# Optimally Biased Expertise111We are grateful to Jean-Michel Benkert,
Alessandro Ispano, Meg Meyer, Allen Vong, seminar participants at Bern,
Copenhagen, EPFL, CERGE-EI, NRU HSE, and THEMA Cergy, as well as 12th
Conference on Economic Design, Oligo Workshop 2022, SAET 2022, and YEM 2022
conference participants for helpful comments and valuable feedback. This
project has received funding from the European Research Council under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 740369) and from the Grant Agency of the Charles University in
Prague (GAUK project No. 323221). Funding by the German Research Foundation
(DFG) through CRC TR 224 (project B03) is gratefully acknowledged.

Pavel Ilinov222Corresponding author, email address<EMAIL_ADDRESS>CERGE-EI  Andrei Matveenko  University of Mannheim, Department of Economics
Maxim Senkov  CERGE-EI  Egor Starkov  University of Copenhagen, Department of
Economics

###### Abstract

This paper shows that the principal can strictly benefit from delegating a
decision to an agent whose opinion differs from that of the principal. We
consider a “delegated expertise” problem, in which the agent has an advantage
in information acquisition relative to the principal, as opposed to having
preexisting private information. When the principal is ex ante predisposed
towards some action, it is optimal for her to hire an agent who is predisposed
towards the same action, but to a smaller extent, since such an agent would
acquire more information, which outweighs the bias stemming from misalignment.
We show that belief misalignment between an agent and a principal is a viable
instrument in delegation, performing on par with contracting and communication
in a class of problems.

JEL-codes: D82, D83, D91, M51.

Keywords: delegation, rational inattention, heterogeneous beliefs, discrete
choice.

##  1 Introduction

Delegation is a valuable management tool widely used in business structures,
political organizations, and other economic contexts. Firm owners delegate
operational decisions to managers, politicians delegate to their advisors,
grant funders delegate award decisions to experts in the field, people rely on
advice from financial and tax advisors. It is also true that in many of these
scenarios, the experts do not have much preexisting knowledge about the case
they consider, but rather use their expertise to more easily acquire
additional information to make the best decision.333E.g., Graham et al.,
(2015) show that delegation tends to be used when the decision-making demands
more evidence that the delegatee can provide. Alternatively, the choice to
delegate a decision is often associated with a volatile environment that a
delegator faces (Foss and Laursen,, 2005; Ekinci and Theodoropoulos,, 2021),
so any knowledge quickly becomes obsolete.

What happens if the preferences, or even the perceptions about the
fundamentals are not fully aligned between the principal and the agent? The
common wisdom (see, e.g., Holmstrom, (1980)) suggests that this leads to a
conflict, since from the principal’s point of view, the agent then makes
suboptimal decisions.444This is supported by the empirical evidence: e.g.,
Hoffman et al., (2018) find that inefficiencies in the HR managers’ hiring
decisions can be a result of their biased preferences. Kennedy, (2016)
presents evidence that the principals take the conflict of interest into
account when selecting the expert. However, we show in this paper that when
the agent has to actively acquire information, as opposed to having a
preexisting informational advantage, there is more to this story. In
particular, such a misalignment can then benefit the principal by encouraging
the agent to acquire more information than an aligned agent would.

We consider a delegation model, in which the payoffs from different actions
depend on the unknown state of the world. Our main focus is on the case, in
which the principal and the agent have common preferences regarding states and
actions but misaligned beliefs about the state of the world. The agent has no
private information about the state, but can acquire costly information about
it, which would improve the quality of the decision made (this setting has
been labeled by Demski and Sappington, (1987) as “delegated expertise”). The
cost of learning is not internalized by the principal.

We show that the principal benefits the most from delegating to an agent who
ex ante is more uncertain than she is about what the best course of action is
(shown in different contexts by Propositions 1, 2, and 3). This is because the
more uncertain the agent is, the more he learns about the state, and the
better his action fits the conditions – which benefits the principal. This,
however, has to be balanced against the channel described above: any kind of
misalignment between the principal and the agent leads to a bias in the
agent’s decisions, compared to what the principal would prefer. Therefore, the
principal ends up hiring an agent, who is more uncertain than she is and thus
conducts a more thorough investigation than an aligned agent would, – but who
still shares her action predispositions to some extent. This result holds
regardless of who has the final decision rights: the optimal delegation
strategy is the same whether the principal delegates the decision rights to
the agent or merely expects a recommendation on the optimal course of action
(Proposition 10).

The presence of the principal’s trade-off between the amount of information
acquired by an agent and the bias in his resulting decisions (Section 4)
relies upon the flexible information acquisition technology. We use the
Shannon model of discrete rational inattention (see, e.g., Matějka and McKay,
(2015), Caplin et al., (2019)) to provide this dimension of richness.
According to the Shannon model, an agent can choose any signal structure but
has to pay a cost proportional to the expected entropy reduction.555The
entropy parametrization leads to information cost being dependent on the prior
belief, even keeping the signal structure constant. This has been one of the
critiques of the Shannon model (see Mensch, (2018)). Such a cost function has,
however, been rationalized in both the information theory as a cost function
arising from the optimal encoding problem (see Cover and Thomas, (2012)), and
decision theory as arising naturally from Wald’s sequential sampling model
(see Hébert and Woodford, (2019)). In turn, the Shannon model has been shown
to work as a microfoundation of the logit choice rule commonly used in choice
estimation (Matějka and McKay, (2015)). The choice of the signal in this model
depends on the agent’s prior belief: an agent, whose prior is skewed towards
some state of the world, chooses a signal which is relatively more informative
regarding that state. This dimension of flexibility is what leads to bias in
the final decisions when the agent’s prior belief is not aligned with the
principal’s.

We also show that the principal can equivalently use the misalignment in
preferences rather than the misalignment in beliefs (see Theorem 1). Namely,
Proposition 5 states that the second-best outcome can be implemented by hiring
an agent with either optimally misaligned beliefs, or optimally misaligned
preferences (or, equivalently, offering action-contingent compensation). This
result has a mirror implication for the empirical literature estimating
discrete choice models: Theorem 1 implies that the observed choice
probabilities alone do not allow an external observer to jointly identify the
decision maker’s beliefs and preferences in our setting.

The main conclusion of our paper is that delegation to an agent with
misaligned beliefs is an instrument that is available – and valuable – to the
principal. Not only that, but in our setting it can perform as well as action-
contingent payments, while bearing no cost for the principal (Proposition 5),
and cannot be improved on by using outcome-contingent payments (Proposition
7). Further, it is typically better than restricting the agent’s choice set
(Proposition 8). This benefit of misalignment challenges the opinion that
disagreement between the principal and the agent inevitably leads to a
conflict, and thus the principal should seek to hire an agent who is most
aligned with her preferences and beliefs (see Holmstrom, (1980); Crawford and
Sobel, (1982); Prendergast, (1993); Alonso and Matouschek, (2008); Egorov and
Sonin, (2011); Che et al., (2013) for some examples of such a message).

Our paper mainly connects to the literature on delegation, mainly to the
problems of “delegated expertise”, in which delegation takes place not to an
agent with some preexisting private information, but rather to an agent, whose
goal is to _acquire_ relevant information. The assumption is that the agent’s
expertise allows him to gather information at lower cost than what the
principal would have to incur. The seminal paper in the field is that by
Demski and Sappington, (1987), who explore a contracting problem in a setting,
in which the agent chooses between a finite number of signal structures.
Lindbeck and Weibull, (2020) extend this analysis to a rationally inattentive
agent (who can acquire any information, subject to entropy costs). Szalay,
(2005) shows that restricting the agent’s action set could be a useful tool in
such a setting, since banning an ex ante optimal “safe” action can nudge the
agent to acquire more information about which of the risky actions is the
best. Our grand message is similar: the principal is willing to sacrifice
something in exchange for the agent acquiring more information, but we present
a different channel through which the principal can achieve this.

The closest to our paper is contemporary work by Ball and Gao, (2021). They
consider a model of delegated expertise and demonstrate a result similar to
that of Szalay, (2005): that banning the ex ante safe actions can lead to more
information acquisition by the agent, which benefits the principal. However,
where Szalay, (2005) looks at the scenario where the principal’s and the
agent’s preferences coincide ex post (i.e., net of information costs), Ball
and Gao, (2021) explore a model with misaligned preferences and show that the
principal may benefit from some misalignment between her preferences and those
of the agent. In their setting, this is due to divergence between the
principal’s and the agent’s ex ante optimal actions (due to preference
misalignment), which makes banning the ex ante agent-preferred action less
costly for the principal. Our paper suggests a different channel through which
misalignment may incentivize the agent’s information acquisition: we show,
using a flexible information acquisition framework, that misalignment can lead
to more information acquisition by the mere virtue of the agent being more
uncertain about what the optimal action is.

The misalignment in prior beliefs was also studied by Che and Kartik, (2009).
They analyze a delegated expertise game, in which the principal retains the
decision rights: i.e., after acquiring the relevant information, the agent
must communicate it to the principal, who then makes a decision she believes
is optimal. They show that the need to communicate may also incentivize an
agent with misaligned preferences to acquire more information, in order to
more effectively persuade the principal about which action needs to be taken.
Their conclusion does, however, rely on an inflexible information acquisition
structure (the agent’s probability of observing the true state of the world is
increasing in his effort). It appears likely that in a more flexible model, a
misaligned agent would acquire not _more_ , but rather _different_ information
in order to be persuasive.

The remainder of the paper is organized as follows: we present a simple
example that demonstrates the main effect in Section 2. Section 3 formulates
the main model, which is then analyzed in Section 4 for the special case of
binary states and actions, while Section 5 analyzes the general problem.
Section 6 compares misaligned beliefs as a delegation tool to other tools,
such as misaligned preferences, payments, and restricting the action set.
Section 7 explores a number of extensions of the baseline model, and Section 8
concludes.

##  2 Illustrative example

This section presents a simple example with inflexible learning and
demonstrates how a misalignment in beliefs between a principal and an agent
may benefit the principal. The full model is introduced in Section 3.

Consider a president (a principal, she) who needs to appoint a minister or a
head of the government agency (an agent/expert, he) to solve a particular
policy issue, e.g., “the green transition” policy, or the antitrust policy
regarding the big tech companies. Suppose that there are two policies to
choose from: $a\in\\{L,R\\}$.666This binary model is common in the delegation
literature, see e.g. Li and Suen, (2004) with slightly different informal
story. Which policy is optimal depends on the state of the world
$\omega\in\\{l,r\\}$, which is initially unknown to both the principal and the
agent(s). Suppose further that the principal and the agent have a common
interest in implementing the correct policy.777In this example we focus on
misalignment in beliefs; see Section 6.1 for the discussion of misalignment in
preferences. In particular, the utility $u(a,\omega)$ all players derive from
policy $a$ in state $\omega$ is given by $u(L,l)=u(R,r)=1$ and
$u(L,r)=u(R,l)=0$. We denote the probability that the principal’s prior belief
assigns to state $\omega=r$ as $\mu_{p}\in[0.5,1]$. The agent can choose
whether to learn the state at cost $c$ or not, and we assume that this cost is
neither internalized, nor compensated for by the principal.

The principal’s problem is to select the best agent to delegate the decision
to. There are many experts available to the principal, who differ in their
prior beliefs about the state of the world: $\mu\in[0,1]$. For example, she
can delegate a decision to an expert who is certain that $\omega=r$ (i.e.,
$\mu=1$), or to an agent who has the same opinion as her ($\mu=\mu_{p}$), or
to the most uncertain agent ($\mu=1/2$). The expert’s prior belief is
observable by the principal, which could be due to him having an established
reputation for having a certain position on the question at hand. We assume
that the principal can not use monetary transfers and/or restrict the set of
actions available to the agent (see Section 3 for a discussion of this
assumption).

The timing is as follows: the principal chooses an agent based on his prior
belief $\mu$ from a pool of agents $\mathcal{M}=[0,1]$; then the agent chooses
whether to learn the state $\omega$ at cost $c$, and subsequently implements
the policy $a$ preferred given his prior belief and the acquired signal.

The agent’s expected utility when he learns the state and when he does not is
given by, respectively:

$\displaystyle\mathbb{E}_{\mu}[u(a,\omega)]-c=1-c,$
$\displaystyle\mathbb{E}_{\mu}[u(a,\omega)]=\max\\{\mu,1-\mu\\}.$

Hence the willingness to pay for information of an agent with prior belief
$\mu\geq 1/2$ is given by

$\underbrace{1-c}_{\text{with
info}}-\underbrace{\max\\{\mu,1-\mu\\}.}_{\text{without info}}$

Since this amount is decreasing in $\mu$ when $\mu\geq 1/2$ and increasing
when $\mu\leq 1/2$, it is maximized at $\mu=1/2$. Namely, the most uncertain
agent is willing to pay the most for information: if an agent with some prior
$\mu$ chooses to obtain information, then an agent with prior $\mu=1/2$ would
also choose to learn the state. Therefore, the principal (weakly) prefers to
delegate to the most uncertain agent. The next simple proposition formalizes
the result.

######  Proposition 1.

In the equilibrium of the described model, it is (weakly) optimal for the
principal to delegate decision to the most uncertain agent: $\mu^{*}=1/2$.

The proposition above demonstrates that hiring an agent with misaligned
beliefs may lead to the agent acquiring more information, which is beneficial
for the principal. This restricted setting, however, cannot illustrate the
bias introduced in the agent’s decisions by the misalignment. In the next
section, we proceed to the full model, which demonstrates that there also
exists a countervailing preference for less misaligned agent due to the bias
in actions that this misalignment introduces. Such a preference arises due to
the agent being able to acquire information flexibly.

##  3 Model

###  3.1 Concepts and Definitions

Consider a principal (she) who would like to implement an optimal decision
that depends on the unknown state of the world. To choose the best course of
action, the principal delegates the decision to an expert (an agent, he), who
can acquire information about what the optimal decision is.888An alternative
would be to ask the agent to learn about the state and report the findings to
the principal, who then makes the decision. This version is explored in
Section 7.2, which demonstrates that communication is equivalent to delegation
in our setting (barring the equilibrium multiplicity). There are many experts
available to the principal, and all experts have common interest with the
principal, but differ in their opinions on the issue.999In the “green
transition” policy example, the experts would differ in their stance on the
severity of the climate threat. Experts with different initial opinions would
acquire different information, and thus possibly make different final
decisions. The principal is thus concerned with finding the best agent for the
job.101010Our results can, alternatively, be interpreted as comparative
statics for a game between a principal and a given an agent with some fixed
misalignment, w.r.t. the degree of misalignment.

The above can be modeled as a game played between a principal and a population
of agents. In particular, let $\mathcal{A}\equiv\\{a_{1},...,a_{N}\\}$ denote
the set of actions and $\Omega\equiv\\{\omega_{1},...,\omega_{M}\\}$ the set
of states. The principal has a prior belief $\mu_{p}\in\varDelta(\Omega)$,
where $\varDelta(\Omega)$ denotes the set of all probability distributions on
$\Omega$. Every agent in the population has some prior belief
$\mu\in\varDelta(\Omega)$, which is observable and verifiable, e.g., due to
the reputation concerns (i.e., agents needing to publicly establish a
particular stance on the broad policy question for sake of earning, and
subsequently capitalizing on, a specific reputation).111111To clarify, we work
with a model of non-common prior beliefs about $\omega$, and we take this
assumption at face value. Such settings are not uncommon in economic theory
(see Morris, (1995); Alonso and Câmara, (2016); Che and Kartik, (2009) for
some examples). It is well known (see Aumann, (1976) and Bonanno and Nehring,
(1997)) that agents starting with a common prior can not commonly know that
they hold differing beliefs. We allow the agents to have heterogeneous prior
beliefs, and, thus, to “agree to disagree”. While it may be possible to
replicate our results in a common-prior model with asymmetric information,
such a model would feature signaling concerns (e.g., an agent learning
something about the principal’s information about the state from the fact that
he was chosen for the job, and the principal then exploiting this inference
channel). We prefer to abstract from such signaling and simply assume non-
common priors from the start. In what follows, we refer to an agent according
to his prior belief. Let $\mathcal{M}\subseteq\varDelta(\Omega)$ denote the
set of prior beliefs of all agents in the population.121212For most of the
results we assume that the population of agents is rich enough to represent
the whole spectrum of viewpoints: $\mathcal{M}=\varDelta(\Omega)$.,

The terminal payoff that both the principal and the agent selected by the
principal receive when action $a$ is chosen in the state $\omega$ is given by
$u(a,\omega)$. Prior to making the decision, the selected agent can acquire
additional information about the realized state. We assume that the agent can
choose any signal structure defined by the respective conditional distribution
$\phi(s|\omega)$ of signals $s\in\mathcal{S}$ given the state
$\omega\in\Omega$, where $\mathcal{S}$ is arbitrarily rich. Signals are
costly: when choosing a signal structure $\phi$, the agent must incur cost
$c(\phi,\mu)$ that may depend on the informativeness of the signal and the
agent’s prior belief.131313Similar to, e.g., Alonso and Câmara, (2016), we
assume that the agent and the principal share the understanding of the signal
structure. Combined with them having different (subjective) prior beliefs over
states, this implies they would also have different (subjective) posterior
beliefs if both observed the signal realization.

