#  Attention Elasticities
and Invariant Information Costs††thanks: I thank Andrew Caplin for helpful
comments and discussions.

Dániel Csaba QuantCo. Email<EMAIL_ADDRESS>
(April, 2021)

###### Abstract

We consider a generalization of rational inattention problems by measuring
costs of information through the information radius (Sibson, 1969; Verdú,
2015) of statistical experiments. We introduce a notion of attention
elasticity measuring the sensitivity of attention strategies with respect to
changes in incentives. We show how the introduced class of cost functions
controls attention elasticities while the Shannon model restricts attention
elasticity to be unity. We explore further differences and similarities
relative to the Shannon model in relation to invariance, posterior
separability, consideration sets, and the ability to learn events with
certainty. Lastly, we provide an efficient alternating minimization
method—analogous to the Blahut-Arimoto algorithm—to obtain optimal attention
strategies.

JEL Codes: D83, D90
Keywords: information costs, rational inattention, $\alpha$-mutual information

##  1 Introduction

The pivotal role of information has long been recognized in economic
decisions. Understanding the constraints that shape the beliefs decision
makers hold helps us understand and predict their behavior in different
environments. The model of rational inattention (Sims, 1998, 2003) has become
a widely adopted framework for the analysis of beliefs and behavior when
constraints on attention and information acquisition are of primary
importance.111For an extensive overview of the rational inattention literature
see Maćkowiak, Matějka, and Wiederholt (2020).

A central question of the rational inattention (RI) literature has been
identifying and understanding the behavioral implications of different classes
of information cost functions. Our assumptions on the shape of information
costs fully determine patterns in behavior under the assumption of Bayesian
rationality and allowing unrestricted preferences.

In the present paper we introduce a notion of attention elasticity, a simple
behavioral implication of information costs. Attention elasticity captures how
sensitively attention strategies change with changes in incentives. One
expects that increasing the payoff-difference between two actions in a given
state increases the conditional odds of the better action being chosen as the
decision maker pays more attention. Similarly, we expect that such a local
change in payoff-differences is also reflected on the unconditional odds. The
posterior odds links these quantities and is equal to the ratio of the
conditional odds over the unconditional odds. Our notion of attention
elasticity captures the relative rate of change in posterior odds as we change
incentives.

We show that mutual information costs imply an attention elasticity of unity
irrespective of the payoff structure and prior beliefs. This puts significant
restrictions on the types of beliefs and choice probabilities that one can
attain within the Shannon model as introduced in the seminal works of Sims
(1998, 2003) and widely used in applications. We show that such patterns in
attention elasticity can be understood through a perspective of the mutual
information as the average divergence of signal probabilities relative to the
barycenter of the Blackwell experiment. This motivates our generalization of
the Shannon model.

In particular, we analyze a novel class of information cost functions that
relaxes the restrictive elasticity patterns of the Shannon model while
maintaining many of its desirable features. The introduced class generalizes
mutual information by characterizing costs of statistical experiments based on
the average Rényi divergence of conditional signal probabilities relative to
the barycenter of the experiment (Sibson, 1969; Verdú, 2015), a class named
$\alpha$-mutual information.222The recent work of Hoiles, Krishnamurthy, and
Pattanayak (2020) in the computer science literature has used the same class
of information costs in an inverse reinforcement learning task without
exploring behavioral implications. We show that such a view gives rise to
information costs that are analogous to constant elasticity of substitution
production functions widely used in other domains of economics. This gives
clear behavioral meaning to information cost parameters affecting the changes
in beliefs and behavior due to changes in incentives. We refer to the
corresponding class as $\alpha$-rational inattention ($\alpha$-RI). In general
symmetric decision problems we provide conditions under which attention
strategies exhibit elastic or inelastic patterns in $\alpha$-RI problems.

We show that $\alpha$-mutual information satisfies invariance but is not
posterior separable providing a novel direction for generalizing the Shannon
model. Invariance can be a desirable property in several economic contexts.
For instance, in large games with incomplete information invariance of
information costs is a prerequisite for the efficiency of equilibria and it
prevents non-fundamental volatility as shown by the recent work of Hébert and
La’O (2020) and Angeletos and Sastry (2021).

Our approach thus complements several recent papers which have provided
generalizations of the Shannon model that result in various posterior
separable classes of information cost functions. Caplin, Dean, and Leahy
(2019b) show that mutual information is uniquely characterized within the
class of uniformly posterior separable cost functions by the invariance under
compression property, which is a general implication of invariant cost
functions. On the other hand, invariance implies a strong form of symmetry
across states and its implications can be restrictive in other contexts.
Neighborhood-based cost functions, such as the ones proposed by Hébert and
Woodford (2021), and Pomatto, Strack, and Tamuz (2019), aim to accommodate
scenarios in which the costs of learning are asymmetric across states.
Posterior separable cost functions are natural in sequential learning problems
as shown by Morris and Strack (2017), Hébert and Woodford (2019), and Bloedel
and Zhong (2021) for instance.

We explore further behavioral implications. We show that under $\alpha$-mutual
information costs the rationally inattentive decision maker might learn
payoff-relevant events with certainty, a behavior that is impossible in the
Shannon model. We also characterize consideration sets and provide a
generalization of the conditions corresponding to the Shannon case (Caplin,
Dean, and Leahy, 2019a).

Lastly, the introduced class of rational inattention problems are tractable
and can be efficiently solved through a generalized Blahut-Arimoto-type
algorithm, hence have great appeal for empirical applications. We provide
guarantees for convergence and discuss details of implementation.

The rest of the paper is organized as follows. Section 2 introduces the notion
of attention elasticity and discusses elasticity patterns under the Shannon
model. Section 3 introduces $\alpha$-mutual information costs and presents its
main properties. Section 4 derives elasticity patterns in $\alpha$-RI
problems. Section 5 discusses the invariance properties of $\alpha$-mutual
information and its corresponding behavioral implications. Section 6 shows
that under the introduced class, learning payoff-relevant events with
certainty is possible. Section 7 characterizes consideration sets. Section 8
presents a modification of the Blahut-Arimoto algorithm providing efficient
solutions for optimal attention strategies in $\alpha$-RI problems. Section 9
concludes.

##  2 Attention Elasticities and the Shannon Model

We consider general rational inattention problems with finite domains.
Specifically, we consider decision problems described by a finite set of
actions, $a\in\mathcal{A}$, and a finite set of states, $\omega\in\Omega$. The
prior belief over states is denoted $\mu\in\Delta(\Omega)$, and the utility
function is denoted by $u\colon\mathcal{A}\times\Omega\to\mathbb{R}$. The
decision maker (DM) can acquire information and base her action on the
realization of a signal from a statistical experiment. A statistical
experiment is a Markov kernel, $P\colon\Omega\to\Delta(\mathcal{S})$, defining
a collection of probability distributions over signals $s\in\mathcal{S}$ for
each realization of the state. The conditional signal distributions are
denoted as $P(\cdot\mid\omega)$ or as $P_{\omega}$. A simple experiment has a
finite signal support. Keeping the state space fixed, we denote the set of
simple experiments as $\mathcal{P}$. For a given experiment, $P$, and prior
distribution over states, $\mu$, denote the joint distribution as $\mu
P\in\Delta\left(\Omega\times\mathcal{S}\right)$; the marginal distribution
over signals—derived from the joint distribution $\mu P$—as $\mu
P_{\mathcal{S}}$; and the joint distribution obtained from the product of two
marginals, $\mu\in\Delta(\Omega)$ and $q\in\Delta(\mathcal{S})$, as
$\mu\otimes q$.

We allow information costs to depend on the statistical experiment and the
prior, and denote
$K\colon\mathcal{P}\times\Delta(\Omega)\to\overline{\mathbb{R}}$. We only
consider information cost functions that satisfy the convexity property such
that, without loss of generality, we can identify the set of signals with the
set of actions, $\mathcal{S}\equiv\mathcal{A}$, and use the two sets
interchangeably.

The problem of the rationally inattentive DM is then to choose a statistical
experiment that maximizes expected utility net of information costs,333We
assume $K$ is proper and lower-semicontinuous so the optimum is attained.

$\max_{P\in\mathcal{P}}\
\sum_{\omega,s}\mu(\omega)P(s\mid\omega)u(s,\omega)-K(P,\mu).\ $ (2.1)

Following the seminal works of Sims (1998, 2003) mutual information has been
the most widely used information cost in the RI literature—for a large part
due to its tractability. The mutual information is defined as the Kullback-
Leibler (KL) divergence between the joint distribution and the product
distribution derived from the marginals,

$I(P,\mu):=D\left(\mu P\|\mu\otimes\mu
P_{\mathcal{S}}\right):=\sum_{\omega,s}P(s\mid\omega)\mu(\omega)\log\frac{P(s\mid\omega)\mu(\omega)}{\mu
P_{\mathcal{S}}(s)\mu(\omega)}.$ (2.2)

We convert units of information costs (nats) to units of expected utility
through a generic scalar, $\kappa$, to obtain

$K(P,\mu)=\kappa I(P,\mu).$ (2.3)

With these preliminaries in hand we introduce our notion of attention
elasticity.

###  2.1 Attention Elasticity

The attention strategy of the DM is described by the statistical experiment,
$P^{*}$, the solution to the RI problem (2.1). This specifies the conditional
probability of choosing a certain action for each possible realization of the
state. Loosely speaking, we expect that if stakes are higher the inattentive
DM pays more attention and chooses the action that maximizes her utility with
higher probability for each realization of the state.

Following this reasoning, a fundamental characteristic of attention strategies
is how the conditional odds of choosing action $a$ over $b$,
$P^{*}(a\mid\omega)/P^{*}(b\mid\omega)$, change as we change corresponding
payoff differences, $u(a,\omega)-u(b,\omega)$. Such local changes in payoff
differences are also reflected in changes on the unconditional odds, $\mu
P_{\mathcal{S}}^{*}(a)/\mu P_{\mathcal{S}}^{*}(b)$. The posterior odds induced
by the optimal experiment links these quantities and is given by the ratio of
conditional and unconditional odds,

$\frac{\gamma^{a}(\omega)}{\gamma^{b}(\omega)}=\frac{P^{*}(a\mid\omega)}{P^{*}(b\mid\omega)}\frac{\mu
P_{\mathcal{S}}^{*}(b)}{\mu P_{\mathcal{S}}^{*}(a)}.$ (2.4)

This connects to an equivalent way of modeling attention strategies through
distributions over posteriors such that the posteriors satisfy the martingale
property of averaging back to the prior. We capture the local sensitivity of
attention strategies by analyzing the sensitivity of posterior odds with
changes in incentives. We introduce our notion of attention elasticity under
arbitrary changes to the payoff structure.

######  Definition 1 (Attention elasticity).

The attention elasticity between actions $a$ and $b$ in state $\omega$ is
defined as the relative rate of change in posterior odds as we change
incentives. Letting the posterior odds be denoted by

$\Gamma^{a,b}_{\omega}:=\frac{\gamma^{a}(\omega)}{\gamma^{b}(\omega)}=\frac{P^{*}(a\mid\omega)}{\mu
P^{*}_{\mathcal{S}}(a)}\frac{\mu P^{*}_{\mathcal{S}}(b)}{P^{*}(b\mid\omega)},$
(2.5)

and letting $\theta$ be a parameter of the incentive structure the attention
elasticity is given by,

$\epsilon^{a,b}_{\omega,\theta}:=\frac{\partial\Gamma^{a,b}_{\omega}}{\partial\theta}\frac{1}{\Gamma^{a,b}_{\omega}}.$
(2.6)

We consider only $a,b$ and $\omega$ such that conditional choice probabilities
are interior, $P^{*}\left(a\mid\omega\right),\linebreak
P^{*}\left(b\mid\omega\right)>0$.

The relative rate of change in the posterior odds is equal to the relative
rate of change in conditional odds minus that in unconditional odds. Under
certain parametrization of the incentive structure, changes in the incentives
affecting all actions leave unconditional choice probabilities intact, only
affecting conditional choice probabilities. In such cases our notion of
attention elasticity is equivalent to the relative rate of change in
conditional choice probability odds.

Given a fixed information cost function and a prior the attention strategy is
dependent only on the differences in payoffs per units of information costs of
the form, $(u(a,\omega)-u(b,\omega))/\kappa$. Any additive constant applied
uniformly for each state-dependent payoff has no impact on the attention
strategy. Correspondingly, we will consider changes in the incentive structure
by changing absolute payoff differences per units of information costs across
some actions. A scenario of key interest is in which we change the payoff per
unit of information cost of a single action, $\theta=u(a,\omega)/\kappa$, and
analyze the changes in the corresponding posterior ratio,
$\Gamma^{a,b}_{\omega}$.

It is worth noting the similarities in our definition of attention elasticity
and that of elasticity of substitution applied in the context of a profit-
maximizing firm. The optimal production plan of the profit-maximizing firm is
only dependent on relative price differences across available inputs and the
price of the output. We correspondingly analyze the relative rate of change in
input ratios with relative changes in price ratios. On the other hand, the
attention strategy of a rationally inattentive DM is only dependent on
absolute payoff differences per units of information costs. Correspondingly we
analyze the relative rate of change in posterior odds with changes in absolute
payoff differences.

The Shannon model puts stringent behavioral restrictions on attention
elasticities. While immediate we state it as a proposition to emphasize its
importance.

######  Proposition 1.

Under Shannon mutual information costs, for any two actions, $a,b$, that are
chosen with strictly positive probability under the optimal attention strategy
of the rationally inattentive DM, the attention elasticity in state $\omega$
with $\theta=u(a,\omega)/\kappa$ is identically equal to one.

$\epsilon^{a,b}_{\omega,\theta}=1\qquad\forall\omega\in\Omega.$ (2.7)

For any incentive change under the parametrization
$\theta=u(c,\tilde{\omega})/\kappa$ with
$(c,\tilde{\omega})\notin\\{(a,\omega),(b,\omega)\\}$ the attention elasticity
is zero.

Proof. This is an immediate consequence of the invariant likelihood ratio
property (Caplin and Dean, 2013) of the Shannon model,

$\frac{\gamma^{a}(\omega)}{\gamma^{b}(\omega)}=\exp\left(\frac{u(a,\omega)-u(b,\omega)}{\kappa}\right).$
(2.8)

∎

The restrictions on attention elasticities under the Shannon model is quite
strong—the absolute value of elasticity is either one or zero. For instance,
in the case where the Shannon model is used in a discrete choice context
proposition 1 has significant implications for elasticity patterns in the
corresponding demand system.

