}(\theta)=\theta$ if and only if
$J(\theta|q_{f})=1-\frac{\int_{\theta}^{\bar{\theta}}1/x\mathop{}\\!\mathrm{d}F(x)}{f(\theta)}$
is increasing.

######  Corollary 7.1 (Cf. Zubrickas2015, Propositions 2).

Assume $y(q,\theta)=q$. If $q_{f}(\theta)$ is convex (or equivalently,
$c^{\prime}(q_{f}(\theta))/\theta$ is decreasing in $\theta$) on
$[\theta_{L},\bar{\theta}]$ for sufficiently large $\theta_{L}<\bar{\theta}$,
then the optimal rating induces full separation on
$[\theta_{L},\bar{\theta}]$.

###### Proof sketch.

$q_{f}(\theta)$ is convex if and only if the first term of $J(\theta|q_{f})$,
$\theta/c^{\prime}(q_{f}(\theta))$, is increasing in $\theta$. For the second
term
$J_{2}(\theta)=-\frac{\int_{\theta}^{\bar{\theta}}1/c^{\prime}(q_{f}(x))\mathop{}\\!\mathrm{d}F(x)}{f(\theta)}$,
$J_{2}^{\prime}(\theta)=\frac{c^{\prime}(q_{f}(\theta))f(\theta)+f^{\prime}(\theta)\int_{\theta}^{\bar{\theta}}1/c^{\prime}(q_{f}(x))\mathop{}\\!\mathrm{d}F(x)}{f(\theta)^{2}}$
is increasing in $\theta$ for sufficiently large $\theta_{L}<\bar{\theta}$. ∎

###  5.3 Optimal Ratings with a Fixed Transfer

Now assume there is a fixed transfer $P\geq 0$ from the agents who take the
test to the principal, which can be interpreted as a certification fee. With
transfers, the agent of type $\theta>\underline{\theta}$ may not take the
test. If so, he will invest no effort but is still endowed with some ability
$\theta>\underline{\theta}$. Thus, in contrast to the case where the market
values the endogenous quality, multiple equilibria can arise.474747 In that
case, because the market values quality (effort), it sets $w_{\varnothing}=0$
as whoever does not take the test chooses $q=0$.  I will focus on the
principal-optimal equilibrium.

Denote the decision dummy by $\sigma(\theta)=\mathbf{1}[\text{type $\theta$
takes the test}]$.

######  Lemma 14.

There exists a cutoff type $\theta_{0}$ such that the agent of type $\theta$
takes the test ($\sigma(\theta)=1$) if and only if $\theta\geq\theta_{0}$.
Consequently,
$w_{\varnothing}=\operatorname{\mathbf{E}}[\theta\mid\theta\leq\theta_{0}]$

Assume the principal maximizes a weighted sum of the certification fee and the
expected quality, that is, $\operatorname{\mathbf{E}}[\sigma(\theta)(\rho
P+(1-\rho)q(\theta))]$ where $\rho\in[0,1]$. The principal’s problem is

$\displaystyle\max_{q,w,\theta_{0}}\int_{\theta_{0}}^{\bar{\theta}}\sigma(\theta)[(1-\rho)q(\theta)+\rho
P]\mathop{}\\!\mathrm{d}F(\theta)$ (37)

subject to (MPS’) and (BP’) on $\theta\in[\theta_{0},\bar{\theta}]$, and

$\displaystyle
U(\theta)\equiv[w(\theta)-c(q(\theta),\theta)-P]\sigma(\theta)+w_{\varnothing}(1-\sigma(\theta))=-\int_{\underline{\theta}}^{\theta}\sigma(x)c_{\theta}(q(x),x)\mathop{}\\!\mathrm{d}x$
(IC-Env) $\displaystyle q(\theta)\mbox{ increasing}$ (IC-Mon) $\displaystyle
U(\theta)=[w(\theta)-c(q(\theta),\theta)-P]\sigma(\theta)+w_{\varnothing}(1-\sigma(\theta))\geq
0$ (IR)

######  Proposition 8.

Assume $\rho$ is sufficiently large or $c(q,\theta)$ is sufficiently convex in
$q$. Then, the optimal rating induces the quality scheme $q^{*}(\theta)$ given
by

$\displaystyle\begin{cases}c_{q}(q^{*}(\theta),\theta)=\frac{1-\rho}{\rho}+\frac{1-F(\theta)}{f(\theta)}c_{\theta
q}(q^{*}(\theta),\theta)\mbox{ (``$\geq$'' if
$c(q^{*}(\theta),\theta)=0$)},&\text{ if }\theta\geq\theta_{0}^{*},\\\
c(q^{*}(\theta),\theta)=0,&\text{ otherwise},\end{cases}$

where the optimal cutoff is

$\theta_{0}^{*}=\sup\\{\theta\in[\underline{\theta},\bar{\theta}]:(1-\rho)q^{*}(\theta)+(\rho-1)\theta-\rho
c(q^{*}(\theta),\theta)+\rho\frac{1-F(\theta)}{f(\theta)}c_{\theta}(q^{*}(\theta),\theta)\leq
0\\}.$