The cost function we consider is the Shannon entropy cost function used in
rational inattention models (Matějka and McKay,, 2015). In this specification,
the cost is proportional to the expected reduction in the entropy of the
agent’s belief resulting from receiving the signal. Namely, let
$\eta:\mathcal{S}\to\varDelta(\Omega)$ denote the agent’s posterior belief
after observing signal realization $s$, obtained from $\mu$ and $\phi$ using
the Bayes’ rule. The cost can be defined as

$\displaystyle c(\phi,\mu)\equiv\lambda\Bigg{(}$
$\displaystyle-\sum_{\omega\in\Omega}\mu(\omega)\ln\mu(\omega)+$
$\displaystyle+\sum_{\omega\in\Omega}\sum_{s\in\mathcal{S}}\left(\sum_{\omega^{\prime}\in\Omega}\mu(\omega^{\prime})\phi(s|\omega^{\prime})\right)\eta(\omega|s)\ln\eta(\omega|s)\Bigg{)},$
(1)

where $\lambda\in\mathbb{R}_{++}$ is a cost parameter.141414We also follow the
standard convention and let $0\ln 0=0$. We assume that the principal does not
internalize the cost of learning, and the agent fully bears this cost. The
main interpretation (shared by, e.g., Lipnowski et al., (2020)) of this
assumption is that the cost reflects the cognitive process of the agent.
Information acquisition costs thus lead to moral hazard, with the agent
potentially not willing to acquire the amount of information desired by the
principal. This is the main conflict between the two parties in our model.

In line with the delegation literature, we assume that the principal can not
use monetary or other kinds of transfers to manage the agent’s incentives.
This is primarily because learning is non-contractible in most settings –
indeed, it is hard to think of a setting, in which a learning-based contract
could be enforceable, i.e., either the principal or the agent could
demonstrate beyond reasonable doubt exactly how much effort the agent has put
into learning the relevant information, and what kind of conclusions he has
arrived at. A simpler justification of the no-transfer assumption could be
that such transfers are institutionally prohibited in some settings.151515See
Laffont and Triole, (1990); Armstrong and Sappington, (2007); Alonso and
Matouschek, (2008) for some examples and discussion of such settings. We do
allow for some classes of transfers in Section 7 and show that even in those
settings where contracting is feasible, it does not necessarily perform better
than hiring an agent with a misaligned belief.

The game proceeds as follows. In the first stage, the principal selects an
agent from the population based on the agent’s prior belief $\mu$. In the
second stage, the selected agent chooses signal structure $\phi$ and pays cost
$c(\phi,\mu$). In the third stage, the agent receives signal $s$ according to
the chosen $\phi$ and selects action $a$ given $s$. Payoffs $u(a,\omega)$ are
then realized for the principal and the agent.

The following subsections describe the respective optimization problems faced
by the principal and her selected agent, and define the equilibrium concept.

###  3.2 The Agent’s Problem

The agent selected by the principal chooses a signal structure
$\phi:\Omega\to\varDelta(\mathcal{S})$ and a choice rule
$\sigma:\mathcal{S}\to\mathcal{A}$ to maximize his expected payoff net of the
information costs. The agent’s objective function is

$\displaystyle\mathbb{E}_{\mu}[u(a,\omega)]-c(\phi,\mu)=\sum_{\omega\in\Omega}\mu(\omega)\sum_{s\in\mathcal{S}}\phi(s|\omega)u(\sigma(s),\omega)-c(\phi,\mu),$

where $\mathbb{E}_{\mu}$ denotes the expectation w.r.t. the agent’s belief
$\mu$. The agent’s problem can then be written down as

$\displaystyle\
\max_{\phi,\sigma}\left\\{\sum_{\omega\in\Omega}\mu(\omega)\sum_{s\in\mathcal{S}}\phi(s|\omega)u(\sigma(s),\omega)-c(\phi,\mu)\right\\},$
(2) $\displaystyle\text{s.t.
}\forall\omega\in\Omega:\phi(\cdot|\omega)\in\varDelta(\mathcal{S}).$

Lemma 1 in Matějka and McKay, (2015) shows that problem (2) with entropy cost
function can be reframed as a problem of selecting a collection of conditional
choice probabilities. This reformulation is presented in Section 3.5.

###  3.3 The Principal’s Problem

The principal’s problem is to choose an agent based on his prior belief
$\mu\in\mathcal{M}$ in order to maximize her expected utility from the action
eventually chosen by the agent (where $\mathbb{E}_{p}$ denotes the expectation
w.r.t. the principal’s belief $\mu_{p}$). Her objective function is

$\displaystyle\mathbb{E}_{p}[u(a,\omega)]=\sum_{\omega\in\Omega}\mu_{p}(\omega)\sum_{s\in\mathcal{S}}\phi(s|\omega)u(\sigma(s),\omega),$

so her optimization problem can be written down as

$\displaystyle\max_{\mu}$
$\displaystyle\left\\{\sum_{\omega\in\Omega}\mu_{p}(\omega)\sum_{s\in\mathcal{S}}\phi(s|\omega)u(\sigma(s),\omega)\right\\},$
(3)

where the signal structure $\phi$ and the decision rule $\sigma$ are given by
the agent’s equilibrium strategy, and this strategy depends on his prior
belief $\mu$. Therefore, the principal’s choice of agent indirectly determines
$\phi$ and $\sigma$.

###  3.4 Equilibrium Definition

We now present the equilibrium notion used throughout the paper; the
discussion follows.

######  Definition (Equilibrium).

An equilibrium of the game is given by
$(\mu^{*},\\{\phi^{*}_{\mu},\sigma^{*}_{\mu}\\}_{\mu\in\mathcal{M}})$: the
principal’s choice $\mu^{*}\in\mathcal{M}$ of the agent who the task is
delegated to and a collection of the agents’ information acquisition
strategies $\phi^{*}_{\mu}:\Omega\to\varDelta(\mathcal{S})$ and decision rules
$\sigma^{*}_{\mu}:\mathcal{S}\to\mathcal{A}$ for all $\mu\in\mathcal{M}$, such
that:

  1. 1.

$\phi^{*}_{\mu}$ and $\sigma^{*}_{\mu}$ constitute a solution to (2) for every
$\mu\in\mathcal{M}$;

  2. 2.

$\mu^{*}$ is a solution to (3) given $(\phi^{*}_{\mu},\sigma^{*}_{\mu})$.

Note that the above effectively defines a Subgame-Perfect Nash Equilibrium.
While our game features incomplete information (about the state of the world
chosen by Nature), and the players’ beliefs play a central role in the
analysis, problem formulations (2) and (3) allow us to treat these beliefs as
just some exogenous functions entering the terminal payoff functions. This is
primarily due to the fact that one player’s actions do not affect another
player’s beliefs in this game, hence a belief consistency requirement is not
needed (however, we do require internal consistency in that the agent’s
posterior belief $\eta$ is obtained by updating his prior belief $\mu$ via
Bayes’ rule given his requested signal structure $\phi$).

###  3.5 Preliminary Analysis

Matějka and McKay, (2015) show that with entropy costs, the agent’s problem of
choosing information structure and choice rule can be reduced to the problem
of choosing the conditional action probabilities. Namely, the maximization
problem of the agent can be rewritten as that of choosing a single state-
contingent action distribution $\pi:\Omega\to\varDelta(\mathcal{A})$ (as
opposed to the combination of a signal strategy
$\phi:\Omega\to\varDelta(\mathcal{S})$ and a decision rule
$\sigma:\mathcal{S}\to\mathcal{A}$):

$\max_{\pi}\Bigg{\\{}\sum_{j=1}^{N}\mu(\omega_{j})\left(\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})u(a_{i},\omega_{j})\right)-\\\
-\lambda\left(\sum_{j=1}^{N}\mu(\omega_{j})\left(\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})\ln\pi(a_{i}|w_{j})\right)-\sum_{i=1}^{N}\beta(a_{i})\ln\beta(a_{i})\right)\Bigg{\\}},$
(4)

where $\pi(a_{i}|\omega_{j})$ stands for the conditional probability of
choosing alternative $a_{i}$ in state $\omega_{j}$, and $\beta(a_{i})$ denotes
the respective unconditional probability of choosing alternative $a_{i}$
(calculated using the agent’s own prior belief $\mu$):

$\beta(a_{i})\equiv\sum_{j=1}^{N}\mu(\omega_{j})\pi(a_{i}|\omega_{j}).$ (5)

The principal’s problem can then be rewritten as choosing $\mu\in\mathcal{M}$
that solves

$\displaystyle\max_{\mu}\left\\{\sum_{j=1}^{N}\mu_{p}(\omega_{j})\left(\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})u(a_{i},\omega_{j})\right)\right\\}$
(6) $\displaystyle\text{s.t. }\pi\text{ solves \eqref{eq:ri_problem_full}}.$

In what follows, we refer to problem (6) as the principal’s full problem. Note
that the solution of the full problem together with the corresponding $\pi$
constitute an equilibrium as defined above, when complemented with the optimal
strategies of the non-chosen agents (those with
$\mu\in\mathcal{M}\backslash\\{\mu^{*}\\}$).

We now proceed to analyze the model described above.

##  4 Binary Case

In this section we return to the binary-state, binary-action version of the
model set up in Section 2 to see how it behaves when the agent has flexibility
in his learning technology (as opposed to binary all-or-nothing learning). We
show that with the entropy cost function, the principal has to balance off the
amount of information acquired against the _nature_ of information acquired –
since agents with different prior beliefs bias their learning towards
different states. This makes the principal favor agents who are somewhat more
uncertain than her regarding the state, but not necessarily have a uniform
prior belief (Proposition 2).

To remind, Section 2 assumed that the state space is $\Omega=\\{l,r\\}$, the
action set is $\mathcal{A}=\\{L,R\\}$, and the common utility function net of
information costs is such that $u(L|l)=u(R|r)=1$ and $u(L|r)=u(R|l)=0$. We
proceed by the backward induction, looking at the agent’s problem first, and
then using the agent’s optimal behavior to solve the principal’s problem of
choosing the best agent.

The agent is allowed to choose any informational structure (Blackwell
experiment) he wants, paying the cost which is proportional to expected
reduction of the Shannon entropy of his belief. Using the result presented in
Section 3.5, the agent’s problem can be reformulated as the problem of
choosing a stochastic choice rule $\pi:\Omega\to\varDelta(\mathcal{A})$, which
solves

$\displaystyle\max_{\pi}\Big{\\{}\mu\pi(R|r)+(1-\mu)\pi(L|l)-c(\pi,\mu)\Big{\\}},$
(7) where $\displaystyle
c(\pi,\mu)=\lambda\left(\sum_{a\in\\{L,R\\}}\beta(a)\ln\beta(a)-\sum_{\omega\in\\{l,r\\}}\mu(\omega)\sum_{a\in\\{L,R\\}}\pi(a|\omega)\ln\pi(a|\omega)\right).$

The solution to this problem can be summarized by the two precisions
$\\{\pi(R|r),\pi(L|l)\\}$ or, alternatively, the two unconditional
probabilities $\\{\beta(R),\beta(L)\\}$. Using Theorem 1 in Matějka and McKay,
(2015), we get that

$\displaystyle\pi(L|l)$
$\displaystyle=\frac{\beta(L)e^{\frac{1}{\lambda}}}{\beta(L)e^{\frac{1}{\lambda}}+\beta(R)},$
$\displaystyle\pi(R|r)$
$\displaystyle=\frac{\beta(R)e^{\frac{1}{\lambda}}}{\beta(L)e^{\frac{1}{\lambda}}+\beta(R)},$
(8)

and their Corollary 2 then adds the conditions

$\displaystyle\frac{\mu}{\beta(L)+\beta(R)e^{\frac{1}{\lambda}}}+\frac{(1-\mu)e^{\frac{1}{\lambda}}}{\beta(L)e^{\frac{1}{\lambda}}+\beta(R)}$
$\displaystyle=1,$ (9) $\displaystyle\frac{\mu
e^{\frac{1}{\lambda}}}{\beta(L)+\beta(R)e^{\frac{1}{\lambda}}}+\frac{(1-\mu)}{\beta(L)e^{\frac{1}{\lambda}}+\beta(R)}$
$\displaystyle=1.$ (10)

Combining (8)–(10), we get161616This solution takes the form of the so-called
rational inattention (RI) logit. In comparison to the standard logit behavior,
under RI-logit the decision-maker (the agent in our case) has a stronger
tendency to select the ex ante optimal alternatives more frequently.

$\displaystyle\pi(R|r)$ $\displaystyle=\frac{\left(\mu
e^{\frac{1}{\lambda}}-(1-\mu)\right)e^{\frac{1}{\lambda}}}{\left(e^{\frac{2}{\lambda}}-1\right)\mu},$
(11) $\displaystyle\pi(L|l)$
$\displaystyle=\frac{\left((1-\mu)e^{\frac{1}{\lambda}}-\mu\right)e^{\frac{1}{\lambda}}}{\left(e^{\frac{2}{\lambda}}-1\right)(1-\mu)},$

cropped to $[0,1]$. In turn, the principal’s problem is the same as in Section
2:

$\displaystyle\max_{\mu}\left\\{\mu_{p}\pi(R|r)+(1-\mu_{p})\pi(L|l)\right\\}$
(12) $\displaystyle\text{s.t. }\pi\text{ solves problem \eqref{eq:ri_problem}
given }\mu.$

It is easy to see by comparing the payoffs in (7) and (12) that the principal
benefits from higher precisions $\pi(R|r)$ and $\pi(L|l)$, same as the agent.
However, the relative weights the principal and the agent assign to these
precisions depend on their respective priors $\mu_{p}$ and $\mu$, and hence
differ between the two. Hence in order to understand the trade-offs that the
principal faces in hiring agents with different priors, we need to explore how
the agent’s optimal strategy (11) depends on his prior belief $\mu$.

Solving the problem (7), the agent faces a trade-off between increasing the
precision of his decisions, $\pi(R|r)$ and $\pi(L|l)$, and the cost of
information. With the flexible acquisition technology, the agent focuses on
the more important event: namely, the higher is the probability that the
agent’s prior belief assigns to $\omega=r$, the more important is precision
$\pi(R|r)$ for his payoff, compared to $\pi(L|l)$. Therefore, two agents with
different beliefs would acquire _different_ information, leading to different
precisions $\pi(R|r)$ and $\pi(L|l)$.171717This feature of the flexible
information acquisition model was analyzed in the application to belief
polarization by Nimark and Sundaresan, (2019), as well as in the marketing
literature (see Jerath and Ren, (2021)). At the same time, as the prior belief
$\mu$ gets close to the extremes ($\mu=0$ or $\mu=1$), the agent becomes so
confident about the state that the precision in the other state becomes
irrelevant, and the agent gravitates to the ex ante optimal action. The
marginal benefit of additional information for the agent in this scenario is
low, and the agent consumes less information in total (in bits).

To summarize, the agent’s belief $\mu$ affects his optimal decision precisions
in two ways: a more uncertain agent acquires more information (and hence makes
a better decision on average) than an agent who believes one state is more
likely. However, the latter is more concerned with choosing the correct action
in the ex ante more likely state, while neglecting the other state. Figure 1
demonstrates how the agent’s action precisions choice depends on his prior
belief.

Figure 1: Solution to the problem (7) with different prior beliefs.

The principal prefers, ceteris paribus, to hire an agent who acquires more
information and hence makes better choices – i.e., a more uncertain agent
($\mu$ close to $0.5$). However, if she believes that, e.g., state $r$ is ex
ante more likely ($\mu_{p}>0.5$), then she, for all the same reasons as the
agent, cares more about the agent choosing the optimal action in state $r$
than in state $l$. The latter leads her to prefer an agent who is not
completely uncertain ($\mu\neq 0.5$), favoring those who agree with her in
terms of which state is more likely ($\mu>0.5$). Balancing the two issues
leads to the principal optimally hiring an agent who has a belief different
from hers: $\mu\neq\mu_{p}$, yet who fundamentally agrees with her on the ex
ante optimal action: $\mu\geq 0.5\iff\mu_{p}\geq 0.5$.