We note however, that typically the scalar $\kappa$ plays a dual role. First,
it converts units of information costs (nats) to units of expected utility and
allows us to measure all quantities in the same metric. Second, it affects the
relative difficulty of acquiring information and hence should affect attention
elasticities. With a redefinition of parameters we can consider costs of
information where these two roles are treated separately,

$K(P,\mu)=\kappa\sum_{\omega,s}P(s\mid\omega)\mu(\omega)\log\left(\frac{P(s\mid\omega)\mu(\omega)}{\mu
P_{\mathcal{S}}(s)\mu(\omega)}\right)^{\frac{1}{\sigma}}=\frac{\kappa}{\sigma}I(P,\mu).$
(2.9)

It is immediate to see that under the above scaled mutual information costs
the attention elasticity is constant, $\epsilon^{a,b}_{\omega,\theta}=\sigma$
under the parametrization $\theta=u(a,\omega)/\kappa$. Hence, mutual
information implies constant attention elasticity and is the analogue of
constant elasticity of substitution (CES) cost functions widely used in
production theory. The Shannon model has the distinctive feature that the
relative rate of change in posterior odds with changes in incentives is the
same irrespective of the initial payoff difference between actions being $1 or
$1M.

We can gain further insight to this remarkable elasticity property of mutual
information costs through the modified logit rule (Matějka and McKay, 2015)
that the interior conditional choice probabilities satisfy:

$P^{*}(a\mid\omega)\propto\mu
P_{\mathcal{S}}^{*}(a)\exp\left(\frac{u(a,\omega)}{\kappa}\right).$ (2.10)

Conditional choice probabilities are tilted relative to unconditional choice
probabilities through an exponential tilting function dependent on the payoff
parameters. This link between the barycenter of the optimal experiment, $\mu
P^{*}_{\mathcal{S}}$, and conditional choice probabilities is not unique to
the Shannon model. In the following section we introduce a novel class of
information cost functions that maintains this relationship while relaxing
functional forms.

##  3 Information Radius and $\alpha$-Mutual Information

Intuitively, without reference to a decision problem, a signal $s$
corresponding to a statistical experiment is informative in as much as the
posterior after observing $s$ deviates from the prior. Since the posterior is
proportional to the prior times the evidence,
$\gamma^{s}(\omega)\propto\mu(\omega)P(s\mid\omega)$, if the likelihood is
constant at $s$ such that $P(s\mid\cdot)=c$ for some $c$, the corresponding
posterior coincides with the prior and we obtain no information at $s$.

In fact, most characterizations of information costs can be understood as
generalizing how much the likelihoods deviate from a reference constant
vector. This requires defining the reference point and a measure of
dispersion. In the case of mutual information the reference point is defined
by the vector of unconditional signal probabilities, $\mu
P_{\mathcal{S}}(s)=\sum_{\omega}\mu(\omega)P(s\mid\omega)$, while the
dispersion is captured by the relative entropy, or KL divergence:

$\displaystyle I(P,\mu)=$
$\displaystyle\sum_{\omega,s}P(s\mid\omega)\mu(\omega)\log\frac{P(s\mid\omega)\mu(\omega)}{\mu
P_{\mathcal{S}}(s)\mu(\omega)}$ (3.1) $\displaystyle=$
$\displaystyle\sum_{\omega}\mu(\omega)D\left(P_{\omega}\|\mu
P_{\mathcal{S}}\right).$ (3.2)

As seen by equation 3.2 the mutual information is equal to the average KL
divergence between the conditional distributions, $P_{\omega}$, and the vector
of unconditional signal probabilities, $\mu P_{\mathcal{S}}$.

A useful way to think about the reference point is as the barycenter of the
experiment under the KL divergence and the prior. That is, the reference point
is the center of the experiment under the prior in the sense that it is the
distribution that minimizes the average KL divergence:

$\displaystyle\mu P_{\mathcal{S}}$
$\displaystyle=\arg\min_{q\in\Delta(\mathcal{S})}\sum_{\omega}\mu(\omega)D\left(P_{\omega}\|q\right)$
$\displaystyle=\arg\min_{q\in\Delta(\mathcal{S})}D\left(\mu P\|\mu\otimes
q\right).$ (3.3)

Correspondingly, the mutual information is given by,

$I(P,\mu)=\min_{q\in\Delta(\mathcal{S})}D\left(\mu P\|\mu\otimes q\right).$
(3.4)

We can obtain generalizations of the mutual information by considering
different divergence measures and calculating the corresponding deviation from
the barycenter. The information radius or $\alpha$-mutual information is a
special case of the analogous definition using the Rényi divergence (Rényi,
1961) instead of the KL divergence.

######  Definition 2 (Rényi divergence).

For two probability distributions, $P,Q$, defined over the same finite sample
space $\mathcal{X}$, the Rényi divergence of order
$\alpha\in(0,1)\cup(1,\infty)$ is defined as,

$D_{\alpha}\left(P\|Q\right):=\frac{1}{\alpha-1}\log\left(\sum_{x\in\mathcal{X}}P(x)^{\alpha}Q(x)^{(1-\alpha)}\right).$
(3.5)

For $\alpha>1$, we interpret $P(x)^{\alpha}Q(x)^{(1-\alpha)}$ as
$P(x)^{\alpha}/Q(x)^{(\alpha-1)}$ and adopt the conventions that $0/0=0$ and
$y/0=\infty$ for $y>0$.

######  Definition 3 ($\alpha$-mutual information).

For $\alpha\in(0,1)\cup(1,\infty)$ the $\alpha$-mutual information is defined
as,

$\displaystyle I_{\alpha}\left(P,\mu\right):=$
$\displaystyle\min_{q\in\Delta(\mathcal{S})}D_{\alpha}\left(\mu P\|\mu\otimes
q\right)$ (3.6) $\displaystyle=$
$\displaystyle\min_{q\in\Delta(\mathcal{S})}\frac{1}{\alpha-1}\log\left(\sum_{\omega,s}\mu(\omega)P(s\mid\omega)^{\alpha}q(s)^{1-\alpha}\right).$

The solution for the minimization problem in (3.6), the barycenter of the
experiment, can be expressed in closed form. Suppressing the dependence on
$P,\mu$, and $\alpha$, the barycenter is given by,

$q^{*}(s)=\frac{\left(\sum_{\omega}\mu(\omega)P(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}}{\sum_{s^{\prime}}\left(\sum_{\omega}\mu(\omega)P(s^{\prime}\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}}\qquad\forall
s\in\mathcal{S}.$ (3.7)

Substituting the above expression for the barycenter the $\alpha$-mutual
information is given by,

$I_{\alpha}\left(P,\mu\right)=\frac{\alpha}{\alpha-1}\log\sum_{s}\left(\sum_{\omega}\mu(\omega)P(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}.$
(3.8)

The information radius was first introduced by Sibson (1969) as a means to
generalize measures of dissimilarity among a finite set of weighted
probability measures. Verdú (2015) pointed out the connection between the
information radius and the generalization of mutual information through Rényi
divergences coining the class $\alpha$-mutual information.

###  3.1 Properties of $\alpha$-Mutual Information

It is immediate that for any $\alpha$, $I_{\alpha}\left(P,\mu\right)\geq 0$
for all $P,\mu$, and is equal to zero for only non-informative experiments—for
a non-informative experiment all likelihoods $P(s\mid\cdot)$ are constant and
resulting posteriors coincide with the prior. As $\alpha\to 1$ the
$\alpha$-mutual information converges to the Shannon mutual information, just
as the Rényi-divergence converges to the KL divergence. For a fixed prior and
experiment the $\alpha$-mutual information is increasing in $\alpha$.444For
further details on $\alpha$-mutual information see the comprehensive reviews
of Verdú (2015) and Ho and Verdú (2015).

Looking at the statistical experiment as a distribution over posteriors
provides another potentially illuminating perspective. How much the posterior
at a signal $s$ deviates from the prior only depends on the likelihood ratios
over states—how different the vector $P(s\mid\cdot)$ is from a constant
vector. The cost of an information structure can thus be decomposed to two
parts: (i) how much posteriors deviate from the prior, and (ii) what are the
probabilities of ending up with different posteriors. With the unconditional
signal probabilities being denoted by $\mu P_{\mathcal{S}}$ we can rewrite the
cost function as,

$\displaystyle I_{\alpha}(P,\mu)$
$\displaystyle=\frac{\alpha}{\alpha-1}\log\sum_{s}\left(\sum_{\omega}\mu(\omega)P(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}$
$\displaystyle=\frac{\alpha}{\alpha-1}\log\sum_{s}\mu
P_{\mathcal{S}}(s)\left(\sum_{\omega}\mu(\omega)\left(\frac{P(s\mid\omega)}{\mu
P_{\mathcal{S}}(s)}\right)^{\alpha}\right)^{\frac{1}{\alpha}}.$ (3.9)

Since the posterior is as $\gamma^{s}(\omega)\propto
P(s\mid\omega)\mu(\omega)$, constant scaling of the likelihood $P(s\mid\cdot)$
for a given signal $s$ has no impact on the posterior. Hence, from equation
(3.9) we see that we price the information structure based on the posterior
deviating from the prior and the probability with which we obtain the
posterior under consideration. The parameter $\alpha$ determines how costly it
is to twist the likelihoods away from the constant vector.

Applying the monotone increasing transformation
$t\mapsto\frac{1}{\alpha-1}\exp\left(t\frac{\alpha-1}{\alpha}\right)$ on the
$\alpha$-mutual information we obtain a cost function that is convex in $P$
and that is posterior separable.

$\frac{1}{\alpha-1}\sum_{s}\left(\sum_{\omega}\mu(\omega)P(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}=\frac{1}{\alpha-1}\sum_{s}\mu
P_{\mathcal{S}}(s)\left(\sum_{\omega}\gamma^{s}(\omega)^{\alpha}\mu(\omega)^{1-\alpha}\right)^{\frac{1}{\alpha}}$
(3.10)

For any interior $\mu$ the convexity of the map
$\phi_{\mu}\colon\Delta\left(\Omega\right)\to\mathbb{R}$ defined by,

$\phi_{\mu}(\gamma):=\frac{1}{\alpha-1}\left(\sum_{\omega}\gamma(\omega)^{\alpha}\mu(\omega)^{1-\alpha}\right)^{\frac{1}{\alpha}},$
(3.11)

implies that the $\alpha$-mutual information is monotone in the Blackwell
order.555For a detailed argument showing Blackwell-monotonicity see appendix
A. On the other hand, by the results of Denti, Marinacci, and Rustichini
(2020) we have that the $\alpha$-mutual information is not an experimental
cost of information as it approaches zero as the prior becomes more dogmatic.
Following Denti et al. (2020) it is immediate to form the experimental version
of the $\alpha$-mutual information by considering the corresponding
information cost under the uniform prior.

As seen from equation (3.8) the $\alpha$-mutual information function has the
CES property (Arrow, Chenery, Minhas, and Solow, 1961) across states for a
given signal—this, however, does not translate to an analogous constant
attention elasticity property as remarked in the previous section.

Next, we analyze the class of rational inattention problems under
$\alpha$-mutual information costs and refer to the class as $\alpha$-RI.

##  4 $\alpha$-RI and Attention Elasticities

We take the information costs to be the scaled $\alpha$-mutual information,
$K(P,\mu)=\kappa I_{\alpha}(P,\mu)$, and pose the rational inattention
problem.

$\min_{P\in\mathcal{P}}\ \kappa
I_{\alpha}(P,\mu)-\sum_{\omega,s}\mu(\omega)P(s\mid\omega)u(s,w).$ (4.1)

Since the $\alpha$-mutual information is quasi-convex in $P$ (Ho and Verdú,
2015, Theorem 10) and the constraints are convex, the Karush-Kuhn-Tucker (KKT)
conditions are necessary and sufficient for characterizing the solution.
Denoting the dual variables corresponding to the primal feasibility
conditions, $\sum_{s}P(s\mid\omega)=1,\forall\omega\in\Omega$, by
$\lambda_{\omega}$, and the ones corresponding to the positivity constraints,
$P(s\mid\omega)\geq 0,\forall\omega,s\in\Omega\times\mathcal{S}$, by
$\delta_{\omega,s}$ the KKT conditions are given by,

$\displaystyle\frac{\alpha}{\alpha-1}$
$\displaystyle\Bigg{(}\underbrace{\sum_{\tilde{s}}\left(\sum_{\tilde{\omega}}P^{*}(\tilde{s}\mid\tilde{\omega})^{\alpha}\mu(\tilde{\omega})\right)^{\frac{1}{\alpha}}}_{:=C^{*}}\Bigg{)}^{-1}\Bigg{(}\underbrace{\sum_{\tilde{\omega}}P^{*}(s\mid\tilde{\omega})^{\alpha}\mu(\tilde{\omega})}_{:=\tilde{q}^{*}(s)^{(1-\alpha)}}\Bigg{)}^{\frac{1-\alpha}{\alpha}}\mu(\omega)P^{*}(s\mid\omega)^{(\alpha-1)}$
$\displaystyle-$
$\displaystyle\frac{\mu(\omega)u(s,\omega)}{\kappa}+\lambda^{*}_{\omega}-\delta^{*}_{\omega,s}=0,$
(4.2) $\displaystyle\sum_{s}P^{*}(s\mid\omega)=1\qquad\forall\omega,$ (4.3)
$\displaystyle P^{*}(s\mid\omega)\geq 0,\quad\delta^{*}_{\omega,s}\geq 0,\quad
P^{*}(s\mid\omega)\delta^{*}_{\omega,s}=0\qquad\forall\omega,s.$ (4.4)

For interior points, $P^{*}(s\mid\omega)>0$,

$\displaystyle
P^{*}(s\mid\omega)=\left(C^{*^{\alpha}}\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right)^{\frac{1}{\alpha-1}}\frac{\tilde{q}^{*}(s)}{C^{*}}.$
(4.5)

Note that $q^{*}(s):=\tilde{q}^{*}(s)/C^{*}$ is the barycenter of the optimal
experiment as given by equation (3.7) and is analogous to the unconditional
choice probabilities $\mu P_{\mathcal{S}}^{*}(s)$ in the Shannon case. Unlike
in the Shannon case, we can not express the optimal dual variables
$\lambda^{*}_{\omega}$ in closed form, however, we can efficiently obtain them
numerically as we discuss in section 8 describing the modified Blahut-Arimoto
algortihm.

We state the analogy with the Shannon model explicitly. At interior points, in
the Shannon model,

$P^{*}(a\mid\omega)\propto\mu
P_{\mathcal{S}}^{*}(a)\exp\left(\frac{u(a,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right);$
(4.6)

while in the $\alpha$-RI model,

$P^{*}(a\mid\omega)\propto
q^{*}(a)\left(\frac{\alpha-1}{\alpha}\left(\frac{u(a,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right)^{\frac{1}{\alpha-1}}.$
(4.7)

We see that in both cases the optimal conditional choice probabilities are
tilted relative to the barycenter of the optimal experiment. In the Shannon
model we can express the dual variables in closed form using the primal
feasibility conditions,

$\lambda^{*}_{\omega}=\mu(\omega)\log\left(\sum_{s\in\mathcal{S}}\mu
P^{*}_{\mathcal{S}}(a)\exp\left(\frac{u(s,\omega)}{\kappa}\right)\right).$
(4.8)

For ease of notation we denote,

$x^{*}(s,\omega):=\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)},$
(4.9)

which determines the amount by which conditional choice probabilities are
tilted relative to the barycenter given the optimal dual variables and payoff
parameters.