The optimal certification fee is

$\displaystyle
P^{*}=\operatorname{\mathbf{E}}[\theta]-\operatorname{\mathbf{E}}\left[c(q^{*}(\theta),\theta)-\frac{1-F(\theta)}{f(\theta)}c_{\theta}(q^{*}(\theta),\theta)\mid\theta\geq\theta_{0}^{*}\right],$

and the optimal rating is a noisy test characterized by the interim wage

$w^{*}(\theta)=\begin{cases}\int_{\theta_{0}^{*}}^{\theta}c_{q}(q^{*}(\theta),\theta)q^{*\prime}(\theta)\mathop{}\\!\mathrm{d}\theta+P^{*},&\text{
if }\theta\in[\theta_{0}^{*},\bar{\theta}],\\\
\operatorname{\mathbf{E}}[\theta\mid\theta\leq\theta_{0}^{*}],&\text{
otherwise,}\end{cases}$

which satisfies
$w^{*\prime}(\theta)=c_{q}(q^{*}(\theta),\theta)q^{*\prime}(\theta)<1$ when
$\rho$ is sufficiently large or $c(q,\theta)$ is sufficiently convex in $q$.

######  Corollary 8.1.

When $\rho=1$, that is, the principal (e.g., a monopoly certifier) cares only
about the transfer (certification fee), the optimal rating is to give the same
score to every participant, thereby inducing no effort
($c(q^{*}(\theta),\theta)=0$) and full participation ($\sigma^{*}(\theta)=1$
for all $\theta\in[\underline{\theta},\bar{\theta}]$). The optimal fee is
$P^{*}=\operatorname{\mathbf{E}}[\theta]$.

######  Remark 13.

The monopoly certifier extracts the entire surplus by setting a certification
fee $P^{*}=\operatorname{\mathbf{E}}[\theta]$, which is reminiscent of the
result of Lizzeri1999.

In this case, both $\theta_{0}^{*}=\bar{\theta}$ and
$\theta_{0}^{*}=\underline{\theta}$ satisfies the condition for the optimal
cutoff. Therefore, there also exists an equilibrium where agents invest no
effort and do not take the test ($\sigma^{*}(\theta)=0$ for all
$\theta\in[\underline{\theta},\bar{\theta}]$) (Cf. Lizzeri1999;
AliHaghpanahLin2022, where there is no manipulative effort to signal
abilities). Nevertheless, the equilibrium with full participation is unique
under the D1 refinement (see Rayo2013; CamboniNiuPai2024).

##  6 Conclusion

Ratings are often used to motivate agents’ performance or the firm’s
investments in product quality, particularly when monetary transfers are
limited or prohibited. When the market rewards agents based on their
perception of their endogenous quality or exogenous abilities, ratings can
provide reputational incentives in place of monetary incentives. In this
paper, I study the optimal rating to incentivize agents’ quality choices, when
they have private information on their costs of effort (or quality provision).
I provide necessary and sufficient conditions under which lower censorship is
optimal within deterministic ratings and solve for the optimal deterministic
ratings in general. In particular, when the principal elicits maximal effort
(quality), lower censorship is optimal if the density is unimodal, and a
pass/fail test is optimal if the density is increasing. In addition, I provide
sufficient conditions under which lower censorship remains optimal. If the
principal elicits maximal effort from agents, a pass/fail test is optimal if
the density is increasing.

When ratings are deterministic, the rating design problem is equivalent to a
delegation problem with participation constraints (AmadorBagwell2022). Lower
censorship in the rating design problem corresponds to a price-cap allocation
in the delegation problem. My result on lower censorship improves upon the
existing results by providing weaker sufficient conditions that are also
necessary in the linear delegation case. Additionally, through the equivalence
established by KolotilinZapechelnyuk2019, the results also have potential
applications to persuasion literature, especially in cases where the sender’s
payoffs are nonlinear in the state.

By defining an interim wage function (interim posterior mean) and
characterizing the necessary and sufficient condition for an incentive-
compatible (and thus increasing) interim wage to be inducible by a rating
(i.e., feasible), I provide an interim approach to the rating design problem.
The interim approach is particularly useful in solving for the optimal general
ratings as it reduces the rating design problem to the optimization over
interim wage functions rather than ratings themselves.

Nevertheless, while I provide sufficient conditions for lower censorship to
remain optimal, I have not characterized the optimal ratings in general when
stochastic ratings are allowed. Further, in the current model, the market
either values endogenous quality or exogenous abilities of agents, whereas a
combination of both cases is not considered. Moreover, I have focused on the
case where agents can choose quality deterministically. A more general case
where investing effort increases quality stochastically is also worth
exploring. In addition, competition among certifiers (i.e., test designers) is
also a direction for future research.