Figure 2 plots the principal’s expected utility from hiring an agent as a
function of the agent’s belief $\mu$ when $\mu_{p}=0.7$. We can see the
principal with a prior belief $\mu_{p}=0.7$ would prefer to hire an agent with
a prior belief $\mu\approx 0.6$. Note that the graph is flat for very high and
very low $\mu$, which corresponds to the agents who do not learn anything, and
simply always choose the ex ante optimal action. Further, there exist agents
with low $\mu\sim(0.15,0.25)$ who acquire non-trivial information, but hiring
whom is worse for the principal than taking the ex ante optimal action
(equivalent to hiring an agent with $\mu=1$). The latter demonstrates that if
an agent is too biased, the information he acquires does not benefit the
principal due to the bias in the agent’s actions resulting from his initial
predisposition.

Figure 2: Expected utility of principal with prior belief $\mu_{p}=0.7$ as a
function of the agent’s prior belief $\mu$.

The proposition 2 below formalizes this intuition and provides a closed-form
solution for the optimal delegation strategy given the principal’s prior
belief $\mu_{p}$. Figure 3 visualizes the optimal delegation strategy as a
function of $\mu_{p}$.

######  Proposition 2.

If $\mathcal{M}=[0,1]$, then the principal’s optimal delegation strategy is
given by

$\displaystyle\mu^{*}=\frac{\sqrt{\mu_{p}}}{\sqrt{\mu_{p}}+\sqrt{1-\mu_{p}}}.$
(13)

Therefore, if $\mu_{p}\in\left(\frac{1}{2},1\right)$, the principal optimally
delegates to an agent with belief
$\mu_{A}\in\left(\frac{1}{2},\mu_{p}\right)$.

Figure 3: The optimal delegation strategy $\mu^{*}$ as a function of the
principal’s prior belief $\mu_{p}$.

One thing to note about Proposition 2 is that the optimal delegation strategy
(13) does not depend on the agent’s information cost factor, $\lambda$. While
it is immediate that the higher is $\lambda$, the less information the agent
with any given prior $\mu$ collects, Proposition 2 serves to show that the
trade-off between the quantity of information and the bias in the decisions
does not depend on the absolute quantity of information the agent acquires.
Further, for a moderately high $\lambda$ (and/or a moderately extreme
$\mu_{p}$), an agent with belief $\mu^{*}$ acquires no information, while an
agent with belief $\mu=1/2$ would acquire some. In this case, it is
nonetheless optimal for the principal to hire the $\mu^{*}$ agent (or,
alternatively, simply choose the ex ante optimal action himself), since a
learning agent would be too misaligned, and the benefits of information would
be outweighed by the bias in the decisions.

Figure 4 demonstrates the difference in the action precisions between
delegating to a perfectly aligned agent ($\mu=\mu_{p}$) and the optimally
misaligned agent as given by (13). Optimal delegation leads to the agent
consuming more information, lowers the probability of correctly matching the
ex ante more likely (according to the principal’s belief $\mu_{p}$) state,
$\pi(R|r)$, and increases $\pi(L|l)$, thereby bringing the two closer
together. Overall, under the optimal delegation the less attractive ex-ante
option is implemented relatively more frequently as compared to the case of
the aligned delegation.

Figure 4: Action precisions under optimal delegation and delegating to the
aligned agent.

An interesting connection can be made here to prospect theory (see Barberis,
(2013) for a review). In particular, Tversky and Kahneman, (1992) suggest that
in problems of choice under risk, individual decision-makers tend to succumb
to cognitive biases such as overweighing small probabilities and underweighing
large probabilities. They propose a probability weighting function that
decision makers unconsciously use, which is reminiscent of our optimal
delegation strategy (13), with $\mu_{p}$ being the objective probability and
$\mu^{*}$ being the decision-maker’s perceived probability. Our result can
thus be interpreted as one possible evolutionary explanation of the
probability weighting functions. Namely, suppose that “Nature” (evolutionary
pressure) is the principal and “Human” is the agent. They both have common
utility function $u(a,\omega)$ representing the survival probability of the
individual/population, but natural selection is indifferent towards the
human’s cognitive costs $c(\phi,\mu)$ involved in the decision-making process.
In this setting, natural selection would lead humans to develop probabilistic
misperceptions according to (13), since these maximize the survival
probability.181818Steiner and Stewart, (2016) suggest an alternative
explanation of probabilistic misperceptions using a similar nature-as-a-
principal approach, but a different source of conflict between Nature and
Human.

In the next section we generalize the binary model, assuming more available
alternatives, while keeping the structure of the payoffs the same.

##  5 General Case

In this section we extend the analysis to a general problem of finding the
best alternative, allowing for $N>2$ actions and states. We show that the
principal’s optimal delegation strategy is qualitatively the same as in the
binary case, i.e., it is optimal to hire a “more uncertain” agent, who
considers more actions in search of the best one, relative to a fully aligned
agent. Further, we characterize the whole set of decision rules that can be
achieved by selecting the agent’s prior belief and show that it coincides with
what can be achieved by selecting action-contingent subsidies for the agent.

We are now looking at the model with $\mathcal{A}\equiv\\{a_{1},...,a_{N}\\}$
and $\Omega\equiv\\{\omega_{1},...,\omega_{N}\\}$ for some $N$, and the
preferences are given by $u(a_{i},\omega_{i})=1$ and $u(a_{i},\omega_{j})=0$
if $i\neq j$. Without loss of generality we assume that the principal’s belief
$\mu_{p}$ is such that
$\mu_{p}(\omega_{1})\geq\mu_{p}(\omega_{2})\geq\ldots\geq\mu_{p}(\omega_{N})$
(otherwise relabel states and actions as necessary). Same as before, the
results from Section 3.5 apply, meaning that the agent’s problem is equivalent
to choosing the action distribution $\pi:\Omega\to\varDelta(\mathcal{A})$ to
maximize (4), and the principal selects an agent according to his prior
$\mu\in\mathcal{M}$ to maximize (6). We do not restrict the choice of agents
and let $\mathcal{M}=\varDelta(\Omega)$ (i.e., for any probability
distribution $\mu\in\varDelta(\Omega)$, the principal can find and hire an
agent with prior belief $\mu$).

###  5.1 Agent’s Problem

Proceeding by backward induction, we start by looking at the problem of an
agent with some prior belief $\mu$. Invoking Theorem 1 from Matějka and McKay,
(2015), as we did in the binary case, we can obtain the agent’s optimal
decision rule:

$\displaystyle\pi(a_{i}|\omega_{j})=\frac{\beta(a_{i})e^{\frac{u(a_{i},\omega_{j})}{\lambda}}}{\sum_{k=1}^{N}\beta(a_{k})e^{\frac{u(a_{k},\omega_{j})}{\lambda}}},$
(14)

where $\beta(a_{i})$, defined in (5), is the unconditional choice probability
according to the agent’s prior belief $\mu$. Corollary 2 from Matějka and
McKay, (2015) shows that these probabilities solve the system of equations
given by

$\displaystyle\sum_{j=1}^{N}\mu(\omega_{j})\frac{e^{\frac{u(a_{i},\omega_{j})}{\lambda}}}{\sum_{k=1}^{N}\beta(a_{k})e^{\frac{u(a_{k},\omega_{j})}{\lambda}}}=1,$
(15)

for every $i\in\\{1,...,N\\}$ such that $\beta(a_{i})>0$.

###  5.2 Principal’s Relaxed Problem

Note that (14) implies that a collection of the unconditional choice
probabilities $\beta$ pins down the whole decision rule. Let us then consider
a relaxed problem for the principal, in which instead of choosing agent’s
prior $\mu$, she is free to select the unconditional choice probabilities
$\beta\in\varDelta(\mathcal{A})$ directly:

$\max_{\beta}\left\\{\sum_{j=1}^{N}\mu_{p}(\omega_{j})\left(\sum_{i=1}^{N}\frac{\beta(a_{i})e^{\frac{u(a_{i},\omega_{j})}{\lambda}}}{\sum_{k=1}^{N}\beta(a_{k})e^{\frac{u(a_{k},\omega_{j})}{\lambda}}}u(a_{i},\omega_{j})\right)\right\\}.$
(16)

In the above, we used (14) to represent the conditional probabilities
$\pi(a_{i}|\omega_{j})$ in (6) in terms of unconditional $\beta(a_{i})$. We
show in Section 5.3 that the solution to this relaxed problem is implementable
in the full problem.

Note that $\beta(a_{i})$ in the above represents the probability with which
_an agent_ expects to select action $a_{i}$. The principal’s expected
probability of $a_{i}$ being selected,
$\sum_{j=1}^{N}\mu_{p}(\omega_{j})\pi(a_{i}|\omega_{j})$, would generically be
different, since her prior belief $\mu_{p}$ is different. Despite the
potential confusion this enables, analyzing the principal’s problem through
the prism of choosing $\beta$ is the most convenient approach due to the RI-
logit structure of the solution to the agent’s problem.

Given the state-matching preferences $u(a_{j},\omega_{j})=1$,
$u(a_{i},\omega_{j})=0$ if $i\neq j$, we can simplify (16) to

$\max_{\beta}\left\\{\sum_{j=1}^{N}\mu_{p}(\omega_{j})\frac{\beta(a_{j})e^{\frac{1}{\lambda}}}{1+\delta\beta(a_{j})}\right\\},$
(17)

where $\delta\equiv e^{\frac{1}{\lambda}}-1$. To characterize the solution to
this relaxed problem, we need to introduce two new pieces of notation. For a
given decision rule $\pi$, let
$C(\beta)\equiv\left\\{i\in\\{1,...,N\\}:\beta(a_{i})>0\right\\}$ denote the
consideration set, i.e., the set of actions that are chosen with strictly
positive probabilities. Further, let $K(\beta)\equiv|C(\beta)|$ denote the
power (number of actions in) this set. We can now state the solution to the
principal’s problem as follows.

######  Lemma 1.

The solution to the principal’s relaxed problem (17) is given by

$\displaystyle\beta^{*}(a_{i})=\max\left\\{0,\frac{1}{\delta}\left(\frac{(K(\beta^{*})+\delta)\sqrt{\mu_{p}(\omega_{i})}}{\sum\limits_{j\in
C(\beta^{*})}\sqrt{\mu_{p}(\omega_{j})}}-1\right)\right\\},$

where $\delta\equiv e^{\frac{1}{\lambda}}-1$.

Lemma 1 describes the solution in terms of the action choice probabilities,
which do not necessarily give the reader a good idea of its features and the
intuition behind this solution. We explore these in more detail in Section
5.4. Before that, however, we need to ensure that this solution is attainable
in the principal’s full problem, which is done in the following section.

###  5.3 Principal’s Full Problem

The question this section explores is: can the principal generate choice
probabilities $\beta^{*}$ by appropriately choosing the agent’s prior belief
$\mu$? In the binary case, the answer was trivially “yes”: due to continuity
of the agent’s strategy, by varying the agent’s belief $\mu(r)$ between $0$
and $1$, the principal could induce any unconditional probability $\beta(R)$.
In the multidimensional case, this is not immediate. However, the following
theorem shows that the result still holds with $N$ actions and states under
state-matching preferences.

######  Theorem 1.

In the principal’s full problem (6), any vector
$\beta\in\varDelta(\mathcal{A})$ of unconditional choice probabilities is
implementable: there exists a prior belief $\mu\in\varDelta(\Omega)$ such that
$\beta(a_{i})=\sum_{j=1}^{N}\mu(\omega_{j})\pi^{*}_{\mu}(a_{i}|\omega_{j})$,
where $\pi^{*}_{\mu}$ solves the agent’s problem (4) given $\mu$.

The theorem states that if $\mathcal{M}=\varDelta(\Omega)$, then the principal
can generate any vector of unconditional action probabilities. Note that this
does not imply that she is able to select any decision rule
$\pi(a_{i}|\omega_{j})$ – if this were the case, under the state-matching
preferences she would simply choose to have $\pi(a_{i}|\omega_{i})=1$ for all
$i$. However, Theorem 1 does imply that the choice probabilities described in
Lemma 1 – those that solve the principal’s relaxed problem, – are
implementable and thus also solve her full problem.

The result does, however, rely on the state-matching preferences: we show in
Section 6.1 that it does not hold for arbitrary payoffs.

###  5.4 Properties of the Optimal Delegation Strategy

While Lemma 1 presents the solution of the principal’s problem in terms of the
unconditional choice probabilities, this representation is not the most
visual. We now demonstrate some implications of this solution in terms of
other variables. Namely, Proposition 3 extends Proposition 2 and shows how the
chosen agent’s prior belief relates to that of the principal. Proposition 4
then compares actions taken under optimal delegation vs aligned delegation.

We begin by looking at the optimal agent choice in terms of the agent’s belief
$\mu^{*}$.

######  Proposition 3.

The principal’s equilibrium delegation strategy $\mu^{*}$ is such that for all
$i,j\in\\{1,...,N\\}$:

$\displaystyle\frac{\mu^{*}(\omega_{i})}{\mu^{*}(\omega_{j})}=\frac{\sqrt{\mu_{p}(\omega_{i})}}{\sqrt{\mu_{p}(\omega_{j})}}.$

In particular, $\mu^{*}(\omega_{1})\geq...\geq\mu^{*}(\omega_{N})$. Further,
$\mu^{*}(\omega_{1})\leq\mu_{p}(\omega_{1})$ and
$\mu^{*}(\omega_{N})\geq\mu_{p}(\omega_{N})$, with equalities if and only if
$\mu_{p}(\omega_{1})=...=\mu_{p}(\omega_{j})$.

The intuition behind the proposition above is the same as behind Proposition
2: the optimally chosen agent is more uncertain than the principal between any
given pair of states. To see this, note that if
$\mu_{p}(\omega_{i})>\mu_{p}(\omega_{j})$ then
$1<\frac{\mu^{*}(\omega_{i})}{\mu^{*}(\omega_{j})}<\frac{\mu_{p}(\omega_{i})}{\mu_{p}(\omega_{j})}$
– i.e., the agent believes state $\omega_{i}$ is ex ante more likely than
$\omega_{j}$, as the principal does, but he assigns relatively less weight to
$\omega_{i}$. This applies to any pair of states, so the implication is that
the optimal agent must assign a lower ex ante probability to $\omega_{1}$, the
most likely state according to the principal, than she does, and vice versa
for $\omega_{N}$. Note further that the result in Proposition 3 is again
independent of $\lambda$, implying that the optimal delegation strategy is
determined by the relative trade-off between the quantity of information
acquired and the bias introduced in actions by the misalignment in beliefs,
but the absolute quantity of information acquired is irrelevant. In
particular, hiring an agent with $\mu^{*}$ is optimal even when he acquires no
information, and another agent $\mu$ is available, who would be willing invest
effort in learning $\omega$ (since such a $\mu$-agent would be too biased
relative to the principal).

We now switch to comparing the choices made under optimal delegation to those
that would arise under aligned delegation – i.e., if the principal selected an
agent with $\mu=\mu_{p}$. Let $\bar{\beta}$ denote the choice probabilities
that would be generated under aligned delegation. Caplin et al., (2019) show
that these probabilities $\bar{\beta}$, as a function of the agent’s prior
$\mu$, are given by (see their Theorem 1)

$\displaystyle\bar{\beta}(a_{i})=\max\left\\{0,\frac{1}{\delta}\left(\frac{(K(\bar{\beta})+\delta){\mu(\omega_{i})}}{\sum\limits_{j\in
C(\bar{\beta})}{\mu(\omega_{j})}}-1\right)\right\\}.$ (18)

Since $\mu_{p}(\omega_{1})>...>\mu_{p}(\omega_{N})$, the consideration set in
the aligned problem is then simply $C(\bar{\beta})=\\{1,...,\bar{K}\\}$, and
its size $\bar{K}\equiv K(\bar{\beta})$ is the unique solution of

$\displaystyle\mu_{p}(\omega_{\bar{K}})>\frac{1}{\bar{K}+\delta}\sum_{j=1}^{\bar{K}}\mu_{p}(\omega_{j})\geq\mu_{p}(\omega_{\bar{K}+1}).$
(19)

In turn, we can see from Lemma 1 that under optimal delegation, size
$K^{*}=K(\beta^{*})$ of the consideration set under optimal choice is

$\displaystyle\sqrt{\mu_{p}(\omega_{K^{*}})}>\frac{1}{K^{*}+\delta}\sum_{j=1}^{K^{*}}\sqrt{\mu_{p}(\omega_{j})}\geq\sqrt{\mu_{p}(\omega_{K^{*}+1})}.$
(20)

These two conditions allow us to compare $K^{*}$ and $\bar{K}$ directly, which
is done by the following proposition.