The restrictive elasticity patterns of the Shannon model are relaxed in
$\alpha$-RI problems. The attention elasticities depend on the entire payoff
structure and in particular on the baseline payoff difference between the
considered actions. To highlight the mechanisms underlying attention
elasticities in the $\alpha$-RI model we analyze the class of symmetric
tracking problems.

###  4.1 Symmetric Problem with Global Incentive Changes

We consider general symmetric tracking problems defined as follows. Given a
finite set of states $n:=|\Omega|$ consider the same number of available
actions, $\mathcal{A}:=\\{\ldots,a_{\omega},\ldots\\}$ with $|\mathcal{A}|=n$,
such that the payoff structure is given by

$u(a,\omega)=\begin{cases}h\qquad&\text{if }a=a_{\omega},\\\
l\qquad&\text{otherwise},\end{cases}$ (4.10)

with $h>l$. It is without loss of generality to normalize the low payoff to
zero, $l=0$. Let the prior be uniform over the state space.

Denote $h^{*}:=x^{*}\left(a_{\omega},\omega\right)$ corresponding to action-
state pairs that yield high payoff, and
$l^{*}:=x^{*}\left(a_{\omega},\tilde{\omega}\right)$ corresponding to pairs
that yield low payoff—note that the dual variables $\lambda^{*}_{\omega}$ are
constant across $\omega$ due to symmetry.

Arguably, one of the simplest possible change to the incentive scheme is
increasing the payoff per unit of information cost for the high-payoff actions
in all states simultaneously, $\theta=h/\kappa$. Since the payoff structure is
symmetric and the prior is uniform, from the symmetry of the cost function we
know that the optimal barycenter, $q^{*}$, and the unconditional choice
probabilities over actions, $\mu P^{*}_{\mathcal{S}}$, are both uniform over
the action/signal space both before and after the change to the incentive
scheme. This implies that changes in the posterior ratios are due only to
changes in the conditional choice probabilities while unconditional choice
probabilities are left intact. That is, in this case the attention elasticity
captures the relative rate of change in conditional choice probabilities.

This global incentive change exhibits a “two-regime” elasticity pattern as
stated by the next proposition.

######  Proposition 2.

In the general symmetric tracking problem consider any state $\omega$ and two
distinct actions, $a_{\omega},b$, that are chosen with strictly positive
probability conditional on $\omega$ under the optimal strategy of the
$\alpha$-RI problem. The attention elasticity under the incentive change
$\theta=h/\kappa$ satisfies the following.

For any $\alpha>0$,

$\displaystyle\epsilon^{a_{\omega},b}_{\omega,\theta}\ \begin{cases}\
>1\qquad&\text{if }(\alpha-1)l^{*}<1,\\\\[8.99994pt] \ <1\qquad&\text{if
}(\alpha-1)l^{*}>1.\end{cases}$ (4.11)

Furthermore, for any $\alpha<1$,

$0<(\alpha-1)h^{*}<\alpha<(\alpha-1)l^{*},$ (4.12)

while for $\alpha>1$,

$0<(\alpha-1)l^{*}<\alpha<(\alpha-1)h^{*}.$ (4.13)

Proof. Appendix B.

Note that the magnitude of
$(\alpha-1)l^{*}=(1-\alpha)\lambda^{*}_{\omega}/\mu(\omega)$ depends on the
payoff difference between the high- and low-payoff state-action pairs, $h$.
While we don’t have a closed-form solution for $h^{*}$ or $l^{*}$, by
definition we know that their difference is $h^{*}-l^{*}=h$. This payoff
difference and the curvature of the information cost function determined by
$\alpha$ define whether or not the response in attention strategies is elastic
or inelastic. In contrast, under the Shannon model the attention elasticity is
constant irrespective of the payoff difference.

For cost parameters $\alpha<1$ the cost of acquiring information is relatively
low. When the initial payoff difference is relatively low, the attention
elasticity is elastic, that is the relative rate of change in the posterior
odds as we increase payoff differences is higher than that under the Shannon
case. However, if the initial payoff difference is above a certain threshold
the attention elasticity drops below one. We can express the threshold
directly in terms of the dual variable, $\lambda^{*}_{\omega}$, which is
increasing in the payoff difference $h$.

For cost parameters $\alpha>1$ the cost of acquiring information is relatively
high and the elasticity patterns are flipped in an analogous manner. When the
initial payoff difference is low, the attention elasticity is inelastic, that
is the relative rate of change in the posterior odds is less than that under
the Shannon model. For initial payoff difference above a certain threshold the
attention elasticity is greater than 1.

Figure 1: Attentional elasticity, $\epsilon^{a_{\omega},b}_{\omega,\theta}$

Figure 1 illustrates the attention elasticity for various levels of initial
payoff differences, $h/\kappa$, for different values of the cost parameter.
The left-hand-side panel shows attention elasticities when acquiring
information is “easy”—low values of $\alpha$. The right-hand-side panel shows
attention elasticities acquiring information is “hard”.

We require the two actions to be chosen with strictly positive conditional
probability to rule out the scenario under $\alpha>1$ in which the payoff
difference is large enough that the optimal attention strategy is the fully
informative statistical experiment, under which posterior ratios are not well-
defined. Note that this can never happen for $\alpha<1$ as we discuss in
detail in section 6.

###  4.2 Invariant Likelihood Ratio Property

As implied by the first-order conditions for interior points (4.5) the
invariant likelihood ratio property (Caplin and Dean, 2013) does not hold
under $\alpha$-mutual information costs—except for $\alpha\to 1$ as we fall
back to the Shannon case.

For signals associated with actions $a$ and $b$ that are chosen with positive
probability conditional on a given state under the optimal attention strategy
we have,

$\frac{P^{*}(a\mid\omega)}{P^{*}(b\mid\omega)}=\left(\frac{u(a,\omega)/\kappa-\lambda^{*}_{\omega}/\mu(\omega)}{u(b,\omega)/\kappa-\lambda^{*}_{\omega}/\mu(\omega)}\right)^{\frac{1}{\alpha-1}}\frac{q^{*}(a)}{q^{*}(b)}.$
(4.14)

We see that the likelihood ratios are twisted relative to the barycenter
ratios by a quantity dependent on the payoff parameters (and the optimal dual
variables which also depend on the payoff parameters).

###  4.3 Locally Invariant Posteriors

The locally invariant posteriors condition of Caplin et al. (2019b) states
that local changes in the prior—conditional on the set of supported actions
not changing—do not lead to changes in the optimal posteriors. Caplin et al.
(2019b) show that locally invariant posteriors is a characteristic feature of
uniformly posterior separable cost functions.

The $\alpha$-mutual information is not posterior separable nor uniformly
posterior separable and hence the locally invariant posteriors property does
not hold. However, as noted in equation (3.10) with a monotone increasing
transformation we can transform the $\alpha$-mutual information to a posterior
separable cost. If we consider the problem of minimizing information costs
under the constraint of obtaining a given level of expected utility this
transformation should be innocuous (analogous to the expenditure minimization
problem).

##  5 Invariant Cost Functions

Invariant information costs have important behavioral implications in various
economic contexts. In recent work Caplin et al. (2019b) show that invariance
is the defining feature of the Shannon model within the class of uniformly
posterior separable cost functions. Hébert and La’O (2020) and Angeletos and
Sastry (2021) show that invariant cost functions play a crucial role in the
efficiency of equilibria in large games with incomplete information.

While much of the literature has so far focused on generalizations of the
Shannon model within posterior separable classes, the above examples
demonstrate that invariance can be a desirable feature in many economic
contexts. In this section we show that the $\alpha$-mutual information is an
invariant class, which is not posterior separable. This provides a novel
direction for potential further generalization of the Shannon model.

Invariance imposes two conditions on information cost functions. The first
condition—often referred to as information monotonicity—requires that
coarsening the information acquired within an event should always weakly
reduce information costs. The second condition requires a form of independence
from the prior. The least costly way to learn about an event should be
invariant to priors that only differ in their relative likelihoods over the
event but not in the overall probability mass they place on the event. We
follow Caplin et al. (2019b) and restate the conditions in terms of
statistical experiments.

Take a subset of the state space $\bar{\Omega}\subset\Omega$ and a partition
of $\bar{\Omega}$, $\left\\{\bar{\Omega}_{z}\right\\}_{z\in Z}$. Take any
prior $\mu\in\text{int}\Delta(\bar{\Omega})$. Next, given experiment $P$ we
construct another experiment $P^{\prime}$ that carries the same information
about the events in the partition as $P$ but no information within any event
of the partition.

Specifically, given $P$ and $\mu$ define $P^{\prime}$ as follows:

  1. (i)

The marginal signal distributions coincide:

$\text{supp}\mu P^{\prime}_{\mathcal{S}}=\text{supp}\mu
P_{\mathcal{S}}\quad\text{and}\quad\mu P^{\prime}_{\mathcal{S}}(s)=\mu
P_{\mathcal{S}}(s)\quad\forall s\in\text{supp}\mu P_{\mathcal{S}}.$ (5.1)

  2. (ii)

For each $z\in Z$,

     * •

The corresponding posteriors induced by $P$ and $P^{\prime}$ put the same mass
on events in the partition:

$\sum_{\omega\in\bar{\Omega}_{z}}\frac{P(s\mid\omega)\mu(\omega)}{\mu
P_{\mathcal{S}}(s)}=\sum_{\omega\in\bar{\Omega}_{z}}\frac{P^{\prime}(s\mid\omega)\mu(\omega)}{\mu
P^{\prime}_{\mathcal{S}}(s)}.$ (5.2)

     * •

Conditional on a given event in the partition the posterior induced by
$P^{\prime}$ coincides with the prior. This is equivalent to the likelihood
being constant over the event:

$P^{\prime}(s\mid\omega)=c_{z}\quad\forall\omega\in\bar{\Omega}_{z}.$ (5.3)

Note that taken together equations (5.1) and (5.3), the condition of (5.2)
implies that the constant likelihood has to satisfy:

$c_{z}=\frac{\sum_{\omega\in\bar{\Omega}_{z}}P(s\mid\omega)\mu(\omega)}{\sum_{\omega\in\bar{\Omega}_{z}}\mu(\omega)}.$

It is immediate to check that $\sum_{s}P^{\prime}(s\mid\omega)=1$ for all
$\omega\in\bar{\Omega}$ and all entries are weakly positive hence $P^{\prime}$
is a valid statistical experiment.

Next take any prior $\bar{\mu}$ such that the priors put equal weight on
events in the partition, i.e.
$\bar{\mu}\left(\bar{\Omega}_{z}\right)=\mu\left(\bar{\Omega}_{z}\right)$ for
all $z\in Z$. Define a statistical experiment $\bar{P}$ in an analogous way as
we defined $P^{\prime}$ but under $\bar{\mu}$. Specifically, given $P$, $\mu$,
and $\bar{\mu}$ define $\bar{P}$ as

  1. (i)

The marginal signal distributions coincide:

$\text{supp}\bar{\mu}\bar{P}_{\mathcal{S}}=\text{supp}\mu
P_{\mathcal{S}}\quad\text{and}\quad\bar{\mu}\bar{P}_{\mathcal{S}}(s)=\mu
P_{\mathcal{S}}(s)\quad\forall s\in\text{supp}\mu P_{\mathcal{S}}.$ (5.4)

  2. (ii)

For each $z\in Z$,

     * •

The corresponding posteriors induced by $(P,\mu)$ and $(\bar{P},\bar{\mu})$
put the same mass on events in the partition:

$\sum_{\omega\in\bar{\Omega}_{z}}\frac{P(s\mid\omega)\mu(\omega)}{\mu
P_{\mathcal{S}}(s)}=\sum_{\omega\in\bar{\Omega}_{z}}\frac{\bar{P}(s\mid\omega)\bar{\mu}(\omega)}{\bar{\mu}\bar{P}_{\mathcal{S}}(s)}$
(5.5)

     * •

Conditional on a given event in the partition the posterior induced by
$(\bar{P},\bar{\mu})$ coincides with the prior. This is equivalent to the
likelihood being constant over the event:

$\bar{P}(s\mid\omega)=\bar{c}_{z}\quad\forall\omega\in\bar{\Omega}_{z}$ (5.6)

Similarly, taken together equations (5.4) and (5.6), the condition of (5.5)
implies that the constant likelihood has to satisfy:

$\bar{c}_{z}=\frac{\sum_{\omega\in\bar{\Omega}_{z}}P(s\mid\omega)\mu(\omega)}{\sum_{\omega\in\bar{\Omega}_{z}}\bar{\mu}(\omega)}.$

With the above transformations in hand we can define invariant cost functions
as in Caplin et al. (2019b).

######  Definition 4 (Invariance).

A cost function $K$ is invariant if for all finite subsets
$\bar{\Omega}\subset\Omega$, all partitions of $\bar{\Omega}$, and all pairs
$\mu$ and $\bar{\mu}$ that put equal mass on each event of the partition we
have,

$K\left(P,\mu\right)\geq K\left(P^{\prime},\mu\right)$

and

$K\left(P^{\prime},\mu\right)=K\left(\bar{P},\bar{\mu}\right).$

Our next proposition states that the $\alpha$-mutual information is an
invariant cost function.

######  Proposition 3.

With the convention of $\alpha=1$ denoting the Shannon mutual information, the
$\alpha$-mutual information cost function is invariant for any $\alpha>0$.

Proof. Appendix C.