######  Proposition 4.

Optimal delegation weakly expands the consideration set relative to aligned
delegation:

$\displaystyle K(\beta^{*})\geq K(\bar{\beta}).$

In other words, delegating to an optimally misaligned agent leads to a wider
variety of actions played in equilibrium. This is a direct consequence of
delegation to a more uncertain agent – since he is less sure than the
principal of what the optimal action is ex ante, he considers more actions as
worth investigating. Every action has some positive probability of actually
being optimal, thus a more uncertain agent plays a wider range of different
actions ex post. We could already see this effect at play in the binary case,
where if $\mu_{p}$ is extreme, then an aligned agent takes the ex ante optimal
action without acquiring any additional information, whereas the optimally
chosen agent could investigate both actions.

##  6 Misaligned Beliefs Versus Other Instruments

The preceding analysis above explored the problem of selecting an agent
according to their prior belief. It gave grounds for using the misaligned
beliefs as an instrument in delegation, yet it still worth studying how this
instrument compares to the other instruments, such as contracting or
restricting the delegation set. In this section, we draw this comparison. We
keep the overall structure of the problem the same as in Section 5, but now
consider three separate versions, in which the principal has different tools
at her disposal, and compare outcomes to those in the baseline problem of
choosing agent’s beliefs.

###  6.1 Contracting on Actions/Misaligned Preferences

The most basic delegation tool is contracting: if the principal could offer
the agent a contract that specifies contingent payments, this would be the
most direct way to provide incentives.191919See Laffont and Martimort, (2009)
for many examples. We can think of two main options here: contracting on
outcomes (where “outcome” is understood in the sense of “was the agent’s
action correct?”) and contracting on actions. The former requires that both
outcomes are contractible (i.e., observable and verifiable), the latter
imposes such requirement on actions. Both options require that the principal
has the freedom to design the payments, which is a strong assumption in
itself.

We begin by exploring the contracting on actions in our framework, in which
the principal must design a payment schedule $\\{\tau(a_{i})\\}$ to be paid to
the agent. We assume that all agents and the principal have a common prior
belief $\mu_{p}$, all players’ preferences are quasilinear in payments, and
the principal’s marginal utility of money is $\rho$, and the agent’s marginal
utility of money is $1$. For some results, we additionally impose the limited
liability assumption ($\tau(a_{i})\geq 0$ for all $i$).202020In line with the
baseline problem, we do not impose any participation constraints on the agent.
The implicit assumption here is that the agent is being paid some non-
negotiable unconditional salary if he is hired, which is sufficient to ensure
participation. Payments $\\{\tau(a_{i})\\}$ should then be treated as premia,
with the limited liability assumption implying they must be non-negative.

Note that instead of contracting, we can interpret this setup as a problem of
selecting an agent with misaligned preferences by setting $\rho=0$. Collection
$\\{\tau(a_{i})\\}$ then represents an agent’s “biases”, i.e., inherent
preferences towards certain actions on top of the “unbiased” utility
$u(a,\omega)$. Such a problem of selecting an agent with optimally misaligned
preferences is a natural counterpart to our baseline problem of selecting an
agent with optimally misaligned beliefs.

The agent’s problem (again using the equivalence presented in 3.5) is then
given by

$\displaystyle\max_{\pi}\Bigg{\\{}\sum_{j=1}^{N}\mu_{p}(\omega_{j})\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})\big{(}u(a_{i},\omega_{j})+\tau(a_{i})\big{)}-c(\phi,\mu_{p})\Bigg{\\}},$
(21)

given $\tau$, and the principal’s contracting problem is

$\displaystyle\max_{\tau}$
$\displaystyle\left\\{\sum_{j=1}^{N}\mu_{p}(\omega_{j})\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})\big{(}u(a_{i},\omega_{j})-\rho\tau(a_{i})\big{)}\right\\},$
(22)

subject to $\pi$ solving (21) given $\tau$.

Instead of providing a closed-form solution to this problem, we appeal to
Theorem 1 to argue that regardless of $\rho$, the principal cannot obtain
higher expected utility than in the baseline problem of choosing an agent with
a misaligned belief $\mu$. In particular, Theorem 1 implies that any
unconditional choice probabilities $\beta\in\varDelta(\mathcal{A})$ generated
by an agent, who is incentivized by payments or misaligned preferences, can
also be obtained by selecting an agent with appropriately misaligned beliefs.
Moreover, if we relax limited liability and allow negative payments, then by
using results of Matveenko and Mikhalishchev, (2021) we can also show the
converse – that any choice rule achievable with misaligned beliefs can be
replicated with payments $\tau$ (or by setting the quotas – i.e., imposing
specific unconditional choice probabilities for different action). These
results are formalized in the following proposition.212121The result regarding
quotas is not included in the proposition, yet it follows immediately from
Lemma 1 of Matveenko and Mikhalishchev, (2021).

######  Proposition 5.

The principal’s problem of contracting on actions (22) is equivalent to her
full (delegation) problem (6):

  1. 1.

For any vector $\tau:\mathcal{A}\to\mathbb{R}$ of payments/biases and a
corresponding $\beta:\Omega\to\varDelta(\mathcal{A})$ that solves (21) given
$\tau$, there exists a prior belief $\mu\in\varDelta(\Omega)$ such that
$\beta$ also solves (4) given $\mu$.

  2. 2.

For any $\mu\in\varDelta(\Omega)$ and the corresponding
$\beta:\Omega\to\varDelta(\mathcal{A})$ that solves (4) given $\mu$, there
exist payments $\tau:\mathcal{A}\to\mathbb{R}$ such that $\beta$ also solves
(21) given $\tau$.

The proposition above directly implies that neither of the two instruments
(contracting on actions or searching for an agent with stronger/weaker
preferences for specific actions) can yield better results than hiring an
agent with an optimally misaligned belief. If limited liability is in place
($\tau(a_{i})\geq 0$ for all $i$), then contracting on actions is strictly
worse, since it cannot yield a better decision rule, but requires payments
from the principal – payments which are avoidable if she instead hires an
agent who is intrinsically motivated by his beliefs over states or preferences
towards specific actions.

Further, Theorem 1 and the results of Matveenko and Mikhalishchev, (2021) also
imply that the space of solutions to either problem covers the whole universe
of RI-logit choice rules given fixed preferences _between states_. This
implies that no combination of misaligned beliefs, misaligned preferences, and
payments for actions can perform better than any individual instrument.
Moreover, it also implies that suboptimal misalignment along any dimension can
be amended using other instruments. I.e., if a given agent holds a non-optimal
prior belief (that does not coincide with the principal’s either), the optimal
behavior might be induced via action-contingent transfers. Conversely, if all
agents are biased towards certain actions, this misalignment can be
compensated for by selecting an agent with the right prior belief. The
following proposition presents one example of such equivalence, in the context
of a model with $N=2$.

######  Proposition 6.

Consider the binary setting of Section 4. Consider the principal’s problem of
contracting on actions (22), where $\rho=0$ and the agent holds prior belief
$\mu\neq\mu_{p}$. Then:

  1. 1.

for any $\mu$, there exist payments/biases $\\{\tau^{*}(L),\tau^{*}(R)\\}$
that implement the optimal conditional choice probabilities from Section 4;

  2. 2.

these payments/biases are such that222222The closed-form expressions are
available in the proof in the Appendix.

$\displaystyle\tau^{*}(R)\geq\tau^{*}(L)\iff\mu\leq\mu^{*}=\frac{\sqrt{\mu_{p}}}{\sqrt{\mu_{p}}+\sqrt{1-\mu_{p}}}.$

It is easy to see the intuition behind the proposition: if the agent’s prior
belief $\mu$ assigns lower probability to state $\omega=r$ compared to the
principal-optimal prior $\mu^{*}$ given in Proposition 2, such an agent is ex
ante too biased towards action $L$ for the principal’s taste, even though he
potentially acquires more information than an agent with belief $\mu^{*}$.
Therefore, the principal can nudge the agent towards action $R$ by offering
higher payment if he selects $R$ (or find an agent, whose preference bias
towards $R$ offsets his belief bias towards state $l$).

###  6.2 Contracting on Outcomes

We now turn to contracting on outcomes. The outcome in our model is
effectively binary: whether a correct action was chosen ($a=a_{j}$ when
$\omega=\omega_{j}$) or not. We thus let the principal select payments
$\bar{\tau},\underline{\tau}$ that the agent receives, so that
$\tau(a_{i},\omega_{i})=\bar{\tau}$ and
$\tau(a_{i},\omega_{j})=\underline{\tau}$ if $i\neq j$.232323If the principal
could contract on both actions and outcomes, she would have the freedom to
select any payment schedule $\\{\tau(a_{i},\omega_{j})\\}$. Lindbeck and
Weibull, (2020) study such a problem with $N$ states and two actions. We again
assume limited liability ($\bar{\tau},\underline{\tau}\geq 0$), preferences
that are quasilinear in payments for all agents, and let the agent’s marginal
utility of money to be $1$, and the principal’s marginal utility of money to
be $\rho$.

The agent’s problem is then choosing $\pi:\Omega\to\varDelta(\mathcal{A})$
that solves242424While it is more common in the literature to consider an
agent who yields no intrinsic utility from actions and is motivated
exclusively via payments, for sake of consistency, we maintain the assumption
that the agent enjoys the same intrinsic utility $u(a,\omega)$ as the
principal, albeit possibly to a different magnitude.

$\displaystyle\max_{\pi}\Bigg{\\{}\sum_{j=1}^{N}\mu(\omega_{j})\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})\big{(}u(a_{i},\omega_{j})+\tau(a_{i},\omega_{j})\big{)}-c(\phi,\mu)\Bigg{\\}},$
(23)

and the principal’s contracting problem is

$\displaystyle\max_{\bar{\tau},\underline{\tau}}$
$\displaystyle\left\\{\sum_{j=1}^{N}\mu_{p}(\omega_{j})\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})\big{(}u(a_{i},\omega_{j})-\rho\tau(a_{i},\omega_{j})\big{)}\right\\},$
(24) s.t. $\displaystyle\tau(a_{i},\omega_{i})=\bar{\tau}\text{ for all }i,$
$\displaystyle\tau(a_{i},\omega_{j})=\underline{\tau}\text{ for all }i,j\neq
i,$

and subject to $\beta$ corresponding to a solution of (23) given
$\bar{\tau},\underline{\tau}$.

It is trivially optimal for the principal to set $\underline{\tau}=0$, since
her objective is to provide incentives for the agent to match the state. Then,
however, the agent’s (ex post) payoff net of information cost becomes
$u(a_{i},\omega_{j})+\tau(a_{i},\omega_{j})=(1+\bar{\tau})u(a_{i},\omega_{j})$,
and the principal’s payoff is
$u(a_{i},\omega_{j})-\tau(a_{i},\omega_{j})=(1-\rho\bar{\tau})u(a_{i},\omega_{j})$.
In other words, by increasing the incentive payment $\bar{\tau}$, the
principal effectively lowers the relative cost of information for the agent,
at the cost of decreasing her own payoff. It then appears like an instrument
that could be universally useful for the principal – even when she chooses an
agent with the optimal prior belief, she could still benefit from reducing the
agent’s information cost, which would result in him acquiring more
information. The following proposition shows, however, that this is not the
case: while contracting on outcomes may be a useful instrument, it cannot
improve on delegating to the optimally biased agent when payments are costly
to the principal.

######  Proposition 7.

Consider the principal’s contracting problem (24) in the binary setting of
Section 4 and suppose $\rho=1$ and $\mu_{p}>1/2$. Then there exist
$\bar{\mu}_{1},\bar{\mu}_{2}$: $\bar{\mu}_{1}<\mu^{*}<\mu_{p}<\bar{\mu}_{2}$
(where $\mu^{*}$ is as in Proposition 2), such that if either
$\frac{1}{1+e^{\frac{1}{\lambda}}}\leq\mu\leq\bar{\mu}_{1}$, or
$\bar{\mu}_{2}\leq\mu\leq\frac{e^{\frac{1}{\lambda}}}{1+e^{\frac{1}{\lambda}}}$,
then $\bar{\tau}>0$ solves (24).

The proposition states that the principal uses the incentive payments,
$\bar{\tau}>0$, when there is an intermediate degree of misalignment in
opinions with the agent. If the agent has a very extreme prior belief and
acquires no information on his own, it may be too costly for the principal to
incentivize such an agent to acquire any information. The requirements that
$\frac{1}{1+e^{\frac{1}{\lambda}}}\leq\mu\leq\frac{e^{\frac{1}{\lambda}}}{1+e^{\frac{1}{\lambda}}}$
ensure that the agent acquires information voluntarily so that this problem
does not arise.252525Note that there may still be $\mu$ outside of this
interval for which offering positive incentive payments is optimal for the
principal. In that sense, Proposition 7 only provides a condition on $\mu$
that is sufficient for $\bar{\tau}>0$. A necessary and sufficient condition
would look similarly, but feature different outer boundaries for $\mu$. On the
other hand, if there is too little misalignment
($\bar{\mu}_{1}<\mu<\bar{\mu}_{2}$), then the agent’s information acquisition
choice is already in line with the principal’s wishes, and any further
payments would not have a significant enough effect on the agent’s incentives
to justify the cost they incur on the principal.

###  6.3 Restricting the Delegation Set

Another instrument commonly explored in the delegation literature is
restricting the delegation set – i.e., the set of actions that the agent may
take (see, e.g., Holmstrom, (1980)). In particular, in the context of
“delegated expertise” problems, Szalay, (2005) and Ball and Gao, (2021) show
that it may be optimal to rule out an ex ante optimal action in order to force
the agent to exert effort and learn which of the ex post optimal (but ex ante
risky) actions is best. Lipnowski et al., (2020) show a similar result in a
Bayesian Persuasion setting, where the receiver is rationally inattentive to
the sender’s message.

In our setting, however, there are no “safe” actions that the principal could
rule out, as Propostion 4 suggests. Assuming that the principal and the agent
hold the same prior belief $\mu_{p}$, and
$\mu_{p}(\omega_{1})>...>\mu_{p}(\omega_{N})$, action $a_{1}$ is the “safest”
in the sense of being the most likely ex ante to be optimal. Yet, it would be
trivially suboptimal for the principal to ban $a_{1}$ – since, indeed, this is
the action that is ex ante most likely to be ex post optimal! In other words,
while excluding $a_{1}$ from the delegation set would lead the agent to
acquire more information, it would also lead to larger ex post losses due to
the agent being unable to select action $a_{1}$ in cases when it is optimal to
do so. So while the general idea of the principal being willing to nudge the
agent to acquire more information/information about ex ante suboptimal actions
holds true in our setting, restricting the delegation set is not an instrument
that lends any value to the principal.

Proposition 8 below summarizes this logic. Consider the agent’s problem as
given by

$\displaystyle\max_{\pi}\Bigg{\\{}\sum_{j=1}^{N}\mu_{p}(\omega_{j})\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})u(a_{i},\omega_{j})-c(\phi,\mu_{p})\Bigg{\\}},$
(25)

given $A^{*}\subseteq\mathcal{A}$ (and the maximization is w.r.t. a mapping
$\pi:\Omega\to\varDelta(A^{*})$), and the principal’s restriction problem

$\displaystyle\max_{A^{*}}$
$\displaystyle\left\\{\sum_{j=1}^{N}\mu_{p}(\omega_{j})\sum_{i=1}^{N}\pi(a_{i}|\omega_{j})u(a_{i},\omega_{j})\right\\},$
(26)

subject to $\pi:\Omega\to\varDelta(A^{*})$ solving (25) given $A^{*}$. Then we
can state the result as follows.

######  Proposition 8.

The unrestricted delegation set $A^{*}=\mathcal{A}$ is always a solution to
the principal’s restriction problem (26).

##  7 Extensions

###  7.1 Alternative Preference Specifications

The analysis in Sections 5 and 6.1 is heavily reliant on state-matching
preferences that we assume are shared by both the principal and the agent(s).
It is reasonable to ask whether our conclusions hold under other preference
specifications. Since the utility function $u(a,\omega)$ is shared by both the
principal and the agent, it is reasonable to generalize one at a time.