###  5.1 Invariance Under Compression

There are important behavioral implications of the cost function being
invariant. If there are events for which all actions that have strictly
positive probability of being chosen under the optimal policy also have
identical payoffs for each state within the events, then conditional action
probabilities are constant within such events under the optimal attention
strategy. Formally, suppose there is an event $E$ such that,

$u(a,\omega_{i})=u(a,\omega_{j})\qquad\text{for all }\omega_{i},\omega_{j}\in
E,a\in\text{supp}\mu P^{*}_{\mathcal{S}}.$ (5.7)

Clearly, any two attention strategies, $P$ and $P^{\prime}$, that satisfy

$\displaystyle\sum_{\omega\in E}\mu(\omega)P(a\mid\omega)$
$\displaystyle=\sum_{\omega\in
E}\mu(\omega)P^{\prime}(a\mid\omega)\qquad\forall a\in\text{supp}\mu
P^{*}_{\mathcal{S}},$ (5.8) $\displaystyle P(a\mid\omega)$
$\displaystyle=P^{\prime}(a\mid\omega)\qquad\forall\omega\notin E,$ (5.9)

yield the same expected utility. By invariance of the $\alpha$-mutual
information, among all such attention strategies the one that has minimal cost
is such that conditional signal/action probabilities are constant over $E$,

$P(a\mid\omega)=c_{a}\qquad\forall\omega\in E,a\in\text{supp}\mu
P^{*}_{\mathcal{S}}.$ (5.10)

This in turn implies that posteriors over $E$ are proportional to the prior,

$\frac{\gamma^{a}\left(\omega_{i}\right)}{\mu\left(\omega_{i}\right)}=\frac{\gamma^{a}\left(\omega_{j}\right)}{\mu\left(\omega_{j}\right)}\qquad\text{for
all }\omega_{i},\omega_{j}\in E,a\in\text{supp}\mu P^{*}_{\mathcal{S}}.$
(5.11)

Invariance under compression states that if two decision problems have the
same basic form (Caplin et al., 2019b), in which we collapse all events like
$E$ above, then corresponding attention strategies must coincide. Invariance
under compression is the defining feature of the Shannon model (Caplin et al.,
2019b) within the class of uniformly posterior separable cost functions. The
$\alpha$-mutual information is not posterior separable yet it satisfies
invariance under compression.

######  Proposition 4.

The solution to the $\alpha$-RI problem satisfies invariance under compression
for any cost parameter $\alpha>0$.

Proof. Immediate from the fact that $\alpha$-mutual information class is
invariant for all $\alpha>0$.

##  6 The Utility Cost Curve

In Caplin, Csaba, Leahy, and Nov (2020) we introduce the utility cost curve
(UCC) to capture the attention costs of producing different levels of expected
utility in a given decision problem. These curves give a succinct visual
demonstration of the difficulty of learning in $\alpha$-RI models in relation
to the standard Shannon model.

Fixing the payoff structure and the prior, denote the expected utility of
decisions made based on experiment $P$ as,

$U(P):=\sum_{\omega,a}P(a\mid\omega)\mu(\omega)u(a,\omega).$ (6.1)

The minimum and maximum attainable expected utility levels correspond to the
non-informative and perfectly informative experiments, respectively. They are
given by,

$\displaystyle\bar{U}^{\min}$
$\displaystyle:=\max_{a\in\mathcal{A}}\sum_{\omega}\mu(\omega)u(a,\omega);$
(6.2) $\displaystyle\bar{U}^{\max}$
$\displaystyle:=\sum_{\omega}\mu(\omega)\max_{a\in\mathcal{A}}u(a,\omega).$
(6.3)

For any potentially attainable expected utility level
$v\in[\bar{U}^{\min},\bar{U}^{\max}]$ the UCC in $\alpha$-RI problems is
defined as,

$\bar{K}_{\alpha}(v):=\min_{P\in\mathcal{P}:U(P)\geq v}I_{\alpha}(P,\mu).$
(6.4)

Consider the binary symmetric decision problem with uniform prior and payoffs
as

| $a$ | $b$
---|---|---
$\omega_{1}$ | 1 | 0
$\omega_{2}$ | 0 | 1

,

Table 1: Binary symmetric problem

which implies that the ex ante expected utility must be between $1/2$ (no
information) and $1$ (perfectly informative experiment).

The utility cost curve of producing expected utility $v\in[1/2,1]$ under
$\alpha$-mutual information costs is

$\bar{K}_{\alpha}(v)=\frac{\alpha}{\alpha-1}\log\left(2\left(\frac{1}{2}\left(v^{\alpha}+(1-v)^{\alpha}\right)\right)^{\frac{1}{\alpha}}\right),$
(6.5)

which we can derive from the symmetry of the optimal attention strategy.

Figure 2: The utility cost curve under different values of $\alpha$

Figure 2 depicts the utility cost curve for different values of $\alpha$. For
each level of attainable expected utility $v\in[1/2,1]$ the utility cost is
increasing in $\alpha$. In general, the cost of the perfectly informative
experiment is equal to the Rényi entropy of order $1/\alpha$ of the prior,
which for a uniform prior of $n$ dimensions is equal to $\log n$ independent
of $\alpha$. In the binary symmetric example this implies that the perfectly
informative experiments uniformly cost $\log 2$ which we see on figure 2 at
$v=1$.

Figure 2 shows that as we approach the maximum attentional output, the utility
cost curves get steeper for the Shannon case and for $\alpha<1$. In fact, it
is well known that in the Shannon model the DM never chooses an attention
strategy that resolves all uncertainty.

Formally, the derivative of the utility cost curve at $v\in\left[1/2,1\right)$
is,

$\bar{K}_{\alpha}^{\prime}(v)=\frac{\alpha}{\alpha-1}\frac{v^{(\alpha-1)}-(1-v)^{(\alpha-1)}}{v^{\alpha}+(1-v)^{\alpha}}.$
(6.6)

For any $\alpha<1$ we have

$\lim_{v\to 1}\bar{K}_{\alpha}^{\prime}(v)=\infty.$

Furthermore, noting that as $\alpha\to 1$ we obtain the Shannon case for
which,

$\bar{K}_{1}(v)=v\log v+(1-v)\log(1-v)+\log
2,\quad\text{and}\quad\bar{K}_{1}^{\prime}(v)=\log\frac{v}{1-v},$

we also have $\lim_{v\to 1}\bar{K}_{1}^{\prime}(v)=\infty$. On the other hand,
for $\alpha>1$ we have,

$\lim_{v\to 1}\bar{K}_{\alpha}^{\prime}(v)=\frac{\alpha}{\alpha-1}.$

The above features of the cost function imply that in the binary example under
$\alpha$-mutual information costs with parameter values $\alpha\leq 1$ a
rationally inattentive DM never learns the realization or non-realization of a
state with certainty under the optimal attention strategy. This does not
necessarily hold for cost parameters $\alpha>1$. The implications for learning
payoff-relevant events with certainty hold in general.

######  Proposition 5.

For any $\alpha$-RI problem with bounded payoffs we have that:

  * •

for $\alpha\leq 1$ there exists no non-trivial event
$E\subsetneq\text{supp}(\mu)$ that the DM optimally learns with certainty;

  * •

for $\alpha>1$, and if all payoffs are distinct, there exists a positive
scalar $\pi>0$ such that multiplying all payoffs by $\pi$ the DM optimally
learns all events of the form $E=\\{\omega:\exists a\text{ s.t.
}u(a,\omega)>u(b,\omega)\ \forall b\neq a\\}$ with certainty.

Proof. Appendix D.

Proposition 5 states that under Shannon- or $\alpha$-mutual information costs
with parameter $\alpha<1$, a rationally inattentive agent never learns to the
point where some of her optimal posteriors put zero weight on states that are
in the support of the prior. On the other hand, for $\alpha>1$ we can always
increase the stakes by multiplying the payoffs by a positive scalar such that
in the resulting decision problem the DM’s optimal posteriors only put weight
on events under which a single action is optimal, i.e. for all
$a\in\text{supp}(\mu P^{*}_{\mathcal{S}})$ we have $\gamma^{a}(E_{a})=1$ where
$E_{a}:=\\{\omega:u(a,\omega)>u(b,\omega)\ \forall b\neq a\\}$.

###  6.1 Incentive-based Psychometric Curve

From the utility cost curve we can directly obtain the incentive-based
psychometric curve (IPC) introduced in Caplin et al. (2020). This captures the
attentional output as we scale the incentives linearly. Formally, fixing the
prior and the payoff structure, denote the optimal information structure in
the modified $\alpha$-RI problem where all the payoffs are scaled by a
positive constant $\pi$ as $P^{*}_{\pi}$. We capture the changes in the
attentional output as we vary the scaling factor $\pi$ by obtaining the
expected utility resulting from $P^{*}_{\pi}$ in the original problem. The
incentive-based psychometric curve captures the increase in expected utility
only due to the changes in the attention strategy but not the increase in
payoffs.

$\bar{U}(\pi):=U\left(P^{*}_{\pi}\right)=\sum_{a,\omega}P^{*}_{\pi}(a\mid\omega)\mu(\omega)u(a,\omega).$
(6.7)

Figure 3: The IPC for different values of $\alpha$

As seen in figure 3 scaling the payoffs the attentional output of the DM is
weakly increasing. For $\alpha\leq 1$ under no circumstances does the DM
produce the maximum attentional output, $\bar{U}^{\max}=1$. For $\alpha>1$,
there is always a threshold such that scaling the payoffs above the threshold,
the DM optimally chooses the perfectly informative experiment and the
attentional output attains its maximum in line with the general claims of
proposition 5.

##  7 Optimal Consideration Sets

Proposition 5 characterizes optimal posterior beliefs that put zero
probability mass on certain events even though the events are ex ante deemed
possible under the prior. The present section characterizes the conditions
under which certain actions are never chosen under the optimal attention
strategy even though they could be the optimal action under the realization of
some states. In this sense, RI models provide a framework to study the
formation of optimal consideration sets which arise endogenously dependent on
the choice problem. In the Shannon model the formation of optimal
consideration sets is analyzed by Caplin et al. (2019a).

Note that in the $\alpha$-RI model the unconditional choice probability is
strictly positive if and only if the corresponding barycenter is strictly
positive,

$\mu P^{*}_{\mathcal{S}}(a)>0\quad\iff\quad q^{*}(a)>0.$ (7.1)

Correspondingly, whether or not an action is in the optimal consideration set
can be expressed in terms of the barycenter of the experiment. Denote the
function $x\mapsto\max\\{x,0\\}=:\left[x\right]^{+}$ for any $x\in\mathbb{R}$.

######  Proposition 6.

Under the optimal attention strategy,

$\displaystyle\frac{\sum_{\omega}\mu(\omega)\left(\left[\frac{\alpha-1}{\alpha}\left(\frac{u(a,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}}{\sum_{\tilde{s}}q^{*}(\tilde{s})\sum_{\tilde{\omega}}\mu(\tilde{\omega})\left(\left[\frac{\alpha-1}{\alpha}\left(\frac{u(\tilde{s},\tilde{\omega})}{\kappa}-\frac{\lambda^{*}_{\tilde{\omega}}}{\mu(\tilde{\omega})}\right)\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}}\leq
1\qquad\text{for all }a\in\mathcal{A},$ (7.2)

with equality for actions that are chosen with strictly positive probability,
that is for $a$ such that $\mu P^{*}_{\mathcal{S}}(a)>0$. Note that $q^{*}$ is
the barycenter of the optimal experiment $P^{*}$, while the dual variables
$\lambda^{*}$ are obtained from the primal feasibility constraints on $P^{*}$.

Proof. Appendix E.

As we discuss in appendix E in detail the conditions characterizing
consideration sets can be obtained from the KKT conditions. The function
thresholding at zero, $[\cdot]^{+}$, appears in the expression to handle
corner-solutions under $\alpha>1$ discussed in the previous section. For cost
parameters $\alpha\leq 1$ strictly positive unconditional choice probabilities
imply strictly positive conditional choice probabilities by the first claim of
proposition 5.

$\mu P^{*}_{\mathcal{S}}(a)>0\quad\implies\quad
P^{*}(a\mid\omega)>0\quad\forall\omega\in\text{supp}\mu\qquad\left(\text{for
}\alpha\leq 1\right).$ (7.3)

However, this is not necessarily the case for cost parameters $\alpha>1$ as
stated by the second claim of proposition 5, which states that the DM might
learn some payoff-relevant events with certainty, necessarily contradicting
the above condition (7.3).

As we show in section 8 the conditions characterizing the consideration sets
are also tightly related to the updating rules in the alternating minimization
algorithm that we use to obtain the solution to the $\alpha$-RI problem.

###  7.1 Example

To illustrate the formation of consideration sets consider again the binary
decision problem with symmetric payoffs as in section 6 under varying the
prior probability of state $\omega_{1}$. As we increase $\mu(\omega_{1})$
starting from the uniform prior, action $a$ becomes ex ante optimal with
expected utility equal to $\mu(\omega_{1})$. The rationally inattentive DM
chooses $a$ with higher unconditional probability, while for some threshold
value of $\mu(\omega_{1})$ the DM exclusively chooses action $a$, that is
action $b$ is not in the consideration set.

(a) Probability of choosing $a$, $\mu P^{*}_{\mathcal{S}}(a)$

(b) Expected utility net of information costs

Figure 4: Optimal formation of consideration sets

Panel 4(a) illustrates the unconditional probability of choosing action $a$ as
the prior puts more weight on the state in which it is the better action. We
see that the higher the value of $\alpha$ the lower the threshold value of the
prior $\mu(\omega_{1})$ above which the DM exclusively chooses $a$. That is,
the costlier the attention the earlier the DM drops actions from the
consideration set.

On panel 4(b) we see the evolution of the net expected utility the DM achieves
relative to the no information benchmark. With only two actions, once the only
action that is considered is $a$ there is no information acquisition as the
corresponding lines join with the dotted line on the diagonal.

The formation of optimal consideration sets in $\alpha$-RI problems is a
natural generalization of the Shannon case where besides beliefs and payoffs
the curvature of the information cost function controlled by $\alpha$
determines which actions are chosen under the optimal attention strategy.

##  8 The Modified Blahut-Arimoto Algorithm

In practice we obtain the solution to the $\alpha$-RI problem through an
alternating minimization procedure analogous to the Blahut-Arimoto algorithm
(Blahut, 1972; Arimoto, 1972) which is used to obtain solutions for the
rational inattention problem under mutual information costs. In the present
section we discuss details of implementation and provide proof of convergence.

Anticipating the alternating minimization approach we write the problem using
the formulation for the cost function as in equation (3.6).

$\displaystyle\min_{P\in\mathcal{P}}\ \min_{q\in\Delta(\mathcal{S})}\kappa
D_{\alpha}(\mu P\|\mu\otimes
q)-\sum_{\omega,s}\mu(\omega)P(s\mid\omega)u(s,w)$ (8.1)

The Rényi-divergence is jointly quasi-convex in its two arguments666For
reference see for instance (Van Erven and Harremoës, 2014, Theorem 13). and
the constraint sets are both convex so the KKT conditions are necessary and
sufficient for obtaining the solution.

Letting $q^{0}$ be the uniform distribution over all potential signals
identified with the action set $\mathcal{S}=\mathcal{A}$ we obtain the
sequence $\left\\{(P^{t},q^{t})\right\\}_{t\geq 1}$ through alternating
minimization,

$\displaystyle P^{t}\in\arg\min_{P\in\mathcal{P}}\kappa D_{\alpha}(\mu
P\|\mu\otimes q^{t-1})-\sum_{\omega,s}\mu(\omega)P(s\mid\omega)u(s,w);$ (8.2)
$\displaystyle q^{t}\in\arg\min_{q\in\Delta(\mathcal{S})}\kappa D_{\alpha}(\mu
P^{t}\|\mu\otimes q)-\sum_{\omega,s}\mu(\omega)P^{t}(s\mid\omega)u(s,w).$
(8.3)

The above alternating-minimization method is known as the block coordinate
descent method or Gauss-Seidel method. Its convergence properties are well
studied and we can verify the conditions of the general results in Tseng
(2001) to prove convergence.