We begin by generalizing the principal’s utility function $u_{p}(a,\omega)$
while maintaining the agent’s intrinsic preference for matching the state:
$u_{A}(a_{i},\omega_{i})=1$, $u_{A}(a_{i},\omega_{j})=0$ if $i\neq j$.
Naturally, the specific functional forms of the optimal delegation strategies
(such as those presented in Propositions 2 and 3, and Lemma 1) depend on the
specific form of the principal’s utility function. However, Theorem 1 only
depends on the agent’s utility function, meaning that Proposition 5 still
holds: any outcome that can be achieved by contracting on actions or hiring an
agent with the misaligned intrinsic preferences, can also be achieved by
hiring an agent with misaligned beliefs (and vice versa). Meaning that
regardless of the principal’s objective function, hiring an agent with state-
matching preferences and a suitable belief is as good as hiring an agent with
aligned prior belief, state-matching preferences, and either some additional
preference over actions, or action-contingent payments on top of that.

The above does, however, hinge on the agent having state-matching preferences
as a baseline. Once we allow arbitrary preferences for the agent – even if
they align with the principal’s preferences net of the information cost – the
equivalence stated in Proposition 5 breaks down. In such a general case,
finding an agent with optimally misaligned preferences may yield strictly
better results for the principal than hiring an agent with an optimally
misaligned belief, and hence contracting on actions may, in principle, yield
better results too. An example (for misaligned preferences) is presented in
the proof of the following proposition.

######  Proposition 9.

There exists a utility function $u(a_{i},\omega_{j})$ with
$u(a_{i},\omega_{i})=\max\limits_{k}\\{u(a_{k},\omega_{i})\\}$ such that the
solution of the principal’s relaxed problem (17) can not be attained as a
solution of the full problem (6).

The optimal choice rule is never in the relative interior, since mixing
between two options dominates using mixing from three. However, according to
Proposition 3 and Lemma 1 from Matveenko and Mikhalishchev, (2021), the
principal may implement the optimal solution via action-contingent
payments/biased preferences.

###  7.2 Communication

In this section we consider the importance of decision rights in our model
with misaligned beliefs. In particular, we juxtapose the _delegation_ scheme
explored so far, under which the agent has the power to make the final
decision, to _communication_ , where an agent must instead communicate his
findings to the principal, who then chooses the action. A large literature in
organizational economics is devoted to comparing delegation and communication
in various settings (see Dessein, (2002); Alonso et al., (2008), and
Rantakari, (2008) for some examples). We show that in our setting,
communication performs as well as delegation – i.e., the principal will always
find it optimal to follow the agent’s recommendation. This is, perhaps,
unsuprising, since Holmstrom, (1980) showed that communication is equivalent
to restricting the agent’s action set, and this latter problem was shown in
Section 6.3 to be an irrelevant instrument in our setting, as long as the
principal can select an agent with the prior belief she prefers.

Although the principal and an agent have the same preferences, it is generally
unclear whether it is optimal for the principal to follow a recommendation of
an agent due to the misalignment in their beliefs. Namely, since the principal
and the agent start from different prior beliefs, the same is true for
posteriors: if the principal could observe the information that the agent
obtained, her posterior belief would be different from that of the agent. This
implies that ex post, the principal could prefer an action different from the
agent’s choice, and could benefit ex post from overruling the agent’s decision
if she had the power to do so. However, this would mean that the agent’s
incentives to acquire information are different from the baseline model, and
could lead to the agent acquiring either more or less information than in the
baseline, with the principal having some influence over the agent’s learning
strategy via her final decision rule.262626Argenziano et al., (2016) provide
one example of how the principal can manipulate the agent’s information
acquisition incentives under cheap talk communication. We show below that in
the end, none of these effects come into play, and there exists a
communication equilibrium that replicates the delegation equilibrium.

The setup follows the baseline model from Section 3, with the exception that
the final stage (“agent selecting action $a\in\mathcal{A}$”) is replaced by
two. First, after observing signal $s\in\mathcal{S}$ generated by his signal
structure $\phi$, the agent selects a recommendation (message)
$\tilde{a}\in\mathcal{A}$ to the principal. After that, the principal observes
the recommendation $\tilde{a}$, uses it to update her belief
$\mu_{p}(\omega|\tilde{a})$ about the state of the world, and then selects an
action $a\in\mathcal{A}$ that determines the both parties’ payoffs.272727For
simplicity, we assume that the principal only observes the recommendation made
by the agent, and not the signal he received or the signal structure he
requested. These appear to be reasonable, yet substantial assumptions, and the
results would be different if we assumed the principal could observe either
the learning strategy, or the realized signal. The equilibrium of the
communication game is then defined as follows.

######  Definition (Communication Equilibrium).

An equilibrium of the cheap talk game is characterized by
$(\mu^{*},\\{\phi^{*}_{\mu},\tilde{\sigma}^{*}_{\mu}\\}_{\mu\in\mathcal{M}},\sigma^{*},\mu_{p})$,
which consists of the following:

  1. 1.

the principal’s posterior beliefs $\mu_{p}:\mathcal{A}\to\varDelta(\Omega)$
that are consistent with $(\phi^{*}_{\mu},\sigma^{*}_{\mu})$ (i.e., satisfy
Bayes’ rule on the equilibrium path);

  2. 2.

the principal’s decision rule $\sigma^{*}:\mathcal{A}\to\mathcal{A}$, which
solves the following for every $\tilde{a}\in\mathcal{A}$, given the posterior
$\mu_{p}$:

$\displaystyle\max_{\sigma(\tilde{a})}\left\\{\sum_{\omega\in\Omega}\mu_{p}(\omega|\tilde{a})u(\sigma(\tilde{a})|\omega)\right\\};$

  3. 3.

a collection of the agents’ information acquisition strategies
$\phi^{*}_{\mu}:\Omega\to\varDelta(\mathcal{S})$ and communication strategies
$\tilde{\sigma}^{*}_{\mu}:\mathcal{S}\to\mathcal{A}$ that solve the following
given $\sigma$ for every $\mu\in\mathcal{M}$:

$\displaystyle\max_{\phi,\tilde{\sigma}}\left\\{\sum_{\omega\in\Omega}\mu(\omega)\sum_{s\in\mathcal{S}}\phi(s|\omega)u(\sigma(\tilde{\sigma}(s)),\omega)-c(\phi,\mu)\right\\};$

  4. 4.

the principal’s choice $\mu^{*}\in\mathcal{M}$ of the agent to whom the task
is delegated, which solves the following given
$(\phi^{*}_{\mu},\sigma^{*}_{\mu})$, $\sigma^{*}$, and $\mu_{p}$:

$\displaystyle\max_{\mu}$
$\displaystyle\left\\{\sum_{\omega\in\Omega}\mu_{p}(\omega)\sum_{s\in\mathcal{S}}\phi(s|\omega)u(\sigma(\tilde{\sigma}(s)),\omega)\right\\}.$

We can then state the result as follows.

######  Proposition 10.

There exists a communication equilibrium that is outcome-equivalent to the
equilibrium of the original game, in the sense that
$(\mu^{*},\phi^{*}_{\mu^{*}},\tilde{\sigma}^{*}_{\mu^{*}})$ are the same in
both equilibria, and $\sigma^{*}$ is an identity mapping.

This result, however, is subject to a few caveats. First, cheap talk models
are plagued by equilibrium multiplicity: for any informative equilibrium,
there exist equilibria with less informative communication, up to completely
uninformative (babbling) equilibria. In our setting this means that in
addition to the equilibrium outlined in Proposition 10 above, there also
exists a babbling equilibrium, in which the agent acquires no information and
makes a random recommendation, and the principal always ignores it and selects
the ex ante optimal action.282828If an agent makes uninformed recommendations,
it is optimal for the principal to ignore it. If the principal ignores the
recommendation, it is optimal for the agent to not acquire any information.
Neither agent in this situation can unilaterally deviate to informative
communication. There would also likely exist multiple equilibria of
intermediate informativeness – e.g., equilibria with a limited vocabulary,
where only some actions $\tilde{\mathcal{A}}\subset\mathcal{A}$ are
recommended on equilibrium path. In practice, this means that under
communication, there is a risk of miscoordination on uninformative equilibria,
whereas under delegation the equilibrium is unique. The same force may also
work the other way, and there may be equilibria that are preferred by the
principal to the delegation equilibrium, that can only be sustained under
cheap talk (see Argenziano et al., (2016) for an example of how such
equilibria may arise). However, the question of whether such equilibria exist
is beyond the scope of this paper.

The second caveat lies in the fact that Proposition 10 relies on the state-
matching preferences. In our setting (with the exception of Section 7.1), any
action is either “right” or “wrong”, without any degrees of correctness. The
misalignment of beliefs across the principal and the agent is thus small
enough to not warrant the principal overriding the agent’s suggested action.
In contrast, in a uniform-quadratic framework, both states and actions lie in
a continuum, and the principal’s loss is proportional to the distance between
the realized state and the chosen action. In such a setting, any misalignment
(be it in preferences or beliefs) between the principal and the agent would
lead to the principal being willing to override the agent’s recommendation,
leading to the delegation equilibrium being no longer directly sustainable
under communication.

##  8 Conclusion

We show that hiring an agent with beliefs that are misaligned with those of
the principal can be beneficial for the principal, contrary to popular belief.
In particular, if the agent needs to acquire information to make a decision,
delegation to an agent who is ex ante more uncertain about what the best
action is but shares some of the principal’s predispositions is optimal for
her. This is mainly due to a more uncertain agent being willing to acquire
more information about the state, which enables more efficient actions being
taken. We show that exploiting belief misalignment can be a valid instrument
that the principal can use in delegation, which in some settings is on par
with or better than state-contingent transfers or restriction of the action
set from which the agent can choose.

In the analysis, we use the workhorse rational inattention model for discrete
choice, the Shannon model. It allows us to provide a richer demonstration of
the consequences of delegation to a biased agent by allowing the agent to
acquire information flexibly, which introduces additional bias into the
decisions of an agent with misaligned beliefs. We show that misaligned
delegation is optimal _despite_ the bias introduced by this flexibility.
Hence, while the exact trade-offs do, obviously, depend on the particular cost
function specification, the main takeaways would persist regardless of the
form of the costs of information.

Our analysis focuses on misalignment and away from contracting. A potentially
fruitful avenue for further research would be to consider more carefully the
contracting problem in a setting where the principal and the agent have
misaligned beliefs and/or preferences, and to investigate how contracting can
build upon the inherent benefits of misalignment.

Further, due to the added complexity of rational inattention models, we
confine our exploration to a discrete state-matching model, which strays away
from the continuous models more commonly used in delegation problems. In a
model with continuous actions, the scope for an agent to manifest his bias is
much larger, hence the trade-off between the agent’s information acquisition
and biased decision-making would again be different. Exploration of the
effects of misalignment in a continuous model of delegated expertise could be
an interesting direction for further research as well.

Yet another assumption that may feel excessively strong in our analysis is the
common knowledge of all agents’ and the principal’s prior beliefs. It may be
more reasonable to assume that agents are strategic in presenting their
viewpoints to the principal, as well as making inferences from the fact that
they were chosen for the job. Such signaling concerns could yield an
economically meaningful effect, yet we abstract from them completely in our
paper. A more careful investigation is in order.

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##  Appendix A Main Proofs

###  A.1 Proof of Proposition 2

Throughout this proof, we will refer to the delegation rule under
consideration,

$\displaystyle\mu^{*}=\frac{\sqrt{\mu_{p}}}{\sqrt{\mu_{p}}+\sqrt{1-\mu_{p}}},$

as the candidate rule. It is straightforward that under the candidate rule, if
$\mu_{p}>\frac{1}{2}$ then $\mu^{*}\in\left(\frac{1}{2},\mu_{p}\right)$, since

$\displaystyle\frac{\mu^{*}}{1-\mu^{*}}=\frac{\sqrt{\mu_{p}}}{\sqrt{1-\mu_{p}}}<\frac{\mu_{p}}{1-\mu_{p}}$

when $\mu_{p}>\frac{1}{2}$, so $\mu^{*}<\mu_{p}$, and also
$\sqrt{\mu_{p}}>\sqrt{1-\mu_{p}}$ in that case, so $\mu^{*}>\frac{1}{2}$. It
thus remains to show that the candidate rule is indeed optimal for the
principal. While a shorter proof exists that invokes Lemma 1 that derives an
optimal strategy for the case of $N$ states and actions, we choose to present
a more direct, albeit a somewhat longer, proof.

Plugging the solution of the agent’s problem (11) (assuming this solution is
interior for now) into the principal’s problem (12), we get that the
principal’s payoff looks as follows:

$\displaystyle\mu_{p}\pi(R|r)+(1-\mu_{p})\pi(L|l)$
$\displaystyle=\mu_{p}\frac{\left(\mu
e^{\frac{1}{\lambda}}-(1-\mu)\right)e^{\frac{1}{\lambda}}}{\left(e^{\frac{2}{\lambda}}-1\right)\mu}+(1-\mu_{p})\frac{\left((1-\mu)e^{\frac{1}{\lambda}}-\mu\right)e^{\frac{1}{\lambda}}}{\left(e^{\frac{2}{\lambda}}-1\right)(1-\mu)}$
$\displaystyle=\frac{e^{\frac{1}{\lambda}}}{e^{\frac{2}{\lambda}}-1}\left[\mu_{p}\left(e^{\frac{1}{\lambda}}-\frac{1-\mu}{\mu}\right)+(1-\mu_{p})\left(e^{\frac{1}{\lambda}}-\frac{\mu}{1-\mu}\right)\right]$
$\displaystyle\propto
e^{\frac{1}{\lambda}}-\mu_{p}\frac{1-\mu}{\mu}-(1-\mu_{p})\frac{\mu}{1-\mu}.$

The FOC for the principal’s maximization problem above w.r.t. $\mu$ is

$\displaystyle\frac{\mu_{p}}{\mu^{2}}-\frac{1-\mu_{p}}{(1-\mu_{p})^{2}}=0$
$\displaystyle\iff$
$\displaystyle\frac{\mu}{1-\mu}=\frac{\sqrt{\mu_{p}}}{\sqrt{1-\mu_{p}}}.$ (27)

It is trivial to verify that the second-order condition holds as well, hence
as long as (27) yields an interior solution (i.e., the probabilities in (11)
are in $[0,1]$), the candidate solution is indeed optimal among all such
interior solutions.

We now check for which $\mu$ the solution (11) is interior. Using the
expressions (11), one can easily verify that $\pi(R|r)\geq
0\iff\frac{\mu}{1-\mu}\geq e^{-\frac{1}{\lambda}}$ and $\pi(R|r)\leq
1\iff\frac{\mu}{1-\mu}\leq e^{\frac{1}{\lambda}}$, and the conditions
$\pi(L|l)\in[0,1]$ yield the same two interiority conditions. This implies
that if
$\frac{\mu}{1-\mu}\in\left[e^{-\frac{1}{\lambda}},e^{\frac{1}{\lambda}}\right]$,
then the agent acquires some information and selects both actions with
positive probabilities, and otherwise
$(\pi(R|r),\pi(L|l))\in\\{(1,0),(0,1)\\}$, meaning that the agent simply
chooses the ex ante optimal action for sure without acquiring any information
about the state.

The candidate rule then suggests that the principal delegates to a learning
agent iff
$\frac{\mu_{p}}{1-\mu_{p}}\in\left[e^{-\frac{2}{\lambda}},e^{\frac{2}{\lambda}}\right]$,
and otherwise delegates to an agent who plays the ex ante optimal action. We
have shown that the candidate rule selects the optimal among the learning
agents; it is left to verify that such a choice rule between learning and non-
learning agents is optimal for the principal.