######  Proposition 7.

The sequence $\\{(P^{t},q^{t})\\}_{t\geq 1}$ generated by the alternating
minimization algorithm is converging to $(P^{*},q^{*})$, the critical point of
the $\alpha$-RI problem.

Proof. Appendix F.

###  8.1 Practical Implementation

Next, we consider details of the modified Blahut-Arimoto algorithm. The
algorithm provides and efficient way for obtaining solutions to $\alpha$-RI
problems, making their use appealing in applications.777An implementation of
the algorithm in Python is available at
https://github.com/danielcsaba/alpha_RI.

####  8.1.1 Minimization with respect to $q$

For a given information structure $P^{t}$ the barycenter of the information
structure minimizing the Rényi divergence of order $\alpha$ is given by,

$q^{t}(s)=\frac{\left(\sum_{\omega}\mu(\omega)P^{t}(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}}{\sum_{\tilde{s}}\left(\sum_{\omega}\mu(\omega)P^{t}(\tilde{s}\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}}.$
(8.4)

For $q^{t}(s)>0$ the KKT conditions for the $q$-minimization step imply the
above expression. $q^{t}(s)=0$ only if $P^{t}(s\mid\omega)=0,\
\forall\omega\in\Omega$, hence the expression is valid for cornersolutions as
well.

Obtaining the barycenter under the Rényi divergence is computationally as
efficient as under the KL divergence.

####  8.1.2 Minimization with respect to $P$

Fixing $q^{t-1}$ the KKT conditions characterizing the minimizer $P^{t}$ are
as follows—$\lambda^{t}$ and $\delta^{t}$ are the dual variables corresponding
to the linear and positivity constraints on the constraint set.

$\displaystyle\frac{\alpha}{\alpha-1}\Bigg{(}\underbrace{\sum_{\omega,s}\mu(\omega)P^{t}(s\mid\omega)^{\alpha}q^{t-1}(s)^{(1-\alpha)}}_{\text{$:=C^{t}$}}\Bigg{)}^{-1}\mu(\omega)\left(\frac{P^{t}(s\mid\omega)}{q^{t-1}(s)}\right)^{(\alpha-1)}-$
$\displaystyle\frac{\mu(\omega)u(s,\omega)}{\kappa}+\lambda^{t}_{\omega}-\delta^{t}_{\omega,s}=0\qquad\forall\omega,s,$
(8.5) $\displaystyle\sum_{s}P^{t}(s\mid\omega)=1\qquad\forall\omega,$ (8.6)
$\displaystyle P^{t}(s\mid\omega)\geq 0,\quad\delta^{t}_{\omega,s}\geq 0,\quad
P^{t}(s\mid\omega)\delta^{t}_{\omega,s}=0\qquad\forall\omega,s$ (8.7)

Note that if $q^{t-1}(s)=0$ then we necessarily have $P^{t}(s\mid\omega)=0$
for all $\omega\in\Omega$.

Suppose that $q^{t-1}(s)>0$.

  * •

If $\alpha<1$ then necessarily $P^{t}(s\mid\omega)>0$ for all
$\omega\in\Omega$ in line with the first claim of proposition 5. By the
complementary slackness condition $\delta^{t}_{\omega,s}=0$.

  * •

If $\alpha>1$ and $P^{t}(s\mid\omega)=0$ then
$\lambda^{t}_{\omega}-\frac{\mu(\omega)u(s,\omega)}{\kappa}=\delta^{t}_{\omega,s}>0$.
Otherwise $\delta^{t}_{\omega,s}=0$ by complementary slackness.

Taking the above arguments together we have,

$P^{t}(s\mid\omega)=\left(C^{t}\left[\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{t}_{\omega}}{\mu(\omega)}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{1}{\alpha-1}}q^{t-1}(s).$
(8.8)

Thresholding the inner term at zero is necessary to take care of the case
under $\alpha>1$ in which $q^{t-1}(s)>0$ and $P^{t}(s\mid\omega)=0$. This can
only happen if the term
$\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{t}_{\omega}}{\mu(\omega)}\right)\frac{\alpha-1}{\alpha}<0$,
so in this case the thresholding function $[\cdot]^{+}$ returns the correct
solution for $P^{t}(s\mid\omega)$.

While the constant term $C^{t}$ is a function of $P^{t}$, it is homogeneous of
degree $\alpha$ in $P^{t}$ if we define $\bar{C}^{t}$ by the mapping
$P\mapsto\sum_{\omega,s}\mu(\omega)P(s\mid\omega)^{\alpha}q^{t-1}(s)^{(1-\alpha)}$.
We can use this fact to factor it out. Define $P_{-C}^{t}$ which is
proportional to $P^{t}$ as,

$P_{-C^{t}}^{t}(s\mid\omega):=\left(\left[\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{t}_{\omega}}{\mu(\omega)}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{1}{\alpha-1}}q^{t-1}(s).$
(8.9)

Then by homogeneity of degree $\alpha$ of $\bar{C}^{t}$ we have

$\displaystyle C^{t}$
$\displaystyle=\bar{C}^{t}\left(P^{t}\right)=\bar{C}^{t}\left(C^{t}P^{t}_{-C^{t}}\right)={C^{t}}^{\alpha}\bar{C}^{t}\left(P^{t}_{-C^{t}}\right),$
$\displaystyle C^{t}$
$\displaystyle=\left(\bar{C}^{t}\left(P^{t}_{-C^{t}}\right)\right)^{\frac{1}{1-\alpha}}.$
(8.10)

Finally,

$P^{t}(s\mid\omega)=\frac{\left(\left[\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{t}_{\omega}}{\mu(\omega)}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{1}{\alpha-1}}}{\sum_{\tilde{s},\tilde{\omega}}\mu(\tilde{\omega})\left(\left[\left(\frac{u(\tilde{s},\tilde{\omega})}{\kappa}-\frac{\lambda^{t}_{\tilde{\omega}}}{\mu(\tilde{\omega})}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}q^{t-1}(\tilde{s})}\
q^{t-1}(s).$ (8.11)

It is instructive to compare this expression with proposition 6 characterizing
consideration sets. Applying the formula for $q^{t}$ as in (8.4),

$q^{t}(s)=\frac{\left(\sum_{\omega}\mu(\omega)\left(\left[\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{t}_{\omega}}{\mu(\omega)}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}\right)^{\frac{1}{\alpha}}}{\sum_{\tilde{s}}\left(\sum_{\tilde{\omega}}\mu(\tilde{\omega})\left(\left[\left(\frac{u(\tilde{s},\tilde{\omega})}{\kappa}-\frac{\lambda^{t}_{\tilde{\omega}}}{\mu(\tilde{\omega})}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}\right)^{\frac{1}{\alpha}}q^{t-1}(\tilde{s})}\
q^{t-1}(s).$ (8.12)

We see that either the ratio is one in which case $q^{t}(s)=q^{t-1}(s)$ or it
is less than one in which case $q^{t}(s)$ converges to zero and the action
corresponding to $s$ is not in the consideration set.

####  8.1.3 Obtaining Dual Variables

Equation (8.11) describes the update step for the information structure given
$q^{t-1}$. The expression depends on the dual variables $\lambda^{t}_{\omega}$
corresponding to the primal feasibility constraints,
$\sum_{s}P^{t}(s\mid\omega)=1,\quad\forall\omega$. Using these conditions we
can obtain dual variables $\lambda^{t}$ through root-finding. To make the
dependence on the dual variables explicit write $P^{t}_{\lambda}$ and let the
primal feasibility conditions be denoted
$R^{t}_{\omega}(\lambda):=\sum_{s}P^{t}_{\lambda}(s\mid\omega)-1$ where the
information structure is defined as in (8.11) for a candidate $\lambda$. Since
we have a closed form expression for the Jacobian we can efficiently apply
Newton’s method.

$\frac{\partial
R^{t}_{\omega_{i}}}{\partial\lambda_{\omega_{k}}}=\begin{cases}\left(\sum_{s}P^{t}_{\lambda}(s\mid\omega_{i})\right)\left(\sum_{s}P^{t}_{\lambda}(s\mid\omega_{k})\right)\qquad&\text{for}\quad
i\neq k,\\\\[15.00002pt]
\left(\sum_{s}P^{t}_{\lambda}(s\mid\omega_{k})\right)^{2}-\frac{\sum_{s}\left(\left[\left(\frac{u(s,\omega_{k})}{\kappa}-\frac{\lambda_{\omega_{k}}}{\mu(\omega_{k})}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{2-\alpha}{\alpha-1}}\frac{q^{t-1}(s)}{\alpha\mu(\omega_{k})}}{\sum_{\omega,s}\mu(\omega)\left(\left[\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda_{\omega}}{\mu(\omega)}\right)\frac{\alpha-1}{\alpha}\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}q^{t-1}(s)}\qquad&\text{for}\quad
i=k.\end{cases}$ (8.13)

Furthermore, we have natural bounds on the optimal dual variables depending on
the value of $\alpha$ that we enforce in our custom root-finding algorithm.
Without enforcing these bounds out-of-the-box root-finding algorithms will
typically overshoot and lead to invalid or suboptimal attention strategies.
Specifically, in order to satisfy the KKT conditions we need,

  * •

for $\alpha<1$

$\lambda^{t}_{\omega}>\max_{a\in\mathcal{A}}\frac{\mu(\omega)u(a,\omega)}{\kappa}\qquad\forall\omega\in\Omega;$
(8.14)

  * •

for $\alpha>1$

$\lambda^{t}_{\omega}<\max_{a\in\mathcal{A}}\frac{\mu(\omega)u(a,\omega)}{\kappa}\qquad\forall\omega\in\Omega.$
(8.15)

Taken together the modified Blahut-Arimoto algorithm takes the following form:

Input : $(\Omega,\mathcal{A},u,\mu,\kappa,\alpha)$ Result:
$P^{*},q^{*},\lambda^{*}$ 1 $q^{0}\leftarrow
ones(|\mathcal{A}|)/|\mathcal{A}|$;  2 while  _$distance >tol$_ do 3
$\lambda^{t}\leftarrow$ via root-finding on primal feasibility given
$q^{t-1}$,  4 obeying bounds (8.14) and (8.15);  5 $P^{t}\leftarrow$ via
equation (8.11) given $q^{t-1}$ and $\lambda^{t}$;  6 $q^{t}\leftarrow$ via
equation (8.4) given $P^{t}$;  7 $distance\leftarrow\|q^{t}-q^{t-1}\|$ 8 end
while  Algorithm 1 Modified Blahut-Arimoto algorithm

##  9 Conclusion

We consider a generalization of the rational inattention model of Sims (1998,
2003) by measuring costs of information via the information radius of
statistical experiments. We show that such costs control elasticities of
attention, measuring the sensitivity of attention strategies with changes in
incentives. In contrast, the Shannon model restricts attention elasticities to
be constant irrespective of the incentives. The resulting class of information
costs is invariant but not posterior separable. This provides a novel
direction for generalizations of the Shannon model. We explore further
behavioral implications relative to the Shannon model and provide an efficient
alternating minimization method for obtaining optimal attention strategies.

## References

  * Angeletos and Sastry (2021) George-Marios Angeletos and Karthik Sastry.  Inattentive economies.  Working paper, 2021.
  * Arimoto (1972) Suguru Arimoto.  An algorithm for computing the capacity of arbitrary discrete memoryless channels.  _IEEE Transactions on Information Theory_ , 18(1):14–20, 1972.
  * Arrow et al. (1961) Kenneth J. Arrow, Hollis B. Chenery, Bagicha S. Minhas, and Robert M. Solow.  Capital-Labor Substitution and Economic Efficiency.  _The Review of Economics and Statistics_ , pages 225–250, 1961.
  * Blackwell (1953) David Blackwell.  Equivalent comparisons of experiments.  _The Annals of Mathematical Statistics_ , pages 265–272, 1953.
  * Blahut (1972) Richard Blahut.  Computation of channel capacity and rate-distortion functions.  _IEEE transactions on Information Theory_ , 18(4):460–473, 1972.
  * Bloedel and Zhong (2021) Alexander W. Bloedel and Weijie Zhong.  The cost of optimally-acquired information.  Working paper, 2021.
  * Caplin and Dean (2013) Andrew Caplin and Mark Dean.  Behavioral Implications of Rational Inattention with Shannon Entropy.  Technical report, National Bureau of Economic Research, 2013.
  * Caplin et al. (2019a) Andrew Caplin, Mark Dean, and John Leahy.  Rational Inattention, Optimal Consideration Sets, and Stochastic Choice.  _The Review of Economic Studies_ , 86(3):1061–1094, 2019a.
  * Caplin et al. (2019b) Andrew Caplin, Mark Dean, and John Leahy.  Rationally Inattentive Behavior: Characterizing and Generalizing Shannon Entropy.  Technical report, National Bureau of Economic Research, 2019b.
  * Caplin et al. (2020) Andrew Caplin, Dániel Csaba, John Leahy, and Oded Nov.  Rational Inattention, Competitive Supply, and Psychometrics.  _The Quarterly Journal of Economics_ , 135:1681–1724, August 2020.
  * Denti et al. (2020) Tommaso Denti, Massimo Marinacci, and Aldo Rustichini.  Experimental cost of information.  2020\.
  * Hébert and La’O (2020) Benjamin M. Hébert and Jennifer La’O.  Information acquisition, efficiency, and non-fundamental volatility.  Technical report, National Bureau of Economic Research, 2020.
  * Hébert and Woodford (2019) Benjamin M. Hébert and Michael Woodford.  Rational inattention when decisions take time.  Technical report, National Bureau of Economic Research, 2019.
  * Hébert and Woodford (2021) Benjamin M. Hébert and Michael Woodford.  Neighborhood-based information costs.  Technical report, National Bureau of Economic Research, 2021.
  * Ho and Verdú (2015) Siu-Wai Ho and Sergio Verdú.  Convexity/Concavity of Rényi Entropy and $\alpha$-mutual Information.  In _2015 IEEE International Symposium on Information Theory (ISIT)_ , pages 745–749, 2015.
  * Hoiles et al. (2020) William Hoiles, Vikram Krishnamurthy, and Kunal Pattanayak.  Rationally inattentive inverse reinforcement learning explains youtube commenting behavior.  _Journal of Machine Learning Research_ , 21(170):1–39, 2020.
  * Maćkowiak et al. (2020) Bartosz Maćkowiak, Filip Matějka, and Mirko Wiederholt.  Rational inattention: A review.  _CEPR Discussion Papers_ , (15408), 2020.
  * Matějka and McKay (2015) Filip Matějka and Alisdair McKay.  Rational inattention to discrete choices: A new foundation for the multinomial logit model.  _American Economic Review_ , 105(1):272–98, 2015\.
  * Morris and Strack (2017) Stephen Morris and Philipp Strack.  The Wald problem and the equivalence of sequential sampling and static information costs.  _Unpublished manuscript, June_ , 2017.
  * Pomatto et al. (2019) Luciano Pomatto, Philipp Strack, and Omer Tamuz.  The Cost of Information.  _arXiv preprint arXiv:1812.04211_ , 2019.
  * Rényi (1961) Alfréd Rényi.  On Measures of Entropy and Information.  In _Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics_ , 1961.
  * Sibson (1969) Robin Sibson.  Information Radius.  _Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete_ , 14(2):149–160, 1969.
  * Sims (1998) Christopher A. Sims.  Stickiness.  _Carnegie-Rochester Conference Series on Public Policy_ , 49:317 – 356, 1998.
  * Sims (2003) Christopher A. Sims.  Implications of Rational Inattention.  _Journal of Monetary Economics_ , 50(3):665–690, 2003.
  * Torgersen (1991) Erik Torgersen.  _Comparison of Statistical Experiments_.  Cambridge University Press, 1991.
  * Tseng (2001) Paul Tseng.  Convergence of a block coordinate descent method for nondifferentiable minimization.  _Journal of Optimization Theory and Applications_ , 109(3):475–494, 2001.
  * Van Erven and Harremoës (2014) Tim Van Erven and Peter Harremoës.  Rényi divergence and Kullback-Leibler divergence.  _IEEE Transactions on Information Theory_ , 60(7):3797–3820, 2014.
  * Verdú (2015) Sergio Verdú.  $\alpha$-Mutual Information.  In _2015 Information Theory and Applications Workshop (ITA)_ , pages 1–6. IEEE, 2015.

##  Appendix A Monotonicity in the Blackwell order

The following Bayesian criterion is a standard characterization of the
Blackwell order—for further reference see Blackwell (1953) and Torgersen
(1991).

######  Definition 5 (Blackwell order).

For two simple statistical experiments, $P,Q$, defined over the same finite
state space $\Omega$, experiment $P$ is more informative than experiment $Q$
if for all $\mu\in\text{int}\Delta(\Omega)$ we have,

$\sum_{s}\phi\left(\gamma^{s}\right)\mu
P_{\mathcal{S}}(s)\geq\sum_{s}\phi\left(\gamma^{s}\right)\mu
Q_{\mathcal{S}}(s),$ (A.1)

for all convex functions $\phi\colon\Delta\left(\Omega\right)\to\mathbb{R}$.

######  Proposition 8.

The $\alpha$-mutual information is monotone in the Blackwell order.

Proof. As introduced in equation (3.10), applying the monotone increasing
transformation
$t\mapsto\frac{1}{\alpha-1}\exp\left(t\frac{\alpha-1}{\alpha}\right)$ on the
$\alpha$-mutual information we get,

$\frac{1}{\alpha-1}\sum_{s}\mu
P_{\mathcal{S}}(s)\left(\sum_{\omega}\gamma^{s}(\omega)^{\alpha}\mu(\omega)^{1-\alpha}\right)^{\frac{1}{\alpha}}.$
(A.2)

For monotonicity in the Blackwell order we need to show that the map
$\phi_{\mu}\colon\Delta\left(\Omega\right)\to\mathbb{R}$ defined by,

$\phi_{\mu}(\gamma):=\frac{1}{\alpha-1}\left(\sum_{\omega}\gamma(\omega)^{\alpha}\mu(\omega)^{1-\alpha}\right)^{\frac{1}{\alpha}},$
(A.3)

for any $\mu\in\text{int}\Delta(\Omega)$ is convex.

We show that the Hessian of $\phi_{\mu}$ is positive semi-definite. Given an
enumeration of the finite states denote $\gamma_{i}=\gamma(\omega_{i})$. Given
$\alpha,\mu$ and $\gamma$, denote
$c:=\sum_{k}\gamma_{k}^{\alpha}\mu_{k}^{1-\alpha}$. The Hessian of
$\phi_{\mu}$ is,

$H_{ij}:=\frac{\partial^{2}\phi_{\mu}(\gamma)}{\partial\gamma_{i}\partial\gamma_{j}}=\begin{cases}-c^{\frac{1}{\alpha}-2}\left(\frac{\gamma_{i}}{\mu_{i}}\right)^{\alpha-1}\left(\frac{\gamma_{j}}{\mu_{j}}\right)^{\alpha-1}\qquad&\text{if}\quad
i\neq j;\\\\[10.00002pt]
-c^{\frac{1}{\alpha}-2}\left(\frac{\gamma_{i}}{\mu_{i}}\right)^{2(\alpha-1)}+c^{\frac{1}{\alpha}-1}\left(\frac{\gamma_{i}}{\mu_{i}}\right)^{\alpha-1}\frac{1}{\gamma_{i}}\qquad&\text{if}\quad
i=j.\end{cases}$ (A.4)

For any $x\in\mathbb{R}^{|\Omega|}\setminus\\{\mathbf{0}\\}$ we have,

$x^{\prime}Hx=c^{\frac{1}{\alpha}-1}\left(\sum_{j}\left(\frac{\gamma_{j}}{\mu_{j}}\right)^{\alpha-1}\frac{1}{\gamma_{j}}x_{j}^{2}-\frac{1}{c}\left(\sum_{j}\left(\frac{\gamma_{j}}{\mu_{j}}\right)^{\alpha-1}x_{j}\right)^{2}\right).$
(A.5)

As $c$ is strictly positive the sign of $x^{\prime}Hx$ only depends on the
term in the parentheses. Multiplying by $c$ we get,

$\left(\sum_{j}\mu_{j}^{1-\alpha}\gamma_{j}^{\alpha}\left(\frac{x_{j}}{\gamma_{j}}\right)^{2}\right)\left(\sum_{j}\mu_{j}^{1-\alpha}\gamma_{j}^{\alpha}\right)-\left(\sum_{j}\mu_{j}^{1-\alpha}\gamma_{j}^{\alpha}\left(\frac{x_{j}}{\gamma_{j}}\right)\right)^{2}.$
(A.6)

For fixed $\mu,\gamma$, defining the inner product on $\mathbb{R}^{|\Omega|}$
by,

$\langle x,y\rangle:=\sum_{j}\mu_{j}^{1-\alpha}\gamma_{j}^{\alpha}x_{j}y_{j},$
(A.7)

we have,

$\text{sign}\left(x^{\prime}Hx\right)=\text{sign}\left(\left\langle\frac{x}{\gamma},\frac{x}{\gamma}\right\rangle\left\langle\mathbf{1}_{|\Omega|},\mathbf{1}_{|\Omega|}\right\rangle-\left\langle\frac{x}{\gamma},\mathbf{1}_{|\Omega|}\right\rangle^{2}\right),$
(A.8)

with $\mathbf{1}_{|\Omega|}$ denoting the vector of ones of length $|\Omega|$.
By the Cauchy-Schwarz inequality the right hand side is weakly positive,
$\phi_{\mu}$ is positive semi-definite.

∎

##  Appendix B Proof of Proposition 2

We first derive the formula for the attention elasticity in terms of changes
in the dual variables, $\lambda^{*}$, then analyze the change in the dual via
an application of the implicit function theorem. Taking these together we
obtain the proof of proposition 2.

To ease notation—keeping the underlying $\alpha$ parameter fixed—we denote the
tilting function,

$\displaystyle f\colon\mathbb{R}$ $\displaystyle\to\mathbb{R}$ $\displaystyle
x$
$\displaystyle\mapsto\left(\frac{\alpha-1}{\alpha}x\right)^{\frac{1}{\alpha-1}}.$
(B.1)

$f\left(x^{*}(s,\omega)\right)$ determines the optimal tilting of the
conditional signal probabilities relative to the barycenter,

$P^{*}(s\mid\omega)\propto f\left(x^{*}(s,\omega)\right)q^{*}(s).$ (B.2)

At interior points the the quantity $(\alpha-1)x^{*}(s,\omega))>0$ so the
function $f$ is well-defined. We deal with boundary points in later sections.

The posterior ratio for a fixed state under the $\alpha$-mutual information is
given by,

$\Gamma^{a,b}_{\omega}=\frac{\gamma^{a}(\omega)}{\gamma^{b}(\omega)}=\frac{f\left(x^{*}(a,\omega)\right)}{f\left(x^{*}(b,\omega)\right)}\frac{\sum_{\tilde{\omega}}\mu(\tilde{\omega})f\left(x^{*}(b,\tilde{\omega})\right)}{\sum_{\tilde{\omega}}\mu(\tilde{\omega})f\left(x^{*}(a,\tilde{\omega})\right)}.$
(B.3)

With these notations in hand the general formula for the attention elasticity
is given by,

$\displaystyle\epsilon^{a_{\omega},b}_{\omega,\theta}=$
$\displaystyle\frac{f^{\prime}\left(x^{*}(a_{\omega},\omega)\right){x^{*}}^{\prime}(a_{\omega},\omega)}{f\left(x^{*}(a_{\omega},\omega)\right)}-\frac{f^{\prime}\left(x^{*}(b,\omega)\right){x^{*}}^{\prime}(b,\omega)}{f\left(x^{*}(b,\omega)\right)}+$
$\displaystyle\frac{\sum_{\tilde{\omega}}\mu(\tilde{\omega})f^{\prime}\left(x^{*}(b,\tilde{\omega})\right){x^{*}}^{\prime}(b,\tilde{\omega})}{\sum_{\tilde{\omega}}\mu(\tilde{\omega})f\left(x^{*}(b,\tilde{\omega})\right)}-\frac{\sum_{\tilde{\omega}}\mu(\tilde{\omega})f^{\prime}\left(x^{*}(a_{\omega},\tilde{\omega})\right){x^{*}}^{\prime}(a_{\omega},\tilde{\omega})}{\sum_{\tilde{\omega}}\mu(\tilde{\omega})f\left(x^{*}(a_{\omega},\tilde{\omega})\right)},$
(B.4)

with,

$\displaystyle f^{\prime}(x)$
$\displaystyle:=\frac{\mathrm{d}f}{\mathrm{d}z}\Big{|}_{z=x}=\frac{1}{\alpha}\left(\frac{\alpha-1}{\alpha}x\right)^{\frac{2-\alpha}{\alpha-1}},$
(B.5) $\displaystyle{x^{*}}^{\prime}(s,\omega)$ $\displaystyle:=\frac{\partial
x^{*}}{\partial\theta}=\begin{cases}1-\frac{{\lambda^{*}}^{\prime}_{\omega}}{\mu(\omega)}\qquad&\text{if
}s=s_{\omega},\\\
-\frac{{\lambda^{*}}^{\prime}_{\omega}}{\mu(\omega)}\qquad&\text{otherwise}.\end{cases}$
(B.6)

The derivative of the dual is with respect to the incentive change $\theta$,

${\lambda^{*}}^{\prime}_{\omega}:=\frac{\partial\lambda^{*}_{\omega}}{\partial\theta},$

and we write $D_{\theta}\boldsymbol{\lambda^{*}}$ to denote the vector of
derivatives for all states.

Note that in the general symmetric tracking problem with the global incentive
change the last two term of equation B.4 representing changes in the
unconditional choice probabilities cancel out—as the unconditional choice
probabilities stay uniform.

Note further that,

$\frac{f^{\prime}(x)}{f(x)}=\frac{1}{(\alpha-1)x}.$ (B.7)

Correspondingly we have,

$\epsilon^{a_{\omega},b}_{\omega,\theta}=\frac{{\lambda^{*}}^{\prime}_{\omega}}{\mu(\omega)}\left(\frac{1}{(\alpha-1)x^{*}(b,\omega)}-\frac{1}{(\alpha-1)x^{*}(a_{\omega},\omega)}\right)+\frac{1}{(\alpha-1)x^{*}(a_{\omega},\omega)}.$
(B.8)

We derive the change in the dual variable via an application of the implicit
function theorem on the primal feasibility constraints. We analyze this under
a general problem with potentially asymmetric incentive structure.

###  B.1 Expressing Changes in the Dual Variables

The primal feasibility constraints are given by888We can use the first-order
conditions (4.5) and the fact that $C^{*}$ is homogeneous of degree 1 in the
information structure. ,

$R^{*}_{\omega}:=\frac{\sum_{s}f\left(x^{*}(s,\omega)\right)q^{*}(s)}{\left(\sum_{s}\left(\sum_{\tilde{\omega}}f\left(x^{*}(s,\tilde{\omega})\right)^{\alpha}\mu(\tilde{\omega})\right)^{\frac{1}{\alpha}}q^{*}(s)\right)^{\alpha}}=1\qquad\forall\omega\in\Omega.$
(B.9)

To ease notation we denote the constant in the denominator,

$\xi:=\left(\sum_{s}\left(\sum_{\tilde{\omega}}f\left(x^{*}(s,\tilde{\omega})\right)^{\alpha}\mu(\tilde{\omega})\right)^{\frac{1}{\alpha}}q^{*}(s)\right)^{\alpha},$

and note that for any $\omega\in\Omega$ the primal feasibility implies,

$\sum_{s}f\left(x^{*}(s,\omega)\right)q^{*}(s)=\xi.$

To make the dependence on the parameter of interest, $\theta$, explicit we
rewrite the vector of primal feasibility constraints as,

$\boldsymbol{R^{*}}(\theta,\boldsymbol{\lambda^{*}}(\theta),\boldsymbol{q^{*}}(\theta))=\boldsymbol{1}.$

By the implicit function theorem, we have that,

$D_{\theta}\boldsymbol{\lambda^{*}}=\left[D_{\boldsymbol{\lambda^{*}}}\boldsymbol{R^{*}}\right]^{-1}_{|\Omega|\times|\Omega|}\left(-\left[D_{\boldsymbol{q^{*}}}\boldsymbol{R^{*}}\right]_{|\Omega|\times|\mathcal{S}|}D_{\theta}\boldsymbol{q^{*}}-D_{\theta}\boldsymbol{R^{*}}\right).$
(B.10)

We can exploit the structure of the Jacobian with respect to
$\boldsymbol{\lambda^{*}}$ to obtain the inverse. Note that,

$D_{\boldsymbol{\lambda^{*}}}\boldsymbol{R^{*}}_{\omega,\tilde{\omega}}=\begin{cases}1\qquad&\text{if
}\omega\neq\tilde{\omega},\\\
1-\frac{1}{\mu(\omega)}\frac{\sum_{s}f^{\prime}(x^{*}(s,\omega))q^{*}(s)}{\xi}\qquad&\text{if
}\omega=\tilde{\omega}.\end{cases}$ (B.11)

Since all off-diagonal elements are constant we can apply the Sherman-Morrison
formula to obtain the inverse of the Jacobian,

$\left[D_{\boldsymbol{\lambda^{*}}}\boldsymbol{R^{*}}\right]^{-1}=I_{|\Omega|}\boldsymbol{d}-\frac{1}{1+\langle\boldsymbol{d},\boldsymbol{1}_{|\Omega|}\rangle}\boldsymbol{d}\boldsymbol{d}^{\prime},$
(B.12)

where $I_{|\Omega|}$ is the identity matrix of size $|\Omega|\times|\Omega|$,
$\boldsymbol{1}_{|\Omega|}$ is the vector of ones of length $|\Omega|$, and
$\boldsymbol{d}$ is the vector with entries,

$\boldsymbol{d}_{\omega}=-\frac{\mu(\omega)\xi}{\sum_{s}f^{\prime}(x^{*}(s,\omega))q^{*}(s)}.$
(B.13)

Further,

$D_{\boldsymbol{q^{*}}}\boldsymbol{R^{*}}_{\omega,s}=\frac{f\left(x^{*}(s,\omega)\right)}{\xi}-A,$
(B.14)

for some constant $A$. Since $D_{\theta}\boldsymbol{q^{*}}$ sums to zero we
can ignore $A$.