Consider $\mu_{p}\geq\frac{1}{2}$; then among the non-learning agents the
principal would obviously choose the one who plays $a=R$ (rather than $a=L$),
and such a choice yields the principal expected payoff $\mu_{p}\cdot
1+(1-\mu_{p})\cdot 0=\mu_{p}$. Optimal delegation to a learning agent yields
(by plugging the candidate rule into the principal’s payoff obtained above)

$\displaystyle\frac{e^{\frac{1}{\lambda}}}{e^{\frac{2}{\lambda}}-1}\left[e^{\frac{1}{\lambda}}-\mu_{p}\frac{1-\mu^{*}}{\mu^{*}}-(1-\mu_{p})\frac{\mu^{*}}{1-\mu^{*}}\right]=$
$\displaystyle\frac{e^{\frac{1}{\lambda}}}{e^{\frac{2}{\lambda}}-1}\left[e^{\frac{1}{\lambda}}-2\sqrt{\mu_{p}(1-\mu_{p})}\right].$
(28)

Taking the difference between (28) and $\mu_{p}$, the payoff from delegating
to a non-learning agent, lets us find belief $\mu_{p}$ of a principal who
would be indifferent between the two:

$\displaystyle\frac{e^{\frac{1}{\lambda}}}{e^{\frac{2}{\lambda}}-1}\left[e^{\frac{1}{\lambda}}-2\sqrt{\mu_{p}(1-\mu_{p})}\right]-\mu_{p}=0$
$\displaystyle\iff$ $\displaystyle
e^{\frac{2}{\lambda}}-2e^{\frac{1}{\lambda}}\sqrt{\mu_{p}(1-\mu_{p})}=\mu_{p}e^{\frac{2}{\lambda}}-\mu_{p}$
$\displaystyle\iff$
$\displaystyle\left(e^{\frac{1}{\lambda}}\sqrt{1-\mu_{p}}-\sqrt{\mu_{p}}\right)^{2}=0$
$\displaystyle\iff$
$\displaystyle\frac{\sqrt{\mu_{p}}}{\sqrt{1-\mu_{p}}}=e^{\frac{1}{\lambda}}.$

Hence, the principal prefers a learning agent when
$\frac{\sqrt{\mu_{p}}}{\sqrt{1-\mu_{p}}}<e^{\frac{1}{\lambda}}$ and a non-
learning agent when
$\frac{\sqrt{\mu_{p}}}{\sqrt{1-\mu_{p}}}>e^{\frac{1}{\lambda}}$. Therefore,
the candidate rule is indeed optimal for $\mu_{p}\geq\frac{1}{2}$. A mirror
argument can be used to establish optimality for $\mu_{p}\leq\frac{1}{2}$.
This concludes the proof of Proposition 2.

###  A.2 Proof of Lemma 1

The goal is to find the optimal choice probabilities
$\beta^{*}\in\varDelta(\mathcal{A})$ which maximize the principal’s expected
utility (17). First, let us rewrite expression (17) using $\delta\equiv
e^{\frac{1}{\lambda}}-1$:

$\displaystyle\sum_{j=1}^{N}\mu_{p}(\omega_{j})\frac{\beta(a_{j})e^{\frac{1}{\lambda}}}{1+\delta\beta(a_{j})}$
$\displaystyle=\sum_{j\in
C(\beta)}e^{\frac{1}{\lambda}}\frac{\frac{\mu_{p}(\omega_{j})}{\delta}(1+\delta\beta(a_{j}))-\frac{\mu_{p}(\omega_{j})}{\delta}}{1+\delta\beta(a_{j})}$
$\displaystyle=\sum_{j\in
C(\beta)}e^{\frac{1}{\lambda}}\left(\frac{\mu_{p}(\omega_{j})}{\delta}-\frac{\mu_{p}(\omega_{j})}{\delta\left(1+\delta\beta(a_{j})\right)}\right).$

The first term in the brackets above is independent of $\beta$, so the
principal’s maximization problem is equivalent to

$\min_{\beta}\sum_{j\in
C(\beta)}\frac{\mu_{p}(\omega_{j})}{1+\delta\beta(a_{j})}.$ (29)

Let $\xi$ denote the Lagrange multiplier corresponding to the constraint
$\sum\limits_{j=1}^{N}\beta(a_{j})=1$. Then the first-order condition for
$\beta(a_{i})$ with $i\in C(\beta)$ is

$\displaystyle(1+\delta\beta(a_{i}))^{2}=-\frac{\mu_{p}(\omega_{i})}{\xi}.$
(30)

Summing up these equalities over all $j\in C(\beta)$, we get that

$\displaystyle\sum\limits_{j\in
C(\beta)}(1+\delta\beta(a_{j}))^{2}=-\frac{\sum_{j\in
C(\beta)}\mu_{p}(\omega_{j})}{\xi}$ (31)

Combining (30) and (31):

$\displaystyle
1+\delta\beta(a_{i})=\frac{\sqrt{\mu_{p}(\omega_{i})}}{\sqrt{\sum_{j\in
C(\beta)}\mu_{p}(\omega_{j})}}\sqrt{\sum\limits_{j\in
C(\beta)}(1+\delta\beta(a_{j}))^{2}}$ (32)

Once again summing up these equalities over all $j\in C(\beta)$, we get that

$\displaystyle K(\beta)+\delta=\frac{\sum_{j\in
C(\beta)}\sqrt{\mu_{p}(\omega_{j})}}{\sqrt{\sum_{j\in
C(\beta)}\mu_{p}(\omega_{j})}}\sqrt{\sum\limits_{j\in
C(\beta)}(1+\delta\beta(a_{j}))^{2}}.$

Expressing $\sqrt{\sum_{j\in C(\beta)}(1+\delta\beta(a_{j}))^{2}}$ from this
expression and plugging it into (32) allows us to express $\beta(a_{i})$ (for
$i\in C(\beta)$) in closed form as

$\displaystyle\beta(a_{i})=\frac{1}{\delta}\left(\frac{(K(\beta)+\delta)\sqrt{\mu_{p}(\omega_{i})}}{\sum_{j\in
C(\beta)}\sqrt{\mu_{p}(\omega_{j})}}-1\right).$ (33)

The necessary condition for option $i$ to be in a consideration set ($i\in
C(\beta)$) is $\beta(a_{i})\geq 0$ or, equivalently,

$\displaystyle\sqrt{\mu_{p}(\omega_{i})}>\frac{1}{K(\beta)+\delta}\sum\limits_{j\in
C(\beta)}\sqrt{\mu_{p}(\omega_{j})}.$

Now let $\xi_{k}$ denote the Lagrange multiplier for the constraint
$\beta(a_{k})\geq 0$. Then the first-order condition for an alternative
$k\notin C(\beta)$ that is not chosen is

$\displaystyle\mu_{p}(\omega_{k})$ $\displaystyle=-\xi-\xi_{k}$
$\displaystyle\Rightarrow$ $\displaystyle\mu_{p}(\omega_{k})$
$\displaystyle\leq-\xi.$

Plugging in $\xi$ from (30) into the inequality above yields

$\displaystyle\mu_{p}(\omega_{k})$ $\displaystyle\leq\frac{\sum_{j\in
C(\beta)}\mu_{p}(\omega_{j})}{\sum_{j\in C(\beta)}(1+\delta\beta(a_{j}))^{2}}$
$\displaystyle\Leftrightarrow$ $\displaystyle\sqrt{\mu_{p}(\omega_{k})}$
$\displaystyle\leq\frac{1}{K(\beta)+\delta}\sum\limits_{j\in
C(\beta)}\sqrt{\mu_{p}(\omega_{j})}$

for all $k\notin C(\beta)$.

Since the minimization problem has a convex objective function and linear
constraints, the Kuhn-Tucker conditions are necessary and sufficient. Thus the
necessary and sufficient conditions that the solution $\beta^{*}$ must satisfy
are given by:

$\displaystyle\begin{cases}\sqrt{\mu_{p}(\omega_{i})}>\frac{1}{K(\beta^{*})+\delta}\sum\limits_{j\in
C(\beta^{*})}\sqrt{\mu_{p}(\omega_{j})}&\text{ for all }i\in C(\beta^{*}),\\\
\sqrt{\mu_{p}(\omega_{k})}\leq\frac{1}{K(\beta^{*})+\delta}\sum\limits_{j\in
C(\beta^{*})}\sqrt{\mu_{p}(\omega_{j})}&\text{ for all }k\notin
C(\beta^{*}).\end{cases}$

Recall that we assumed, without loss of generality, that
$\mu_{p}(\omega_{1})\geq\mu_{p}(\omega_{2})\geq\ldots\geq\mu_{p}(\omega_{N})$.
Suppose that the solution $\beta^{*}$ is such that $K(\beta^{*})=K^{\prime}$.
Clearly then, in the optimum, the consideration set $C(\beta^{*})$ will
consist of the first $K^{\prime}$ alternatives.

Denote
$\Delta_{L}\equiv(L+\delta)\sqrt{\mu_{p}(\omega_{L})}-\sum\limits_{j=1}^{L}\sqrt{\mu_{p}(\omega_{j})}$.
Notice that for all $L>1$:

$\displaystyle\Delta_{L}\equiv$
$\displaystyle(L+\delta)\sqrt{\mu_{p}(\omega_{L})}-\sum\limits_{j=1}^{L}\sqrt{\mu_{p}(\omega_{j})}$
$\displaystyle=$
$\displaystyle(L-1+\delta)\sqrt{\mu_{p}(\omega_{L-1})}-\sum\limits_{j=1}^{L-1}\sqrt{\mu_{p}(\omega_{j})}-\sqrt{\mu_{p}(\omega_{L})}$
$\displaystyle-(L-1+\delta)\sqrt{\mu_{p}(\omega_{L-1})}+(L+\delta)\sqrt{\mu_{p}(\omega_{L})}$
$\displaystyle=$
$\displaystyle\Delta_{L-1}-(L-1+\delta)\left(\sqrt{\mu_{p}(\omega_{L-1})}-\sqrt{\mu_{p}(\omega_{L})}\right).$

Therefore, $\Delta_{L}$ decreases in $L$. Since $\Delta_{1}>0$, there either
exists unique $K^{\prime}$ such that $\Delta_{K^{\prime}}>0$ and
$\Delta_{K^{\prime}+1}\leq 0$, or $\Delta_{L}>0$ for all $L$. In the former
case, $K(\beta^{*})=K^{\prime}$, and in the latter case, $K(\beta^{*})=N$.

In the end, the solution to the principal’s problem is given by
$\beta^{*}(a_{i})$ as in (33) if $i\in C(\beta^{*})$, $\beta^{*}(a_{i})=0$ if
$i\notin C(\beta^{*})$, and $C(\beta^{*})={1,...,K(\beta^{*})}$, where
$K(\beta^{*})$ is as described above.

###  A.3 Proof of Theorem 1

As stated in the text, the results in Matějka and McKay, (2015) and Caplin et
al., (2019) state that $\beta\in\varDelta(\mathcal{A})$ can only be a solution
of (4) if (15) holds for all $i\in C(\beta)$. The question then is: given a
vector $\beta\in\Delta(\mathcal{A})$ of unconditional choice probabilities,
can we find $\mu\in\mathbb{R}^{N}_{+}$ that solves the following system:

$\begin{cases}\mu(\omega_{1})+\mu(\omega_{2})+\ldots+\mu(\omega_{N})=1,&\\\
\sum\limits_{j=1}^{N}\mu(\omega_{j})\frac{e^{\frac{u(a_{i},\omega_{j})}{\lambda}}}{\sum\limits_{k=1}^{N}\beta(a_{k})e^{\frac{u(a_{k},\omega_{j})}{\lambda}}}=1&\forall
i\in C(\beta).\end{cases}$ (34)

The system above is a linear system of $K(\beta)+1$ equations with $N$
unknowns. To prove the solution exists, we use the Farkas’ lemma (see, e.g.,
Corollary 5.85 in Aliprantis and Border, (2006)). It states that given some
matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^{m}$, the
linear system $Ax=b$ has a non-negative root $x\in\mathbb{R}^{n}_{+}$ if and
only if there exists no vector $y\in\mathbb{R}^{m}$ such that $A^{\prime}y\geq
0$ with $b^{\prime}y<0$. The two latter inequalities applied to our case form
the following system:

$\begin{cases}y_{0}\left(\sum\limits_{k=1}^{N}\beta(a_{k})e^{\frac{u(a_{k},\omega_{j})}{\lambda}}\right)+\left(\sum\limits_{i\in
C(\beta)}y_{i}e^{\frac{u(a_{i},\omega_{j})}{\lambda}}\right)\geq 0&\forall
j\in\\{1,...,N\\},\\\ y_{0}+\sum\limits_{i\in C(\beta)}y_{i}<0.&\end{cases}$
(35)

We need to show there exists no $y\in\mathbb{R}^{K(\beta)+1}$ that solves the
system above. Let us define $z_{i}\equiv y_{i}+y_{0}\beta(a_{i})$ for $i\in
C(\beta)$. Then recalling that
$e^{\frac{u(a_{i},\omega_{i})}{\lambda}}=e^{\frac{1}{\lambda}}$ and
$e^{\frac{u(a_{i},\omega_{j})}{\lambda}}=1$ for $i\neq j$, system (35)
transforms to

$\begin{cases}z_{j}e^{\frac{1}{\lambda}}+\sum\limits_{i\in
C(\beta)\backslash\\{j\\}}z_{i}\geq 0&\forall j\in C(\beta),\\\
\sum\limits_{i\in C(\beta)}z_{i}\geq 0&\forall j\in\\{1,...,N\\}\backslash
C(\beta),\\\ \sum\limits_{i\in C(\beta)}z_{i}<0.&\end{cases}$ (36)

System (36) above does not have a solution. Indeed, if
$C(\beta)\subsetneq\\{1,...,N\\}$, then the middle set of inequalities
directly contradicts the latter inequality. If $C(\beta)=\\{1,...,N\\}$, then
subtracting the latter inequality from the former, for a given $j\in
C(\beta)$, yields $z_{j}\delta\geq 0\iff z_{j}\geq 0$. Since this must hold
for all $j\in C(\beta)$, we obtain a contradiction with the latter inequality,
$\sum\limits_{i\in C(\beta)}z_{i}<0$.

By the Farkas’ lemma, we then conclude that for any vector
$\beta\in\Delta(\mathcal{A})$ there exists a belief $\mu\in\varDelta(\Omega)$
that solves system (34). This concludes the proof.

###  A.4 Proof of Proposition 3

This proof proceeds in two parts. First we show that the delegation strategy
introduced in the proposition (hereinafter referred to as “the candidate
strategy”) is optimal for the principal. Then we establish that it does indeed
possess the stated properties.

Consider an agent with a prior belief

$\displaystyle\mu(\omega_{i})=\frac{\sqrt{\mu_{p}(\omega_{i})}}{\sum\limits_{j=1}^{N}\sqrt{\mu_{p}(\omega_{j})}}.$
(37)

It is trivial to verify that prior belief $\mu$ defined this way satisfies the
candidate strategy in the statement of the proposition, and hence represents
the candidate strategy. Caplin et al., (2019) show (see their Theorem 1) that
an agent with a prior belief $\mu$ optimally chooses a decision rule that
generates unconditional choice probabilities

$\displaystyle\beta(a_{i})=\max\left\\{0,\frac{1}{\delta}\left(\frac{(K(\beta)+\delta)\mu(\omega_{i})}{\sum\limits_{j\in
C(\beta)}\mu(\omega_{j})}-1\right)\right\\}.$ (38)

Consider an agent hired in accordance with the candidate rule. Substituting
(37) into (38) yields

$\displaystyle\beta(a_{i})=\max\left\\{0,\frac{1}{\delta}\left(\frac{(K(\beta^{*})+\delta)\sqrt{\mu_{p}(\omega_{i})}}{\sum\limits_{j\in
C(\beta^{*})}\sqrt{\mu_{p}(\omega_{j})}}-1\right)\right\\},$ (39)

which are exactly the probabilities stated in Lemma 1. Therefore, an agent
hired according to the candidate strategy makes decisions in such a way that
generates the principal-optimal unconditional choice probabilities. Therefore,
delegation according to the candidate strategy is indeed optimal for the
principal.

Now we show that the candidate strategy satisfies the properties stated in the
proposition. Firstly, it follows clearly from (37) that
$\mu^{*}(\omega_{1})\geq\mu^{*}(\omega_{2})\geq\ldots\geq\mu^{*}(\omega_{N})$.
It remains to show that $\mu^{*}(\omega_{1})\leq\mu_{p}(\omega_{1})$ and
$\mu^{*}(\omega_{N})\geq\mu_{p}(\omega_{N})$. The former inequality can be
shown as follows:

$\displaystyle\mu^{*}(\omega_{1})\leq\mu_{p}(\omega_{1})$ $\displaystyle\iff$
$\displaystyle\frac{\sqrt{\mu_{p}(\omega_{i})}}{\sum\limits_{j=1}^{N}\sqrt{\mu_{p}(\omega_{j})}}\leq\mu_{p}(\omega_{1})$
$\displaystyle\iff$ $\displaystyle
1\leq\sqrt{\mu_{p}(\omega_{1})}\cdot\left(\sum\limits_{j=1}^{N}\sqrt{\mu_{p}(\omega_{j})}\right)$
$\displaystyle\iff$ $\displaystyle
1\leq\mu_{p}(\omega_{1})+\sqrt{\mu_{p}(\omega_{1})\mu_{p}(\omega_{2})}+...+\sqrt{\mu_{p}(\omega_{1})\mu_{p}(\omega_{N})},$

and the latter inequality holds because
$\mu_{p}(\omega_{1})+...+\mu_{p}(\omega_{N})=1$ and
$\sqrt{\mu_{p}(\omega_{1})\mu_{p}(\omega_{j})}\geq\mu_{p}(\omega_{j})$ for all
$j\in\\{1,...,N\\}$, since $\mu_{p}(\omega_{1})\geq\mu_{p}(\omega_{j})$. Note
that $\mu^{*}(\omega_{1})=\mu_{p}(\omega_{1})$ only if
$\mu_{p}(\omega_{1})=...=\mu_{p}(\omega_{N})$.