Analogous derivations pertaining to the implicit function theorem hold for
arbitrary incentive structures.

Continuing with the proof of the current section for the case of the general
symmetric tracking problem with $\theta=h/\kappa$, denote $n:=|\Omega|$ and

$x^{*}(s,\omega)=\begin{cases}h^{*}\qquad&\text{if }s=a_{\omega},\\\
l^{*}\qquad&\text{otherwise}.\end{cases}$ (B.15)

We have,

$D_{\theta}\boldsymbol{R^{*}}=\boldsymbol{1}_{|\Omega|}\frac{\left(f^{\prime}\left(h^{*}\right)-f\left(h^{*}\right)\right)q^{*}(a_{\omega})}{\xi}.$
(B.16)

Note that in the current case $D_{\theta}\boldsymbol{q^{*}}=\boldsymbol{0}$
and $\boldsymbol{d}_{\omega}$ is constant across $\omega$ so we have,

$\left[D_{\boldsymbol{\lambda^{*}}}\boldsymbol{R^{*}}\right]^{-1}\boldsymbol{1}_{|\Omega|}=\boldsymbol{1}_{|\Omega|}\frac{\xi}{\left(f\left(h^{*}\right)-f^{\prime}\left(h^{*}\right)\right)+\left(n-1\right)\left(f\left(l^{*}\right)-f^{\prime}\left(l^{*}\right)\right)}.$
(B.17)

Combining we have,

${\lambda^{*}}^{\prime}_{\omega}=\frac{\left(f\left(h^{*}\right)-f^{\prime}\left(h^{*}\right)\right)q^{*}(a_{\omega})}{\left(f\left(h^{*}\right)-f^{\prime}\left(h^{*}\right)\right)+\left(n-1\right)\left(f\left(l^{*}\right)-f^{\prime}\left(l^{*}\right)\right)}.$
(B.18)

We can express the difference,

$f(x)-f^{\prime}(x)=f(x)\left(1-\frac{1}{(\alpha-1)x}\right)$ (B.19)

Noting that both $q^{*}$ and $\mu$ are uniform we arrive at,

$\displaystyle\epsilon^{a_{\omega},b}_{\omega,\theta}$
$\displaystyle=\frac{f(h^{*})\left(1-\frac{1}{(\alpha-1)h^{*}}\right)\left(\frac{1}{(\alpha-1)l^{*}}-\frac{1}{(\alpha-1)h^{*}}\right)}{f(h^{*})\left(1-\frac{1}{(\alpha-1)h^{*}}\right)+(n-1)f(l^{*})\left(1-\frac{1}{(\alpha-1)l^{*}}\right)}+\frac{1}{(\alpha-1)h^{*}}$
$\displaystyle=\frac{f(h^{*})\left(1-\frac{1}{(\alpha-1)h^{*}}\right)\frac{1}{(\alpha-1)l^{*}}+(n-1)f(l^{*})\left(1-\frac{1}{(\alpha-1)l^{*}}\right)\frac{1}{(\alpha-1)h^{*}}}{f(h^{*})\left(1-\frac{1}{(\alpha-1)h^{*}}\right)+(n-1)f(l^{*})\left(1-\frac{1}{(\alpha-1)l^{*}}\right)}$
(B.20)

We next prove a lemma on the magnitudes of $(\alpha-1)h^{*}$ and
$(\alpha-1)l^{*}$ and then finish the proof.

######  Lemma 1.

In the general symmetric tracking problem for $\alpha<1$ the optimal solution
satisfies,

$(\alpha-1)h^{*}<\alpha<(\alpha-1)l^{*};$ (B.21)

for $\alpha>1$ the optimal interior‘ solution satisfies,

$(\alpha-1)h^{*}>\alpha>(\alpha-1)l^{*}.$ (B.22)

Proof. First, take the stationarity first-order condition (4.2) and rearrange,

$\frac{\left(\sum_{\tilde{\omega}}P^{*}\left(s\mid\tilde{\omega}\right)^{\alpha}\mu(\tilde{\omega})\right)^{-1}}{\left(\sum_{\tilde{s}}\left(\sum_{\tilde{\omega}}P^{*}\left(\tilde{s}\mid\tilde{\omega}\right)^{\alpha}\mu(\tilde{\omega})\right)^{\frac{1}{\alpha}}\right)^{\frac{\alpha}{\alpha-1}}}P^{*}(\omega)^{\alpha}=\left(\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}+\frac{\delta^{*}_{\omega,s}}{\mu(\omega)}\right)\right)^{\frac{\alpha}{\alpha-1}}.$
(B.23)

Multiplying by $\mu(\omega)$ and summing across all states yields,

$\displaystyle\Psi:=$
$\displaystyle\left(\sum_{\tilde{s}}\left(\sum_{\tilde{\omega}}P^{*}\left(\tilde{s}\mid\tilde{\omega}\right)^{\alpha}\mu(\tilde{\omega})\right)^{\frac{1}{\alpha}}\right)^{\frac{\alpha}{1-\alpha}}=$
$\displaystyle\sum_{\omega}\mu(\omega)\left(\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}+\frac{\delta^{*}_{\omega,s}}{\mu(\omega)}\right)\right)^{\frac{\alpha}{\alpha-1}}\quad\forall
s\in\mathcal{S}.$ (B.24)

Note that the above condition holds for all actions $s$ irrespective of
whether or not they are chosen with strictly positive probability under the
optimal strategy. The constant $\Psi$ is strictly positive and real—it is
equal to $\Psi=\exp\left(-I_{\alpha}(P^{*},\mu)\right)$.

Using the primal feasibility constraints and noting that as $q^{*}$ is uniform
$\xi$ is equal to the right-hand side of equation $\eqref{eq: foc_constant}$
we have that,

$\displaystyle\sum_{s}f\left(x^{*}(s,\omega)\right)q^{*}(s)$
$\displaystyle=\xi=\exp\left(-I_{\alpha}(P^{*},\mu)\right)\leq
1\qquad\forall\omega\in\Omega;$ (B.25)
$\displaystyle\sum_{\omega}f\left(x^{*}(s,\omega)\right)^{\alpha}\mu(\omega)$
$\displaystyle=\xi=\exp\left(-I_{\alpha}(P^{*},\mu)\right)\leq 1\qquad\forall
s\ \text{with}\ q^{*}(s)>0.$ (B.26)

We obtain the inequalities by the fact that the $\alpha$-mutual information is
always weakly positive, $I_{\alpha}(P^{*},\mu)\geq 0$.

In the symmetric problem the uniform barycenter $q^{*}$ in (B.25) implies,

$\displaystyle\frac{1}{n}f\left(h^{*}\right)+\frac{n-1}{n}f\left(l^{*}\right)\leq
1$ $\displaystyle
f\left(l^{*}\right)\left(\underbrace{\frac{f\left(h^{*}\right)}{f\left(l^{*}\right)}}_{>1}+(n-1)\right)\leq
n$ (B.27)

Next, use the equality,

$\sum_{\tilde{s}}f\left(x^{*}(\tilde{s},\omega)\right)q^{*}(\tilde{s})=\sum_{\tilde{\omega}}f\left(x^{*}(s,\tilde{\omega})\right)^{\alpha}\mu(\tilde{\omega})\quad\text{for
any }\omega\ \text{and }s\ \text{with}\ q^{*}(s)>0,$ (B.28)

which in the symmetric tracking problem implies,

$\underbrace{\left(\frac{\alpha-1}{\alpha}h^{*}\right)^{\frac{\alpha}{\alpha-1}}}_{>0}\left(1-\left(\frac{\alpha-1}{\alpha}h^{*}\right)^{-1}\right)=\underbrace{(n-1)\left(\frac{\alpha-1}{\alpha}l^{*}\right)^{\frac{\alpha}{\alpha-1}}}_{>0}\left(\left(\frac{\alpha-1}{\alpha}l^{*}\right)^{-1}-1\right).$
(B.29)

If $\alpha<1$ then necessarily $(\alpha-1)l^{*}>(\alpha-1)h^{*}>0$ since $l<h$
and conditional probabilities are interior. This implies
$f\left(h^{*}\right)>f\left(l^{*}\right)>0$ and the term in the bracket of
equation (B.27) is greater than $n$. To satisfy the equation we need

$f\left(l^{*}\right)<1\quad\implies\quad\frac{\alpha-1}{\alpha}l^{*}>1.$
(B.30)

And to match the signs of the two sides of equation (B.29) we need,

$\frac{\alpha-1}{\alpha}h^{*}<1.$ (B.31)

If $\alpha>1$ then necessarily $(\alpha-1)h^{*}>(\alpha-1)l^{*}>0$ since $h<l$
and conditional probabilities are interior. This implies
$f\left(h^{*}\right)>f\left(l^{*}\right)>0$ and the term in the bracket of
equation (B.27) is greater than $n$. To satisfy the equation we need

$f\left(l^{*}\right)<1\quad\implies\quad\frac{\alpha-1}{\alpha}l^{*}\leq 1.$
(B.32)

And to match the signs of the two sides of equation (B.29) we need,

$\frac{\alpha-1}{\alpha}h^{*}>1.$ (B.33)

∎

Lastly, note that equation (B.29) implies that under $\alpha<1$ the
denominator of equation (B.20) is always negative, while under $\alpha>1$ it
is always positive.

To complete the proof of proposition 2 we use lemma 1 and equation (B.20).

Case I. $\alpha<1$.

By lemma 1 we have $(\alpha-1)l^{*}>\alpha>(\alpha-1)h^{*}>0$ and
$f\left(h^{*}\right)>f\left(l^{*}\right)>0$. Given the denominator of (B.20)
being negative,

$\displaystyle\epsilon^{a_{\omega},b}_{\omega,\theta}>1\qquad\text{if}\quad(\alpha-1)l^{*}<1,$
$\displaystyle\epsilon^{a_{\omega},b}_{\omega,\theta}<1\qquad\text{if}\quad(\alpha-1)l^{*}>1.$
(B.34)

Case II. $\alpha>1$.

By lemma 1 we have $(\alpha-1)h^{*}>\alpha>(\alpha-1)l^{*}>0$ and
$f\left(h^{*}\right)>f\left(l^{*}\right)>0$. Given the denominator of (B.20)
being positive,

$\displaystyle\epsilon^{a_{\omega},b}_{\omega,\theta}>1\qquad\text{if}\quad(\alpha-1)l^{*}<1,$
$\displaystyle\epsilon^{a_{\omega},b}_{\omega,\theta}<1\qquad\text{if}\quad(\alpha-1)l^{*}>1.$
(B.35)

∎

##  Appendix C Proof of Proposition 3

Note that the $\alpha$-mutual information for $\alpha\in(0,1)\cup(1,\infty)$
is given by,

$I_{\alpha}\left(P,\mu\right)=\frac{\alpha}{\alpha-1}\log\sum_{s}\left(\sum_{\omega}\mu(\omega)P(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}.$
(C.1)

First, we prove information monotonicity. With the transformations introduced
in section 5 defining $P^{\prime},\bar{P}$, and $\bar{\mu}$, we need to show,

$I_{\alpha}\left(P,\mu\right)\geq I_{\alpha}\left(P^{\prime},\mu\right).$

Take the monotone increasing transformation $\phi\colon
t\mapsto\frac{1}{\alpha-1}\exp\left(t\frac{\alpha-1}{\alpha}\right)$,
information monotonicity is equivalent to,

$\displaystyle\phi\left(I_{\alpha}\left(P,\mu\right)\right)$
$\displaystyle\geq\phi\left(I_{\alpha}\left(P^{\prime},\mu\right)\right)$
$\displaystyle\frac{1}{\alpha-1}\sum_{s}\left(\sum_{\omega}\mu(\omega)P(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}$
$\displaystyle\geq\frac{1}{\alpha-1}\sum_{s}\left(\sum_{\omega}\mu(\omega)P^{\prime}(s\mid\omega)^{\alpha}\right)^{\frac{1}{\alpha}}.$

Note that the transformed $\alpha$-mutual information is posterior separable
as previously remarked in (3.10).

Given that likelihoods of $P^{\prime}$ are constant over elements of the
partition we can write,

$\displaystyle\phi\left(I_{\alpha}\left(P^{\prime},\mu\right)\right)$
$\displaystyle=\frac{1}{\alpha-1}\sum_{s}\left(\sum_{z\in
Z}\left(c_{z}^{\alpha}\sum_{\omega\in\bar{\Omega}_{z}}\mu(\omega)\right)\right)^{\frac{1}{\alpha}}$
(C.2) $\displaystyle=\frac{1}{\alpha-1}\sum_{s}\left(\sum_{z\in
Z}\left(\sum_{\omega\in\bar{\Omega}_{z}}P(s\mid\omega)\mu(\omega)\right)^{\alpha}\left(\sum_{\omega\in\bar{\Omega}_{z}}\mu(\omega)\right)^{(1-\alpha)}\right)^{\frac{1}{\alpha}}.$
(C.3)

On the other hand,

$\phi\left(I_{\alpha}\left(P,\mu\right)\right)=\frac{1}{\alpha-1}\sum_{s}\left(\sum_{z\in
Z}\sum_{\omega\in\bar{\Omega}_{z}}\left(P(s\mid\omega)\mu(\omega)\right)^{\alpha}\mu(\omega)^{(1-\alpha)}\right)^{\frac{1}{\alpha}}.$
(C.4)

Denote the inner function in the parentheses as
$f_{\alpha}(x,y):=x^{\alpha}y^{(1-\alpha)}$. Note that $f_{\alpha}(x,y)$ is
the “conic transform” of $\tilde{f}_{\alpha}(x):=x^{\alpha}$, that is
$f_{\alpha}(x,y)=\tilde{f}_{\alpha}(x/y)y$ for $y>0$. Since the prior is
$\mu\in\text{int}\Delta\left(\bar{\Omega}\right)$ we have no issues applying
the transformation. Next, we will use the fact that the conic transform
preserves convexity/concavity.

We have two cases to consider. If $\alpha<1$,

  * •

$\tilde{f}_{\alpha}$ is concave, so the conic transform $f_{\alpha}$ is also
concave. By Jensen’s inequality we have,

$\frac{1}{|\bar{\Omega}_{z}|}\sum_{\omega\in\bar{\Omega}_{z}}\left(P(s\mid\omega)\mu(\omega)\right)^{\alpha}\mu(\omega)^{(1-\alpha)}\leq\frac{1}{|\bar{\Omega}_{z}|}\left(\sum_{\omega\in\bar{\Omega}_{z}}P(s\mid\omega)\mu(\omega)\right)^{\alpha}\left(\sum_{\omega\in\bar{\Omega}_{z}}\mu(\omega)\right)^{(1-\alpha)}.$

  * •

The inequality holds for all $z\in Z$ so it is preserved for the sum over the
partitions.

  * •

Since the mapping $x\mapsto x^{\frac{1}{\alpha}}$ is monotone increasing for
$x>0$ and the scalar $\frac{1}{\alpha-1}<0$, we have that information
monotonicity holds,

$I_{\alpha}\left(P,\mu\right)\geq I_{\alpha}\left(P^{\prime},\mu\right).$

For $\alpha>1$ we can apply an analogous argument with $\tilde{f}_{\alpha}$
being convex, and so $f_{\alpha}$ also being convex. Applying Jensen’s
inequality and noting that $\frac{1}{\alpha-1}>0$ we have,

$I_{\alpha}\left(P,\mu\right)\geq I_{\alpha}\left(P^{\prime},\mu\right).$

Next we show that

$I_{\alpha}\left(P^{\prime},\mu\right)=I_{\alpha}\left(\bar{P},\bar{\mu}\right).$

Note, that since by construction
$\mu\left(\bar{\Omega}_{z}\right)=\bar{\mu}\left(\bar{\Omega}_{z}\right)$ for
all $z\in Z$ we have that,

$c_{z}=\frac{\sum_{\omega\in\bar{\Omega}_{z}}P(s\mid\omega)\mu(\omega)}{\sum_{\omega\in\bar{\Omega}_{z}}\mu(\omega)}=\frac{\sum_{\omega\in\bar{\Omega}_{z}}P(s\mid\omega)\mu(\omega)}{\sum_{\omega\in\bar{\Omega}_{z}}\bar{\mu}(\omega)}=\bar{c}_{z}\qquad\forall
z\in Z.$

Now use the decomposition as in equation (C.2),

$\displaystyle\phi\left(I_{\alpha}\left(P^{\prime},\mu\right)\right)$
$\displaystyle=\frac{1}{\alpha-1}\sum_{s}\left(\sum_{z\in
Z}\left(c_{z}^{\alpha}\sum_{\omega\in\bar{\Omega}_{z}}\mu(\omega)\right)\right)^{\frac{1}{\alpha}}$
$\displaystyle=\frac{1}{\alpha-1}\sum_{s}\left(\sum_{z\in
Z}\left(\bar{c}_{z}^{\alpha}\sum_{\omega\in\bar{\Omega}_{z}}\bar{\mu}(\omega)\right)\right)^{\frac{1}{\alpha}}=\phi\left(I_{\alpha}\left(\bar{P},\bar{\mu}\right)\right).$

This completes our proof. ∎

##  Appendix D Proof of Proposition 5

The proof for the claims of proposition 5 hinges on the directional
derivatives of the $\alpha$-mutual information evaluated at points on the
boundary of the constraint set.

Note that the partial derivative of $I_{\alpha}$ with respect to
$P(a\mid\omega)$ is

$\frac{\partial I_{\alpha}(P,\mu)}{\partial
P(a\mid\omega)}=\frac{\alpha}{\alpha-1}\frac{\left(\sum_{\tilde{\omega}}\mu(\tilde{\omega})P(a\mid\tilde{\omega})^{\alpha}\right)^{\frac{1-\alpha}{\alpha}}}{\sum_{\tilde{s}}\left(\sum_{\tilde{\omega}}\mu(\tilde{\omega})P(\tilde{s}\mid\tilde{\omega})^{\alpha}\right)^{\frac{1}{\alpha}}}\mu(\omega)P(a\mid\omega)^{\alpha-1}.$
(D.1)

We treat the two claims separately.

Claim 1. For $\alpha\leq 1$ there exists no non-trivial event
$E\subsetneq\text{supp}(\mu)$ that the DM optimally learns with certainty.

Suppose there exists such a set. Then necessarily there exists
$a\in\mathcal{A}$ such that $P^{*}(a\mid\omega_{i})=0$ for some $i$ while
$P^{*}(a\mid\omega_{j})>0$ for some $j$. Furthermore, since signal
probabilities have to add up to one, there exists some $b$ such that
$P^{*}(b\mid\omega_{i})>0$. Moving in the feasible direction that results in a
new experiment
$(P^{*}(a\mid\omega_{i})+\epsilon,P^{*}(b\mid\omega_{i})-\epsilon)$ with all
other conditional signal probabilities being the same as in $P^{*}$ reduces
information costs by infinity at the margin as shown by equation (D.1) while
only affecting payoffs by a finite amount. This contradicts the optimality of
$P^{*}$.

Claim 2. For $\alpha>1$ take an event of the form
$E_{a}:=\left\\{\omega:u(a,\omega)>u(b,\omega)\ \forall b\neq a\right\\}$.
Suppose that the optimal attention strategy under the scaled payoff, $\pi
u(\cdot,\cdot)$, is such that $\gamma^{a}\left(E_{a}\right)<1$. This implies
that $P^{*}(a\mid\omega_{k})>0$ for some $\omega_{k}\notin E_{a}$. Since by
assumption all payoffs are distinct, $\omega_{k}\in E_{b}$ for some other
action $b\neq a$ which implies $u(b,\omega_{k})>u(a,\omega_{k})$. Moving in
the feasible direction that results in a new experiment
$(P^{*}(b\mid\omega_{k})+\epsilon,P^{*}(a\mid\omega_{k})-\epsilon)$ with all
other conditional signal probabilities being the same as in $P^{*}$ changes
information costs by a finite amount on the margin as shown by equation (D.1),
so there must be $\pi$ such that the considered move has greater utility
increase on the margin $\pi(u(b,\omega_{k})-u(a,\omega_{k}))$. This would
contradict the optimality of the information structure inducing
$\gamma^{a}\left(E_{a}\right)<1$.

∎

##  Appendix E Proof of Proposition 6

First, take the stationarity first-order condition (4.2) and rearrange,

$\frac{\left(\sum_{\tilde{\omega}}P^{*}\left(s\mid\tilde{\omega}\right)^{\alpha}\mu(\tilde{\omega})\right)^{-1}}{\left(\sum_{\tilde{s}}\left(\sum_{\tilde{\omega}}P^{*}\left(\tilde{s}\mid\tilde{\omega}\right)^{\alpha}\mu(\tilde{\omega})\right)^{\frac{1}{\alpha}}\right)^{\frac{\alpha}{\alpha-1}}}P^{*}(\omega)^{\alpha}=\left(\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}+\frac{\delta^{*}_{\omega,s}}{\mu(\omega)}\right)\right)^{\frac{\alpha}{\alpha-1}}.$
(E.1)

Multiplying by $\mu(\omega)$ and summing across all states yields,

$\displaystyle\Psi:=$
$\displaystyle\left(\sum_{\tilde{s}}\left(\sum_{\tilde{\omega}}P^{*}\left(\tilde{s}\mid\tilde{\omega}\right)^{\alpha}\mu(\tilde{\omega})\right)^{\frac{1}{\alpha}}\right)^{\frac{\alpha}{1-\alpha}}=$
$\displaystyle\sum_{\omega}\mu(\omega)\left(\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}+\frac{\delta^{*}_{\omega,s}}{\mu(\omega)}\right)\right)^{\frac{\alpha}{\alpha-1}}\quad\forall
s\in\mathcal{S}.$ (E.2)

Note that the above condition holds for all actions $s$ irrespective of
whether or not they are chosen with strictly positive probability under the
optimal strategy. The constant $\Psi$ is strictly positive and real.

We consider two cases depending on the value of $\alpha$.

Case 1.  $\alpha<1$.

If $\mu P^{*}_{\mathcal{S}}(a)>0$ (or equivalently $q^{*}(a)>0$) then
$P^{*}(a\mid\omega)>0$ for all $\omega$ by proposition 5. By the complementary
slackness conditions we have $\delta^{*}_{\omega,a}=0$ for all $\omega$. Hence
we have,

$\sum_{\omega}\mu(\omega)\left(\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right)^{\frac{\alpha}{\alpha-1}}=\Psi\quad\forall
s\quad\text{such that }q^{*}(s)>0.$ (E.3)

If $\mu P^{*}_{\mathcal{S}}(a)=0$ (or equivalently $q^{*}(a)=0$) then
$P^{*}(a\mid\omega)=0$ for all $\omega$. By the complementary slackness and
dual feasibility conditions $\delta^{*}_{\omega,a}>0$. Since $\alpha-1<0$ we
have that,

$\sum_{\omega}\mu(\omega)\left(\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right)^{\frac{\alpha}{\alpha-1}}<\Psi\quad\forall
s\quad\text{such that }q^{*}(s)=0.$ (E.4)

Note that irrespective of $P^{*}(s\mid\omega)$ being strictly positive
$\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)$
is positive otherwise the stationarity condition would be violated.

Case 2.  $\alpha>1$.

Under $\alpha>1$ we might have $P^{*}(a\mid\omega)=0$ while $\mu P^{*}(a)>0$.
Consider such a case. By the stationarity first-order condition we have,

$\frac{u(a,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}=-\frac{\delta^{*}_{\omega,a}}{\mu(\omega)},$
(E.5)

and since $\alpha-1>0$,

$\left[\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right]^{+}=\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}+\frac{\delta^{*}_{\omega,s}}{\mu(\omega)}\right).$
(E.6)

Correspondingly we arrive at,

$\sum_{\omega}\mu(\omega)\left(\left[\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}=\Psi\quad\forall
s\quad\text{such that }q^{*}(s)>0.$ (E.7)

Note that without thresholding the expression at zero for the terms where
$P^{*}(s\mid\omega)=0$ would render the exponentiation ill-defined as we would
be taking the fractional power of a negative number.

If $\mu P^{*}_{\mathcal{S}}(a)=0$ (or equivalently $q^{*}(a)=0$) then
$P^{*}(a\mid\omega)=0$ for all $\omega$. By the complementary slackness and
dual feasibility conditions $\delta^{*}_{\omega,a}>0$. Since $\alpha-1>0$ we
have that,

$\sum_{\omega}\mu(\omega)\left(\left[\frac{\alpha-1}{\alpha}\left(\frac{u(s,\omega)}{\kappa}-\frac{\lambda^{*}_{\omega}}{\mu(\omega)}\right)\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}<\Psi\quad\forall
s\quad\text{such that }q^{*}(s)=0.$ (E.8)

Now taking the convex combination of these expressions with the convex weights
given by the barycenter of the optimal experiment,

$\sum_{\tilde{s}}q^{*}(\tilde{s})\sum_{\tilde{\omega}}\mu(\tilde{\omega})\left(\left[\frac{\alpha-1}{\alpha}\left(\frac{u(\tilde{s},\tilde{\omega})}{\kappa}-\frac{\lambda^{*}_{\tilde{\omega}}}{\mu(\tilde{\omega})}\right)\right]^{+}\right)^{\frac{\alpha}{\alpha-1}}=\Psi.$
(E.9)

By the fact that $\mu P^{*}_{\mathcal{S}}(s)>0\iff q^{*}(s)>0$ proposition 6
holds if we substitute the unconditional signal probabilities for the
barycenter. ∎

##  Appendix F Proof of Proposition 7

Since the objective is bounded from below and the objective function is
decreasing at each step the algorithm is converging. That does not necessarily
imply that the limit point is the global optimum of the problem. We can use
the fact that algorithm 1 is a special case of the block non-linear Gauss-
Seidel method—also called block coordinate descent method—with quasi-convex
objective function, two blocks, and convex constraints. We can directly apply
theorem 5.1 of Tseng (2001).

The conditions that we need to verify in order to apply the results of Tseng
(2001) are:

  * •

The Rényi divergence, $D_{\alpha}$ is continuous on its effective domain,
which holds for finite distributions.

  * •

$D_{\alpha}$ is quasi-convex in both arguments—in fact it is jointly quasi-
convex (Van Erven and Harremoës, 2014, Theorem 13)—and hemivariate in both
arguments.

  * •

$D_{\alpha}$ is lower semi-continuous (Van Erven and Harremoës, 2014, Theorem.
15).

  * •

The effective domain of $D_{\alpha}$ is a product of subsets of Euclidean
spaces.

The only property that we need to show is that $D_{\alpha}$ is hemivariate in
both arguments separately on its effective domain. A function is hemivariate
in an argument if it is not constant on any line segment belonging to the
effective domain. That is for a given $Q$ take distinct $P_{0}$ and $P_{1}$
such that the line segment $\lambda(P_{0},Q)+(1-\lambda)(P_{1},Q)$ for any
$\lambda\in[0,1]$ is in the effective domain of $D_{\alpha}$. Suppose that
$D_{\alpha}$ is not hemivariate. Then with $P_{\lambda}:=\lambda
P_{0}+(1-\lambda)P_{1}$,

$D_{\alpha}(P_{\lambda}\|Q):=\frac{1}{\alpha-1}\log\left(\sum_{x\in\mathcal{X}}P_{\lambda}(x)^{\alpha}Q(x)^{(1-\alpha)}\right)=C\qquad\forall\lambda\in[0,1].$
(F.1)

We can derive a contradiction by considering

$\sum_{x\in\mathcal{X}}P_{\lambda}(x)^{\alpha}Q(x)^{(1-\alpha)}=\exp\left((\alpha-1)C\right)\qquad\forall\lambda\in[0,1],$
(F.2)

and using the strict convexity or strict concavity of the function $z\mapsto
z^{\alpha}$ depending on the value of $\alpha$. The case for $Q$ is analogous.

As a result theorem 5.1 of Tseng (2001) applies and the modified Blahut-
Arimoto algorithm converges to a stationary point of the objective function at
which all KKT conditions are satisfied.

∎