Similarly, the inequality $\mu^{*}(\omega_{N})\geq\mu_{p}(\omega_{N})$ is
equivalent to

$\displaystyle
1\geq\sqrt{\mu_{p}(\omega_{1})\mu_{p}(\omega_{N})}+...+\sqrt{\mu_{p}(\omega_{N-1})\mu_{p}(\omega_{N})}+\mu_{p}(\omega_{N}),$

which holds because
$\sqrt{\mu_{p}(\omega_{j})\mu_{p}(\omega_{N})}\leq\mu_{p}(\omega_{j})$ for all
$j\in\\{1,...,N\\}$, with equalities only if
$\mu_{p}(\omega_{1})=...=\mu_{p}(\omega_{N})$. This concludes the proof of
Proposition 3.

###  A.5 Proof of Proposition 4

It follows from (19) that the size of the consideration set in the aligned
problem, $\bar{K}$, is such that

$\displaystyle\sum\limits_{j=1}^{\bar{K}}\frac{\mu_{p}(\omega_{j})}{\mu_{p}(\omega_{\bar{K}})}<\bar{K}+\delta\leq\sum\limits_{j=1}^{\bar{K}}\frac{\mu_{p}(\omega_{j})}{\mu_{p}(\omega_{\bar{K}+1})}$

Since $\frac{\mu_{p}(\omega_{i})}{\mu_{p}(\omega_{\bar{K}})}>1$ for all
$i<\bar{K}$, we have that
$\frac{\mu_{p}(\omega_{i})}{\mu_{p}(\omega_{\bar{K}})}>\frac{\sqrt{\mu_{p}(\omega_{i})}}{\sqrt{\mu_{p}(\omega_{\bar{K}})}}>1$
holds for all $i<K$. Therefore,

$\displaystyle\sum\limits_{j=1}^{\bar{K}}\frac{\sqrt{\mu_{p}(\omega_{j})}}{\sqrt{\mu_{p}(\omega_{\bar{K}})}}<\bar{K}+\delta.$
(40)

From (20), $K^{*}$ is the unique solution of

$\displaystyle\sum\limits_{j=1}^{K^{*}}\frac{\sqrt{\mu_{p}(\omega_{j})}}{\sqrt{\mu_{p}(\omega_{K^{*}})}}<K^{*}+\delta\leq\sum\limits_{j=1}^{K^{*}}\frac{\sqrt{\mu_{p}(\omega_{j})}}{\sqrt{\mu_{p}(\omega_{K^{*}+1})}}.$
(41)

Two cases are possible, depending on whether

$\displaystyle\bar{K}+\delta\gtreqless\sum\limits_{j=1}^{\bar{K}}\frac{\sqrt{\mu_{p}(\omega_{j})}}{\sqrt{\mu_{p}(\omega_{\bar{K}+1})}}.$
(42)

If $\bar{K}+\delta\leq RHS$ in (42) (where RHS refers to the right-hand side),
then together with (40) this implies that $\bar{K}$ solves (41), and thus
$\bar{K}=K^{*}$, which satisfies that statement of the proposition.

If, however, $\bar{K}+\delta>RHS$ in (42), then $\bar{K}$ does not solve (41).
In this case, note that going from $K$ by $K+1$ increases the LHS of (42) by
$1$ and increases the RHS by the amount strictly greater than $1$, since a new
term $\frac{\sqrt{\mu_{p}(\omega_{K+1})}}{\sqrt{\mu_{p}(\omega_{K+2})}}>1$ is
added to the sum, and all existing terms increase because
$\mu_{p}(\omega_{K+1})<\mu_{p}(\omega_{K})$. This holds for all $K$, meaning
that if $\bar{K}+\delta>RHS$ in (42), then
$K+\delta>\sum\limits_{j=1}^{K}\frac{\sqrt{\mu_{p}(\omega_{j})}}{\sqrt{\mu_{p}(\omega_{K+1})}}$
for all $K<\bar{K}$. Therefore, the unique solution $K^{*}$ of (41) must be
such that $K_{M}>\bar{K}$. This concludes the proof.

###  A.6 Proof of Proposition 5

Part 2 of the statement follows immediately from Proposition 3 of Matveenko
and Mikhalishchev, (2021).

To show part 1, we invoke Theorem 1 from Matějka and McKay, (2015) stated in
(14), which claims that in the contracting problem, the
$\beta:\Omega\to\varDelta(\mathcal{A})$ that solves the agent’s problem (21)
is given by

$\displaystyle\pi(a_{i}|\omega_{j})$
$\displaystyle=\frac{\beta(a_{i})e^{\frac{u(a_{i},\omega_{j})+\tau(a_{i})}{\lambda}}}{\sum_{k=1}^{N}\beta(a_{k})e^{\frac{u(a_{k},\omega_{j})+\tau(a_{k})}{\lambda}}}$
$\displaystyle=\frac{\beta^{\prime}(a_{i})e^{\frac{u(a_{i},\omega_{j})}{\lambda}}}{\sum_{k=1}^{N}\beta^{\prime}(a_{k})e^{\frac{u(a_{k},\omega_{j})}{\lambda}}},$
(43) $\displaystyle\text{where }\beta(a_{i})$
$\displaystyle=\sum_{j=1}^{N}\mu(\omega_{j})\pi(a_{i}|\omega_{j}).$
$\displaystyle\text{and }\beta^{\prime}(a_{i})$
$\displaystyle\equiv\frac{\beta(a_{i})e^{\frac{\tau(a_{i})}{\lambda}}}{\sum_{k=1}^{N}\beta(a_{k})e^{\frac{\tau(a_{k})}{\lambda}}}.$

Since $\beta^{\prime}$ is a valid probability distribution on $\mathcal{A}$,
representation (43) together with (14) imply that such a collection of
conditional probabilities $\pi$ is a valid solution of the agent’s problem (4)
when the agent’s preferences net of information costs are given by
$u(a_{i},\omega_{j})$. I.e., the principal can implement the desired
conditional choice probabilities $\pi$ by choosing an agent with unbiased
preferences and some belief $\mu$, such that the unconditional choice
probabilities selected by this agent are given by $\beta^{\prime}$. Theorem 1
implies that such a belief $\mu\in\varDelta(\Omega)$ does indeed exist.

###  A.7 Proof of Proposition 6

Plugging (13) in (11) yields the optimal conditional choice probabilities for
the binary model, given by

$\displaystyle\pi^{*}(R|r)$
$\displaystyle=\left(e^{\frac{2}{\lambda}}-1\right)^{-1}e^{\frac{1}{\lambda}}\left(e^{\frac{1}{\lambda}}-\sqrt{\frac{1-\mu_{p}}{\mu_{p}}}\right),$
(44) $\displaystyle\pi^{*}(L|l)$
$\displaystyle=\left(e^{\frac{2}{\lambda}}-1\right)^{-1}e^{\frac{1}{\lambda}}\left(e^{\frac{1}{\lambda}}-\sqrt{\frac{\mu_{p}}{1-\mu_{p}}}\right),$

cropped to $[0,1]$.

The agent’s preferences only depend on the difference $\tau(R)-\tau(L)$.
Assuming all $\tau(R)\in\mathbb{R}$ are available to the principal (no limited
liability), it is without loss to set $\tau(L)=0$. The agent’s problem is
given by (21). Solving it given $\tau=(\tau(R),0)$ yields

$\displaystyle\pi(R|r)$
$\displaystyle=1-\frac{e^{\frac{2}{\lambda}}(1-\mu)-e^{\frac{1+\tau(R)}{\lambda}}+\mu}{\left(e^{\frac{2}{\lambda}}-1\right)\left(e^{\frac{1+\tau(R)}{\lambda}}-1\right)\mu},$
(45) $\displaystyle\pi(L|l)$
$\displaystyle=\frac{e^{\frac{1}{\lambda}}\left(e^{\frac{2}{\lambda}}(1-\mu)-e^{\frac{1+\tau(R)}{\lambda}}+\mu\right)}{\left(e^{\frac{2}{\lambda}}-1\right)\left(e^{\frac{1}{\lambda}}-e^{\frac{\tau(R)}{\lambda}}\right)(1-\mu)},$

cropped to $[0,1]$.

The principal’s contracting problem (22) in the binary setting with $\rho=0$
is similar to (12):

$\displaystyle\max_{\tau(R)}$
$\displaystyle\left\\{\mu_{p}\pi(R|r)+\left(1-\mu_{p}\right)\pi(L|l)\right\\}$
(46) s.t. $\displaystyle\pi(R|r),\pi(L|l)\text{ are given by
\eqref{eq:cond_t_R}}.$

Assuming the probabilities in (45) are interior, the F.O.C. for (46) yields
the candidate solution $\tau(R)$ given by

$\displaystyle\tau^{*}(R)$
$\displaystyle=\lambda\ln\left[\frac{\frac{1-\mu}{\mu}e^{\frac{1}{\lambda}}+\sqrt{\frac{1-\mu_{p}}{\mu_{p}}}}{\frac{1-\mu}{\mu}+e^{\frac{1}{\lambda}}\sqrt{\frac{1-\mu_{p}}{\mu_{p}}}}\right],$
(47)

where the expression under the $\ln(\cdot)$ is non-negative for any
$\lambda,\mu_{p},\mu$, thus the candidate $\tau(R)$ exists for any $\mu$ that
yields interior probabilities (45).

Plugging (47) into (45) yields, after some routine manipulations, the
conditional choice probabilities that coincide with (44) (hence, the
probabilities (45) are interior given $\mu$ and $\tau^{*}(R)$ if and only if
the probabilities (44) are interior). Thus, the condition (47) is not only
necessary, but also sufficient. Hence, for any $\mu_{p}$ for which (44) are
interior, $\tau^{*}(R)$ as given by (47) solves the principal’s problem (46),
and this solution exists for any $\mu$.

If $\lambda$ and $\mu_{p}$ are such that probabilities (44) are not interior,
then the principal would like the agent to take the ex ante
(principal-)preferred action (it can be verified that the expressions in (44)
are such that $\pi^{*}(R|r)\geq 1\iff\pi^{*}(L|l)\leq 0$ and vice versa). The
candidate transfers (47) yield exactly such non-interior probabilities (when
plugged into (45)), hence they still solve the principal’s problem (46) for
any respective $\mu$.292929 Note that $\tau^{*}(R)$ is not the unique solution
in this case. If $\pi^{*}(R|r)=1,\pi^{*}(L|l)=0$, then any
$\tau(R)\geq\lambda\ln\left(\mu+(1-\mu)e^{\frac{2}{\lambda}}\right)-1$ yields
the optimal choice probabilities, and if $\pi^{*}(R|r)=0,\pi^{*}(L|l)=1$, then
any $\tau(R)\leq 1-\lambda\ln\left(\mu e^{\frac{2}{\lambda}}+(1-\mu)\right)$
solves the principal’s problem.  This concludes the proof of part 1 of the
proposition.

To show part 2, consider (47) as a function of $\mu$. It is strictly
decreasing in $\mu$ on $[0,1]$, and the equation $\tau^{*}(R)(\mu)=0$ has a
unique root in $[0,1]$ equal to

$\displaystyle\mu^{*}=\frac{\sqrt{\mu_{p}}}{\sqrt{\mu_{p}}+\sqrt{1-\mu_{p}}},$

meaning that $\tau(R)\geq 0\iff\mu\leq\mu^{*}$.

###  A.8 Proof of Proposition 7

Proceeding analogously to Section 4, we obtain that the agent’s problem (23)
given incentive payment $\bar{\tau}$ (assuming $\underline{\tau}=0$) is solved
by

$\displaystyle\pi\left(R|r\right)$
$\displaystyle=\frac{e^{\frac{1+\bar{\tau}}{\lambda}}\left(e^{\frac{1+\bar{\tau}}{\lambda}}\mu-(1-\mu)\right)}{\left(e^{\frac{2\left(1+\bar{\tau}\right)}{\lambda}}-1\right)\mu}=\frac{e^{\frac{1+\bar{\tau}}{\lambda}}}{e^{\frac{2(1+\bar{\tau})}{\lambda}}-1}\left(e^{\frac{1+\bar{\tau}}{\lambda}}-\frac{1-\mu}{\mu}\right),$
(48) $\displaystyle\pi\left(L|l\right)$
$\displaystyle=\frac{e^{\frac{1+\bar{\tau}}{\lambda}}\left(e^{\frac{1+\bar{\tau}}{\lambda}}(1-\mu)-\mu\right)}{\left(e^{\frac{2\left(1+\bar{\tau}\right)}{\lambda}}-1\right)(1-\mu)}=\frac{e^{\frac{1+\bar{\tau}}{\lambda}}}{e^{\frac{2(1+\bar{\tau})}{\lambda}}-1}\left(e^{\frac{1+\bar{\tau}}{\lambda}}-\frac{\mu}{1-\mu}\right),$

each cropped to $\left[0,1\right]$.

The principal’s problem (24) can be rewritten as

$\displaystyle\max_{\bar{\tau}}$
$\displaystyle\left\\{\left(1-\bar{\tau}\right)\left(\mu_{p}\pi(R|r)+(1-\mu_{p})\pi(L|l)\right)\right\\},$
(49) s.t. $\displaystyle\pi(R|r),\pi(L|l)\text{ are given by
\eqref{eq:cond_prob_tbar}}.$

To begin with, note that the principal would never choose $\bar{\tau}<0$ (due
to limited liability) or $\bar{\tau}\geq 1$ (since this would render the
principal’s payoff negative). Further, if the solution (48) yields values
outside $[0,1]$ for some $\bar{\tau}>0$, then such a $\bar{\tau}$ is clearly
suboptimal for the principal, as she can then reduce her costs without
decreasing the precision of the agent’s choice. Thus, in the optimum, if
$\bar{\tau}>0$, then the probabilities (48) are interior. Assuming the latter
and plugging (48) into the principal’s problem (49), the F.O.C. of this
problem is given by

$\mu_{p}\frac{1-\mu}{\mu}+(1-\mu_{p})\frac{\mu}{1-\mu}=e^{\frac{1+\bar{\tau}}{\lambda}}\cdot\frac{\lambda\left(e^{2\frac{1+\bar{\tau}}{\lambda}}-1\right)+2(1-\bar{\tau})}{\lambda\left(e^{2\frac{1+\bar{\tau}}{\lambda}}-1\right)+\left(e^{2\frac{1+\bar{\tau}}{\lambda}}+1\right)(1-\bar{\tau})}.$
(50)

Let $\gamma(\mu,\mu_{p})$ denote the LHS and $\chi(\bar{\tau},\lambda)$ the
RHS of (50), respectively. Then the necessary condition for an interior
$\bar{\tau}$ to be optimal is $\gamma(\mu,\mu_{p})=\chi(\bar{\tau},\lambda)$.
Note that $\gamma(\mu,\mu_{p})$ is minimized for a given $\mu_{p}$ by
$\mu=\mu^{*}(\mu_{p})$ given by (13), and
$\gamma(\mu^{*}(\mu_{p}),\mu_{p})=2\sqrt{\mu_{p}\left(1-\mu_{p}\right)}<1$.
Further, it can be shown that $\chi(\bar{\tau},\lambda)$ is continuous and
increasing in $\bar{\tau}$ for all $\lambda$. This means that a solution to
(50) exists for a given $\mu,\mu_{p},\lambda$ if and only if
$\chi(0,\lambda)\leq\gamma(\mu,\mu_{p})\leq\chi(1,\lambda)$ (where the “only
if” part follows from the intermediate value theorem). Since
$\chi(\bar{\tau},\lambda)$ is increasing in $\bar{\tau}$, the second-order
condition holds, meaning that any $\bar{\tau}$ that solves (50) is a local
maximizer of (49); further, there is at most one local maximizer.

Suppose $\mu<\mu^{*}(\mu_{p})$. As $1\leq\chi(0,\lambda)<+\infty$ for all
$\lambda>0$, $\gamma(0,\mu_{p})=+\infty$,
$\gamma(\mu^{*}(\mu_{p}),\mu_{p})<1$, and $\gamma(\mu,\mu_{p})$ is continuous
in $\mu\in(0,1)$, there exists $\bar{\mu}_{1}<\mu^{*}(\mu_{p})$ such that
$\gamma(\bar{\mu}_{1},\mu_{p})=\chi(0,\lambda)$. As
$\chi(1,\lambda)=e^{\frac{2}{\lambda}}>1$, there also exists
$\tilde{\mu}_{1}<\mu^{*}(\mu_{p})$ such that
$\gamma(\tilde{\mu}_{1},\mu_{p})=\chi(1,\lambda)$. As
$\chi(0,\mu)<\chi(1,\lambda)$ for all $\lambda$, it follows that
$\tilde{\mu}_{1}<\bar{\mu}_{1}$. By a mirror argument, there also exist
$\tilde{\mu}_{2}>\bar{\mu}_{2}>\mu^{*}(\mu_{p})$ that satisfy the analogous
properties. Further, $\gamma\left(\mu_{p},\mu_{p}\right)=1$, so that
$\bar{\mu}_{2}>\mu_{p}.$ In the end,
$\chi(0,\lambda)\leq\gamma(\mu,\mu_{p})\leq\chi(1,\lambda)$ and, hence, a
solution $\bar{\tau}$ to (50) exists (and thus constitutes a local maximum of
(49)) if and only if

$\mu\in[\tilde{\mu}_{1},\bar{\mu}_{1}]\cup[\bar{\mu}_{2},\tilde{\mu}_{2}].$
(51)

If $\mu$ does not satisfy (51) (which is the case in the “aligned” case,
$\mu=\mu_{p}$), no interior solution exists to (49), hence $\bar{\tau}=0$ is
optimal. On the other hand, if $\mu$ satisfies (51), then the optimal
$\bar{\tau}$ (that solves (49)) can be given by either the solution to (50),
or the corner solution $\bar{\tau}=0$. As argued previously, the latter can
only be a candidate solution if the probabilities (48) are non-interior given
$\bar{\tau}=1$, which is the case if
$\mu\notin\left[\frac{1}{1+e^{\frac{1}{\lambda}}},\frac{e^{\frac{1}{\lambda}}}{1+e^{\frac{1}{\lambda}}}\right]$.
Further, it can be shown that
$\tilde{\mu}_{1}<\frac{1}{1+e^{\frac{1}{\lambda}}}$ and
$\frac{e^{\frac{1}{\lambda}}}{1+e^{\frac{1}{\lambda}}}<\tilde{\mu}_{2}$. Thus,
we conclude that if

$\mu\in\left[\frac{1}{1+e^{\frac{1}{\lambda}}},\bar{\mu}_{1}\right]\cup\left[\bar{\mu}_{2},\frac{e^{\frac{1}{\lambda}}}{1+e^{\frac{1}{\lambda}}}\right],$
(52)

then a $\bar{\tau}\in(0,1)$ that solves (50) is a global maximizer of (49),
which proves the statement of the proposition. It is, however, worth noting
that (52) is only a sufficient condition and not a necessary one. Further, one
of the invervals in (52) may be empty for extreme enough $\mu_{p}$.

###  A.9 Proof of Proposition 8

Using Theorem 1 of Caplin et al., (2019), the agent’s problem (25) given some
restriction set $A^{*}$ is solved by $\pi$ such that the corresponding
$\beta\in\varDelta(A^{*})$ satisfies (18) for all $a_{i}\in A^{*}$. Further,
recall from Section 5 that $\pi$ and $\beta$ are connected in the optimum by
relation (14) (where we set $\pi(a_{i}|\omega_{j})\equiv\beta(a_{i})\equiv 0$
for all $a_{i}\notin A^{*}$ and all $\omega_{j}\in\Omega$). Then by plugging
(14) and the state-matching utility into the principal’s expected payoff, it
can be rewritten as in (17):

$\sum_{i=1}^{N}\mu_{p}(\omega_{i})\frac{\beta(a_{i})e^{\frac{1}{\lambda}}}{1+\delta\beta(a_{i})}=\sum_{i\in
C(\beta)}\mu_{p}(\omega_{i})\frac{(1+\delta)\beta(a_{i})}{1+\delta\beta(a_{i})}.$

Plugging in (18) for $\beta$ in the expression above transforms it to

$\sum_{i\in
C(\beta)}\frac{\frac{1+\delta}{\delta}\mu_{p}(\omega_{i})\left[\left(K(\beta)+\delta\right)\mu_{p}(\omega_{i})-\sum\limits_{j\in
C(\beta)}{\mu(\omega_{j})}\right]}{(K(\beta)+\delta)\mu_{p}(\omega_{i})}\\\
=\frac{1+\delta}{\delta}\left[\sum_{i\in
C(\beta)}\mu_{p}(\omega_{i})-\sum_{i\in C(\beta)}\frac{\sum\limits_{j\in
C(\beta)}{\mu(\omega_{j})}}{(K(\beta)+\delta)}\right]\\\
=\frac{1+\delta}{K(\beta)+\delta}\sum_{i\in C(\beta)}\mu_{p}(\omega_{i}).$
(53)

To prove the proposition statement, we proceed by induction. Consider some
arbitrary action set $A_{-}\subset\mathcal{A}$ such that $a_{k}\notin A_{-}$
for some $k\in\\{1,...,N\\}$ and another action set $A_{+}\equiv
A_{-}\cup\\{a_{k}\\}$. Let $\beta_{+}$ denote the unconditional choice
probabilities corresponding to the solution of (25) given $A_{+}$, let
$C_{+}\equiv C(\beta_{+})$ and $K_{+}\equiv K(\beta_{+})$, and define
$\beta_{-},C_{-},K_{-}$ analogously given $A_{-}$.

Our goal is to show that that selecting $A^{*}=A_{+}$ is weakly better for the
principal than $A^{*}=A_{-}$. If $a_{k}\notin C_{+}$, then the payoffs in the
two cases are equal, and the statement is trivially true. Otherwise, using
(53) for the principal’s expected payoff, the statement amounts to:

$\displaystyle 0\leq$
$\displaystyle\left(\frac{1+\delta}{K_{+}+\delta}\sum_{i\in
C_{+}}\mu_{p}(\omega_{i})\right)-\left(\frac{1+\delta}{K_{-}+\delta}\sum_{i\in
C_{-}}\mu_{p}(\omega_{i})\right)$ $\displaystyle\iff 0\leq$
$\displaystyle\left((K_{-}+\delta)\sum_{i\in
C_{+}}\mu_{p}(\omega_{i})\right)-\left((K_{+}+\delta)\sum_{i\in
C_{-}}\mu_{p}(\omega_{i})\right)$ $\displaystyle\iff 0\leq$
$\displaystyle(K_{-}+\delta)\mu_{p}(\omega_{k})-\left(\sum_{i\in
C_{-}}\mu_{p}(\omega_{i})\right).$ (54)

Since $a_{k}\in C_{+}$ by assumption, $\beta_{+}(a_{k})>0$, which, from (18),
implies that

$\displaystyle 0<$
$\displaystyle\frac{(K(\bar{\beta})+\delta){\mu(\omega_{i})}}{\sum\limits_{j\in
C(\bar{\beta})}{\mu(\omega_{j})}}-1$ $\displaystyle\iff 0<$
$\displaystyle(K_{+}+\delta)\mu_{p}(\omega_{k})-\left(\sum_{i\in
C_{+}}\mu_{p}(\omega_{i})\right)$ $\displaystyle\iff 0<$
$\displaystyle(K_{-}+1+\delta)\mu_{p}(\omega_{k})-\left(\sum_{i\in
C_{-}}\mu_{p}(\omega_{i})+\mu_{p}(\omega_{k})\right)$ $\displaystyle\iff 0<$
$\displaystyle(K_{-}+\delta)\mu_{p}(\omega_{k})-\left(\sum_{i\in
C_{-}}\mu_{p}(\omega_{i})\right),$

which immediately implies that (54) holds. Therefore, it is indeed better for
the principal to choose $A_{+}$ over $A_{-}$. Since $A_{-}$ was arbitrary,
this proves by induction that allowing a larger action set is always weakly
better for the principal, and hence proves the original proposition.

###  A.10 Proof of Proposition 9

We provide an example for $N=3$. We use the same version of the Farkas’ Lemma
as in the proof of Theorem 1. To show that there is no prior belief that
solves the system of the first-order conditions for the problem, it is
sufficient to show that there is a solution to the following dual inequality
system

$\begin{cases}z_{1}e^{\frac{u(a_{1},\omega_{1})}{\lambda}}+z_{2}e^{\frac{u(a_{1},\omega_{2})}{\lambda}}+z_{3}e^{\frac{u(a_{1},\omega_{3})}{\lambda}}\geq
0,\\\
z_{1}e^{\frac{u(a_{2},\omega_{1})}{\lambda}}+z_{2}e^{\frac{u(a_{2},\omega_{2})}{\lambda}}+z_{3}e^{\frac{u(a_{2},\omega_{3})}{\lambda}}\geq
0,\\\
z_{1}e^{\frac{u(a_{3},\omega_{1})}{\lambda}}+z_{2}e^{\frac{u(a_{3},\omega_{2})}{\lambda}}+z_{3}e^{\frac{u(a_{3},\omega_{3})}{\lambda}}\geq
0,\\\ z_{1}+z_{2}+z_{3}<0.\end{cases}$ (55)

Let us normalize $\lambda=1$ and consider payoffs given by the following
matrix:

$\begin{pmatrix}u(a_{1},\omega_{1})&u(a_{2},\omega_{1})&u(a_{3},\omega_{1})\\\
u(a_{1},\omega_{2})&u(a_{2},\omega_{2})&u(a_{3},\omega_{2})\\\
u(a_{1},\omega_{3})&u(a_{2},\omega_{3})&u(a_{3},\omega_{3})\end{pmatrix}=\begin{pmatrix}\ln
3&0&\ln(2+\varepsilon)\\\ 0&\ln 3&\ln(2+\varepsilon)\\\
0&0&\ln(2+\varepsilon)\end{pmatrix}$

Notice that vector $(z_{1},z_{2},z_{3})=(-1-\delta,-1-\delta,2)$ for small
enough $\delta,\varepsilon\geq 0$ solves system (55): the two latter
inequalities hold trivially for all such $z$, and the two former inequalities
hold if $\varepsilon\geq 3^{\frac{1+\delta}{2}}-2$. Therefore, there exists no
$\mu$ that solves system (34) given $\beta\in\varDelta(\Theta)$.

###  A.11 Proof of Proposition 10

We first show that there exists an equilibrium that replicates the deletation
scenario: the optimal agent acquires the same information, makes a truthful
action recommendation, and the principal follows the recommendation.

Suppose that under delegation, the optimally chosen agent follows a decision
rule $\beta^{*}$ that yields a consideration set
$C(\beta^{*})=\\{1,...,K^{*}\\}$. By Lemma 1, we have that

$\displaystyle\sqrt{\mu(\omega_{K^{*}})}$
$\displaystyle\geq\frac{1}{K^{*}+\delta}\sum_{i=1}^{K^{*}}\sqrt{\mu(\omega_{i})}$
$\displaystyle\iff\delta\sqrt{\mu(\omega_{K^{*}})}$
$\displaystyle\geq\sum_{i=1}^{K^{*}-1}\left(\sqrt{\mu(\omega_{i})}-\sqrt{\mu(\omega_{K^{*}})}\right)$
(56)

Suppose the agent reports truthfully. Given the state-matching payoffs, for
the principal to follow recommendation $\tilde{a}=\tilde{a}_{K^{*}}$ whenever
it is issued, it must hold that

$\displaystyle\mu_{p}(\omega_{K^{*}}|\tilde{a}_{K^{*}})=\max_{i}\mu_{p}(\omega_{i}|\tilde{a}_{K^{*}}),$
(57)

where $\mu_{p}(\omega|\tilde{a})$ is the probability that the principal’s
posterior belief assigns to state $\omega$ after hearing recommendation
$\tilde{a}$ from the agent. In equilibrium, the principal’s posterior
$\mu_{p}(\omega_{K^{*}}|\tilde{a}_{K^{*}})$ must satisfy the Bayes’ rule:

$\displaystyle\mu_{p}(\omega_{K^{*}}|\tilde{a}_{K^{*}})$
$\displaystyle=\frac{\pi(a_{K^{*}}|\omega_{K^{*}})\mu_{p}(\omega_{K^{*}})}{\sum_{i=1}^{N}\mu_{p}(\omega_{i})\pi(a_{K^{*}}|\omega_{i})}$
$\displaystyle=\frac{\beta(a_{K^{*}})e^{\frac{1}{\lambda}}}{\beta(a_{1})+...+\beta(a_{K^{*}-1})+\beta(a_{K^{*}})e^{\frac{1}{\lambda}}}\cdot\frac{\mu_{p}(\omega_{K^{*}})}{\sum_{i=1}^{N}\mu_{p}(\omega_{i})\pi(a_{K^{*}}|\omega_{i})}$
$\displaystyle=\frac{\beta(a_{K^{*}})e^{\frac{1}{\lambda}}}{1+\delta\beta(a_{K^{*}})}\cdot\frac{\mu_{p}(\omega_{K^{*}})}{\sum_{i=1}^{N}\mu_{p}(\omega_{i})\pi(a_{K^{*}}|\omega_{i})}$
$\displaystyle=\frac{\sum_{i=1}^{K^{*}}\sqrt{\mu_{p}(\omega_{i})}}{K^{*}+\delta}\cdot\beta(a_{K^{*}})e^{\frac{1}{\lambda}}\cdot\frac{\sqrt{\mu(\omega_{K^{*}})}}{\sum_{i=1}^{N}\mu_{p}(\omega_{i})\pi(a_{K^{*}}|\omega_{i})},$

Where the last line is obtained by plugging the expression for
$\beta(a_{K^{*}})$ from Lemma 1 in the denominator of the preceding line.
Similarly, we can calculate the probability that the principal’s posterior
assigns to any other state $\omega_{j}$:

$\displaystyle\mu_{p}(\omega_{j}|\tilde{a}_{K^{*}})$
$\displaystyle=\begin{cases}\frac{\sum_{i=1}^{K^{*}}\sqrt{\mu_{p}(\omega_{i})}}{K^{*}+\delta}\cdot\beta(a_{K^{*}})e^{\frac{1}{\lambda}}\cdot\frac{\sqrt{\mu(\omega_{j})}}{\sum_{i=1}^{N}\mu_{p}(\omega_{i})\pi(a_{K^{*}}|\omega_{i})}&\text{
if }j<K^{*},\\\ 0&\text{ if }j>K^{*}.\end{cases}$

For condition (57) to hold, it is then enough for

$\displaystyle e^{\frac{1}{\lambda}}\sqrt{\mu(\omega_{K^{*}})}$
$\displaystyle\geq\sqrt{\mu(\omega_{1})}$ $\displaystyle\iff$
$\displaystyle\delta\sqrt{\mu(\omega_{K^{*}})}$
$\displaystyle\geq\sqrt{\mu(\omega_{1})}-\sqrt{\mu(\omega_{K^{*}})},$ (58)

to be satisfied. Note, however, that it is strictly weaker than (56), since

$\displaystyle\sqrt{\mu(\omega_{1})}-\sqrt{\mu(\omega_{K^{*}})}<\sum_{i=1}^{K^{*}-1}\left(\sqrt{\mu(\omega_{i})}-\sqrt{\mu(\omega_{K^{*}})}\right).$

Therefore, we conclude that (58) holds, and thus it is optimal for the
principal to choose action $a_{K^{*}}$ when the agent with prior belief
$\mu^{*}$ recommends it.

Following the same argument, we can show the same for any other recommendation
$\tilde{a}_{i}$ for $i\in C(\beta^{*})$: the necessary and sufficient
condition for the principal to find it optimal to follow the recommendation
would be

$\displaystyle e^{\frac{1}{\lambda}}\sqrt{\mu(\omega_{i})}$
$\displaystyle\geq\sqrt{\mu(\omega_{1})},$

which is implied by (57), since $\mu(\omega_{i})\geq\mu(\omega_{K^{*}})$ for
$i\in C(\beta^{*})$. This concludes the proof.